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Chapter 26
Cosmology
Douglas Scott, Joseph Silk,
Edward W. Kolb, and Michael S. Turner
26.1
Friedmann–Robertson–Walker Metric and
Distance Measures . . . . . . . . . . . . . . . . . . . . . 644
26.2
The Age of the Universe . . . . . . . . . . . . . . . . . 646
26.3
Conversion Factors for the Early Universe . . . . . . . 647
26.4
Other Useful Conversion Factors . . . . . . . . . . . . 648
26.5
Cosmological Parameters . . . . . . . . . . . . . . . . . 649
26.6
Friedmann–Lemaı̂tre Model . . . . . . . . . . . . . . . 650
26.7
Epochs of Interest . . . . . . . . . . . . . . . . . . . . . 650
26.8
Age Limits . . . . . . . . . . . . . . . . . . . . . . . . . 652
26.9
Cosmological Tests: H0 . . . . . . . . . . . . . . . . . 653
26.10
Cosmological Tests: q0 . . . . . . . . . . . . . . . . . . 653
26.11
Other Cosmological Parameters . . . . . . . . . . . . . 654
26.12
Primordial Nucleosynthesis and Neutrinos . . . . . . 654
26.13
Power Spectrum of Density Fluctuations . . . . . . . 655
26.14
Structure Formation Scales . . . . . . . . . . . . . . . . 656
26.15
Cosmic Microwave Background Anisotropies . . . . 658
26.16
Large-Scale Structure . . . . . . . . . . . . . . . . . . . 659
26.17
Densities . . . . . . . . . . . . . . . . . . . . . . . . . . 660
26.18
Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . 661
26.19
Intergalactic Medium . . . . . . . . . . . . . . . . . . . 662
26.20
Extragalactic Diffuse Backgrounds . . . . . . . . . . . 663
643
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C OSMOLOGY
26.1 FRIEDMANN–ROBERTSON–WALKER METRIC AND
DISTANCE MEASURES
The standard metric for homogeneous and isotropic spaces can be written in many forms [1–5],
including this version for spherical curvature:
ds 2 = c2 dt 2 − R 2 (t) dr 2 + R2 sin2 (r/R)(dθ 2 + sin2 θ dφ 2 ) .
Here t is cosmic time, R(t) is the scale factor, r is the comoving radial distance coordinate, and R
is the radius of curvature at the present epoch [for more realistic spaces of negative curvature (open
models), we can write R → iR and sin → sinh]. We can then define an effective distance [6] (also
known as transverse comoving or proper motion distance)
D = R sin(r/R).
The Hubble and deceleration parameters can be defined at any epoch as
H=
Ṙ
R
q=−
and
R̈ R
.
Ṙ 2
The subscript 0 is used to denote the present value of a quantity, so that, for example, H0 is the Hubble
constant today.
The Einstein field equations lead to the Friedmann equation:
Ṙ 2 = 83 π Gρ R 2 − K R02 c2 +
1
3
R2,
where ρ(t) is the density, K is the spatial curvature, and is the cosmological constant.
For a universe dominated by pressureless matter and with = 0, the Friedmann equation gives
dz
= −H (z)(1 + z) = −H0 (1 + z)2 (1 +
dt
1/2
,
0 z)
where the definition of redshift is
1 + z ≡ λobs /λem = R0 /R,
curvature can be written
and the density parameter,
K ≡ (R0 R)−2 = H02 (
0
− 1)/c2 ,
= 8π Gρ/3H 2 , scales as
(z) =
0 (1 +
z)/(1 +
0 z).
Deceleration is related to density and the cosmological constant through
q=
/2 −
/3H 2 ,
and a spatially flat (K = 0) model means
+
/3H 2 = 1.
Because of this, the dimensionless cosmological constant is often written λ =
/3H 2 .
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26.1 F RIEDMANN –ROBERTSON –WALKER M ETRIC / 645
= 0)
Using the Friedmann equation and the definition of D we can derive (for
D =
where
2c
2 (1 +
0
H0
0z
z)
+(
D =
2c 1 − (1 + z)−1/2
H0
D→
2c
H0
0
− 2) (1 +
for
0
0 z)
1/2
−1
,
=1
and
(
for
0
0 z)
1/2
1.
Conventional distance measures can then be defined in terms of D; e.g., angular diameter distance
d A = D/(1+z) and luminosity distance d L = D(1+z). More useful, however, are specific examples of
how observed quantities depend on redshift and cosmological parameters. The angular size subtended
by a physical scale is
(1 + z)
θ=
,
D
for small angles. Bolometric flux is related to bolometric luminosity through
Sbol =
L bol
2
4π D (1 +
z)2
,
whereas flux density or monochromatic flux
Sν (νobs ) =
L ν (νem )
,
4π D 2 (1 + z)
with νobs = νem /(1 + z), and if L ν ∝ ν −α , then
Sν (νobs ) =
L ν (νobs )
.
4π D 2 (1 + z)1+α
The distance modulus for this quantity is
m − M = 25 + 5 log[3000(1 + z)H0 D] − 5 log h + K ,
where h = H0 /100 km s−1 Mpc−1 and the K -correction [7] is
K = −2.5 log[(1 + z)L ν (νem )/L ν (νobs )].
The solid angle ω subtended by the proper area A is
ω=
A(1 + z)2
,
D2
so that the bolometric surface brightness
Ibol (observed) =
Sbol
Ibol (emitted)
L bol
=
.
=
4
ω
4π A(1 + z)
(1 + z)4
And for monochromatic surface brightness
Iν (νobs ) =
Iν (νem )
.
(1 + z)3
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C OSMOLOGY
Comoving volume can be expressed as
d V = 4π D 2 dr = 4π D 2
c dz
.
H (z)
So number counts per steradian, for a comoving number density n 0 , are given by d N /dz =
n 0 D 2 c/H (z). Note that neither D, nor d A , d L , etc., are additive quantities, although r (the radial
distance if it could be measured at fixed time today) is.
26.2
THE AGE OF THE UNIVERSE
For a matter-dominated model, with = 0 and negligible pressure, the age is
1
(1 + 0 z)1/2
0
−1 2(1 − 0 )
t (z) =
cosh
−
+
1
H0 (1 − 0 )(1 + z) 2(1 − 0 )3/2
0 (1 + z)
=
2
(1 + z)−3/2
3H0
for
0
for
0
<1
=1
The look-back time at a number of different redshifts is given in Table 26.1 in units of 109 h −1 years
(note the dependence on the Hubble constant) for different values of 0 .
Table 26.1. Look-back time in h −1 Gyr.
0
Redshift
2.0
1.0
0.5
0.3
0.2
0.1
0.0
1
2
3
4
5
10
∞
3.81
4.65
4.98
5.16
5.26
5.45
5.58
4.21
5.26
5.70
5.94
6.07
6.34
6.52
4.50
5.75
6.29
6.59
6.77
7.12
7.37
4.64
6.00
6.62
6.96
7.18
7.60
7.91
4.71
6.15
6.82
7.20
7.43
7.91
8.28
4.80
6.32
7.05
7.47
7.75
8.31
8.78
4.89
6.52
7.33
7.82
8.15
8.89
9.78
The age of the Universe today is given in Table 26.2 in units of 109 years.
