AST4320: LECTURE 10
M. DIJKSTRA
1. The Mass Power Spectrum P (k)
1.1. Introduction: the Power Spectrum & Transfer Function. The power spectrum
P (k) emerged in several of our previous lectures:
• It fully characterised the properties of a Gaussian random (density) field.
• It determined σ(M ) in Press-Schechter theory.
• We constrain(ed) the slope d log P (k)/d log k from the observed galaxy two-point
correlation function in assignment 5.
Now we will discuss some key properties of P (k). Linear theory allowed us to describe
the time evolution of a density perturbation δ with wavenumber k. This time evolution
was
(1)
δ(k, t) = δ(k, t = 0)D+ (t),
where we derived that D+ (t) ∝ t2/3 ∝ a in the Einstein-de Sitter Universe (Ωm = 1.0,
ΩΛ = 0.0). Therefore,
(2)
2
2
P (k, t) = P (k, t = 0)D+
(t) ≡ P0 (k)D+
(t).
One of the goals of modern cosmology is to calculate P0 (k).
There are no preferred length scales in the very early Universe, and the only functional
form for P0 (k) with no scale length is a power law:
(3)
P0 (k) = Ak n .
We may think of this P0 (k) as the ‘primordial’ power spectrum as it described density fluctuations at t = 0. As we will see next, perturbations with different wave numbers evolved
differently in the very early Universe. This modifies the matter power spectrum from the
power law form given above. These modifications are encoded in the so-called ‘transfer
function’ T (k). This transfer function T (k) encodes the information on the evolution of
some density perturbation δ(k), and therefore affects the power spectrum as
(4)
P0 (k) = Ak n T 2 (k) .
Our goal is now to obtain some intuition for T (k), and expressions for some limiting
behaviour of T (k). Before proceeding it is useful to discuss the concept of a ‘horizon’, as
it plays a key role in shaping the transfer function.
1
2
M. DIJKSTRA
1.2. The Horizon. There are many horizons in cosmology. We focus on the so called
’particle horizon’, which corresponds to the ‘maximum proper distance over which there can
be causal communication at time t’ (Quote lifted from M. Longair’s book). The particle
horizon thus corresponds to the maximum proper distance a photon could have travelled
between t = 0 and t = t. This distance corresponds to
Z t
cdt0
.
(5)
rH (t) = a(t)
0
0 a(t )
The integral gives the total comoving distance traversed by the photon. The term a(t)
converts that into a proper distance at time t. During radiation domination the scale factor a(t) ∝ t1/2 ≡ Ct1/2 , where C is a constant. We can solve the integral to give rH = 2ct.
This is a factor of two larger than the maximum distance a photon could travel in a static
medium. The factor of 2 accounts for the fact that space itself is expanding during the
photon’s flight.
Now consider the evolution of a perturbation of some proper length/wavenumber L.
The time evolution of the wavelength of this perturbation is given by L = L0 (a/a0 ) =
L0 (t/t0 )1/2 . Consider a perturbation for which L > rH : because L > rH there is no way to
know what lies outside of L. It is therefore impossible to determine what the mean density
of the background should be, and therefore whether the perturbation L is overdense or
under dense. We need general relativity to describe the time evolution of perturbations
larger than the horizon scale. I will comment on this later.
While we currently cannot say much about the expected time evolution of the perturbation L, we can predict that L will become smaller than the horizon in the future: If L > rH
at some time t1 , then L2 = L(t2 /t1 )1/2 at some later time t2 while the horizon at this time
rH,2 = rH (t2 /t1 ). The perturbation size equals the horizon scale when
(6)
rH (t2 /t1 ) = L(t2 /t1 )1/2 ⇒ rH (t2 /t1 )1/2 = L ⇒ t2 = t1 (L/rH )2 > t1 .
After the perturbation enters the horizon we can apply our classical (non-relativistic) perturbation theory.
1.3. The Transfer Function T (k). Figure 1 shows the time evolution of a perturbation
δ(k) (with corresponding wavelength or ‘size’ of the perturbation L = 2π/k) that enters
the horizon during the radiation dominated era at scale factor aenter ). There are three key
events during the evolution of this perturbation:
• When the matter density starts to dominate the Universal energy density - this
happens at the redshift of matter-radiation equality zeq ∼ 24000 - the dark matter
perturbation grows as δ ∝ a. This is what we derived in previous lectures.
