Anistropic Invariant FRW Cosmology
AAnistropic Invariant FRW Cosmology
J. F. Chagoya ∗ and M. Sabido † Departamento de Física de la Universidad de Guanajuato,A.P. E-143, C.P. 37150, León, Guanajuato, México (Dated: October 22, 2018)In this paper we study the effects of including anisotropic scaling invariance in the minisuperspaceLagrangian for a universe modelled by the Friedman-Robertson-Walker metric, a massless scalarfield and cosmological constant. We find that canonical quantization of this system leads to aSchroedinger type equation, thus avoiding the frozen time problem of the usual Wheeler-DeWittequation. Furthermore, we find numerical solutions for the classical equations of motion, and wealso find evidence that under some conditions the big bang singularity is avoided in this model.
PACS numbers: 04.60.-m; 04.50.Kd; 98.80.Qc; 04.60.Kz
I. INTRODUCTION
The most widely accepted description of the universeis given by the Λ CDM model, which makes use of theFriedman-Robertson-Walker (FRW) metric in the frame-work of General Relativity (GR) supplemented with acosmological constant Λ . However, there are theoreti-cal problems concerning the inclusion of the cosmologicalconstant (see [1–3] for a review). One of these problems isthat GR and Quantum Field Theory —the theory thatdescribes the other fundamental interactions —are in-compatible [4]. The tools of quantum field theory givea theory with an ill ultraviolet (UV) behaviour whenapplied to GR. This is one of the reasons for the in-terest in the search for alternative descriptions of grav-ity at high energies. Several modifications to GR havebeen proposed to describe gravitational physics at (pre-)inflationary and accelerated expansion epochs [for a re-view of models of modified gravity see, for instance 5].The lack of a fundamental physical principle to constructthe ultraviolet theory of gravity makes the search for sucha theory a complicated quest. A pragmatic approach isto consider GR as the low energy limit of a more fun-damental and so far unknown theory, the quantum the-ory of gravity. Any formulation of quantum gravity canhave new symmetries and degrees of freedom in the UVregime, but should recover or be compatible with GR inthe low energy limit.An UV completion to GR was proposed in [6] bymaking the theory invariant under anisotropic scalingtransformations. The resulting theory is power countingrenormalizable, at the expense of losing Lorentz invari-ance in the UV limit as a consequence of the asymmetryin the transformations that are enforced, t → b z t, (cid:126)x → b(cid:126)x. The critical exponent z is adjusted to have a renormal-izable theory in the UV region, but in the IR flows to ∗ jfchagoya@fisica.ugto.mx † msabido@fisica.ugto.mx z = 1 and Lorentz invariance is recovered. Although thistheory has been considered as a real candidate for theUV region of GR, it is plagued by the existence of newdegrees of freedom which make it incompatible with GRat low energies [7].The cosmological implications have been studied inHořava gravity by solving the equations of motion for thefull theory (see [8, 9]) and the quantum regime in [10].In this paper we take a different approach, we considerjust one of the key ingredients of Hořava’s proposal, theanisotropic scaling invariance and apply it on a minisu-perspace having only two degrees of freedom correspond-ing to a metric function and a scalar field. This approachis similar to that of several works about minisuperspacenoncommutativity [11] or quantum cosmology, a well es-tablished line of research that deals with quantization inthe minisuperspace [12] and is likely to capture qualita-tive features of the full, non-symmetry reduced theory[13]. The resulting theory will be an effective theory,that recovers GR results when the anisotropy is elim-inated. For the minisuperspace model we choose theflat FRW metric, so we can easily contrast our resultswith standard cosmology and the deviations obtainedwill be produced only by the modification to GR dueto the invariance under anisotropic scaling. We will con-struct an action for this model where the minisuperspacevariables are compatible with the