7.4. Generating a small cosmological constant from Inflationary particle production
A novel means of generating a small at the present epoch was suggested by Sahni & Habib (1998).
Massive scalar fields in curved spacetime satisfy the wave equation
(85) |
where R is the Ricci scalar and parametrizes the coupling to gravity. In a spatially flat FRW universe the field variables separate so that
for each wave mode. The comoving wavenumber k = 2a / where is the physical wavelength of scalar field quanta. Defining the conformal field _{k} = a_{k} and substituting R = 6 / a^{3} into Eq. (85) leads to
(86) |
where differentiation is carried out with respect to the conformal time = dt / a. Equation (86) closely resembles the one dimensional Schrödinger equation in quantum mechanics
(87) |
Comparing (87) and (86) we find that the role of the "potential barrier in space" V(x) is played by the time dependent term V() = -m^{2} a^{2} + (1 - 6) / a which may be thought of as a "potential barrier in time" [82, 178, 84]. (The form of the barrier is shown in Fig. 14 assuming that Inflation is succeeded by radiative and matter dominated eras.) In quantum mechanics the presence of a barrier leads to particles being reflected and transmitted so that _{in}(x) = exp(ikx) + R(k)exp(-ikx) in the incoming region, and _{out}(x) = T(k)exp(ikx) in the outgoing region. Similarly, the presence of the time-like barrier V() will lead to particles moving forwards in time as well as backwards, after being reflected off the barrier. The scalar field at late times will therefore not be in its vacuum state _{k}^{+} but will be described by a linear superposition of positive and negative frequency states
(88) |
The role of reflection and transmission coefficients R, T is now played by the Bogoliubov coefficients , which quantify particle production and vacuum polarization effects and are obtained by matching `in modes' during Inflation with `out modes' defined during the radiation or matter dominated eras.
Due to the existence of space-time curvature, positive and negative frequencies can be defined only in the limiting case of small wavelengths, lim_{k} _{k}^{±} (1 / [2 k]^{1/2} a)exp(-/+ik ), for which effects of curvature can be neglected. The value of , is obtained by matching modes corresponding to the `out' vacuum with those of the `in' vacuum just after Inflation. (The `in' and `out' vacua are defined during Inflation and radiation/matter domination respectively.)
Figure 14. The process of super-adiabatic amplification of zero-point fluctuations (particle production) is illustrated. The amplitude of modes having wavelengths smaller than the Hubble radius decreases conformally with the expansion of the universe, whereas that of larger-than Hubble radius modes freezes (if = 0) or grows with time ( < 0). Consequently, modes with 0 have their amplitude super-adiabatically amplified on re-entering the Hubble radius after inflation (from Sahni & Habib 1998) (the case = 0 also describes quantum mechanical production of gravity waves in a FRW model [82].) |
The net effect of particle creation and vacuum polarization is quantified by the vacuum expectation value of the energy-momentum tensor <T_{ik}>. For < 0,|| << 1 and m/H 1 the leading order contribution to <T_{ik}> is given by
(89) |
We immediately see that the first term is simply proportional to the Einstein tensor and the second has the covariant form usually associated with a cosmological constant (i.e. T_{ik} = g_{ik} ). Substituting for <T_{ik}> in the semiclassical Einstein equations
(90) |
we find
(91) |
where
(92) (93) |
The term proportional to H^{2} <^{2}> in (92) may be absorbed into the left hand side of (91) leading to
(94) |
where G / (1 + 8G || <^{2}>) is the new, time dependent gravitational constant. (Observational bounds on the rate of change of set the constraint || << 1.) As shown in [171] for < 0 the value of <^{2}> can be very large, so that 1 / (8 || <^{2}>) and
(95) |
We therefore find that the energy density of created particles defines an effective cosmological constant which can contribute significantly to the total density of the universe at late times leading to _{m} + _{} 1 [171].
However, it should be noted that this result was obtained in the Hatree-Fock (or. semiclassical gravity) approximation (90) which is not exact in considerations of a single quantum field, since metric and field fluctuations may significantly deviate from their rms values. So, further study of this problem using stochastic methods (similar to those used in stochastic inflation [182, 183, 196] and stochastic reheating after inflation Einstein is quoted as saying [115]) is desirable.