Table 26.2. Age of the Universe in Gyr (if
= 0)
0
H0 (km s−1 Mpc−1 )
2.0
1.0
0.5
0.3
0.2
0.1
0.0
100
90
80
70
60
50
40
5.58
6.20
6.98
7.97
9.30
11.16
13.95
6.52
7.24
8.15
9.31
10.86
13.03
16.29
7.37
8.19
9.21
10.52
12.28
14.73
18.41
7.91
8.79
9.88
11.29
13.17
15.81
19.75
8.28
9.19
10.34
11.82
13.79
16.54
20.67
8.78
9.75
10.97
12.54
14.62
17.54
21.90
9.78
10.86
12.22
13.97
16.30
19.56
24.44
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26.3 C ONVERSION FACTORS FOR THE E ARLY U NIVERSE / 647
Including a cosmological constant, the age of a matter-dominated Universe is given by
t (z) =
1
H0
∞
z
dz
(1 + z ) (
0z
For a flat Universe with nonzero λ0 (i.e., λ0 = 1 −
t=
2
1
√
3H0 1 −
+ 1 − λ0 )(1 + z )2 + λ0
1/2
.
0)
the age today is
√
1+ 1− 0
,
ln
√
0
0
and is given in Table 26.3 in units of 109 years.
Table 26.3. Age of the Universe in Gyr (if
0 + λ0 = 1)
0 (or 1 − λ0 )
26.3
H0 (km s−1 Mpc−1 )
1.0
0.5
0.3
0.2
0.1
100
90
80
70
60
50
40
6.52
7.24
8.15
9.31
10.86
13.03
16.29
8.13
9.03
10.16
11.61
13.54
16.24
20.31
9.43
10.47
11.78
13.46
15.70
18.84
23.54
10.52
11.69
13.15
15.02
17.52
21.02
26.26
12.49
13.87
15.60
17.83
20.79
24.93
31.13
CONVERSION FACTORS FOR THE EARLY UNIVERSE
Numerical factors in this and subsequent sections are given to five significant figures, which will
almost always be more than sufficient for cosmological calculations. Anyone interested in the
uncertainties associated with such values will be able to trace them through the earlier chapter on
constants and units, or through, e.g., [8]. Less well-determined quantities will be quoted using an
appropriate number of significant figures. Any quantity involving the gravitational constant G will
be known with a little less accuracy than implied by the five figures. For definiteness we have used
G = 6.672 59 × 10−8 cm3 g−1 s−2 ≡ m −2
Pl . On the other hand, quantities involving GM (e.g., the
gravitational length scale of the Sun, or the critical density in M Mpc−3 ) will be known to many
more than five figures. The following conversions are useful for early Universe work, where “natural”
units are used, so that h̄ = c = k = 1, and there is one fundamental dimension, energy, with the
conventional unit of GeV [3]. Note that going from Tesla to Gauss is not a conversion in the normal
sense, since they come from systems of units which differ by factors of 4π , etc. [9].
Energy:
Temperature:
Mass:
Length:
Time:
Power:
Number density:
Mass density:
1 GeV = 1.6022 × 10−3 erg
1 GeV = 1.1604 × 1013 K
1 GeV = 1.7827 × 10−24 g
1 GeV−1 = 1.9733 × 10−14 cm
1 GeV−1 = 6.5821 × 10−25 s
1 GeV2 = 2.4341 × 1021 erg s−1
1 GeV3 = 1.3015 × 1041 cm−3
1 GeV4 = 2.3201 × 1017 g cm−3
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C OSMOLOGY
Energy density:
Volume emissivity:
Cross section:
Magnetic field:
Field energy:
26.4
1 GeV4 = 2.0852 × 1038 erg cm−3
1 GeV5 = 3.1680 × 1062 erg cm−3 s−1
1 barn = 103 mb = 10−24 cm2
1 mb = 2.5681 GeV−2
1 Tesla = 104 Gauss
1 (Gauss)2 /8π = 1.9081 × 10−40 GeV4
1 (Tesla)2 /2 = 1.9081 × 10−32 GeV4
OTHER USEFUL CONVERSION FACTORS
This is an assortment of conversion factors which are sometimes useful in various branches of
cosmology.
Wavelength/energy:
Energy of 1 µm photon:
Astronomical unit:
Parsec:
Megaparsec:
Grav. scale of 1M :
Energy in 1M :
Proton. equiv. of 1M :
Surface density:
Mass density:
Sidereal day:
Sidereal year:
Speed:
λ = 12 398 Å/E(eV) = 1239.8 nm/E(eV)
hν = 1.9864 × 10−19 W s
1 AU = 1.4960 × 1013 cm
= 214.94 R = 7.5812 × 1026 GeV−1
1 pc ≡ 648 000/π AU
= 3.2615 (sidereal) light-yr
= 3.0857 × 1018 cm
1 Mpc = 106 pc = 3.0857 × 1024 cm
= 1.5637 × 1038 GeV−1
GM /c2 = 1.4766 km
M c2 = 1.1157 × 1057 GeV
M = 1.1891 × 1057 protons
1 M pc−2 = 1.1718 × 1020 GeV cm−2
1 M pc−3 = 37.975 GeV cm−3
1 day (sidereal) = 86 164 s
= 1.3091 × 1029 GeV−1
1 yr (sidereal) = 3.1558 × 107 s
= 4.7945 × 1031 GeV−1 = 10−9 Gyr
−1
1 km s = 1.0227 kpc Gyr−1
There are many conventions for measurements of brightness, intensity, etc., in different wavebands.
We list some useful conversions below, where m is an apparent magnitude, dω is here an element of
solid angle, 1 Jy = 10−23 erg cm−2 s−1 Hz−1 , and one S10 unit corresponds to one magnitude 10 star
per square degree or 27.78 magnitudes per square arcsecond [10].
Solid angle:
Specific intensity:
Solar magnitudes:
Bolometric flux:
4π steradians ≡ 3602 /π degree2
= 5.3464 × 1011 arcsec2
Iλ = Sλ /dω = (ν/λ)Sν /dω
m bol, = −26.83; Mbol, = 4.74
m B, = −26.09; M B, = 5.48
Sbol = 2.52 × 10−5−0.4m bol erg cm−2 s−1
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26.5 C OSMOLOGICAL PARAMETERS / 649
Flux density (Blue):
Bolometric brightness:
Blue brightness:
Night sky:
26.5
−1
Sλ (λ 4400 Å) = 6.76 × 10−9−0.4m B erg cm−2 Å s−1
Sν (λ 4400 Å) = 4.37 × 103−0.4m B Jy
m bol mag. arcsec−2 = m bol µ
= 2.52 × 10−5−0.4m bol erg cm−2 arcsec−2 s−1
= 3.35 × 1016−0.4µ L kpc−2
m B mag. arcsec−2 = m B µ B
−1
= 6.76 × 10−9−0.4m B erg cm−2 arcsec−2 Å s−1
= 6.62 × 1016−0.4µ B L kpc−2
−1
22µ B 1.1 × 10−17 erg cm−2 arcsec−2 Å s−1 = 205S10
COSMOLOGICAL PARAMETERS
Many parameters depend on the precise value of the cosmic microwave background (CMB) temperature, so we scale by the quantity T2.73 = Tγ 0 /2.73 K. Current observations indicate that T2.73 =
0.9993 ± 0.0007 [11, 12].
