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2.1. The Expanding Universe

The modern physical description of the universe as a whole can be traced back to Einstein, who argued theoretically for the so-called ``cosmological principle'': that the distribution of matter and energy must be homogeneous and isotropic on the largest scales. Today isotropy is well established (see the review by Wu, Lahav, & Rees 1999) for the distribution of faint radio sources, optically-selected galaxies, the X-ray background, and most importantly the cosmic microwave background (henceforth, CMB; see, e.g., Bennett et al. 1996). The constraints on homogeneity are less strict, but a cosmological model in which the universe is isotropic but significantly inhomogeneous in spherical shells around our special location is also excluded (Goodman 1995).

In General Relativity, the metric for a space which is spatially homogeneous and isotropic is the Robertson-Walker metric, which can be written in the form

Equation 2   (2)

where a(t) is the cosmic scale factor which describes expansion in time, and (R, theta, phi) are spherical comoving coordinates. The constant k determines the geometry of the metric; it is positive in a closed universe, zero in a flat universe, and negative in an open universe. Observers at rest remain at rest, at fixed (R, theta, phi), with their physical separation increasing with time in proportion to a(t). A given observer sees a nearby observer at physical distance D receding at the Hubble velocity H(t)D, where the Hubble constant at time t is H(t) = d ln a(t) / dt. Light emitted by a source at time t is observed at t = 0 with a redshift z = 1 / a(t) - 1, where we set a(t = 0) ident 1.

The Einstein field equations of General Relativity yield the Friedmann equation (e.g., Weinberg 1972; Kolb & Turner 1990)

Equation 3   (3)

which relates the expansion of the universe to its matter-energy content. For each component of the energy density rho, with an equation of state p = p(rho), the density rho varies with a(t) according to the equation of energy conservation

Equation 4   (4)

With the critical density

Equation 5   (5)

defined as the density needed for k = 0, we define the ratio of the total density to the critical density as

Equation 6   (6)

With Omegam, OmegaLambda, and Omegar denoting the present contributions to Omega from matter (including cold dark matter as well as a contribution Omegab from baryons), vacuum density (cosmological constant), and radiation, respectively, the Friedmann equation becomes

Equation 7   (7)

where we define H0 and Omega0 = Omegam + OmegaLambda + Omegar to be the present values of H and Omega, respectively, and we let

Equation 8   (8)

In the particularly simple Einstein-de Sitter model (Omegam = 1, OmegaLambda = Omegar = Omegak = 0), the scale factor varies as a(t) propto t2/3. Even models with non-zero OmegaLambda or Omegak approach the Einstein-de Sitter behavior at high redshifts, i.e., when

Equation 9   (9)

(as long as Omegar can be neglected). The Friedmann equation implies that models with Omegak = 0 converge to the Einstein-de Sitter limit faster than do open models. E.g., for Omegam = 0.3 and OmegaLambda = 0.7 equation (9) corresponds to the condition z >> 1.3, which is easily satisfied by the reionization redshift. In this high-z regime, H(t) approx 2 / (3t), and the age of the universe is

Equation 10   (10)

where in the last expression we assumed our standard cosmological parameters (see the end of Section 1).

In the standard hot Big Bang model, the universe is initially hot and the energy density is dominated by radiation. The transition to matter domination occurs at z ~ 104, but the universe remains hot enough that the gas is ionized, and electron-photon scattering effectively couples the matter and radiation. At z ~ 1200 the temperature drops below ~ 3300 K and protons and electrons recombine to form neutral hydrogen. The photons then decouple and travel freely until the present, when they are observed as the CMB.

2.2. Linear Gravitational Growth

Observations of the CMB (e.g., Bennett et al. 1996) show that the universe at recombination was extremely uniform, but with spatial fluctuations in the energy density and gravitational potential of roughly one part in 105. Such small fluctuations, generated in the early universe, grow over time due to gravitational instability, and eventually lead to the formation of galaxies and the large-scale structure observed in the present universe.

