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Open AccessFeature PaperReview

In the Quest for Cosmic Rotation

1
Mathematics Department, Higher Military Institute of Air Defence, Moskovsky prospect 28, 150001 Yaroslavl, Russia
2
Department of Theoretical Physics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina 6280, 842 48 Bratislava, Slovakia
3
Nuclear Safety Institute (IBRAE), Russian Academy of Sciences, B. Tulskaya 52, 115191 Moscow, Russia
*
Author to whom correspondence should be addressed.
Universe 2020, 6(1), 14; https://doi.org/10.3390/universe6010014
Received: 27 November 2019 / Revised: 7 January 2020 / Accepted: 8 January 2020 / Published: 15 January 2020
(This article belongs to the Special Issue Rotation Effects in Relativity)

Abstract

This paper analyzes the problem of global rotation in general relativity (GR) theory. Simple cosmological models with rotation and expansion are presented, which give a natural explanation of the modern values of the acceleration parameter at different red shifts without involving the concepts of “dark energy” and “dark matter”. It is shown that due to the smallness of the cosmological rotation, for its detection one should use observations that do not depend on the magnitude of the angular velocity of the Universe. Such tests include the effects of the cosmic mirror and the cosmic lens. For the first time on the basis of modern electronic catalogs the search on the celestial sphere of images of our Galaxy and other galaxies is made. Viable candidates for both effects have been found.
Keywords: cosmology; cosmological effects; rotation of the Universe cosmology; cosmological effects; rotation of the Universe

1. Introduction

Rotational, vortex, and chiral phenomena are widely observed in physics on all scales including astrophysics, high-energy physics, and the heavy-ion collisions. Quoting E.T. Whittaker [1]: “Rotation is a universal phenomenon; the earth and all the other members of the solar system rotate on their axes, the satellites revolve round the planets, the planets revolve round the Sun, and the Sun himself is a member of the galaxy or Milky Way system which revolves in a very remarkable way. How did all these rotary motions come into being? What secures their permanence or brings about their modifications? And what part do they play in the system of the world?” Currently, the search for the similarities of vortex structures encountered in the heavy-ion physics and in astrophysical conditions attracts a lot of attention in the literature.
The hypothesis of a universe’s rotation, in our opinion, belongs to the most intriguing issues of the modern cosmology. This idea is quite natural from the physical viewpoint and it does not contradict any of the astrophysical observations and still remains a challenging unsolved mystery.
In observational cosmology, the main difficulty for detecting a global rotation is its smallness—less than 10−13 yr−1 according to the generally accepted assessment. It is impossible in the Universe to distinguish the direction corresponding to the axis of rotation, with respect to which one could notice deviations (in the standard tests) from the Friedman standard cosmology.
In theoretical cosmology, the main difficulties are related, on the one hand, to the lack of simple models of an expanding and rotating Universe in general relativity (GR) similar to Friedman–Robertson–Walker models. On the other hand, there are no convincing predictive effects of cosmic rotation that are consistent with the capabilities of the equipment of modern astronomical observatories.
Observing rotating planets, stars, galaxies, and clusters of galaxies, we naturally come to two hypotheses. The first, the rotation was originally inherent in the content of “embryonic singularity” (“primeval atom” [2]). Following Einstein’s assumption which linked properties of matter and properties of a space-time, it is natural to assume that physical rotation of matter (orbital and intrinsic) generates geometrical rotation of space-time (universal and local). Eventually, the rotation slows down and global orbital rotation becomes barely noticeable. Rotating cosmic objects are traces of that initial global rotation.
It is difficult to build a model of matter with orbital and inner angular momentum. The best candidate is the model of the perfect Weyssenhoff–Raabe fluid [3], whose elements have a “classical” spin. It remains to add the global rotation of the fluid as a whole (as a solid body for example) to the energy-momentum tensor of the Weyssenhoff fluid to obtain a realistic source for cosmology with rotation. One could attempt to do it directly, in the form of global rotation energy but this is hindered by a determination of the (local) orbital moment of inertia. Then one should look for an indirect introduction of orbital rotation.
This can be done in several ways. One of them is to consider a fluid with additional properties beyond a perfect fluid (for example, with viscosity, heat flow, or electric charge), taking into account the spin-spin and spin-electromagnetic interaction [4]. The next step is the consistent introduction of known physical fields into the energy-momentum tensor of the Universe matter. A promising generalization of the Weissenhoff fluid model from the point of view of adequacy to cosmological models with rotation, in our opinion, is the antisymmetric third rank tensor field, the sources of which are extended objects (cosmic strings, shells, or bags) in a perfect fluid (see [5] and the literature cited there) [4].
According to the second hypothesis, after the explosion of the “embryonic singularity”, the matter particles run in different directions and gravitational interaction (within the framework of the Newtonian theory) inevitably leads to rotation relative to center of mass (and relative to each other (orbital rotation)) and to their own intrinsic rotation (in GR in harmonic coordinate system [6,7]). As a result, there are no non-rotating objects in the Universe. The latter is an instructive proof of the validity of GR and the Big Bang theory. The angular velocity of global and local rotation in this case increases from zero to modern values.
Let us start with the requirement of spatial homogeneity, since the fact of large-scale homogeneity of the Universe is considered to be proven in observational astronomy. We restrict ourselves to metrics of the form (t—cosmological time; xi, i = 1,2,3—three spatial coordinates)
d s 2 = d t 2 2 R n i d x i d t R 2 γ i j d x i d x j ,    i , j = 1 , 2 , 3 .
We assume the same law of changing of scale factor R = R(t) in all directions, therefore, all metrics (1) are shear-free. In all other aspects, (1) is space-time interval of the most general form. Here
n i = ν a e i ( a ) , γ i j = β a b e i ( a ) e j ( b ) ,
where ν a , β a b ( det β a b 0 ) , a, b = 1,2,3 are constant coefficients, n i = n i ( x k ) , γ i j = γ i j ( x k ) are functions of spatial coordinates xi on the t = const hypersurfaces, and
e ( a ) = e i ( a ) ( x ) d x i
are the invariant 1-forms with respect to the action of a three-parameter group of motion which is admitted by the space-time (1). One can prove that the isotropic expansion in (1) guarantees the isotropy of microwave background radiation (MBR) in such models [8]. Accordingly, the angular velocity of rotation should not be estimated on the basis of models with shear effects, contrary to the statements of [9].
The spatial homogeneity condition imposes a restriction on the group of motions which should act simplyransitively on the spatial (t = const) hypersurfaces. Such metrics are called Bianchi metrics, there are nine types of them. These manifolds are classified according to the Killing vectors ξ ( a ) and their commutators [ ξ ( a ) , ξ ( b ) ] = C a b c ξ ( c ) . The invariant forms (3) solve the Lie equations L ξ ( b ) e ( a ) = 0 for each Bianchi type, so models n (1) are spatially homogeneous.
Besides the three Killing vector fields, space-times (1) admit a nontrivial conformal Killing vector ξ c o n f = R t . The explicit form for all Bianchi metrics are given in [10]. Standard cosmologies (with ν a = 0 in (2)) are known to belong to types I, V, and IX.
The rotation tensor ω μ ν and the volume expansion scalar θ for (1) are determined in the co-moving matter characterized by (average) 4-velocity vector u μ = δ 0 μ . They read as follows:
ω μ ν = { ω 0 j = 0 , ω i j = R 2 C i j k n k , θ = 3 R ˙ R ,        μ , ν = 0 , 1 , 2 , 3 ,
here, C i j k = e ( a ) k ( i e j a j e i a ) . The explicit form of the structure constants C b c a , of the anholonomity objects C i j k and also of ξ ( a ) and e ( а ) are given in [10]. It is easy to notice that ω μ ν = 0 for Bianchi-I models.
The acceleration vector aμ for (1) is nontrivial
a μ = { a 0 = 0 , a i = n i R ˙ ,
for non-zero R ˙ . It is easy to see that uμ is not orthogonal to the hypersurface of homogeneity (“tilted” models).
Cosmological models (1) are parallax-free [10] (in terminology of [11]), and hence velocity of rotation cannot be estimated from parallax effects, contrary to the statements of [12,13].
It is worthwhile to mention an unusual physical property of metrics (1): there is no shear despite the non-zero rotation. In other words, despite the rotation, the expansion has the same magnitude in all directions. One usually thinks of a two-dimensional expanding and rotating elastic sphere and expects that at the poles the sphere should expand more slowly than at the equator. However imagine a two-dimensional expanding and rotating cylindrical surface: it is obvious that relative to any point on the axis of rotation, the expansion velocity is the same. The expansion along the axis of rotation can be arbitrary in magnitude; the observed MBR isotropy requires that the expansion along the axis of rotation is the same as in directions perpendicular to the axis of rotation. Note that this line of reasoning gives preference to models with non-zero initial spin in “embryonic singularity”.
The correct causal structure of space-time, that is, the absence of closed time-like curves is considered as a serious restriction on the properties of a space-time manifold with rotation. K. Gödel drew attention to the “non-physicality” of such models with rotation in his classical work [14]. S. Maitra found a simple and elegant way to describe such curves [15]. For (1) the absence of closed time-like geodesics is guaranteed when the matrix βab is positive-definite. Indeed, let a closed curve x μ ( s ) , 0 s 1 , x μ ( 0 ) = x μ ( 1 ) , be everywhere time-like, i.e., for arbitrary s ,
g μ ν d x μ d s d x ν d s > 0 .
Choose s 0 as the value of the parameter s for which d t d s = 0 . Such a point necessarily exists by assumption that the curve is closed: with the growth of s, the coordinate t at first increases (decreases), and then decreases (increases). We compute the square of the modulus of the 4-velocity vector tangent to x μ ( s ) at point s 0 for (1):
g μ ν d x μ d s d x ν d s | s = s 0 = R 2 β a b e i ( a ) e j ( b ) d x i d s d x j d s .
The right-hand side expression in (5) is always negative for a positive-definite β a b , thus contradicting the initial condition (4) that x μ ( s ) is time-like. It is important to emphasize that the presence or absence of causality is not connected in any way (except for the historic aspect [14]) with the presence or absence of cosmic rotation.
There are no other restrictions for (1).

2. Metric and Matter vs. Dark Energy and Oscillating Physics

Different cosmological scenarios can be described by the following special cases of metrics (1)
d s 2 = d t 2 2 R ( t ) B ( d x z d y ) d t R 2 ( t ) ( A ( d x z d y ) 2 + d y 2 + d x 2 ) ,
d s 2 = d t 2 2 R ( t ) C e m x d y d t R 2 ( t ) ( d x 2 + D e 2 m x d y 2 + d z 2 ) ,
d s 2 = d t 2 2 R ( t ) K e ( 1 ) d t R 2 ( t ) ( L ( e ( 1 ) ) 2 + ( e ( 2 ) ) 2 + ( e ( 3 ) ) 2 ) .
In (6)–(8) the following notation is introduced: A, B, C, D, m, K, L – const; e ( 1 ) = cos y cos z d x sin z d y ;     e ( 2 ) = cos y sin z d x + cos z d y ;     e ( 3 ) = sin y d x + d z . These metrics belong to Bianchi-II, Bianchi-III; Bianchi-IX classes respectively. In all metrics the rotation slows down with the growth of the scale factor R(t). The modified spinning fluid [4] describes the material source in these cosmological models. This is a continuum whose elements are charged particles with spin: in the modern epoch these are galaxies or clusters of galaxies [16,17], whereas in the early cosmological stages these are fundamental particles with spin [18]. In general, the spin-electromagnetic and spin-spin coupling between fluid’s elements is mediated by the scalar fields and antisymmetric tensor fields. As a result, the energy-momentum tensor Tμν encompasses the usual contributions from the electromagnetic and scalar fields and different types of interactions between the matter elements are described by antisymmetric tensors: the electromagnetic field Fμν; the third-rank tensor field Fμνα whose sources are extended objects (strings, shells, and bags) [5]; and Fμναβ that is dynamically equivalent to the scalar field. The coupled system of the gravitational field equations and the equations of motion of physical fields and their sources [19,20,21,22] can be consistently solved for the class of cosmological metrics with expansion and rotation. These models take into account the cosmological constant (Λ-term) of either Einstein type (Λ > 0, which corresponds to negative pressure and repulsion) or Gödel type (Λ < 0, which corresponds to positive pressure and attraction).
Let us consider the nearest non-stationary cosmological generalization of Gödel’s solution [19,21], when the gravitational field is described by the metrics (6) and (8).
In particular, for (6) we have [19]: for scale factor evolution (0 < A << 1, A/B << 1–constants)
R ˙ 2 = c R 2 B 4 ,
(where c is an integration constant) and for matter state equation
p =    ε = 1 κ Λ ,
where κ = 8 π G Einstein’s gravitational constant of GR and rotation is
ω 2 = ω 2 ^ 3 ^ ω 2 ^ 3 ^ = B 4 R 2 .
Analyzing (9), we see that c cannot have a negative value since B > 0. Differentiating (9) over t, we obtain
R ¨ R = c > 0 ,
consequently, this solution describes the accelerated expansion of Metagalaxy, which is apparently currently observed. The change of acceleration is of the Hubble type: the farther is the galaxy, the greater is its acceleration. It is assumed that scale factor is proportional to the distance between galaxies. The presence of acceleration and repulsion forces in a rotating Universe are absolutely natural physical factors to explain the accelerated scattering of galaxies. (Although it is more common to introduce dark matter and dark energy paradigm.) Rewrite (9) with (11) as
R ˙ 2 + ω 2 R 2 = c R 2 .
Equation (13) is very similar to the equality of kinetic energy (the sum of translational expansion and rotation) and the potential energy of the Universe (as in a tensile and rotating spring). Relation (10) reproduces the equation of state of matter in the Gödel model: Λ < 0; ε and p remain constant (as time goes on) despite the expansion and rotation. The relations (10) and (13) agree perfectly with each other in physical sense if we interpret the left-hand side of (13) as ε and the right-hand side as minus p. That is, at any time at small R and at large R the sum of kinetic and potential energy is zero. In this sense, this solution can be called stationary. It is also clear why the potential energy of gravity is proportional to R2: this is because the potential energy is computed inside a homogeneous cylinder of a perfect fluid.
A completely similar solution exists for the metric (8) [21], in this sense, such a solution can be considered as typical one for models of the Universe with rotation and expansion.
Let us analyze the cosmological model (6) in which the source of the gravitational field is a neutral spinning Weyssenhoff-Raabe fluid and a scalar field φ = φ(t) which determines the spin-spin coupling between elements of the fluid [4,19]. This model can be considered as the nearest generalization of the model (9) and (10). Indeed, in order to make the model (9) and (10) look more realistic, it is necessary to take into account that all the elements of the cosmological fluid (galaxies) are rotating. Therefore, Weyssenhoff-Raabe fluid provides an appropriate description for the cosmological matter. However, since one should also take into account the spin of galaxies, we cannot ignore their spin-spin interaction. It is known [6] that in GR rotating spherically symmetric gravitational bodies affect the rotation of each other. Such an interaction can be effectively modeled by means of a scalar field. Now instead of (9) and (10) we find (0 < А << 1, А/B << 1)
R ˙ 2 = c R 2 B 4 + 2 α R ,
p = 1 κ Λ ,      ε = 2 χ 2 S φ 2 1 κ Λ ,
where φ ˙ 2 = 12 α / R 3 , α = c o n s t > 0 , S = S 0 R 3 spin density of matter, S 0 , c o n s t , χ 2 constant of spin-spin interaction (for more details see [4,23]); it is clear that (11) is the same; c and B are positive similarly to (9).
Differentiating (14) by time, we find instead of (12)
R ¨ R = c α R 3 > 0 ,
that is, if R < ( α / c ) 1 / 3 then the acceleration is negative (deceleration); if R = ( α / c ) 1 / 3 then the acceleration is zero, finally, if R > ( α / c ) 1 / 3 the acceleration is positive; so (16) is in a full agreement with modern observations on the dependence of the acceleration parameter on the red shift (in 1998–1999 [24,25] the conclusion about the accelerated expansion of the universe was made; over the past decade, the results [24,25] have been repeatedly tested with ever-improving statistics but the main result remained unchanged: relatively recently at z < 0.5, the Universe had the transition from decelerated expansion to accelerated one).
Let us rewrite (14) with (11) as
R ˙ 2 + ω 2 R 2 = c R 2 + 2 α R .
We see that the equation again has the form of the law of conservation of energy. As compared to (13), a term corresponding to the potential energy of the gravitational field (there is no electromagnetic field) appeared in the ratio (17). However, this is exactly what was expected: to simulate (“effectively”) the gravitational interaction of spins of galaxies. The structure of (17) fully confirms the validity of this approach.
A remark is in order about the equation of state of matter. (i) For Gödel-type Λ-term we have: p < 0; ε > 0; p + ε > 0 which is to a large extent similar (10). (ii) For Λ = 0 we find: p = 0; ε > 0 – dust. (iii) For Einstein-type Λ-term: p > 0; ε ≥ 0. As one can see, there are several variants of possible states of matter, and different signs of the Λ-term are allowed. However, following the logic of gradual complication of the model the first variant looks more preferable.
In 1990 [26] an apparently periodic structure of the number of sources was demonstrated as a function of red shift in the large-scale distribution of galaxies. A possible explanation of this fact was discussed in [27]. In particular, it was noticed that the periodic structure of the universe may arise from the “oscillating physics”, e.g., oscillations of the gravitational or fine structure constants, or of a dark matter (scalar) fields. In [28] we pointed on global cosmological rotation as a possible reason of this observational effect.
Along with many known exotic and beautiful hypotheses in the history of cosmology, the cosmological rotation on the one hand does not contradict any of the known observations, on the other hand, it gives natural explanations to new discoveries in astrophysics and therefore it deserves closer attention from observational astronomy.