Hubble constant:
Hubble time:
Hubble distance:
Critical density:
Photons:
H0 = 100h km s−1 Mpc−1
= 2.1331 × 10−42 h GeV (0.5 h 0.85) [4]
H0−1 = 3.0857 × 1017 h −1 s = 9.7778h −1 Gyr
cH0−1 = 2997.9h −1 Mpc = 9.2506 × 1027 h −1 cm
ρc ≡ 3H02 /8π G = 1.8788 × 10−29 h 2 g cm−3
= 8.0980 × 10−47 h 2 GeV4 = 1.0539 × 104 h 2 eV cm−3
= 1.1233 × 10−5 h 2 protons cm−3
= 2.7754 × 1011 h 2 M Mpc−3
Tγ 0 ≡ 2.73 T2.73 K = 2.3525 × 10−13 T2.73 GeV
3 cm−3
n γ 0 = 412.77 T2.73
4 GeV4
ργ 0 = 2.0154 × 10−51 T2.73
4 g cm−3
= 4.6760 × 10−34 T2.73
γ0
Neutrinos:
4
= 2.4888 × 10−5 h −2 T2.73
Tν0 = (4/11)1/3 Tγ 0 = 1.9486 T2.73 K
= 1.6792 × 10−13 T2.73 GeV
3
3 cm−3 (per species)
n ν0 = 11
n γ 0 = 112.57 T2.73
4/3 ρ
ρν0 = 21
γ 0 = 0.681 32ργ 0 (3 species)
8 (4/11)
−5 h −2 T 4
=
1.6957
×
10
ν0
2.73 (3 massless species)
ν0
Entropy:
Cosmological constant:
Age of Universe:
3
= (m ν /93.625 eV)h −2 T2.73
[43π 4 /11 × 45 × ζ (3)]n
s0 =
3 cm−3
= 2905.7 T2.73
γ0
(1 massive species)
= 7.0394 n γ 0
−2
3H02 = 9.1556 × 10−122 h 2 tPl
t0 23 H0−1 = 3.8161 × 1060 h −1 tPl
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26.6
C OSMOLOGY
FRIEDMANN–LEMAÎTRE MODEL
For the standard, homogeneous, isotropic, expanding cosmologies the following relationships may
be useful (expressed below in early Universe units). Here the temperature of some species X which
decoupled at temperature TD is denoted by TX . The subscript “eq” will refer to the equality epoch
when ρ(radiation) = ρ(matter). The baryon-to-photon number density ratio is denoted by η. Note
that an accurate conversion between B and n B or η will depend on the fraction of helium, electron
mass, binding energy, etc. Here we quote numbers using simply the proton mass.
We first define the factors g∗ and g∗s to be the total number of effectively massless degrees of
freedom which contribute to the radiation density and entropy density, respectively. At high energies
g∗s → g∗ . Again subscript 0 refers to the present-day value.
Relativistic degrees
of freedom (ρ R ):
Relativistic degrees
of freedom (s):
Radiation density:
Entropy density:
Decoupled species:
Scale factor:
(relativistic):
(baryons):
Entropy in the
horizon:
Baryons in the
horizon:
26.7
g∗ =
7
4
4
i=boson (Ti /T ) gi + 8
i=fermion (Ti /T ) gi
4/3 = 3.3626 (3 ν species)
g∗0 = 2 + 42
8 (4/11)
g∗s = i=boson (Ti /T )3 gi + 78 i=fermion (Ti /T )3 gi
g∗s0 = 2 + 42
8 (4/11) = 3.9091 (3 ν species)
2
ρ R = (π /30)g∗ T 4
s = (2π 2 /45)g∗s T 3
TX 0 = [3.9091/g∗s (TD )]1/3 Tγ 0
−1/3
R/R0 = 3.7059 × 10−13 T2.73 g∗s (GeV/T )
2
−5 4
γ ν ν̄ h = 4.1844 × 10 T2.73
2
7 3
B h = 3.67 × 10 T2.73 η
3 n /s
= 2.59 × 108 T2.73
B
Shor ≡ (4π/3) t 3 s
−3/2
= 5.0196 × 10−2 g∗s g∗ (m Pl /T )3 (t teq )
= 2.8548 × 1087 ( 0 h 2 )−3/2 (1 + z)−3/2 (t teq )
N B−hor ≡ (n B /s)Shor
−3/2
= 1.94 × 10−10 ( B h 2 )g∗s g∗ (m Pl /T )3 (t teq )
79
2
2
= 1.10 × 10 ( B h )( 0 h )−3/2 (1 + z)−3/2 (t teq )
EPOCHS OF INTEREST
We can define several quantities at the Planck epoch. The Planck redshift is computed assuming there
has been no inflationary period.
Planck mass:
Planck energy:
Planck time:
Planck length:
Planck density:
Planck redshift:
cH0−1 at Planck epoch:
m Pl ≡ (h̄c/G)1/2 = 2.1767 × 10−5 g
m Pl c2 ≡ (h̄c5 /G)1/2 = 1.2210 × 1019 GeV
tPl ≡ (h̄G/c5 )1/2 = 5.3906 × 10−44 s
lPl ≡ (h̄G/c3 )1/2 = 1.6160 × 10−33 cm
ρPl ≡ c5 /h̄G 2 = 5.1575 × 1093 g cm−3
1/3 −1
T (z Pl ) ≡ m Pl at 1 + z Pl = 3.2948 × 1031 g∗s T2.73
−1
cH0−1 z Pl
= 2.8076 × 10−4 cm
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26.7 E POCHS OF I NTEREST / 651
The age of the Universe can be followed analytically through the equality epoch (at least to
O(Req /R0 )). Under the assumption that we live in a flat, matter-dominated Universe today, with
both matter and radiation contributing in the past, we can rewrite the Friedmann equation as ȧ =
H0 (a + aeq )1/2 /a, where, for conciseness, we have used a ≡ R/R0 . From this can be derived
3/2
H0 t = 23 (a + aeq )1/2 (a − 2aeq ) + 2aeq ,
which is an exact expression for t (R), although it cannot be easily inverted to give R(t).
For conformal time (defined by dη ≡ dt/a, not to be confused with the baryon-to-photon ratio) we
also have
1/2
H0 η = 2 (a + aeq )1/2 − aeq ,
√
which can, in fact, be inverted to give a = 14 H02 η2 + aeq H0 η. The conformal time can be thought of
as the comoving size of the horizon.
Similarly, for the expansion time, texp ≡ R/ Ṙ ≡ H −1 , we have
H0 texp = a 2 (a + aeq )−1/2 .
At the epoch of equality we have
√ 3/2
√
1/2
H0 teq = 23 (2 − 2)aeq ,
H0 ηeq = 2( 2 − 1)aeq ,
and
3/2 √
H0 texp,eq = aeq / 2,
−1/2
again with a ≡ R/R0 ≡ (1 + z)−1 . The approximate scaling t ∝ 0
can also be used for 0 z 1.
The time scales of the Universe can then be written in many useful ways in either the radiation- or
matter-dominated limits:
Age of Universe:
−1/2
t = 0.301 18g∗
=
=
t=
=
Conformal time:
=
η=
(T Teq ):
=
=
(T Teq ):
η=
=
Expansion time:
−1/2
= 2.4206 × 10−6 g∗
(T Teq ):
(T Teq ):
m Pl /T 2
texp
=
=
=
(GeV/T )2 s
−1/2
2.4206g∗ (MeV/T )2 s
2/3 −1/2
1.7625 × 1019 g∗s g∗ (1 + z)−2 s
−1/2
3/2 1/2
0.401 57g∗ m Pl /(T Teq )
−1/2
3/2
7.4228 × 1011 ( 0 h 2 )
T2.73 (eV/T )3/2 s
2.0571 × 1017 ( 0 h 2 )−1/2 (1 + z)−3/2 s
−1 1/3 −1/2
1.9847 × 1031 T2.73
g∗s g∗ m Pl /T
−1 1/3 −1/2
7
1.2984 × 10 T2.73 g∗s g∗ (GeV/T ) s
−2 2/3 −1/2
3.4247 × 105 T2.73
g∗s g∗ (1 + z)−1 Mpc
1/2 1/2
−1 1/3 −1/2
3.9693 × 1031 T2.73
g∗s g∗ m Pl /(T Teq )
−1/2 1/2 1/3
6.0088 × 1015 ( 0 h 2 )
T2.73 g∗s (eV/T )1/2 s
5995.8( 0 h 2 )−1/2 (1 + z)−1/2 Mpc
−1/2
(1 + z eq )1/2 (2 + z + z eq )−1/2 (1 + z)−3/2 H0−1 0
−2
4.7702 × 1019 (2 + z + z eq )−1/2 (1 + z)−3/2 T2.73
s
Several epochs relevant for structure formation and the microwave background spectrum can
also be useful. Note that the definition of z eq depends on the number of neutrino species included.
