A Scale-Dependent Distance Functional between Past Light Cones in Cosmology

We discuss a rigorous procedure for quantifying the difference between our past lightcone and the past lightcone of the fiducial Friedmann-Lemaitre-Robertson-Walker spacetime modeling the large-scale description of cosmological data in the standard $\Lambda\mathrm{CDM}$ scenario. This result is made possible by exploiting the scale-dependent distance functional between past lightcones recently introduced by us. We express this harmonic map type functional in terms of the physical quantities that characterize the actual measurements along our past lightcone, namely the area distance and the lensing distortion, also addressing the very delicate problem of the presence of lightcone caustics. This analysis works beautifully and seems to remove several of the difficulties encountered in comparing the actual geometry of our past lightcone with the geometry of the fiducial FLRW lightcone of choice. We also discuss how, from the point of view of the FLRW geometry, this distance functional may be interpreted as a scale-dependent effective field, the pre-homogeneity field, that may be of relevance in selecting the FLRW model that best fits the observational data.


INTRODUCTION
It is a pleasure to dedicate this paper to Maurizio Gasperini who has always liked it best on the past light cone even if the routes are tough, but in such a rugged landscape that is to be expected The ΛCDM model and the Friedman-Lemaitre-Robertson-Walker (FLRW) spacetimes provide a rather accurate physical and geometrical representation of the universe in the present era 1 and over spatial scales ranging from 2 ≈ 100 h −1 Mpc to the visual horizon of our past light cone [27], [34], [50], where h is the dimensionless parameter describing the relative uncertainty of the true value of the present-epoch Hubble-Lemaitre constant. Within such observational range, and on scales significantly smaller than the Hubble scale 3 , we have a testable ground for statistical isotropy in the distribution of the dark and visible matter components on our past light cone. Homogeneity of this distribution is difficult to test directly via astronomical surveys, but a number of observational results [41] and in particular the kinematic Sunyaev-Zeldovich effect [55], [18] imply that fluctuations around spatial homogeneity cannot be too large. Thus, without resorting to an axiomatic use of the Copernican principle, we have an observational ground for assuming that spatial homogeneity holds, in a statistically averaged sense, over large scales. It must be stressed that it is in a statistical sense and only over large scales that this weak form of the cosmological principle provides observational support for best fitting the description of spacetime geometry in terms of a member of the FLRW family of solutions of the Einstein equations. In particular, to whatever degree one accepts this FLRW scenario, one has to address the fact that the role of FLRW spacetime geometry becomes delicate to interpret when past light cone data are gathered in our 1 Characterized by the actual temperature of the cosmic microwave background TCMB = 2.725 K as measured in the frame centered on us but stationary with respect to the CMB. 2 The actual averaging scale marking the statistical onset of isotropy and homogeneity is still much debated. For the sake of the argument presented in this paper, we adopt the rather conservative estimate of the scales over which an average isotropic expansion is seen to emerge, namely 70 − 120 h −1 M pc, and ideally extending to a few times this scale [54]. 3 At the Hubble scale, the problem of cosmic variance may alter the statistical significance of the data samples we gather.
cosmological neighborhood. As we probe spatial regions in the range 100h −1 Mpc, the actual distribution of matter (dark and visible) becomes extremely anisotropic with a high density contrast. In particular, gravitational clustering gives rise to a complex network of structures, characterized by the presence of a foam-like web of voids and galaxy filaments often extending well into the 100h −1 Mpc range. At these scales, the Einstein evolution of the FLRW geometry uncouples from the dynamics of the matter sources and survives more as a useful computational assumption (often assisted by Newtonian theory) rather than as a bona fide perturbative background gravitationally determined by the actual matter distribution. FLRW is thus a very strong assumption and not a correct representation of spacetime geometry at the pre-homogeneity scales, not even in a statistical sense. If we want or need to go beyond FLRW perturbation theory and enter into a fully relativistic regime, it is fair to say that we have little mathematical control over the actual spacetime at these pre-homogeneity scales. In particular, the transition from the large-scale FLRW to the actual inhomogeneous and anisotropic spacetime geometry emergent at these local scales is poorly understood in a model-independent way, and the idea that around 100h −1 Mpc we have a gradual and smooth transition between these two regimes is somewhat illusive. To wit, we may have non-perturbative correction terms due to the coupling between gravitationally bound structures and the emergent spacetime geometry (e.g. structure formation-induced curvature) that can be significant in cosmological modeling. For instance, they can back-react, in a top-down causation way [52], on the choice of the large-scale FLRW spacetime that best fits the observational data. This complex scenario gives rise to a number of delicate and to some extent controversial issues that are currently much debated in discussing the existence of possible tensions between cosmological observations and the standard ΛCDM model and in preparation to the coming era of high-precision cosmology [11]. Some of the very delicate reasons 4 motivating this tension is that large-scale isotropy can hold for a much wider class of models, the so-called effective model [30], [31] that need not even be a solution of Einstein's equations. As an illustrative example, one may consider inhomogeneous spatial sections that can be smoothed into a constant-curvature space, e.g. with Ricci flow deformation techniques [4][5][6][8][9][10]. While spatially, such slices can be identified with spatial sections of a FLRW model, their Einstein time-evolution in general does not follow the FLRW class of solutions, a backreaction is present [4][5][6]. Thus, at least in principle, one may actually deal with an effective model with global backreaction that can be large-scale isotropic and homogeneous, or almost so, and it is not necessarily perturbatively away from a FLRW model. Thus, restricting a priori the "best-fit" to the class of FLRW models is indeed a strong assumption. By its very nature, a discussion of this very complex scenario should be related, as far as possible, to a model-independent direct observational cosmology approach, namely to the analysis of data determined on our past lightcone without using any theory of gravity. Since the dark matter and dark energy components cannot be measured yet via direct observations, it must be stressed that a full model-independent cosmographic approach is not actually possible [21]. Model hypotheses must be imposed for the dark components, in particular on how they interact with observed matter. The simplest assumptions made are that the dark matter component follows the baryonic component, namely that: i) we know the primordial ratio of cold dark matter (CDM) density to baryonic density; ii) they have the same 4-velocity; iii) we know their relative concentration in matter clusters. To these, one typically adds the working assumption that the dark energy component is described in the form of a cosmological constant Λ, the value of which should be known from noncosmological physics and independently from cosmological observations (for a thorough discussion of the implications of these assumptions in cosmography see Chap. 8 of [21]). However, although there are efforts to derive Λ from non-cosmological physics, it remains a fitting parameter of the model. The appropriate cosmographical framework was put forward in the '80s by G.F.R. Ellis, R. Maartens, W. Stoeger, and A. Whitman [20] (see also [21]) by characterizing the set of cosmological observables on the past lightcone which, together with the Einstein field equations, allows to reconstruct the spacetime geometry in a way adapted to the process of observation [20], [15], [16].
In this paper we address an important step in this cosmographical framework. In particular we discuss a rigorous procedure for quantifying the difference between our past lightcone and the reference past lightcone that, for consistency, we associate with the fiducial large-scale FLRW spacetime. This result is made possible by exploiting the scale-dependent (harmonic map type) distance functional between past lightcones recently introduced by us in [12], and which extended the light-cone theorem [14]. We express this functional in terms of the physical quantities that characterize measurements along our past lightcone, namely the area distance and the lensing distortion, also briefly addressing the very delicate problem of the presence of lightcone caustics. This analysis works beautifully and seems to remove several of the difficulties encountered in comparing the actual geometry of our past lightcone with the geometry of a fiducial FLRW lightcone of choice. We also discuss how, from the point of view of the FLRW geometry, this distance functional may be interpreted as a scale-dependent effective field that may be of relevance in selecting the FLRW model that best fits the observative data. In this connection and in line with the introductory remarks above its worthwhile to stress that our choice of a reference FLRW spacetime is strictly related to the prevalence of this family of metrics in discussing the ΛCDM model. The results presented here can be easily extended to more general reference metrics. It is also important to make clear that in this paper we are not addressing the extremely delicate averaging problem on the past lightcone, a problem to which Maurizio Gasperini has significantly contributed with the seminal paper [24], and that has seen importat recent progress in [7]... but the past lighcone routes are still tough and the landscape rugged ... .

2.