3. In the Theoretical Quest for Cosmic Rotation

The idea of cosmic rotation is revisited (perhaps less often than it deserves) when a certain anisotropy is detected in the Universe. The most famous attempt to explain such observations by a global rotation was due to P. Birch [29] in 1982. As it turned out later after a long discussion [30,31,32] the discovered effect, if it really exists, is not related to Universe rotation but this interpretation had nevertheless caused an explosion of interest in the theoretical development of this subject (particularly in the group of D. Ivanenko in Moscow).
Classical cosmological tests, such as apparent magnitude-red shift (m-z), number counts-red shift (N-z), angular size-red shift relations, and some other, reveal specific dependence of astrophysical observables on the angular coordinates (θ, φ) in a rotating world. Thus, a careful analysis of the angular variations of empirical data over the whole celestial sphere is necessary. The knowledge of null geodesics makes it possible to obtain the explicit form of the area distance r between an observer at a point P and a distant star S, which is a crucial step in deriving formulas for classical cosmological tests [33]. For metric (7) we derive an apparent magnitude-red shift relation (m-z).
m = M 5 log 10 H 0 + 5 log 10 z + 5 2 ( log 10 e ) ( 1 q 0 ) z    5 log 10 ( 1 + C C + D sin θ sin φ )    5 2 ( log 10 e ) ω 0 H 0 sin θ cos φ ( C C + D + sin θ sin φ ) ( 1 + C C + D sin θ sin φ ) 2 z + o ( z 2 ) ,
and number of sources-red shift relation (N-z)
d N d Ω = n 0 z 3 3 H 0 3 ( 1 + C C + D sin θ sin φ ) 3 ( 1 3 2 ( 1 + q 0 ) z 3 ω 0 H 0 sin θ cos φ ( C C + D + sin θ sin φ ) ( 1 + C C + D sin θ sin φ ) 2 z + o ( z 2 ) ) ,
here M = 5 2 log 10 L s is the absolute magnitude of a light source with an intrinsic luminosity L s and n 0 is the modern value of number density of n = n ( t ) (as usual, (19) is derived under the assumption of the absence of source evolution). The (N-z) relation describes the number of sources observed in a solid angle dΩ up to the value z of red shift. One can estimate the global difference of the number of sources visible in two hemispheres of the sky, N + , N by integrating (19):
N + N N + + N = 1 2 C C + D ( 3 C C + D ) + o ( z 2 ) .
Note that (20) is independent of z and (18)–(20) obtained for nearby galaxies with z < 1. Similar formulas are not difficult to obtain for (6) and (8).
The main difficulty of using classical cosmological tests for the discovery of space rotation is that the angular velocity of the Universe is apparently very small and it is not easy to separate the effects of the evolution of the physical properties of the sources from the effects of rotation. The only possibility for observational astronomy is to detect some angular dependencies like in (18)–(20) or in similar metrics of Bianchi-type with rotation.
An experiment similar to Foucault’s pendulum, which proves the rotation of the Earth, carried out in space rocket would be also inconclusive because of the very slow rotation.
Therefore, it is necessary to look for such effects of cosmic rotation, which would not depend on how great the velocity of rotation is, but would be possible only in the rotating Universe. Such effects were predicted in 1999 [34] for Bianchi-II (6) cosmological model.
In the geometric optics approximation light rays are null geodesics, i.e., curves xµ(λ) with an affine parameter λ and tangent vector kµ:
k μ = d x μ / d λ ,     k μ k ν ; μ = 0 ,     k μ k μ = 0 .
The metric (6) has three Killing and one conformal Killing vectors:
ξ ( 1 ) μ = ( 0 , 1 , 0 , 0 ) ,    ξ ( 2 ) μ = ( 0 , 0 , 1 , 0 ) ,    ξ ( 3 ) μ = ( 0 , y , 0 , 1 ) ,    ξ ( c o n f ) μ ξ ( 4 ) μ = ( R , 0 , 0 , 0 ) .
Therefore, we easily get four first integrals of null geodesics (21):
k μ ξ ( J ) μ = q J ,
where q J are constants along the null geodesics (J = 1, 2, 3, 4). From (22) and (23) we obtain
k 0 ^ = q 4 R ,     k 1 ^ = q 1 + B q 4 A + B R ,     k 2 ^ = q 2 + q 1 z R ,     k 3 ^ = q 3 q 1 y R .
It is convenient to use the local orthonormal tetrad h μ a (Greek indices µ, ν, ... = 0, 1, 2, 3 refer to the local coordinates; a, b, ... = 0 ^ , 1 ^ , 2 ^ , 3 ^ are tetrad indices):
h 0 0 ^ = 1 ,    h 1 0 ^ = B R ,    h 2 0 ^ = B R z ,    h 1 1 ^ = A + B R ,    h 2 1 ^ = A + B R z ,    h 2 2 ^ = R ,    h 3 3 ^ = R .
We assume that an observer has coordinates P(t = t0, x = 0, y = 0, z = 0). The null geodesics through P are labeled with the help of spherical angles θ and φ which define direction of a light ray in the local Lorentz basis of the observer at P:
k P 0 ^ = 1 ,    k P 1 ^ = sin θ cos φ ,    k P 2 ^ = sin θ sin φ ,    k P 3 ^ = cos θ .
From (25) and (26) we find the constants qJ:
q 4 = R 0 ,    q 1 = R 0 ( B + A + B sin θ cos φ ) ,    q 2 = R 0 sin θ sin φ ,    q 3 = R 0 cos θ ,
where R 0 R ( t 0 ) . Using (21), k 0 = d t / d λ , we can eliminate λ for t and convert (24) to the following system
k 0 = d t d λ = R 0 R ( 1 + B A + B sin θ cos φ ) q 0 R ,
d x d τ = q 1 R 0 B q 0 ( A + B ) + z ( q 2 + q 1 z ) q 0 ,
d y d τ = q 2 + q 1 z q 0 ,
d z d τ = q 3 q 1 y q 0 ,
where d τ d t R . Solving (29)–(31), we get the exact form of null geodesics equations for q1 ≠ 0:
x = q 2 q 3 2 q 1 2 + τ ( q 1 R 0 B q 0 ( A + B ) + q 2 2 + q 3 2 2 q 1 q 0 ) + q 2 q 3 q 1 2 cos ( q 1 q 0 τ ) q 2 2 q 1 2 sin ( q 1 q 0 τ ) q 2 q 3 2 q 1 2 cos ( 2 q 1 τ q 0 ) + q 2 2 q 3 2 4 q 1 2 sin ( 2 q 1 τ q 0 ) ,
y = q 3 q 1 cos ( q 1 q 0 τ ) + q 2 q 1 sin ( q 1 q 0 τ ) + q 3 q 1 ,
z = q 2 q 1 cos ( q 1 q 0 τ ) + q 3 q 1 sin ( q 1 q 0 τ ) q 2 q 1 .
From (33) and (34) we get
( y q 3 q 1 ) 2 + ( z + q 2 q 1 ) 2 = q 3 2 + q 2 2 q 1 2 ,
i.e., null geodesics in Bianchi-II metric (6) are circular helix and pitch of these helices that may be found from (32) (Figure 1). It is easy to see that all the loops of (35) touch the x-axis.
Time-like geodesics are circular helices too and all massive test particles in the Bianchi-II space-time (6) move along spirals [34].
Here we come to the particular case of the null-geodesics set in Bianchi-II rotating world: the existence of closed 3-curves. The necessary condition of their existence is
q 1 R 0 B q 0 ( A + B ) + q 2 2 + q 3 2 2 q 1 q 0 = 0 .
Then (32)–(34) describe periodic functions of τ. The condition (36) gives the directions (θ, φ) on the celestial sphere, along which the light rays are closed (compare (36), (27)–(28)):
sin θ cos φ = A B A + B < 0 ,
(negative values of A are acceptable for correct causality space-time structure [32]). From (32)–(34) it is clear that at the point of observation we have: τ = τ0 = 0 and t = t0, x = 0, y = 0, z = 0. Light returns to the same spatial point x = 0, y = 0, z = 0 at the moments of time
τ n = 2 π q 0 q 1 n ;      n = 1 , 2 , ...
The existence of the closed null 3-curves gives rise to a number of new observational effects in rotating and expanding models of the Universe independent on the magnitude of the angular velocity of the Universe.
In what can be considered as a cosmological lens effect, let us assume that some galaxy is located at a point on the closed light ray (null geodesic). Then an observer can see the same galaxy in the two opposite directions (Figure 2b). In a generic case the observed galaxy is asymmetrically located relative to the opposite directions of observation. As a result, a detection of the two images of identical galaxies at different distances (with different redshifts) visible at the opposite semi-spheres of the sky would likely mean that this is the same galaxy, and we therefore observe the lens effect due to rotation of the Metagalaxy. Thus, the Universe as a whole becomes a lens.
In a different situation which could be described as a cosmological shadow effect, let us assume that two galaxies (or more) are located on the closed light ray (Figure 2c). Then the observer can see only the two closest galaxies, which screen one another and the other galaxies on this ray.
The lens (which was described above) is “broken” and the image of a galaxy is not duplicated.
This would also mean (though unlikely) that we may not be able to observe many galaxies that are in shadow of other galaxies. This effect can qualitatively (at least partially) contribute to the problem of the hidden mass.
Finally, one can think of cosmological mirror effect as follows. Suppose, that there are no galaxies on a closed light ray (Figure 2a). Then an observer would be able to see his own galaxy from the different sides. Thus, provided we are living in a rotating world, we would have a chance to look at ourselves from aside. The Universe as a whole becomes a mirror (similar local mirror effect can take place in the Kerr gravitational field of a rotating compact source). An astronomer on the Earth, by discovering absolutely identical images of galaxies in the pairwise opposite directions on the sky sphere, may in fact happen to observe just one and the same native Galaxy. Additionally, there could be many such mirror reflections! This would be an observational evidence of the existence of the universal rotation, even for a small value of the cosmic vorticity. In practice of course all the closed light paths may be blocked by other galaxies, which breaks the cosmic mirror.
These effects are due to the geometry and topology of our physical space and are inherent only to the rotating (no matter how fast) world. In contrast, the standard Friedman type universe with expansion does not have closed light rays, and the same is true for general expanding non-rotating models. It is worthwhile to clarify the understanding of “the galaxy is on the closed null-geodesic”. Here we consider a null-geodesic, satisfying the condition (37) and labeled by angles θ and φ on the celestial sphere. The set of such curves forms a cone with a vertex at the observation point and intersecting with the celestial sphere along a circumference with the radius 2 / ( 1 B / A ) and the center on x-axis: x = ( A B ) / A + B the points of which satisfy the condition n (37).
The real galaxy is an extended object, its image is a spot—not a point. The observer receives a bundle of light rays from the object of observation. In geometric optics approximation, rays of light are null-geodesics: the telescope receives a truncated light cone with one base at the galaxy and another base at the telescope lens (Figure 3).
Let us call the middle ray GT in Figure 3 central, let it correspond to the central null-geodesic. The path of the rays in the formation of the image in the telescope lens with a closed central null-geodesic is shown on Figure 4a. It is clear that only a central geodesic “returns” exactly to the point from which it “started”.
Other geodesics, taking part in the formation of the image in the telescope lens, have labels which do not satisfy (37), so they are not closed. Indeed, consider the “inverse” problem: imagine that an observer on Earth does not have a telescope, but a flashlight to detect the Galaxy in the darkness of the Universe (Figure 4b). Taking into account a continuity property, null-geodesics continuously fill the light cone around the central null-geodesic.
Some of them fall on the Galaxy and they form an image on the photo-plate of the telescope (light rays coming back); the rest is scattered in the Universe; and only a bunch of null-geodesics passing through the points (37) of the segment crossing the Galaxy will return exactly to the original places.
Figure 4a,b looks like the Ouroboros—a snake eating its own tail (Figure 4c).
It is thus reasonable to search for the effects of the cosmic lens and cosmic mirror on our sky. Clearly there are many galaxies and many other opaque objects in the Universe that can block the Milky Way image. However, the snapshots of the Galaxy are sent out along the (formally infinite) set of null-geodesics (37), so there is a non-zero possibility that at least one of them can return to us.
Analyzing the catalogs of galaxies and quasars in the quest for cosmic rotation, it is necessary to take into account that the equations of null-geodesics in the real world may differ from the equations of ideal homogeneous models. Local gravitational inhomogeneities can disrupt the smoothness of curves (the observer on the Earth is located at the edge (at the tail of spiral arm) of a spiral Galaxy, in the center of which there is the strongest center of gravity, possibly a black hole).
Important, though indirect, evidence of the rotation of the Universe could be the detection of anisotropy in: (i) the distribution of the directions of the axes of galaxies, i.e., the existence of some predominant direction of orientation of the axes; and (ii) the directions of rotation of galaxies: clockwise or counterclockwise. Theoretically, the existence of such anisotropy follows from the use of the Weyssenhoff–Raabe fluid model [3] as a cosmological material medium [17,35].

4. In the Experimental Quest for Cosmic Rotation

Let us now turn to the analysis of the experimental data accumulated in several online electronic catalogs of galaxies and quasars. We focus on looking for two effects, namely the cosmic mirror effect and the cosmic lens effect, described above. These effects will show up themselves if the Earth crosses a closed null geodesic and if we manage to find two similar object images, located on two opposite sides of this null geodesic. Every pair of candidate objects must fulfill the condition of being mutually opposite on the celestial sphere.
For the cosmic lens effect, both objects in the pair must be of the same type (an elliptical galaxy or a spiral galaxy or a quasar). Their redshift may be different as those objects can actually be a double image of the same object located at an arbitrary point of the closed null geodesic.
However when looking for the cosmic mirror effect, we require both objects in the pair to have also the same redshift values (ideally, all other astrometric parameters should be the same as well). According to modern ideas, most of elliptical galaxies are created as a result of spiral galaxies collisions. On the celestial sphere we observe galaxies younger than their present state. For that reason, when searching for the effect of cosmic mirror, we limit ourselves to spiral galaxies and quasars.
For every catalog we obtain several series of pair sets characterized by their positional accuracy with which the locations of the two objects in the pair are opposite on the celestial sphere. The largest set in every series is always in the table for the cosmic lens effect. Tables for the cosmic mirror effect are subsets of tables for the lens effect.

4.1. RCSED (Reference Catalog of Spectral Energy Distributions of Galaxies) Catalog

We begin with the data of the electronic catalogue RCSED (Reference Catalog of Spectral Energy Distributions of galaxies) [36], which became public in 2016. This catalog contains spectroscopic and photometric data for 800,299 galaxies with low and intermediate redshift (0.007 < z < 0.6) selected from the Sloan Digital Sky Survey DR7 spectroscopic sample. A special program written to work with this catalog compares declination (dec), right ascension (ra), and redshift of galaxies (z). It turned out that the information about galaxies in the opposite parts of the sky with respect to celestial equator is extremely uneven. Only 65,610 galaxies (8.2%) out of 800,299 have a negative declination. That is, despite the impressive size of the catalog, we are able to use for our purpose only a small part of its volume which, of course, dramatically reduces the chances of detecting the effects of the cosmic mirror and lens. A second restriction comes from uneven angular distribution of observations done in northern and southern celestial hemispheres. Minimal value of the declination angle in this catalog is only −11.25° what means that all possible candidates for pairs are distributed close to the celestial equator. The last restriction is that for the effect of the cosmic mirror, only spiral galaxies may be taken into account.
First we look for the cosmic mirror effect. With an accuracy of 0.01° for ra and dec and 0.01 for z, we have found 187 pairs of galaxies, composed of 371 galaxies, where three galaxies are found in more than one pair (Appendix A, Table A1). To reduce the number for pair candidates we can require a better accuracy. Within an accuracy of 0.001° for ra and dec and 0.01 for z, we have three pairs of galaxies (Table 1).
All three pairs of galaxies in Table 1 may be interpreted not as real galaxies, but only as images of the Galaxy on the celestial sphere. All three pairs are not too far from us (z < 0.4), so they can be quite “fresh” pictures of the Galaxy (from the recent past), provided they are of spiral type.
Note that the accuracy in z in all three pairs in Table 1 is actually better than 0.006. The first pair has an even better match for z with accuracy value 0.001 (Table 2).
RCSED catalog [36] contains also redshift error values zerr. For example, the first and second galaxy listed in Table 2 have zerr values of 0.000106854 and 0.000167721, respectively. Other objects in this catalog have similar zerr values so our search for galaxy pairs requiring they match in redshift z up to 0.001 is reasonable. These redshift values are from SDSS DR7 and in newer data releases they may change. For example, in SDSS DR15 the first galaxy in Table 2 has a listed redshift value z of 0.0591773 with zerr value of 0.0000133428. We see that these data are mutually consistent and that newer data are more precise.
However, the spectroscopic observations are not always available to determine an object’s redshift with good precision. For faint objects it is very difficult to measure good spectra. In those cases the redshift may be estimated from colors, it is then called the photometric redshift. To illustrate possible differences in precision between outcomes of these two methods, let us take as an example the first galaxy from Table 2 (named SDSS J215545.02-070347.5). Its photometric redshift in DR7 is 0.074914 with error value zerr of 0.019358. In DR15 we find photomeric redshift of 0.076662 with zerr value of 0.017785. zerr/z ratio for photometric redshift is 0.26 in DR7 and 0.23 in DR15. This is 100–1000 times greater than the same ratio computed for spectroscopic redshift values. Our next example is the galaxy with maximum listed redshift value of z = 0.599891 (zerr = 0.0002402) in RCSED catalog. Its name is SDSS J125400.43+523835.4 and the listed photometric redshifts in DR7 and DR15 are z = 0.554839 with zerr = 0.018274 and z = 0.559419 with zerr = 0.026632, respectively. Although zerr/z ratio here is slightly better, the spectroscopic redshift value (0.6) is not even in the interval given by z and zerr values.
Next we will work also with objects having only photometric redshifts. To be safe, when filtering photometric redshifts, we should not consider the listed redshift values to be “good data” in the sense that at least several most significant digits in z values are valid. Instead, the listed photometric z values should be considered as only estimates of unknown true redshift values. We will work also with some other catalogs, which are collections of objects from various sources. In these catalogs, the situation is even worse, because the order of zerr estimations may vary considerably from one source to another and at the same time, we are not sure how good these estimates of z and zerr are.
This is why we use also an alternative approach for filtering objects when looking for the effect of cosmic mirror. In this approach for all objects with the listed redshift values we use the same estimate for zerr value equal to z/2. Redshift z then represents an interval of values from 0.5z to 1.5z, and we consider two objects to be possible pair candidates if their redshift intervals overlap. There are cases when this more conservative approach may actually prove to be more realistic compared to our first approach in which z values are considered to be accurate numbers.
Filtering RCSED catalog [36] in this way with ra and dec accuracy of 0.001° gives 18 candidate pairs for the effect of cosmic mirror (Appendix A, Table A3). Among them, there are also all three galaxy pairs from Table 1 (pairs 7,9,18).
Requiring a relaxed z accuracy, one can then require better positional accuracy for the effect of cosmic mirror. In RCSED, one has three galaxy pairs with ra and dec accuracy of 0.0005° (Table 3).
Let us note that when filtering galaxies which are not too distant (like those in the RCSED catalog) in the search for opposite pairs there is no sense trying to get better positional (angular) accuracy than 0.0001° for ra and dec parameters. For example, the galaxy SDSS J215545.02-070347.5 (first galaxy in Table 2) has ra, dec coordinates of 328.93761990, −7.06320875 in DR7 and of 328.937618974, −7.063200482 in DR15. These are actually the same coordinates because their decimal parts have only five significant digits. Although in newer catalogs decimal parts of ra, dec may have more significant digits, for extended objects like galaxies it has no meaning. For example, when looking at the image of SDSS J215545.02-070347.5 we see its angular size is about 20 arcseconds = 0.0056°. Even the most distant galaxy in the RCSED catalog (SDSS J125400.43+523835.4 mentioned above) has the angular size of several arcseconds and 1 arcsecond = 0.00028°.
When searching for the effect of cosmic lens, it is natural to remove the restriction (filter) on the equality of galaxies redshifts z. With an accuracy of 0.01° for ra and dec, we get 2127 pairs of galaxies, composed of 4174 galaxies (76 galaxies are found in more than one pair). If we restrict ourselves to the accuracy of 0.001° for ra and dec, we have 22 pairs of galaxies (Appendix A, Table A2). These are all pairs from Table A3 plus four new pairs (1,14,19,22). Within accuracy of 0.0005° for ra and dec, we get four pairs of galaxies (Table 4).
Of course, Table 4 must list all pairs from Table 3 plus those pairs which do not fulfill the condition for the redshift values from Table 3. There is only one such pair—number 4.
Note that (1) all pairs in Table 4 are relatively close to us (z < 0.3), so each pair can be actually composed of doubled pictures of one nearby galaxy (different in each case); (2) any type of galaxies may be taken into account, the only limitation is that in every pair, both galaxies should be of the same type; so all the four pairs are still possible candidates because their type is still unknown, for now we only know that these eight objects are galaxies.

