1 Introduction

We consider a bouncing Universe model which explains the flatness of the primordial scalar spectrum via complex scalar field that rolls down its negative quartic potential and dominates in the Universe. We show that in this model, there exists a rapid contraction regime of classical evolution. We calculate the power spectrum of tensor modes in this scenario. We find that it is blue and its amplitude is typically small, leading to mild constraints on the parameters of the model.

INR-TH-2012-72

Pseudo-conformal Universe: late-time

[0.3cm] contraction and generation of tensor modes

S. Mironov

Institute for Nuclear Research of the Russian Academy of Sciences,

60th October Anniversary Prospect, 7a, 117312 Moscow, Russia;

Institute of Theoretical and Experimental Physics

Bolshaya Cheremushkinskaya 25, 117218, Moscow, Russia

1 Introduction

Inflation is the best studied and very plausible scenario for the cosmological epoch preceding the hot Big Bang era. Nevertheless, it is of interest to consider alternative scenarios. One of them is Ekpyrosis [1, 2], understood in a broad sense as a period of slow contraction with equation of state . Its particular version, pseudo-conformal Universe [3, 4] makes use of a nearly conformal evolution to generate scalar perturbations with nearly flat power spectrum, thus building upon the idea that the flat spectrum may be due to conformal invariance [5, 6, 7]. The adiabatic perturbations generated by conformal mechanisms have interesting non-linear properties, such as non-Gaussianity and statistical anisotropy [8].

The concrete example of the pseudo-conformal Universe [3] is based on the Einstein gravity interacting minimally with a complex scalar field with negative quartic potential. The action is (signature )

The contracting epoch is assumed to begin with the asymptotically Minkowski space-time and . Since the potential is negative, this state is unstable, and under the assumption of the homogeneous background evolution, the field rolls down as follows,

(1)

At this stage perturbations of the phase are generated, which, to the linear order, automatically have flat power spectrum [6]. These entropy perturbations are assumed to get reprocessed into adiabatic ones by one or another mechanism, see, e.g., Refs. [9, 10, 11]. It has been pointed out [3] that tensor perturbations are generated as well, and that they have blue power spectrum. The latter is similar to the tensor spectra generated in other ekpyrotic models [12].

In this paper our focus is on the late-time contraction in this model. The classical behavior turns out to be rather interesting: the initial slow contraction with the effective equation of state is followed by the rapid contraction regime in which the equation of state asymptotes to . As in all ekpyrosis-like models, one assumes that the latter regime ends up in a bounce and defrosting [13] whose mechanism we do not specify.

We also consider the generation of tensor modes, both at slow and rapid contraction stages. We find that their power spectrum is always blue, and the amplitude is small enough, so that no strong constraints on the parameters of the model can be placed. Thus, the model is safe from this viewpoint.

This paper is organized as follows. In Section 2 we consider the classical contraction. In Section 3 we study the tensor perturbations. We conclude in Section 4.

2 Classical solution

The homogeneous contracting Universe is governed by the equation of motion for the field ,

(2)

and the Friedmann equation

(3)

At early times, , one neglects the Hubble friction term in the first equation and obtains the solution (1), where we take real. To this order, the energy density of the field vanishes. The Hubble parameter and then the scale factor can be obtained from the Raychaudhuri equation

One finds

(4)

This corresponds to the equation of state

From eq.(4) we get the scale factor

(5)

where we keep the overall normalization factor as a free parameter. So, the Universe contracts slowly at early times.

We are mainly interested in the solution at late times. For large we neglect the scalar potential and solve the simplified equation of motion and Friedmann equation:

(6)

Upon substituting into the first equation we get

The solution is

(7)

where and are to be determined by matching this solution to eq. (1). We can indeed neglect the scalar potential in eqs. (2), (3), provided that and . Using eq. (7) we see that these conditions are satisfied at late times, such that

(8)

Note that for small , there exists a time interval where the evolution is still classical, , while the inequality (8) is satisfied.

It follows from eqs. (6) and (7) that at late times, the Hubble parameter and scale factor are

(9)
(10)

where is yet undetermined parameter. They correspond to the equation of state

Let us match the two regimes and express the coefficients , and in terms of . For estimates, we require that , and their time derivatives are equal for the solutions (1), (5) and (7), (10) at some time . We write

(11)
(12)
(13)
(14)

As a cross check, the potential and Hubble friction terms in eq. (2) are of the same order at time . Likewise, the kinetic and potential terms in the Friedmann equation (3) are of the same order at that time, .

By solving eqs. (11) – (14), we obtain the time of the transition from slow to fast contraction,

(15)

Of course, if the bounce happens at , the fast contraction regime does not occur at all. Assuming that this is not the case, we find the parameters of the fast contraction stage

(16)
(17)
(18)

An example of numerical solution is shown in Fig. 1. One observes that the two regimes (1) and (7) indeed nicely match at intermediate time.

In what follows we use the solution in the second regime, written in terms of conformal time (in the first regime conformal time equals ):

(19)

Making use of eqs. (17), (18) and requiring that , we get

(20)
(21)

Figure 1: An example of numerical solution to eqs. (2), (3) (solid line). Short dashed line is . Long dashed line is .

To end up this Section, we express the parameters of our solutions in terms of physical quantities. We consider the simpliest possibility that the contracting stage terminates abruptly by the bounce and instantaneous defrosting. We therefore assume that both the scale factor and energy density (and hence ) coincide at the end of the contracting stage and at the beginning of the hot epoch. At the beginning of the hot epoch just after the bounce we have

(22)
(23)

where is the defrosting temperature, and refers to any moment of time at the radiation dominated epoch. On the other hand, just before the bounce the Hubble parameter and scale factor are given by

(24)

We thus obtain

(25)
(26)

Finally, the scale factor at the beginning of contraction is

(27)

Let us now turn to the calculation of the power spectrum of tensor perturbations.