The past light cone and the celestial sphere Throughout this paper (M, g) denotes a cosmological spacetime where g is a Lorentzian metric, and where M is a smooth 4-dimensional manifold which for our purposes we can assume diffeomorphic to R 4 (or to V 3 × R, for some smooth compact or complete 3-manifold V 3 ). In local coordinates {x i } 4 i=1 , we write g = g ik dx i ⊗ dx k , where the metric components g ik := g(∂ i , ∂ k ) in the coordinate basis have the Lorentzian signature (+, +, +, −), and the Einstein summation convention is in effect 5 . We assume that (M, g) is associated with the evolution of a universe which is (statistically) isotropic and homogeneous on sufficiently large scales L > L 0 where, according to the introductory remarks, we indicatively assume L 0 ∼ = 100h −1 Mpc, and let local inhomogeneities dominate for L < L 0 . The matter content in (M, g) is phenomenologically described by a (multi-component) energy-momentum tensor T = T ik dx i ⊗ dx k , (typically in the form of a perfect fluid, dust, and radiation). If not otherwise stated, the explicit expression of T is not needed for our analysis. We assume that in (M, g) the motion of the matter components characterize a phenomenological Hubble flow that generates a family of preferred world-lines parametrized by proper time τ and labeled by suitable comoving (Lagrangian) coordinates s adapted to the flow. We denote bẏ γ s := dγs(τ ) dτ , g(γ s ,γ s ) = −1, the corresponding 4-velocity field. For simplicity, we assume that at the present era these worldlines are geodesics, i.e. ∇γ sγs = 0. This phenomenological Hubble flow is strongly affected by the peculiar motion of the astrophysical sources and by the complex spacetime geometry that dominates on the pre-homogeneity scales. In particular, it exhibits a complex pattern of fluctuations with respect to the linear FLRW Hubble flow that sets in, relatively to the standard of rest provided by the cosmic microwave background (CMB), when we probe the homogeneity scales, L 100h −1 Mpc. Again, we stress that the transitional region between the phenomenological Hubble flow and the statistical onset of the large-scale FLRW linear Hubble flow is quite uncertain and still actively debated [54]. If we adopt the weak form of the cosmological principle described in the introduction, (M, g, γ s ) can be identified with the phenomenological background spacetime or Phenomenological Background Solution (PBS) [39] associated with the actual cosmological data gathered from our past lightcone observations. In the same vein, we define Phenomenological Observers the collection of observers {γ s } comoving with the phenomenological Hubble flow (1). Since in our analysis we fix our attention on a given observer, we drop the subscript s in (1), and describe a finite portion of the observer's world-line with the timelike geodesic segment τ −→ γ(τ ), −δ < τ < δ, for some δ > 0, where p := γ(τ = 0) is the selected event corresponding to which the cosmological data are gathered. To organize and describe these data in the local rest frame of the observer p := γ(τ = 0), let T p M, g p , {E (i) } be the tangent space to M at p endowed with a g-orthonormal frame Since we have the distinguished choice E (4) :=γ(τ )| τ =0 for the timelike basis vector E (4) , we can where δ ik denote the components of the standard Euclidean metric. As discussed in detail by Chen and LeFloch [13], this reference metric comes in handy in the characterization of the functional Lipschitz and Banach space norms of tensor fields defined on the past lightcone 6 .
2.1. The celestial sphere. Let respectively denote the set of past-directed null vectors and the set of past-directed causal vectors in (T p M, is the radial coordinate in the hyperplane X 4 = 0 ⊂ T p M parametrizing the one-parameter family of 2-spheres 6 The indefinite character of a Lorentzian metric makes it unsuitable for defining integral norms of tensor fields, and for such a purpose one is forced to introduce a reference positive definite metric. In particular, by exploiting the Nash embedding theorem, one typically uses the Euclidean metric and the associated definitions of the functional space of choice, say a Sobolev space of tensor fields. Different choices of reference metrics, as long as they are of controlled geometry, induce equivalent Banach space norms. In our case, we can exploit the natural choice provided by (3) by using normal coordinates and identifying (TpM, . can be thought of as providing a representation of the sky directions, at a given value of r, in the rest space T p M, {E (i) } of the (instantaneous) observer (p,γ(0)). In particular, the 2-sphere S 2 r (T p M ) r=1 or, equivalently, its projection on the hyperplane X 4 = 0 in T p M , can be used to parametrize the (spatial) past directions of sight constituting the field of vision of the observer (p,γ(0)). In the sense described by R. Penrose [46], this is a representation of the abstract sphere S − (p) of past null directions parameterizing the past-directed null geodesics through p. Explicitly, let denote the spatial direction in T p M associated with the point (θ, φ) ∈ S 2 (T p M ), (by abusing notation, we often write n(θ, φ) ∈ S 2 (T p M )). Any such spatial direction characterizes a corresponding past-directed null vector ℓ(θ, φ) ∈ T p M, {E (i) } , normalized according to (11) g p (ℓ(θ, φ),γ(τ )| τ =0 ) = g p ℓ(θ, φ), E (4) = 1 .
The corresponding past-directed null rays Note that in such a kinematical setup for the instantaneous rest space T p M, {E (a) } of the observer (p,γ(0)), a photon reaching p from the past-directed null direction ℓ(θ, φ), is characterized by the (future-pointing) wave vector (13) k(θ, φ) := − ν ℓ(θ, φ) ∈ T p M , where ν = − g p k, E (4) is the photon frequency as measured by the instantaneous observer γ(τ )| τ =0 . The spherical surface S 2 (T p M ) endowed with the standard round metric (14) h(S 2 ) = dθ 2 + sin 2 θ dφ 2 , and the associated area form dµ S 2 = det( h(S 2 )) dθdφ = sin θ dθdφ, defines [46] the celestial sphere providing, in the instantaneous rest space T p M, {E (i) } , the geometrical representation of the set of all directions towards which the observer can look at astrophysical sources from her instantaneous location in (M, g). In this connection, dµ S 2 can be interpreted as the element of solid angle subtended on the celestial sphere C S(p) by the observed astrophysical sources. It is also useful to keep track of the radial coordinate 7 r as a possible parametrization of the past-directed null 7 To avoid any misunderstanding we stress that r is not a distance parameter on the past light cone with vertex in p ∈ (M, g).
geodesics, and introduce a celestial sphere that provides also this information according to Lacking a better name, we shall refer to C S r (p) as the celestial sphere at radius r in T p M, The celestial sphere C S(p) plays a basic role in what follows since it provides the logbook where astrophysical data are recorded. Let m (α) (θ, φ) ∈ T p M , with α = 2, 3, denote two spatial g p -orthonormal vectors spanning the tangent space The tetrad (18) n, m (2) , m (3) , ℓ(n) provides a basis for T p M (the Sachs basis), and the pair T (θ,φ) S 2 (T p M ) , m (α) (θ, φ) defines the screen plane T n C S(p) associated with the direction of sight n(θ, φ) ∈ C S(p) in the celestial sphere C S(p), i.e.
In the instantaneous rest space of the observer, the screen T (θ,φ) C S(p) is the (spatial) 2-plane on which the apparent image of the astrophysical source, pointed by the direction n ∈ C S(p), is by convention displayed.

2.2.
Sky sections and observational coordinates on the past light cone. We transfer the above kinematical setup from T p M to (M, g) by using the exponential map based at p, where λ X : I W −→ (M, g), for some maximal interval I W ⊆ R ≥0 , is the past-directed causal geodesic emanating from the point p with initial tangent vectorλ X (0) = X ∈ W p , and where W p ⊆ T p M is the maximal domain of exp p . Thus, the past lightcone C − (p, g) ∈ (M, g) with the vertex at p, i.e. the set of all events q ∈ (M, g) that can be reached from p along the past-pointing null geodesics r −→ exp p (rℓ(n(θ, φ))), r ∈ I W , (θ, φ) ∈ C S(p), can be represented as and the portion of C − (p, g) accessible to observations for a given value r 0 ∈ I W of the affine parameter r is given by The exponential map representation, on the celestial spheres C S(p) and C S r (p), provides a natural setup for a description of observational data gathered from C − (p, g). It emphasizes the basic role of past-directed null geodesics and provides the framework for interpreting the physical data in the local rest frame of the observer at p. In particular, it allows us to represent on C S(p) and C S r (p) the actual geometry of the observed sky at a given length scale. This role is quite effective in a neighborhood of p, where we can introduce normal coordinates associated with exp p , but it is delicate to handle in regions where exp p is not a diffeomorphism of W p ∩ C − (T p M, g p ) onto its image. To set notation, our strategy is to start with the standard description [20], [21] of observational coordinates on C − (p, g) associated with the usual assumption that the exponential map is a diffeomorphism 8 in a sufficiently small neighborhood of p, and then we move to the more 8 From an observational point of view, this is the geometrical set-up proper of the weak lensing regime describing the alteration, due to the effect of gravity, of the apparent shape and brightness of astrophysical sources.
general, low regularity, Lipschitz case. In this connection, it is worthwhile to stress that the standard normal coordinates description is strictly associated with the assumption that the metric of (M, g) is sufficiently regular, with components g ij (x ℓ ) which are at least twice continuously differentiable, i.e. g ij (x ℓ ) ∈ C k (R 4 , R), for k ≥ 2. Under this hypothesis, there is a star-shaped neighborhood N 0 (g) of 0 in W p ⊆ T p M and a corresponding geodesically convex neighborhood of p, U p ⊆ (M, g), restricted to which exp p : In such U p we can introduce geodesic normal coordinates (x i ) according to are the components, in the g-orthonormal frame {E (i) }, (or with respect to the corresponding basis (18)), of the vector exp −1 p (q) ∈ W p ⊆ T p M . Thus, in C − (p, g) ∩ U p we can write, According to (21) and to the Gauss lemma applied to exp p : is foliated by the r-dependent family of 2-dimensional surfaces Σ(p, r), the cosmological sky sections, defined by and g-orthogonal to all null geodesics originating at p, i.e.
