Astrophysics II
Prof. M. Carollo
Spring 2015
Solution 7
Problem 1: Horizon scale at the last scattering surface
a) The comoving particle horizon at zeq is given by
Z zeq
c
R0 ωH (zeq ) =
dz
H(z)
∞
Z Req
cR0
=
dR
H(R)R2
0
Z Req
1
c
dR
r
=
−4
2
−3 R0
H0 0
R
R
R
Ω
+
Ω
r
m
R0
R0
R0
Z Req
1
c
dR
p
=
,
H0 0
Ωr + Ωm R/R0 R0
(1)
(2)
(3)
(4)
where we made the variable substitution R(z) = R0 /(1+z) and we neglected (1−Ωtot )(1+
z)2 and ΩΛ within H(z) as for zeq > zrec ' 1100 they are completely neglectable relative
to Ωm (1 + z)3 and Ωr (1 + z)4 . Since ρm (t) ∝ R−3 (t) and ρr (t) ∝ R−4 (t), the condition of
matter-radiation equality, i.e. ρm (teq ) = ρr (teq ), leads to
ρm0
ρm (teq ) R0
R0
Ωm
=
=
=
= 1 + zeq .
Ωr
ρr0
ρr (teq ) Req
Req
(5)
Thus making another variable substitution y = RΩm /(R0 Ωr ) we find
√ Z
c Ωr 1
1
√
R0 ωH (zeq ) =
dy ,
(6)
H0 Ωm 0
1+y
√
since
y
=
R
Ω
/(R
Ω
)
=
1.
With
F
(y)
=
2
1 + y being the privitive of f (y) =
eq
eq
m
0
r
√
1/ 1 + y this becomes
√
c Ωr √
R0 ωH (zeq ) =
2( 2 − 1) .
(7)
H0 Ωm
Using Eq. (5) we can experessed this as
√
2( 2 − 1) c
R0 ωH (zeq ) ' p
,
Ωm zeq H0
(8)
where we set 1 + z ' z as zeq ' Ωm /Ωr ' 3500 (for h = 0.7) is very large. Moreover, for
Ωr ' 4.2 · 10−5 h−2 Eq. (7) immediately leads to the numerical expression
R0 ωH (zeq ) =
16 Mpc
.
Ωm h2
(9)
b) The comoving particle horizon for zrec can be computed by dividing up the integral as
follows:
Z Rrec
cR0
dR
(10)
R0 ωH (zrec ) =
H(R)R2
0
Z Rrec
Z Req
cR0
cR0
dR +
dR
(11)
=
2
2
H(R)R
Req H(R)R
0
√
Z Rrec
cR0
2( 2 − 1) c
+
= p
dR ,
(12)
2
Ωm zeq H0
Req H(R)R
(13)
where we inserted Eq. (8) in the last step. To compute the second term in Eq. (12) we
make the approximation that the universe is essentially matter dominated for τ > τeq .
For a matter dominated universe, the particle horizon at redshift z is given by
Z R(z)
cR0
dR
(14)
R0 ΩH (z) =
H(R)R2
0
Z R(z)
1
q
= cR0
dR
(15)
3
0
Ωm RR0 R2
cR0
R(z)H(z)
1+z
= 2c
H(z)
c
1+z
p
=2
H0 Ωm (1 + z)3
c
1
√
'2
H0 Ωm z
=2
(16)
(17)
(18)
(19)
for z 1. Thus, we obtain for the second term in Eq. (12)
"
#
Z Rrec
cR0
c
1
1
√
dR = 2
−p
2
H0
Ωm zrec
Ωm zeq
Req H(R)R
and the total horizion at zrec becomes
"
R0 ωH (zrec ) = 2
√
c 1
1
2−2
√
− √
√
H0 Ωm
zrec
zeq
(20)
#
.
(21)
For zrec ' 1100 and zeq ' 3500 the second term is about a factor of 3 smaller than the
first term. If we neglect the second term we obtain for a given zrec the approximate
dependence of the horizion on the cosmology as
1
R0 ωH (zrec ) ∝ √
.
(22)
Ωm h
Note that we have neglected the dependence of zrec on the cosmology.
c) The angle subtended by the horizon at zrec is
θ=
R0 ωH (zrec )
,
D(z)
(23)
where the particle horizon can be numerically computed using Eq. (10) and D(z) is the
effective distance given by

