Time reversal symmetry in cosmology and the quantum creation of universes

The classical evolution of the universe can be seen as a parametrised worldline of the minisuperspace, with the time variable $t$ being the parameter that parametrises the worldline. The time reversal symmetry of the classical action implies then that for any positive oriented solution it must be a symmetric negative oriented one that, in terms of the same time variable, represent an expanding and a contracting universe, respectively. However, the choice of the time variable induced by the correct value of the Schr\"odinger equation in the two universes makes that their physical time variables are reversely related. Thus, the two universes turn out to be expanding universes from the point of view of their internal inhabitants, who identify matter with the particles that move in their spacetimes and antimatter with the particles that move in the symmetric universe. This might explain two of the main problems in cosmology: the time asymmetry observed in the context of a single universe and the matter-antimatter asymmetry of the primordial universe.


I. INTRODUCTION
There is a formal analogy between the classical description of the universe and the trajectory of a test particle in a curved spacetime.The former is given, in a homogeneous and isotropic universe, by the solutions of the field equation a(t) and ϕ(t) = (ϕ 1 (t), . . ., ϕ n (t)), where a is the scale factor and ϕ i are n scalar fields that represent the matter content of the universe.The evolution of the universe can then be seen as a parametrised trajectory in the n + 1 dimensional space formed by the coordinates a and ϕ, which is called the minisuperspace.The trajectory is the worldline that extremizes the Hilbert-Einstein action, the parameter that parametrises the worldline is precisely the time variable t, and the parametric coordinates along the worldline are the classical solutions, (a(t), ϕ(t)) .
From that point of view, the time reversal invariance of the laws of physics translates into the minisuperspace in the invariance that we have in running the worldline in the two possible directions, forward and backward, along the worldline.This is similar to what happens with the trajectory of a test particle in the spacetime.In particle physics, Feynman interpreted the time forward and the time backward solutions of the trajectory of a test particle as the trajectories of particles and antiparticles of the Dirac's theory [1].In the universe, however, a forward oriented trajectory with respect to the scale factor component means an increasing value of the scale factor so it represents an expanding universe.Similarly, a backward solution represents a contracting universe.Nevertheless, it is worth noticing that the positive or the negative character of the time variable has always a relative meaning with respect to the partner component so it is a matter of taste to choose what component of the pair is the expanding one and which component is the contracting, much in a similar way as particles and antiparticles have always a relative meaning in the context of particle physics.
In fact, it could well happen that the time variables of the two universes would be reversely related [2,3] and, in that case, the two components of the symmetric pair would describe two universes, both expanding or both contracting.The creation of two contracting universe becomes meaningless because the two newborn universes would rapidly delve again into the gravitational vacuum from which they emerged.Therefore, the most plausible scenario is the creation of two expanding branches.
The time reversal relation between the time variables of the two universes is not only suggested by the analogy with particle physics, it is also derived from the semiclassical description of the universe in quantum cosmology.Let us notice that, from the point of view of quantum cosmology, the semiclassical picture of quantum matter fields propagating in a classical background spacetime is an emergent feature that appears, after some decoherence process, in the semiclassical regime [4,5].In that case, we shall see in this paper that in order to obtain the correct value of the Schrdinger equation in the two universes their time variables must necessarily be reversely related.Then, the time variables measured by the internal observers in their particle physics experiments, i.e. the time variables that appear in the Schrdinger equation of their physical experiments, are reversely related and, from the point of view of those physical time variables, both universes are expanding universes.
The paper is outlined as follows.In Sect.II, we present the analogy between the classical evolution of the universe and the trajectory of a test particle in a curved spacetime.It is shown that the time reversal symmetry of the action and the conservation of the total momentum in the minisuperspace imply that the universes should be created in pairs with opposite values of their momenta so that the total momentum is conserved.In Sect.III, it is analysed the symmetry of the Wheeler-DeWitt under complex conjugation of its solutions, which is eventually rooted in the time reversal invariance of the classical action, too.Then, it is shown that in order to obtain the correct value of the Schrdinger equation in the universes described by two symmetric solutions theirs physical time variables must be reversely related.Thus, the two newborn universes are both expanding universes from the point of view of their internal observers.In Sect.IV, we summarise and make some conclusions.

