Model for a timesymmetric Milnelike universe without big bang curvature singularity
Abstract
We present a physics model for a timesymmetric Milnelike universe.
The model is
based on the theory approach to the cosmological constant problem,
supplemented by an assumed vacuummatter energy exchange
possibly due to quantumdissipative effects.
Without finetuning of the initial vacuum energy density,
we obtain a harmless big bang singularity (with finite values of the
Ricci and Kretschmann curvature scalars)
and attractor behavior towards Minkowski spacetime.
The timesymmetric spacetime manifold of our model may provide the
proper setting for a possible CPTsymmetric universe.
pacs:
04.20.Cv, 98.80.Bp, 98.80.JkarXiv:1909.05816 KA–TP–18–2019 (v2)
I Introduction
Certain singular versions of the Friedmann equation have recently been shown to have nonsingular solutions, i.e., solutions without physical big bang singularities Ling2018 ; Klinkhamer2019 ; Klinkhamer2019revisited .
Here, we revisit the Milnelike universe of Ref. Ling2018 , where the spacetime near the putative big bang singularity resembles the spacetime of the socalled Milne universe Milne1932 ; BirrellDavies1982 ; Mukhanov2005 , which is just a slice of Minkowski spacetime in expanding coordinates and, hence, has no physical singularity. In this way, we are able to improve upon the metric used in Ref. BoyleFinnTurok2018 for a “CPTsymmetric universe,” as that metric does have a physical big bang singularity with, for example, a diverging Kretschmann curvature scalar.
The goal, now, is to construct an explicit physics model for a timesymmetric Milnelike universe. For the particular model, we will use theory (see Refs. KlinkhamerVolovik2008a ; KlinkhamerVolovik2008b ; KlinkhamerVolovik2016 and references therein), because it is, then, possible to describe the change of vacuum energy in a natural and consistent way.
Specifically, the model has two characteristics. First, there is, in the simplest version of the model, only an arbitrary positive cosmological constant operative at , which transforms the big bang singularity of an appropriate spacetime metric into a coordinate singularity Ling2018 . Second, there is the field which ultimately compensates the cosmological constant, so that Minkowski spacetime is approached at late (and early) times. The focus of our model is on rendering the big bang singularity at harmless and not on getting an entirely realistic description at later (and earlier) times.
Ii Model for a Milnelike universe
ii.1 Preliminary remarks and metric
The metric obtained in this article improves upon the one used in Ref. BoyleFinnTurok2018 , as it does not have a big bang curvature singularity Ling2018 : the spacetime near the apparent big bang singularity resembles the spacetime of the Milne universe Milne1932 ; BirrellDavies1982 ; Mukhanov2005 , which corresponds to a slice of Minkowski spacetime in expanding coordinates and, therefore, has no curvature singularity. An alternative way to control the big bang singularity has been presented in Refs. Klinkhamer2019 ; Klinkhamer2019revisited , but will not be discussed further here.
We now start with the construction of our model for a Milnelike universe. In the framework of theory KlinkhamerVolovik2008a , we use the extended model of Ref. KlinkhamerVolovik2016 with minor changes. The main inputs are as follows:

the spatiallyhyperbolic Robertson–Walker metric,

a conserved relativistic field,

a relativisticmatter component with constant equationofstate parameter ,

a positive cosmological constant (),

special vacuummatter energy exchange,

special boundary conditions at , which give a positive vacuum energy density, , and a vanishing matter energy density, .
The metric will be discussed in this subsection and further details will be given in Sec. II.2.