4.2. Kuminski and Shamir Catalogs

Next we turn to catalogs created by Kuminski and Shamir [37,38,39,40] in 2016. These catalogs contain about 3,000,000 galaxies. Catalogs [38,40] are almost of the same size (2,911,899 vs. 2,912,341 objects) and contain mostly the same objects, differing only by morphological type parameters, calculated by computer algorithm. For our purpose, there is an angular restriction for pair candidates as the minimum declination angle is only −24.9° with only 22.03% of objects having negative declination angles. In the catalog of [39], the possible angular distribution of our pair candidates is even narrower as the minimum declination angle is only −19.7° with only 11.36% from 2,638,883 objects having negative declination angles. Moreover, a lot of objects in the catalog of [39] are classified as stars. The next complication is that many celestial objects (defined by their angular position ra and dec on the sky) are contained in these catalogs several times with different IDs. As for our purpose the position of an object is a key feature, we need to remove those redundant objects. For the catalogs of [38,39,40] we found 222,647, 18,739, and 222,655 redundant objects, respectively.
Our search in the catalogs of [38,40] gave the same resulting pair objects. Accordingly, we report only results of searching in the catalog of [40]. Without the redshift parameter in this catalog, we were looking for pair candidates with possible lens effect. With an accuracy of 0.001° for both ra and dec parameters, we find 1222 pairs of galaxies, composed of 725 galaxies (385 galaxies are found in more than one pair)1. If we restrict ourselves to the accuracy of 0.0005° for ra and dec, we have 605 pairs of galaxies, composed of 265 galaxies (176 galaxies are found in more than one pair). This is still too much uncertainty, so we require even better positional accuracy. Within an accuracy of 0.0002° for ra and dec, we get 32 pairs of galaxies, composed of 30 galaxies, where 12 galaxies are found in more than one pair (Appendix B, Table A4). For the lens effect both galaxies in a pair should be of the same type. Let us define, that a galaxy is elliptical when parameter “elliptical” > 0.5 and spiral when parameter “spiral” > 0.5. Then from 32 pairs in Table A4, only pairs 3, 6, and 8 are not composed from galaxies of the same type. We are then left with 29 valid candidates for the cosmic lens effect in Table A4.
Finally, searching in [40] for pairs with a positional accuracy of 0.00015° for ra and dec, we get five pairs of galaxies, composed of eight different galaxies and only two galaxies are found in more than one pair (Table 5).
It is interesting that every pair in Table 5 has a high probability of both galaxies being of the same type. These five galaxy pairs are thus quite realistic candidates for discovering the cosmic lens effect.
The catalog of [41] adds a photometric redshift parameter zphot for objects in [40] and this is what we need to search for possible pair candidates for cosmic mirror effect. With an accuracy of 0.001° for ra and dec and 0.01 for zphot we get 115 pairs of galaxies with 52 galaxies in more than one pair. However, for cosmic mirror effect, both galaxies in pair should be spiral. Again, let us consider galaxy to be spiral if parameter “spiral” > 0.5. Then, keeping only spiral galaxies, we are left with 77 candidate pairs for cosmic mirror effect, with 36 objects in more than one pair (Appendix B, Table A5).
If we search the catalog [41] with a better accuracy for zphot, namely 0.001, we find seven pairs of galaxies with two galaxies in more than one pair (Table 6).
Here we searched [41] without filtering out elliptical galaxies. For the cosmic mirror effect we need both galaxies in the pair to be spiral. As we can see, only the seventh pair does not fulfil this condition. However the best probability for both galaxies be of spiral type has the sixth pair. Actually this is the third pair from Table 5 and at the same time it is the only pair from [41] with opposite position accuracy of 0.00015° for ra and dec and zphot accuracy of 0.001 (Table 7).
An alternative approach (described in Section 4.1) for searching candidate pairs for the effect of cosmic mirror, using only spiral galaxies in the catalog of [41] with positional accuracy of 0.001° for both ra and dec, gives 171 pairs, with 58 galaxies in more than one pair (Appendix B, Table A6). Three pairs with the best positional accuracy of 0.00015° for both ra and dec within this approach are collected in Table 8.
This table should be the subset of Table 5 and indeed it is2.
Finally we searched the catalog of [39]. In this catalog, parameters whose values are expressed by floating-point numbers have only six significant digits. Therefore, the best accuracy we can reasonably require in right ascension angle is 0.001°. Requiring accuracy 0.001° in ra and dec with z arbitrary we obtain 513 galaxy pairs (25 galaxies in more than one pair) as candidates for the cosmic lens effect, providing both galaxy types in the pair are the same. This is too many candidates to be useful, so we have to make some refinements in our search. We begin by selecting only elliptical galaxies first which we define as objects with parameter “elliptical” > 0.5. Searching for pairs of opposite galaxies in this selection with 1,185,705 objects requiring accuracy 0.001° for both ra and dec gives 106 pairs with two galaxies in more than one pair (Appendix B, Table A7). Then we select only spiral galaxies, defined as objects with parameter “spiral” > 0.5. Searching in this selection with 274,416 objects with accuracy 0.001° for ra and dec gives zero number of pairs. Searching with accuracy of 0.005° for ra and dec gives 58 pairs with two galaxies in more than one pair (Appendix B, Table A8). One can do the same also for stars (we define them as objects with parameter “star” > 0.5). There are 569,643 stars in the catalog. Although stars are too close to us and their angular velocities may be too large to be seriously considered as members of pair candidates, possibly lying on one null-geodesic, it is interesting to do such search just to compare the number of star pairs in catalog with the number of galaxy pairs. With an accuracy of 0.001° for ra and dec one finds 30 star pairs with two stars in more than one pair (Appendix B, Table A9). One can ask why searching among spiral galaxies with their half count compared to number of stars in this catalog returned no pairs. This is explained by the difference in distribution of stars and galaxies on the celestial sphere. While galaxies are distributed evenly across the whole celestial sphere, most of the stars are located near the celestial equator. With higher density near the celestial equator, stars have a better chance to participate in pairs of opposite objects.
When searching for the cosmic mirror effect, the two opposite galaxies in our pair should be spiral. We have already seen that positional accuracy of opposite spiral pairs in the catalog of [39] is within the limit of 0.005° in ra and dec. If we add requirement for equal redshift z with accuracy of 0.01, we get six pairs (Table 9).
Some pairs in Table 9 have quite a high probability ratio spiral/elliptical to be considered as serious candidates for the effect of cosmic mirror, even if positional accuracy is not very good. If one requires a better accuracy in z, eventually one obtains the best pair in parameters ra, dec, and z (Table 10).
When using an alternative approach for filtering redshifts of spiral galaxies in catalog [39] for the effect of cosmic mirror, with the same accuracy of 0.005° for ra and dec one gets 36 candidate pairs (Appendix B, Table A10). This is of course a subset of pair candidates for the effect of cosmic lens in Table A8.

4.3. GAIA Data Release 2 Quasar Catalog

GAIA Data Release 2 [42,43,44] is based on 22 months of space observations (25 July 2014–23 May 2016). This catalog is known for its most precise astrometric parameters to date. Errors of right ascension, declination, and parallax are given in miliarcsecond units and typical error values are only fractions of mas. Specific to GAIA Data Release 2 celestial object database (compared to other catalogs we searched) is the use of the ICRS reference system, implemented by the International Celestial Reference Frame through the coordinates of a defining set of quasars. Resulting objects coordinates are given in reference epoch J2015.5 unlike old reference standard J2000.0. However, the difference between ICRS coordinates and mean J2000.0 equatorial coordinates is only ≈25 mas = 0.0000069°. This is sufficient for our purpose in the sense that it allows us to identify the same celestial objects across different catalogs, using their position coordinates. Then we can directly compare pairs found in GAIA to pairs from other catalogs.
Although the main objective of GAIA space observatory is to create detailed 3D map of objects in the Milky Way, its database contains many extragalactic objects as well. We are interested in GAIA DR2 quasar catalog database, to which we will refer as GAIADR2Q catalog. A list of 555,934 GAIA DR2 objects (their source_id parameters), matched to the AllWISE AGN catalog can be downloaded from http://cdn.gea.esac.esa.int/Gaia/gdr2/, together with the list of 2880 objects matched to the ICRF3-prototype. However, in the second list, only 935 records are unique, the rest are already contained in the first list. This gives together 556,869 unique GAIA DR2 sources, matched to quasars or AGNs. The corresponding quasar database (GAIADR2Q) with astrometric and photometric parameters (97 parameters in total) can be downloaded from GAIA archive https://gea.esac.esa.int/archive/.
Being located in space, GAIA observatory produces almost symmetrical data with respect to declination sign. This is in big contrast to the asymmetry of data that is present in all catalogs of the ground-based observatories. Therefore, with a seemingly small volume of 556,869 quasars, we have almost the same number of quasars with negative (271,063 or 48.68%) and positive (285,806 or 51.32%) declinations. Angular distribution of quasars in declination angle is from −89.77° to 89.96°. All this significantly increases the chance of detecting the effects of the cosmic mirror and lens. Unfortunately, the database does not contain redshifts yet.
According to modern ideas, quasar is a pre-galaxy, i.e., a galaxy at an early stage of evolution. Therefore, when searching for opposite pairs of quasars assuming the cosmic mirror effect is real, then the young image of our Galaxy may well be found among these pairs. With the improvement of the quality of observation equipment in the future we would see our Earth and everything that happened on it in the distant past, recorded on null-geodesic as on a film.
Searching in GAIADR2Q catalog for pairs of opposite quasars with positional accuracy of 0.01° in ra and dec coordinate gives 1945 pairs, with 22 quasars in more than one pair. Searching with accuracy 0.005° in ra and dec gives 485 pairs, with two quasars in more than one pair. To reduce the count of candidate pairs to a reasonable number, we require a still better positional accuracy. In Table A11 (Appendix C), 17 pairs of opposite quasars with accuracy of opposite positions equal to 0.001° for ra and dec are given. Unfortunately, the data do not contain the information on the redshifts of quasars, so one cannot separate the quasars involved in the effect of the cosmic mirror from the quasars in the lens effect.
Finally, with an accuracy of 0.0005° in ra and dec, we get four pairs of quasars for mirror and lens effect (Table 11). The absence of redshift slightly reduces the effect of the result, but we can say that all of these four pairs can participate in the effect of the cosmic lens.
As quasars are point-like objects on the celestial sphere, it is important that their positions in the GAIADR2Q catalog are determined with an error (of order of 1 mas = 2.78 × 10−7 degrees), which is much smaller compared to the accuracy (0.0005°) with which objects in pairs are opposite.

4.4. Milliquas 6.3 Catalog

The Million Quasars (MILLIQUAS) catalog is maintained by Eric W. Flesch [45]. We have used the version 6.3 (16 June 2019) [46,47]. This version already uses GAIA-DR2 astrometry where available (approx. 63% of all objects in database). The number of used decimals in right ascension and declination coordinates is seven, which can then (for those 63% of all objects) be considered as a number of significant digits in the fraction part of coordinates.
Milliquas 6.3 database contains 1,986,800 objects, 525,349 (26.44%) with negative declination and 1,461,451 (73.56%) with positive declination. Positional distribution is over the whole sky, with declination values from −89.77° to 89.96°, the same as in GAIADR2Q.
We first search for pair candidates for the effect of cosmic lens. It means that we are interested only in positional accuracy with which two objects in pair are opposite and ignore their redshifts. Search with accuracy 0.01° for ra and dec gives 23,024 pairs with 1037 objects in more than one pair. This is nearly 12× more compared to the same search in GAIADR2Q catalog. Next we require the accuracy of 0.001° for ra and dec. This gives 219 pairs with every object in only one pair (Appendix D, Table A12). One can now compare Table A12 with Table A11 where corresponding 17 pairs from GAIA DR2 are given. We see that pairs 1, 2 in Table A11 are the same as pairs 2, 3 in Table A12. Pair 3 from Table A11 is missing in Table A12. Then pairs 4, 5 in Table A11 are the same as pairs 8, 9 in Table A12. Pairs 6, 7 in Table A11 looks almost like pairs 27, 29 in Table A12, but they are not the same. Thus, we see that not all quasars from GAIADR2Q catalog are contained in Milliquas 6.3 catalog as well.
Search with an accuracy 0.0005° for ra and dec gives 50 candidate pairs for the cosmic lens effect (Appendix D, Table A13). Comparing Table A13 with Table 11 we see that number of corresponding pairs in GAIADR2Q is only four. Pairs 1, 2, 4 in Table 11 are the same as pairs 1, 21, 29 in Table A13. Pair 3 from Table 11 is similar to pair 23 in Table A13, but they are not the same.
Eventually we searched Milliquas 6.3 with an accuracy 0.0001° for ra and dec which gives two pairs (Table 12).
These two pairs are not found in the GAIADR2Q catalog, because it contains only the first and the third quasar from Table 12. However, the number of significant digits in coordinates of second and fourth quasar in Table 12 may be questioned. Hopefully, the GAIA DR3 database will have also these two quasars so their coordinates will be known with better accuracy.
We now turn to the search for pair candidates for the effect of cosmic mirror. Objects in Milliquas 6.3 catalog are collected from various sources so redshift values are computed by various methods and with different relative errors. Thus, some redshifts in catalog are rounded to 0.1z, while others may have error up to 0.5z. Many redshift values in Milliquas 6.3 are estimated by Eric W. Flesch using the four-color based method described in his original HMQ article [45] Appendix 2. Therefore, although Milliquas 6.3 gives redshift values for 1,906,535 objects, it is hard to compare them using some kind of fixed filter value. We used 0.2 as the best redshift accuracy we can reasonably have and even this value may be too optimistic.
We begin with an accuracy of 0.001° for ra and dec and 0.2 for z. Search in Milliquas 6.3 then gives 27 pairs (Appendix D, Table A14). Search with an accuracy of 0.0005° for ra and dec and 0.2 for z gives four pairs (Table 13).
Here we added two columns Rmag, Bmag to show also red and blue colors magnitudes to see striking similarity in these parameters in the first pair, which has also the best match in redshift.
We now repeat our search in Milliquas 6.3 database for the effect of cosmic mirror with an alternative filtering of redshift, described in Section 4.1. With an accuracy of 0.001° for ra and dec we found 135 pairs (Appendix D, Table A15). Requiring the accuracy of 0.0005° for ra and dec we found 30 pairs (Appendix D, Table A16). The pair with the best positional accuracy of 0.0001° for ra and dec is listed in Table 14. As always, every table created by alternative filtering of redshift is a subset of corresponding set of pairs for the lens effect with the same positional accuracy.

4.5. KQCG (Known Quasars Catalog for GAIA Mission)

Authors of catalog [48] compiled QSOs and AGNs from several sources, resulting in 1,842,076 objects in total. According to the authors, the purpose of this compilation is to provide positions of known QSOs, which can be used for cross matching with the GAIA observations.
After removing two objects with duplicate positions, the KQCG catalog has actually 1,842,074 unique objects, 595,535 of them with redshift value. There are 721,960 objects (39.19%) with negative declination and 1,120,114 objects (60.81%) with positive declinaton. The catalog has good declination angle distribution (from −89.82° to 89.97°).
When searching for pair candidates for the effect of cosmic lens, we first search KQCG catalog with the accuracy of 0.01° for ra and dec which gives 19,250 pairs with 849 objects in more than one pair. Search with an accuracy of 0.001° for ra and dec gives 167 pairs with three objects in more than one pair (Appendix E, Table A17). When searching with accuracy of 0.0005° for ra and dec one gets 36 pairs (Appendix E, Table A18). The best positional accuracy we can achieve here is 0.0001° for ra and dec which gives two pairs (Table 15).
Next we are looking for opposite quasar pair candidates for the effect of cosmic mirror. With only one-third of objects in KQCG database having nonempty redshift value, we can not expect big numbers of pairs. Search with an accuracy of 0.01° for ra and dec and 0.1 for z gives 109 pairs (Appendix E, Table A19). Search with an accuracy of 0.001° for ra and dec and 0.1 for z gives only three pairs (Table 16).
The last quasar in Table 16 may not have a required precision in the dec coordinate so the third pair in that table should be taken with care.
An alternative filtering of redshift for quasar pairs with a positional accuracy of 0.001° for ra and dec gives 16 pairs (Appendix E, Table A20). Three pairs with the best possible positional accuracy of 0.0005° in ra and dec are listed in Table 17.

5. Numerical Experiments

We searched several databases of galaxies and quasars for the presence of pairs of objects, with opposite location on the sky sphere and with certain angular accuracy. This is main feature of the null geodesics which reveals their existence in our part of the Universe. However, in a large sample of celestial objects, pairs of opposite objects may exist by pure chance also in a non-rotating Universe. With rotation, their number just should be greater and the difference should be due to the presence of closed null geodesics.
Therefore it would be useful if we could estimate number of pair objects in our part of the Universe considering the standard three-dimensional space without rotation using randomly generated objects positions on the celestial sphere. For each real global catalog, we can create a set of randomized counterparts with the same number of objects. Then, their filtering with a certain angular precision in dec and ra coordinates, will give us a set of counts of opposite pairs, from which we can estimate the corresponding values for the real world catalogs in the case of a non-rotating Universe.
Let us begin with the RCSED catalog. Although it contains 800,299 galaxies, they are located mostly on the northern part of the celestial sphere and only 221,502 of them, located within the interval from −11.25° to 11.25° in dec angle have a chance to participate in opposite pairs. Therefore it makes sense to create random catalogs of only 221,502 objects, located randomly in dec interval from −11.25° to 11.25°. Results of this simulation with four random catalogs are collected in Table 18.
Apparently the RCSED catalog contains a suspiciously large number of opposite pairs. However this has nothing to do with closed null geodesics. It only means that objects in the RCSED catalog within dec interval from −11.25° to 11.25° are not evenly randomly distributed in the dec coordinate. Of course this is due to the fact that data in the RCSED catalog were obtained in observatories located mostly on the northern hemisphere. The value of eps = 0.1 is not suitable for the search of closed null geodesics, because it corresponds to cone with too wide solid angle. However, it is useful for testing if our random catalogs correspond to the real world catalog3. Objects located on the null geodesic should be opposite with good angular accuracy so for our purpose values of eps from 0.001 to 0.0001 are more useful. For such eps values, our random catalogs and RCSED catalog give similar numbers of pairs. However, we should consider this with caution, because we know from results for eps = 0.1 that our random catalogs do not emulate real world data in the RCSED catalog.
Our simple experiment with evenly distributed random catalogs does not help to decide if the data in the RCSED catalog could indicate the possible presence of null geodesics. The situation is similar for other catalogs with the data from the Earth-based observatories. Fortunately, the GAIA observatory is located in space so we can expect better angular distribution of quasars in the GAIA DR2 catalog.
Let us then repeat our numerical experiment with evenly distributed random catalogs for the GAIA DR2 quasar catalog. Objects in this catalog have dec positions from −89.77° to 89.96° so our random catalogs can use the full range of dec coordinates and they should have the same number of objects as the GAIA DR2 catalog (Table 19).
Again we see that, although for small eps values, opposite pairs counts in the GAIA DR2 catalog are similar to those from random catalogs, results for higher eps values are different. This is due to the fact that objects positions in the GAIA DR2 catalog are not quite evenly distributed across the celestial sphere. Indeed, the histogram of GAIA DR2 dec angular position values shows the difference from an ideal angular distribution in dec angle, represented by the cosine function (Figure 5). We should expect also some deviation from an ideal (linear) distribution of the ra coordinate since GAIA cannot see quasars hidden behind the Milky way.
This means that our test catalogs with evenly distributed random positions on the celestial sphere are not good emulations of GAIA DR2 positions data.
Since in all real world catalogs which we processed in Section 4, the number of opposite pairs for small eps values is very small, to find some hints for the presence of closed null geodesics, we should use as good random catalogs as possible. On a large angular scale, positions in a reasonably good test catalog should copy deviations of the real world catalog from ideal angular distribution. On a small scale, its angular positions should be distributed randomly.
The following two tables summarize the results of such a simulation for two different angular distributions. The first (Table 20) is simulating an uneven angular distribution in the RCSED catalog. The second (Table 21) is a simulation of an almost evenly distributed quasars in the GAIA DR2 catalog. Every random catalog named RandomN_x where x = 1, …, 5 contains the same number of positions as its real world counterpart, distributed to an Nx2N angular mesh in dec and ra coordinates with number of objects in every cell which copies the corresponding real world catalog. In order to provide a sufficient space for local randomness, we limited the range of N to a maximum of 180 mesh cells in dec coordinate (and 360 cells in ra coordinate).
As expected, with higher number of mesh cells, the number of opposite pairs for larger eps values in random catalogs is closer to the corresponding values of the real world catalog. Thus random catalog with a more detailed angular mesh is indeed a better simulation of the real world catalog. Next, we see that the better the uniform angular distribution of the real world catalog, the less detailed mesh is needed for a good reproduction of its opposite pairs number for given eps value.
Tables with filtering results for real world catalogs together with data from of our random catalogs have a twofold purpose. Firstly, they show that our simulation catalogs can be useful as imitations of real world catalogs. Secondly, for large N values they show how the numbers of opposite pairs for the given real world catalog should look like in the case of a non-rotating Universe. To give a better prediction for the case of a non-rotating Universe, we attached two lines with results of 20 simulations using random catalogs with N = 180 for small eps values, together with corresponding standard deviation values.
In Table 17 and Table 18 we see that, in RCSED and GAIA DR2 catalogs, the observed number of pairs for small eps value does not exceed predicted values for a non-rotating Universe. However, it is not clear, if those eps values are small enough for our task, because observing null geodesics may require a very narrow observation cone. To register the presence of closed null geodesics, we need to estimate number of pairs in a non-rotating Universe for as small eps values as possible. From our random catalogs we see the feature shared with the real world catalogs, namely that number of opposite pairs for small eps values can be predicted from the number of pairs for greater eps values. If, for example, number of pairs for eps = 0.1 is 200,000, then we can expect about 20 pairs for eps = 0.001. This feature is expected for evenly distributed objects and from our simulations we see it remains valid for non-even angular distributions as well.
To observe several pairs for eps = 0.0001, we need hundreds of pairs for eps = 0.001 and millions for eps = 0.1. Therefore, let us proceed to emulate larger databases.
The biggest database we are dealing here with is Kuminski and Shamir [38,39,40]. We chose the catalog of [40], cleaned from repeated objects, containing only galaxies (and not stars like in [39]). The number of objects is 26,899,686 and dec values start at −24.93. Here we are filtering it without distinguishing galaxy type (Table 22).
From Table 22 we see that according to our random catalogs, number of pairs for eps = 0.0001 should be about seven. However, we found zero pairs in [40], although for close eps value of 0.0002 (not in the table) we found 32 pairs. On the other hand, the number of pairs for eps = 0.0005, 0.001 is too large compared to our random catalogs. Finally, while data in the first row for eps = 0.1–0.005 follow the approximate square rule dependence of pairs numbers on eps (like 7,612,266/76,867 ≈ (0.1/0.01)2), data for eps = 0.001–0.0001 do not follow this rule.
We have no explanation for these strange results of catalog [40] filtering. Our random catalogs have the same number of objects and the same angular distribution. Nevertheless, the number of opposite objects for eps values 0.001, 0.0005, 0.0001 is quite different even for detailed angular mesh. While it may be just large statistical deviations, we did not see such behavior in other catalogs.
Next we will emulate the quasar catalog Milliquas 6.3. Its angular distribution is not far from uniform so it should be easy to reproduce outcomes of its filtration with our random catalogs. The results are in Table 23.
According to our simulations, the number of opposite pairs in Milliquas 6.3 catalog for small eps values is standard for a non-rotating Universe so there is no evidence for the presence of closed null geodesics.
Our last studied database is the the KQCG quasar catalog (Table 24).
Again numbers of observed opposite pairs in KQCG catalog for small eps values are not greater than corresponding estimations from the analysis of similar random databases. We should therefore conclude that observations cannot be definitely interpreted as a consequence of the presence of closed null geodesics in our part of Universe.
We conclude this section with two remarks.
Firstly, in order to get better statistics for small eps values, we need bigger catalogs than are available today. Only then there is a hope to see them on the background of random coincidence where two objects are opposite just by chance.
Secondly, perhaps one does not need statistical methods at all. When the number of closed null geodesics is very small, say one or two, what we actually need is to observe the opposite pair in which both objects are astrophysically identified as being the images of the same cosmic object, possibly from different time of its evolution. Astrophysical analysis is even more important than the angular accuracy with which the two objects are opposite, as a light path across the Universe may be diverted by gravitational lensing of other cosmic objects. For example, let us consider the first quasar pair in Table 13. These two quasars are not only opposite with a good accuracy, but also have close values of Rmag, Bmag and redshift z parameters as well. This may well be a pure coincidence and a further astrophysical analysis may eventually prove that they are different objects. However, if one finds a similar pair and the two images are indeed from the same object, then one should seriously consider the existence of a closed null geodesic even for a single pair. If a small number of closed null geodesics is a reality, then we need to catalogize a very large number of distant cosmic objects and be lucky to find two of them located precisely on such geodesic.