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C OSMOLOGY
Recombination through the visibility function (or e−τ dτ for Thomson scattering), and is centered
around z 1100 largely independent of cosmological parameters. We avoid using the term
“decoupling” to refer to the last scattering epoch, since it might more physically be applied to the
epoch when the matter temperature can depart from the radiation, i.e., when the Compton cooling time
is short. Other epochs are defined in terms of the redshift when the rate (generally |˙ |/) is equal to the
expansion rate, although definitions differing by constant factors could also have been used. Compton
cooling and Compton drag depend on the evolution of the ionized fraction, xe (z). At redshifts much
above z C , Compton scattering will relax spectral distortions to a Bose–Einstein form, whereas much
above the smaller of z DC or z Br there will be complete thermalization. Reviews of these topics have
been presented in, e.g., [16–18]. Here the factor (1 − Y p /2) accounts for the helium contribution,
assuming it is fully ionized.
End of neutrino
free-streaming:
Radiation-matter
equality:
Req = 4.1845 × 10−5 (
2 −1 4
0 h ) T2.73 R0
−4
1 + z eq = 23 898( 0 h 2 )T2.73
(for γ s + 3ν s)
−4
2
1 + z eq = 40 180( 0 h )T2.73 (for γ s only)
−3
Teq = 5.6222( 0 h 2 )T2.73
eV
6 s
10
2
teq = 3.2618 × 10 ( 0 h )−2 T2.73
2
2
−1
ηeq = 16.066( 0 h ) T2.73 Mpc
Recombination:
1 + z rec ≡ 1100; Rrec ≡ R0 /1100
Trec = 0.259 T2.73 eV
trec = 5.64 × 1012 ( 0 h 2 )−1/2 s
4 )
(valid if 0 h 2 0.04 T2.73
Compton cooling:
tcool (z) ≡ m e cn B /4σT aT 4 n e = texp (z) at
1 + z cool 500( B h 2 )2/5 (for standard rec.)
−2/5
1 + z cool 5.7( 0 h 2 )1/5 xe (1 − Y p /2)−2/5
(for xe ≡ n e /n B = const.)
tdrag (z) ≡ (m p /m e )tcool = texp (z) at
−2/5
1 + z drag 120( 0 h 2 )1/5 xe (1 − Y p /2)−2/5
(for xe = const.) [19]
1/5
1 + z DC 2.8 × 105 (1 − Y p /2)−2/5 ( B h 2 )−2/5 T2.73
(when tDC = texp , for z z eq )
13/5
1 + z Br 4.7 × 104 (1 − Y p /2)−4/5 ( B h 2 )−6/5 T2.73
(when tBr = texp , for z z eq )
1/2
1 + z C 7.1 × 103 (1 − Y p /2)−1/2 ( B h 2 )−1/2 T2.73
Radiation drag:
Double-Compton
thermalization:
Bremsstrahlung
thermalization:
Compton
thermalization:
26.8
3Tν (z nr ) ≡ m ν at
−1
1 + z nr = 59 553(m ν /30 eV)T2.73
(valid for
Bh
2
0.09(1 − Y p /2)−1/2 (
0h
2 )2 T
2.73 )
AGE LIMITS
In this and subsequent sections, observationally determined quantities are quoted with the 1σ , 2σ , or
95% Confidence Limit error bars taken from the references. Quantities without error bars should be
considered as more generic or approximate. For nuclear chronology and for the ages of the oldest
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26.10 C OSMOLOGICAL T ESTS : q0 / 653
globular clusters we give two representative estimates of the range, to indicate that different authors
can be more or less conservative.
Oldest Earth rocks:
Oldest meteorites:
Oldest lunar rock:
Nuclear chronology:
Oldest globular clusters:
White dwarfs:
26.9
t⊕ = 3.962 ± 0.003 Gyr (1σ ) [20]
tmeteor = 4.53 ± 0.02 Gyr (1σ ) [21]
tmoon = 4.6 ± 0.1 Gyr (1σ ) t [22]
tGalaxy = 5.4 ± 1.5 Gyr (1σ ) + t [23, 24]
10 Gyr < tGalaxy < 20 Gyr [25]
tglob = 11.5 ± 1.3 Gyr (1σ ) [26, 27]
12 Gyr < tglob < 20 Gyr (2σ ) [28]
tdisk = 9.5 ± 1 Gyr (1σ ) [29, 30]
COSMOLOGICAL TESTS: H0
This list is not intended to be the ultimate authority on the subject, but should be taken as an indication
of values derived by different methods, based largely upon some recent reviews [31–35]. Note that
most of these methods require more than one step, e.g., many use the calibration of Cepheid variables.
Two distinct estimates are given for the SNe Ia standard candle method. The lensing estimate from
quasar 0957+561 [36] has uncertainty dominated by the lens model. The Sunyaev–Zel’dovich (S–Z)
value includes some estimate of possible systematic errors. Estimates from, e.g., [31] use the derived
distances to Virgo, an assumed ratio of the Coma to Virgo distance of 5.6 ± 0.5 and a Coma velocity of
7160 ± 200 km s−1 after correction for Virgocentric infall. All of the methods are actually attempts to
measure the distance to some distant object (e.g., Virgo or Coma), and so would change if a different
velocity or distance ratio was adopted.
Type Ia supernovae:
Type II supernovae:
Tully–Fisher:
Planetary nebula lum. fn.:
Globular cluster lum. fn.:
Novae:
Surface brightness fluctns.:
Dn − σ :
Gravitational lensing:
Sunyaev–Zel’dovich:
26.10
H0
H0
H0
H0
H0
H0
H0
H0
H0
H0
H0
= 58 ± 8 km s−1 Mpc−1
= 64 ± 6 km s−1 Mpc−1
= 73 ± 9 km s−1 Mpc−1
= 81 ± 11 km s−1 Mpc−1
= 83 ± 10 km s−1 Mpc−1
= 68 ± 15 km s−1 Mpc−1
= 61 ± 13 km s−1 Mpc−1
= 80 ± 8 km s−1 Mpc−1
= 76 ± 13 km s−1 Mpc−1
= 64 ± 13 km s−1 Mpc−1
= 55 ± 17 km s−1 Mpc−1
(1σ ) [37, 38]
(1σ ) [39, 40]
(1σ ) [41]
(1σ ) [31]
(1σ ) [31]
(1σ ) [31]
(1σ ) [31]
(1σ ) [31]
(1σ ) [31]
(95% CL) [36]
(1σ ) [42–45]
COSMOLOGICAL TESTS: q0
“Classical” cosmology has dealt with methods of determining the deceleration parameter q0 as well
as H0 . Here it is still conventional to use q0 in place of 0 /2, although occasionally 0 /2 − λ0 is
meant. The variation of two observable quantities is used to determine the best fit q0 , specifically
apparent magnitude versus redshift (m, z), number versus redshift (N , z), number versus apparent
magnitude (N , m), angular diameter versus redshift (θ, z), and other geometrical methods of distance
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C OSMOLOGY
determination. However, all of these tests are subject to large evolutionary corrections and other
redshift-dependent effects. They therefore require much interpretation in order to estimate q0 .