The metric Ansatz is given by the spatiallyhyperbolic () Robertson–Walker metric in terms of comoving spatial coordinates ,
(1) 
The resulting expressions for the Ricci curvature scalar and the Kretschmann curvature scalar are
(2a)  
(2b) 
where the prime stands for differentiation with respect to . Observe that these curvature scalars vanish for , which is precisely the scale factor of the Milne universe Milne1932 ; BirrellDavies1982 ; Mukhanov2005 .
ii.2 Theory
For the theory considered, we use the fourformfieldstrength realization of the field KlinkhamerVolovik2008a ; KlinkhamerVolovik2008b and neglect any possible modification of Einstein’s general relativity [which would, for example, have a gravitational coupling parameter ]. The classical action is given by:
(3a)  
(3b)  
(3c) 
where is the 4form field strength from a threeform gauge field (Ref. KlinkhamerVolovik2008a contains an extensive list of references) and is a generic even function of . The symbol in (3b) stands for the covariant derivative and a pair of square brackets around spacetime indices denotes complete antisymmetrization. The symbol in (3c) corresponds to the Levi–Civita symbol, which makes a pseudoscalar and a scalar. Hence, in the action (3a) is a scalar but not a fundamental scalar, . The gauge field , the metric , and the generic matter field are the fundamental fields of the theory. Here, and in the following, we use natural units with and .
The field equations from (3) are the Einstein equation with the gravitating vacuum energy density and the generalized Maxwell equation,
(4a)  
(4b)  
(4c) 
where is the matter energymomentum tensor from (3a).
We now turn to the metric Ansatz (1) and consider a homogenous field,
(5) 
The matter content is assumed to be given by a homogenous perfect fluid with the following energy density, pressure, and equationofstate parameter,
(6a)  
(6b)  
(6c) 
where is a nonnegative constant. For this setup, the field equations produce reduced field equations which correspond to ordinary differential equations (ODEs). These ODEs will be given in Sec. II.3 in terms of dimensionless variables.
At this moment, we need to mention one last important property of the theory considered. This concerns possible quantumdissipative effects that produce vacuummatter energy exchange KlinkhamerVolovik2016 . In practical terms, these effects are modelled by a source term on the righthand side of the generalized Maxwell equation (4b), with the corresponding opposite term in the matter energyconservation equation. In the cosmological context, we have
(7a)  
(7b) 
with the first equality in (7a) resulting from the definition (4c) and the Hubble parameter
(8) 
The explicit calculation KlinkhamerVolovik2016 of particle production by spacetime curvature ZeldovichStarobinsky1977 gives a particular result for the source term: with Hubble parameter and Ricci curvature scalar . Here, we assume a somewhat different functional dependence on the cosmic scale factor ,
(9) 
with the Ricci curvature scalar from (2a), a positive decay constant , and a length scale . In the following, we assume that is related to the Planck length scale,
(10) 
with and temporarily displayed (the numerical value of the reduced Planck energy is ).
ii.3 ODEs
We now introduce dimensionless variables () as in Sec. IV B of Ref. KlinkhamerVolovik2008b or in Sec. 6 of Ref. KlinkhamerVolovik2016 . Then, is the dimensionless variable corresponding to , the dimensionless variable corresponding to , and the dimensionless variable corresponding to , where is the dimensionless cosmic time coordinate. The ODEs for , , and are essentially the same as those in Ref. KlinkhamerVolovik2016 :
(11a)  
(11b)  
(11c)  
(11d)  
(11e)  
(11f) 
where, from now on, the overdot stands for differentiation with respect to and the prime for differentiation with respect to . As mentioned in Sec. II.2, the crucial new input of these model equations is the righthand side of (11c), where the numerator makes for a vanishing if vanishes (as happens for the putative big bang singularity at , see Sec. II.4) and where the term makes for a decreasing vacuummatter energy exchange as Minkowski spacetime is approached [ for , as will be shown in Sec. II.4]. A further characteristic of (11c) is its timereversal invariance under , , and .
