Density fluctuations grow too slowly to explain structures such as
galaxies and clusters in an expanding universe *unless* some
source of `primordial' perturbations can be identified. The simplest
available hypothesis is that the primordial fluctuation spectrum
produces bound structures with length and mass scales growing in
direct proportion to the cosmological horizon and the mass contained
therein. As a high-density universe expands and cools, the changing
equation of state imprints a characteristic scale of `~10^16`
solar masses on the primordial spectrum; fluctuations on smaller
scales are highly damped in a `hot' dark matter universe and survive
in modified form in a `cold' dark matter universe. The latter choice
seems closer to the observations but a unique and well-specified model
eludes us.

The mass distribution in the universe is *highly* irregular on
small scales but much smoother on larger scales. It seems natural to
imagine that the structures we see evolved through a process of
*hierarchical clustering* in which small objects, forming first,
were then incorporated into successively larger objects (Layzer
1954, Peebles 1980, Ch. 26).

A clustering process in which small structures form first will
develop if density fluctuations are larger on small scales than they
are on large scales. The details of this process depend on the
relative amplitude of fluctuations on different scales. As in the
last lecture, we expand `delta(x,t)` in a Fourier series. It
is plausible to assume that the Fourier components `delta_k`,
where `k` is the wave-vector, have random phases. The central
limit theorem then implies that `delta(x)` has a Gaussian
distribution about its mean value of `0`. The statistical
properties of `delta(x)` are *completely*
specified by the power-spectrum, `P(k) = |delta_k|^2`.

Although `P(k)` can theoretically be an arbitrary function
of `k`, it is convenient to adopt a power-law model, in which
the statistical properties of the fluctuation field are invariant --
up to a constant factor -- under a change of scale. As long as the
fluctuations on a given scale are linear (*i.e.*, `|delta_k|
< 1`), they will simply grow in place. Thus the power spectrum
has the form

2 2 n (1) |delta_k| ~ D(t) k , -3 < n < 4 ,where the linear growth factor, calculated last time, is

2/3 (2) D(t) ~ a ~ t .

To quantify the density variations, imagine measuring the mass
within a sphere of proper radius `r`. If the density is
perfectly uniform, the expected mass within the sphere is `M =
(4pi/3) r^3 rho_b`. Let `delta M` be the difference
between the *actual* mass and the expected mass `M`. Then
the squared value of the relative density variation is

/ 2 | 3 2 2 (3) < (delta M/M) > = | d k W_M(k) |delta_k| , | /where

2 -1-n/3 (4) < (delta M/M) > ~ D Mfor fluctuations on a mass scale

When the density fluctuations on a particular scale have grown to
nonlinear amplitude (`< (delta M/M)^2 > > 1`) they `break
away' from the general expansion and separate out as bound objects.
From Eq. (4), the characteristic mass scale of bound objects forming
at time `t` is

6/(3+n) 4/(3+n) (5) M*(t) ~ D(t) ~ t .Note that the clustering scale grows rapidly for

In the hierarchical clustering model above the power-law index
`n` is a free parameter. Without knowing much about the source
of fluctuations in the very early universe, we can pick out a value of
`n` by demanding that the clustering scale grow in proportion
to the horizon scale (Harrison 1970, Zel'dovich 1972). The proper
radius `r_H` of the horizon grows in proportion to the proper
time `t`, while the proper density scales like `rho_b ~
t^-2` in a matter-dominated universe, so the mass within the
horizon is

3 (6) M_H(t) ~ rho_b(t) r_H ~ t .Comparing with Eq. (5), we see that

(7) M* ~ M_H => n = 1 .This is known as the

This spectrum is *predicted* by theoretical explanations for
the origin of fluctuations in the early universe precisely because it
does not pick out any later time as special. For example, if `n
< 1` then the clustering mass `M*` grows faster than the
horizon mass `M_H`, and at some point in the future the two
scales will become comparable, while if `n > 1` then the two
scales must have been comparable at some point in the not-too-distant
past. In either case, when `M*` is comparable to `M_H`
bound objects collapse at the speed of light and a substantial
fraction of the universe goes into black holes. There is nothing in
general relativity to say that this *cannot* happen, but any
theory in which it *does* happen carries the burden of explaining
*when* it happens.

Observations of temperature fluctuations in the microwave background
are consistent with a scale-invariant spectrum with an amplitude on
the horizon scale of `~10^-4`. An amplitude of this order also
emerges fairly naturally from the inflationary picture (*e.g.*
Peebles 1993, Ch. 17). But the galaxy clustering pattern we observe
is *not* consistent with a simple extrapolation of the
scale-invariant spectrum to smaller scales. However, it is plausible
that processes at relatively `low' redshifts (*e.g.* `z ~
10^4`) have modified the spectrum on small scales; some possible
models are outlined in the next section.

