Quantum cosmology of a classically constrained nonsingular Universe

The quantum cosmological version of a nonsingular Universe presented by Mukhanov and Brandenberger in the early nineties has been developed and the Hamilton Jacobi equation has been found under semiclassical (WKB) approximation. It has been pointed out that, parameterization of classical trajectories with semiclassical time parameter, for such a classically constrained system, is a nontrivial task and requires Lagrangian formulation rather than the Hamiltonian formalism.


Introduction
It transpires from Hawking-Penrose energy condition that the singularities of the general theory of relativity are unavoidable classically. In the absence of a complete and satisfactory theory of quantum gravity, it is not clear what would be the nature or the fate of singularities in the quantum domain. However, it might be possible in principle to get rid of the singularities classically by imposing some quantum mechanical bounds on certain quantities. At the singularity, some of the curvatures, eg., energy momentum and even the Riemann tensors diverge. Further, quantum field theory predicts that, the scattering cross sections become infinity when all radiative corrections are taken into account. Renormalization makes individual terms manageable, but the entire series diverges. So in order to get rid of the singularities, curvature invariant terms must be constrained to take some upper limits. If Planck's length l pl is assumed to be the fundamental length, below which no length is measurable, then from dimensional argument, curvature invariants must be bounded as, |R| < l −2 pl , |R µν R µν | < l −4 pl , etc. However, curvature invariants are infinite in number and there is no guarantee that imposing constraint on some lower order curvature invariant terms, all the higher order terms will be bounded. Hence, 'Limiting Curvature Hypothesis' (LCH) [1] had been invoked. LCH states that a finite number of curvature invariants should be bounded and one of those, say I 2 , should have the property that I 2 = 0 singles out a particular nonsingular solution as the only one. In the process all the infinite set of curvature invariants are automatically bounded. If the limiting space is de-Sitter, then for a isotropic Universe, LCH implies that the initial and the final stages of cosmological evolution for a closed Universe will give de-Sitter phase, while baby Universes will arise at the centre of the Black-Hole and eventually gravitational collapse is avoided. Since Weyl tensor vanishes for an isotropic Universe, so in its simplest form, LCH can be realized by constraining only a couple of lower order curvature invariants, viz., I 1 = 4 R and I 2 = 4R µν R µν − R 2 . Thus in its simplest form, ie., for homogeneous and isotropic space-time, Mukhanov and Brandenberger [2] followed by Brandenberger, Mukhanov and Sornborger [3] had constructed the following action (1), which preserves LCH.
Here, the scalar field φ 2 has been introduced to obtain nonsingular solutions, while φ 1 bounds the curvature invariants for these nonsingular solutions. The potentials V 1 (φ 1 ) and V 2 (φ 2 ) should be chosen in a manner such that (i) for φ i ≪ 1 , the leading terms in the action gives back Einstein's theory and (ii) their asymptotic behaviour should be such that LCH may be realized, which requires V 1 (φ 1 ) → constant as φ 1 → ∞, and V 2 (φ 2 ) → 0 as φ 2 → ∞. For a flat Robertson-Walker metric, they ( [2] & [3]) had obtained a first order differential equation containing fields and the potentials and had drawn a phase diagram in φ 1 − φ 2 plane, for some particular choice of the potentials. The phase diagram indicates nonsingular solutions in all regions and de-Sitter phase in the asymptotic region. In a nut-shell, the results obtained [4] are the following. Firstly, all the homogeneous and