In order to be specific, we choose the following Ansatz function:
(12a)  
(12b) 
whose properties have been discussed in the sentence below Eq. (6.3b) of Ref. KlinkhamerVolovik2016 . For the Ansatz (12a), the dimensionless gravitating vacuum energy density (11e) is given by
(13a)  
which vanishes for the following equilibrium value (real and positive):  
(13b) 
ii.4 Analytic results
We can now use the same procedure as in Ref. Klinkhamer2019revisited for the regularized Friedmann singularity. Specifically, we obtain, for a small interval , an approximate analytic solution of the ODEs (11) and, for outside this interval, the numerical solution with matching boundary conditions at . Analytic results are discussed in this subsection and numerical results in Sec. II.5.
The zerothorder analytic solution is given by
(15a)  
(15b)  
(15c) 
where the particular value in (15b) makes that the corresponding value of the gravitating vacuum energy density (13a) is solely given by the cosmological constant, . Observe that, for and any fixed positive value of , the scale factor function (15a) approaches the function of the Milne universe Milne1932 ; BirrellDavies1982 ; Mukhanov2005 , which solves the ODEs (11a), (11b), and (11d) for .
Two remarks on the zerothorder analytic solution (15) are in order. First, we get the following expressions for the Ricci curvature scalar and the Kretschmann curvature scalar from (2):
(16a)  
(16b) 
which are constant, as we really have a part of deSitter spacetime Ling2018 . Second, the functions (15) provide, for , the exact solution of the ODEs (11) over the whole cosmic time range () and, for this reason, we have called (15) the zerothorder analytic solution.
We can push the analytic calculation somewhat further. Inserting the perturbative Ansatz
(17a)  
(17b) 
into the ODEs (11a) and (11c) gives the following results for the first few coefficients:
(18a)  
(18b)  
(18c)  
(18d)  
(18e)  
(18f) 
where the coefficients , , and are somewhat bulky and are not given explicitly here.
With and in hand, we obtain from (13a) and from (11b),
(19a)  
(19b) 
Considering the terms in (19), we note that energy transferred from the vacuum to the matter component does two things thermodynamically: it increases the matter energy density and it lets the extra matter perform work. With , we can also obtain the series expansions of the dimensionless Ricci curvature scalar and the dimensionless Kretschmann curvature scalar from (2):
(20a)  
(20b) 
which show that and are nonsingular and have a local maximum at .
The ODEs (11) are relatively simple and we obtain the following asymptotic results for :
(21a)  
(21b)  
(21c) 
with the sign function
(22) 
From (21a) without further correction terms, we have an inflationtype expansion,
(23) 
similar to what was found in Ref. KlinkhamerVolovik2016 .
ii.5 Numerical results
Numerical results are shown in Figs. 2 and 2, where the procedure is described in the caption of the first figure. We remark that the curves of Fig. 2, in particular, are not perfectly smooth near , but the behavior can be improved if we add perturbative corrections (17) to the zerothorder results (15). The asymptotic behavior of the numerical results is displayed in Fig. 3 and agrees with the analytic behavior from (21).
ii.6 Generalizations and attractor behavior
For the boundary conditions and setup as given in Sec. II.3, we have obtained two results: the regular behavior at , locally corresponding to deSitter spacetime, and the attractor behavior for , approaching Minkowski spacetime. Both results hold for any positive value of the cosmological constant, . The Minkowskispacetime attractor behavior has been established numerically, but also follows from the analytic results (21b) and (21c), which do not depend on the actual value of the boundary condition (see Ref. KlinkhamerVolovik2016 for further discussion of this type of attractor behavior).
Similar results are obtained for in (12a) and (13a), as long as the boundary conditions (14) have a value , so that . In fact, the general requirement for obtaining the regular behavior at and the Minkowskispacetime attractor is simply that the boundary conditions (14) have
(24) 
which gives a violation of the strong energy condition at (cf. Ref. Ling2018 ). The requirement (24) may even be satisfied for the case of a negative value of the cosmological constant , provided there is an appropriate value of , according to the expression (13a) for the explicit Ansatz (12a).