Here d (r,θ,φ) exp p (...) denotes the tangent mapping associated to exp p evaluated at the point (θ, φ) ∈ S 2 r (p), and v ∈ T θ,φ S 2 r (p) is the generic vector tangent to S 2 r (p). In C − (p, g) ∩ U p \{p}, each surface Σ(p, r) is topologically a 2-sphere endowed with the r-dependent two-dimensional Riemannian metric (26) g| Σ(p,r) := ι * r g| C − (p,g) induced by the inclusion ι r : Σ(p, r) ֒→ C − (p, g) of Σ(p, r) into C − (p, g) ∩ U p \ {p}. We can pull back this metric to the celestial sphere C S r (p) := S 2 r (T p M ) , r 2 h(S 2 ) by using the exponential map according to (27) h(r, θ, φ) This metric can be profitably compared with the pre-existing round metric r 2 h(S 2 ) on C S r (p) (see (14) and (16)). To this end, let r n(θ, φ) ∈ C S r (p) be the direction of sight pointing, in the celestial sphere C S r (p), to the (extended) astrophysical source located around the point q ∈ Σ(p, r). If rℓ(n(θ, φ)) = r n a (θ, φ)E (a) − E (4) is the corresponding null direction in C − T p M, {E (i) } , then according to (23) we have exp p (rℓ(n)) = q and, via the exponential map along the past-directed null geodesic reaching the observer located at p from the astrophysical source located at q, we can pull-back the area element of Σ(p, r), g| Σ(p,r) on the celestial sphere C S r (p) of the observer at p. We have (28) dµ h(r) (p, n(θ, φ), r) := exp * p dµ g| Σ(p,r) • exp p (rℓ(n)) = det(h(r, θ, φ)) dθdφ . This defines the area element associated with the metric (27), and can be interpreted [21] as the cross-sectional area element at the source location as seen by the observer at p. Since the round measure dµ S 2 r = r 2 dµ S 2 = r 2 sin θ dθ dϕ and the actual physical measure dµ h(r) are both defined over the celestial sphere C S r (p) ∈ T p M , we can introduce the relative density of dµ h(r) with respect to the Euclidean solid angle measure dµ S 2 , viz. the function D(r, θ, φ) defined by the relation ). The function D(r, θ, φ) is the observer area distance [20], [21], [33]. By definition, it provides the ratio of an object's cross sectional area to its (apparent) angular size as seen on the celestial sphere S 2 (p) ⊂ T p M . Roughly speaking, it converts the angular separations as seen in the images of an astrophysical source, gathered by the observer at p, into proper separations at the source. In general, D(r) := D(r, θ, φ)| θ,φ=const. cannot be used as an affine parameter along the past-directed null geodesic r → exp p (k(r, n)) since it is not a monotonic function of r, (for instance in FLRW models, monotonicity fails around z ∼ 1). However, if we have an accurate knowledge of the brightness and of the spectrum of the astrophysical source seen at the past light cone location q := exp p (ℓ(r, n)) ∈ C − (p, g), then D(r, θ, φ) is, at least in principle, a measurable quantity (see paragraph 4.3 of [20] and 7.4.3 of [21] for a discussion of this point 9 ). As stressed above, we can also compare the physical metric (27), h(r, θ, φ) := exp * p g| Σ(p,r) αβ dx α dx β r , with the round metric r 2 h(S 2 ) pre-existing on the celestial sphere C S r (p), and introduce [20], [21] the set of functions L αβ (r, θ, φ), α, β = 2, 3, implicitly defined by representing (27) in the distorted polar form We normalize this representation by imposing [20] that, in the limit r ց 0, the distortion, , of the normalized metric h(r)/D 2 (r) with respect to the round metric h(S 2 ) goes to zero uniformly, i.e., (31) lim where, for rising indexes, we used the inverse round metric h µβ (S 2 ) to write L µ α := h µβ (S 2 ) L αβ , and where we have exploited the relation det (29)). Since (33) det from relation (32) it follows that which implies that L µ α cannot be trace-free. Roughly speaking, L αβ (r) can be interpreted as the image distortion of the sources on (Σ(p, r), h(r)) as seen by the observer at p on her celestial sphere. It can in principle be directly observed and it can be related to the gravitational lensing shear [20], (see also chap. 8 of [21]). Explicitly, let us compute the deformation tensor Θ αβ defined by the rate of variation of the metric tensor h(r) as r varies. Dropping the angular dependence for notational ease, we get where we exploited d h αβ (S 2 )/dr = 0 and rewrote D(r)dD(r)/dr as D 2 (r)d ln D(r)/dr. Similarly, from the defining relation det(h(r, θ, φ)) = D 2 (r, θ, φ) det( h(S 2 )), (see (29)), we compute Inserting this relation in (35) we obtain The shear σ αβ is the trace-free part of this expression, σ αβ : we eventually get as might have been expected. Note that, in contrast to L αβ , σ αβ is trace-free (but with respect to the physical metric h αβ ). Now, let us introduce the other basic player of our narrative.
3. The background FLRW past light cone.
As already pointed out, the standard ΛCDM model is built on the assumption that over scales L > 100 h −1 Mpc, the phenomenological background spacetime (M, g, γ s ) follows on average the dynamics of a FLRW model with a (linear) Hubble expansion law. It is also assumed that below the scale of statistical homogeneity, deviations from this average scenario can be described by FLRW perturbation theory. Since there is no smooth transition between the large-scale FLRW Hubble flow and the phenomenological Hubble flow, this latter assumption rests on quite delicate ground. For instance, the field of peculiar velocities {γ s (τ )} of the phenomenological observers {τ −→ γ s (τ )} shows a significant statistical variance [53] with respect to the average FLRW Hubble flow and the standard of rest provided by the cosmic microwave background (CMB). This remark has an important effect on the relation between the celestial sphere C S r (p) of the phenomenological observer (p,γ(0)) and the corresponding celestial sphere C S r (p) of the idealized FLRW observer (p, γ(0)). They cannot be identified and must be connected by a Lorentz boost that takes into account the origin of this statistical variance. The actual scenario is significantly constrained by the coupling of the matter inhomogeneities with a spacetime geometry that is no longer Friedmannian. As a consequence, the peculiar velocity field of the phenomenological observer may have a rather complex origin, and its variance with respect to the FLRW average expansion may become a variable of relevance in cosmography. This scenario naturally calls into play a delicate comparison between the geometry of C − (p, g) and the geometry of the associated FLRW past light cone that sets in at scales L > 100 h −1 Mpc. For this purpose, along with the physical metric g, we consider on the spacetime manifold M a reference FLRW metricĝ and the associated family of global Friedmannian observersτ −→γ s (τ ). Strictly speaking, the FLRW model (M,ĝ,γ s (τ )) should be used only over the scales L > L 0 ≃ 100 h −1 Mpc. We need to consider it also over the inhomogeneity scales L < L 0 where it plays the role of the geometrical background used to interpret the data according to the standard perturbative FLRW point of view recalled above. In such an extended role, the chosen FLRW is the Global Background Solution (GBS according to [39]) we need to check against the physical metric g representing the phenomenological background solution. In this section, we set up the kinematical aspects for such a comparison. First some standard verbiage for introducing the FLRW model (M,ĝ,γ s (τ )). In terms of the radial, and angular FRLW coordinates y α := r,θ,φ , and of the proper time of the comoving fundamental observers y 4 :=τ , we set where a(τ ) is the time-dependent scale factor, k is the normalized dimensionless spatial curvature constant, and γ h are the components of the 4-velocity γ of the fundamental FLRW observers. According to the above remarks, the geodesics τ −→ γ(τ ), andτ −→γ(τ ), −δ < τ,τ < δ, associated with the corresponding Hubble flow in (M, g, γ) and (M,ĝ,γ), are assumed to be distinct, but in line with the scale-dependent cosmographic approach adopted here we assume that they share a common observational event p ∈ M . We denote by C − (p,ĝ) the associated FLRW past light cone, and normalize the proper times τ andτ along γ(τ ) andγ(τ ) so that at τ = 0 =τ we have γ(0) = p =γ(0). As stressed, the two instantaneous observers (p,γ(0)) and (p, γ(0)) have different 4-velocities,γ(0) = γ(0), and their respective celestial spheres, C S(p) and C S 2 (p) are quite distinct. They are related by a Lorentz trasformation describing the aberration of the sky mapping of one instantaneous observer with respect to the other. This mapping will play a basic role in our analysis, and to provide an explicit description of its properties, we start by adapting to the FLRW instantaneous observer (p, γ(0)) ∈ (M,ĝ,γ) the setup characterizing the celestial spheres C S(p) and C S r (p) of the instantaneous observer (p,γ(0)) ∈ (M, g, γ).
