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The galaxy correlation function and power spectrum
Galaxies are not spread randomly in space. They form groups, clusters, walls
and voids. To measure the deviation from a uniform, random distribution we
introduce the galaxy correlation function ξg (r) as follows: Make a random
choice of two small volumes ∆V1 and ∆V2 . Let the average number density of
galaxies be n = N/V , where N is the total number of galaxies in the volume
V . The probability of finding a galaxy in ∆V1 if the galaxies are uniformly
distributed is
n∆V1
n∆V1
∆V1
=
=
,
(1)
N
nV
V
and in that case the joint probability of finding a galaxy in ∆V1 and another
in ∆V2 is
∆V1 ∆V2
×
.
(2)
∆Puniform =
V
V
If the galaxies cluster, the joint probabilty will differ from this value, and we
decribe the deviation from a uniform distribution by writing
∆P = [1 + ξg (r)]
∆V1 ∆V2
,
V V
(3)
where r is the distance measured from the galaxy in ∆V1 . The galaxy correlation function ξg (r) is positive when galaxies cluster, since the probability of
finding two galaxies within a distance r of each other is then enhanced compared to the case of a uniform distribution. When ξg (r) is negative, galaxies
tend to avoid each other.
The correlation function is measured in galaxy redshift surveys where
one measures the positions and redshifts for a large number of galaxies in a
considerable volume. On scales smaller than 10 h−1 Mpc ξg seems to be well
described by a power law
r
ξg (r) =
r0
−γ
,
(4)
with γ > 0, and where r0 is the so-called correlation length. For r < r0 , the
probability of finding one galaxy at a distance r from another is significantly
larger than for a random distribution.
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But since ξg (r) is the deviation from the average, it must at some point
become negative. That is, if we sit at a fixed galaxy and sum the probabilities
of finding another galaxy at r for all values of r, the probability must be unity,
the same as for a uniform distribution. Looking back at the definition of ξg ,
this means that its integral over all space must vanish:
Z
3
ξg (r)d r = 4π
Z
∞
ξg (r)r 2 dr = 0.
0
(5)
which can only be fulfilled if ξg (r) < 0 over some range in r.
The 2-degree-Field Galaxy Redshift Survey (2dFGRS) carried out in the
early 2000s found r0 ≈ 5 h−1 Mpc and γ ≈ 1.7 for a sample of more than 200
000 galaxies. On the scales of the largest voids and walls, about 50 h−1 Mpc,
the correlation function oscillates around 0, meaning that the galaxy distribution becomes uniform on very large scales.
The power spectrum of the galaxy distribution is the Fourier transform of
the correlation function:
Pg (k) =
Z
d3 rξg (r)e−ik·r .
(6)
Since ξg (r) is independent of the direction of r, at least by assumption, we
can carry out the angular integrals, taking k to be along the z-axis so that
k · r = kr cos θ:
Pg (k) =
Z
2π
0
= 2π
Z
dφ
∞
Z
0
π
sin θdθ
2
drr ξg (r)
0
Z
Z
0
0
π
∞
drr 2 ξg (r)e−ikr cos θ
dθ sin θe−ikr cos θ
ikr
1
dxe−x
ikr −ikr
0
Z ∞
sin(kr)
,
= 4π
drr 2ξg (r)
kr
0
= 2π
Z
∞
drr 2ξg (r)
Z
(7)
where I have used the substitution x = ikr cos θ and the relation sin v =
(eiv − e−iv )/2i. Note that in the limit k → 0, sin(kr)/kr → 1, and by
equation (5) Pg (k) vanishes:
Pg (k = 0) = 4π
Z
0
2
∞
drr 2 ξg (r) = 0.
