Inflationary constraints on cosmological field theory

Abstract

We consider a non-singular scalar field cosmology proposed by Starkovich & Cooperstock, in which the matter- and radiation-dominated eras are preceded by a ‘prematter’ era characterized by negative pressure and rapid inflation. The form of the scalar potential during this period is of approximately power-law form, and is consistent with COBE and Tenerife data on the spectral index of density perturbations. However, the predicted density contrast is considerably larger than that observed. The theory also violates constraints on the energy at the end of inflation. Prospects for addressing these difficulties are briefly discussed.

cosmic microwave background, cosmology: theory, early Universe

1 Introduction

Starkovich & Cooperstock (1992; hereafter ‘SC’) presented a simple but surprisingly rich classical field theory of the Universe, inspired by an earlier singularity-free model of Israelit & Rosen (1989). Matter is represented by a minimally coupled scalar field φ. (The extension to a non-minimally coupled scalar has subsequently been made by Bayin, Cooperstock & Faraoni 1994.) The Universe passes through three successive phases (‘prematter,’ radiation and dust), each defined by an appropriate equation of state. The prematter phase is characterized by negative pressure and hence inflation. Expansion begins from an initial state with finite size, Planck density, and low temperature. It heats up adiabatically as it inflates, however — a consequence of the usual laws of thermodynamics, assuming that prematter dominates the entropy, as well as the energy density of the Universe in this early phase (Israelit & Rosen 1989, SC). This removes the need for the finely tuned reheating mechanisms of traditional theory. Inflation ends gracefully (but suddenly) when the Planck temperature is attained. A discontinuous change in equation of state (i.e., phase transition) then ushers in the radiation era and the rest of the standard model (with continued expansion now cooling the cosmological fluid). Besides the ideas of limiting density and temperature (Gliner 1970; Markov 1982; Rosen 1985), the theory makes only standard assumptions: general relativity with spatial homogeneity, isotropy and no cosmological term. The rest is accomplished by identifying and fitting together the equations of state appropriate to each epoch.

SC theory makes several testable predictions. First, it postulates a closed universe. This is required by the Friedmann—Lemai^tre—Robertson—Walker equations since there is no cosmological term Λ and the universe is initially in a state with positive density ρ₀ and zero expansion velocity (, where a is the cosmological scale factor). Secondly, it predicts a present value of the Hubble parameter between 33 and 44 km s⁻¹ Mpc⁻¹ (SC, table 8). (This rises to 47 km s⁻¹ Mpc⁻¹ in the non-minimal version of the theory by Bayin et al. 1994.) Both these predictions go somewhat against the grain of current observational work, which favours an open (or possibly flat) universe (Coles & Ellis 1997) and a value of H₀ in the range (69±10) km s⁻¹ Mpc⁻¹ (Ferrarese et al. 1999). The possibility that the Universe might be closed is, however, not ruled out observationally (White & Scott 1996), and some observers continue to report lower values of H₀ in the range (55±10) km s⁻¹ Mpc⁻¹ (Sandage & Tammann 1997). So it is probably premature to rule out the theory on these grounds alone.

In this paper we take up a suggestion by SC to test the theory based on its inflationary properties. We investigate specifically the spectral index of density perturbations, density contrast, and energy scale at the end of inflation.

2 Spectral index of density perturbations

In SC theory the scalar field potential does not have to be guessed at, but is fixed uniquely by the equation of state and the boundary conditions of the theory. Its form turns out to be given by (SC, equation 3.17)  

V_scφC^−αφβ^αφ1/b

(1)

where C is a constant. The parameters α, β and b are defined as follows (during the inflationary prematter era)  

(2)

where φ_o is the value of the scalar field at the moment when inflation begins. φ(t) decreases monotonically, vanishing when expansion halts, so that at all times of interest. It is thus convenient to put equation (1) in the form  

(3)