3.1. The FLRW celestial sphere and the associated sky sections.
For ease of notation, we shall often use the shorthand T p M when referring to the tangent space to (M,ĝ,γ) at p. Let respectively denote the set of past-directed null vectors and the set of past-directed causal vectors (6)) in the hyperplane Y 4 = 0 ⊂ T p M parametrizing the one-parameter family of 2-spheres can be thought of as providing a representation of the sky, at a given value of the radial coordinate r, in the instantaneous rest space T p M, { E (i) } of the FLRW observer. In analogy with the characterization (8) of the celestial sphere C S(p), we use the projection of S 2 parametrizing the directions of sight In full analogy with (16), With a straightforward adaptation to the FLRW geometry of the definitions (10), (18), and (19), we also introduce in T p M the tetrad (48) n, m (2) , m (3) , ℓ( n) and associate with the pair T ( θ, φ) S 2 T p M , m (α) ( θ, φ) the screen plane T n C S(p) associated with the direction of sight n( θ, φ) in the FLRW celestial sphere C S(p), Together with the observational normal coordinates {X i } in (M, g, γ), describing the local geometry on the past lightcone C − (p, g) ∩ U p , we introduce corresponding (normal) coordinates {Y k } on the past light cone C − (p,ĝ) in the reference FLRW spacetime (M,ĝ,γ). To begin with, let exp p denote the exponential mapping based at the event p =γ(0), i.e.
where W p is the maximal domain of exp p . To keep on with the notation set by (21) and (22), we characterize the past lightcone C − (p,ĝ) ∈ (M, g), with vertex at p, according to and we denote by the portion of C − (p,ĝ) accessible to observations for a given value r 0 of the radial parameter r. That said, ifÛ p ⊂ (M,ĝ) denotes the region of injectivity of exp p , then normal coordinates are defined by (52) where Y i are the components of the vectors Y ∈ T p M with respect to aĝ-orthonormal frame endowed with the metric induced by the inclusion of Σ(p,r) into C − (p,ĝ), i. e.
where a( τ ( r)) is the FLRW expansion factor a( τ ) (see (40)) evaluated in correspondence of the given value of the radial coordinate r ∈ T p M . We proceed as in Subsection 2.2, and exploit the exponential map exp p to pull back g| Σ(p,r) on the celestial sphere C S r (p), This pull-back can be explicitly computed. To wit, let y i q = ( r q , θ q , φ q , τ q ) the normal coordinates of the event q ∈ C − (p,ĝ) associated with the observation of a given astrophysical source. The equation for the radial, past-directed, null geodesic connecting q to the observation event p reduces in the FLRW case to [19] (56) that integrates to the expression providing the (matter-comoving) radial coordinate distance between p and q Thus, the metric (55), evaluated at exp p −1 (q), can be written in terms of τ q as If we introduce the dimensionless FLRW cosmological redshift corresponding to the event q, where a 0 := a( τ = 0), then we can rewrite h( r q , θ q , φ q ) as Note that the area element associated with the metric h q , characterizes the FLRW observer area distance (see (29)) of the event q ∈ C − (p,ĝ) according to

Comparing the celestial spheres C S(p) and C S(p)
As stressed in the previous Section, the celestial sphere C S(p) of the phenomenological observer (p,γ(0)), and the celestial sphere C S(p) of the FLRW ideal observer (p, γ(0)) cannot be directly identified as they stand. The velocity fieldsγ(0) and γ(0) are distinct and to compensate for the induced aberration, the celestial spheres C S(p) and C S(p) can be identified only up to Lorentz boosts. In the standard FLRW view, this is the familiar global boost taking care of the kinematical dipole component in the CMB spectrum due to our peculiar motion with respect to the standard of rest provided by the CMB. However, in a cosmographic setting and presence of a complex pattern of local inhomogeneities coupled with a non-FLRW spacetime geometry over scales 100h −1 Mpc, the peculiar motion of the phenomenological observer has a dynamical origin, driven by the gravitational interaction and not just by a kinematical velocity effect. Even if we factor out the effect of coherent bulk flows due to the non-linear local gravitational dynamics, and average the rate of expansion over spherical shells at increasing distances from (p,γ(0)), the variance in the peculiar velocity of (p,γ(0)) with respect to the average rate of expansion is significant [54]. These remarks imply that the Lorentz boosts connecting C S(p) and C S(p) acquire a dynamical meaning that plays a basic role in what follows. As a first step, we describe the Lorentz boost in the idealized pure kinematical situation where we need to compensate for a well-defined velocity field of the celestial sphere C S(p) with respect to the celestial sphere C S(p) taken as providing a well-defined standard of rest. As a second step, we move to the more general setting required in the pre-homogeneity region where we sample scales 100h −1 Mpc. In this latter case, a pure kinematical Lorentz boost will not suffice, the large fluctuations in the sources distribution require a suitable localization of the Lorentz boosts to compare the data on C S(p) with those on C S(p).

4.1.
The kinematical setting. To describe a kinematical Lorentz boost acting between C S(p) and C S(p), we find it convenient to use in this section the well-known correspondence between the restricted Lorentz group and the six-dimensional projective special linear group PSL(2, C) describing the automorphisms of the Riemann sphere S 2 ≃ C ∪ {∞}. More expressively, PSL(2, C) can be viewed as the group of the conformal transformations of the celestial spheres that correspond to the restricted Lorentz transformations connecting C S(p) to C S(p). In oder to set notation, let us recall that the elements of PSL(2, C) can be identified with the set of the Möbius transformations of the Riemann sphere S 2 ≃ C ∪ {∞}, i.e. the fractional linear transformations of the form where, to avoid a notational conflict with the redshift parameter z, we have labeled the complex coordinate in C ∪ {∞} with w rather than with the standard z. Let Y = n( θ, φ) denote a point on the celestial sphere C S(p), and let w denote its stereographic projection 10 on the Riemann sphere C ∪ {∞}, i.e., with 0 < θ ≤ π, 0 ≤ φ < 2π. It is worthwhile to stress once more that the celestial spheres C S(p) and C S(p) play the role of a mapping frame, a celestial globe where astrophysical positions are registered, and where the Lorentz boost C S(p) −→ C S(p) must be interpreted actively as affecting only the recorded astrophysical data. In other words, the Lorentz boost affects the null directions in C S(p), mapping them in the corresponding directions in C S(p). To quote a few illustrative examples [46] of the PSL(2, C) transformations associated to the Lorentz group action between the celestial spheres C S(p) and C S(p), let v denote the modulus of the relative 3-velocity of the FLRW ideal observer (p, γ(0)) with respect to the phenomenological observer (p,γ(0)), (where E 4 is identified with the observer's 4-velocityγ(0)). If the map between C S(p) and C S(p) is a pure Lorentz boost in a common direction, say E 3 , then the associated PSL(2, C) transformation is provided by − v is the relativistic Doppler factor and w is the point in the Riemann sphere corresponding, under stereographic projection, to the direction n(θ, φ) ∈ C S(p). Similarly, if C S(p) and C S(p) differ by a pure rotation through an angle α about the E 3 direction, then the associated PSL(2, C) transformation is given by By composing them, e. g. by considering a rotation through an angle α about the E 3 direction, followed by a boost with rapidity β := log 1 + v 1 − v along the E 3 axis, we get describing the general fractional linear transformation mapping C S(p) and C S(p). From the physical point of view, this corresponds to the composition of the adjustment of the relative orientation of the spatial bases {E (α) } with respect to { E (α) }, α = 1, 2, 3, followed by a Lorentz boost adjusting for the relative velocity of (p,γ(0)) with respect to (p, γ(0)). Since the spatial directions n(θ, φ) ∈ C S(p) and n( θ, φ) ∈ C S(p) characterize corresponding past-directed null vectors (10) and (48)), we can associate with the spatial directions {E (α) } and { E (α) } the respective null directions

4.2.