There may be some evidence from (N , m) that q0 0.15 [46], although other values can also
fit [47], while (θ, z) may indicate q0 0.5 [48], and (m, z) for distant supernovae prefers q0 (actually
( 0 −λ)/2) 0 [49, 50]. More concretely, age constraints give, for example, q0 ≤ 0.08 if t0 ≥ 13 Gyr
and h ≥ 0.65 (and assuming = 0).
26.11
OTHER COSMOLOGICAL PARAMETERS
The best limit on comes from considerations of gravitational lensing, with distant supernovae now
competing. Ġ limits come from laser and radar ranging experiments in the solar system, from the
constancy of neutron star masses, and from Big Bang Nucleosynthesis. Brans–Dicke theories are also
constrained by lunar ranging data. The global rotation ωrot and global shear σ of the Universe are
constrained by CMB anisotropy measurements.
Cosmological constant:
Variable G:
Brans–Dicke
coupling constant:
Global rotation:
Global shear:
26.12
λ0 < 0.66 (95% CL, assuming 0 + λ0 = 1) [51, 52]
|Ġ|/G ≤ 4 × 10−12 yr−1 (1σ ) or |Ġ|/(G H0 ) ≤ 0.04h −1
(from ranging measurement) [53]
Ġ/G = (−0.6 ± 4.2) × 10−12 yr−1 or |Ġ|/(G H0 ) ≤ 0.05h −1
(95% CL, from neutron star masses) [54]
|Ġ|/G ≤ 9 × 10−13 yr−1 (2σ ) or |Ġ|/(G H0 ) ≤ 0.009h −1
(assuming G ∝ t −x from BBN) [55]
ωBD > 600 (1σ ) [53]
(ωBD → ∞ gives General Relativity)
(ωrot /H )0 < 6 × 10−8 (95% CL, for 0.1 ≤ 0 ≤ 1) [56, 57]
(σ/H )0 < 10−9 (95% CL) [56]
PRIMORDIAL NUCLEOSYNTHESIS AND NEUTRINOS
The following quantities relate to the formation of the light elements during Big Bang Nucleosynthesis
(BBN), and are a combination of observational limits and the results of computer simulations.
Nucleon-to-photon ratio:
Baryon density:
Primordial 4 He:
Primordial D + 3 He:
Primordial 7 Li:
2.0 × 10−10 ≤ η ≤ 6.5 × 10−10 (95% CL) [58–60]
0.007 ≤ B h 2 ≤ 0.024 (95% CL) [59]
0.221 ≤ Y p ≤ 0.243 (95% CL) [58]
(D + 3 He)/H ≤ 1.1 × 10−4 (95% CL) [58]
0.7 × 10−10 ≤ 7 Li/H ≤ 3.5 × 10−10 (95% CL) [58]
The number and masses of neutrinos can potentially be important for cosmology:
ν0
3 . Limits on the number of light neutrino species come from both par= ( m ν /93.625 eV)h −2 T2.73
ticle accelerators (e.g., LEP at CERN), where “light” means m(Z 0 )/2 46 GeV, and from BBN
where “light” means 1 MeV.
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26.13 P OWER S PECTRUM OF D ENSITY F LUCTUATIONS / 655
Nν < 4 (conservative limit) [59, 60]
Nν = 2.993 ± 0.011 (1σ ) [8]
m νe < 15 eV (conservative limit) [8]
m νµ < 170 keV (90% CL) [8, 61]
m ντ < 24 MeV (95% CL) [62, 8]
Nν from BBN:
Nν from LEP:
Electron neutrino:
Muon neutrino:
Tau neutrino:
26.13
POWER SPECTRUM OF DENSITY FLUCTUATIONS
We begin with some definitions and conventions (note that these can vary significantly between
authors).
Definition:
Fourier transform convention:
Harrison–Zel’dovich:
Two-point correlation fn.:
First-moment:
Second-moment:
Density variance:
P(k) = |δk |2
δk (t) = (1/V ) δx e−ik.x d 3 x
P(k) = Ak n with n = 1
∞
ξ(x) = (V /2π 2 ) 0 P(k) j0 (kx)k 2 dk
∞
J2 (x) = (V /2π 2 ) 0 P(k)(1 − cos(kx)) dk
∞
J3 (x) = (V /2π 2 ) 0 P(k)(kx)2 j1 (kx) dk/k
∞
σρ2 (x) = (V /2π 2 ) 0 P(k)(3 j1 (kx)/kx)2 k 2 dk
Here V is the (large) volume used to conveniently apply periodic boundary conditions for the
Fourier transform; it should be considered as merely a bookkeeping device.
A useful parametrization of Cold Dark Matter (CDM) power spectra is given by
P(k) =
Ak
ν
1 + ak + (bk)3/2 + (ck)2
2/ν
,
where a = (6.4/ )h −1 Mpc, b = (3.0/ )h −1 Mpc, c = (1.7/ )h −1 Mpc, ν = 1.13, and the fit is
specifically for B = 0.03 [63]. There are other parametrizations of the transfer function which differ
slightly at large k [64–66]. The normalization given by COBE is [67]
A = (5.9 ± 1.1) × 105
(h −1 Mpc)4
.
V
For the standard, scale-invariant, adiabatic CDM model 0 = 1 and h above, with a favored
value around 0.5. Flat CDM models with nonzero are well fitted using h. Generally can be
used as a shape parameter and 0.2 provides the best fit to large-scale structure observations [68].
Explicit parametrizations of other theoretical power spectra can be found in, e.g., [64, 69], and can be
obtained using codes such as cmbfast [70].
Figure 26.1 indicates the approximate status and range of scales spanned by current data relating to
the matter power spectrum, and also highlights the relative differences between “reasonable” models.
The solid points are from the compilation of [68], which represents an average of several surveys
(dominated by the APM galaxy survey [71]), and is explicitly presented here for 0 = 1 models with
IRAS galaxies assumed to be unbiased. It is worth stressing that the normalization and the redshiftspace corrections would change in open or -dominated cosmologies, and that there are normalization
uncertainties due to different bias factors between, e.g., IRAS and optically selected galaxies. The
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C OSMOLOGY
Figure 26.1. Measurements of the power spectrum of density fluctuations.
triangle is derived from peculiar velocities [72], and the square from cluster abundance [73], with error
bars indicating 1σ ranges. Both these points are independent of bias, but would move somewhat for
different values of 0 . An approximation to the COBE error box is plotted at low k. The arrows along
the top indicate the angles probed by CMB experiments with given FWHM beams. The curves plotted
are the standard CDM (solid) mass power spectrum, together with three representative examples of
changes brought about by varying a single cosmological parameter: lowering H0 (long dashes); tilting
the initial conditions (short dashes); and introducing some massive neutrinos (dots). These models are
in no sense fits to the data, but demonstrate that variations in several parameters (including 0 and ,
which would require separate figures) can lead to acceptable fits.
26.14
STRUCTURE FORMATION SCALES
There are many physical scales which can be important for the formation of structure in different
cosmological scenarios, and which are collected below. Here Thor (λ) and thor (λ) refer to the
temperature and age of the Universe when the comoving scale λ had a physical size equal to c times
that of the age of the Universe (R λ/R0 = ct). The comoving scale which crosses the horizon at
2 (
2 −1 Mpc, as computed using the exact expression
matter-radiation equality is λeq = 7.5733 T2.73
0h )
for teq . Using either the matter- or radiation-dominated expressions would give significantly different
values [3].