Formalism
The action presented by Mukhanov and Brandenberger [2], corresponding to a homogeneous and isotropic nonsingular Universe, for which Weyl tensor vanishes, is given by, where, V 1 (φ 1 ) and V 2 (φ 2 ) are the potentials corresponding to the pair of apparently non-dynamical scalar fields φ 1 and φ 2 respectively. LCH may be realized from the above action (1), only for some appropriate choice of the potentials. Action (1) leads to Einstein's theory at small curvature, provided, Further, to realize LCH, first requirement is to bound the curvature and the next is to obtain nonsingular solutions. Curvature is bounded provided, V 1 (φ 1 ) ∼ φ 1 at |φ 1 | ≫ 1 and de Sitter solution is obtained in the asymptotic region, provided, V 2 (φ 2 ) ∼ constant, at |φ 2 | ≫ 1 . Now, taking Robertson-Walker line element, where N (t) is the lapse function, the Ricci scalar is given by, and the next higher order curvature invariant term is, So the action takes the following form, where, the surface term is Nȧ a , being the trace of the extrinsic curvature. The surface term is clearly different from the one that appears in Einstein-Hilbert action. However, the two are the same, if φ 2 vanishes, which has got a kinetic term in the action, and acts as a dynamical variable. It is important to notice that the surface term is not affected by the presence of the scalar field φ 1 , since it acts as Lagrange multiplier only. In the above action N and φ 1 act as Lagrange multipliers, resulting in a pair of constraint equations. Variation with respect to N gives the Hamiltonian constraint equation, but one can fix the gauge, N = 1 , without loss of generality. However, we are not going to fix the other Lagrange multiplier φ 1 , since it has been invoked to bound the curvature invariants for the nonsingular universe guaranteed by the field variable φ 2 . So, altogether we get four equations (not all independent), two of which are constraint equations and those obtained under the variation with respect to φ 2 and the scale factor a are the two field equations, where we set, N = 1 , at the end. The constraint equation appearing under the variation of φ 1 is, while the φ 2 variation equation is, The Hamiltonian constraint equation, obtained by varying the action (3) with respect to N is, Finally, varying the action with respect to the scale factor a, one gets, In the above field equations V ′ 1 (φ 1 ) and V ′ 2 (φ 2 ) denote the derivatives of the potentials with respect to φ 1 and φ 2 respectively. For spatially flat, k = 0 case, the above field equations can be combined to yield a first order differential equation, The phase diagram for a particular choice of the potentials was plotted by Mukhanov and Brandenberger [2], showing four different classes of trajectories. In the first, the trajectory starts from the de-Sitter phase as φ 2 → −∞ and evolves through to de-Sitter as φ 2 → ∞. In the second, for small initial values of φ 1 trajectory starts at φ 2 = −∞ reaches a turning point and returns to φ 2 = −∞. In the third, the trajectory shows periodic solutions about Minkowski space-time φ 1 = φ 2 = 0 . Finally, in the fourth class, trajectories starting with small φ 1 and φ1 φ2 , along with φ 2 ≥ 0 , evolve towards de-Sitter at φ 2 = ∞. In a nutshell, all the phase trajectories are either periodic about Minkowski space-time or else they asymptotically approach to de Sitter space. Hence all solutions are nonsingular. It is important to note that if V ′′ 1 vanishes, φ 2 turns out to be a constant and as a result V ′ 2 vanishes and the resulting solutions again become singular. However, the condition V ′′ 1 > 0 is satisfied by all the potentials chosen by Mukhanov and Brandenberger [2].

