Solutions to Midterm Exam problems
Problem I If matter density ρ = 4×10−27 kg/m3 , what is the curvature radius R0 of the space in Einstein’
static Universe ? How long it will take a photon to circumnavigate such a universe ?
Solution: Einstein static universe contains matter and Λ-term. With such combination it is possible
to arrange for zero expansion rate ȧ = 0 and zero acceleration ä = 0, thus to have a static solution.
2
ȧ
a
3
= 8πGρ + Λ −
3k
c2 R2 a2
=0
(1)
ä
= −4πGρ + Λ = 0
(2)
a
As we see, the static solution is possible if Λ > 0 and k > 0 (i.e the space is positively curved). Of
course mass density of matter is always positive.
3
In Einstein static Universe
c2
R2
q
= 4πGρ. Thus, R = c
1
4πGρ
= 1.34 × 1026 m = 4331 M pc.
Travelling light satisfies r = c tte dt/a(t), however in static universe a(t) = const = 1. Therefore, the
time needed to travel the distance 2πR is just 2πR/c = 2.80 × 1018 s = 88.7 Gyr.
R
Problem II In a single component flat universe with the equation of state given by parameter w.
Solution: The expansion law of such Universe is given by (here I use dimensional time units)
1
H02
2
ȧ
a
= a−3(1+w)
(3)
i.e
ȧ = H0 a−(1+3w)/2
(4)
1. The proper distance to the galaxy observed at the present time t0 with redshift z is
Z t0
dP (t0 , z) = c
t(z)
dt
=c
a(t)
Z 1
a(z)
Z 1
1
1+z
c
2
1
1−
1 + 3w H0
(1 + z)(1+3w)/2
=
da
c
=
ȧa
H0
daa(3w−1)/2
(5)
where I have switched the integration variable to a normalized so that a(t0 ) = 1 and, therefore,
a(z) = 1/(1 + z).
2. The luminosity distance in the flat Universe is dL (t0 , z) = dP (1 + z), thus
i
c h
2
dL (t0 , z) =
(1 + z) − (1 + z)(1−3w)/2
(6)
1 + 3w H0
3. The anglular-diameter distance is dA (t0 , z) = dP /(1 + z), thus
c
1
1
2
dA (t0 , z) =
−
1 + 3w H0 1 + z (1 + z)(3+3w)/2
(7)
4. The maximum of the angular diameter distance (i.e the redshift at which a fixed size object
will have the smallest angular extend) is obtained by differentiation
1
3
1
0
dA (z) ∝ −
+ (1 + w)
=0
(1 + z)2 2
(1 + z)(5+3w)/2
and gives 1 + zmax =
3+3w
2
2/(1+3w)
.
5. the value of dA (z) at zmax is
3+3w
1+3w
2
dA (zmax ) = c/H0
3 + 3w
In particular, for the matter-dominated flat universe dA (zmax ) ≈ 0.3c/H0
(8)