The pre-homogeneity setting. From the above remarks, it follows that the Lorentz mapping from C S(p) to C S(p) is fully determined if we specify the three distinct null directions on the FLRW celestial sphere C S(p) that are the images, under the PSL(2, C)-transformation, of three chosen distinct sources on C S(p). The selection of these three distinct sources of choice and of the corresponding null directions on C S(p) will depend on the scale L we are probing in our cosmological observations. This is a particularly delicate matter when looking at the pre-homogeneity scales L 100 h −1 Mpc, where astrophysical sources are characterized by a complex distribution of peculiar velocities with respect to the assumed Hubble flow. To keep track of this scale dependence, let us consider the celestial spheres C S r (p) and C S r (p) defined by (16) and (47), respectively. For L > 0, let r(L) be the value of r such that the FLRW sky section (53) (70) Σ(p,r(L)) := exp p C S r(L) (p) = exp p r(L) ℓ( n( θ, φ)) ( θ, φ) ∈ C S(p) , probes the length scale L. Similarly, we let r(L) denote the value of r such that the physical sky section (71) probes the length scale L. Since the FLRW area distance (62), is isotropic and may be directly expressed in terms of z, we may well use the redshift parameter z as the reference L. Given z, we denote by L(z) the corresponding length-scale of choice. As long as D( r) is an increasing function, we can identify L(z) with the area distance D( r), but in general, we leave the selection of the most appropriate L(z) to the nature of the cosmographical observations one wants to perform. Given ζ ∈ PSL(2, C) and a value of the redshift z, we have a corresponding relation between the "radial" variables r(L(z)) and r(L(z)) in (70) and (71). We can take advantage of this relation to simplify the notation for the celestial spheres and the associated sky sections according to (73) C S z (p) := C S r(L(z)) (p) =⇒ Σ z := Σ(p,r(L(z))) := exp p C S z (p) ,

and
(74) C S z (p) := C S r(L(z)) (p) =⇒ Σ z := Σ(p, r(L(z))) := exp p [C S z (p)] , a notation that, if not otherwise stated, we adopt henceforth. Since in the pre-homogeneity region L(z) 100 h −1 Mpc, the large variance in peculiar velocities of the astrophysical sources implies a great variability in the selection of the three reference null directions that fix the PSL(2, C) action, we localize this action according to the following construction.
where d S 2 (y ′ , y (I) ) denotes the distance in the round unit metric on S 2 . We also assume that any such B(y (I) , δ) contains the images of three reference astrophysical sources of choice, call them A (I, k) , k = 1, 2, 3,, with celestial coordinates in C S z (p) given by y (I, k) =: n (I, k) ( θ, φ). • We adopt a similar partition on the celestial sphere C S z (p), to the effect that associated with each disk B(y (I) , δ) there is, in C S z (p), a corresponding metric disk We require that the images A (I, k) of the three reference astrophysical sources of choice, that in B(y (I) , δ) have celestial coordinates y (I, k) , are represented in B(x(y (I) ), δ) by three distinct points with celestial coordinates x (I, k) =: n (I, k) (θ, φ). • We further assume that the past null directions ℓ (I, k) = n (I, k) ( θ, φ) − γ(0), associated with the location of the reference sources A (I, k) in the portion of the celestial sphere B(y (I) , δ) ∩ C S z (p), are related to the corresponding null directions ℓ (I, k) = n (I, k) (θ, φ) −γ(0), locating the sources A (I, k) in B(x(y (I) ), δ) ∩ C S z (p), by the PSL(2, C) map k) ) is the composition of the Lorentz boost (v being the relative 3velocity ofγ(0) with respect to γ(0)) and of the spatial rotation that, at the given scale L(z), allow us to align the portion of the celestial sphere C S z (p) described by B(x(y (I) ), δ) with the portion of the FLRW celestial sphere C S z (p) described by B(y (I) , δ).
• Finally, we require that the finite collections of celestial coordinate bins { B(y (I) , δ)} and B(x(y (I) ), δ) cover the respective celestial spheres C S z (p) and C S z (p).
It is worthwhile to stress that the collections of bins { B(y (I) , δ)} and B(x(y (I) ), δ) can be chosen in many distinct ways, according to the cosmographic observations one wishes to carry out (we use disks for mathematical convenience). Whatever choice of the above type we make, we can extend the localized PSL(2, C) maps (77) where ζ (I) (w) is provided by (77). Note that, when necessary, this localized PSL(2, C) map can be further generalized by completing it in the Sobolev space of maps which together with their derivatives are square-summable over C S z (p). This completion requires some care which we do not enter here (see [12] for details), and it is needed when discussing the distance between the FLRW and the cosmographic lightcones.
It is worthwhile to stress that in the pre-homogeneity region L(z) 100 h −1 Mpc, the large variance in peculiar velocities of the astrophysical sources implies a great variability in the selection of the three reference null directions that fix the local PSL(2, C) action characterizing the map ζ (z) . This implies that ζ (z) may vary considerably with L(z). Recall that the role of the celestial spheres C S z (p) and C S z (p) is simply that of representing past null directions at the observational event p ∈ M , directions that respectively point to the astrophysical sources on the sky section Σ z , as seen by (p,γ(0)), and on Σ z , as seen according to (p, γ(0)). These data are transferred from these sky sections to the respective celestial spheres through null geodesics, thus we can associate with the localized PSL(2, C) action the map between the sky sections Σ z and Σ z given by for any point q ∈ Σ z .

5.
The comparison between the screen planes T n C S z (p) and T n C S z (p) The localized PSL(2, C) map ζ (z) induces a corresponding map between the screen plane T n C S(p) z associated with the direction of sight n( θ, φ) in the FLRW celestial sphere C S z (p) (see (49)), and the screen plane T n C S z (p) associated with the direction of sight n(θ, φ) = ζ (z) n( θ, φ) in the celestial sphere C S z (p) (see (19)). The geometry of this correspondence is quite sophisticated since it is strictly related to harmonic map theory and it will be described here in some detail. To begin with, we denote by T C S z and by T C S z the screen bundles associated with the screen planes on C S z (p) and C S z (p), respectively. These are just two copies of the usual tangent bundle T S 2 of the 2-sphere. If there is no danger of confusion, we use both notations in what follows. Under such notational assumptions, we can associate with the map (78), , are the vector fields over C S z (p) covering the map ζ (z) . In physical terms, the vectors v are the tangent vector on the celestial sphere C S z (p) that describe the (active) effect of the combination of rotation and Lorentz boost induced by ζ (z) on the null direction ℓ( n). More expressively, let us remark that for a given direction of sight ζ (z) ( n) = n(θ, φ) ∈ C S z (p), the vectors V ∈ T n C S z (p) can be used to describe the geometrical characteristics of the astrophysical images on the screen T n C S z (p), for instance, the apparent diameters of the source. Thus, the vectors v ≡ ζ −1 (z) V := V • ζ (z) , sections of the pull-back bundle ζ −1 (z) T C S z , can be interpreted as transferring the "images" of the screens in T C S z back to C S z (p) so as to be able to compare them with the reference screen-shots in T C S z . In terms of the local coordinates y a := θ, φ , a = 1, 2, on C S z (p) (see (52)) 11 , we can write the where ζ b (z) (y), b = 1, 2, are the coordinates of the point (direction of sight) in ζ (z) (y) ∈ C S z (p) given, in terms of the y a by (64). In particular, if T * C S z denotes the cotangent bundle to C S z (p), we can locally introduce the differential , and interpret it as a section of the product bundle To provide a comparison between the geometrical information gathered from the astrophysical data, let us recall that on the screens T C S z and T C S z we have the inner products respectively defined by the pull-back metrics (55) and (27), i.e. , a, b = 1, 2, x 1 := θ, x 2 := φ . 11 In what follows the ( θ, φ), corresponding to (y 2 , y 2 ) in the normal coordinates string {y α }, are relabelled as {y a }, with a = 1, 2; a similar relabeling is also adopted for the normal coordinates (θ, φ) on C Sz(p). 12 In what follows we freely refer to the excellent [22], [32], and [36] for a detailed analysis of the geometry of the computations involved in harmonic map theory.