6. Conclusions

The study of the rotational and chiral phenomena is of fundamental importance in astrophysics, high-energy physics, and the heavy-ion collisions [49,50,51]. In particular, it is worthwhile to mention the recent discovery of observational evidence for the coherence between galaxy rotation and neighbor rotational motions on the scale of several megaparsecs [16,17]. Moreover, the search for the similarities of vortex structures in the heavy-ion physics and in astrophysical conditions attracts considerable attention in the current research [52,53,54,55].
Here we considered possible observational manifestations of the global cosmic rotation. The goal of our analysis of modern electronic catalogues of galaxies and quasars was to find, among the observed galaxies and quasars, pairs of images that might not be real different objects, but instead that could be several images of a single object. This might have been a demonstration that the Universe is not only expanding, but also rotating. Our analysis of the number of opposite pairs did not find a clear statistical evidence of the global rotation. However, currently available catalogs are far from being perfect. They collect only a small number of very distant celestial objects with an uneven angular distribution (except for the GAIA DR2 quasar catalog). Therefore further work is needed in order to verify and improve our current results for bigger catalogs with better distribution of objects. One cannot exclude a possibility that an appearance of closed null geodesics, as a manifestation of the cosmic rotation, is rare and is not statistically significant. In that case, one needs more thorough astrophysical observations which could reveal that two opposite objects are actually images of the same object, possibly in different epoch of its evolution.

Author Contributions

Conceptualization, methodology: Y.N.O. and V.A.K.; validation, formal analysis, investigation: V.A.K. (analytical part) and E.M. (numerical part); software and data processing: E.M.; writing—original draft preparation: V.A.K. (analytical part), Y.N.O. (analytical part), E.M. (numerical part); writing—review and editing: V.A.K., E.M., and Y.N.O.; supervision, project administration: Y.N.O. All authors have read and agreed to the published version of the manuscript.

Funding

For Y.N.O. this work was partially supported by the Russian Foundation for Basic Research (Grant No. 18-02-40056-mega).

Acknowledgments

Y.N.O. and V.A.K. are grateful to Inna Sandina for the fruitful discussions and Danil Kuznetsov for the stimulating interest in the work and helpful advice. The authors thank the anonymous referees for the valuable comments and suggestions.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

Table A1. Pairs of galaxies for the effect of cosmic mirror [36]. Id SDSS DR7 is objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.01° for ra and dec and 0.01 for z. Pair is a valid candidate for the effect only if both galaxies in the pair are spiral.
Table A1. Pairs of galaxies for the effect of cosmic mirror [36]. Id SDSS DR7 is objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.01° for ra and dec and 0.01 for z. Pair is a valid candidate for the effect only if both galaxies in the pair are spiral.
Id SDSS DR7RaDecz
(1)5877271779153019896.31801788−11.128996740.164466
587734894366425177186.321595711.129454610.167687
(2)587727177911304300356.9463884−11.11316640.11336
587734894362427519176.936208211.121831860.10569
(3)5877271779153675676.473508−11.089789670.109483
587734894366490714186.475218311.090988060.103002
(4)5877272251574518229.78490157−10.975861230.0656358
587732772669292660189.780389210.981782720.0667346
(5)58772717792106912519.65932903−10.933686090.141083
588017569241432335199.668954710.930442050.145941
(6)58772717792106911919.64324854−10.902078660.140991
588017569241432309199.642177610.898854530.149119
(7)58772722515777957510.47400922−10.893148550.115176
587732772669554736190.469376710.883494830.113392
(8)587727873696006248354.4140938−10.767287710.0980759
587732772662673520174.407839110.765807670.0941737
(9)5877271784506656332.84320186−10.720608490.0626986
587734893828112436182.83476810.729013710.0684232
(10)5877271784534835859.28275768−10.701005630.130024
587734893830864981189.282124510.705623350.128155
(11)5877271784514520244.57186615−10.671350560.061156
587734893828833380184.56840710.666314340.0513871
(12)5877272256935363617.83605949−10.599583840.0905357
587732772131569821187.825996410.594699040.0927822
(13)58772722944796276923.72675428−10.511705480.0992561
587736543623839915203.734794310.505104640.103054
(14)58772717845833326720.61377325−10.473150780.109546
587736543622529169200.616495210.481165330.10888
(15)58772722944802828623.8634114−10.44203830.0872344
587736543623905435203.864496510.449233290.083543
(16)587727874233335894355.4913684−10.413562560.0980471
587732772126261484175.496781910.4197770.106774
(17)58772717845964409923.56363216−10.306766410.087484
588017992299970676203.56036410.298263660.0842322
(18)587727874232942672354.5407195−10.29051620.115404
587732772125802696174.531194110.299067880.112148
(19)5877271789892404636.75089715−10.284526720.141096
587734893292879988186.754157210.29058590.138833
(20)58772717845977509523.99049618−10.246623290.16844
588017992300167307203.985496910.236708310.159882
(21)58772717899232061813.92268478−10.237982630.0545347
588017991758970941193.928692510.239196180.0576316
(22)587726877275324641341.4198226−10.227753390.0854893
587732772657037548161.425025810.22886030.0864014
(23)58772717845977505823.95379218−10.208922270.170874
588017992300167290203.960532910.212632850.16155
(24)58772717845977505923.9542841−10.208634180.167602
588017992300167290203.960532910.212632850.16155
(25)58772717899238615214.00945119−10.169388340.0582089
588017991758971000194.015463210.170117850.0547655
(26)587727178983473249353.3035011−10.113140130.175384
587734893287112784173.305433710.108025980.171464
(27)587730815751291175336.940361−9.953736770.0563963
587734864295166164156.93945999.945074020.046747
(28)5877271795267010908.03597019−9.911947470.0790208
587734892756598903188.03616629.915731240.0763644
(29)58772717899533525320.82821245−9.861580210.128453
588017991761920153200.82554289.860313090.125208
(30)587730815750045949334.0325931−9.853536580.349603
588017702382403743154.02775869.851577640.356585
(31)58772717899644930423.51060112−9.774145380.0408057
588017991763099797203.52069789.783985720.0398321
(32)587730815749980360334.0089931−9.763519380.0919012
588017702382403698154.0168969.763747580.0834979
(33)58772717953135412218.84193042−9.691971990.0623454
588017991224197282198.85069249.70029040.0665447
(34)58772723052131131022.64387404−9.636323120.121776
587736542549704781202.6450849.643601080.123518
(35)587730815749128409331.9847156−9.623538450.105015
587735342655471787151.98401699.632332160.114253
(36)587730815748866357331.395196−9.606404270.0800119
587735342655209578151.38702559.606937880.0783352
(37)587726877811934142340.8324856−9.591919860.439336
587732772119904870160.83940929.597420670.433453
(38)587730816827326649342.3741793−9.565745010.0821777
587746210520367211162.36999379.566091920.0862479
(39)58772723052216336224.74103715−9.563254330.0862579
587736542550556886204.74963869.556919820.0868169
(40)58772717792945777639.13882791−9.519910560.081008
588017702410453175219.14576229.518369980.088222
(41)5877271800618680494.08337588−9.463681270.0872652
587734892218023997184.08192299.458840630.094822
(42)58772723052242549525.33108177−9.421300980.104163
587736542550818989205.33722849.418746750.0996079
(43)58772717792978559139.92521168−9.338311910.0532149
588017702410846353219.91757129.343915730.0516048
(44)58772717793011315440.58369451−9.326300040.155765
588017702411108409220.59083959.32010340.1517
(45)587730816826212465339.7809475−9.284731910.0631319
587734863222669402159.77262179.277767670.0686803
(46)58772723052426048129.54831241−9.280658850.0912357
587736542552653969209.54054739.274955930.0982227
(47)587730816287113330334.562248−9.279263770.097067
587732772117217405154.56033129.275253840.1006
(48)587726877808787613333.5596377−9.108730.125766
587732772116758696153.55479959.106775810.117325
(49)58772722730880215518.62293278−9.091167390.0828624
587736541474193627198.62596199.086560380.0928012
(50)58772723105883764024.24960403−9.072208040.116431
587736542013489445204.2589429.070938910.122971
(51)58772723105903422124.70990827−9.0507730.0766974
587736542013685982204.7076679.056882150.0775408
(52)58772424068838203438.55751214−8.988336280.111355
588017992306458874218.5636338.992732530.116682
(53)587727180595134577355.7997977−8.9877120.0753556
587734891677548628175.80780128.986223820.0829316
(54)587727180594479192354.2791585−8.987258680.0806326
587732770515124373174.2775348.996256890.0760123
(55)587726878886723739343.2696496−8.985475790.170464
587732771047211211163.27578348.992390730.166468
(56)587726879426150556349.2821756−8.91390550.0837956
587732770512961595169.27656438.909519150.0850834
(57)587726877269229794327.3372251−8.884898790.0911886
587732772650942640147.33419828.88461790.0977782
(58)58772722784351034913.58226852−8.826691940.0775205
587732769986576485193.57764338.823246460.0802786
(59)5877272278415443349.17385956−8.813433670.161788
587732769984675893189.18008298.815706190.165148
(60)5877272278408233647.41474543−8.787397860.134174
587732769983889559187.41737968.792603190.141713
(61)587726879424774256346.0556579−8.748182090.097193
587732770511519918166.05666858.7403580.0875037
(62)58772718007150190226.40085465−8.731072120.121164
588017990690603214206.39247888.733867690.120342
(63)587726877268902222326.6199672−8.720260280.135262
587734863753838722146.62459958.712667020.134986
(64)58772722784298612212.49783122−8.701985550.0739487
587732769986117760192.50437838.696103530.0837418
(65)587726878345265364332.6514376−8.662532290.103393
587732771579494537152.64587068.664857510.0973786
(66)587730817361182802335.363148−8.647166090.0373209
587734862683832453155.35975558.644389590.0450276
(67)58772424068910296940.22878711−8.643695170.0380482
587736543094177958220.22066798.646751970.0308106
(68)587726877268312182325.2592388−8.637772960.0863781
587732772650025128145.25090338.639710770.0846345
(69)58772722784351031613.66287605−8.632553570.0778701
588017730839642176193.66265038.639586890.0827405
(70)587730818439512260346.092724−8.505600840.172473
587734891673354322166.0946458.513690650.175036
(71)58772723159747800428.3056497−8.420972870.0471675
587736541478387896208.31271378.414979840.0450439
(72)587730818439053410344.9876466−8.397451080.0779055
587734891672895587164.98948618.395645620.073904
(73)587727212272943399320.8338911−8.394852180.117779
587734948049912071140.84223138.396613840.11207
(74)587727212273271140321.5992953−8.375442290.158414
587734948050239774141.60726988.38329730.152886
(75)58772424015570550748.22814118−8.335813160.0735155
587736477058990249228.23781538.339402350.0804487
(76)58772717793430745350.24970865−8.299513750.0330458
588017702952173868230.24884428.3084330.0393475
(77)587726878343823623329.4183427−8.276734360.148898
587732771578052774149.41344998.275279560.15852
(78)587726877804855542324.4594709−8.212099110.0871732
587732772112826583144.45746748.202282440.0867666
(79)58772717846980214546.9844432−8.182249690.0744629
588017992310128884226.99323758.190299980.0765801
(80)587726878882332761333.2058322−8.140839530.0837303
587732771042820325153.21514818.13694310.0842566
(81)58772723213487317729.37707349−7.946395680.10361
587736526444560520209.37935867.944878030.103126
(82)58772723213474206429.16828094−7.913039270.0989567
587736526444429478209.16353517.917499150.103407
(83)587730817895629069329.9017868−7.787375550.0856975
587732771041378462149.89295097.786689110.0914809
(84)587726878341726580324.6422074−7.751100020.0880603
587732771576021174144.63790177.742391990.0931418
(85)587730818435317875336.3522559−7.75086160.109669
587732769970454629156.34341737.741771210.103487
(86)587726877264839112317.3101898−7.738548530.0732018
587734948048404618137.31475167.729458070.0821823
(87)587730818433155251331.3679425−7.456236430.0592957
587734861608321127151.37135187.464006060.0621482
(88)58772424069454249152.76892686−7.356457150.138503
587736543099617547232.76010597.347224260.129012
(89)58772424176500748045.13170845−7.346407250.161412
587736542022598934225.12856797.344915870.154524
(90)58772717900831133750.7121024−7.327391040.0845821
588017991774896410230.71903417.33497470.0783759
(91)58772424069467350052.96494777−7.255339890.133988
587736543099683339232.9592737.247277060.128557
(92)587730818432041321328.9376199−7.063208750.0592138
587732579378724931148.93835867.062511790.0602015
(93)587726879954370763329.3517007−7.046377150.0859399
587732579378921673149.34403197.039887610.0784532
(94)58772424230253369946.63534829−6.837865110.028473
587736541486383232226.62840516.839054190.0376736
(95)587726879953322317327.0174505−6.788483360.0898924
587732579377873009147.01283836.797907070.0932444
(96)58772718008166002649.76721128−6.654629970.171731
588017990700826690229.77513326.652652760.17779
(97)587726879952732589325.6644648−6.534242170.0882166
587732703396430084145.67180536.528028640.0933982
(98)58772717901119509257.28409909−6.51468570.0656059
588017991777780111237.29167356.517113380.0727241
(99)587726879413371300319.8134098−6.41539010.0633028
587732770500116746139.81423446.409378910.0725347
(100)587726879950045526319.5050187−5.899923850.0887226
587732578837725451139.50352745.893147520.090063
(101)58772424123469020260.1469532−5.822466070.128101
587736542566023562240.14425995.815441370.120623
(102)58772424177110239159.12398332−5.67499180.0597927
587736542028693952239.11633885.665559380.0621059
(103)58772424177070904158.1813494−5.664028690.124912
587736542028300451238.18149685.668963830.132543
(104)58772424230744895057.81327563−5.441936490.12508
587736541491298661237.81394525.43203150.117586
(105)58772718008591988259.45274909−5.377769390.112835
588017990705086634239.4596865.37307170.114617
(106)587726879409635950311.3361796−5.130252470.100322
587732703390204231131.33587945.129422690.0944183
(107)587727214952710451310.6951216−5.009573180.0533286
587732578833859013130.70398915.015903730.0516006
(108)587726879408915112309.7014626−5.003786110.0822278
587732703389483300129.71023735.00382620.0759225
(109)58772718062252872558.85069515−4.882133140.0739501
587730023336902902238.84273524.883953620.0681825
(110)587724650869686415184.6727131−1.256201330.0803352
5877311878165955284.666794871.260855560.0876818
(111)587724650869424248184.0892593−1.253719140.108284
5877311878163334174.093251261.257893160.105123
(112)58801550767108958128.14765501−1.220611060.174438
587726031187148932208.14792721.225780360.174331
(113)587729971792052457228.9119379−1.220295570.120609
58773151422958810648.917975591.215845660.112363
(114)587748927626608776172.1310313−1.215042230.113913
587731187811090545352.12252991.219741340.119537
(115)587734303270764663341.5351663−1.205034730.0583548
587726031166701712161.52518991.203288150.0671257
(116)587722981747458159196.8323221−1.104951170.0858605
58773151421556341816.840685081.103836950.0941228
(117)58801550767698755041.62918519−1.097996910.0740636
587726014013440008221.62643221.104137370.0735006
(118)587722981750931657204.8306575−1.063698370.0707877
58773151421903679524.831548181.067104080.078975
(119)58773151153245404319.75110345−0.996749320.044761
587722984433057983199.75369730.996689310.0541101
(120)587731185125359869349.0427567−0.987457570.091398
587748930309587102169.05249290.99315950.0962012
(121)587731185116315784328.3852669−0.924914130.0954053
587728950120612023148.38996180.923610810.0948823
(122)588848898860056883228.2742186−0.908325710.129659
58801551036422579748.279570410.914912840.131263
(123)587731185119723924336.1672691−0.897117790.0991819
587728950124019803156.16663190.893734350.0938088
(124)588848898847539378199.7024399−0.89268270.0818081
58801551035170832219.707760730.899087920.0800928
(125)588848898830172339159.9730114−0.855679020.0908891
587734305954463933339.96684260.863194130.0873762
(126)588848898843279472189.8915184−0.853662770.0242445
5880155103474483399.895146850.859968260.0146933
(127)587728947975291011153.3264231−0.83272210.0490744
587731187265962312333.32130350.838532450.0571919
(128)587729778526781840234.5183544−0.82621510.144058
58773151369514208054.526636340.824508860.139076
(129)587722982293176752217.0007899−0.822915350.399234
58773151368747443736.997224150.817552090.397598
(130)587734303804031262333.3022362−0.805199860.129767
588848900974706861153.30350570.809915030.120938
(131)587722982300123441232.9101284−0.791679340.0766653
58773151369442119652.905492750.791020860.0856506
(132)587722982285508856199.6286217−0.787457390.0882834
58773151367987213819.623758170.787982760.0898079
(133)587734303805866210337.4814454−0.75171740.0903008
588848900976541910157.48561250.758983640.099725
(134)588015508192624827353.1832021−0.748267340.0574768
588848900983423150173.17674120.756308830.0674753
(135)58801550820324162017.35657948−0.734516210.103116
588848900994039948197.36141640.728668730.0944896
(136)58801550821687323748.61535843−0.716405990.0873679
588848901007671746228.60742050.709484510.0915592
(137)58801550821890463453.23858074−0.704144710.0331694
588848901009703095233.24196170.710688080.0378785
(138)587722982280593611188.3839313−0.692064060.0716765
5877311872812975238.387611850.692659650.0666569
(139)587748928166887621179.9836371−0.683067590.0800753
587731187277627630359.98492220.675102420.0848031
140)58801550821228557638.07654238−0.660919860.152935
588848901003084008218.08338980.657823540.148348
(141)587722982275154175175.8949958−0.631549720.0777659
587731187275858096355.90251560.634556710.0830219
(142)588848899399090470233.1265588−0.603185610.0834699
58801550982951744653.124241420.608993840.0888991
(143)588848899394830608223.4164629−0.597939010.0757585
58801550982525758243.42424850.597796670.0674929
(144)587731172768809842312.9796387−0.59579540.0542642
587725075526189422132.97351060.588603980.0512173
(145)588848899367698514161.4291165−0.593213240.0850623
587734305418182948341.43533930.598693450.0853401
(146)588848899377791164184.5422801−0.549417290.0806642
5880155098082182044.533342220.543204360.0883403
(147)588848899381395678192.8322052−0.543590220.0823546
58801550981182266112.826614960.549381950.0838967
(148)58773151207057007422.52506877−0.538868510.0752843
587722983897432095202.52923890.531445610.0753058
(149)588848899365142729155.7201062−0.529031440.161499
587734305415692542335.72668320.527496150.161789
(150)587731185656922427336.925539−0.521880820.142554
587728949587476580156.93185510.514208340.147027
(151)58773151208570896357.24298571−0.521099910.0398141
587722983912636747237.24017980.519483360.0326624
(152)588848899365208074155.7345006−0.520484830.152984
587734305415692542335.72668320.527496150.161789
(153)58773151207502662332.7941736−0.516971220.104186
587722983901888776212.78746430.517576550.10494
(154)58773151207057008022.53092957−0.513012860.0754378
587722983897432249202.52900720.509605050.0842453
(155)588848899367567503161.1736846−0.503545580.116059
587734305418117322341.17134090.498875290.10686
(156)587731185663738041352.414892−0.495597340.0610491
587748929774223562172.42125360.491962920.0617746
(157)587731185657315605337.8171047−0.467148830.089706
587728949587869853157.81655830.468753880.0972739
(158)588848899365011498155.3428317−0.466218610.0616145
587734305415561419335.34041110.467821040.0585452
(159)587731185654825160332.0329757−0.453054310.0966575
587728949585314077152.03162550.445155990.096859
(160)588848899369599194165.777933−0.432321610.408335
588015509800026315345.76814310.428890550.406645
(161)588848899392012585217.0954067−0.429219540.13692
58801550982250507837.100560050.422632470.13819
(162)587748928964460711162.3131562−0.41267410.0631013
587731186733023422342.30724560.409604750.0687452
(163)58801550873847410213.66692127−0.407111970.0431046
588848900455530621193.65877780.409779960.048432
(164)58801550874424126826.76827379−0.396582620.0921257
588848900461232473206.77141060.399992810.0904585
(165)587730846891377212322.2164035−0.365306060.0524633
588848900432986349142.22016170.36295340.0564073
(166)587722982818971844191.8125016−0.358987760.126037
58773118674593395611.820365410.358180730.117239
(167)58801550874103000119.45674285−0.340824640.0468492
588848900458086553199.45996980.349948190.048473
(168)588015508727529597348.5866992−0.332494660.0806627
588848900444520624168.58703720.323654690.0784033
(169)588015508727136416347.7652686−0.325764890.0983488
588848900444192846167.77332210.325337210.0968191
(170)587730846890132061319.3488484−0.312747780.0580555
588848900431741191139.34146360.321914690.0540347
(171)587734304341426385334.4289933−0.290725370.0946837
588848900438360204154.4348980.29515350.0948182
(172)587734304340640020332.7387428−0.263015950.180004
588848900437573950152.73594840.256331470.186205
(173)587722982822772953200.5231175−0.257127760.0755962
58773151314339437720.525390830.261063390.0739661
(174)587730846892097872323.7534775−0.255803720.23578
588848900433641580143.75088710.250872060.230907
(175)587722982818578564190.8538024−0.255747110.108047
58773118674554078110.852309430.252720550.108846
(176)587728948512882891155.0284918−0.248939390.0946879
587731186729877696335.02728750.251188870.0976836
(177)587722982829588620215.9538986−0.248349380.0845643
58773151315014467035.963092740.244299160.0878976
(178)587728948515438760160.8665986−0.244083330.0612075
587731186732433592340.86555530.236434330.0602342
(179)587728948515438760160.8665986−0.244083330.0612075
587731186732433605340.87501380.242610570.0586656
(180)587722982817202308187.6635646−0.215130050.112303
5877311867441644497.65533430.215550440.114242
(181)58773151260999696628.42414238−0.207666320.116707
587722983363117368208.42291630.209472980.116208
(182)58773151262035162652.0486704−0.189382220.0851471
587722983373472062232.05813740.183686290.0798633
(183)588848899901030743153.4421478−0.133584120.096212
587734304877838573333.44618830.141018770.0959779
(184)588848899931963795224.1292204−0.101507060.37695
58801550928864896944.128540880.100907970.37109
(185)58773151262061391952.66581732−0.083917740.0747399
587722983373733982232.65674340.091447940.0712474
(186)58773151261655055543.4068315−0.029227210.0437379
587722983369670840223.40848830.020993280.0447694
(187)587731186188353997324.4826276−0.020633340.0912575
587725074994364633144.48020560.021658780.0908592
Table A2. Pairs of galaxies for the effect of cosmic lens [36]. Id SDSS DR7 is objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.001° for ra and dec. The redshift z is indicated to make it easier to imagine the degree of asymmetry of the galaxy’s location on the circumference (null-geodesic) with respect to the observer on Earth. The candidate pair is valid if both galaxies in a pair are of the same type.
Table A2. Pairs of galaxies for the effect of cosmic lens [36]. Id SDSS DR7 is objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.001° for ra and dec. The redshift z is indicated to make it easier to imagine the degree of asymmetry of the galaxy’s location on the circumference (null-geodesic) with respect to the observer on Earth. The candidate pair is valid if both galaxies in a pair are of the same type.
Id SDSS DR7RaDecz
(1)5877272278395782544.62339359−8.827764690.104797
587732769982709974184.62306098.8268890.481948
(2)587726877268312182325.2592388−8.637772960.0863781
587732772650025129145.25841888.637479870.182619
(3)58772722784580415119.00360312−8.538416520.125065
587736540937519274199.0032748.537676480.0523744
(4)587726879422152915340.0250784−8.325768710.211179
587732770508898457160.02433958.325799670.46663
(5)587727212272943498320.9004948−8.317340320.118795
587734948049977496140.9007378.316613740.132243
(6)58772424069185539546.59481629−8.022870190.119974
587736543096930620226.59463688.02348980.141641
(7)587726879416648036327.4332687−7.244825580.120794
587732770503458957147.43340357.245107240.172101
(8)587730818432041321328.9376199−7.063208750.0592138
587732579378724931148.93835867.062511790.0602015
(9)587726879951618320323.0326676−6.245427160.0835999
587732703395315898143.03246896.24639790.152482
(10)587726879409635950311.3361796−5.130252470.100322
587732703390204231131.33587945.129422690.0944183
(11)58772424230843209360.09682211−5.06705920.139483
587736541492281504240.09613975.067986050.161674
(12)587722981749620931201.8110903−1.152908910.0672574
58773151421772611521.810947881.153326550.0825304
(13)588848898840002840182.4290969−0.990102770.101162
5880155103441716062.429459510.990833290.0595599
(14)58773151154234998042.40717742−0.842458130.023485
587722984443019391222.40757110.842799220.211561
(15)588848899373138039173.8651882−0.554996350.105208
588015509803565082353.86519820.554708510.241575
(16)5880155087361148248.29352396−0.350228690.0803753
588848900453171355188.29374590.351191130.091289
(17)587722982819365056192.6996089−0.240231070.082066
58773118674632719712.699232440.239657120.243192
(18)588848899913482481181.8731715−0.202868980.086595
5880155092701677181.873154520.202275520.139049
(19)587731186189599168327.3151749−0.174729730.475598
587725074995609777147.3144630.174848140.0847248
(20)58773151260881725325.67986817−0.108625770.114301
587722983361937622205.67992420.109440990.0763704
(21)588848899931963795224.1292204−0.101507060.37695
58801550928864896944.128540880.100907970.37109
(22)588848899903258801158.5313323−0.048173150.0649146
587734304880066822338.53052290.048195630.212747
Table A3. Pairs of opposite galaxies for the effect of cosmic mirror [36]—alternative approach. Id SDSS DR7 is objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.001° for ra and dec and z/2 for z.
Table A3. Pairs of opposite galaxies for the effect of cosmic mirror [36]—alternative approach. Id SDSS DR7 is objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.001° for ra and dec and z/2 for z.
Id SDSS DR7RaDecz
(1)587726877268312182325.2592388−8.637772960.0863781
587732772650025129145.25841888.637479870.182619
(2)58772722784580415119.00360312−8.538416520.125065
587736540937519274199.0032748.537676480.0523744
(3)587726879422152915340.0250784−8.325768710.211179
587732770508898457160.02433958.325799670.46663
(4)587727212272943498320.9004948−8.317340320.118795
587734948049977496140.9007378.316613740.132243
(5)58772424069185539546.59481629−8.022870190.119974
587736543096930620226.59463688.02348980.141641
(6)587726879416648036327.4332687−7.244825580.120794
587732770503458957147.43340357.245107240.172101
(7)587730818432041321328.9376199−7.063208750.0592138
587732579378724931148.93835867.062511790.0602015
(8)587726879951618320323.0326676−6.245427160.0835999
587732703395315898143.03246896.24639790.152482
(9)587726879409635950311.3361796−5.130252470.100322
587732703390204231131.33587945.129422690.0944183
(10)58772424230843209360.09682211−5.06705920.139483
587736541492281504240.09613975.067986050.161674
(11)587722981749620931201.8110903−1.152908910.0672574
58773151421772611521.810947881.153326550.0825304
(12)588848898840002840182.4290969−0.990102770.101162
5880155103441716062.429459510.990833290.0595599
(13)588848899373138039173.8651882−0.554996350.105208
588015509803565082353.86519820.554708510.241575
(14)5880155087361148248.29352396−0.350228690.0803753
588848900453171355188.29374590.351191130.091289
(15)587722982819365056192.6996089−0.240231070.082066
58773118674632719712.699232440.239657120.243192
(16)588848899913482481181.8731715−0.202868980.086595
5880155092701677181.873154520.202275520.139049
(17)58773151260881725325.67986817−0.108625770.114301
587722983361937622205.67992420.109440990.0763704
(18)588848899931963795224.1292204−0.101507060.37695
58801550928864896944.128540880.100907970.37109

