Cosmology of the very early universe
E. Galindo Dellavalle, G. Germán^{*}^{*}* Email: g.german1@physics.ox.ac.uk, A. de la Macorra^{†}^{†}†Email: macorra@fisica.unam.mx
[0.3cm] The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, UK
[0.3cm] Abacus College, Threeways House, George Street, Oxford, OX1 2BJ, UK
[0.3cm] Instituto de Física, Universidad Nacional Autónoma de México,
Apdo. Postal 20364, 01000 México D.F. México
July 13, 2019
Abstract
We study cosmological solutions for the very early universe beginning at the Planck scale for a universe containing radiation, curvature and, as a simplification of a possible scalar field potential, a cosmological constant term. The solutions are the natural counterpart of the well known results for a postinflationary universe of nonrelativistic matter, curvature and a cosmological constant. Contrary to the common belief that inflation arises independently of the initial curvature we show that in the positive curvature case the universe collapses again into a Big Crunch without allowing the cosmological term to dominate and to produce inflation. There is a critical value for the cosmological constant which divide the regions where inflation is allowed from those where inflation cannot occur. One can also have loitering solutions where the scale factor remains almost constant growing to produce inflation (or decreasing to a Big Crunch) after a time which depends on the amount of energy above (or below) the critical energy. At the critical energy the solution approaches asymptotically a particular value for the scale factor (Einstein’s static preinflationary universe). The cases where the cosmological term vanishes or becomes negative are also studied providing a complete discussion of Friedmann models.
1 Introduction
Studies of the evolution of the universe typically deal with the radiation and matter dominated eras in the old cosmology and with inflation and quintessence in the modern standard cosmology scenario. However, the preinflationary epoch is clearly also of great interest since it may contain the link between more fundamental theories (strings, branes) and low energy physics. With this in mind we have made a simplified attempt to understand the evolution of the universe in a preinflationary era. We study the problem of the evolution of the very early universe in the presence of radiation, curvature and a cosmological constant. The cosmological term is a very natural simplification of the more realistic and difficult problem of dealing with the evolution of a scalar field. The problem we discuss below is based on an exact solution to Friedmann equations. This solution is closely related to the one originally obtained by Harrison [1]. However to the best of our knowledge no further study of such solution has been published perhaps because interest has centred mostly in inflationary and postinflationary solutions. This solution is the natural counterpart of the classical postinflationary problem which has been discussed since long ago [2] and which deals with a universe containing nonrelativistic matter, curvature and a cosmological term.
2 A Solution to Friedmann equations
We begin by writing the equations of Friedmann and acceleration in a convenient way
(1) 
(2) 
where
(3) 
are the densities of radiation, cosmological constant and curvature, respectively. The obvious time dependence has been dropped for clarity of notation. As usual denotes Newton’s constant and is the velocity of light. Adding Eq.(1) and Eq.(2) and multiplying by we get
(4) 
We notice that the of Eq.(4) can be written as
(5) 
Introducing the new variable
(6) 
where
(7) 
We see that Eq.(4) can be rewritten in the very simple form
(8) 
Thus the solution for is given by
(9) 
The arbitrary constants and are, as usual, fixed by the initial conditions on and on by using Friedmann equation. Requiring that , where the asterisk signals some initial condition, gives
(10) 
From the Friedmann equation Eq.(1) evaluated at we get the second constant
(11) 
Thus the solution is given by
(12) 
where we have introduced the new variable defined in terms of by
(13) 
We can simplify Eq.(12) by introducing the (constant) density parameters
(14) 
where is the critical density at and the energy density of the substance at . We also use the Friedmann equation written in terms of density parameters evaluated at
(15) 
We then have the final expression for our solution which is valid for any and
(16) 
Closely related solutions were originally obtained by Harrison [1] case by case. Here we concentrate in the cosmological consequences of such a solution. The Hubble parameter is
(17) 
The velocity is given by
(18) 
and the acceleration is then
(19) 
The case when follows from an expansion of Eq.(16) on and trowing away terms proportional to (contained in , see Eq.(7)). We can also follow the previous procedure. In any case the solution is
(20) 
where
(21) 
Finally the solution for a negative cosmological constant also follows from Eq.(16)
(22) 
From now on we particularize to the case where the initial values correspond to the Planck era
(23) 
where and are Planck time, Planck lenght and Planck density, respectively. We further work in units where so that
(24) 
Thus the densities are given by
(25) 
We introduce the quantity which by analogy with the usual is here defined by
(26) 
where . The notation reminds us that refers to an epoch ”before” inflation while the usual refers to the evolution of the scale factor ”after” inflation.
3 Friedmann models
A typical value for the inflationary scale is thus, to illustrate the behaviour of and all the other quantities of interest we take .
Positive cosmological term. In we show the behaviour of as given by Eq.(16) for the three possible values of the curvature.