anisotropic transforma-tions and time reparametrization invariance is preserved.We argue that the quantization of this model leads to aSchrödinger type equation for z (cid:54) = 1 , as we have shownfor the Kantowski-Sachs model in [14]. This quantum dy-namical equation with a first order time derivative allowsus to construct a conserved probability current. In thecase z = 1 , the system becomes singular and we returnto the original WDW equation. We also obtain the clas-sical equations of motion and find numerical solutions fordifferent values of the anisotropy index z . The analysisis done by comparing with the solution obtained in GR.The paper is organized as follows, in section II the La-grangian invariant under anisotropic scaling for the FRWmodel is presented as well as the symmetries of the the-ory. Then we use canonical formalism to compute theHamiltonian of the model and to obtain the modified a r X i v : . [ g r- q c ] D ec dispersion relation and the Friedmann equations. In sec-tion III we present the exact solutions for the particularcases Λ = 0 and z = 1 , and we analyze the numericalsolutions for different values of the critical exponent z – also known as anisotropic index – and the cosmologi-cal constant. Section IV is devoted for conclusions andoutlook. II. ANISOTROPIC SCALING INVARIANT FRWCOSMOLOGY
The flat Friedman-Robertson-Walker (FRW) cosmo-logical model is described, in the ADM decomposition,by the line element ds = − N dt + e α ( t ) (cid:2) dr + r ( dθ + sin θdϕ ) (cid:3) , (1)where N ( t ) is the lapse function and e α ( t ) = a ( t ) is thescale factor. The Einstein-Hilbert action minimally cou-pled to a massless scalar field φ and a cosmological con-stant Λ for this flat FRW metric is S ( α, φ ) = (cid:90) dt (cid:34) − a ˙ a k N + a (cid:32) ˙ φ N + N Λ (cid:33)(cid:35) , (2)where the dot represents derivatives with respect to t .The first goal in this work is to look for generalizations ofthis action incorporating the invariance under anisotropicscale transformations, (cid:126)x → b(cid:126)x , t → b z t , where z charac-terizes the anisotropy of the system. Besides this invari-ance, we also demand the action to be invariant undertime reparametrizations, which means that if we see thetime coordinate as a function of a parameter τ , then theaction has to be invariant under any choice of τ . A finalrequirement that we impose on the action is that in thelimit z = 1 it has to reduce to the usual Lorentz invariantaction for the FRW model in GR. The only action thatwe could find satisfying the above conditions is S ( a, φ, t ) = (cid:90) dτ (cid:40) − a ( ˙ a ) z k N z ˙ t z − + a ( ˙ φ ) z N z ˙ t z − + (cid:18) a − z N Λ˙ t (cid:19) z z − ˙ t (cid:41) , (3)where now dots denote derivatives with respect to τ . Theappearance of ˙ t in the action is necessary to achieve ex-plicit invariance under time reparametrizations. It isstraightforward to see that when z = 1 the standardFRW action is recovered. The invariances that we arerequiring are satisfied if under time reparametrization τ = τ ( f ) we have N → ˙ φN and under anisotropic scaling N → b z − z z N, Λ → b z − z − z Λ , k → b − k . (4)These transformations for N might seem to restrictive,however we will see that this is not a problem when try-ing to solve the equations of motion resulting from (3). Indeed, even the gauge N = 1 that is usually chosenin order to simplify the equations in the z = 1 case iscompatible with (4).Now we proceed to apply the canonical formalism tothe anisotropic scaling invariant action (3) in order toobtain the respective Hamiltonian and the equations ofmotion. A. Canonical formalism for the anisotropic scalinginvariant FRW action