Appendix B

Table A4. Pairs of galaxies for the effect of cosmic lens [40]. Id SDSS DR8—objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.0002° for ra and dec. Candidate pair is valid if both galaxies in pair are of the same type (elliptical or spiral).
Table A4. Pairs of galaxies for the effect of cosmic lens [40]. Id SDSS DR8—objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.0002° for ra and dec. Candidate pair is valid if both galaxies in pair are of the same type (elliptical or spiral).
Id SDSS DR8RaDecEllSp
(1)123767888852123704726.56961575−3.0366908820.0096710.990329
1237674469000413487206.5697433.0367617040.1472950.852705
(2)1237655693017940456231.1268834−1.2373627860.1539290.846071
123764658823397420051.127070111.2373920670.4712050.528795
(3)1237655693017940456231.1268834−1.2373627860.1539290.846071
123765781527547115351.127068871.2374004960.523820.47618
(4)1237655693017940456231.1268834−1.2373627860.1539290.846071
123765300061392121951.127077461.2374022140.4862340.513766
(5)1237655693017940456231.1268834−1.2373627860.1539290.846071
123765991551172641351.127075851.2374040450.4840720.515928
(6)1237655693017940456231.1268834−1.2373627860.1539290.846071
123766651688183400051.127043691.2374125660.5459120.454088
(7)1237655693017940456231.1268834−1.2373627860.1539290.846071
123766666293778879951.127066961.237419680.4929490.507051
(8)1237655693017940456231.1268834−1.2373627860.1539290.846071
123766333001125104251.1270831.2374271340.5020220.497978
(9)1237655693017940456231.1268834−1.2373627860.1539290.846071
123766640953851948951.127081461.2374452280.4615940.538406
(10)1237655693017940456231.1268834−1.2373627860.1539290.846071
123766634081471695051.127081411.2374512080.4357340.564266
(11)1237655693017940456231.1268834−1.2373627860.1539290.846071
123765758763938643851.127082431.2374576950.4322670.567733
(12)1237655693017940456231.1268834−1.2373627860.1539290.846071
123765758763932109651.127078841.2374579190.4579140.542086
(13)123764674874002664262.17470211−1.0167921470.0956110.904389
1237648705676575427242.17488811.0167164510.0929520.907048
(14)123764674874002664262.17470211−1.0167921470.0956110.904389
1237648705676575428242.17488451.016720360.0782520.921748
(15)123765299793441622562.1747088−1.0167873970.1081730.891827
1237648705676575427242.17488811.0167164510.0929520.907048
(16)123765299793441622562.1747088−1.0167873970.1081730.891827
1237648705676575428242.17488451.016720360.0782520.921748
(17)123765776104816668362.17471976−1.0167865180.1143830.885617
1237648705676575427242.17488811.0167164510.0929520.907048
(18)123765776104816668362.17471976−1.0167865180.1143830.885617
1237648705676575428242.17488451.016720360.0782520.921748
(19)123764674874002664362.17475111−1.0167814190.0976820.902318
1237655551815123559242.17494571.016715260.0830470.916953
(20)123764674874002664362.17475111−1.0167814190.0976820.902318
1237648705676575427242.17488811.0167164510.0929520.907048
(21)123764674874002664362.17475111−1.0167814190.0976820.902318
1237648705676575428242.17488451.016720360.0782520.921748
(22)123764982393798679762.17470476−1.0167781570.1097120.890288
1237648705676575427242.17488811.0167164510.0929520.907048
(23)123764982393798679762.17470476−1.0167781570.1097120.890288
1237648705676575428242.17488451.016720360.0782520.921748
(24)123764658555446939462.17471262−1.0167595570.0879260.912074
1237648705676575427242.17488811.0167164510.0929520.907048
(25)123764658555446939462.17471262−1.0167595570.0879260.912074
1237648705676575428242.17488451.016720360.0782520.921748
(26)123766649702809641162.17470133−1.0167243430.069740.93026
1237648705676575427242.17488811.0167164510.0929520.907048
(27)123766649702809641162.17470133−1.0167243430.069740.93026
1237648705676575428242.17488451.016720360.0782520.921748
(28)123766649702809641262.17471675−1.0167112220.076330.92367
1237648705676575427242.17488811.0167164510.0929520.907048
(29)123766649702809641262.17471675−1.0167112220.076330.92367
1237648705676575428242.17488451.016720360.0782520.921748
(30)1237655177615638991128.668235−0.9265472640.8832250.116775
1237649942587703356308.66814110.9264188980.8616310.138369
(31)1237648720134537650128.6682518−0.9265383680.9026990.097301
1237649942587703356308.66814110.9264188980.8616310.138369
(32)1237648720166060380200.6434022−0.8661109510.9891740.010826
123766348035325973220.643393860.866310350.9583590.041641
Table A5. Pairs of galaxies for the effect of cosmic mirror from catalog [41]. Id SDSS DR8—objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.001° for ra and dec and 0.01 for zphot. All galaxies are spiral (defined as those with parameter spiral (sp) > 0.5).
Table A5. Pairs of galaxies for the effect of cosmic mirror from catalog [41]. Id SDSS DR8—objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.001° for ra and dec and 0.01 for zphot. All galaxies are spiral (defined as those with parameter spiral (sp) > 0.5).
Id SDSS DR8RaDecSpZphot
(1)1237650372102127690201.8110906−1.15293320.800.061
123766666292494350221.810933251.153321430.560.070
(2)1237650372102127690201.8110906−1.15293320.800.061
123766023765262349921.810933661.153333170.640.071
(3)1237650372102127690201.8110906−1.15293320.800.061
123766320546060708121.810925281.153335620.610.070
(4)1237650372102127690201.8110906−1.15293320.800.061
123766001002106069521.81093531.153338280.580.069
(5)1237650372102127690201.8110906−1.15293320.800.061
123766001002106069321.81094261.15333830.580.069
(6)1237650372102127690201.8110906−1.15293320.800.061
123764658822112887421.810952511.153343380.540.071
(7)1237651709428367652201.8110888−1.15291420.820.079
123765690904897142421.810940411.15331580.590.072
(8)1237651709428367652201.8110888−1.15291420.820.079
123766666292494350221.810933251.153321430.560.070
(9)1237651709428367652201.8110888−1.15291420.820.079
123765723544263086021.81094741.153324660.570.078
(10)1237651709428367652201.8110888−1.15291420.820.079
123765781526262589921.810925831.153330020.570.075
(11)1237651709428367652201.8110888−1.15291420.820.079
123764658822112887621.810945991.153332210.670.074
(12)1237651709428367652201.8110888−1.15291420.820.079
123766634080187198821.810938091.153332610.590.072
(13)1237651709428367652201.8110888−1.15291420.820.079
123766023765262349921.810933661.153333170.640.071
(14)1237651709428367652201.8110888−1.15291420.820.079
123766296922049757921.810936061.153333650.670.073
(15)1237651709428367652201.8110888−1.15291420.820.079
123766320546060708121.810925281.153335620.610.070
(16)1237651709428367652201.8110888−1.15291420.820.079
123765710660606379221.810926121.153337050.650.077
(17)1237651709428367652201.8110888−1.15291420.820.079
123765991549888121121.810938811.153337490.640.076
(18)1237651709428367652201.8110888−1.15291420.820.079
123766001002106069321.81094261.15333830.580.069
(19)1237651709428367652201.8110888−1.15291420.820.079
123766354477927648121.81093811.153343310.590.074
(20)1237651709428367652201.8110888−1.15291420.820.079
123764658822112887421.810952511.153343380.540.071
(21)1237651709428367652201.8110888−1.15291420.820.079
123765773795078972521.810924831.153345340.640.073
(22)1237651709428367652201.8110888−1.15291420.820.079
123766640952573958821.810911241.153346130.600.080
(23)1237651709428367652201.8110888−1.15291420.820.079
123765651388989458721.810929481.153347930.590.072
(24)1237651709428367652201.8110888−1.15291420.820.079
123765651388989458621.810935781.153348620.640.073
(25)1237651709428367652201.8110888−1.15291420.820.079
123765707223040022021.810932591.153348730.610.076
(26)1237651709428367652201.8110888−1.15291420.820.079
123767861742737839921.810925451.153356050.660.079
(27)1237651709428367652201.8110888−1.15291420.820.079
123767861742737839721.810931581.153362410.590.076
(28)1237651279931506991201.8110746−1.15291180.760.085
123765723544263086021.81094741.153324660.570.078
(29)1237651279931506991201.8110746−1.15291180.760.085
123765781526262589921.810925831.153330020.570.075
(30)1237651279931506991201.8110746−1.15291180.760.085
123765710660606379221.810926121.153337050.650.077
(31)1237651279931506991201.8110746−1.15291180.760.085
123765991549888121121.810938811.153337490.640.076
(32)1237651279931506991201.8110746−1.15291180.760.085
123766640952573958821.810911241.153346130.600.080
(33)1237651279931506991201.8110746−1.15291180.760.085
123765707223040022021.810932591.153348730.610.076
(34)1237651279931506991201.8110746−1.15291180.760.085
123767861742737839921.810925451.153356050.660.079
(35)1237651279931506991201.8110746−1.15291180.760.085
123767861742737839721.810931581.153362410.590.076
(36)1237648702974525748201.8110905−1.15290690.820.071
123765690904897142421.810940411.15331580.590.072
(37)1237648702974525748201.8110905−1.15290690.820.071
123766666292494350221.810933251.153321430.560.070
(38)1237648702974525748201.8110905−1.15290690.820.071
123765723544263086021.81094741.153324660.570.078
(39)1237648702974525748201.8110905−1.15290690.820.071
123765781526262589921.810925831.153330020.570.075
(40)1237648702974525748201.8110905−1.15290690.820.071
123764658822112887621.810945991.153332210.670.074
(41)1237648702974525748201.8110905−1.15290690.820.071
123766634080187198821.810938091.153332610.590.072
(42)1237648702974525748201.8110905−1.15290690.820.071
123766023765262349921.810933661.153333170.640.071
(43)1237648702974525748201.8110905−1.15290690.820.071
123766296922049757921.810936061.153333650.670.073
(44)1237648702974525748201.8110905−1.15290690.820.071
123766320546060708121.810925281.153335620.610.070
(45)1237648702974525748201.8110905−1.15290690.820.071
123765710660606379221.810926121.153337050.650.077
(46)1237648702974525748201.8110905−1.15290690.820.071
123765991549888121121.810938811.153337490.640.076
(47)1237648702974525748201.8110905−1.15290690.820.071
123766001002106069521.81093531.153338280.580.069
(48)1237648702974525748201.8110905−1.15290690.820.071
123766001002106069321.81094261.15333830.580.069
(49)1237648702974525748201.8110905−1.15290690.820.071
123766354477927648121.81093811.153343310.590.074
(50)1237648702974525748201.8110905−1.15290690.820.071
123764658822112887421.810952511.153343380.540.071
(51)1237648702974525748201.8110905−1.15290690.820.071
123765773795078972521.810924831.153345340.640.073
(52)1237648702974525748201.8110905−1.15290690.820.071
123766640952573958821.810911241.153346130.600.080
(53)1237648702974525748201.8110905−1.15290690.820.071
123765651388989458721.810929481.153347930.590.072
(54)1237648702974525748201.8110905−1.15290690.820.071
123765651388989458621.810935781.153348620.640.073
(55)1237648702974525748201.8110905−1.15290690.820.071
123765707223040022021.810932591.153348730.610.076
(56)1237648702974525748201.8110905−1.15290690.820.071
123767861742737839921.810925451.153356050.660.079
(57)1237648702974525748201.8110905−1.15290690.820.071
123767861742737839721.810931581.153362410.590.076
(58)123766640685901466662.17467447−1.0167960.900.081
1237655551815123559242.17494571.016715260.920.088
(59)123766640685901466662.17467447−1.0167960.900.081
1237648705676575427242.17488811.016716450.910.084
(60)123766640685901466662.17467447−1.0167960.900.081
1237648705676575428242.17488451.016720360.920.076
(61)123765299793441622562.1747088−1.01678740.890.074
1237648705676575427242.17488811.016716450.910.084
(62)123765299793441622562.1747088−1.01678740.890.074
1237648705676575428242.17488451.016720360.920.076
(63)123764674874002664362.17475111−1.01678140.900.076
1237648705676575427242.17488811.016716450.910.084
(64)123764674874002664362.17475111−1.01678140.900.076
1237648705676575428242.17488451.016720360.920.076
(65)123764982393798679762.17470476−1.01677820.890.071
1237648705676575428242.17488451.016720360.920.076
(66)123764658555446939462.17471262−1.01675960.910.083
1237655551815123559242.17494571.016715260.920.088
(67)123764658555446939462.17471262−1.01675960.910.083
1237648705676575427242.17488811.016716450.910.084
(68)123764658555446939462.17471262−1.01675960.910.083
1237648705676575428242.17488451.016720360.920.076
(69)123766649702809641162.17470133−1.01672430.930.085
1237655551815123559242.17494571.016715260.920.088
(70)123766649702809641162.17470133−1.01672430.930.085
1237648705676575427242.17488811.016716450.910.084
(71)123766649702809641162.17470133−1.01672430.930.085
1237648705676575428242.17488451.016720360.920.076
(72)123766000735433581462.17467871−1.0167170.890.068
1237648705676575428242.17488451.016720360.920.076
(73)123766649702809641262.17471675−1.01671120.920.089
1237655551815123559242.17494571.016715260.920.088
(74)123766649702809641262.17471675−1.01671120.920.089
1237648705676575427242.17488811.016716450.910.084
(75)1237648721222828169161.910801−0.19314210.690.123
1237653012428554879341.91176680.192385540.590.131
(76)1237648721222828169161.910801−0.19314210.690.123
1237663479262544284341.91176930.192406270.630.119
(77)1237648721222828169161.910801−0.19314210.690.123
1237663479262544283341.91176930.192406270.620.116
Table A6. Pairs of galaxies for the effect of cosmic mirror from catalog [41]—alternative approach. Id SDSS DR8—objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.001° for ra and dec and zphot/2 for zphot. All galaxies are spiral (defined as those with parameter spiral (sp) > 0.5).
Table A6. Pairs of galaxies for the effect of cosmic mirror from catalog [41]—alternative approach. Id SDSS DR8—objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.001° for ra and dec and zphot/2 for zphot. All galaxies are spiral (defined as those with parameter spiral (sp) > 0.5).
Id SDSS DR8RaDecSpZphot
(1)1237671139859628503164.797490−21.9364030.990.103
1237680273114923318344.79682021.9367820.880.142
(2)1237671139859628503164.797490−21.9364030.990.103
1237679478537912674344.79683121.9367890.850.123
(3)1237671239182254464165.440684−20.3978810.560.079
1237679476390560006345.43984820.3980630.960.059
(4)1237671239182254464165.440684−20.3978810.560.079
1237680270967636262345.43985920.3980650.950.058
(5)1237671239182254464165.440684−20.3978810.560.079
1237680270967636263345.43985520.3980730.930.064
(6)1237671239182254464165.440684−20.3978810.560.079
1237679476390560005345.43985020.3980830.960.058
(7)1237668756705771673313.351245−17.1574700.990.092
1237667539074023810133.35150317.1572390.620.210
(8)1237668756705771673313.351245−17.1574700.990.092
1237667292113011156133.35151917.1572530.560.154
(9)123766724702260453055.572846−11.9627900.960.099
1237668350292328878235.57291511.9630250.760.072
(10)123766724702260453055.572846−11.9627900.960.099
1237668270839038355235.57292511.9630540.780.058
(11)123766772644053009383.727284−6.1670190.630.274
1237671693914146439263.7265486.1671520.990.092
(12)1237680066953806059334.767374−3.5118910.990.067
1237654600491729036154.7668003.5112640.970.117
(13)1237679079114932401334.767352−3.5118790.990.076
1237654600491729036154.7668003.5112640.970.117
(14)1237679433981362251359.485290−3.3151960.650.063
1237651737367347360179.4856473.3149650.840.086
(15)1237679433981362251359.485290−3.3151960.650.063
1237651737367347361179.4856413.3149730.800.079
(16)1237679433981362253359.485278−3.3151920.740.066
1237651737367347360179.4856473.3149650.840.086
(17)1237679433981362253359.485278−3.3151920.740.066
1237651737367347361179.4856413.3149730.800.079
(18)1237672836917362790359.485272−3.3151900.710.061
1237651737367347360179.4856473.3149650.840.086
(19)1237672836917362790359.485272−3.3151900.710.061
1237651737367347361179.4856413.3149730.800.079
(20)123767888852123704626.569616−3.0366910.990.076
1237674469000413486206.5697433.0367620.850.115
(21)1237650372102127690201.811091−1.1529330.800.061
123765690904897142421.8109401.1533160.590.072
(22)1237650372102127690201.811091−1.1529330.800.061
123766666292494350221.8109331.1533210.560.070
(23)1237650372102127690201.811091−1.1529330.800.061
123765723544263086021.8109471.1533250.570.078
(24)1237650372102127690201.811091−1.1529330.800.061
123765781526262589921.8109261.1533300.570.075
(25)1237650372102127690201.811091−1.1529330.800.061
123764658822112887621.8109461.1533320.670.074
(26)1237650372102127690201.811091−1.1529330.800.061
123766634080187198821.8109381.1533330.590.072
(27)1237650372102127690201.811091−1.1529330.800.061
123766023765262349921.8109341.1533330.640.071
(28)1237650372102127690201.811091−1.1529330.800.061
123766296922049757921.8109361.1533340.670.073
(29)1237650372102127690201.811091−1.1529330.800.061
123766320546060708121.8109251.1533360.610.070
(30)1237650372102127690201.811091−1.1529330.800.061
123765710660606379221.8109261.1533370.650.077
(31)1237650372102127690201.811091−1.1529330.800.061
123765991549888121121.8109391.1533370.640.076
(32)1237650372102127690201.811091−1.1529330.800.061
123766001002106069521.8109351.1533380.580.069
(33)1237650372102127690201.811091−1.1529330.800.061
123766001002106069321.8109431.1533380.580.069
(34)1237650372102127690201.811091−1.1529330.800.061
123766354477927648121.8109381.1533430.590.074
(35)1237650372102127690201.811091−1.1529330.800.061
123764658822112887421.8109531.1533430.540.071
(36)1237650372102127690201.811091−1.1529330.800.061
123765773795078972521.8109251.1533450.640.073
(37)1237650372102127690201.811091−1.1529330.800.061
123766640952573958821.8109111.1533460.600.080
(38)1237650372102127690201.811091−1.1529330.800.061
123765651388989458721.8109291.1533480.590.072
(39)1237650372102127690201.811091−1.1529330.800.061
123765651388989458621.8109361.1533490.640.073
(40)1237650372102127690201.811091−1.1529330.800.061
123765707223040022021.8109331.1533490.610.076
(41)1237650372102127690201.811091−1.1529330.800.061
123767861742737839921.8109251.1533560.660.079
(42)1237650372102127690201.811091−1.1529330.800.061
123767861742737839721.8109321.1533620.590.076
(43)1237655500274270455201.811109−1.1529260.760.116
123765690904897142421.8109401.1533160.590.072
(44)1237655500274270455201.811109−1.1529260.760.116
123766666292494350221.8109331.1533210.560.070