The Riemannian metric in the pull-back screen ζ −1 (z) T C S z y over y ∈ C S z (p) is provided by h(ζ (z) (y)), hence the tensor bundle T * C S z ⊗ ζ −1 (z) T C S z over the celestial sphere C S z (p) is endowed with the pointwise inner product where h −1 (y) := h ab (y) ∂ a ⊗ ∂ b is the metric tensor in T * y C S z . The corresponding Levi-Civita connection will be denoted by where ∇ denotes the Levi-Civita connection on ( C S z (p), h), and ∇ * is the pull back on ζ −1 (z) T C S z of the Levi-Civita connection of (C S z , h). If Γ a bc ( h) and Γ a bc (h) respectively denote the Christoffel symbols of ( C S z (p), h) and , and one computes . These remarks on the geometry of the map (80) allow us to compare the data on the screens T C S z and T C S z . For this purpose, the relevant quantity is the norm, evaluated with respect to the inner product (85), of the differential (82) of the PSL(2, C) map ζ (z) . Direct computation provides where tr h(y) (ζ * (z) h) denotes the trace, with respect to the metric h of the pull-back metric ζ * (z) h. In other words, at any point y, e( h, ζ (z) ; h)(y) is the sum of the eigenvalues of the metric ζ * (z) h, thus providing the sum of the squares of the length stretching generated by the (pull-back of) the physical metric ζ * (z) h along the orthogonal directions ( θ, φ). To such stretching, we can associate the tension field of the map ζ (z) , defined by To provide some intuition on these geometrical quantities, we can adapt to our case a nice heuristic remark by J. Eells and L. Lemaire described in their classical paper on harmonic map theory [22]. Let us imagine the FLRW celestial sphere ( C S z (p), h) as a rubber balloon, decorated with dots representing the astrophysical sources recorded from the sky section Σ z . This balloon has the geometry described by the round metric h(z, θ, φ) defined by (83), explicitly (see (60)) where z L is the redshift associated with the length scale L. Conversely, let us imagine the physical celestial sphere (C S z (p), h) as a rigid surface with the geometry induced by the metric h(r(z), θ, φ) defined by (84), i.e., (see (30)), providing the geometric landscape of the astrophysical sources reaching us along null geodesics from the physical sky section Σ z . We can think of the P SL(2, C) map ζ (z) as stretching the elastic surface ( C S z (p), h) on the rigid surface (C S z (p), h). The purpose of this stretching is to overlap the images of the astrophysical sources recorded on ( C S z (p), h) with the images of the corresponding sources as registered on (C S z (p), h). In general, this overlap is not successful without stretching the surface, and to any point y ∈ ( C S z (p), h) we can associate a corresponding vector measuring the stretch necessary for connecting the images of the same source on the two celestial spheres 13 ( C S z (p), h) and (C S z (p), h). To leading order, the required stretching is provided by the tension vector τ i (ζ (z) , y) at y. Both the Hilbert-Schmidt norm (88) and the tension vector field (89) of the map ζ (z) are basic quantities in harmonic map theory, and to understand the strategy we will follow in comparing, at a given length scale L, the FLRW past light cone C(p, g) with the physical observational past light cone C(p, g) we need to look into the harmonic map theory associated with ζ (z) . Let us start by associating with dζ (z) , dζ (z) T * [C S]z⊗ζ −1 (z) T C Sz the density where dµ h is the volume element defined by the metric h on the FLRW celestial sphere C S z (p). An important property of the density e( h, ζ (z) ; h) dµ h is that it is invariant under the two-dimensional conformal transformations where f is a smooth function on C S z (p). In this connection, it is worthwhile to recall that conformal invariance is strictly related to the action of the Lorentz group on the celestial spheres (and it is ultimately the rationale for the relation between Lorentz transformations and the fractional linear transformations of PSL(2, C)). The expression 1 2 e( h, ζ (z) ; h) dµ h characterizes the harmonic map energy functional associated to the map ζ (z) , viz.
It is worthwhile to put forward a more explicit characterization of the nature of the harmonic map functional E[ h, ζ (z) ; h] by making explicit, together with the celestial spheres C S z (p) and C S z (p), 13 This is not to be confused with the phenomenon of strong gravitational lensing that occurs in a given celestial sphere. It is simply a mismatch due to the comparison between the description of the same astrophysical source on two distinct celestial spheres.
the role of the corresponding sky sections Σ z and Σ z . To this end, let us consider the map (79) acting between the sky sections Σ z and Σ z , . The corresponding harmonic map functional is provided by where, for notational ease, we have set g (z) := gΣ z and g (z) := g Σz . We can equivalently write E g (z) , ψ (z) , g (z) in terms of pull-backs of the relevant maps, and get the following chain of relations from which it follows that the harmonic map energy functional associated with the localized PSL(2, C) map ζ (z) and with the map ψ (z) , defined by (95), can be identified. This is not surprising since ψ (z) := exp p • ζ (z) • exp −1 p can be seen as the representation of ζ (z) on the sky sections Σ z := exp p C S z (p) and Σ z := exp p (C S z (p)). From the conformal nature of the map ζ (z) : C S z (p) −→ C S z (p), it follows that ψ (z) acts as a conformal diffeomorphism betweenΣ z and Σ z as long as the exponential maps are diffeomorphisms from C S z (p) and C S z (p) onto their respective images Σ z and Σ z . Later we shall see how this result can be extended, under suitable hypotheses, to the less regular case of Lipschitzian exponential map. Here, we restrict our attention to the stated regularity assumptions on the exponential maps exp p and exp p . They imply that the sky sectionsΣ z and Σ z have the topology of a 2-sphere. Moreover, we can take advantage of the fact that ( Σ z , g (z) ) is a (rescaled) round sphere, thus we can apply the Poincaré-Koebe uniformization theorem, to the effect that there is a positive scalar function Φ Σ Σ ∈ C ∞ ( Σ z , R >0 ) such that The required conformal factor Φ Σ Σ ∈ C ∞ ( Σ z , R >0 ) is the solution, (unique up to the PSL(2, C) action on ( Σ z , g (z) )), of the elliptic partial differential equation on ( Σ z ,ĝ (z) ) defined by [2] (99) − ∆ g (z) ln(Φ 2 ΣΣ ) + R( g (z) ) = R(ψ * (z) g (z) ) Φ 2 ΣΣ , where ∆ g (z) := g ab (z) ∇ a ∇ b is the Laplace-Beltrami operator on ( Σ z ,ĝ (z) ), and where we respectively denoted by R( g (z) ) and R(ψ * (z) g (z) ) the scalar curvature of the metrics g (z) and ψ * (z) g (z) . Notice that the scalar curvature R(ĝ (z) ) is associated with the metric (60) evaluated for r = r(L) and hence is given by the constant R(ĝ (z) ) = Similarly, R(g (z) ) is associated with the metric (30) evaluated for r = r(z), and as such it depends on the area distance D 2 (r(z), θ, φ) and the lensing distortion L ab .
As the PSL(2, C)-localized map ζ (z) varies with the scale L(z), relation (102) shows that E h, ζ (z) , h describes the ζ (z) -dependent total "energy" associated with the conformal stretching of ( C S z (p), h) over (C S z (p), h).
In terms of the FLRW area distance we can equivalently write (106) in the simpler form (where, to have handy the formula for later use, we have taken the square root) (108) Φ ΣΣ ( r(z), θ, φ) = Jac ζ (z) ( r(z), θ, φ) This clearly shows that the conformal factor Φ ΣΣ is an explicit and, at least in principle, measurable quantity associated with the local Lorentz mapping (described by the localized PSL(2, C) map ζ (z) ) needed for adjusting the three reference null directions in the chosen celestial coordinates bin B(y (I) , δ) in the celestial sphere C S z (p). This adjustment allows to transfer to B(y (I) , δ) the actual area distance, namely, compute D(ζ (z) ( r(z), θ, φ)), and compare its distribution on the FLRW celestial sphere C S z (p) with respect to the isotropic FLRW area distance D( r(z)). The anisotropies in the angular distribution with respect to D( r(z)) give rise to fluctuations in Φ ΣΣ . It may appear somewhat surprising that, after all, the conformal factor does not explicitly depend also from the distortion tensor L ab defined by (30). This dependence is implicit in the definition of the area distance (29) and of the coordinate parametrization (30) characterizing L ab . These definitions give rise to the relation (34) that, as can be easily checked, remove the explicit L ab dependence from Φ ΣΣ . As we shall see, this fact will turn to our advantage when extending our analysis to the more general case of fractal-like sky sections.
6. The sky section comparison functional at scale L The harmonic energy E h, ζ (z) , h , or equivalently E g (z) , ψ (z) , g (z) , associated with the maps ζ (z) and ψ (z) , cannot be used directly as comparison functional between the sky sections ( Σ z , g (z) ) and (Σ z , g (z) ). This follows directly as a consequence of the conformal invariance (93) which implies where, as usual, h(S 2 )) is the round metric on the unit 2-sphere S 2 . From the above relation it follows that E g (z) , ψ (z) , g (z) , (and similarly for E h, ζ (z) , h ), does not depend from the area distance a 2 0 (1 + z) 2 f 2 ( r(z)) on the FLRW past lightcone C − (p). Thus, E g (z) , ψ (z) , g (z) cannot be a good candidate for the role of the functional that compares the sky sections ( Σ z , g (z) ) and (Σ z , g (z) ). For this role, we introduced in [12] a functional whose structure was suggested by the rich repertoire of functionals used in the problem of comparing shapes of surfaces in relation to computer graphic and visualization problems (see e.g. [35] and [28], to quote two relevant papers in a vast literature). In particular, we were inspired by an energy functional introduced, under the name of elastic energy, in a remarkable paper by J. Hass and P. Koehl [29], who use it as a powerful means of comparing the shapes of genus-zero surfaces in problems relevant to surface visualization.
