174 Inflationary cosmology and creation of matter in the universe
Hence the coherent oscillations of the homogeneous scalar field correspond to the
matter-dominated effective equation of state with vanishing pressure.
We will assume thatg> 10 −^5 [16], which impliesgMP> 102 mfor the
realistic value of the massm ∼ 10 −^6 MP. Thus, immediately after the end of
inflation, whenφ∼MP/3, the effective massg|φ|of the fieldχis much greater
thanm. It decreases when the fieldφmoves down, but initially this process
remains adiabatic,| ̇mχ|m^2 χ.
Particle production occurs at the time when the adiabaticity condition
becomes violated, i.e. when| ̇mχ|∼g|φ ̇|becomes greater thanm^2 χ =g^2 φ^2.
This happens only when the fieldφrolls close toφ=0. The velocity of the field
at that time was|φ ̇ 0 |≈mMP/ 10 ≈ 10 −^7 MP. The process becomes non-adiabatic
forg^2 φ^2 <g|φ ̇ 0 |,i.e.for−φ∗<φ<φ∗,whereφ∗∼
√
|φ ̇ 0 |/g[16]. Note that
forg 10 −^5 the interval−φ∗<φ<φ∗is very narrow:φ∗MP/10. As a
result, the process of particle production occurs nearly instantaneously, within the
time
t∗∼φ∗
|φ ̇ 0 |∼(g|φ ̇ 0 |)−^1 /^2. (4.29)This time interval is much smaller than the age of the universe, so all effects
related to the expansion of the universe can be neglected during the process of
particle production. The uncertainty principle implies in this case that the created
particles will have typical momentak∼(t∗)−^1 ∼(g|φ ̇ 0 |)^1 /^2. The occupation
numbernkofχparticles with momentumkis equal to zero all the time when it
moves towardφ=0. When it reachesφ=0 (or, more exactly, after it moves
through the small region−φ∗<φ<φ∗) the occupation number suddenly (within
the timet∗) acquires the value [16]
nk=exp(
−
πk^2
g|φ ̇ 0 |)
, (4.30)
and this value does not change until the fieldφrolls to the pointφ=0again.
To derive this equation one should first represent quantum fluctuations of the
scalar fieldχˆminimally interacting with gravity in the following way:
χ(ˆ t,x)=1
( 2 π)^3 /^2∫
d^3 k(aˆkχk(t)e−ikx+ˆak+χk∗(t)eikx), (4.31)whereaˆk andaˆk+are annihilation and creation operators. In general, one
should write equations for these fluctuations taking into account expansion of
the universe. However, in the beginning we will neglect expansion. Then the
functionsχkobey the following equation:
χ ̈k+(k^2 +g^2 φ^2 (t))χk= 0. (4.32)Equation (4.32) describes an oscillator with a variable frequencyω^2 k =k^2 +
g^2 φ^2 (t).Ifφdoes not change in time, then one has the usual solutionχk =