(45)1237655500274270455201.811109−1.1529260.760.116
123765723544263086021.8109471.1533250.570.078
(46)1237655500274270455201.811109−1.1529260.760.116
123765781526262589921.8109261.1533300.570.075
(47)1237655500274270455201.811109−1.1529260.760.116
123764658822112887621.8109461.1533320.670.074
(48)1237655500274270455201.811109−1.1529260.760.116
123766634080187198821.8109381.1533330.590.072
(49)1237655500274270455201.811109−1.1529260.760.116
123766023765262349921.8109341.1533330.640.071
(50)1237655500274270455201.811109−1.1529260.760.116
123766296922049757921.8109361.1533340.670.073
(51)1237655500274270455201.811109−1.1529260.760.116
123766320546060708121.8109251.1533360.610.070
(52)1237655500274270455201.811109−1.1529260.760.116
123765710660606379221.8109261.1533370.650.077
(53)1237655500274270455201.811109−1.1529260.760.116
123765991549888121121.8109391.1533370.640.076
(54)1237655500274270455201.811109−1.1529260.760.116
123766001002106069521.8109351.1533380.580.069
(55)1237655500274270455201.811109−1.1529260.760.116
123766001002106069321.8109431.1533380.580.069
(56)1237655500274270455201.811109−1.1529260.760.116
123766354477927648121.8109381.1533430.590.074
(57)1237655500274270455201.811109−1.1529260.760.116
123764658822112887421.8109531.1533430.540.071
(58)1237655500274270455201.811109−1.1529260.760.116
123765773795078972521.8109251.1533450.640.073
(59)1237655500274270455201.811109−1.1529260.760.116
123766640952573958821.8109111.1533460.600.080
(60)1237655500274270455201.811109−1.1529260.760.116
123765651388989458721.8109291.1533480.590.072
(61)1237655500274270455201.811109−1.1529260.760.116
123765651388989458621.8109361.1533490.640.073
(62)1237655500274270455201.811109−1.1529260.760.116
123765707223040022021.8109331.1533490.610.076
(63)1237655500274270455201.811109−1.1529260.760.116
123767861742737839921.8109251.1533560.660.079
(64)1237655500274270455201.811109−1.1529260.760.116
123767861742737839721.8109321.1533620.590.076
(65)1237651709428367652201.811089−1.1529140.820.079
123765690904897142421.8109401.1533160.590.072
(66)1237651709428367652201.811089−1.1529140.820.079
123766666292494350221.8109331.1533210.560.070
(67)1237651709428367652201.811089−1.1529140.820.079
123765723544263086021.8109471.1533250.570.078
(68)1237651709428367652201.811089−1.1529140.820.079
123765781526262589921.8109261.1533300.570.075
(69)1237651709428367652201.811089−1.1529140.820.079
123764658822112887621.8109461.1533320.670.074
(70)1237651709428367652201.811089−1.1529140.820.079
123766634080187198821.8109381.1533330.590.072
(71)1237651709428367652201.811089−1.1529140.820.079
123766023765262349921.8109341.1533330.640.071
(72)1237651709428367652201.811089−1.1529140.820.079
123766296922049757921.8109361.1533340.670.073
(73)1237651709428367652201.811089−1.1529140.820.079
123766320546060708121.8109251.1533360.610.070
(74)1237651709428367652201.811089−1.1529140.820.079
123765710660606379221.8109261.1533370.650.077
(75)1237651709428367652201.811089−1.1529140.820.079
123765991549888121121.8109391.1533370.640.076
(76)1237651709428367652201.811089−1.1529140.820.079
123766001002106069521.8109351.1533380.580.069
(77)1237651709428367652201.811089−1.1529140.820.079
123766001002106069321.8109431.1533380.580.069
(78)1237651709428367652201.811089−1.1529140.820.079
123766354477927648121.8109381.1533430.590.074
(79)1237651709428367652201.811089−1.1529140.820.079
123764658822112887421.8109531.1533430.540.071
(80)1237651709428367652201.811089−1.1529140.820.079
123765773795078972521.8109251.1533450.640.073
(81)1237651709428367652201.811089−1.1529140.820.079
123766640952573958821.8109111.1533460.600.080
(82)1237651709428367652201.811089−1.1529140.820.079
123765651388989458721.8109291.1533480.590.072
(83)1237651709428367652201.811089−1.1529140.820.079
123765651388989458621.8109361.1533490.640.073
(84)1237651709428367652201.811089−1.1529140.820.079
123765707223040022021.8109331.1533490.610.076
(85)1237651709428367652201.811089−1.1529140.820.079
123767861742737839921.8109251.1533560.660.079
(86)1237651709428367652201.811089−1.1529140.820.079
123767861742737839721.8109321.1533620.590.076
(87)1237651279931506991201.811075−1.1529120.760.085
123765690904897142421.8109401.1533160.590.072
(88)1237651279931506991201.811075−1.1529120.760.085
123766666292494350221.8109331.1533210.560.070
(89)1237651279931506991201.811075−1.1529120.760.085
123765723544263086021.8109471.1533250.570.078
(90)1237651279931506991201.811075−1.1529120.760.085
123765781526262589921.8109261.1533300.570.075
(91)1237651279931506991201.811075−1.1529120.760.085
123764658822112887621.8109461.1533320.670.074
(92)1237651279931506991201.811075−1.1529120.760.085
123766634080187198821.8109381.1533330.590.072
(93)1237651279931506991201.811075−1.1529120.760.085
123766023765262349921.8109341.1533330.640.071
(94)1237651279931506991201.811075−1.1529120.760.085
123766296922049757921.8109361.1533340.670.073
(95)1237651279931506991201.811075−1.1529120.760.085
123766320546060708121.8109251.1533360.610.070
(96)1237651279931506991201.811075−1.1529120.760.085
123765710660606379221.8109261.1533370.650.077
(97)1237651279931506991201.811075−1.1529120.760.085
123765991549888121121.8109391.1533370.640.076
(98)1237651279931506991201.811075−1.1529120.760.085
123766001002106069521.8109351.1533380.580.069
(99)1237651279931506991201.811075−1.1529120.760.085
123766001002106069321.8109431.1533380.580.069
(100)1237651279931506991201.811075−1.1529120.760.085
123766354477927648121.8109381.1533430.590.074
(101)1237651279931506991201.811075−1.1529120.760.085
123764658822112887421.8109531.1533430.540.071
(102)1237651279931506991201.811075−1.1529120.760.085
123765773795078972521.8109251.1533450.640.073
(103)1237651279931506991201.811075−1.1529120.760.085
123766640952573958821.8109111.1533460.600.080
(104)1237651279931506991201.811075−1.1529120.760.085
123765651388989458721.8109291.1533480.590.072
(105)1237651279931506991201.811075−1.1529120.760.085
123765651388989458621.8109361.1533490.640.073
(106)1237651279931506991201.811075−1.1529120.760.085
123765707223040022021.8109331.1533490.610.076
(107)1237651279931506991201.811075−1.1529120.760.085
123767861742737839921.8109251.1533560.660.079
(108)1237651279931506991201.811075−1.1529120.760.085
123767861742737839721.8109321.1533620.590.076
(109)1237648702974525748201.811090−1.1529070.820.071
123765690904897142421.8109401.1533160.590.072
(110)1237648702974525748201.811090−1.1529070.820.071
123766666292494350221.8109331.1533210.560.070
(111)1237648702974525748201.811090−1.1529070.820.071
123765723544263086021.8109471.1533250.570.078
(112)1237648702974525748201.811090−1.1529070.820.071
123765781526262589921.8109261.1533300.570.075
(113)1237648702974525748201.811090−1.1529070.820.071
123764658822112887621.8109461.1533320.670.074
(114)1237648702974525748201.811090−1.1529070.820.071
123766634080187198821.8109381.1533330.590.072
(115)1237648702974525748201.811090−1.1529070.820.071
123766023765262349921.8109341.1533330.640.071
(116)1237648702974525748201.811090−1.1529070.820.071
123766296922049757921.8109361.1533340.670.073
(117)1237648702974525748201.811090−1.1529070.820.071
123766320546060708121.8109251.1533360.610.070
(118)1237648702974525748201.811090−1.1529070.820.071
123765710660606379221.8109261.1533370.650.077
(119)1237648702974525748201.811090−1.1529070.820.071
123765991549888121121.8109391.1533370.640.076
(120)1237648702974525748201.811090−1.1529070.820.071
123766001002106069521.8109351.1533380.580.069
(121)1237648702974525748201.811090−1.1529070.820.071
123766001002106069321.8109431.1533380.580.069
(122)1237648702974525748201.811090−1.1529070.820.071
123766354477927648121.8109381.1533430.590.074
(123)1237648702974525748201.811090−1.1529070.820.071
123764658822112887421.8109531.1533430.540.071
(124)1237648702974525748201.811090−1.1529070.820.071
123765773795078972521.8109251.1533450.640.073
(125)1237648702974525748201.811090−1.1529070.820.071
123766640952573958821.8109111.1533460.600.080
(126)1237648702974525748201.811090−1.1529070.820.071
123765651388989458721.8109291.1533480.590.072
(127)1237648702974525748201.811090−1.1529070.820.071
123765651388989458621.8109361.1533490.640.073
(128)1237648702974525748201.811090−1.1529070.820.071
123765707223040022021.8109331.1533490.610.076
(129)1237648702974525748201.811090−1.1529070.820.071
123767861742737839921.8109251.1533560.660.079
(130)1237648702974525748201.811090−1.1529070.820.071
123767861742737839721.8109321.1533620.590.076
(131)123766640685901466662.174674−1.0167960.900.081
1237655551815123559242.1749461.0167150.920.088
(132)123766640685901466662.174674−1.0167960.900.081
1237648705676575427242.1748881.0167160.910.084
(133)123766640685901466662.174674−1.0167960.900.081
1237648705676575428242.1748841.0167200.920.076
(134)123764674874002664262.174702−1.0167920.900.099
1237655551815123559242.1749461.0167150.920.088
(135)123764674874002664262.174702−1.0167920.900.099
1237648705676575427242.1748881.0167160.910.084
(136)123764674874002664262.174702−1.0167920.900.099
1237648705676575428242.1748841.0167200.920.076
(137)123765299793441622562.174709−1.0167870.890.074
1237655551815123559242.1749461.0167150.920.088
(138)123765299793441622562.174709−1.0167870.890.074
1237648705676575427242.1748881.0167160.910.084
(139)123765299793441622562.174709−1.0167870.890.074
1237648705676575428242.1748841.0167200.920.076
(140)123765776104816668362.174720−1.0167870.890.113
1237655551815123559242.1749461.0167150.920.088
(141)123765776104816668362.174720−1.0167870.890.113
1237648705676575427242.1748881.0167160.910.084
(142)123765776104816668362.174720−1.0167870.890.113
1237648705676575428242.1748841.0167200.920.076
(143)123764674874002664362.174751−1.0167810.900.076
1237655551815123559242.1749461.0167150.920.088
(144)123764674874002664362.174751−1.0167810.900.076
1237648705676575427242.1748881.0167160.910.084
(145)123764674874002664362.174751−1.0167810.900.076
1237648705676575428242.1748841.0167200.920.076
(146)123764982393798679762.174705−1.0167780.890.071
1237655551815123559242.1749461.0167150.920.088
(147)123764982393798679762.174705−1.0167780.890.071
1237648705676575427242.1748881.0167160.910.084
(148)123764982393798679762.174705−1.0167780.890.071
1237648705676575428242.1748841.0167200.920.076
(149)123764658555446939462.174713−1.0167600.910.083
1237655551815123559242.1749461.0167150.920.088
(150)123764658555446939462.174713−1.0167600.910.083
1237648705676575427242.1748881.0167160.910.084
(151)123764658555446939462.174713−1.0167600.910.083
1237648705676575428242.1748841.0167200.920.076
(152)123766649702809641162.174701−1.0167240.930.085
1237655551815123559242.1749461.0167150.920.088
(153)123766649702809641162.174701−1.0167240.930.085
1237648705676575427242.1748881.0167160.910.084
(154)123766649702809641162.174701−1.0167240.930.085
1237648705676575428242.1748841.0167200.920.076
(155)123766000735433581462.174679−1.0167170.890.068
1237655551815123559242.1749461.0167150.920.088
(156)123766000735433581462.174679−1.0167170.890.068
1237648705676575427242.1748881.0167160.910.084
(157)123766000735433581462.174679−1.0167170.890.068
1237648705676575428242.1748841.0167200.920.076
(158)123766649702809641262.174717−1.0167110.920.089
1237655551815123559242.1749461.0167150.920.088
(159)123766649702809641262.174717−1.0167110.920.089
1237648705676575427242.1748881.0167160.910.084
(160)123766649702809641262.174717−1.0167110.920.089
1237648705676575428242.1748841.0167200.920.076
(161)12376664975388838992.523832−0.4446200.660.260
1237648705113555092182.5248140.4445270.570.110
(162)12376664975388838992.523832−0.4446200.660.260
1237674651003519212182.5248300.4445530.680.088
(163)12376662999977825942.523821−0.4446070.720.310
1237648705113555092182.5248140.4445270.570.110
(164)1237648673959378974161.910785−0.1931580.790.150
1237653012428554879341.9117670.1923860.590.131
(165)1237648673959378974161.910785−0.1931580.790.150
1237663479262544284341.9117690.1924060.630.119
(166)1237648673959378974161.910785−0.1931580.790.150
1237663479262544283341.9117690.1924060.620.116
(167)1237648673959378974161.910785−0.1931580.790.150
1237663526510395758341.9117520.1924090.740.135
(168)1237648721222828169161.910801−0.1931420.690.123
1237653012428554879341.9117670.1923860.590.131
(169)1237648721222828169161.910801−0.1931420.690.123
1237663479262544284341.9117690.1924060.630.119
(170)1237648721222828169161.910801−0.1931420.690.123
1237663479262544283341.9117690.1924060.620.116
(171)1237648721222828169161.910801−0.1931420.690.123
1237663526510395758341.9117520.1924090.740.135
Table A7. Pairs of elliptical galaxies (with parameter elliptical > 0.5) for the effect of cosmic lens [39]. Accuracy is of 0.001° for ra and dec. SpecObjID is ID of optical spectroscopic object in SDSS.
Table A7. Pairs of elliptical galaxies (with parameter elliptical > 0.5) for the effect of cosmic lens [39]. Accuracy is of 0.001° for ra and dec. SpecObjID is ID of optical spectroscopic object in SDSS.
SpecObjIDRaDeczElliptical
(1)48392086017786511362.669321.371620.26780.935
4329151476691107840182.669−1.37220.58090.892
(2)49145883706696663042.99248−3.24950.00030.861
5346040858121076736182.9933.249960.60520.958
(3)7342244889624842244.30274−9.45650.11700.663
6075454403388112896184.3039.456392.47960.730
(4)7342420811485286404.62338−8.82780.10470.686
1384895138274240512184.6238.826870.48190.997
(5)16800068123104645125.919660.546680.05960.709
4331440114590285824185.92−0.5460.51160.995
(6)7353781516553687046.86606−8.82170.18880.550
1830869162549864448186.8668.8220.08550.911
(7)48448647644195061768.161932.995060.45950.708
376056649636407296188.161−2.99540.09910.792
(8)49191733389893304329.19367−2.89090.47210.999
5351416371190693888189.1932.891510.53330.999
(9)475361107067640217610.3652−0.56980.45950.994
4333736164478943232190.3650.56890.00060.661
(10)403978970074238156810.3652−0.56980.45940.995
4333736164478943232190.3650.56890.00060.661
(11)492027834325952102411.2634−2.73180.34790.998
5353712700990423040191.2632.731840.49500.636
(12)168325339547070668811.3174−0.12290.47620.913
4333669644025462784191.3180.121990.00040.788
(13)77812304148066508811.54281.112250.21600.885
4270781151322832896191.542−1.11280.57370.877
(14)492024673230022246411.7652−2.40010.55870.995
5354974390096756736191.7662.400770.43180.966
(15)77921870485129830412.16670.063390.33150.827
4333606696984772608192.167−0.06290.64600.988
(16)484837193290927308812.78622.315580.39160.995
4254850052604297216192.786−2.31570.36380.927
(17)475588788500037632013.4472−0.28160.65210.988
4334910722825256960193.4480.281420.63790.998
(18)492151447417310412813.5581−1.42650.64100.891
4334919244040372224193.5591.426842.73790.512
(19)121944007622055731213.80811.197830.15010.630
4334772459238064128193.809−1.19780.00080.784
(20)44586958771666329616.9041−1.21010.12780.744
4511605535782993920196.9041.210510.63760.920
(21)485843136343284121617.78682.552290.11870.975
4563366149601361920197.787−2.55220.47480.999
(22)78137759606665420818.153−0.80430.43440.825
4511537915817885696198.1520.80380.45380.743
(23)44713265132997222418.63710.233770.04490.812
3294404095737096192198.638−0.2330.00030.524
(24)74319320596456448019.0036−8.53840.12510.834
2023364803757631488199.0038.537690.05240.910
(25)78371378339777126419.6607−0.12020.07740.815
4508350431508299776199.660.120220.49570.989
(26)490014683509397913621.7069−1.67540.35760.814
4554394684994617344201.7071.675420.52820.995
(27)489895413993963520022.8196−1.76040.60820.997
4554317994058579968202.821.76101−0.00060.797
(28)324035732411150540822.9715−9.98911.75560.784
2028969828722698240202.9729.989440.07080.740
(29)215610360488318771225.3766−10.3010.00010.616
6125025335974559744205.37610.30150.09760.768
(30)480885273493608857628.54181.698380.08350.889
1029209999458461696208.541−1.69740.97820.728
(31)495068424430682112030.6002−9.32230.28580.994
2036853600411478016210.69.321520.11290.612
(32)476709520697196544031.0675−1.13780.54280.819
599056024569145344211.0671.137880.24130.941
(33)495062789433589760031.3118−9.5680.56540.996
6136202696354955264211.3129.568280.52250.994
(34)494629691841996390432.0863−3.33370.50130.994
599121995266811904212.0863.333960.18040.941
(35)175314440783556812833.08440.479450.12080.788
4541919901161881600213.085−0.48030.35920.973
(36)494278040542353817633.8174−4.7570.29310.973
5384229646118682624213.8174.757230.53770.731
(37)494856740798516428834.7932−7.00720.30110.990
5473179862867378176214.7937.006450.30600.977
(38)477062738807619584035.11630.110264.23780.744
4538720871560773632215.117−0.10950.30800.869
(39)480219491711844352036.50912.82070.51390.999
1033647348191881216216.509−2.82060.10830.759
(40)79383671192580300836.9665−1.16940.07050.937
4533999294172626944216.9671.168830.58980.998
(41)494163030939900313638.565−9.39223.15850.552
6155479883750834176218.5659.392721.79610.540
(42)170018615022167040039.1133−1.15980.13250.765
4530637812533821440219.1131.15965−0.00010.768
(43)493823226873013862439.5952−5.03230.38330.999
5470806016826802176219.5965.0330.31470.747
(44)120487017395417292840.37190.943660.31240.984
4533100713194553344220.371−0.94450.90010.959
(45)170129143351374028841.581−1.06970.15360.707
604667932487739392221.5821.06882.72130.593
(46)175755014641499545642.5052−1.24790.11070.895
4523926393155223552222.5051.247160.53120.998
(47)477611120228617420844.2673−0.05680.47660.923
4525024526048165888224.2680.056480.63590.870
(48)91319228040219238446.5123−0.15530.10950.768
4521668546195357696226.5120.155690.35290.986
(49)90975795393700864046.6698−0.13220.33710.977
4521657276201172992226.6690.133060.50100.994
(50)51794862404327219250.282−8.03890.07230.852
6181271678262706176230.2818.038590.43350.896
(51)183537057999830425653.5805−6.2290.25500.979
2049174459813750784233.5816.22840.13940.787
(52)297025788849424998455.46570.62570.57090.991
354686810980378624235.466−0.62650.08090.870
(53)52145853976860467258.4107−4.97950.11340.770
5491059571152125952238.414.978620.90570.767