Analyzing the constraint and the Wheeler-deWitt equation
It is important to note that in the process of developing an action that might produce nonsingular cosmological solutions, the determinant of the Hessian, W ij = ∂ 2 L ∂qi∂qj , corresponding to the Lagrangian of the above action (3) vanishes, and so the action becomes singular. Vanishing of the determinant of Hessian, signals the presence of constraint in the theory which should be analyzed step by step carefully. Dirac algorithm [8] is the best known technique to handle such constrained system and to construct the Hamiltonian. Before we proceed, let us recapitulate a few important artefact of Dirac's algorithm. 1. Firstly we remember that the existence of an infinite invariance group leads to first class constraints, while singular Lagrangians which do not possess a local gauge invariance leads to second class constraints. 2. The rank of W ij for a system having 2N phase space variables (that does not contain time explicitly) is N . For a singular Lagrangian it is R < N . under this circumstances there exists a non-degenerate In the process one finds, r = N − R primary constraints, p r = g r (q, p α ), which originate from the definition of momentum. 3. Now if a function F (q, p) be defined on the primary phase space (this will be defined shortly) Γ p < Γ, then the restriction on F to Γ p is achieved by replacing p r by g r (q, p α ). If F vanishes identically after this replacement, then it is called weakly vanishing and is denoted by F ≈ 0 . If in addition the gradient of F also vanishes, then it is called strongly vanishing and is denoted by F ≃ 0 . 4. Primary constraints are expressed as φ r (q, p) ≈ 0, and the constrained Hamiltonian, defined on the constrained phase space Γ c , as, One can write down the primary Hamiltonian defined on the primary phase space Γ p , as, where, λ r are the Lagrange multipliers. 5. Next one has to check if the constraints are preserved in time, i.e., If the Poisson bracket of a constraint vanishes with all other at least weakly, then it turns out to be first class constraint. In that case, {φ j , H c } ≈ 0 , and all these first class constraints are exhausted. The constraints φ i which do not vanish modulo the constraint, leads to secondary constraints χ k . 6. One needs to check the consistency condition of the secondary constraints again, i.e., if the constraints are preserved in time. Again for the secondary constraints, if the poisson brackets of some (say l ), vanish with all other, then they are first class and for them {χ l , H c } ≈ 0 . Otherwise they are second class in nature and the Lagrange multipliers are determined in the process. There are as many undetermined Lagrange multipliers as there are first class primary constraints. If all the constraints are second class then the primary Hamiltonian together with the primary and secondary constraints constitute the field equations, which may be solved in principle, leaving no arbitrariness in the solutions. 7. Finally, one can try to find the true degrees of freedom and express the Hamiltonian in the reduced phase space Γ R . However, it is a very difficult tusk, if not impossible in most of the situations. Even without finding the reduced Hamiltonian it is possible to quantize the theory. There is no unique prescription in this regard. Any quantum theory may be considered to be the correct one if it can go over to the classical counterpart through a suitable correspondence principle. With this background knowledge, let us proceed to construct the Hamiltonian of the theory. We understand that in order to construct the primary Hamiltonian, one has to introduce all the constraints of the theory in the Hamiltonian through Lagrange multiplier. Choosing π 8G = M , M being the square of the Planck mass, one finds in view of the action (3) under the gauge choice N = 1 , whileφ 1 is not invertible due to the constraint, Since, ∂C1 ∂p φ 1 = 0, so the above constraint vanishes weakly. Hence the primary Hamiltonian may be expressed as, where, α is the Lagrange multiplier. Since, {C 1 , H c } does not vanish even weakly, so C 1 is a second class primary constraint. As there are no first class primary constraint, so there will be no undetermined Lagrange multiplier of the theory. Now the constraint must be preserved in time, i.e., This is a new second class constraint as long as As already pointed out that for the existence of nonsingular solutions V ′′ 1 must not vanish, here we again observe that V ′′ 1 must not vanish to remove arbitrariness from the primary Hamiltonian. However no such restriction is required for V 2 . Again the condition that the constraint should be preserved in time leads to, modulo the constraint (10). This is not a new constraint, rather it fixes the Lagrange multiplier α = − Thus both the primary and the secondary constraints are second class. Hence the primary Hamiltonian, being free from arbitrariness can now be expressed as, One can now easily check that we have obtained the correct Hamiltonian in the usual manner. The only exception appears to be withφ 1 = ∂H , which looks like a new equation. However, this is the one, we were in search of, sinceφ 1 can now be inverted through this equation. In any case, it is not an independent equation, since taking time derivative of the first (constraint) equation (4) and using the second one (5), one can arrive at it. Thus the Hamiltonian, (12) being free from constraints and being able to produce all the field equations, is the correct one. To make a comparison, let us take Einstein-Hilbert action minimal coupled to a scalar field, and write down the corresponding Hamiltonian, It is interesting to note that the classical field equations (4) through (7) reduce to the vacuum Einstein's equations at any stage of cosmic evolution as φ 1 = φ 2 = 0 . However, the primary Hamiltonian (12) once constructed in view of the constrained system under investigation, is different from (13) and does not ever reduce to the vacuum Einstein's equation. This fact has been manifested in the linear appearance of p a in H p1 . The fact that here p a appears linearly in the Hamiltonian, clearly differentiates all actions (corresponding to minimal and standard non-minimal coupling) with the present one.