The constant γ characterizes the equation of state of the cosmological fluid, which reads p=(γ −1) ρ. During inflation, this parameter must be small () in order for the universe to evolve later to a state like the one we observe today (SC, table 1). It follows that −67/m_pl <α <−64/m_pl. Therefore the exponential term inside the square brackets in equation (3) becomes insignificant almost immediately after the onset of inflation. This approximation, moreover, becomes increasingly exact as inflation progresses and the difference between φ and φ_o grows (Appendix A). Equation (3) may therefore be approximated (especially at late times) by  

(4)

where . This is useful because we are particularly interested in the later stages of the inflationary era, when the observable universe leaves the horizon, setting the scale for the density perturbations we observe today. (This happens near the end of inflation in most models; Liddle & Lyth 1993, hereafter ‘LL’.)

The potential (4) belongs to the class of power-law inflationary (PLI) potentials, the general form of which can be written as (Lucchin & Matarrese 1985)  

(5)

where V_o and ℘ are constants, the latter of which is related to the expansion of the universe via a(t)∝t^℘ (hence ‘power-law’), and must be greater than unity for inflation. In our case the exponent in the potential is positive. The initial value φ_o of the scalar field, in other words, does not correspond to the minimum of its potential, as is the case in ‘standard’ inflationary scenarios (see LL for review). Equation (4) is an example of a chaotic power-law potential (Linde 1985; Ellis & Madsen 1991).

The PLI parameter ℘ may be simply related to the spectral index n of density perturbations (LL) by  

n

(6)

where current 1σ lower limits on combined data from the COBE and Tenerife experiments (Hancock et al. 1994; Bennett et al. 1996) imply that . The power-law behaviour of the scale factor depends only on the magnitude of ℘, and does not distinguish between the standard and chaotic versions of the potential. Two conditions are necessary (though not sufficient) for viable inflation (Linde 1985): (1) V(φ) must grow more slowly than ; and (2) the initial value φ_o of the scalar field must be greater than about a Planck mass (although this latter constraint can be relaxed in some cases). It is straightforward to show that equation (4) meets the first of these criteria, since γ ∼2×10⁻³, implying that the potential goes as . The second criterion is also satisfied since φ_o ∼30m_pl in the theory (SC, table 1).

By comparing equations (4) and (5) we find immediately that ℘_sc=2/3γ in SC theory. This, from equation (6), implies a spectral index of density perturbations n_sc=0.994 for all values of γ in the allowed range. SC theory therefore satisfies the COBE and Tenerife constraint on the spectral index of density perturbations.

In many versions of PLI, the strongest constraints on the theory come from demanding that the universe re-heat to sufficiently high temperatures for baryogenesis after inflation (Burd & Barrow 1988). This typically requires (Lucchin & Matarrese 1985), a constraint which would easily be met by SC theory since (with γ ∼2×10⁻³) at all times. This is of only incidental interest in our case, however, as reheating is not required in SC theory.

3 Density contrast

We turn next to the magnitude of the density perturbations, given in terms of the inflationary potential by (LL section 5.2)  

(7)

where the asterisk denotes a quantity evaluated at the epoch of horizon crossing; that is, at the time when the scale of interest left the horizon during inflation.

Equation (7) applies if three slow-roll conditions are met (LL). We now give these constraints and demonstrate that each one is satisfied in SC theory. The first is  

(8)

With the help of equations (A1) and (A3) in Appendix A, one finds that in SC theory  

(9)

which satisfies the required condition (8) since γ ≪1.

The second condition reads  

(10)

Differentiating equation (1), it is straightforward to show with the help of equations (2) that  

(11)

where ξ ≡{1 − exp[2α(φ₀ −φ)]}/{1 + exp[2α(φ₀ −φ)]}. Recalling that −67/m_pl <α <−64/m_pl, it is clear that at all times of interest, since inflation begins with φφ_o (i.e., ξ=0) and stops at or before the limit in which φ →0 (i.e., ξ →1). Equation (10) is therefore satisfied since γ ≪1.