II. TIME REVERSAL SYMMETRY IN CLASSICAL COSMOLOGY
Let us consider a homogeneous and isotropic spacetime and a scalar field that propagates minimally coupled to gravity and that represents the matter content of the universe.Because the homogeneity of the space the metric components and the scalar field only depend on the time variable so, ϕ(t, x) = ϕ(t), and where a(t) is the scale factor and N (t) is the lapse function that parametrises the time variables (N = 1 corresponds to cosmic time).Small inhomogeneities around this homogeneous and isotropic background can also be considered [6,7] but as far as the inhomogeneities remain small, the dynamics of the background essentially depends on the values of the scale factor and the homogeneous mode of the scalar field, a(t) and ϕ(t).From this point of view, the evolution of the universe in then given by these two functions, which are the solutions of the field equations and the Friedmann equation These are coupled equations because the evolution of the scale factor depends on the matter content of the universe and the evolution of the scalar field depends on the variable of the spacetime where it propagates.They can be difficult to solve but, in any case, the exact or the approximate solutions of the field equations essentially give the evolution of the universe.It is also worth noticing that these equations are invariant under the reversal change in the time variable, t → −t.It means that the solutions of the field equations come always in symmetric pairs, one for t and the other for −t.This will be important later on.An alternative, equivalent point of view for the evolution of the universe is considering that the time dependent solutions of the scale factor and the scalar field, a(t) and ϕ(t), are the parametric equations of a trajectory in the minisuperspace, where the time variable acts as the (non-affine) parameter in terms of which it is described the trajectory of a 'test universe' (see, Fig. 1).Generally speaking we call superspace to the space of all possible geometries, modulo diffeomorphisms, and all the matter field configurations that can be fitted in those spacetime [8,9].However, when we restrict to a few degrees of freedom because the existence of some symmetries, like the homogeneity and isotropy that we are considering here, then, we call it minisuperspace.Therefore, the scale factor and the scalar field are, in the present case, the coordinates of the minisuperspace and the classical evolution of the universe is then a parametrised trajectory of the minisuperspace.
The trajectory of a 'test universe' in the minisuperspace, i.e. the evolution of the universe, is obtained from the variational principle of the Hilbert-Einstein action, which for the present case can be written as where, q A = {a, ϕ}, are the coordinates of the minisuperspace 2 , the minisupermetric G AB is given by and V(q) contains all the potential terms of the spacetime and the scalar field, which for the present analysis is not relevant but, for the shake of concreteness, one can assume it to be where, κ = 0, ±1 for flat, closed and open spacetimes, and V (ϕ) is the potential of the scalar field.An explicit term for a cosmological constant is implicitly included in the case of a constant value of the potential, V (ϕ) = Λ.The action (4) and the minisupermetric (5) clearly reveal the geometric character of the minisuperspace.The scale factor formally plays the role of the time like variable of the minisuperspace and the scalar field formally plays the role of the spatial like variable 3 .The trajectory that extremizes the action ( 4) is the non-affinely parametrised geodesic of the minisuperspace, given by the equations which are nothing more than the field equations ( 2).The momentum conjugated to the minisuperspace variables can be directly obtained from the Lagrangian of the action (4), and the Hamiltonian constraint, δH δN = 0, then reads where for convenience 4 we have defined, m 2 eff (q) = 2V (q).In the present case, it yields which is the Friedmann equation (3) expressed in terms of the momenta instead of in terms of the time derivatives of the unscaled minisuperspace variables.As it was pointed out before, it is worth noticing that the action (4), and therefore the geodesic equation ( 7) and the momentum constraints (9-10) are invariant under a reversal change of the time variable.It means that the solutions come in pairs with opposite values of the associated momenta (let us notice that the momenta given in (8) are not invariant under the same change).From ( 8) and (10), it is easy to see that in terms of the cosmological time (N = 1) the two symmetric solutions are 2 For convenience, the initial scalar field ϕ has been rescaled according to ϕ → 1 √ 2 ϕ. 3 Let us recall, however, that this is just a formal analogy, and let us also notice that in the case of considering n scalar field minimally coupled to gravity, then, the line element of the minisuperspace would be, so the scalar fields would parallel the role of n spatial variables in a n + 1 dimensional spacetime. 