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 W_M(k) is the Fourier transform `window function' for a sphere of proper radius r (e.g. Peebles 1980). Using Eq. (1) for the power spectrum, we then have
2 -1-n/3 (4) < (delta M/M) > ~ D Mfor fluctuations on a mass scale M.
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 n < 0, diverging as n -> -3, while for n > 0 the mass scale grows more slowly. This and other aspects of hierarchical clustering are well-illustrated in recent numerical simulations (e.g. Efstathiou et al. 1988).
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 scale-invariant spectrum because the clustering pattern it yields is statistically invariant when the expanding universe is scaled to a constant horizon radius.
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.
Last modified: April 10, 1995