The third and final slow-roll condition is given by  

(12)

Differentiating equation (1) twice, and using equations (2) as above, we find that  

(13)

Since γ ≪1, η ≈− (1−ξ²). As inflation begins, ξ=0 and eq. (12) is not satisfied. However, as argued in the previous section, the exponential terms in ξ rapidly become negligible with inflation, so that ξ →1 and |η |≪1 as required.

Having satisfied ourselves that SC inflation meets the slow-roll criteria, we return to the density contrast (7). Inserting equations (3) and (11), this reads  

(14)

where φ_* is the value of the scalar field at horizon crossing. We have argued above that the exponential terms inside the square brackets can be neglected throughout most of the inflationary period, and in particular near the end of inflation. Fortunately, it is precisely this regime which is of interest to us, since we would like to find the size of the perturbations that were frozen into the CMB when the currently observable universe left the horizon (LL). We therefore write  

(15)

Let us assume for the moment that the the observable universe crossed the horizon at precisely the end of inflation. In this case φ_*φ_pr where values of φ_pr are specified by SC (table 1), along with the values of γ,C and φ_o. It is thus simple to evaluate equation (15), using the definitions (2). The predicted matter density contrast turns out to be δ=0.53 for all cases, corresponding to CMB temperature fluctuations at the 0.18 level — some 16 000 times larger than the actual fluctuations detected by the COBE satellite (Bennett et al. 1996). This raises serious doubts about the viability of the SC scenario as it stands.

4 Prematter equation of state

One might attempt to evade this constraint by supposing that the observable universe left the horizon some time before the end of inflation in SC theory. However, since φ(t) is a monotonically decreasing function of time, this will push φ_* closer to φ_o and worsen the discrepany between theory and experiment. Indeed, in the limit φ_* →φ_o, one obtains  

(16)

which yields δ=0.65 for all cases, disagreeing with the COBE data by a factor of more than 20 000 times.

Another suggestion might be to let γ take values outside the range specified in SC theory. Let us attempt to estimate the value of γ that would be required to bring the theory into line with observation. With the help of the definitions (2), equation (15) shows that  

(17)

In SC theory, the values of φ_o (and presumably φ_*) depend somewhat on the value chosen for γ. Neglecting this dependence as a first approximation, we absorb these quantities into a ‘constant’ζ and take the ratio  

(18)

where and the subscript denotes the SC values. Inverting equation (18), we see that an improved value of γ (corresponding to a density contrast δ) would be given by  

(19)

Inserting the numerical values for φ_o and φ_*(=φ_pr) corresponding to a typical case γ_sc=1.95×10⁻³, and imposing δδ_sc/16,000, we obtain γ ≈1.26×10⁻², which is almost seven times the original SC value. With this value of γ, the PLI parameter ℘ drops from 330 to ∼53. (This, by the way, indicates that the problem with SC theory in its original form is that it inflates too quickly.) The drop in ℘ in turn affects the spectral index n of density perturbations via equation (6), with the result that n=0.96. Although this is lower than the original SC value, n_sc=0.994, it is still safely above the observational lower limit ().

So it might appear feasible to improve the agreement with observation by moving to higher values of γ. This remedy, however, has a drastic side effect. The small values of γ in SC are not chosen arbitrarily, but are required in order to lead to realistic values of ρ and H in the present day, subject to the boundary conditions of the theory. If γ is increased by the amount suggested above, then the theory will no longer match the present Universe. The only way to avoid this would be to alter some of the other boundary conditions in the theory, such as the requirement that it begin with precisely the Planck density, or that prematter go over to radiation at precisely the Planck temperature.

5 Energy at the end of inflation

The results of the previous section indicate that the difficulties with SC theory are, in a sense, inherent in its boundary conditions, and this can be confirmed by looking at the energy bound up in the scalar field potential at the end of inflation. According to LL (section 5.2.4) this cannot exceed  

V_eoi¹⁶

(20)

To check whether inflation in the SC scenario satisfies this requirement, we could evaluate the SC potential directly, using φφ_pr. The approximate expression (4) will be nearly exact at the end of inflation, as we have remarked.