R0 A sin(ωH (z)/A) if k > 0,





R0 ωH (z) if k = 0,
(24)
D(z) =





R0 A sinh(ωH (z)/A) if k < 0
with
c
R0 ωH (z) =
H0
Z
0
z
1
p
dz .
ΩΛ + Ωm (1 + z)3 + (1 − Ωtot )(1 + z)2
(25)
To be able to compute numerical values for D(z) we need an expression for A being the
present day curvature radius of the universe. From the Friedmann equation we obtain
for Ωtot 6= 1.
c
1
p
.
(26)
AR0 =
H0 |1 − Ωtot |
Now, we have everything at hand to compute the angle θ for different cosmologies (for a
fixed redshift zrec ). In the following we compute θ for a three dimensional cosmological
grid with 0 ≤ Ωm ≤ 1, 0 ≤ ΩΛ ≤ 1 and 0.5 ≤ h ≤ 1, where the step size is 0.1 for the
density parameters and 0.05 for h. The result is shown in the following figure:
2
Ωtot
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
θ
Figure 1: Subtended angle θ for a three dimensional grid of cosmological parameters.
It is clearly visible that the curvature has a strong effect on the obtained θ. This is a
main reason why Boomerang and WMAP were able to determine the curvature of the
15.5 Intermediate Angular Scales – the Acoustic Peaks
449
Some examples of the results of detailed predictions by Challinor are shown in
universe
unprecedented
of the other parameters
not
Fig.
15.7,toindicating
howaccuracy.
differentHowever,
featuresthe
of effect
the temperature
anisotropyispower
negligible. Figure 2 (taken from Longair’s “galaxy evolution”) shows the dependence of
spectrum are sensitive to variations of the cosmological parameters (Challinor, 2005).
the first acoustic peak in the CMB on different cosmological parameters (the parameters
Itvary
is aasuseful
exercise
study
the power
spectra
in Fig. of
15.7
some detail
indicated
withintothe
panels).
While the
dependence
the in
l-coordinate
(i.e.and
the to
use
the ofresults
we ashave
established
this section
the there
dependences
angle)
the peak
a function
of the in
curvature
Ωk = 1to−understand
Ωtot is obvious,
is also
2
upon
cosmological
Forpeak
example,
the left-hand
Fig.as15.7a
shows
a dependence
of the parameters.
position of the
as a function
of Ωm h plot
(andof
even
a function
2
of ΩΛ ). the
Butstrong
as the enhancement
amplitude of the
is also
very sensitive
to Ω
, this effect
is
m h baryon
clearly
of peak
the first
acoustic
peak as
the
density
clearly
distinguishable
from
the
dependence
of
Ω
.
k
increases.
Fig. 15.7a,b. The dependence of the temperature-anisotropy power spectrum on different
Figure
2: Dependence
of the first
acoustic
peakIn
inthese
the CMB
on different
cosmological
paramcosmological
parameters
(Challinor,
2005).
examples,
scale-invariant
adiabatic
initial
eters.perturbations are assumed. a Top pair of diagrams: dependence on the density parameter in
baryons (left) and total matter density parameter Ω0 (right). Top to bottom at first peak: the
baryon density parameter varies linearly in the range 0.06 ≥ ΩB h 2 ≥ 0.005 (left) and the
matter density parameter in the range 0.05 ≤ ΩB h 2 ≤ 0.5 (right). b Bottom pair of diagrams:
The dependence on the curvature density parameter Ωκ (left) and the dark energy density
parameter ΩΛ (right). In both cases, the density parameters in baryons and matter were held
constant, thus preserving the conditions on the last scattering layer. The curvature density
parameter varies (left to right) in the range −0.15 ≤ Ωκ ≤ 0.15 and the dark matter density
parameter in the range 0.9 ≥ ΩΛ ≥ 0.0