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Vacuum-driven Metamorphosis

We show that nonperturbative vacuum effects can produce a vacuum-driven transition from a matter-dominated universe to one in which the effective equation of state is that of radiation plus cosmological constant. The actual material content of the universe after the transition remains that of non-relativistic matter. This metamorphosis of the equation of state can be traced to nonperturbative vacuum effects that cause the scalar curvature to remain nearly constant at a well-defined value after the transition, and is responsible for the observed acceleration of the recent expansion of the universe.


I. INTRODUCTION
Recent observations of high-redshift Type Ia supernovae (SNe-Ia) [1,2] indicate that the expansion of the universe is accelerating at the present time. Attempts to explain this effect include a revival of the cosmological constant term in Einstein's equations [3,4] as well as the hypothesized existence of a classical scalar field with a potential term (quintessence [5]). In previous work [6,7], we show that a free quantized massive scalar field can account for the SNe-Ia observations in a new way. Nonperturbative vacuum contributions to the effective action of such a field [8] lead to a cosmological solution in which the scalar curvature has a constant value after a cosmic time t j that depends on the mass of the scalar field and its curvature coupling. This solution implies an accelerating universe at the present time and gives a good fit to SNe-Ia data for a particle mass parameter of about 10 −33 eV, without the necessity of introducing a cosmological constant term into Einstein's equations. Such a low mass parameter may arise from pseudo Nambu-Goldstone bosons [9]. A graviton of such a small non-zero mass would be expected to produce similar effects.
In Ref. [7], we also show that our cosmological model gives light-element abundances in agreement with standard big-bang nucleosynthesis and that it is consistent with present data on the small angular scale fluctuations of the cosmic microwave background radiation (CMBR), which tend to favor spatial flatness. Our model does not suffer from a fine-tuning problem because, for the allowed range of the mass parameter, the probability is high that the matter and vacuum energy densities are of the same order of magnitude at the present time.
The main point of this letter is to elucidate a remarkable feature of the equation of state of our model, namely, that the equation of state undergoes a metamorphosis from an equation of state dominated by pressureless matter (without a cosmological constant) to an effective equation of state that can be described in classical terms as that of mixed radiation and cosmological constant. As we show, this metamorphosis is initiated by the quantum vacuum terms which grow rapidly after time t j and combine with pressureless matter to give an effective stress tensor identical to that of radiation plus cosmological constant.
In the next section we summarize the main features of our model, and in Section III we analyze the equation of state. Our conclusions are given in Section IV.

II. SUMMARY OF OUR MODEL
We consider a free, massive quantized scalar field of inverse Compton wavelength m, and curvature coupling ξ. The effective action for gravity coupled to such a field is obtained by integrating out the vacuum fluctuations of the field [6,8]. This effective action is the simplest one that gives the standard trace anomaly in the masslessconformally-coupled limit, and contains the nonperturbative sum (in arbitrary dimensions) of all terms in the propagator having at least one factor of the scalar curvature, R. The trace of the Einstein equations, obtained by variation of this effective action with respect to the metric tensor takes the following form in a Friedmann-Robertson-Walker (FRW) spacetime (in units such that c = 1), with zero cosmological constant [6,7]: where T cl is the trace of the stress tensor of classical, perfect fluid matter, is a curvature invariant that vanishes in de Sitter space.
As noted earlier, m is the inverse Compton wavelength of the field. It is related to the actual mass of the field by m actual =hm. Equation (1) above is nonperturbative in R because it contains terms that involve an infinite sum of powers of R. However, for a sufficiently low mass, it is possible to treat m 2 actual /m 2 P l ≡hm 2 /(16πκ o ) (where m P l is the Planck mass) as a small parameter and expand perturbatively in this parameter, as we do in obtaining the solution described below.
In Refs. [6] and [7], we show that, for a sufficiently low mass, in an expanding FRW universe the quantum contributions to the Einstein equations become significant at a time t j , when the density of classical matter, ρ m , has decreased to a value given by where We find that the time t j occurs in the matter-dominated stage of the evolution. Furthermore for t > t j we find that the scalar curvature, R, remains constant to excellent approximation at the value m 2 . For t < t j , the quantum contributions to the Einstein equations are negligible and the scale factor is that of a matter-dominated FRW universe. Then, Eq. (2) implies that, in a spatially flat universe with line element ds 2 = −dt 2 + a(t) 2 (dx 2 + dy 2 + dz 2 ), one has where H(t j ) is the Hubble constant at t j . As shown in [7], the condition of the constancy of the scalar curvature after t j leads to a solution for the scale factor that can be joined, with continuous first and second derivatives (i.e., in a C 2 manner), to the matterdominated solution for t < t j . The scale factor is then given by with We find that the above solution also satisfies, up to terms of order m actual /m P l , the one remaining independent Einstein equation in a FRW universe, which can be taken to be the time- To verify that Eq. (5) is indeed a solution of the above equation for t > t j , we note that, when R is very close to the value m 2 , the dominant terms in the right hand side of Eq. (7) are those that have a factor of m 2 + ξR in the denominator. Keeping these terms and substituting for the various curvature quantities derived from Eq. (5), it is straightforward to check that Eq. (5) satisfies Eq. (7) up to terms of order m actual /m P l . The solution in Eq. (5) corresponds to a universe that is accelerating (i.e., has negative deceleration parameter) for t > √ 3m −1 (α + tanh −1 (2 −1/2 )) ≃ 1.50 t j . This solution gives a good fit to the SNe-Ia data, for the mass range 6.40 × 10 −33 eV < m h < 7.25 × 10 −33 eV, where h is the present value of the Hubble constant, measured as a dimensionless fraction of the value 100 km/(s Mpc). In Fig. 1, we give plots from [7] of the difference between apparent and absolute magnitudes of sources as functions of the redshift z. The two dashed curves represent the two extremal values of the range (8). The solid curve represents an intermediate value of m h , and the data points represent SNe-Ia data acquired from Ref. [2]. In [7], we also show that the ratio of matter density to critical density at the present time, Ω 0 , is a function of the single parameter m/h and turns out to have the range 0.58 > Ω 0 > 0.15 for the range of values of Eq. (8). For the same range of values, the age of the universe t 0 lies in the range 8.10 h −1 Gyr < t 0 < 12.2 h −1 Gyr.