It is worth pointing out that there are various definitions for what is meant by “horizon size”,
differing by constant factors. For a matter-dominated Universe the radius of the particle horizon is
given by 3ct or 2/H . So the angle subtended by this scale is twice that subtended by the Hubble
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26.14 S TRUCTURE F ORMATION S CALES / 657
radius, etc. Typically the physically relevant scale is when kη 1, and so there can also be ambiguous
factors of 2π when calculating a scale rather than a wave number.
Baryon mass
in the horizon:
Mass within
comoving scale λ:
Physical size of
comoving scale λ:
Horizon crossing:
Curvature scale:
MB−hor ≡ m N NB−hor
= 2.97 × 10−10 (
Bh
= 9.25 × 1021 (
B/
M(λ) ≡ π λ3 ρnr /6
= 1.4532 × 1011 (
0h
2 )g g −3/2 (GeV/T )3 M
∗s ∗
3/2
−3/2 M
h)(1
+
z)
(t
0
(t teq )
teq )
2 )λ3 M
Mpc
−1/3
12
Rλ/R0 = 1.1435 × 10 λMpc T2.73 g∗s (GeV/T ) cm
−1/3
= 5.7951 × 1025 λMpc T2.73 g∗s (GeV/T ) GeV−1
−1 1/3 −1/2 −1
Thor (λ) = 63.459 T2.73
g∗s g∗ λMpc eV (λ λeq )
2 g −2/3 g 1/2 λ2
thor (λ) = 6.0108 × 108 T2.73
∗s
∗
Mpc s
−2
2
−1
Thor (λ) = 939.71( 0 h ) T2.73 λMpc eV (λ λeq )
thor (λ) = 2.5768 × 107 ( 0 h 2 )λ3Mpc s
L curv ≡ RR0 = 2997.9(1 − 0 )−1/2 h −1 Mpc
The angle subtended on the sky by the Hubble distance H −1 , by a comoving scale λ, and by the
curvature scale L curv , can all be expressed in simple form for redshifts z 1.
Angle subtended by
Hubble radius:
θ (H −1 , z 1) =
1/2 −1/2
/2
0 z
1/2
= 0.◦ 87 0 (z/1100)−1/2
(R0 λ/H0−1 ) 0 /2
Angle subtended by
comoving scale λ:
θ (λ, z 1) =
= 34. 4(
= 65. 4(
Angle subtended by
curvature scale L curv :
θ (L curv , z 1) =
=
0 h)λMpc
2/3 1/3
)(M/1012 M )1/3
0 h
1/2
0 /2(1 − 0 )
◦
28. 6 0 (1 − 0 )−1/2
The following scales are relevant for various “damping” processes. Below the Jeans scale,
baryon fluctuations oscillate, rather than grow, and below the photon damping scale they are
dissipated. Neutrino free-streaming erases fluctuations below the neutrino damping length. Microwave
background fluctuations tend to be damped on scales below that of the last scattering surface thickness.
Sound speed:
Jeans length:
√
−1/2
cs = (c/ 3) (3ρm /4ργ ) + 1
(z z rec )
1/2
(z z rec )
cs = (5kT /3m p )
λ J ≡ cs (π/Gρm )1/2
51( 0 h 2 )−1 Mpc (comoving) just before z rec
10( 0 h 2 )−1/2 kpc (comoving) just after z rec
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C OSMOLOGY
Neutrino rms velocity:
M J ≡ π λ3J ρm /6
2.0 × 1016 ( 0 h 2 )−2 M just before z rec
1.5 × 105 ( 0 h 2 )−1/2 M just after z rec
L D 3.2( B h 2 )−1/2 ( 0 h 2 )−1/4 Mpc [74]
M D 4.7 × 1012 ( B h 2 )−3/2 ( 0 h 2 )1/4 M
2 1/2
vν
5.0(30 eV/m ν )(1 + z)T2.73 km s−1
Neutrino Jeans length:
or damping length:
Neutrino Jeans mass:
L J ν 41 Mpc(30 eV/m ν ) [75, 76]
λν u 13( 0 h 2 )−1 Mpc
M J ν (∼ m 3Pl /m 2ν ) 3 × 1015 M (30 eV/m ν )2
Horizon at teq :
λeq ≡ cteq = 7.5733(
Horizon at trec :
Thickness of last
scattering surface:
λrec ≡ ctrec 57( 0
(comoving)
z lss ≡ 80 (approx. Gaussian σ ) [15]
−1/2
L lss = 6.6 0 h −1 Mpc
Jeans mass:
Photon damping length:
Photon damping mass:
2 −1 2
0 h ) T2.73
2
−1/2
h )
Mpc
(comoving)
1/2
0
θlss = 3.8
26.15
Mpc
COSMIC MICROWAVE BACKGROUND ANISOTROPIES
The temperature fluctuations can be expanded in spherical harmonics
T
am Ym (θ, φ).
=
T
,m
Theory predicts the rms values of the multipole moments C = |am |2 . On large angular scales
C =
1
2π
H0
c
4
if the fluctuations are adiabatic, and assuming
C =
1
4π
∞
0
0
P(k) 2
j (2ck/H0 ) dk,
k2 = 1. For an n = 1 spectrum P(k) = Ak,
H0
c
4
A
.
( + 1)
The dipole is considered to be due mainly to local velocities, i.e., the motion of the Local Group.
The quadrupole coefficient is related to C2 through Q 2rms /Tγ20 = 5C2 /4π . Data from the COBE
satellite’s DMR experiment [77] are summarized below.
Dipole:
Quadrupole:
rms amplitude:
Best-fit amplitude:
Best-fit slope:
Dobs = 3.372 ± 0.007 mK (95% CL)
toward (α, δ) = (11h 12m ± 0.m 4, −7.◦ 22 ± 0.◦ 08) (1σ ) [11, 12, 78–80]
6 µK < Q obs < 17 µK (68% CL) [81, 82]
σobs (10◦ ) = 29 ± 1 µK; σobs (7◦ ) = 35 ± 2 µK (1σ ) [83]
Q = 18 ± 1.6 µK (1σ, assuming n = 1) [82, 84]
n = 1.2 ± 0.3 (1σ ) [82, 84]
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26.16 L ARGE -S CALE S TRUCTURE / 659
Detections and upper limits on smaller angular scales are still in a state of flux, with many
experimental results at the level of about 1–3 × 10−5 over a range of scales roughly 1 –10◦ , and have
been summarized in [85–87].
Experimental results can be quoted in many ways; one approach is to derive the amplitude of a flat
(i.e., ( + 1)C = constant) power spectrum through the experimental window function, and quote
the equivalent quadrupole. On the largest angular scales COBE fixes Q flat 20 µK. For ∼ 100
several experiments indicate Q flat 40 µK. Upper limits at ∼ 1000 are 30 µK [88]. Detailed C
determinations hold great promise for cosmological parameter estimation.
It is worth repeating here that for z 1 the Hubble radius (c/H (z)) subtends an angle of
1/2
◦
0. 87 0 (z/1100)−1/2 ; if there has been a period of reionization leading to optical depth unity at
some z, then this will give approximately the scale up to which fluctuations could be erased. Without
−1/2
a period of reionization the main “acoustic peak” occurs at 220 0 .
