PHY 7097
August 29, 2012
Assignment #2
Reading:
Begin chapter 2 of Mukhanov.
Problems:
Due in class on Wednesday, 9/5/12.
(1) Use the Reconstruction Theorem to find a scalar potential V (φ) such that the stressenergy of just the scalar — with no other source, spatial curvature or cosmological
constant — will support a the power law expansion a(t) = (t/t0 )α , where α is an
arbitrary positive real number.
(2) Suppose the Universe is spatially flat (Ω = 1) and that the total energy density consists
of radiation, matter and vacuum energy, whose current fractions are,
Ωrad = 0.000085
,
Ωmat = 0.28
and
Ωvac = 1 − Ωrad − Ωmat .
Suppose also that the current value of the Hubble parameter is H0 = 73.8 km/s Mpc.
Numerically integrate the Friedman equation to find the following co-moving times (in
years):
(a) The age of the Universe t0 .
(b) The time teq at which the energy densities of matter and radiation are the same.
(c) The time at which the energy densities of matter and vacuum energy are the same.
(d) The time at which the deceleration parameter q(t) became equal to zero.
(3) On a space-time diagram in co-moving coordinates, sketch the particle horizons and
light-cones for the following expansion histories:
1
(a) a(t) = (t/t0 ) 2 .
2
(b) a(t) = (t/t0 ) 3 .
(c) a(t) = (t/t0 )1 .
(d) a(t) = eH0 (t−t0 ) .