4Written in this way, the resemblance between the description of the trajectory in the minisuperspace and the description of a trajectory in the spacetime is quite evident.This analogy can be taken quite far and it is the subject of a current line of research [10].This is a clear reminder of the solutions that we obtain for the trajectory of a test particle moving in the spacetime [10].For instance, in Minkowski spacetime5 , the time component of the geodesics satisfies where τ is an affine parameter and, p t = ±E, is the energy of the test particle.In the spacetime, the two signs in (12) represent the opposite values of the time component of the tangent vector to the geodesic, i.e. the two ways in which the geodesic can be run: forward in time and backward in time.This was used by Feynman to interpret the trajectories of particles and antiparticles of the Dirac theory [1].However, in order to avoid the negative values of the energy, one can make use of the invariance of the geodesics under the reversal change of the affine parameter, τ → −τ , and take the parameter τ for the positive value in (12) and the value −τ for the negative value.It is then obtained two particles moving both forward in time, with positive energy, but because momentum conservation, with opposite values of their spatial momenta.
Similarly, in the case of the universe the two solutions given by (11) represent a universe moving forward in the scale factor component and another universe moving backward in the scale factor component (see, Fig. 2).The reversal change in the time variable reveals the two possible directions in which the geodesic can be run in the minisuperspace: one forward in the scale factor component and the other in the backward directions.Let us notice however that in the minisuperspace, moving forward in the scale factor component means an increasing value of the scale factor so that solution represents an expanding universe, and moving backward in the scale factor component means a decreasing value of the scale factor so the symmetric solution represents a contracting universe.Therefore, the two symmetric solutions form an expanding and contracting pair of universes (see, Fig. 2), which for momentum conservation must always be present in symmetric pairs.Let us notice however that the terms expanding and contracting, like the terms future and past in the spacetime, have always a relative meaning with respect to the partner component.The field equations (2) and the Friedmann equation ( 3) are invariant under a reversal change in the time variable, t → −t, so it is a matter of taste to choose which solution is considered the expanding one and which one is the contracting.As it happens in the case of the particles moving in the spacetime, we can even choose one time variable for one of the universes and the reverse time variable for the other.In that case, the two solutions would represent two universes, both expanding or both contracting in terms of their reversely related time variables.However, considering two contracting universes at the beginning of the universe means that the newborn universes would rapidly delve again into the gravitational vacuum from which they just emerged.Therefore, the solution that seems to be more consistent is the creation of two expanding universes.
What is true is that because momentum conservation both solutions must be present.It means, as it happens in particle physics, that the universes must be created in pairs.In terms of the same time variable one of the universes must be an expanding universe and the other must be a contracting universe.However, as we have said, this is just a relative sign of their time variables with respect to those of the partner universe.For an internal observer of one of the universes, say Alice, her universe is the expanding one and the partner universe is the contracting, being thus conserved the total momentum in the minisuperspace.However, for Bob, an internal observer of the partner universe, things are the other way around, it is his universe the one that is expanding and Alice's universe, from his point of view, the one that is contracting.For both, the total momentum is always conserved in the minisuperspace.Besides, particles moving in the partner universe look like they were propagating backward in time, so they can be identified in the quantum theory with antiparticles.Therefore, primordial antimatter is always created at the onset in the partner universe, and thus the matter-antimatter asymmetry observed in each single universe would be restored in the context of a pair of time reversed related universes.

III. QUANTUM COSMOLOGY AND THE CREATION OF UNIVERSES
The reversal relation between the time variables of the two universes is not only suggested by the analogy with particle physics.As we shall see in this section it is also motivated by the correct definition of the Schrdinger equation in the semiclassical regime of the universes.Let us notice that from the point of view of quantum cosmology the semiclassical picture of quantum matter fields propagating in a classical spacetime background is an emergent feature that appears, in the appropriate limit, from the complete quantum state of the universe.