As in the previous section, we distinguish between fixed and comoving coordinates. Using vector notation, the fixed coordinate r corresponds to a comoving position x = r / a. In a homogeneous universe with density rho, we describe the cosmological expansion in terms of an ideal pressure-less fluid of particles each of which is at fixed x, expanding with the Hubble flow v = H(t)r where v = dr / dt. Onto this uniform expansion we impose small perturbations, given by a relative density perturbation

Equation 11   (11)

where the mean fluid density is rhobar, with a corresponding peculiar velocity u ident v - Hr. Then the fluid is described by the continuity and Euler equations in comoving coordinates (Peebles 1980, 1993):

Equation 12   (12) Equation 13   (13)

The potential phi is given by the Poisson equation, in terms of the density perturbation:

Equation 14   (14)

This fluid description is valid for describing the evolution of collisionless cold dark matter particles until different particle streams cross. This ``shell-crossing'' typically occurs only after perturbations have grown to become non-linear, and at that point the individual particle trajectories must in general be followed. Similarly, baryons can be described as a pressure-less fluid as long as their temperature is negligibly small, but non-linear collapse leads to the formation of shocks in the gas.

For small perturbations delta << 1, the fluid equations can be linearized and combined to yield

Equation 15   (15)

This linear equation has in general two independent solutions, only one of which grows with time. Starting with random initial conditions, this ``growing mode'' comes to dominate the density evolution. Thus, until it becomes non-linear, the density perturbation maintains its shape in comoving coordinates and grows in proportion to a growth factor D(t). The growth factor is in general given by (Peebles 1980)

Equation 16   (16)

where we neglect Omegar when considering halos forming at z << 104. In the Einstein-de Sitter model (or, at high redshift, in other models as well) the growth factor is simply proportional to a(t).

The spatial form of the initial density fluctuations can be described in Fourier space, in terms of Fourier components

Equation 17   (17)

Here we use the comoving wavevector k, whose magnitude k is the comoving wavenumber which is equal to 2pi divided by the wavelength. The Fourier description is particularly simple for fluctuations generated by inflation (e.g., Kolb & Turner 1990). Inflation generates perturbations given by a Gaussian random field, in which different k-modes are statistically independent, each with a random phase. The statistical properties of the fluctuations are determined by the variance of the different k-modes, and the variance is described in terms of the power spectrum P(k) as follows:

Equation 18   (18)

where delta(3) is the three-dimensional Dirac delta function.

In standard models, inflation produces a primordial power-law spectrum P(k) propto kn with n ~ 1. Perturbation growth in the radiation-dominated and then matter-dominated universe results in a modified final power spectrum, characterized by a turnover at a scale of order the horizon cH-1 at matter-radiation equality, and a small-scale asymptotic shape of P(k) propto kn - 4. On large scales the power spectrum evolves in proportion to the square of the growth factor, and this simple evolution is termed linear evolution. On small scales, the power spectrum changes shape due to the additional non-linear gravitational growth of perturbations, yielding the full, non-linear power spectrum. The overall amplitude of the power spectrum is not specified by current models of inflation, and it is usually set observationally using the CMB temperature fluctuations or local measures of large-scale structure.

Since density fluctuations may exist on all scales, in order to determine the formation of objects of a given size or mass it is useful to consider the statistical distribution of the smoothed density field. Using a window function W(y) normalized so that integ d3y W(y) = 1, the smoothed density perturbation field, integ d3y delta(x + y)W(y), itself follows a Gaussian distribution with zero mean. For the particular choice of a spherical top-hat, in which W = 1 in a sphere of radius R and is zero outside, the smoothed perturbation field measures the fluctuations in the mass in spheres of radius R. The normalization of the present power spectrum is often specified by the value of sigma8 ident sigma(R = 8h-1Mpc). For the top-hat, the smoothed perturbation field is denoted deltaR or deltaM, where the mass M is related to the comoving radius R by M = 4pi rhom R3 / 3, in terms of the current mean density of matter rhom. The variance < deltaM >2 is

Equation 19   (19)

where j1(x) = (sin x - x cos x) / x2. The function sigma(M) plays a crucial role in estimates of the abundance of collapsed objects, as described below.

2.3. Formation of Nonlinear Objects

The small density fluctuations evidenced in the CMB grow over time as described in the previous subsection, until the perturbation delta becomes of order unity, and the full non-linear gravitational problem must be considered. The dynamical collapse of a dark matter halo can be solved analytically only in cases of particular symmetry. If we consider a region which is much smaller than the horizon cH-1, then the formation of a halo can be formulated as a problem in Newtonian gravity, in some cases with minor corrections coming from General Relativity. The simplest case is that of spherical symmetry, with an initial (t = ti << t0) top-hat of uniform overdensity deltai inside a sphere of radius R. Although this model is restricted in its direct applicability, the results of spherical collapse have turned out to be surprisingly useful in understanding the properties and distribution of halos in models based on cold dark matter.