Iii Discussion
In this article, we have presented a physics model for a universe which has, at , both a vanishing cosmic scale factor and finite values of the Ricci and Kretschmann curvature scalars and , according to the analytic results in Sec. II.4 and the relevant panels in Figs. 2 and 2. This regular behavior at differs from that of the standard relativisticmatter Friedmann solution, which has and a diverging value of the Kretschmann curvature scalar, (see Ref. Klinkhamer2019 for a possible regularization of this divergence with the introduction of a new length scale , which may or may not result from quantumgravity effects).
The physics model presented in this article relies on the spatiallyhyperbolic () Robertson–Walker metric (1), as suggested in Ref. Ling2018 , and the theory approach to the cosmological constant problem KlinkhamerVolovik2008a , here taken in the fourformfieldstrength realization (3). In order to allow for changing vacuum energy density , it is possible to consider modified gravity KlinkhamerVolovik2008b or to appeal to quantumdissipative effects KlinkhamerVolovik2016 . The latter option is chosen in the present article, with a particular form of vacuummatter energy exchange.
The model spacetime constructed in this article follows from appropriate boundary conditions at the “origin of the universe” [with cosmic time coordinate for the standard Robertson–Walker metric (1)]: the gravitating vacuum energy density can have any positive value, but the matter energy density must be strictly zero. Matter creation must, in fact, take place at a moment different from the big bang coordinate singularity and has, in this article, been modelled by the source term (9). This source term essentially contains the Zeldovich–Starobinsky term for particle production by spacetime curvature ZeldovichStarobinsky1977 and a hypothetical on/off factor . This latter factor also makes for timesymmetric reduced field equations and, hence, for a timesymmetric solution.
A related type of particle production has been discussed in Ref. BoyleFinnTurok2018 , with “particles” appearing to comoving observers in the postbigbang phase and “antiparticles” to comoving observers in the prebigbang phase. The idea of a CPTsymmetric universe and vacuum state is definitely interesting, but the metric used in Ref. BoyleFinnTurok2018 is singular and not derived from the Einstein equation (or an improved version thereof). The hope is that the timesymmetric spacetime manifold of our model, derived from the Einstein equation and the field equation, may provide the proper setting for a CPTsymmetric universe, if at all relevant.
Acknowledgements.
F.R.K. thanks G.E. Volovik for discussions on vacuum energy decay.References
 (1) E. Ling, “The big bang is a coordinate singularity for inflationary FLRW spacetimes,” arXiv:1810.06789.
 (2) F.R. Klinkhamer, “Regularized big bang singularity,” Phys. Rev. D 100, 023536 (2019), arXiv:1903.10450.
 (3) F.R. Klinkhamer, “Nonsingular bounce revisited,” arXiv:1907.06547.
 (4) E.A. Milne, “World structure and the expansion of the universe,” Nature (London) 130, 9 (1932).
 (5) N.D. Birrell and P.C.W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, England, 1982), Sec. 5.3.
 (6) V. Mukhanov, Physical Foundations of Cosmology (Cambridge University Press, Cambridge, England, 2005), Sec 1.3.5.
 (7) L. Boyle, K. Finn, and N. Turok, “CPTsymmetric Universe,” Phys. Rev. Lett. 121, 251301 (2018), arXiv:1803.08928.
 (8) F.R. Klinkhamer and G.E. Volovik, “Selftuning vacuum variable and cosmological constant,” Phys. Rev. D 77, 085015 (2008), arXiv:0711.3170.
 (9) F.R. Klinkhamer and G.E. Volovik, “Dynamic vacuum variable and equilibrium approach in cosmology,” Phys. Rev. D 78, 063528 (2008), arXiv:0806.2805.
 (10) F R. Klinkhamer and G.E. Volovik, “Dynamic cancellation of a cosmological constant and approach to the Minkowski vacuum,” Mod. Phys. Lett. A 31, 1650160 (2016), arXiv:1601.00601.
 (11) Ya.B. Zel’dovich and A.A. Starobinsky, “Rate of particle production in gravitational fields,” JETP Lett. 26, 252 (1977).