In the more complex framework addressed in cosmography, we found it useful to define the sky section comparison functional at scale L(z) according to that can be, more expressively, rewritten as (see (108)) Thus, from the physical point of view, E ΣΣ [ψ (z) ] describes the mean square fluctuations of the physical area distance D ζ (z) ( r(z), θ, φ) (biased by the localized PSL(2, C) mapping ζ (z) ) with respect to the reference FLRW isotropic area distance D( r(z)). Notice that, whereas the harmonic map energy E g (z) , ψ (z) , g (z) is a conformal invariant quantity, the functional E ΣΣ [ψ (z) ] is not conformally invariant. Under a conformal transformationĥ −→ e 2fĥ we get Since we can also write , (see (101)), it is also clear from its definition that corresponding to large linear "stretches" in conformally mapping ψ * (z) g (z) on g (z) , E ΣΣ [ψ (z) ] tends to the harmonic map energy.
In our particular framework, the functional E ΣΣ [ψ (z) ] has many important properties that make it a natural candidate for comparing, at the given length scale L, the sky sections ( Σ z , g (z) ) and (Σ z , g (z) ) and, as the length-scale L varies, the physical lightcone region C − L (p, g) with the FLRW reference region C − L (p,ĝ). These properties are discussed in detail in [12] (see Lemma 8 and Theorem 9), here we recollect them, without presenting their proof, in the following 14 is the comparison functional associated with the inverse map ψ −1 (z) : Σ z −→ Σ z , and Φ Σ Σ is the corresponding conformal factor. Let (M, g) be another member of the FLRW family of spacetimes, distinct from (M,ĝ), that we may wish to use as a control in a best-fitting procedure for the physical spacetime (M, g). Let ( Σ z , g (z) ) denote the sky section on the past lightcone C − L 0 (p,g), with vertex at p, and let ψ (z) : Σ z −→ Σ z , and Φ Σ Σ respectively denote the corresponding diffeomorphism and conformal factor. Then to the composition of maps we can associate the triangular inequality Moreover, iff the sky sections ( Σ,ĝ (z) ) and (Σ, g (z) ) are isometric. Finally, if we denote by W 1,2 ζ (z) ( C S z (p), C S z (p)) the space of localized PSL(2, C)-maps ζ (z) which are of Sobolev class W 1,2 , ( i.e. square summable together with their first derivatives), then defines a scale-dependent distance between the sky sections ( Σ z ,ĝ (z) ) and (Σ z , g (z) ) on the lightcone regions C − L (p,ĝ) and C − L (p, g). We need to conclude our long lightcone journey addressing the real nature of the physical sky section Σ z . This forces us to leave the comfort zone of the assumed smoothness of the past physical lightcone C − (p, g). 14 In [12], the general notation is somehow at variance from the one adopted here, since we address the analysis of E ΣΣ directly on the surfaces Σ and Σ. In particular, we refer to Σ and Σ as celestial spheres rather than sky sections.

The Lipschitz geometry of the cosmological sky sections Σ z
The celestial sphere description of the sky sections Σ z discussed above is inherently vulnerable to the vagaries of the local distribution of astrophysical sources, and the associated strong gravitational lensing phenomena 15 imply that the actual past light cone C − (p, g) is not smooth as we have assumed 16 . In particular, C − (p, g) may fail to be the boundary ∂ I − (p, g) of the chronological past I − (p, g) of p, (the set of all events q ∈ M that can be connected to p by a past-directed timelike curve), because past-directed null geodesics generators of C − (p, g), λ : [0, δ) −→ (M, g), with λ(0) = p, may leave ∂I − (p, g) and, under the action of the local spacetime curvature, plunge into the interior I − (p, g). A spacetime description of this behaviour in connection with the phenomenology of gravitational lensing is discussed in detail in [45], with a rich repertoire of examples of the possible singular structure that C − (p, g) may induce on the cosmological sky sections Σ(p, r). As a matter of fact, the sections Σ z may evolve into fractal-like surfaces, and to describe them from the point of view of geometric analysis, we need to introduce a framework tailored to the low-regularity landscape generated by the local inhomogeneities.
7.1. The Lipschitz landscape. Given a past-directed null geodesic I W ∋ r −→ exp p (rk(n(θ, φ))), issued from p ∈ M in the direction n(θ, φ) ∈ C S z , we follow [37] and define its terminal point as the last-point (121) q(r * , n(θ, φ)) := exp p (rk(n(θ, φ))) that lies on the boundary ∂I − (p, g) of the chronological past of p. Any such terminal point q(r * , n(θ, φ)) is said to be: i) a conjugate terminal point if the exponential map exp p is singular at (r * , n(θ, φ)); ii) a cut locus terminal point if the exponential map exp p is non-singular at (r * , n(θ, φ)) and there exists another null geodesic, issued from p, passing through q(r * , n(θ, φ)), (see also [1], [45]). We denote [37] by T − (p) the set of all terminal points associated with the past null geodesic flow issuing from p. In presence of cut points, C − (p, g) fails to be an embedded submanifold of (M, g). Failure to be an immersed manifold is more directly related to conjugate points along the generators of C − (p, g) and of the associated conjugate locus [45]. It follows that in presence of terminal points the mapping is no longer one-to-one, and the cosmological sky section Σ z fails to be a smooth surface. From the physical point of view, this is the geometrical setting associated with the generation of multiple images of astrophysical sources 17 in the observer celestial sphere C S z . The mathematical framework for handling such a scenario is to assume that the past null cone C − (p, g) has the regularity of a Lipschitz manifold, characterized by a maximal atlas A = {(U α , ϕ α )} such that all transition maps between the coordinate charts (U α , ϕ α ) of C − (p, g), , are locally Lipschitz maps between domains of the Euclidean space (R 3 , δ). On C − (p, g), the condition of being Lipschitz can be viewed as a weakened version of the differentiability. In particular, if f : C − (p, g) ∋ U −→ R 3 is a continuous map between open sets, then f is Lipschitz if and only if it admits distributional partial derivatives that are in L ∞ (U ) with respect to the 15 See [45] for a thorough analysis of the geometry of gravitational lensing. 16 The restrictive nature of the smoothness assumption on the metric g, typically represented by functions g ab ∈ C k (R 4 , R), k ≥ 2, and of the associated light cone, has been pointed out by many authors, mainly in the context of the proof of singularity theorems and in causality theory, (see e.g. [13], [17], [38], [42], [51]). 17 If the sources are not pointlike, we also have the more complex ring patterns typical of strong gravitational lensing.
Lebesgue measure. This statement of Rademacher's theorem [23], [48] implies that the transition maps ϕ αβ on C − (p, g) have differentials dϕ αβ that are defined almost everywhere, and which are locally bounded an measurable on their domains. In such a low-regularity setting the exponential map is quite delicate to handle. However, a key result, geometrically proved by M. Kunzinger, R. Steinbauer, M. Stojkovic [40], (based on work by B.-L. Chen and P. LeFloch [13]), and by E.
Minguzzi [42], implies that the exponential map associated with a C 1, 1 metric can still be defined as a local bi-Lipschitz homeomorphism, namely a bijective map which along with its inverse is Lipschitz continuous in a sufficiently small neighborhood of p. Thus, the exponential map retains an appropriate form of regularity in the sense that locally, for each point p ∈ M , there exist open star-shaped neighborhoods, N 0 (p) of 0 ∈ T p M and U p ⊂ (M, g), such that exp p : N 0 (p) −→ U p is a bi-Lipschitz homeomorphism [40]. In particular, each point p ∈ (M, g) possesses a basis of totally normal neighborhoods. It is worthwhile to stress that geodesic normal coordinates (see (22)) can be still defined, but the transition from the current smooth coordinate systems 18 used around p ∈ M to the normal coordinates associated with exp p is only continuous.

7.2.
The fractal-like sky section Σ z . We are interested in the geometry that such past light cone scenario induces on the cosmological sky section Σ z := exp p [C S z ] of C − (p, g). As long as exp p is bi-Lipschitz, the sky sections Σ z are topological 2-spheres, and the results above seem to suggest that after all there is no such a strong motivation to abandon the comforts of the smooth framework in favor of a Lipschitzian rugged landscape. However, as the length scale L varies, the development of caustics in C − (p, g) generates cusps and crossings in the surfaces Σ z , to the effect that they are no longer homeomorphic to 2-spheres. In such a setting, the restriction of the exponential map to the celestial sphere C S z , characterizing the surface Σ z , (see (71)), is only a Lipschitz map between the metric spaces C S z , d S 2 r and Σ z , d g|Σ , where d S 2 r is the standard distance function on the round 2-sphere S 2 r or radius r, and d g|Σ is the distance function induced (almost everywhere) on Σ z by the metric g| Σz defined 19 by (26). In general, the sky section Σ z can be topologically very complex since it may contain terminal points of the exponential map exp p , giving rise to cusps and swallow-tail points associated with self-intersections of Σ z . Even if this may evolve in a very complex picture of Σ z , we still have quite a geometric control over its metric structure. The Lipschitz regularity of exp p implies that there is a constant c r , depending on the parameter r, such that y), ∀ x, y ∈ S 2 r , and we can define the pull-back on the celestial sphere C S z ∈ T p M of the distance function d Σz according to , ∀ x, y ∈ C S z . We can also pull-back the metric g| g|Σz to C S z . By Rademacher's theorem exp p is differentiable almost everywhere, and (127) h(θ, φ) := exp * p g| Σz αβ dx α dx β , is a metric defined, almost everywhere on the celestial sphere C S z , (by a slight abuse of language, we have used the same notation as for the smooth version (27)). We can also define almost everywhere 18 Recall that M is a smooth manifold, and that the low Lipschitz C 1, 1 regularity is caused by the metric g, and not by the differentiable structure of M . 19 In presence of cut points the inclusion map ιr : Σz ֒→ C − (p, g) of the sky section Σz into C − (p, g) is Lipschitz, thus Rademacher's theorem allows us to define the pull-back metric g|Σ z := ι * r g| C − (p,g) only almost-everywhere the volume element dµ h associated with the metric (127), i.e.