The collapse of a spherical top-hat is described by the Newtonian equation (with a correction for the cosmological constant)

Equation 20   (20)

where r is the radius in a fixed (not comoving) coordinate frame, H0 is the present Hubble constant, M is the total mass enclosed within radius r, and the initial velocity field is given by the Hubble flow dr / dt = H(t)r. The enclosed delta grows initially as deltaL = deltai D(t) / D(ti), in accordance with linear theory, but eventually delta grows above deltaL. If the mass shell at radius r is bound (i.e., if its total Newtonian energy is negative) then it reaches a radius of maximum expansion and subsequently collapses. At the moment when the top-hat collapses to a point, the overdensity predicted by linear theory is (Peebles 1980) deltaL = 1.686 in the Einstein-de Sitter model, with only a weak dependence on Omegam and OmegaLambda. Thus a top-hat collapses at redshift z if its linear overdensity extrapolated to the present day (also termed the critical density of collapse) is

Equation 21   (21)

where we set D(z = 0) = 1.

Even a slight violation of the exact symmetry of the initial perturbation can prevent the top-hat from collapsing to a point. Instead, the halo reaches a state of virial equilibrium by violent relaxation (phase mixing). Using the virial theorem U = - 2K to relate the potential energy U to the kinetic energy K in the final state, the final overdensity relative to the critical density at the collapse redshift is Deltac = 18pi2 appeq 178 in the Einstein-de Sitter model, modified in a universe with Omegam + OmegaLambda = 1 to the fitting formula (Bryan & Norman 1998)

Equation 22   (22)

where d ident Omegamz - 1 is evaluated at the collapse redshift, so that

Equation 23   (23)

A halo of mass M collapsing at redshift z thus has a (physical) virial radius

Equation 24   (24)

and a corresponding circular velocity,

Equation 25   (25)

In these expressions we have assumed a present Hubble constant written in the form H0 = 100 h km s-1Mpc-1. We may also define a virial temperature

Equation 26   (26)

where µ is the mean molecular weight and mp is the proton mass. Note that the value of µ depends on the ionization fraction of the gas; µ = 0.59 for a fully ionized primordial gas, µ = 0.61 for a gas with ionized hydrogen but only singly-ionized helium, and µ = 1.22 for neutral primordial gas. The binding energy of the halo is approximately (3)

Equation 27   (27)

Note that the binding energy of the baryons is smaller by a factor equal to the baryon fraction Omegab / Omegam.

Although spherical collapse captures some of the physics governing the formation of halos, structure formation in cold dark matter models proceeds hierarchically. At early times, most of the dark matter is in low-mass halos, and these halos continuously accrete and merge to form high-mass halos. Numerical simulations of hierarchical halo formation indicate a roughly universal spherically-averaged density profile for the resulting halos (Navarro, Frenk, & White 1997, hereafter NFW), though with considerable scatter among different halos (e.g., Bullock et al. 2001). The NFW profile has the form

Equation 28   (28)

where x = r / rvir, and the characteristic density deltac is related to the concentration parameter cN by

Equation 29   (29)

The concentration parameter itself depends on the halo mass M, at a given redshift z. We note that the dense, cuspy halo profile predicted by CDM models is not apparent in the mass distribution derived from measurements of the rotation curves of dwarf galaxies (e.g., de Blok & McGaugh 1997; Salucci & Burkert 2000), although observational and modeling uncertainties may preclude a firm conclusion at present (van den Bosch et al. 2000; Swaters, Madore, & Trewhella 2000).

2.4. The Abundance of Dark Matter Halos

In addition to characterizing the properties of individual halos, a critical prediction of any theory of structure formation is the abundance of halos, i.e., the number density of halos as a function of mass, at any redshift. This prediction is an important step toward inferring the abundances of galaxies and galaxy clusters. While the number density of halos can be measured for particular cosmologies in numerical simulations, an analytic model helps us gain physical understanding and can be used to explore the dependence of abundances on all the cosmological parameters.