(54)52258031650684313660.0968−5.06710.13950.861
2051397380066011136240.0965.067990.16150.712
(55)2727025575481862144124.03213.33410.20180.588
3532033024394895360304.032−13.334−0.00040.551
(56)5482152155748499456136.7316.569350.57010.995
718431927618529280316.73−6.57030.12580.880
(57)5482200534260121600137.8237.381890.50850.882
718370629845280768317.822−7.38240.13060.874
(58)531483062945474560138.414−0.63260.29410.824
1155301165563406336318.4140.631610.14080.831
(59)4303412737826357248139.3182.84410.40840.997
4934949682486444032319.319−2.84441.26650.589
(60)4302094698111565824139.494−0.17870.45570.997
1155338274080843776319.4930.179350.13780.978
(61)532577351456811008140.0241.075820.08800.845
4719817306338230272320.024−1.07580.53850.970
(62)533708750112974848142.116−0.18281.13830.770
1112513395053062144322.1150.182040.13820.581
(63)4306817925153030144143.71.116280.00020.732
4722047115600592896323.699−1.1160.50330.986
(64)4241523696663527424144.728−0.93670.00010.637
1114734407618422784324.7270.937370.09140.707
(65)4258301419972984832144.786−1.72520.40870.560
5792911249252024320324.7851.725340.18040.908
(66)1468184248975386624145.2588.637480.18260.954
1325221896005904384325.259−8.63780.08620.751
(67)5332337919582011392145.3951.967210.28840.985
4937343042257944576325.394−1.96650.55800.992
(68)4258354471409025024146.058−1.65010.53570.992
5792991238722945024326.0591.649990.53210.980
(69)5340192281129910272147.8551.64960.42770.612
4926010103331028992327.856−1.64980.31520.806
(70)5333521268954628096148.6172.033580.48540.875
4926055458185674752328.617−2.03340.49690.998
(71)300635368191977472149.071−0.10310.08440.660
1162049428530423808329.0710.103940.19140.936
(72)6000020484002414592150.647.27370.55480.996
807373628705368064330.639−7.27310.05850.558
(73)6000135382967517184151.2269.183590.56790.998
808475060082862080331.225−9.18360.11620.960
(74)564246864883378176152.0362.609550.22550.550
4928117316342251520332.035−2.6104−0.00110.525
(75)4312372383706513408152.520.790.49130.962
421174006381570048332.519−0.79060.09540.729
(76)4312233570363506688153.639−0.82490.00030.716
1243118065701382144333.6390.824080.27210.978
(77)567537979422173184155.6311.866630.24920.983
4930524972818366464335.632−1.86751.26070.995
(78)4314694547482869760157.5330.247030.48170.995
4731051565992378368337.533−0.24720.47400.999
(79)4315711870633246720157.6530.025931.42460.742
4732332497072291840337.652−0.0260.50390.669
(80)307417161767348224158.238−0.41380.11820.769
757890377641388032338.2390.41328−0.00030.744
(81)4315788836447191040159.1780.276080.62070.772
4733403696661528576339.177−0.27530.25380.973
(82)570967906565777408162.8932.272020.11580.731
4913639497436495872342.893−2.2713−0.00060.566
(83)4318991988898136064165.716−0.35940.57700.981
4737931210767466496345.7150.359210.41030.999
(84)4318962576962093056166.315−0.01140.46670.999
762409377880631296346.3160.011250.29430.908
(85)4318962576962093056166.315−0.01140.46670.999
4736665947745026048346.3160.011250.29400.894
(86)313068375699908608166.741−1.09690.08960.716
763483055649220608346.7411.097440.26200.826
(87)574323066767697920168.5143.224250.07520.585
4909015221106376704348.514−3.22480.24720.992
(88)5326812049438146560168.7722.645530.51620.895
4908998453554053120348.773−2.64640.30900.975
(89)4321409544887992320169.3570.847780.65570.958
4740153048352423936349.358−0.84680.38190.682
(90)5325629249368358912170.611.435990.28280.962
4907992950069788672350.61−1.43560.46220.977
(91)315397411373606912170.8231.210370.07360.711
1232915690747357184350.822−1.21050.18750.953
(92)4322558259443007488171.3910.98290.59390.998
4741256958060265472351.39−0.98240.31990.850
(93)5325578671833481216171.7031.741190.72310.965
4906813179126349824351.704−1.74130.06390.502
(94)4266153857560084480171.74−2.16580.49190.997
4823580149407023104351.742.165610.08770.694
(95)576595475898066944171.8142.391260.12730.864
4906808781079838720351.814−2.39120.45410.996
(96)316509842296563712171.9780.469550.18360.982
4741226446612594688351.979−0.46870.50930.989
(97)5325776034170667008172.4543.307463.90180.821
4906774146463563776352.455−3.30762.24730.963
(98)4323644302036566016173.0630.778390.28550.966
4742391654076907520353.062−0.77880.34920.979
(99)4266283325054255104173.131−0.36970.40660.946
1673219253177706496353.1310.369080.12080.963
(100)5336938551446077440173.5053.175030.54160.962
4906696356015898624353.504−3.17420.61700.892
(101)368204767274493952173.585−3.59040.00010.668
4822247541330935808353.5843.590060.39670.978
(102)577726048678995968174.1842.676860.45980.998
4905651545024495616354.183−2.67740.53410.999
(103)368325163797735424174.441−1.30750.07740.630
4821148298591600640354.4411.307980.52380.994
(104)6057568923170635776176.50810.6750.50380.601
729620840840194048356.507−10.6760.11510.626
(105)578737599460435968176.7261.685940.17070.866
4904637795303686144356.727−1.68650.08000.745
(106)4326846349808500736179.997−0.61190.19150.757
1676605474079795200359.9980.612590.14770.591
Table A8. Pairs of spiral galaxies (with parameter spiral > 0.5) for the effect of cosmic lens [39]. Accuracy is of 0.005° for ra and dec. SpecObjID is ID of optical spectroscopic object in SDSS.
Table A8. Pairs of spiral galaxies (with parameter spiral > 0.5) for the effect of cosmic lens [39]. Accuracy is of 0.005° for ra and dec. SpecObjID is ID of optical spectroscopic object in SDSS.
SpecObjIDRaDeczSpiral
(1)7341651153345843203.13101−11.19880.10650.667
1383900905740462080183.13211.19740.07820.733
(2)7341651153345843203.13101−11.19880.10650.667
6073191609364119552183.12911.20070.44620.671
(3)74312970916806041619.6432−10.90210.14090.832
1911855335291774976199.64210.89890.14910.844
(4)7318608139245588480.62585−10.90180.02880.743
1382729651062859776180.62210.89790.06160.572
(5)731920737308272640358.864−10.41710.29120.588
1381612822210832384178.86610.4150.11680.544
(6)7352778212193341445.62372−10.38460.08770.634
1385018008698644480185.62810.38720.02160.707
(7)74312503624364236819.8544−10.01950.14200.655
2024520104594663424199.8510.01610.04860.772
(8)74874683904546611228.8179−9.944690.19340.714
2033534720653944832208.8219.942390.15120.721
99)74991946787772211229.7612−9.646190.33740.511
2034618849453697024209.7579.648680.07040.902
(10)73875337752506572812.1847−9.445370.09580.633
6092283803341422592192.189.445410.45490.600
(11)7353319721670021125.53528−9.405610.05300.717
1386048526860969984185.5329.40910.13330.786
(12)7353451663065354246.11665−9.387020.14270.630
1386035332721436672186.1199.388240.07080.749
(13)75212728744440012834.9428−9.368090.06840.760
2039106770269399040214.9399.364460.00420.623
(14)73760191420775833610.3014−9.343220.05590.512
2014319942068889600190.3029.345530.09520.786
(15)494957840894368153631.7084−9.325140.24850.564
2036888509905659904211.7139.32120.02400.602
(16)730872079066359808358.978−9.229650.07560.764
1381502596170147840178.9839.22950.13180.695
(17)7375672795914833929.46966−8.872852.74010.690
1388259919773329408189.4718.876910.18680.607
(18)809694968183547904335.363−8.647170.03730.754
1393917732184942592155.368.64440.04500.871
(19)809695792817268736335.426−8.405690.07050.685
1393918556818663424155.438.409750.08650.599
(20)74995410249399705629.4351−8.392940.07920.737
2033379689514428416209.4398.396840.06290.851
(21)811867328701556736336.686−8.186040.13240.738
1395060674505238528156.6858.182110.03210.696
(22)51910750906406707252.3919−7.939740.12740.780
1939981130342098944232.3957.939930.07490.853
(23)811897015515506688337.822−7.806320.13300.828
1395012845749430272157.8217.807350.06670.557
(24)811928351596898304338.431−7.729590.16690.638
1124917262550067200158.4287.733710.06730.678
(25)808545978582853632332.595−7.525070.05630.742
1392816571306502144152.5987.522290.07070.647
(26)494967461621111193632.8596−7.476620.21860.677
2037965481578620928212.8577.471750.10910.501
(27)51908441931988377652.782−6.981120.20000.591
2048172260535068672232.7786.982770.03480.750
(28)719544634476357632318.955−6.6414−0.00010.822
1343369057658759168138.9546.637370.04530.668
(29)495184807778425241630.0436−6.050430.24590.537
2035663110638954496210.0456.048010.08040.700
(30)717316199754524672315.639−5.416050.09110.781
1342204125038798848135.6445.41390.17660.553
(31)717315649998710784315.543−5.414140.06310.741
1341003733638604800135.545.410150.15140.932
(32)4937148978455642112324.03−3.013850.52870.563
640660174721280000144.0293.011680.01860.644
(33)4557716584164491264202.738−2.646090.36040.507
481454215782046105622.73362.644010.04370.727
(34)4908072664662802432352.216−2.116210.06090.730
576509713991100416172.2122.118880.07430.757
(35)489105112555585536036.2809−2.022190.30700.670
601262744473200640216.2782.017770.02610.716
(36)489230319462331187236.2808−2.021210.89320.607
601262744473200640216.2782.017770.02610.716
(37)1106828368889800704312.725−1.247820.12930.630
525966537325045760132.721.250110.03470.708
(38)304075195369416704151.728−1.233130.03350.700
1244241217248585728331.7331.234240.04950.703
(39)16809944486468915207.38777−1.100370.05910.694
325539587847907328187.3851.10250.07820.504
(40)1659576786771535872330.756−1.09310.43830.697
564131141284554752150.7521.089670.17290.898
(41)336658118953953280206.035−1.073840.08860.630
121274102883079987226.03891.077930.05920.584
(42)7746887169784156165.28659−1.035860.10770.711
324403517626279936185.291.032450.09930.710
(43)422226519742507008336.167−0.897110.09900.505
306369326917642240156.1670.8937170.09370.583
(44)315310824832919552170.316−0.892990.04030.681
1234154023318218752350.3190.8923610.12630.802
(45)427900268781266944344.85−0.782490.12200.874
311974911520303104164.8460.7775270.04030.748
(46)1658520430555719680327.446−0.756490.30440.681
300719755709409280147.450.7580680.07320.850
(47)79495051718796288039.1814−0.713370.08790.724
345754653394233344219.1860.7100530.05620.720
(48)80392940405184307255.401−0.670760.03670.836
354808032137340928235.3990.6743220.08650.760
(49)1659619117969205248329.993−0.634860.12750.595
562961531747198976149.9970.6382090.03290.757
(50)16798880650546647045.2273−0.605550.15950.919
324402692992559104185.2240.6086470.10250.566
(51)306262399411841024157.048−0.542370.22140.805
423454949158971392337.0530.5432070.05840.632
(52)309696723461105664161.174−0.503560.11610.866
425721592429963264341.1710.4988780.10680.780
(53)533738162049017856140.545−0.460010.31970.664
1111393817388410880320.5440.4644960.13440.613
(54)325434309609547776186.904−0.384010.11550.691
16810983524957163526.908590.3875280.16970.874
(55)1239663120406833152339.376−0.19860.18830.616
307543330726635520159.3710.1942650.09650.508
(56)7735361591994224644.18587−0.100270.12720.732
324366134230935552184.190.0994860.07210.599
(57)305206317520283648153.857−0.070520.14430.836
421228432207144960333.8590.0733140.04520.701
(58)385078137475590144237.596−0.006620.08300.536
139962997280449536057.60030.0114090.08390.629
Table A9. Pairs of opposite stars in [39] with accuracy of 0.001° for ra and dec. SpecObjID is ID of optical spectroscopic object in SDSS.
Table A9. Pairs of opposite stars in [39] with accuracy of 0.001° for ra and dec. SpecObjID is ID of optical spectroscopic object in SDSS.
SpecObjIDRaDeczStar
(1)29544232765683404800.694712−5.24687−0.00020.974
948114688929982464180.6945.247620.14230.791
(2)16766403835739770880.94390.28991−0.00020.968
322071453446989824180.944−0.2897461.09260.765
(3)12273628885580779521.525020.6101840.00000.980
322044240534202368181.525−0.6102710.18140.546
(4)12249796975410810885.97699−1.04043−0.00010.997
2891434450627880960185.9771.041390.00010.592
(5)48416132339938099206.058692.552331.66210.928
4269537328760356864186.058−2.55230.00060.758
(6)12656105002264432647.429170.1311640.00010.773
2880060552376248320187.429−0.1318370.00011.000
(7)12768804898812989448.189230.647915−0.00010.875
3259547102262355968188.189−0.6485610.00060.992
(8)12756770802599464969.27467−0.05415190.00000.988
3259523737640265728189.2750.05479−0.00040.990
9)73972562116846592014.4364−10.09521.93250.609
3336213574535112704194.43710.09450.00010.998
(10)121713412564375347218.1146−0.700653−0.00030.961
3294508274463827968198.1150.7000010.00050.814
(11)322460929397833113619.2516−9.867770.00040.997
6109206662252855296199.2529.868212.18200.866
(12)476037141859087155220.3504−0.8824732.53980.621
3723477244270714880200.3510.8823890.00000.599
(13)494069682397668966438.2399−5.232033.45760.582
658818330399893504218.2395.23160.20020.976
(14)170360509419437465645.13140.642292.61160.852
3731245568798859264225.132−0.642758−0.00020.995
(15)232846716703030681646.01270.329006−0.00040.904
4521696583741865984226.013−0.329270.00010.996
(16)531439632236177408139.112−0.7475770.64020.627
1252158529679157248319.1130.74727−0.00021.000
(17)2681979409950337024140.3336.525030.00051.000
719600434691467264320.333−6.525470.00000.995
(18)4306676088153047040142.4450.03282973.06030.627
1112430931680978944322.444−0.0324278−0.00020.805
(19)533850312235051008142.4450.03284013.06110.635
1112430931680978944322.444−0.0324278−0.00020.805
(20)305140071944710144155.122−0.100810.80890.683
1241932791512328192335.1230.101456−0.00010.827
(21)567512965532641280156.5070.8873760.00000.791
1240819811034884096336.506−0.887341−0.00010.995
(22)4320303161044107264167.5370.8585820.62690.566
4739039243643912192347.537−0.857903−0.00030.997
(23)315371023094540288170.3370.3063880.00000.913
1672033429870372864350.336−0.3069940.00020.775
(24)3640122716687802368171.106−0.1554720.00050.999
1672091978864551936351.1060.155463−0.00050.657
(25)315279763629434880171.106−0.1554710.00060.999
1672091978864551936351.1060.155463−0.00050.657
(26)316494998889588736171.4320.434590.00001.000
1232900572462475264351.432−0.433976−0.00040.972
(27)4322575576751144960171.7120.7313122.47680.950
1232886553689221120351.711−0.7305490.09310.798
(28)367067598732421120172.137−3.619070.00040.991
4822344848109993984352.1383.618521.51910.508
(29)369354306480007168175.039−3.432290.00040.930
4820051266384691200355.0393.431410.48100.549
(30)3256182321551796224179.776−0.6589580.00060.999
1228487139180701696359.7770.658661−0.00031.000
Table A10. Pairs of spiral galaxies (with parameter spiral > 0.5) for the effect of cosmic mirror [39]—alternative approach. Accuracy is of 0.005° for ra and dec and z/2 for z. SpecObjID is ID of optical spectroscopic object in SDSS.
Table A10. Pairs of spiral galaxies (with parameter spiral > 0.5) for the effect of cosmic mirror [39]—alternative approach. Accuracy is of 0.005° for ra and dec and z/2 for z. SpecObjID is ID of optical spectroscopic object in SDSS.
SpecObjIDRaDeczSpiral
(1)7341651153345843203.13101−11.19880.10650.667
1383900905740462080183.13211.19740.07820.733
(2)74312970916806041619.6432−10.90210.14090.832
1911855335291774976199.64210.89890.14910.844
(3)7318608139245588480.625854−10.90180.02880.743
1382729651062859776180.62210.89790.06160.572
(4)731920737308272640358.864−10.41710.29120.588
1381612822210832384178.86610.4150.11680.544
(5)74312503624364236819.8544−10.01950.14200.655
2024520104594663424199.8510.01610.04860.772
(6)74874683904546611228.8179−9.944690.19340.714
2033534720653944832208.8219.942390.15120.721
(7)7353319721670021125.53528−9.405610.05300.717
1386048526860969984185.5329.40910.13330.786
(8)7353451663065354246.11665−9.387020.14270.630
1386035332721436672186.1199.388240.07080.749
(9)73760191420775833610.3014−9.343220.05590.512
2014319942068889600190.3029.345530.09520.786
(10)730872079066359808358.978−9.229650.07560.764
1381502596170147840178.9839.22950.13180.695
(11)809694968183547904335.363−8.647170.03730.754
1393917732184942592155.368.64440.04500.871
(12)809695792817268736335.426−8.405690.07050.685
1393918556818663424155.438.409750.08650.599
(13)74995410249399705629.4351−8.392940.07920.737
2033379689514428416209.4398.396840.06290.851
(14)51910750906406707252.3919−7.939740.12740.780
1939981130342098944232.3957.939930.07490.853
(15)811897015515506688337.822−7.806320.13300.828
1395012845749430272157.8217.807350.06670.557
(16)811928351596898304338.431−7.729590.16690.638
1124917262550067200158.4287.733710.06730.678
(17)808545978582853632332.595−7.525070.05630.742
1392816571306502144152.5987.522290.07070.647
(18)494967461621111193632.8596−7.476620.21860.677
2037965481578620928212.8577.471750.10910.501
(19)717316199754524672315.639−5.416050.09110.781
1342204125038798848135.6445.41390.17660.553
(20)717315649998710784315.543−5.414140.06310.741
1341003733638604800135.545.410150.15140.932
(21)4908072664662802432352.216−2.116210.06090.730
576509713991100416172.2122.118880.07430.757
(22)304075195369416704151.728−1.233130.03350.700
1244241217248585728331.7331.234240.04950.703
(23)16809944486468915207.38777−1.100370.05910.694
325539587847907328187.3851.10250.07820.504
(24)1659576786771535872330.756−1.09310.43830.697
564131141284554752150.7521.089670.17290.898
(25)336658118953953280206.035−1.073840.08860.630
121274102883079987226.03891.077930.05920.584
(26)7746887169784156165.28659−1.035860.10770.711
324403517626279936185.291.032450.09930.710
(27)422226519742507008336.167−0.8971120.09900.505
306369326917642240156.1670.8937170.09370.583
(28)79495051718796288039.1814−0.7133690.08790.724
345754653394233344219.1860.7100530.05620.720
(29)80392940405184307255.401−0.6707550.03670.836
354808032137340928235.3990.6743220.08650.760
(30)16798880650546647045.2273−0.6055460.15950.919
324402692992559104185.2240.6086470.10250.566
(31)309696723461105664161.174−0.5035560.11610.866
425721592429963264341.1710.4988780.10680.780
(32)533738162049017856140.545−0.4600140.31970.664
1111393817388410880320.5440.4644960.13440.613
(33)325434309609547776186.904−0.384010.11550.691
16810983524957163526.908590.3875280.16970.874
(34)1239663120406833152339.376−0.19860.18830.616
307543330726635520159.3710.1942650.09650.508
(35)7735361591994224644.18587−0.1002710.12720.732
324366134230935552184.190.09948640.07210.599
(36)385078137475590144237.596−0.006620.08300.536
139962997280449536057.60030.01140880.08390.629