III. EQUATION OF STATE
Although our model incorporates the effective action of a quantum field, it admits a simple, classical representation. Indeed, the scale factor (5) may be used to find the total effective energy density ρ and pressure p of vacuum plus matter by directly computing the Einstein tensor. We obtain, for t > t j , The effective equation of state for t > t j is therefore which is identical to the equation of state for a classical model consisting of radiation plus cosmological constant. In our model the equation of state of pressureless matter and the equation of state of quantum vacuum terms combine in a manner so as to effectively appear as a sum of radiation and cosmological constant equations of state. Our model differs, even at the classical level, from the usual mixed matter-cosmological constant model because (i) for t < t j the effective cosmological constant vanishes, and (ii) for t > t j vacuum contributions transmute the effective equation of state into that of radiation (rather than pressureless matter) plus cosmological constant; this surprising metamorphosis is a result of the near-constancy of the scalar curvature, which causes certain terms in T µν to take the form of an effective cosmological constant term in Einstein's equations. In a general spacetime, these terms do not have the form of a cosmological constant term. The equation of state for the quantum vacuum terms alone may be inferred from Eqs. (9) and (10), and from the fact that the density of pressureless matter is given by where Eqs. (2) and (5) have been used to arrive at the second equality. The quantum vacuum energy density ρ V and pressure p V then follow, for t > t j , as with p(t) given by Eq. (10). The above equations show that for t > t j the vacuum energy density is positive, while the vacuum pressure is negative. As stated earlier, the vacuum terms are negligible for t < t j .
After some straightforward algebra, we find the equation of state for the quantum vacuum terms: This vacuum equation of state joins continously to the equation of state ρ V = p V = 0 at t = t j and asymptotes to the pure cosmological constant equation of state ρ V = −p V as t → ∞. Equation (15) shows that the quantum vacuum stress tensor in our model is parametrized by the single parameter, m, and is different from that of a pure cosmological constant.

IV. CONCLUSIONS
In this letter, we have outlined a cosmological model that includes nonperturbative quantum vacuum effects of a renormalizable free field of very low mass in curved spacetime. The only adjustable parameter, determined by observation, is the mass scale m of the proposed particle. With m in the range of Eq. (8), the model is in agreement with the observed light-element abundances, CMBR fluctuation spectrum, age of the universe, and the magnitude-redshift relation of Type Ia supernovae.
In terms of the equation of state, our model admits a simple description that could also be a generic feature of theories other than the one considered here. At a time t j , m 2 + ξR first becomes small and terms in the quantum stress tensor become large, causing the scalar curvature after t j to take the nearly constant value m 2 = −m 2 /ξ (for ξ < 0). The constancy of R effectively forces the combined equation of state of quantum vacuum terms and pressureless matter to undergo a metamorphosis to that of radiation plus cosmological constant. This effect explains the observations regarding the acceleration of the recent expansion of the universe. Other mechanisms that cause the scalar curvature to remain constant would give rise to a similar metamorphic behavior of the equation of state at the phenomenological level. Nonperturbative terms in the vacuum stress tensor seem to provide the simplest example of such a mechanism.
Finally, we note that the qunatum vacuum terms would also alter density inhomogeneities that existed at time t j because these terms would remain insignificant in regions having average density at t j large with respect to ρ m (t j ) ≡ 2m 2 κ o . This may eventually provide a further observational test of our model.