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7.4. Generating a small cosmological constant from Inflationary particle production

A novel means of generating a small Lambda at the present epoch was suggested by Sahni & Habib (1998).

Massive scalar fields in curved spacetime satisfy the wave equation

Equation 85 (85)

where R is the Ricci scalar and xi 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 = 2pia / lambda where lambda is the physical wavelength of scalar field quanta. Defining the conformal field chik = aphik and substituting R = 6addot / a3 into Eq. (85) leads to

Equation 86 (86)

where differentiation is carried out with respect to the conformal time eta = integdt / a. Equation (86) closely resembles the one dimensional Schrödinger equation in quantum mechanics

Equation 87 (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(eta) = -m2 a2 + (1 - 6xi) addot / 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 Psiin(x) = exp(ikx) + R(k)exp(-ikx) in the incoming region, and Psiout(x) = T(k)exp(ikx) in the outgoing region. Similarly, the presence of the time-like barrier V(eta) 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 phik+ but will be described by a linear superposition of positive and negative frequency states

Equation 88 (88)

The role of reflection and transmission coefficients R, T is now played by the Bogoliubov coefficients alpha, beta 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, limkrightarrowinfty phik± appeq (1 / [2 k]1/2 a)exp(-/+ik eta), for which effects of curvature can be neglected. The value of alpha, beta 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

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 xi = 0) or grows with time (xi < 0). Consequently, modes with xi leq 0 have their amplitude super-adiabatically amplified on re-entering the Hubble radius after inflation (from Sahni & Habib 1998) (the case xi = 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 <Tik>. For xi < 0,|xi| << 1 and m/H ltapprox 1 the leading order contribution to <Tik> is given by

Equation 89 (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. Tik = gik Lambda). Substituting for <Tik> in the semiclassical Einstein equations

Equation 90 (90)

we find

Equation 91 (91)


Equation 92 (92)


The term proportional to H2 <Phi2> in (92) may be absorbed into the left hand side of (91) leading to

Equation 94 (94)

where bar G appeq G / (1 + 8piG |xi| <Phi2>) is the new, time dependent gravitational constant. (Observational bounds on the rate of change of bar G set the constraint |xi| << 1.) As shown in [171] for xi < 0 the value of <Phi2> can be very large, so that bar G appeq 1 / (8pi |xi| <Phi2>) and

Equation 95 (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 Omegam + OmegaLambda appeq 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.

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