• At redshifts z > zeq - i.e. a < aeq - the Universal energy density is dominated
by radiation. During radiation domination the scale factor grows as a ∝ a1/2 (as
opposed to a ∝ t2/3 that we found during matter dominance). This different time
AST4320: LECTURE 10
3
Figure 1. This Figure shows (schematically) the time evolution of an overdensity δ on some scale L that enters the horizon at aenter aeq . The
time evolution goes through three different phases: (i) δ ∝ a2 before horizon entry which follows from general relativistic perturbation theory; (ii)
δ =constant after horizon entry, and up until aeq . This stalling of the growth
of the perturbation is known as the Meszaros effect; (iii) when matter starts
to dominate the Universal energy density δ ∝ a as we derived in previous
lecture. This Figure illustrates that the Meszaros effect suppress the growth
of this perturbation by a factor of (aenter /aeq )2 compared to uninhibited
growth.
dependence of the scale factor - combined with the fact that radiation dominates
the Universal (mass-)energy density - gives rise to a drastically different predicted
time evolution for δ. As we will see next, δ barely grows at all during radiation
dominance. This ‘stalling’ of the growth of density perturbation in the radiationdominated era is known as the ‘Meszaros’ effect. The fluctuations are said to be
frozen in the background. Mathematically, the Meszaros effect is easy to understand. Recall that the density evolution of a perturbation δ was given by the
following differential equation:
(7)
ȧ
δ̈m + 2 δ̇m = 4πGρm δm .
a
Divide both sides by H 2 =
8πGρtot
,
3
where ρtot = ρrad + ρm . Using H =
ȧ
a
we find
4
(8)
(9)
(10)
M. DIJKSTRA
δ̈m
2
3ρm δm
+ δ̇m =
.
H2 H
2[ρm + ρrad ]
If we now use that deep in the radiation dominated era ρrad ρm , then the term
on the RHS can be ignored. This is because we are multiplying a small number
δ with another small number, and can see this term effectively as a second order
term. The differential equation then simplifies to
δ̈m
+ 2δ̇m = 0.
H
If we further use that H = ȧ/a = 1/[2t] then we are left with
δ̇m
= 0 ⇒ δm = A + B log t = A + C log a.
t
The perturbation thus only grows logarithmically with the scale factor. This growth
is represented by the horizontal line in Figure 1.
δ̈m +
• Before the perturbation enters the horizon, at a < aenter , it grows δ ∝ a2 . As I
mentioned in the lecture, this follows from general relativistic perturbation theory.
This is beyond the scope of this lecture. Because the result from GR is not intuitive, we might as well have replaced the words ‘general relativistic perturbation
theory’ with ‘magic’ 1.
Figure 1 shows that for the perturbation that entered the horizon during radiation
2
. This
dominance at aenter the growth was suppressed by a factor of T (k) = aenter
aeq
suggests that T (k) → 1 if aenter → aeq . Indeed, for perturbations that enter the horizon
during matter domination we do not have any inhibition of growth: δ ∝ a2 during radiation
dominance, and δ ∝ a during matter dominance (see footnote). This discussion clearly
suggest that there is a particular scale of interest, namely that for which aenter = aeq .
This corresponds to the smallest scale for which there is no suppression of growth by the
Meszaros effect. The size of this perturbation is therefore equal to the horizon scale at
matter-radiation equality:
Z teq
cdt
c
1
p
(11) L0 = rH (aeq ) = aeq
= ...
=
≈ 80 cMpc,
assignment
5
a(t)
H
2Ωm,0 zeq
0
0
1I found a prescription later today that might provide some more insight than requiring magic: density
perturbation δk generate perturbations in the gravitational potential Φk which correspond to metric perturbation in general relativity (matter warps space time). The equation that described the time evolution
of this metric perturbation corresponds to the perturbed Poisson equation from lecture 2, which in Fourier
space reads −k2 Φk = 4πGa2 ρ̄δk . If we require that the metric perturbation cannot evolve for perturbations
outside the horizon, then we must have that a2 ρ̄δk is independent of time. We therefore must have δk ∝ a2
during radiation domination (ρrad ∝ a−4 ), and δk ∝ a during matter domination (ρrad ∝ a−3 ).