The case (closed universe) presents a Big Crunch (BC) at a time very close to the Planck time not allowing the possibility of an inflationary epoch. For the open and flat universes , respectively inflation is unavoidable due to the dominance of the term at large . The case is particularly interesting. We can calculate, as a function of , the time at which the BC occurs. This is given by
(27) 
From Eq.(27) we see that the BC disappears when
(28) 
or
(29) 
where and the subindex refers to a critical value. We can also solve in terms of with the result
(30) 
With this value of we find that the solution Eq.(16) reduces to
(31) 
This solution is illustrated in , where we see that it approches asymptotically the value . We can recognize as an Einstein solution for a static universe in the preinflationary era. The size of this mini universe is of the order of the Planck lenght.
As discussed long ago for a postinflationary universe, a slightly bigger value for the energy density than the critical one gives rise to a universe which, although closed, expands with an ever increasing velocity due to the dominance of the cosmological term. On the other hand an energy scale less than leads irremediably to a BC. This behaviour is illustrated in . We see that we can keep the solution waiting close to as long as we wish (loitering solution) by tuning the energy density to values close to the critical value. Loitering solutions in standard cosmology for a closed FRW model with matter and a cosmological constant were originally studied by Lemaître [3]. Some other more general studies can be found in [4]. The behaviour of the Hubble parameter is peculiar in loitering solutions. This has been noted by Sahni in the context of loitering braneworld models [5]. There, he notices that loitering solutions are characterized by the fact that the Hubble parameter dips in value during loitering. We illustrate this behaviour in (long dashed curve). The acceleration in a loitering solution always becomes positive for large when the cosmological constant term becomes dominating. The transition from a decelerating to an accelerating universe is signaled by the point at which the acceleration vanishes. For our solution Eq.(16) (with ) this is given by
(32) 
The behaviour of as a function of the energy is shown in
We have also investigated the behaviour of the energy density parameters for the loitering solution. This is illustrated in where we see that it is the curvature term the one which dominates the energy of the universe during loitering, while the cosmological constant and radiation terms remain almost indistinguishable one from the other. This peculiar behaviour during loitering is also consistently shown in where we plot the effective equation of state parameter which takes a value around during loitering. The is given by
(33) 
Vanishing cosmological term. In this case the solution is given by Eq.(20) for the three values of . This solution is illustrated in . Note that this figure is almost identical to . The reason is that for small the cosmological term present in Eq.(16) is irrelevant. Thus solutions Eq.(16) and Eq.(20) become different for larger values of . This is shown in for the cases and only since the case is essentially the same in both solutions whenever .
Negative cosmological term. In this case the solution is given by Eq.(22) for the three values of . This solution is illustrated in .
4 Conclusions
We have studied a model of the very early universe containing radiation, curvature and a cosmological constant. We find that in the cases , and inflation occurs unavoidably while in the case there is no inflation unless the energy density is bigger than a certain critical value. This critical energy signals the limit between contracting solutions which end in a Big Crunch from those which expand forever. At the critical value, the solution approaches asymptotically an Einstein solution for a static universe in the preinflationary regime. The mini universe defined by this solution is of the order of the Planck lenght. Particular emphasis is placed in the closed universe case because there loitering solutions can occur. These solutions have somewhat peculiar properties: the Hubble parameter dips in value during loitering and the energy density parameter associated with curvature dominates over radiation and the cosmological constant. We also solve and briefly discuss the cases with a vanishing and negative cosmological constant thus providing a complete study of the Friedmann models. All of these results have been also obtained in the well studied problem of a universe of matter, curvature and a cosmological constant. They are due to the competition between curvature and the matter terms. Thus, technically, replacing matter by radiation which decays even faster should not change any of these features. However it is most gratifying that in this case one can find closed form analytical solutions to the problem.
5 Acknowledgements
This work was supported by the project PAPIIT IN1149033 and CONACYT: 42096, 45718. G.G. is grateful to Prof. G.G. Ross and Prof. S. Sarkar for the hospitality extended to him at the RPCTP during a sabbatical leave from Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México. Support from DGAPA, UNAM and the RPCTP, Oxford is gratefully acknowledged. This work is part of the Instituto Avanzado de Cosmología (IAC) collaboration.
References
 [1] E.R. Harrison, Mon. Not. R. Astr. Soc. (1967)137, 6979. See also Stephani H., Kramer D., Maccallum M., Hoenselaers C., Herlt E., , Chapter 14, Cambridge U. Press, 2nd Ed. 2003.

[2]
Some general references are:
S. Weinberg, , Oxford University Press, 2008.
S. Dodelson, , Academic Press, 2003.
J.A. Peacock, , Cambridge U. Press, 1999.
P.J.E. Peebles, , Princeton U. Press, 1993.
S. Weinberg, , John Wiley and Sons, 1972.  [3] A.G. Lemaître, Mon. Not. R. Astr. Soc. (1937)91, 438

[4]
V. Sahni and Yu. Shtanov, astroph/0410221.
V. Sahni, H. Feldman and A. Stebbins, Astrophys. J. 385(1992)1.
S. Alenxander, R. Brandenberger, D. Easson, Phys. Rev. D62, 103509(2000).
R. Branderberger, D. Easson, D. Kimberley, Nucl. Phys. B623(2002)421.  [5] V. Sahni, Proceedings of the 14th Workshop on General Relativity and Gravitation (JGRG14), Kyoto University, Japan. W. Hikida, M. Sasaki, T. Tanaka and T. Nakamura Eds., p 95115. astroph/0502032.