3 Tensor perturbations

In this Section we work in terms of the canonically normalized field

(28)

where is the shorthand for the transverse traceless component of the metric perturbation. The field equation has the following form in conformal time:

(29)

In terms of the variable this equation reads:

(30)

In what follows we consider two regimes separately: early times, when the scale factor is varying slowly; and late times, when it evolves as . The former regime is relevant for low momentum modes which exit the horizon early, while the latter is important for high momentum modes.

We begin with the modes of low momenta. In the slow contraction regime, the scale factor behaves as follows:

The low momentum modes are those which exit the horizon, , well before time . Making use of eqs. (4) and (15), we obtain for these modes

(31)

Note that for these modes one has , where is the conformal time of horizon exit.

In the slow contraction regime, eq. (30) has the following explicit form:

(32)

We assume that the field is in its vacuum state at past infinity and hence has the standard asymptotics at large . After crossing out the horizon, but before the onset of rapid contraction, the solution is

(33)

Now we match the two asymptotics and find and in the superhorizon regime. To this end we solve eq. (32) perturbatively near horizon crossing. In the slow contraction regime we have , and the horizon crossing occurs when . Hence, there are two small parameters around the horizon crossing: , and the two linear independent solutions to eq. (32) can be found as series in these parameters:

(34)

Before the horizon crossing, when , these solutions match to and . Since , we get

After horizon crossing, when , this solution reads

(35)

which coincides with eq. (33) with , . So, the low momentum mode after horizon exit is

(36)

where we inserted the standard normalization factor and made use of the fact that . We cross check this calculation in Appendix.

The result (36) is valid also at late times, when the Universe contracts rapidly. Making use of eqs. (5) and (19), we calculate the integral in eq. (36) and find that in the rapid contraction regime, the low momentum modes behave as follows:

It follows from eqs. (15), (20) and the condition of smallness of momentum (31) that the main contribution comes from the first, constant term:

With the standatd definition of the power spectrum,

we get

(37)

So, the power spectrum is blue, in agreement with Ref. [3]. According to eq. (28), the power spectrum of metric perturbations is

where the factor accounts for two polarizations.

Let us now turn to high momentum modes, such that , cf. eq. (31). These modes exit the horizon at the rapid contraction stage, so the field equation is

Its solution that tends to at large negative reads

After horizon crossing, this mode has constant part and logarithmically growing one. The dependence on is only logarithmic:

(38)

Hence, the power spectrum is again blue,

(39)

This power spectrum refers to time before the bounce. To obtain the power spectrum after the bounce, we match the solution (38), valid before the bounce, to the general superhorizon solution after the bounce, i.e., at radiation domination, when . Using the instantaneous bounce approximation, cf. eqs. (24) – (25), we get

The superhorizon solution to eq. (29) at radiation domination is

By matching this expression and its first derivative to eq. (38) at , we get at late times

So, the primordial power spectrum at radiation domination is

(40)

i.e.,

Let us now discuss possible effects of the tensor modes in the present and recent Universe. To this end, we again recall our assumption of the instantaneous bounce and defrosting. We make use of eq.(27) and write the present value of the critical momentum, eq.(31), as follows

where the subscript refers to the present value, and we set . We see that this momentum is fairly high for not too small .

We first consider direct search for gravity waves. Barring the very small value of , the relevant wavenumbers are below the critical value, . Hence, these modes exit the horizon at the slow contraction stage, and we use eq. (37) for their power spectrum. We again recall eq. (26) and find for the energy density

(41)

where is the scale factor at horizon re-entry at radiation domination, . The result (41) is hopelessly small: as an example, for we find that even with and

which is far below the projected sensitivity of gravity wave detectors.

We now turn to the BBN constraint

(42)

We consider the case in which the rapid contraction regime does occur at the contracting stage (in the opposite case one obtains even weaker constraint on the parameters of the model). Since the power spectrum is blue, we make use of the result (40) and obtain

where is the highest momentum of modes that ever exit the horizon, , or . Thus, the constraint (42) translates into a weak bound on the defrosting temperature

Thus, the pseudo-conformal Universe model is viable in a wide range of its parameters.

4 Conclusion

To summarize, we have solved the classical equations of motion and found the evolution of the Universe in the pseudo-conformal model. There are two regimes, one after another. The first is ekpyrotic (slow contraction) and the second is fairly rapid collapse. We then calculated the tensor spectra generated in the two regimes. We have seen that in both regimes there is no considerable production of tensor modes, hence this model is acceptable from the viewpoint of generating the cosmological perturbations.

5 Acknowledgements

The author is grateful to Valery Rubakov for valuable discussions and interest in this work. This work has been supported in part by RFBR grant 12-02-00653 and by RFBR grant for young scientists 12-02-31778, the Federal Agency for Science and Innovations under state contract 16.740.11.0583, the grant of the President of the Russian Federation NS-5590.2012.2 and the Ministry of Education and Science contract 8412. The author acknowledges the support by the Dynasty Foundation.

Appendix

To cross check the calculation leading to the result (35), we explicitly solve eq. (32) in the limit . We write

Then eq. (32) takes the following form:

(43)

For , the correction is small, , so that eq. (43) is

The solution to this equation, which asymptotes as , is

Upon changing the order of integration we get

and after three integrations by parts we come to

This means that in the limit the solution is

This confirms our result (35).

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