AST4320: LECTURE 10
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where Ωm,0 denotes the present-day mass density parameter, H0 is the present-day Hubble constant, and ‘cMpc’ denotes comoving Mpc (just to emphasis that L0 is a comoving
quantity). The corresponding wavenumber is k0 = 0.1 cMpc−1 .
Finally, we would like to express T (k) as a function of k. We found that the suppression
2
. A perturbaT (k) for perturbations entering at a < aeq was given by T (k) = aenter
aeq
tion of length L L0 enters the horizon at aenter (L) ≈ aeq (L/L0 ) = aeq (k0 /k), where in
the first approximation we assumed that the horizon scale evolves as rH ∝ a out to aeq .
2
Under this approximation we therefore have T (k) = kk0
for k k0 . One of the slides
shows the T (k) obtained from a more precise calculation (Bardeen et al. 1986). I have
also explicitly plotted the slope d log T /d log k. The approximate calculation given above
captures the limiting behaviour of T (k) and identifies the physical reason for the turn-over
in the transfer function.
1.4. The Power Spectrum P (k). The power spectrum is given by P0 (k) = Ak n T 2 (k).
We have specified T (k). The slope of the ‘primordial’ power spectrum has been inferred
(from the Cosmic Microwave Background) to be close to 1. This value is predicted naturally
by inflation theories. The power spectrum therefore scales as
k
k k0
(12)
P (k) ∝
−3
k
k k0 ,
with a turn-over at k = k0 .
Several additional comments on P (k)
• The normalisation constant A is obtained by matching to observations of the Cosmic Microwave Background.
• Baryons affect precise shape of P (k) (see paper by Eisenstein & Hu 1998). One
example of how baryons affect the mass power spectrum was given in our discussion
of the acoustic peak in the two-point correlation function (the single acoustic peak
in ξ(r) corresponds to a series of oscillations in the power spectrum).
• The case n = 1 corresponds to a special case which yields scale-invariant fluctuations. This is discussed next.
1.5. Why n = 1 corresponds to Scale Invariance. We derived the relation between
P (k) and the variance in the mass density field averaged over some mass-scale M in previous
lectures. The RMS (root mean square) amplitude of fluctuations smoothed over mass-scale
M is given by
p
σ 2 (M ) ≡ ∆(M ) ∝ M −(n+3/6) ∝ M −2/3 .
(13)
n=1
For a < aenter δ ∝
(14)
a2 ,
so
∆(M ) ∝ a2 M −2/3 .
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M. DIJKSTRA
This equation shows that ∆M is smaller for larger M . However, higher masses correspond
to larger scales. These higher mass fluctuations therefore enter the horizon at a later time,
and the Meszaros effect limits their growth by a smaller factor. We can relate the mass M
of a perturbation to the scale factor at which it entered the horizon as
(15)
3
M = Mh (t) ∝ ρm rH
∝ a−3 t3 ∝ t−3/2 t3 ∝ t3/2 ,
where we used that the mass density ρm ∝ a−3 and that the proper horizon scale scales as
rH ∝ t. We have derived the time-dependence of horizon entry of fluctuations of mass M .
Substituting this into Eq 14 we have
(16)
∆(M ) ∝ a2 M −2/3 ∝ a2 t−1 = constant.
Fluctuations that enter the horizon2 at a < aenter therefore have the same RMS amplitude
at horizon entry. The subsequent growth of these perturbations is stalled until aeq , after
which they all grow as δ ∝ a. This identical growth ensures that the RMS amplitude of
these fluctuations remains independent of scale at all times. Although I have not shown
it in the lecture, you can do the same analysis for perturbations that enter the horizon
at a > aenter and get the same result: namely that ∆(M )=constant! This shows that
all fluctuations enter the horizon with the same RMS amplitude. This remarkable scale
invariance is a special property of the power spectrum with n = 1.
1.6. Some Concluding Remarks. A very brief & broad summary of the processes that
are relevant for the formation of structure in our Universe.