(8)
For a power-law ξg (r) the power spectrum will also be a power law, at
least approximately. To see this, take
r
ξg (r) =
r0
−(3+n)
,
(9)
with n < 0. Then
Pg (k) = 4π
1
r0
−(3+n) Z
∞
0
drr 2r −(3+n)
sin(kr)
.
kr
(10)
As a crude approximation we take sin(kr)/kr ≈ 1 for kr < 1 and sin(kr)/kr ≈
0 outside this range. Then
1
Pg (k) ≈ 4π
r0
−(3+n) Z
0
1/k
drr
−1−n
4π 1
=
|n| r0
−(3+n)
kn.
(11)
For 3 + n = γ = 1.5, that is n = −1.5, we therefore have
Pg (k) ∝ k −1.5 ,
(12)
for scales r < 10 h−1 Mpc, that is, k > 0.1 h Mpc−1 . Note that this corresponds to non-linear scales.
Biased galaxy clustering
Most of the mass int the universe is dark matter. But the visible parts of
galaxies are made of baryouns. Although there clearly has to be some relation
between the clustering properties of dark matter and those of the baryons,
they need not be identical. As a matter of fact, simulations and observations
show that they are not. The relation between the galaxy overdensity field
and the dark matter overdensity field is often parametrized by the bias b,
defined by
δg
b= .
(13)
δ
Generall speaking, b will depend on both the length scale at which we study
the two distributions and on the cosmic epoch. However, for z ≈ 0 and on
large scales we expect b ≈ constant. A value b > 1 means that galaxies
cluster more strongly than the dark matter.
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Peculiar velocities
Local concentrations of matter will exert a gravitational force on their surroundings, resulting in accelerations and deviations from the Hubble flow.
We call the deviations from the Hubble flow peculiar velocities and define
them by
v = Hx + a(t)u = Hx + δv(x, t).
(14)
In the course of deriving the equation for the time evolution of δ(t) in Newtonian perturbation theory in Cosmology I, one finds the relation
δ̇ = −∇c · u,
(15)
where the subscript c stands for comoving coordinates. In Fourier space this
relation can be written as
ik
(16)
δ̇k (t) = − u,
a
where u is the component of u along k. Solving for u we find
u=
i d
a δk .
k dt
(17)
At late times we have seen that we can write δk (t) = T (k)∆1 (t) where T (k)
is the transfer function and D1 (t) is the growth function. This means that
δk /D1 is independent of time, and we can rewrite equation (17) as
i d
u= a
k dt
!
δk
iaδk dD1
D1 =
.
D1
kD1 dt
(18)
It is common to write u in terms of the linear growth rate f , defined as
f=
d(ln D1 )
a dD1
=
.
D1 da
d(ln a)
(19)
By rewriting dD1 /dt we find
dD1
dD1 da
dD1
=
= aH
.
dt
da dt
da
(20)
We then find that we can write
u=
i
a dD1
if aHδk (a)
δk (a)aH
=
.
k
D1 da
k
4
(21)
A good approximation to f is
f = Ωm (a)0.6 ,
(22)
with Ωm (a) = Ωm0 a−3 /H 2 (a). At low redshifts, where a ≈ 1 we get
1
u = if H0 δk .
k
(23)
It can be shown that u has no component perpendicular to k, so we can write
u = if H0 δk
k
,
k2
(24)
by which we have now related the velocity perturbation at low redshifts to the
density perturbation δk . Note that we started from an expression derived in
linear pertubation theory, which means that this expression for the peculiar
velocity field is also only valid to linear order.
Redshift space distortions
The galaxy distribution is, as mentioned earlier, measured in redshift surveys
where one measueres the angular position on the sky (φ, θ) and uses the
redshift z as a proxy for the distance. For low z we can convert the redshift
to a distance through
cz
,
(25)
χs (z) =
H0
resulting in a given galaxy having Cartesian coordinates
xs =
cz
(sin θ cos φ, sin θ sin φ, cos θ).
H0
(26)
This estimate of the distance is off by 50% at z = 1. For deep surveys like
the Euclid satellite scheduled for launch in 2019 this is a problem, but for
older surveys like 2dFGRS and the Sloan Digital Sky Survey (SDSS) which
only went out to z ≈ 0.2–0.3 this is less of a concern. The coordinates in
(26) define what is known as redshift space.