The quantum state of the universe is described by a wave function that depends on the metric components of the spacetime and on the degrees of freedom of the matter fields.It is the solution of the Wheeler-DeWitt equation, which is essentially the canonically quantised version of the classical Hamiltonian constraint.This is in general a very complicated function.However, in the present case, where we are assuming a homogeneous and isotropic background and small inhomogeneities propagating therein, the Hamiltonian can be split into the Hamiltonian of the background and the Hamiltonian of the inhomogeneous degrees of freedom.It reads [6,7] where the Hamiltonian of the background spacetime, H bg , is given by the quantum version 6 of the classical Hamiltonian ( 10) and H m is the Hamiltonian of the inhomogeneous modes of the matter fields.The wave function, φ = φ(q bg , q m ), where q bg are the the variables of the background and q m are the variables of the local matter fields, can then be expressed in the semiclassical regime as a linear combination of WKB solutions, i.e. [4,5] where C = C(q bg ) is a slow-varying function of the background variables, S 0 = S 0 (q bg ) is the action of the background spacetime, ψ = ψ(q bg , q m ) is a complex wave function that contains all the dependence on the matter degrees of freedom, and the sum in (15) extends to all possible classical configurations.A relevant feature to be noticed here is that, because the real character of the Wheeler-DeWitt equation, which in turn is rooted in the time reversal symmetry of the Hamiltonian constraint and the classical action (4), the semiclassical solutions of the Wheeler-DeWitt equations come always in pairs that correspond to the two possible signs in the exponentials of (15).We shall now see that these two signs induce two reversely related time variables for the universes they represent, supporting and enhancing the considerations already made at the classical level.
In order to see how the two branches of (15) represent a particular universe one can insert them into the Wheeler-DeWitt equation ( 13) and solve it term by term in an expansion of powers of [3,7].At zero order in one obtains the following Hamilton-Jacobi equation [7] 6 A particular choice of factor ordering is customary taken to make the Wheeler-DeWitt invariant under rotations in the minisuperspace.Then, the Hamiltonian constraint (10) turns out to be a Klein-Gordon like equation where the Laplace-Beltrami operator is the covariant generalisation of the Laplace operator of the minisuperspace [8,9], given by This equation represents the dynamics of the background spacetime.It can be converted into the Friedmann equation by defining a time variable given by [7] where ∇ is the gradient of the minisuperspace [7].Once again, let us notice the freedom that we have in choosing the sign in (17).As was already pointed out in the previous section, the two signs represent the two ways in which a trajectory can be run in the minisuperspace.In terms of the time variable defined in (17 so that the Hamilton-Jacobi equation ( 16) turns out to be the Friedmann equation (3).Furthermore, at first order in in the expansion of the Wheeler-DeWitt equation, one obtains [3, 7] where the minus sign corresponds to the branch with the positive sign in the exponential in (15) and the positive sign corresponds to the complex conjugated branch.The term in brackets in ( 19) is actually the time variable defined in the background spacetime, given by (17), with the two possible signs, so it means that essentially (19) is the Schrdinger equation for the matter fields that propagate in the classical background spacetime.We then recover the semiclassical picture of quantum matter fields propagating in a classical background.However, in order to have the proper sign in the Schrdinger equation in each single universe we need to choose the positive sign in the definition of the time variable t in (17) for the branch with the negative exponential in (15) and the negative sign of the time variable for the branch with the positive exponential in (15).The time variable involved in the Schrdinger equation can be considered the physical time in the sense that it is the time measured by internal observers in their particle physics experiments, so it is the time variable measured by actual clocks, which are eventually made of matter.It means that the physical time variable measured by internal observers in the two different universes is related by a time reversal change, which is the relation also obtained at the classical level in Sect.II from the analogy with the test particles moving along spacetime trajectories.Therefore, the classical and the quantum symmetries of the minisuperspace naturally impel us: i) to consider the creation of the universes in pairs with opposite values of their momenta as the most natural way in which the universes should be created; and, ii) to consider opposite physical time variables for the single universes of each conjugated pair, from the point of view of internal observers.