At present the microwave background makes only a small contribution
to the total mass-energy density of the universe, but because the
radiation density scales as `a^-4` while the matter density
scales as `a^-3`, radiation must have been dynamically
important in the past.

If radiation is included, two varieties of fluctuations can be
distinguished. In *adiabatic* fluctuations, the densities of
matter and radiation vary in synchrony, while in *isothermal*
fluctuations only the matter density varies, while the radiation
density is constant. Adiabatic fluctuations arise rather naturally in
models of the early universe. To obtain isothermal fluctuations, on
the other hand, requires an effect which can modulate the amount of
mass associated with a given radiation density; this is not very
easily arranged. Thus it seems plausible to assume that the
primordial fluctuations were adiabatic.

In the standard CDM model we assume that the dark matter is
composed of particles with initially low velocity dispersions. The
transition from a radiation-dominated to a matter-dominated universe
at time `t_eq` imprints a characteristic scale on the spectrum
because fluctuations which come within their horizon before
`t_eq` do not grow until after `t_eq`, while
fluctuations which do not come within their horizon until after
`t_eq` enjoy uninterrupted growth. At very small scales, the
effect is to multiply the power spectrum by a transfer function
scaling as `k^-4`, converting a primordial `n=1`
spectrum to one with an index `n=-3`. In fact, the transition
between these regimes is *very* gradual, and the effective
spectral index on the scales associated with galaxies is roughly
`n=-2` (Peebles 1982, Blumenthal *et al.* 1994).

In the HDM model the dark matter is assumed to be composed of
neutrinos with masses of a few tens of `eV`. While the
temperature of the universe exceeds the neutrino rest mass, the
neutrinos have relativistic velocities and effectively smear out
fluctuations on scales smaller than the horizon. Thus fluctuations on
small scales are effectively erased, while those on scales large
enough to come within the horizon only after the neutrinos have become
nonrelativistic are left unchanged (*e.g.* Bond & Szalay
1983). The characteristic scale of the smallest surviving
fluctuations is comparable to that of superclusters; then in a HDM
universe these structures form first as in the `pancake' theory
(Zel'dovich 1970).

Which type of dark matter is preferred? The HDM picture was the
first to predict that the large-scale galaxy distribution might
exhibit a `cellular' structure similar to the filaments, walls, and
voids described in recent redshift surveys. But in this picture
galaxy formation is delayed until redshifts `z < 2`, in
contradiction to the observations. The CDM models can also produce
filaments, walls, and voids which resemble those observed, and many
other predictions can be reconciled with the observations if galaxies
are assumed to form at high peaks of the density field. But recent
measurements of large-scale clustering and streaming motions require
more fluctuation power on these scales than the standard CDM model can
easily provide.

At present there is no *unique* theory for the formation of
galaxies and large-scale structure. With some mix of CDM, HDM, and a
nonzero cosmological constant, it is possible to construct models
which satisfy most of the observational constraints (*e.g.*
Kofman's `barrel diagram', from Einasto *et al.* 1987). But
these models are unsatisfying in that they lack internal harmony or
`beauty'.

Due Date: 4/20/95.

**20.** If a bound object forms at time `t`, its internal
density will be a fixed multiple of the mean cosmic density
`rho_b(t)`. Use this fact together with Eq. (5) to derive
scaling laws for the characteristic radii and internal velocity
dispersions of bound objects as functions of time `t` and the
fluctuation index `n`.

- Blumenthal, G.R.
*et al.*1984,*Nature***311**, 517. - Bond, J.R. & Szalay, A.S. 1983,
*Ap. J.***274**, 443. - Efstathiou, G.P.E.
*et al.*1988,*M.N.R.A.S.***235**, 715. - Einasto, J.
*et al.*1987, in*Dark Matter in the Universe*, eds. J. Kormendy & G.R. Knapp, p. 243. - Harrison, E.R. 1970,
*Phys. Rev. D.***1**, 2726. - Layzer, D. 1954,
*A. J.***59**, 170. - Peebles, P.J.E. 1980,
*The Large-Scale Structure of the Universe*. - Peebles, P.J.E. 1982,
*Ap. J.***263**, L1. - Peebles, P.J.E. 1993,
*Principles of Physical Cosmology*. - Zel'dovich, Ya.B. 1970,
*Astr. Ap.***5**, 84. - Zel'dovich, Ya.B. 1972,
*M.N.R.A.S.***160**, 1P.

Joshua E. Barnes (barnes@zeno.ifa.hawaii.edu)

Last modified: April 10, 1995