The main problem is, however, that we have neglected peculiar velocities.
The clustering of matter will, as we have just seen, cause deviations from the
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Hubble flow. In the present context this means that the redshifts we measure
will have contributions from peculiar velocities, and so if we naively plug
the observed redshift into equation (25) we will over- or underestimate the
distance to a given galaxy. In the linear regime, this will cause an originally
spherical distribution of galaxies to look flattened along the line of sight.
In the non-linear regime, it will cause it to look elongated. These effects
are called redshift space distortions. We will look at the linear scales in the
following and work out the relation between the power spectrum in redshift
space and the underlying true power spectrum in real space.
As a starting point we note that redshift space distortions cannot change
the number of galaxies in a given region, so the redshift space number density
ns and its real-space counterpart n must satisfy
ns (xs )d3 xs = n(x)d3 x.
(27)
The volume elements on each side of this equation are given by
d3 xs = dxs x2s sin θdθdφ
d3 x = dxx2 sin θdθdφ,
(28)
(29)
so since the angular parts are the same we can write
ns (xs ) = n(x)J,
(30)
where the Jacobian of the transformation from redshift space to real space
is given by
dxx2
.
(31)
J=
dxs x2s
We can write the observed redshift as
z = H0 x + u · x̂,
(32)
where x̂ is the unit vector in the direction of x, since only the component
of u along the line of sight will contribute to the measured redshift. Using
equation (25) and setting c = 1 to save some writing, we have
!
u · x̂
u · x̂
xs = x +
=x 1+
,
H0
xH0
6
(33)
and we can write the Jacobian as
"
∂
J = 1+
∂x
u · x̂
H0
!#−1
u · x̂
1+
xH0
!−2
.
(34)
We can now argue that the second term in the first factor is much more
important than the second term in the second. Working in Fourier space,
the spatial derivative will bring down a factor k, so this term is of order
ku/H0 . The other term is of order u/H0x, so their ratio is of order kx. The
value of x will be of order the size of the survey, and the wavenumber k will
range over the Fourier modes measurable by the survey. But modes k of
order 1/x will be poorly measured, because the statistics will be very thin.
There are many modes with k ≫ 1/x, we can get good statistics on those, so
these will be the interesting ones. In other words, the modes we can measure
well have kx ≫ 1, so for these the second term in the first factor is much
more important than the second term in the second. Hence, we can expand
J as
!
∂ u · x̂
J ≈1−
.
(35)
∂x H0
In terms of the overdensities we can write n = n(1 + δ), and ns = n(1 + δs ),
where n is the mean density of galaxies (which is the same in real space and
redshift space, even though the clustering properties vary between the two.)
We therefore have, working to first order,
"
∂
1 + δs (x) = (1 + δ(x)) 1 −
∂x
So
u · x̂
H0
!#
∂
δs (x) = δ(x) −
∂x
∂
=1−
∂x
u(x) · x̂
H0
!
!
u · x̂
+ δ(x).
H0
(36)
(37)
where I in the last term have reminded ourselves that the peculiar velocity
u depends on the position x.
To proceed we now invoke what is known as the distant observer approximation. It rests on the fact that the distance from the observer to the
galaxies in a survey will usually be much larger than the distance between
the galaxies. This means that x will vary little from galaxy to galaxy, and
we can approximate u · x̂ with u · ẑ, where ẑ is a fixed unit vector pointing to
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the center of the survey region. In this approximation we can compute the
Fourier transform of the redshift space overdensity as
δ̃s (k) =
Z
3
−ik·x
d xe
"
∂
δ(x) −
∂x
u(x) · ẑ
H0
!#
.