(128) dµ h := exp * p dµ g| Σz = det(h(r(z), θ, φ)) dθdϕ , in full analogy with its smooth version (28). All this implies that with the proviso of the almost everywhere meaning, the characterization (29) of the angular diameter distance D(r, θ, φ) and of the shear-inducing distortion L αβ defined by (30), carry over to the bi-Lipschitz case.
To put these geometrical remarks at work, let us stress that we cannot have reasonable control over the very complex topological structure of the sky section Σ z induced by a cascade of (strong) lensing events. Moreover, the corresponding caustics and singularities at the terminal points on Σ z provide a level of detail that is not relevant to the present analysis. Thus, as a reasonable compromise, we assume that the exponential map exp p is bi-Lipschitz, that Σ z is topologically a 2-sphere, and we mimic the effect of the many lensing events that may affect Σ z by assuming that the sky section Σ z has the irregularities of a metric surface with the fractal geometry of a 2-sphere with the locally-finite Hausdorff 2-measure associated with (128). Under such assumptions, it can be shown that our smooth analysis can be safely extended, (in particular, we can still exploit the Poincaré-Koebe uniformization theorem [44]), and the results obtained hold also in the more general setting of a Lipschitz description of the cosmographic past lightcone C − (p, g).
8. Concluding remarks: d (z) Σ z , Σ z as a scale-dependent field According to the physical characterization (111) of E ΣΣ [ψ (z) ], and the results described in Theorem 1, the distance function d (z) Σ z , Σ z , (for simplicity, one may work with the E ΣΣ [ψ (z) ] realizing the minimum), can be interpreted as defining a z-dependent field on the FLRW past light cone C − (p, g) describing the mean square fluctuations of the anisotropies of the physical area distance D(ζ (z) ) with respect to the reference FLRW area distance D( r(z)). These fluctuations provide information on how much the local area element on the physical sky section Σ z differs from the corresponding (round) area element on the reference FLRW sky section Σ z . Since for 2-dimensional surfaces the local Riemannian geometry is fully described by the area element, the fluctuations in D(ζ (z) ) give information on how much the geometries of the sky sections Σ z and Σ z differ. When we reach the scale of homogeneity, the physical area distance D(ζ (z) ) becomes isotropic and can be identified with the reference FLRW D( r(z)). The localized null-directions alignment between the corresponding celestial spheres C S z (p) and C S z (p) reduces to a global kinematical Lorentz boost (and a rotation). Thus, corresponding to this homogeneity scale, the distance function d (z) Σ z , Σ z field vanishes.
Thus, we have an interesting scenario whereby it is possible to associate with the distance functional d (z) Σ z , Σ z a scale-dependent field that describes a global effect that the reference FLRW past lightcone C − (p, g) misses in describing the pre-homogeneity anisotropies of the actual past lightcone C − (p, g). This pre-homogeneity field is, in line of principle, measurable since it is the mean-square variation of the physical area distance D(ζ (z) ). The delicate question concerns its possible role in selecting the large-scale FLRW model that best fits the cosmological observations on large scales. A few qualitative indications in this direction, mainly of a perturbative nature, are discussed in [12]. The results presented here are however more precise since they connect directly the distance functional d (z) Σ z , Σ z to the area distance D(ζ (z) ). To describe an important consequence of these results, let us consider the light cone regions C − L (p,ĝ) and C − L (p, g) over a sufficiently small length scale L(z). If ζ (z) and the corresponding ψ (z) denote the minimizing maps characterized in Theorem 1, then we can write [12] where we have exploited the Radon-Nikodyn characterization of Φ 2 ΣΣ , (see (101)), the identification ψ (z) ( Σ z ) = Σ z , and the relation where A(Σ z ) and A Σ z respectively denote the area of the sky sections (Σ z , g (z) ) and (Σ z , g (z) ). Thus, .
To simplify matters, we assume that at the given length scale L(z) the corresponding region C − L (p, g) is caustic free. Let us rewrite Φ Σ Σ as Φ Σ Σ = Φ Σ Σ − 1 + 1 (132) = Jac ζ (z) 1 2 D(ζ (z) ) − D( r(z)) D( r(z)) + 1 , where we have simplified the notation used in (108). By introducing this in (131) we get This expression can be further specialized if we exploit the asymptotic expressions of the area A Σ z and A (Σ z ) of the two surfaces ( Σ z , g (z) ), (Σ z , g (z) ) on the corresponding lightcones C − L (p, g) and C − L (p, g). These asymptotic expressions can be obtained if we consider the associated causal past regions J − L (p, g) and J − L (p, g) sufficiently near the (common) observation point p, in particular when the length scale L(z) we are probing is small with respect to the "cosmological" curvature scale. Under such assumption, there is a unique maximal 3-dimensional region V 3 L (p), embedded in J − L (p, g), having the surface (Σ z , h) as its boundary. This surface intersects the world line γ(τ ) of the observer p at the point q = γ(τ 0 = − L(z)) defined by the given length scale L(z). For the reference FLRW the analogous set up is associated to the constant-time slicing of the FLRW spacetime (M, g) considered. The corresponding 3-dimensional region V 3 L (p), embedded in J − L (p, g), has the surface ( Σ z ,ĥ) as its boundary. The FLRW observer γ( τ ) will intersect V 3 L (p) at the point q = γ( τ 0 = − L(z)). By introducing geodesic normal coordinates {X i } in J − L (p, g) and {Y k } in J − L (p, g), respectively based at the point q and q, we can pull back the metric tensors g and g to T q M and T q M , and obtain the classical normal coordinate development of the metrics g and g valid in a sufficiently small convex neighborhood of q and q. Explicitly, for the (more relevant case of the) metric g, we have (see e. g. Lemma 3.4 (p. 210) of [49] or [47]) where R abcd is the Riemann tensor of the metric g (evaluated at the point q). The induced expansion in the pulled-back measure (exp s(η) ) * dµ g provides the Lorentzian analog of the familiar Bertrand-Puiseux formulas associated with the geometrical interpretation of the sectional, Ricci and scalar curvature for a Riemannian manifold in terms of the length, area, and volume measures of small geodesic balls. In the Lorentzian case the relevant formulas are more delicate to derive, [3], [25], [26], [43]. This asymptotics provides [25], to leading order in L(z), the following expressions for the area of (Σ z , g (z) ) and ( Σ z , g (z) ), Notice that the integral is the average value over the sky section ( Σ z , g (z) ), of the fluctuations of Jac ζ (z) 1 2 D(ζ (z) ) with respect to D( r(z)), average that for notational ease we write as To put these results at work, let us assume the conservative and quite a reasonable scenario where the fluctuations in the area distance D(ζ (z) ), even if locally large in the various celestial coordinates bins, average out to zero over Σ z . However, the corresponding square mean deviation of the fluctuations D(ζ (z) ) D( r(z)) 2 Σz = A −1 ( Σ z ) d (z) Σ z , Σ z can be significantly different from zero, and from (136) we get (139) R(q) = R(q) + 72 π d (z) Σ z , Σ z L 4 (z) + . . . .
The physical scalar curvature we measure (hard to!) in such a scenario is R(q), and if we decide to modeling with a FLRW solution a cosmological spacetime, homogeneous on large scale but highly inhomogeneous at smaller scale, then (139) shows that we cannot identify R(q) with the corresponding FLRW scalar curvature R(q). Such an identification is possible, with a rigorous level of scale dependence precision, only if we take into account the term (140) 72 π d (z) Σ z , Σ z L 4 (z) .
According to Theorem 1, this term vanishes once L(z) probes the homogeneity scales, conversely, it is clear from (139) that in pre-homogeneity region its presence is forced on us and plays the role of a scale-dependent effective positive contribution to the cosmological constant. As long as the local inhomogeneities give rise to significant fluctuations in the area distance D(ζ (z) ), this contribution cannot be considered a priori negligible in high-precision cosmology.