Our next attempt will be to canonically quantize the theory under consideration. There is no standard prescription to quantize a classically constrained system and the only justification of a particular method of quantization lies in its success. The meaning of the last sentence is that one should be able to find a correspondence principle to go over to the classical theory under semiclassical limit. We shall write down the counterpart of the Wheeler-deWitt equationĤ|Ψ >= 0 , for the modified theory of gravity under consideration and show in the following sections how to find the notion of semiclassical time so that classical field equations may be reproduced. Now, in constructing the Wheeler-deWitt equation corresponding to the Hamiltonian (12), operator ordering ambiguities should be taken care off. Some of the operator ordering ambiguities may be removed by expressing,p a → q −1 (a)p a q(a), where, q(a) is an arbitrary function of a. Hence the first term of the Wheeler-deWitt equation corresponding to the Hamiltonian (12), after replacingp by −i ∇, turns out to be Clearly, first order derivative of the wavefunction Ψ with respect to a, (ie., ∂Ψ ∂a ) does not appear, while the same with respect to the scalar field (ie., ∂Ψ ∂φ2 ) appears. Further, operator ordering inp φ1 andp φ2 appearing due to the presence of the second and the third terms in (12), again introduces first derivative terms (ie., ∂Ψ ∂φ1 and ∂Ψ ∂φ2 ) in the Wheeler-deWitt equation, corresponding to the scalars φ i . However, from the regularity argument [9], it follows that ∂Ψ ∂φi can be neglected at sufficiently small values of a, ie, at sufficiently early epoch. Hence, the Wheeler-deWitt equation may be kept free from the first derivative terms even after removing some of the operator ordering ambiguities (unlike the situation encountered in standard and other nonstandard theories of gravity), and is expressed as, which is independent of the operator ordering parameter q(a). Not all quantum states |Ψ > of the Wheeler -deWitt equation (14) are allowed, since they are constrained by the quantum analogue of the classical constraints (10) and (11). However, since our aim is not to find the solution of (14), so we neither, are in search of true degrees of freedom nor incorporate the quantum analogue of the classical constraints. Rather, for the sake of comparison, we write down the Wheeler-deWitt equation corresponding to Einstein's gravity with a minimally coupled scalar field (see Appendix), where, q takes care of some of the operator ordering ambiguities. We observe that, neither ∂ 2 ∂a 2 , nor ∂ ∂a term appears in the Wheeler-deWitt equation (14), which may have some deep significance, not presently known.
Reparametrization invariance of the theory of gravity leads to the Hamiltonian constraint yielding Wheeler-deWitt equation. Despite the fact that the theory under consideration has additional constraints, we have not made any attempt to find the true degrees of freedom to construct the Hamiltonian in the reduced phase space Γ R . Rather, the corresponding quantum equation (14) has been constructed in the same tune of the standard Wheeler-deWitt equationĤ|Ψ >= 0 . Now the obvious question is, "does equation (14) alone represents the quantum version of the classical field equations (4) through (7)"? Other way round one may ask, "is it possible to set up a correspondence between the Wheeler-deWitt equation (14) and the classical equations (4) through (7)"?. Since, the constraint free Hamiltonian represents the correct one to reproduce all the classical field equations, so the answer to the question raised above is positive, if under a suitable semiclassical prescription, one can find a semiclassical notion of time to recover the Hamilton constraint equation (6) from the Wheeler-deWitt equation (14). It is a nontrivial task which we shall take up in this and in the following sections. At energy below Planck scale, the wave function can be expressed as, Ψ(a, φ 1 , φ 2 ) = e i S(a,φ1,φ2) . Expanding, S in the powers of M as, and inserting it in the Wheeler-deWitt equation (14), one obtains Let us now collect expression having same powers in M . For Einstein-Hilbert action with minimally coupled fields one gets to the power of M 2 an expression that states that the Hamilton-Jacobi function S 0 depends on three space only (see Appendix). Here the situation is quite different in the sense that we do not get expressions corresponding to the order M 2 . To the order M 1 , we have, This is the Hamilton-Jacobi equation. It can be identified with the Hamiltonian constraint equation (6) only under an appropriate choice of semiclassical time parameter. Following two sections are devoted to find the semiclassical time parameter.