As it has been already pointed out, the consideration of the creation of conjugated pairs of universes with reversely related time variables could solve some of the global asymmetries observed in the context of a single universe.For instance, the time asymmetry observed in the context of one single universe is automatically restored in the context of the creation of two universes with their time variables related by a reversal change.If the time variables of the two symmetric universes are reversely related, then, assuming the CPT theorem, one would expect that the charge and the parity of the particles that move in the pair of universes would be reversely related too.The primordial matter-antimatter asymmetry, which is one of the biggest problems in cosmology, could then also be restored in the context of a pair of entangled universes [11].
Furthermore, the existence of a partner universe, which, as we have seen, is induced by symmetric considerations, could also have observable consequences on the properties of each single universe is the matter fields of the two universes become entangled.In that case, the important thing is that the state of the matter field in each single universe, which is obtained by tracing out from the composite state the degrees of freedom of the partner universe, turns out to be different depending on whether exists or not a partner universe [3,12].In particular, if the universes are created in pairs the quantum state of the matter fields in each single universe is given by a quasi-thermal state [3], which has very specific thermodynamical properties.For instance, the mode distribution is very specific and it is not reproducible by any other known local effect in the universe 7 .It then becomes a distinguishable and in principle observable effect of the existence of a partner universe, of the compliance of the symmetries of the minisuperspace, and and effect that would make testable the whole multiverse proposal.[3].The time variables of the two symmetric universes are reversely related.It provides us with the correct value of the Schrdinger equation in the two universes.At the onset, primordial matter would be created in the observer's universe and antimatter in the symmetric one.Particles and antiparticles do not collapse because the Euclidean gap that exists between the two newborn universes [3,11].

IV. CONCLUSIONS
The evolution of the universe can be seen as a worldline of the minisuperspace formed by the scale factor, which formally plays the role of a time-like variable, and the scalar fields, which formally play the role of the spatial components.From that point of view, the time reversal symmetry of the Hilbert-Einstein action is equivalent to the invariance of the geodesics of the minisuperspace under a reversal parametrisation of the non-affine parameter.
Positive oriented paths with respect to the scale factor component in the minisuperspace entail an increasing value of the scale factor so they represent expanding universes.On the contrary, negative oriented worldlines with respect to the scale factor component represent contracting universes.However, because the time reversal invariance of the Hilbert-Einstein action, the terms 'expansion' and 'contraction' have a relative meaning with respect to the partner component so it is a matter of taste to choose which branch is the contracting one and which one is the expanding one.
The invariance of the action under time reversal change and the momentum conservation in the minisuperspace variables imply that the universes should be created in pairs, with opposite values of their momenta so that the total momentum is conserved.In terms of the same time variable, one of the universes is expanding and the other is contracting so their corresponding momenta are opposite signed and the total momentum is thus conserved.However, the universes would be both expanding or contracting branches if their time variables are reversely related.Two newborn contracting universes would rapidly delve into the gravitational vacuum from which they emerge so the most consistent solution turns out to be the creation of two expanding universes.The reversal relation between the time variables of the two universes is induced by the choice that provides us with the correct Schrdinger equation in the two universes.Therefore, the physical time measured by internal observers in their particle physics experiments, i.e. the time variable that appears in the Schrdinger equation, is always positive.From the perspective of any observer it is always then the time variable of the partner universe the one that is negative.Particles moving in the observer's universe are thus identified with matter and the particles of the partner universe with antimatter.But this has a relative meaning.For an observer in the partner universe things are the other way around.Therefore, the creation of universes in symmetric pairs with respect to the time reversal symmetry might explain two of the main problems in cosmology: the time asymmetry observed in the context of a single universe and

FIG. 2 .
FIG.2.Left: a trajectory in the minisuperspace that is positively oriented with respect to the scale factor component describes an expanding universe.Similarly, a negatively oriented trajectory describes a contracting universe.Right: in terms of the reversely related time variables the two symmetric solutions represent expanding universes.