The dynamical variables that appear in action (3) arethe scale factor a , the scalar field φ and an explicit timeparametrization t . Their corresponding canonical mo-menta are given, as usual, by Π a = ∂ L ∂ ˙ a = − az ( ˙ a ) z − ˙ ak N z ( ˙ t ) z − ,Π φ = ∂ L ∂ ˙ φ = a z ( ˙ φ ) z − ˙ φN z ( ˙ t ) z − ,Π t = ∂ L ∂ ˙ t = − a ( ˙ a ) z (1 − z ) k N z ( ˙ t ) z + a ( ˙ φ ) z (1 − z )2 N z ( ˙ t ) z + 1 − z − z a z + z z − − z ( N Λ) z z − ˙ t z − z , (5)where the Lagrangian L is read off from the brackets inaction (3). We can try to construct a canonical Hamilto-nian H c , but now not only we have the presence of thenondeterminate function N but also we cannot write theattempted Hamiltonian without explicit reference to thevelocities of the canonical variables, this is a consequenceof the impossibility to invert the canonical momenta tohave, e.g. ˙ a = ˙ a ( Π a ) , so strictly speaking H c is an en-ergy functional rather than a Hamiltonian. As alreadyhappens in GR we must interpret H c as a constraint H c = ˙ aΠ a + ˙ φΠ φ + ˙ tΠ t − L = − az ( ˙ a ) z k N z ( ˙ t ) z − + a z ( ˙ φ ) z N z ( ˙ t ) z − + z − z a z + z z − − z ( N Λ) z z − ˙ t z − z +1 ≈ . (6)One can verify that by choosing ˙ t = 1 then H c canbe written as a true Hamiltonian and it takes the form N z z − ˜ H c ( a, φ, Π a , Π f ) , making evident the roles of N asa Lagrange multiplier and of H c as a constraint. Theweak equality ≈ means that strictly H c is equal to zeroonly on a constraint hypersurface in the phase space.Comparing the last result with the expression for Π t wecan obtain the formal relation H c = z − z ˙ tΠ t . (7)Unlike the weak constraint (6), this last relation is validin the entire phase space. This is a result with remarkableconsequences at the quantum level, since the canonicalquantization of this relation leads to a Schrödinger-typeequation, where a first order derivative in time appears[14], offering a possibility to solve the frozen time prob-lem of the Wheeler-DeWitt equation that appears whencanonical quantization is applied to GR. For a review, seefor example [15]. However, our focus in this work is onthe classical theory. We must note that the relation (7)between H c and Π t does not hold for the FRW model inGR since it is not well defined for z = 1 . This should notbe surprising as for z = 1 the action (3) does not containany factors of ˙ t , so a canonical momenta Π t cannot bedefined.Although we cannot write H c without any explicit ref-erence to the velocities of the dynamical variables, wecan at least remove the dependence on ˙ a and ˙ φ . It isstraightforward to find z − z ˙ tΠ t = − (cid:20) z − k N z Π za a ˙ t − z z z (cid:21) z − + 12 (cid:34) N z Π zφ a ˙ t − z z (cid:35) z − + z − z (cid:20) a − z N Λ˙ t (cid:21) z z − ˙ t. (8)By construction, this model is invariant under timereparametrizations, thus we can choose t = τ . Thischoice is convenient to cast the right hand side of (8)as a true Hamiltonian, which on-shell satisfies the weakconstraint H c ≈ . Let us now derive the Friedmannequations for this modified FRW cosmological model. B. Friedmann equations
In the framework of general relativity it is customaryto obtain the Friedmann equations through the Einsteinequations. Nevertheless, they can also be obtained fromthe Hamiltonian formalism as this is just another way todescribe the same physics. Specifically, they arise fromthe constraint H = 0 and from the Hamilton equationsfor the momenta of the scale factor and of the scalar field.Using this identification, we find that the analogues of theFriedmann equations for arbitrary values of z are az ( ˙ a ) z − a z ( ˙ φ ) z − z − z a z − zz z − Λ z z − =0 , (9) (6 z + A (6 z ) z z − )( ˙ a ) z + 6 z (2 z − a ( ˙ a ) z − ¨ a − Bz z z − a ( ˙ φ ) z − C Λ z z − a z − zz z − − =0 , a ˙ az ( ˙ φ ) z − + a z (2 z − φ ) z − ¨ φ =0 , where we have defined the constants A = 11 − z (cid:18) z − z z (cid:19) z − , B = 32(1 − z ) (cid:18) z (cid:19) z − ,C = z z (4 − z )(1 − z )(2 z − . We will refer to these generalized Friedmann equationssimply as the Friedman equations. For Λ (cid:54) = 0 , analyticalsolutions to (9) cannot be found, so we need to resortto numerical methods. Before showing numerical resultswe present the exact solution for Λ = 0 and we comparethe respective numerical result with it, this is done in thenext section. When z = 1 the Friedmann equations (9)reduce to the ones obtained in GR for a flat FRW metric,a cosmological constant and a massless scalar field: a ˙ a + a Λ = a φ (10) ˙ a + 2 a ¨ a + Λ a = − a ˙ φ a ˙ a ˙ φ + a ¨ φ = 0 . In the next section we explore the properties of the modeldescribed by (9).