(38)
For linear peculiar velocities we found in Fourier space
u(k) = if H0 δ̃(k)
k
.
k2
(39)
We therefore find
δ̃s (k) =
=
=
=
d3 k ′ ik′ ·x ′ k′
e
δ̃(k ) ′2 · ẑ
δ̃(k) − if d xe
(2π)3
k
Z
Z
′
3 ′
d k ik′ ·x ′
′ k
e
ik
·
x̂
δ̃(k
)
· ẑ
δ̃(k) − if d3 xe−ik·x
(2π)3
k ′2
Z
Z
d3 k ′
′
′ ′2
′
2 1
δ̃(k) + f
δ̃(k )k (k̂ · ẑ) ′2 ei(k −k)·x
3
(2π)
k
Z
3 ′
dk
δ̃(k′ )(k̂′ · ẑ)2 (2π)3 δ (3) (k′ − k)
δ̃(k) + f
(2π)3
Z
3
−ik·x
∂
∂x
Z
= δ̃(k) + δ̃(k)f (k̂ · ẑ)2
= [1 + f (k̂ · ẑ)2 ]δ̃(k)
≡ [1 + f µ2k ]δ̃(k),
(40)
where I have used the definition of the three-dimensional delta function
δ (3) (k′ − k) in terms of a Fourier transform, and I have defined µk = k̂ · ẑ
which is the cosine of the angle between the line of sight and the wavevector
k. The growth rate f is always positive, so f µ2k ≥ 0, which means δ̃s ≥ δ̃,
and the enhancement is greatest when the line of sight is parallell to the
wavevector.
Since the power spectrum is proportional to hδ̃ 2 i, we can write the galaxy
power spectrum in redshift space in terms of the galaxy power spectrum in
real space as
Pg,s (k) = Pg (k)[1 + βµ2k ]2 .
(41)
I have here introduced the bias b through
β=
f
Ω0.6
≈ m0
b
b
8
(42)
which follows from the fact that the peculiar velocity is proportional to the
matter overdensity δ, and the bias is defined as b = δg /δ, so u ∝ δ ∝ δg /b.
A consequence of equation (42) is that the redshift space distortion parameter β can be measured by determining the ratio of the quadrupole moment to the monopole moment of the galaxy distribution. A moment of order
2ℓ of a quantity is in this context defined as
2ℓ + 1
A (k) =
2
(ℓ)
Z
1
Pℓ (µk )A(k),
(43)
−1
where Pℓ (x) is the Legendre polynomial of order ℓ. In one of the exercises
you will show that
8
1
4
4
2
β + β 2 P2 (µk ) + β 2 P4 (µk ).
(1 + βµ2k )2 = 1 + β + β 2 P0 (µk ) +
3
5
3
7
35
(44)
The Legendre polynomials are orthogonal:
Z
1
dxPℓ (x)Pℓ′ (x) =
−1
2
δℓℓ′ ,
2ℓ + 1
(45)
and the ones we need are
P0 (x) = 1
3 2 1
x −
P2 (x) =
2
2
35 4 15 2 3
P4 (x) =
x − x + .
8
4
8
(46)
(47)
(48)
Using the orthogonality relation (45) it is then easy to show that the ratio
of the quadrupole moment to the monopole moment is given by
(2)
Pg,s
(0)
Pg,s
=
1
4
β + 47 β 2
3
.
+ 32 β + 15 β 2
(49)
When we estimate the galaxy power spectrum from a redshift survey,
we carry out an average over all directions. This corresponds to taking the
monopole moment of Pg,s (k), so
(0)
Pg,s
(k)
2
1
= 1 + β + β 2 Pg (k).
3
5
9
(50)
Measurements of β indicate that β ∼ 0.7, meaning that the redshift space
galaxy power spectrum overestimates the real space galaxy power spectrum
by 60% . And the relationship between the two may be even more complicated than this. We have made a number of assumptions in deriving equation
(50): Linear perturbation theory, low redshifts, and the distant observer approximation to name a few. And there is still the bias between the galaxy
distribution and the dark matter distribution to account for. Redshift surveys
have been, and are still, and important tool for learning about the matter
perturbations, but the results need to be interpreted with caution and care.
It is therefore important to develop observational techniques which can probe
the dark matter perturbations more directly.
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