Semiclassical time parameter -standard technique
Since in view of the Hamiltonian (12) all the velocities are now invertible and so following standard technique as in Einstein's gravity with minimally coupled scalar field (see Appendix,9.2), the time parameter can be found as, The problem is that, this choice of time parameter does not lead to classical constraint equation (6). This is because, ∂S0 ∂φ1 appearing in the Hamilton-Jacobi equation (16) is not obtainable from it and as a result, remains arbitrary. This is the source of trouble that we encounter in the present situation, to parameterize classical trajectories with semiclassical time parameter following usual procedure. Thus, the standard procedure does not work in the classically constrained system under consideration. One can try to find the same by expressing the time parameter as where, b , c, d, j , l , m, u , v and n are arbitrary functions of a, φ 1 and φ 2 . It is now possible to findȧ,φ 1 anḋ φ 2 , using this expression as, Substituting all these expressions in equation (6) and equating the coefficients of ∇S 0 etc., between the equation thus formed and the Hamilton-Jacobi equation (16), one can finally arrive at the following time parameter, With this technique of parametrization, classical constraint equation (6) is automatically reproduced, but the problem associated with this time parameter is that, there still exists an arbitrariness in the form of the arbitrary parameter n = n(a, φ 1 , φ 2 ). Thus, the time parameter is not unique, and something else should be tried. It is interesting to note that simply by inspection one can choose a time parameter free from such arbitrariness as, This time parameter reproduces equation (6), in view of Hamilton-Jacobi equation (16). However, this is purely intuitive on one hand and is imaginary on the other. So we must find an involved technique for this purpose.

Semiclassical time parameter -an involved technique
As already mentioned, equation (4) is an additional gravitational constraint equation of the theory under consideration, since it does not contain second order derivatives. The standard technique should be to differentiate equation (4) and then to compare it with equation (6). In the process, if the emerging equation is again a constraint equation, then it should be entered into the Lagrangian via Lagrange multiplier. The emerging constraint equation in this case is, Hence, introducing this constraint equation (17) in action (3) via a Lagrange multiplier λ, we obtain, The canonical momenta are found from the action (18) as, We don't write down the field equations since we are not going for classical solutions. However, it is important to note that since variation with respect to λ gives back the constraint equation (17) and in view of (19) the classical constraint equation (6) remains unchanged, so, the primary Hamiltonian (12), the Wheeler-deWitt equation (14) and the Hamilton-Jacobi equation (16) are systematically reproduced. Further, identifying canonical momenta with corresponding derivatives of the Hamilton-Jacobi function, equation (12) can be found again from Hamilton-Jacobi equation (16). Now in order to obtain equation (6) from (16), let us use equations (17) and (19), which gives,ȧ Equations (20), (21) and (22) are now combined to yield the correct and unique semiclassical time parameter, One can now easily obtain ∂S0 ∂a , ∂S0 ∂φ1 and ∂S0 ∂φ2 from above time parameter (23) and substituting these in equation (16), one can reproduce equation (6). To check whether we have found the correct and unique semiclassical time parameter let us continue by including a source term (in the form of a dynamical scalar field) in the model.