26.16
LARGE-SCALE STRUCTURE
We begin with the definition of the two-point correlation function and its moments, and give some
measured quantities for the two- and three-dimensional variants. The three-point function has also
been measured, but estimates for the higher point functions are still crude [4]. Here R is the Abell
richness class [89] of a cluster.
Two-point correlation fn.:
enhanced probability:
N th moment:
J2 normalization:
J3 normalization:
Power-law fit:
galaxy–galaxy (ξgg ):
optical:
IRAS:
galaxy–cluster (ξgc ):
cluster–cluster (ξcc ):
Angular two-point fn.:
faint galaxies:
ξ(x) ≡ δρ(x + x)δρ(x )/ρ̄ 2
δ P ≡ n̄ [1 + ξ(r )] δV
r
J N (r ) ≡ 0 ξ(x)x (N −1) d x
J2 (r → ∞) = 164e±0.15 h −2 Mpc2 [90]
J3 (10h −1 Mpc) 270h −3 Mpc3 [91]
4π J3 (20h −1 Mpc) = 10 000h −3 Mpc3 [92, 93]
J3 (r → ∞) = 596e±0.21 h −3 Mpc3 [90]
ξ(r ) = (r0 /r )γ
r0 = 5.1 ± 0.2(1σ )h −1 Mpc; γ = 1.71 ± 0.05(1σ )
(over 0.2 Mpc hr 20 Mpc) [91–94]
r0 = 3.76 ± 0.22(1σ )h −1 Mpc; γ = 1.66 ± 0.11(1σ )
(for hr 20 Mpc) [95, 96]
r0 = 8.8 ± 0.4(1σ )h −1 Mpc; γ = 2.21 ± 0.04(1σ )
(for R ≥ 1, over 0.2 Mpc < hr < 10 Mpc) [97]
r0 25 ± 6(1σ )h −1 Mpc; γ 1.8 ± 1.8(1σ )
(for R ≥ 1, over 5 Mpc hr 150 Mpc) [98]
r0 13.2e±0.3 (95% CL)h −1 Mpc; γ 1.9 ± 0.3(95% CL)
(for R 0, over 2 Mpc hr 100 Mpc) [99, 100]
w(θ ) = Aθ 1−γ ;
A = 0.0684 ± 0.0057(1σ ); γ = 1.741 ± 0.035(1σ )
(for m B 18.5, θ < 2◦ , steeper for θ 2.◦ 5) [94, 101, 102]
log ω(1◦ ) = (3.3 ± 0.2) − (0.27 ± 0.01)R(1σ )
(for 18 R 25) [103, 104]
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C OSMOLOGY
Three-point corrln. fn.:
Angular three-point fn.:
ζ = Q [ξ(r12 )ξ(r23 ) + ξ(r23 )ξ(r31 ) + ξ(r31 )ξ(r12 )]
with Q = 0.88 ± 0.11(1σ, weighted average)
(for 100 kpc hr 20 Mpc) [105–108]
z(θ1 , θ2 , θ3 ) = P [w(θ1 )w(θ2 ) + w(θ2 )w(θ3 ) + w(θ3 )w(θ1 )]
with P = 1.56 ± 0.22(1σ ) for θ 3◦ [94]
Galaxies may be clustered more or less strongly than the mass, the difference being described by
the “bias” parameter b, the ratio of rms galaxy fluctuations to rms mass fluctuations. The simplest
mathematical model is linear bias, (δρ/ρ)g ≡ b (δρ/ρ)ρ , i.e., ξgg = b2 ξρρ with b = constant.
However, in principle the relation between galaxy and mass fluctuations may be nonlinear, and the
bias parameter may depend on scale. The correlation function of galaxies implies rms fluctuations of
about unity in spheres of radius 8h −1 Mpc, so
σ8,g 1.0 = bσ8,ρ .
This is commonly used to describe the normalization of the power spectrum. One can also normalize
fluctuations using the J2 or J3 observationally determined values listed above. For a standard ( = 1,
h = 0.5) CDM power spectrum, COBE fluctuations imply [67]
σ8,ρ = 1.22 ± 0.11(1σ )
(scaling approximately as h), whereas for optically selected galaxies [92, 109]
σ8,g = 0.94 ± 0.03(1σ ),
or for IRAS-selected galaxies [95]
σ8,IRAS = 0.69 ± 0.04(1σ ),
and from cluster masses and abundances [73]
σ8,c (0.6 ± 0.1)
−α
0 (1σ ),
with α 0.4 for open CDM and α 0.45 with a cosmological constant.
26.17
DENSITIES
This collection of measurements and estimates of various densities may be useful for quick calculations. Here L ∗ ( 8.5 × 109 h −2 L , or Mb J −19.5) is the characteristic scale of the galaxy
luminosity function, which is well fitted by the Schechter form.
Cluster density:
Galaxy density:
Schechter lum. fn.:
Galaxy surface density:
n cluster 6 × 10−6 h 3 Mpc−3 (R ≥ 1) [98]
n gal 0.01h 3 Mpc−3 (for ∼ L ∗ galaxies) [4]
n gal 4.70 × 10−2 h 3 Mpc−3 (± ∼ 5%)
(for − 15 > Mb J > −22) [92, 93]
φ(L) d L = φ ∗ (L/L ∗ )α exp(−L/L ∗ ) d(L/L ∗ )
with Mb∗J = −19.50 ± 0.13 + 5 log h; α = −0.97 ± 0.15;
and φ ∗ = 0.0140 ± 0.0017(h −1 Mpc)−3 (1σ )
d N /dm = 10−5.70±0.10+0.6m mag.−1 sr−1
(for 14 m B 18) [110]
[92, 93]
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26.18 V ELOCITIES / 661
Faint galaxy density:
Radio gal. surf. density:
Quasar surface density:
Quasar density:
AGN density:
Radio galaxy density:
Gamma-ray bursts:
Optical (B band) light:
Mass-to-light ratio:
Stars in galaxies:
Gas fraction in clusters:
Baryonic matter:
Dynamics, 10h −1 Mpc:
Dynamics, 30h −1 Mpc:
26.18
Ngal 1 500 000 (degree)−2 (VAB 29) [111]
Nrg 100 (degree)−2 (> 1 mJy at 20 cm) [112]
NQSO 95e±0.2 (degree)−2 (1σ ) (B 22) [113, 114]
n QSO 2.7(±0.5) × 10−6 (h −1 Mpc)−3 (1σ )
(M B −26) for z 2 [113, 115]
n AGN ≥ 5%n gal (B < 13.4) [116, 117]
n rg 10−6 (h −1 Mpc)−3
(for P2.7 GHz 1025 W Hz−1 sr−1 with 0 = 1) [118]
rate ∼ 10−6 yr−1 per L ∗ galaxy [119]
j B = φ ∗ (α + 2)L ∗B = (1.76 ± 0.21) × 108 hL Mpc−3 [92, 93, 120]
(M/L)crit = 1600h(±10%)M /L
−1 ±0.5 [121, 122]
∗ 0.0050h e
−3/2 [123]
gas / tot = (0.056 ± 0.014)h
−2
(95% CL) [58–60]
B = (0.007–0.024)h
d = 0.24 ± 0.10(1σ ) [4, 120, 121]
∼ 0.2 to 1 [4, 121]
VELOCITIES
Various dipole and other velocity measurements can only be understood as a vector sum of different
relative velocities between the Sun (), the Galactic Center (GC), the Local Standard of Rest (LSR),
the Local Group (LG), and the Cosmic Microwave Background (CMB) [80]. The error bars here are
all 1σ .