However, we can do better in this case by noting from equation (2.12b) of SC that the potential energy of the scalar field is directly related to the density of matter in the Universe according to V(φ)=[(2−γ)/2] ρ. The boundary conditions of the SC model require that the energy density ρ at the end of the prematter period be given by . Therefore it is not necessary to evaluate the potential of the theory at all; we conclude immediately that the potential energy of the scalar field at the end of inflation must be (with γ ≪2)  

(21)

This violates the Liddle—Lyth bound by some 690 times.

One way to repair the problem may be to adopt a different set of boundary conditions at the phase transition between the prematter and radiation era. This is reminiscent of the situation encountered by Israelit (1991, 1994) and Israelit & Rosen (1991) in a similar singularity-free inflationary theory. These authors also found that initial fluctuations grew too large to be reconciled with observation unless the boundary conditions were fine-tuned. As an alternative solution, they introduced transition periods with special equations of state between the original inflationary and radiation eras. The density perturbations were then assumed to develop only as prematter was converted into radiation; the prematter era itself remained perfectly homogeneous and isotropic. It may be that a similar approach is necessary in SC theory.

Another possibility, which we have not studied here, is that the non-minimally coupled version of SC theory proposed by Bayin et al. (1994) might prove to be in better agreement with observational data. The inflationary potential in this theory is not given in closed form, however, so we leave this task for future work.

6 Conclusions

We have investigated the non-singular scalar field cosmology proposed by Starkovich & Cooperstock, and subsequently extended by Bayin et al. Expansion begins with an inflationary prematter stage. The scalar field potential is of chaotic power-law form, and is compatible with COBE and Tenerife data on the spectral index of density perturbations. The predicted density contrast, however, is several orders of magnitude larger than that observed. The model as it stands also violates constraints on the energy at the end of inflation. These problems appear to be inherent in the boundary conditions of the theory, and it may be necessary to modify these if agreement with experiment is to be attained.

Acknowledgments

The author is grateful to F. I. Cooperstock, V. Faraoni and S. P. Starkovich for numerous helpful discussions. Thanks go to the National Science and Engineering Research Council of Canada for financial support and to Gravity Probe B, Stanford University for hospitality.

Appendix

Appendix A: Klein—Gordon equation

To obtain an analytic expression for φ(t), we solve the Klein—Gordon equation in SC theory. For a minimally coupled scalar field this equation reads  

(A1)

where is the Hubble parameter. In SC theory, the scale factor a(φ) is related to the potential V(φ) by the following equation (SC, equation 3.18)  

(A2)

Putting these expressions into equation (A1), and using SC (equation 2.12a), we obtain  

(A3)

This is the most convenient formulation of the Klein—Gordon equation for our purposes. Differentiating the potential (1) and using equations (2), we find  

(A4)

where  

(A5)

and the subscript p denotes the prematter era. Neglecting the exponential terms inside the square brackets in equation (A4), as discussed in Section 2, we are left with  

(A6)

where, since γ ≪2, A_p ≈(γBC/2)exp(−2Bφ_o). It turns out that the same simplification can be made during the radiation- and matter-dominated eras, with A_p replaced by A_r ≡2BCβ⁴ and A_m ≡BCβ⁶ respectively, B given by equation (A5) as usual, and C, β and γ taking the values specified in SC for these two periods. We will not make further use of these expressions, as we are interested here in the inflationary prematter era.

It may be verified by substitution that a solution of equation (A6) is given by  

(A7)

where the boundary condition φ(0)≡φ₀ fixes the value of the integration constant, . This analytic expression for φ(t) may be employed to estimate the value of φ at the end of inflation, and thereby test the self-consistency of SC theory. Results agree to better than 0.4 per cent for all values of γ.