• At time t ∼ 0 some process (inflation) generates the primordial power spectrum
P (k) = Ak n with n ∼ 1.
• Processes like horizon entry of a perturbation combined with the Meszaros effect
then modify the shape of the power spectrum into P (k) = Ak n T 2 (k) at a ≤ aeq
where T (k) = 1 for k k0 ∼ 0.1 cMpc−1 , and T (k) = (k0 /k)2 for k k0 . Baryons
provide further smaller modifications of this power spectrum.
• At aeq < a < arec dark matter perturbations grow as δ ∝ a, while radiation pressure
prevent baryons from collapsing on all scales smaller than the Jeans length (which is
very large during this epoch). Acoustic waves generated by density perturbations
permeate the primordial photon-baryon plasma, which introduce further smaller
corrections to the power spectrum P (k).
• At a > arec the sound speed drops by five orders of magnitude, and the consequently,
the Jeans mass drops by 10 orders of magnitude. Baryons are now free to collapse
into the potential wells generated by the dark matter (i.e δb → δDM ). This is
important: the RMS amplitude of the density fluctuations in baryons at arec is
only δb ∼ 10−5 , which would not be enough to form non-linear objects (recall that
δ ∝ a and that a increases only by a factor of 103 ).
• Press-Schechter theory allows us to take a Gaussian random density field - which
describes the density field post recombination still extremely well - and transform
2Convince yourself that this corresponds to fluctuations on mass-scales that are relevant for astrophysical
objects (galaxies, groups of galaxies, clusters of galaxies).
AST4320: LECTURE 10
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this into predictions for the abundance (number density) and clustering of (nonlinear) collapsed objects. While this theory has many flaws, it predicts the number
density of collapsed objects as a function of mass M and redshift z remarkably well.
It also highlights the important hierarchical aspect of structure formation: namely
that small (i.e. low mass) objects collapse first, and that larger (i.e. more massive)
objects collapse later.
1.7. Shortcomings/Caveats. Our discussion of structure formation does not explain the
most apparent visual appearance of structures in the Universe, namely the walls, filaments,
and nodes that are apparent in observed galaxy distributions and numerical simulations
(see slides). The main reason is that in our discussion of the non-linear growth of structure
we focussed on spherical top-hat model. In this model the evolution of the perturbation
is determined entirely by its radius R(t) and the overdensity inside of it (δ(t)). Most
R1
R2
Figure 2. Geometry for a simple ellipsoidal perturbation.
perturbations in Gaussian random fields are not spherical. In fact, Gaussian random field
theory can be used to study shapes of peaks in the density field. Consider the simple case in
which an overdensity is ellipsoidal instead of spherical, and that there are two characteristic
axes denoted with R1 and R2 . Outside of this ellipsoid the overdensity is δ = 0. Clearly,
the mean overdensity inside the sphere of radius R1 is larger than that inside the sphere
of radius R2 . This implies that less growth in δ is required for the perturbation reach the
critical linear overdensity for collapse δcrit = 1.69. The structure therefore collapses along
its R1 axes while it still has a finite size in its R2 direction. Gravity therefore amplifies
these deviations from spherical symmetry into flattened objects like walls (collapse along
1 axis), filaments (collapse along two axes), and halos (collapse along all three axes). So
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M. DIJKSTRA
while our discussion of structure formation in previous lectures does not produce these
observed & simulated features in the mass distribution, the theory is easily adjusted to be
able to explain these structures.
1.8. Useful Reading. Useful reading material includes
• For the discussion of Transfer functions I used the book by Peter Schneider ’Extragalactic Astronomy and Cosmology: An Introduction’.
• Useful discussions on horizon scales in cosmology, and the difficulty of modelling
super horizon scales are given in Longair ’Galaxy Formation’. Specifically Chapter
12.2 + 12.3.
• The lectures by Frank van den Bosch have some very nice & clear slides. See
http://www.astro.yale.edu/vdbosch/astro610_lecture4.pdf for a very brief
discussion why δ ∝ a2 before horizon entry. His site also has references to his book
(Mo, Van den Bosch & White) which provides much more details to these lectures.
Can be very useful.