Appendix C

Table A11. Pairs of opposite quasars from GAIADR2Q catalog for mirror and lens effects. Source_id—the unique identification number in GAIA DR2 database. Accuracy is of 0.001° for ra and dec. Redshift z is unknown.
Table A11. Pairs of opposite quasars from GAIADR2Q catalog for mirror and lens effects. Source_id—the unique identification number in GAIA DR2 database. Accuracy is of 0.001° for ra and dec. Redshift z is unknown.
Allwise_NameSource_IdRaDec
(1)J210304.14-762224.96368528062846000640315.7673682−76.37360527
J090304.41+762224.01125452302531483520135.768416676.37343393
(2)J000923.83-752826.746852041418646758402.349349915−75.47408959
J120924.56+752830.21692570332734552320182.351539475.47502083
(3)J005511.81-723727.8468898863088273932813.79922983−72.6245651
J125511.51+723731.81689619312245569024193.797822372.62551077
(4)J054056.36-604959.0475933180706003827285.23483284−60.83306433
J174056.57+605000.81435781247493233280265.235723160.83356526
(5)J042637.98-602917.9467830955112347801666.65829152−60.4883202
J162638.53+602917.11624874054648019712246.660044360.48800541
(6)J003634.96-450211.349796637324239023369.145683022−45.03652837
J123635.26+450208.01541170364728499840189.146914345.03558095
(7)J023912.05-424657.5494710750665955430439.80039767−42.78281482
J143911.83+424655.01492995984312364928219.799472842.78193739
(8)J010933.66-415122.0498472138283220800017.39027938−41.85614291
J130933.48+415122.31525961816813896576197.389478541.8561489
(9)J030635.12-342620.5504832885205365427246.6464308−34.43910643
J150634.97+342621.01290665847434238336226.645594134.43914596
(10)J144620.40-322022.36215805347992022144221.5850719−32.33959386
J024620.21+322020.813356417141687372841.5841852432.33907625
(11)J020549.77-245224.7512146266181701171231.45738864−24.87356749
J140549.73+245227.41257473141874860032211.45719924.87428879
(12)J120429.60-215735.53493448195802870400181.1233688−21.95989953
J000429.68+215736.028471619654400821761.12371504321.9600063
(13)J043710.51-215420.2489846686029721753669.29392704−21.90553489
J163710.55+215416.41297817517737857024249.293937721.90458255
(14)J040147.38-180917.4509694925623410880060.4473916−18.15484375
J160147.61+180917.71199646075865168128240.448411618.155046
(15)J124510.71-180851.93522237773904469248191.294554−18.1478664
J004510.62+180851.9278280112029999552011.2945187918.14802663
(16)J150108.76-134935.86311851640047858048225.2865151−13.82676674
J030108.57+134935.12918787618941414445.2857465513.8264156
(17)J035733.19-083041.3319495896384680614459.38814002−8.511587953
J155733.13+083042.94454299557503538688239.38806778.511930041

Appendix D

Table A12. Pairs of opposite quasars from the Milliquas 6.3 catalog for the lens effect. Accuracy is of 0.001° for ra and dec. Descrip column is the classification of object, see [47]. Redshift z is given when available, although is not essential here.
Table A12. Pairs of opposite quasars from the Milliquas 6.3 catalog for the lens effect. Accuracy is of 0.001° for ra and dec. Descrip column is the classification of object, see [47]. Redshift z is given when available, although is not essential here.
RaDecNameDescripZ
(1)255.5777892−77.0249529WISEA J170218.69-770129.2q2.3
75.580996977.0255634WISEA J050219.24+770131.4q2.3
(2)315.7673595−76.3736026WISEA J210304.14-762224.9q1
135.768398376.3734303WISEA J090304.41+762224.0q2.4
(3)2.3493535−75.4740875WISEA J000923.83-752826.7q0.8
182.351593275.4750181WISEA J120924.56+752830.2q0.7
(4)85.679027−68.1421484WISEA J054242.99-680831.9q1.4
265.677374968.1413691SDSS J174242.55+680829.1 q1.5
(5)352.8305884−66.7005686WISEA J233119.35-664201.9q
172.828723966.7011287SDSS J113118.89+664204.0 q1.5
(6)350.1300997−64.0960807WISEA J232031.22-640545.8q0.8
170.129721964.0966844SDSS J112031.12+640548.1 q1.4
(7)77.2847276−61.1581631WISEA J050908.30-610929.5q0.6
257.285168761.1587398SDSS J170908.44+610931.4 q0.9
(8)85.2348298−60.8330635WISEA J054056.36-604959.0q0.6
265.235726760.8335658WISEA J174056.57+605000.8q1.9
(9)66.6582972−60.4883083WISEA J042637.98-602917.9q2.8
246.660037260.4879967WISEA J162638.53+602917.1q
(10)55.6305834−55.3800834J034231.34-552248.3X
235.628933855.3794316SDSS J154230.94+552245.9 Q2.268
(11)12.2865176−55.3358661WISEA J004908.81-552009.3q1.3
192.285476755.3366013SDSS J124908.51+552011.7 q1.9
(12)18.9229286−55.1625237WISEA J011541.50-550945.0qR0.7
198.924174355.16331SDSS J131541.80+550947.9 Q2.518
(13)350.4801742−55.0232756XXLS J232155.23-550123.3 QX1.788
170.478565955.0239884SDSS J112154.85+550126.3 Q1.73
(14)13.9470844−54.2692763WISEA J005547.29-541609.5q2
193.945907654.2689056SDSS J125547.01+541608.0 q0.3
(15)2.4579845−53.9814555WISEA J000949.91-535853.2q0.2
182.457534853.980648SDSS J120949.80+535850.3 q0.3
(16)323.9286029−53.3487347WISEA J213542.85-532055.4q0.9
143.927154553.3484154SDSS J093542.51+532054.2 q0.5
(17)348.3491422−53.3478756XXLS J231323.75-532051.9 QX 0.496
168.350738553.3479691SDSS J111324.17+532052.6 q0.3
(18)80.7328764−52.4533568WISEA J052255.89-522712.0q
260.73269952.4532957SDSS J172255.84+522712.0 q2.1
(19)38.535212−50.9830297WISEA J023408.44-505858.7q
218.534884450.9840003SDSS J143408.35+505902.3 Q3.175
(20)315.5807126−50.5996161WISEA J210219.36-503558.6q2.2
135.579994450.6005463SDSS J090219.19+503601.9 Q2.619
(21)68.3167757−50.5098938WISEA J043316.02-503035.6q1.1
248.316722350.5106667PGC 2375138GR2
(22)27.2491308−50.0692574WISEA J014859.79-500409.4q1.8
207.248649750.0702154SDSS J134859.67+500412.8 Q1.559
(23)70.1774106−49.1605048WISEA J044042.58-490937.9q0.9
250.177943349.1604959SDSS J164042.70+490937.7 q3.2
(24)352.4638668−46.4933229WISEA J232951.30-462936.0q
172.462990246.4929967SDSS J112951.12+462934.8 Q1.874
(25)51.7544779−45.8163472WISEA J032701.07-454859.0q2.1
231.75321445.816666SDSS J152700.77+454859.9 q0.9
(26)13.7156676−45.5401862WISEA J005451.73-453224.7q1
193.714800545.5407984SDSS J125451.55+453226.8 Q0.599
(27)9.1456769−45.036539WISEA J003634.96-450211.3q1.6
189.146917245.0355819SDSS J123635.25+450208.0 Q0.401
(28)312.4572617−43.5311298WISEA J204949.73-433151.7q1.7
132.457871243.5314134SDSS J084949.88+433153.1 q2.6
(29)39.8003988−42.7828145WISEA J023912.05-424657.5q1.8
219.799476442.7819417SDSS J143911.87+424655.0 Q2.376
(30)58.8593889−42.1836389J035526.25-421101.1X
238.85903142.1833947SDSS J155526.16+421100.1 Q1.222
(31)75.7744245−40.7936211WISEA J050305.91-404736.8q2.3
255.774368340.7937126SDSS J170305.84+404737.3 q0.3
(32)327.4360427−38.6604478WISEA J214944.65-383937.5q0.6
147.436128638.659581SDSS J094944.67+383934.5 Q1.326
(33)17.7159333−38.4864674WISEA J011051.83-382911.3q2.2
197.714755838.4865089SDSS J131051.54+382911.4 Q0.817
(34)35.9159531−37.2880557WISEA J022339.81-371716.8q0.7
215.915401837.2873321SDSS J142339.69+371714.3 q1.4
(35)5.4863222−36.9636832WISEA J002156.72-365749.2q1.6
185.486194436.9634266SDSS J122156.67+365748.3 Q2.097
(36)288.7726299−36.2604918WISEA J191505.43-361537.9q1
108.77155336.2600021SDSS J071505.17+361536.0 q0.4
(37)48.2374301−35.2506342WISEA J031256.97-351502.2qX1.9
228.236845735.250557SDSS J151256.83+351502.0 q1.3
(38)71.9969184−34.9231867WISEA J044759.25-345523.3q2.5
251.9980934.922451DEEP J164759.54+345520.8 A0.743
(39)51.0522347−34.8159832WISEA J032412.53-344857.3q0.7
231.05165134.815155SDSS J152412.39+344854.5 q0
(40)64.4053066−34.4443255WISEA J041737.25-342639.2q2.3
244.404924834.4447566SDSS J161737.18+342641.1 q1.9
(41)46.6464316−34.4391069WISEA J030635.12-342620.5q1.5
226.645589934.4391456SDSS J150634.94+342620.9 Q2.275
(42)13.9539794−33.5205216Q 0053-3347Q2.08
193.954330433.519737SDSS J125549.03+333111.0 q0.9
(43)61.3792872−33.3589696WISEA J040531.02-332132.4q1.3
241.380481933.3584594SDSS J160531.31+332130.4 q0.6
(44)52.3380705−33.1624587WISEA J032921.13-330944.7q1
232.337462233.1619242SDSS J152921.00+330942.9 q0.9
(45)35.6523275−32.9501625WISEA J022236.55-325700.5q2
215.651723732.9500411SDSS J142236.40+325700.2 q1.8
(46)24.818008−32.6980218WISEA J013916.33-324152.7q1.6
204.817657532.6980515SDSS J133916.23+324152.9 q0
(47)331.3013548−31.57234592QZ J220512.3-313421Q1.934
151.300445631.5727386SDSS J100512.10+313421.8 q0.9
(48)178.0274651−31.4001968WISEA J115206.58-312400.4q2.4
358.02694731.4004002SDSS J235206.46+312401.4 q1
(49)341.9315476−31.05192466QZ J224743.5-310306Q2.614
161.931481331.0522394SDSS J104743.55+310308.0 q0
(50)309.9034723−30.9013056J203936.83-305404.7X
129.902496930.9010042SDSS J083936.60+305403.6 Q1.1
(51)15.5519167−30.68830562QZ J010212.4-304118N0.201
195.551301930.6890974SDSS J130212.30+304120.7 QX0.897
(52)188.680682−30.5578933WISEA J123443.36-303328.3q1
8.680303630.558761SDSS J003443.27+303331.6 Q1.254
(53)300.1895832−30.1375161WISEA J200045.51-300814.8q0.6
120.189827930.1370937SDSS J080045.55+300813.5 Q1.343
(54)332.1756094−29.98380412QZ J220842.1-295902Q1.023
152.176683229.9847166SDSS J100842.39+295905.0 Q2.56
(55)12.3530442−29.9213175Q 0046-3011Q1.081
192.352991429.9214458SDSS J124924.71+295517.2 q2.2
(56)183.3282018−29.1793456WISEA J121318.77-291045.5q0.3
3.32784129.1787777SDSS J001318.68+291043.5 q0.9
(57)338.8539592−28.71112972QZ J223525.0-284240Q2.567
158.854666728.71175J103525.12+284242.3R
(58)333.8811688−28.33365282QZ J221531.4-282001Q1.909
153.881552428.3342401SDSS J101531.57+282003.3 Q2.395
(59)80.438152−28.1840351WISEA J052145.16-281102.6q0.6
260.43785328.1849529SDSS J172145.08+281105.8 q1.3
(60)65.5058337−27.9944556WISEA J042201.40-275940.0q0.2
245.505137327.993782SDSS J162201.23+275937.6 Q2.29
(61)189.5249477−27.4782258WISEA J123805.97-272841.7q2.3
9.524350727.4772773SDSS J003805.84+272838.2 Q1.937
(62)44.5344194−26.5790235WISEA J025808.26-263444.5q1.5
224.535281826.5780647SDSS J145808.46+263441.0 q2.1
(63)42.3864196−26.4011219WISEA J024932.73-262404.3q3
222.387265826.401885SDSS J144932.94+262406.7 qR0.3
(64)197.0438404−25.791086WISEA J130810.53-254728.0q1.1
17.044298925.7919436SDSS J010810.62+254730.9 Q1.843
(65)47.3495546−25.1085428WISEA J030923.89-250630.5q1
227.349581325.1081434SDSS J150923.90+250629.2 q1.8
(66)31.4573892−24.8735666WISEA J020549.77-245224.7q0.3
211.457200124.8742878SDSS J140549.72+245227.4 QX0.93
(67)48.9227−24.5943481WISEA J031541.43-243539.7qR1
228.922152624.5946659SDSS J151541.31+243540.7 Q3.21
(68)314.8735964−23.9469948WISEA J205929.66-235649.1q0.4
134.873146623.946707SDSS J085929.56+235648.1 Q1.085
(69)211.3857165−23.9417462WISEA J140532.57-235630.1q2.3
31.386459123.9416464SDSS J020532.75+235629.9 Q2.005
(70)9.3024771−23.7730802SDSS J003712.58-234622.9 q1.7
189.302566923.7724481SDSS J123712.61+234620.8 Q2.644
(71)55.1065124−23.5058167WISEA J034025.56-233020.9q2.5
235.107467723.5059795SDSS J154025.79+233021.5 q0.1
(72)325.6254779−23.1632401WISEA J214230.10-230947.5q0.9
145.625285423.1622808SDSS J094230.07+230944.1 q1.4
(73)40.1013553−23.0486453J024024.32-230255.1X
220.101793423.0486683SDSS J144024.43+230255.2 q1.6
(74)330.2260385−23.0089209WISEA J220054.23-230032.0q0.6
150.225994723.0087942SDSS J100054.23+230031.6 Q3.027
(75)15.629173−22.6311972WISEA J010230.99-223752.2qR1.9
195.628696522.6311474SDSS J130230.88+223752.2 Q3.405
(76)152.8034904−22.2785203WISEA J101112.84-221642.2qR0.4
332.80288722.2781582SDSS J221112.69+221641.3 q0
(77)181.1233682−21.9598988WISEA J120429.60-215735.5q0.8
1.123715321.9600067WISEA J000429.68+215736.0q1.4
(78)69.293922−21.9055291WISEA J043710.51-215420.2q0.7
249.29393721.9045811SDSS J163710.55+215416.4 Q1.411
(79)199.0397778−21.5273612J131609.54-213138.5R
19.039073621.5264408SDSS J011609.37+213135.1 Q2.088
(80)22.013869−20.8937502SDSS J012803.33-205337.4 q1.6
202.013458320.8942223SDSS J132803.22+205339.2 q0.3
(81)341.8649478−20.6885193WISEA J224727.58-204119.2q1.2
161.864031520.6894371SDSS J104727.36+204121.9 Q2.262
(82)347.9339167−20.5751112J231144.14-203430.4R
167.933529720.5758125SDSS J111144.05+203432.9 q0
(83)191.7714081−20.2647247SDSS J124705.13-201553.0 q0.3
11.771309920.2641487SDSS J004705.11+201550.9 q4.5
(84)29.9053228−19.4208582WISEA J015937.24-192514.6q0.3
209.904418919.4200344SDSS J135937.06+192512.1 q0
(85)315.432881−19.3894197WISEA J210143.86-192321.5q1.8
135.433480519.3888631SDSS J090144.03+192319.9 q1.4
(86)315.2686772−19.0978465WISEA J210104.48-190552.1q1.1
135.268905619.0972805SDSS J090104.53+190550.2 q0.4
(87)331.1935682−18.7863235QSM3:36QX0.873
151.193039418.7867023SDSS J100446.32+184712.1 q1.5
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