Including a source term
In this section we consider an additional source term in the action (1), in the form of a minimally coupled scalar field with Lagrangian density, where, σ is the scalar field and U (σ) is an arbitrary potential. The action (1) now reads (taking M = π 8G , as before), Corresponding field equations are (under variation of the above action with respect to φ 1 , φ 2 , σ, N and a and setting N = 1 ), where, V ′ 1 (φ 1 ), V ′ 2 (φ 2 ) and U ′ (σ) are the derivatives of the potentials with respect to φ 1 , φ 2 and σ respectively. As before we differentiate equation (25) with respect to time and compare it with equation (26) to get the constraint equatioṅ Next we incorporate this equation (30) into the Lagrangian through the same Lagrange multiplier λ, as before.
In view of the Lagrangian so formed, we obtain the same set of canonical momenta (19) along with an additional one, corresponding to the field σ , viz., Corresponding Wheeler-deWitt equationĤ p |Ψ >= 0 is, As before, we proceed to make semiclassical approximation by expressing Ψ[a, φ 1 , φ 2 , σ] as Ψ = exp[ i S(a, φ 1 , φ 2 , σ)], expanding S in power series of M as, S = M S 0 +S 1 +M −1 S 2 +· · · etc., and then substituting all these in equation (33). Finally, equating terms having different orders of M to zero, we find for M 2 order M 1 order term leads to Hamilton-Jacobi equation for the source free nonsingular gravitational field as, One can easily identify this equation with the Hamilton-Jacobi equation (16). It leads to the classical source free gravitational constraint equation (6) under the same choice of the time parameter (23). Hence, source free Hamiltonian (12) can be found from it by identifying the canonical momenta with corresponding derivatives of S 0 , as before. The next, ie., M 0 order term gives the following equation, Now in order to identify this equation (36) with the functional Schrödinger equation for the matter field σ , let us define a function, In view of (37), equation (36) reduces to Provided, the function D(a, φ i ) satisfies the following equation, Since the right hand side of equation (38) Hence, upto this order of approximation, we have, where, S 0 is obtainable from equation (35), f from (38) and D from (39) or (40). So at the end we observe that under the same choice of time parameter (23), one can parameterize the classical trajectories by reproducing classical source free gravitational equation (6)

Concluding remarks:
In a series of works, Brandenberger, Mukhanov, Sornborger and others [2], [3], [4] and [5] constructed a nonsingular gravitational action by invoking limiting curvature hypothesis (LCH) and studied its classical aspects. The action contains a couple of scalar fields, one of which (φ 2 ) is required to guarantee nonsingular solutions, while the other (φ 1 ) binds the curvature invariants. The corresponding potentials are chosen in such a manner that for |φ i | ≪ 1 , Einstein's theory is recovered at one hand and their asymptotic behaviour should be such that LCH may be realized on the other. In the present work, we have studied the quantum cosmological aspect of the theory and made a connection of the quantum cosmological equation with the Hamilton-Jacobi equation of the classical theory. The answer to the question that, why it is at all necessary to quantize a nonsingular Universe model, lies in fact that some new results have been unveiled in the process and a non-trivial technique of parameterizing the classical trajectories has been found. Summarily, the results are, 1.) Introduction of the scalar field φ 1 in the action (1), only introduces yet another constraint equation (other than the Hamiltonian constraint). Thus it is essentially a classically constrained theory of gravity and so one has to deal with such constraints critically, which we have done using Dirac's algorithm. 2.) In the absence of the scalar fields, the classical field equations reduce to those corresponding to standard theory of gravity. However, the Hamiltonian obtained after analyzing the constraint, does not give Einstein's theory back. Likewise, the Wheeler-deWitt equation is free from first derivative terms ∂ψ ∂a , and thus is quite different from those obtained in view of standard and other nonstandard (nonminimally coupled) theories of gravity. 3.) The wave function is real for k = ±1 , unlike the situation encountered in standard gravity theory (see appendix), where, k = +1 leads to a complex wave function. 4.) The semiclassical time parameter required to parameterize classical trajectories can be found in a unique way only by using the technique of Lagrange undetermined multiplier in the Lagrangian constrained dynamics, rather than using Hamiltonian formalism.