− CMB:
LSR − GC:
− LSR:
− LG:
LG − CMB:
370.6 ± 0.4 km s−1
toward (, b) = (264.◦ 31 ± 0.◦ 17, 48.◦ 05 ± 0.◦ 10) [11, 12, 79, 80]
222.0 ± 5.0 km s−1 toward (, b) = (91.◦ 1 ± 0.◦ 4, 0◦ ) [124]
20.0 ± 1.4 km s−1 toward (, b) = (57◦ ± 4◦ , 23◦ ± 4◦ ) [125]
308 ± 23 km s−1 toward (, b) = (105◦ ± 5◦ , −7◦ ± 4◦ ) [126]
627 ± 22 km s−1 toward (, b) = (276◦ ± 3◦ , 30◦ ± 3◦ ) [78, 80]
Converting from a heliocentric velocity to a galactocentric velocity conventionally involves
VG = VH + 9 cos cos b + 232 sin cos b + 7 sin b
in km s−1 [127].
Data on the relative peculiar velocities of galaxies at projected separation r p indicate
Rel. pec. vel. of galaxies:
540 ± 180(hr p /1 Mpc)(0.13±0.04) km s−1 (1σ )
(for 10 kpc hr p 1 Mpc)
[91, 128]
Pairwise velocities of galaxies on small scales are apparently only consistent with standard Cold
Dark Matter models if the bias parameter is 1.5 b 2.5 [64, 129], but issues relating to biasing are
unresolved [130].
Comparisons of large-scale velocity and density fields tend to measure the parameter β ≡ 0.6 /b.
Current constraints, particularly using IRAS-selected galaxies [131–134], imply β = 0.5–1.2.
Sp.-V/AQuan/1999/10/15:12:47
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662 / 26
26.19
C OSMOLOGY
INTERGALACTIC MEDIUM
The Gunn–Peterson [135] test uses a limit on the integrated Ly α optical depth, shortward of Ly α
emission in quasar spectra, to limit the proper number density:
n H I = 2.42 × 10−11 τGP (z)h(1 + z)(1 +
We quote values for different redshifts using
Gunn–Peterson tests:
(at z 4)
Number density
(comoving):
Helium:
0
1/2
0 z)
cm−3 .
= 1 below:
2 × 10−7 h −1 (1 + z)−2 (1 + 0 z)1/2 [136, 137]
n H I (z = 2.6) < 1.7 × 10−13 h cm−3 (1σ ) [136, 138]
n H I (z = 3.8) < 1.2 × 10−13 h cm−3 (1σ ) [137]
n H I (z = 0.1) < 2.1 × 10−12 h cm−3 (1σ ) [139]
τHe II (z = 3.2) > 1.7 (90%CL) [140]
τHe II (z = 2.4) = 1.0 ± 0.1 (90%CL) [141]
HI
The ionizing background estimates usually derive from the “proximity effect” [142, 143] and are
quoted as the intensity at the Lyman limit
Ionizing flux:
log(Jν )
(for 1.7 < z < 4.1)
= −21.5 ± 0.5 erg cm−2 s−1 Hz−1 sr−1 (1σ )
[143–145]
For the Ly α absorption systems there are several separate classes of systems, with various
parameters determined for each. They tend to be classified by column density into “forest” systems
(1012 cm−2 NH I 1015 cm−2 ), “intermediate” systems (1015 NH I 1020 ) and “damped”
systems (NH I 1020 ). We have attempted to concentrate on some of the more general and robust
quantities, while the specifics have been reviewed elsewhere Note that some quantities may be
extremely sensitive to selection criteria, e.g., the rest equivalent width threshold. Also note that there
is no distinction between a Gunn–Peterson effect and absorption by low NH I clouds.
Ly α cloud
distribution:
Ly α forest
Doppler parameter:
Ly α forest
distribution:
Ly α forest
transverse size:
Ly-limit systems:
Damped Ly α clouds:
−β
d(Prob) = B NH I :
log B = 8–8.5;
cm−2 )
β = 1.5–1.8
(for
NH I [150–154]
√
b ≡ 2σ = (2kT /m)1/2
b 30 km s−1 [151, 155]
dN /dz = A(1 + z)γ :
A = 2–3; γ = 2–3
(for 1014 cm−2 NH I 1016 cm−2 and z 1.6) [153, 156–158]
61h −1 kpc < r < 533h −1 kpc (99% CL)
(assuming spherical clouds) [159]
dN /dz 3.3 at z = 4
(for NH I ≥ 1.6 × 1017 cm−2 ) [160]
−3 −1 (1σ ) [161, 162]
damped (z 2.5) (1.45 ± 0.25) × 10 h
1013
1022
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26.20 E XTRAGALACTIC D IFFUSE BACKGROUNDS / 663
26.20
EXTRAGALACTIC DIFFUSE BACKGROUNDS
Figure 26.2 (adapted from the 1990 overview by [163]) shows a compilation of much of the current data
relevant to the extragalactic background. This should be taken as illustrative of the status at various
wavelengths, and is not meant to be comprehensive. The extreme radio and γ -ray measurements
should be considered as upper limits to a possible extragalactic component. The solid lines on the
plot indicate spectral fits to the microwave, far-infrared and X-ray data. Some convenient estimates for
specific wavebands are listed below. Note that galactic foreground contamination can be considerable,
so that many infrared, optical, and ultraviolet measurements should be treated as upper limits. Another
useful unit here is nW m−2 sr−1 = 10−6 erg cm−2 s−1 sr−1 .
Radio (30 cm):
Sub-mm (400–1000 µm):
Far Infrared (240 µm):
Near Infrared (3.5 µm):
Optical (4400 Å):
Ultraviolet (1600 Å):
X-ray (3 keV):
γ -ray (50 MeV):
ν Iν
ν Iν
ν Iν
ν Iν
ν Iν
ν Iν
ν Iν
ν Iν
6 × 10−11 erg cm−2 s−1 sr−1 [164]
3.4 × 10−6 (λ/400 µm)−3 erg cm−2 s−1 sr−1 [165–167]
= (1.7 ± 0.4) × 10−5 erg cm−2 s−1 sr−1 (95% CL) [10, 168]
< 5.6 × 10−5 erg cm−2 s−1 sr−1 (1σ ) [10, 166]
< 3.2 × 10−5 erg cm−2 s−1 sr−1 (1σ ) [10, 169]
= 0–2.2 × 10−6 erg cm−2 s−1 sr−1 [10, 170]
2.5 × 10−8 erg cm−2 s−1 sr−1 [171]
= (4.5 ± 1.0) × 10−9 erg cm−2 s−1 sr−1 (1σ ) [172]
Figure 26.2. The diffuse extragalactic background.
Sp.-V/AQuan/1999/10/15:12:47
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664 / 26
C OSMOLOGY
The data specifically for the microwave background are listed below. The parameters y, Yff , and µ
describe Compton scattering, free–free emission and Bose–Einstein spectral distortions, respectively.
Microwave background:
Comptonization param.:
Free–free parameter:
Chemical potential:
Tγ 0 = 2.728 ± 0.004 K (95% CL) [11, 12]
4 (1 + z)4 eV cm−3
u γ = 0.26224 T2.73
2 (1 + z)2 Gauss
Bγ (rms) = 3.2496 × 10−6 T2.73
νpeak = 160.50T2.73 GHz
Iν (νpeak ) = 384.94 MJy sr−1 (for T2.73 = 1)
|y| < 1.5 × 10−5 (95% CL) [11, 12]
Yff < 1.9 × 10−5 (95% CL) [173]
|µ| < 9 × 10−5 (95% CL) [11, 12]
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