III. NUMERICAL AND ANALYTICALSOLUTIONS
We can find analytical solutions for
Λ = 0 and for Λ (cid:54) = 0 with z = 1 . Although some properties of the solu-tions for different z can be studied in the first case, themain use of these exact solutions is that we can comparethem to the numerical solutions in order to check theconsistency of our numerical results.For the case Λ = 0 , we combine the first two equationsin (9) obtaining z ( ˙ a ) z + (2 z − a ( ˙ a ) z − ¨ a = 0 . (11)The general solution to this differential equation is a ( t ) = (cid:20) (2 z − (4 z + 1 − z )( c + c t ) (cid:21) (2 z − z − z − . (12)Once we have a ( t ) , the solution for φ is obtained directlyfrom the third equation in (9), i.e. the momentum con-servation for the scalar field, but that is not relevant atthis moment. The quotient of c and c is related to thetime t at which a = 0 , whereas c and z determine thevalue of a at t = 0 . As expected, this solution is wellbehaved in the limit z → and it recovers the usual so-lutions a ( t ) ∼ t / corresponding to the massless scalarfield that we are considering. Nevertheless, it is straight-forward to see that (12) does not reflect the desirablecharacteristics that we would have expected in order tomake the addition of anisotropic invariance relevant ata classical level, as would be the existence of an accel-erating scale factor in the absence of the cosmologicalconstant or the removal of the singularity a = 0 .In Fig(1), we plot the analytical solutions for some val-ues of z as well as the corresponding numerical solutions.The constants in (12) are chosen for each z in order tomake ˙ a | t =0 to coincide for the numerical and exact so-lutions. The thick lines correspond to the analytical so-lutions and the thin lines to the numerical solutions for FIG. 1. Solution to the Friedmann equations with
Λ = 0 , thesame type of line is used both for the numerical and exactsolutions that correspond to the same z , but the thicker linesare for the numerical results. the same z . We can see that numerical numerical andexact solutions are in good agreement. Also from thisplot, we can verify that for z > the exponent in a ( t ) isnegative and therefore there is no acceleration. This canbe verified explicitly from the analytical solution.For the general case, Λ (cid:54) = 0 and z (cid:54) = 1 we will rely onthe numerical results to extract conclusions on the model.We take the system of Hamilton equations derived from(6), which has the advantage of being a first order system,and solve it numerically. The initial conditions are chosensuch that H = 0 . Before describing the results for z (cid:54) = 1 ,we check the consistency of the numerical results for z =1 and Λ positive or negative. We do this by comparingthe numerical results to the exact solution of the equation a ˙ a + 2 a ¨ a + 2Λ a = 0 , (13)which is obtained by taking z = 1 in the Friedman equa-tions(9) and making some algebraic manipulations withthem. A solution (not the most general one) is given by a ( t ) = (cid:34) − C sin( √ √ Λ t + D ) √ √ Λ (cid:35) . (14)The constants C and D are related to the initial condi-tions for a ( t ) and its velocity. Figure (2) shows a compar-ison of (14) with the numerical solutions of the Hamil-ton’s equations for Λ = 0 . and Λ = − . . In both casesthere exist a t such that a ( t ) = 0 , thus these solutionsalways have an initial big bang singularity.Now we turn our attention to the solutions for z (cid:54) = 1 .As already mentioned, here all the results are numerical.Figure (3) shows the results for Λ < . We find that as z becomes larger than 1 the accelerated growth of a ( t ) israpidly suppressed, in fact the scale factor grows almostlinearly with time for large t . In contrast, when z < FIG. 2. Comparison of the numerical and analytical solutionsto the Friedmann equations for
Λ = − . and Λ = 0 . . Inboth cases z = 1 .FIG. 3. Numerical and exact solutions to the Friedmann equa-tions for Λ = − . and different values of z . the accelerated growth of the scale factor is reinforced,so that the universe expands faster than in the solutionfor z = 1 . There are no clear indications that the initialsingularity can be avoided in this scenario.For Λ > the results are shown in Fig. 4. The value of z has remarkable consequences on the behaviour of thescale factor. For example, for z = 5 we see that thereis not a t such that a ( t ) = 0 . This characteristic isalso present and more noticeable for larger values of z ,as shown in Fig.(5), where we can see in more detail theregion where a ( t ) reaches its minimum value, a min . Theprecise value of a min depends on z and Λ . For z < ,all the solutions describe qualitatively the same type ofuniverses, and we cannot say with confidence if all ofthem show a big bang singularity or not. Unlike the FIG. 4. Numerical solutions to the Friedmann equations for