A simple analytic model which successfully matches most of the numerical simulations was developed by Press & Schechter (1974). The model is based on the ideas of a Gaussian random field of density perturbations, linear gravitational growth, and spherical collapse. To determine the abundance of halos at a redshift z, we use deltaM, the density field smoothed on a mass scale M, as defined in Section 2.2. Although the model is based on the initial conditions, it is usually expressed in terms of redshift-zero quantities. Thus, we use the linearly-extrapolated density field, i.e., the initial density field at high redshift extrapolated to the present by simple multiplication by the relative growth factor (see Section 2.2). Similarly, in this section the `present power spectrum' refers to the initial power spectrum, linearly-extrapolated to the present without including non-linear evolution. Since deltaM is distributed as a Gaussian variable with zero mean and standard deviation sigma(M) [which depends only on the present power spectrum, see equation (19)], the probability that deltaM is greater than some delta equals

Equation 30   (30)

The fundamental ansatz is to identify this probability with the fraction of dark matter particles which are part of collapsed halos of mass greater than M, at redshift z. There are two additional ingredients: First, the value used for delta is deltacrit(z) given in equation (21), which is the critical density of collapse found for a spherical top-hat (extrapolated to the present since sigma(M) is calculated using the present power spectrum); and second, the fraction of dark matter in halos above M is multiplied by an additional factor of 2 in order to ensure that every particle ends up as part of some halo with M > 0. Thus, the final formula for the mass fraction in halos above M at redshift z is

Equation 31   (31)

This ad-hoc factor of 2 is necessary, since otherwise only positive fluctuations of deltaM would be included. Bond et al. (1991) found an alternate derivation of this correction factor, using a different ansatz. In their derivation, the factor of 2 has a more satisfactory origin, namely the so-called ``cloud-in-cloud'' problem: For a given mass M, even if deltaM is smaller than deltacrit(z), it is possible that the corresponding region lies inside a region of some larger mass ML > M, with deltaML > deltacrit(z). In this case the original region should be counted as belonging to a halo of mass ML. Thus, the fraction of particles which are part of collapsed halos of mass greater than M is larger than the expression given in equation (30). Bond et al. showed that, under certain assumptions, the additional contribution results precisely in a factor of 2 correction.

Differentiating the fraction of dark matter in halos above M yields the mass distribution. Letting dn be the comoving number density of halos of mass between M and M + dM, we have

Equation 32   (32)

where nuc = deltacrit(z) / sigma(M) is the number of standard deviations which the critical collapse overdensity represents on mass scale M. Thus, the abundance of halos depends on the two functions sigma(M) and deltacrit(z), each of which depends on the energy content of the universe and the values of the other cosmological parameters. We illustrate the abundance of halos for our standard choice of the LambdaCDM model with Omegam = 0.3 (see the end of Section 1).

Figure 5 shows sigma(M) and deltacrit(z), with the input power spectrum computed from Eisenstein & Hu (1999). The solid line is sigma(M) for the cold dark matter model with the parameters specified above. The horizontal dotted lines show the value of deltacrit(z) at z = 0, 2, 5, 10, 20 and 30, as indicated in the figure. From the intersection of these horizontal lines with the solid line we infer, e.g., that at z = 5 a 1 - sigma fluctuation on a mass scale of 2 × 107 Msun will collapse. On the other hand, at z = 5 collapsing halos require a 2 - sigma fluctuation on a mass scale of 3 × 1010 Msun, since sigma(M) on this mass scale equals about half of deltacrit(z = 5). Since at each redshift a fixed fraction (31.7%) of the total dark matter mass lies in halos above the 1 - sigma mass, Figure 5 shows that most of the mass is in small halos at high redshift, but it continuously shifts toward higher characteristic halo masses at lower redshift. Note also that sigma(M) flattens at low masses because of the changing shape of the power spectrum. Since sigma -> infty as M -> 0, in the cold dark matter model all the dark matter is tied up in halos at all redshifts, if sufficiently low-mass halos are considered.

