(5) In the theory of cosmological inflation a quantum scalar field (t) provides
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Question
(5) In the theory of cosmological inflation a quantum scalar field (t) provides an energy source that allows the universe to expand at an exponential rate. The behaviour of the function is determined by the equation: l d2··.d@ c2 dt2 where c is the speed of light, H is the Hubble constant, and ° is the energy density of the quantum field. Answer the following questions in terms of the constants c, H. and . (All are positive in an expanding universe.) (a) Find the critical value of H = Herit above which the scalar field not oscillate but simply decay to zero (b) If H > Herit what is the solution for (t)? (c) If 0Explanation / Answer
the given equation
(1/c^2)d^(phi)/dt^2 + Hd(phi)/dt + epsilon*phi = 0
can be compared to the damped unforced harmonic osscilator
mx" + cx' + kx = 0
a. for the critical H
c^2 - 4mk = 0
i.e.
H^2 - 4(1/c^2)epsilon = 0
H = (2/c)sqroot(epsilon)
b. for H > H crit
this is an overdamped case
the solution is
phi = e^lambda*t
where
lambda = c^2(-H +- sqroot(H^2 - 4*epsilon/c^2))/2
c. for 0 < H < Hcrit
Angular frequency of osscilaton of scalar field is w
where w = sqroot(epsilon*c^2 - H^2*c^4/4)
if H = 0
then
w = c*sqroot(epsilon)
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