Λ = 0 . and different values of z .FIG. 5. Numerical solutions to the Friedmann equations for Λ = 0 . . We see a non zero minimal value for the scale factor. case Λ < , here there is not a relevant difference when z changes from z > to z < , epochs of acceleratedexpansion do not exist for any choice of z . IV. CONCLUSIONS
In this work we have analysed the effects of includ-ing invariance under anisotropic scaling in the minisuper-space as an explicit symmetry of the action. We studya simple cosmological model consisting of the flat FRWmetric plus a massless scalar field and a cosmological con-stant. A first relevant result of our analysis is that theHamiltonian formulation of the model reveals that theinclusion of anisotropic invariance is promising at thequantum level, specifically with respect to the so called problem of frozen time , since the canonical formalism of the model leads to a dispersion relation that includes alinear momenta associated to the time coordinate. Aftercanonical quantization, this relation leads to a dynamicalquantum equation that includes a first derivative in time,providing a way out to the frozen time problem.In the main part of this work, we studied the classi-cal solutions of the model for various values of z > .The reason to study only these values is the following:for z = the last term in the action (3) becomes singu-lar, thus this value of z can be thought as separating theanisotropic index in the ranges z > , which includes theusual Lorentz invariant solutions when z = 1 , and z < .Although we can suppose that the anisotropic invariancewas present at very early stages of the universe and there-fore z could take any value, we want to be able to recoverthe Lorentz invariant ( z = 1 ) solutions at late times. As-suming that the transition from anisotropic to Lorentzinvariance is continuous, this requirement imposes thatwe have to stay in the z > branch.With respect to the solutions that we found to theanisotropic scaling invariant model, some remarks are inorder: • The limit z → is well defined for all of them. • Without a cosmological constant, the inclusion ofanisotropic invariance makes the volume of the uni-verse to grow almost linearly with time. When thislinear regime is reached the velocity ˙ a is larger forlarger z . • With a negative cosmological constant, the acceler-ated growth of a ( t ) is suppressed for z > , whereasfor z < it is reinforced. • With a positive cosmological constant we have themost remarkable modification introduced by theinvariance under anisotropic scaling, namely, wecan remove the Big Bang singularity, obtaining an a min (cid:54) = 0 . The minimum z for which this is attain-able depends on Λ and on the initial conditions.As an overall conclusion, the inclusion of anisotropic in-variance results in desirable characteristics both at thequantum and classical levels. It gives a possible solutionto the problem of time in quantum cosmology as well aresolution of the Big Bang singularity. However, we needto seek further for explicit exact solutions that allow usto establish firmly the existence of a non-vanishing min-imum value for the scale factor, and to find analyticalconditions to determine the relation between the valuesof z , a min and Λ . The existence of a mechanism to changedynamically the anisotropic index in such a way that themodel can flow from an anisotropic invariant epoch athigh energies – close to the Planck scale – to a Lorentzinvariant epoch at lower energies is a non-trivial openquestion that would be interesting to study, since sucha mechanism would allow us to have relevant deviationsfrom classical solutions at high energies while recoveringstandard general relativity results in the infrared regime. ACKNOWLEDGMENTS
This work is supported by CONACYT grants 167335,179208 and DAIP640/2015 and is part of the PROMEPresearch network “Gravitación y Física Matemática". [1] S. Weinberg, Rev. Mod. Phys. , 1 (1989).[2] C. P. Burgess, arXiv:1309.4133 [hep-th].[3] A. Padilla, arXiv:1502.05296 [hep-th].[4] M. H. Goroff and A. Sagnotti, Nucl. Phys. B , 709(1986).[5] T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis,Phys. Rept. , 1 (2012).[6] P. Horava, Phys. Rev. D , 084008 (2009).[7] T. P. Sotiriou, J. Phys. Conf. Ser. , 012034 (2011).[8] S. Mukohyama, Class. Quant. Grav. , 223101 (2010).[9] S. K. Chakrabarti, K. Dutta and A. A. Sen, Phys. Lett.B , 147 (2012).[10] O. Obregon and J. A. Preciado, Phys. Rev. D , 063502(2012) [11] H. Garcia-Compean, O. Obregon and C. Ramirez, Phys.Rev. Lett. , 161301 (2002); B. Vakili, N. Khosraviand H. R. Sepangi, Class. Quant. Grav. (2007) 931;O. Obregon, I. Quiros, Phys. Rev. D84 , 044005 (2011);W. Guzman, M. Sabido, J. Socorro, Phys. Lett.
B697 ,271-274 (2011); B. Vakili, P. Pedram, S. Jalalzadeh,Phys. Lett.