Figure 5

Figure 5. Mass fluctuations and collapse thresholds in cold dark matter models. The horizontal dotted lines show the value of the extrapolated collapse overdensity deltacrit(z) at the indicated redshifts. Also shown is the value of sigma(M) for the cosmological parameters given in the text (solid curve), as well as sigma(M) for a power spectrum with a cutoff below a mass M = 1.7 × 108 Msun (short-dashed curve), or M = 1.7 × 1011 Msun (long-dashed curve). The intersection of the horizontal lines with the other curves indicate, at each redshift z, the mass scale (for each model) at which a 1 - sigma fluctuation is just collapsing at z (see the discussion in the text).

Also shown in Figure 5 is the effect of cutting off the power spectrum on small scales. The short-dashed curve corresponds to the case where the power spectrum is set to zero above a comoving wavenumber k = 10 Mpc-1, which corresponds to a mass M = 1.7 × 108 Msun. The long-dashed curve corresponds to a more radical cutoff above k = 1 Mpc-1, or below M = 1.7 × 1011 Msun. A cutoff severely reduces the abundance of low-mass halos, and the finite value of sigma(M = 0) implies that at all redshifts some fraction of the dark matter does not fall into halos. At high redshifts where deltacrit(z) >> sigma(M = 0), all halos are rare and only a small fraction of the dark matter lies in halos. In particular, this can affect the abundance of halos at the time of reionization, and thus the observed limits on reionization constrain scenarios which include a small-scale cutoff in the power spectrum (Barkana, Haiman, & Ostriker 2000).

In Figures 6 - 9 we show explicitly the properties of collapsing halos which represent 1 - sigma, 2 - sigma, and 3 - sigma fluctuations (corresponding in all cases to the curves in order from bottom to top), as a function of redshift. No cutoff is applied to the power spectrum. Figure 6 shows the halo mass, Figure 7 the virial radius, Figure 8 the virial temperature (with µ in equation (26) set equal to 0.6, although low temperature halos contain neutral gas) as well as circular velocity, and Figure 9 shows the total binding energy of these halos. In Figures 6 and 8, the dashed curves indicate the minimum virial temperature required for efficient cooling (see Section 3.3) with primordial atomic species only (upper curve) or with the addition of molecular hydrogen (lower curve). Figure 9 shows the binding energy of dark matter halos. The binding energy of the baryons is a factor ~ Omegab / Omegam ~ 15% smaller, if they follow the dark matter. Except for this constant factor, the figure shows the minimum amount of energy that needs to be deposited into the gas in order to unbind it from the potential well of the dark matter. For example, the hydrodynamic energy released by a single supernovae, ~ 1051 erg, is sufficient to unbind the gas in all 1 - sigma halos at z gtapprox 5 and in all 2 - sigma halos at z gtapprox 12.

Figure 6

Figure 6. Characteristic properties of collapsing halos: Halo mass. The solid curves show the mass of collapsing halos which correspond to 1 - sigma, 2 - sigma, and 3 - sigma fluctuations (in order from bottom to top). The dashed curves show the mass corresponding to the minimum temperature required for efficient cooling with primordial atomic species only (upper curve) or with the addition of molecular hydrogen (lower curve).

Figure 7

Figure 7. Characteristic properties of collapsing halos: Halo virial radius. The curves show the virial radius of collapsing halos which correspond to 1 - sigma, 2 - sigma, and 3 - sigma fluctuations (in order from bottom to top).

Figure 8

Figure 8. Characteristic properties of collapsing halos: Halo virial temperature and circular velocity. The solid curves show the virial temperature (or, equivalently, the circular velocity) of collapsing halos which correspond to 1 - sigma, 2 - sigma, and 3 - sigma fluctuations (in order from bottom to top). The dashed curves show the minimum temperature required for efficient cooling with primordial atomic species only (upper curve) or with the addition of molecular hydrogen (lower curve).

Figure 9

Figure 9. Characteristic properties of collapsing halos: Halo binding energy. The curves show the total binding energy of collapsing halos which correspond to 1 - sigma, 2 - sigma, and 3 - sigma fluctuations (in order from bottom to top).

At z = 5, the halo masses which correspond to 1 - sigma, 2 - sigma, and 3 - sigma fluctuations are 1.8 × 107 Msun, 3.0 × 1010 Msun, and 7.0 × 1011 Msun, respectively. The corresponding virial temperatures are 2.0 × 103 K, 2.8 × 105 K, and 2.3 × 106 K. The equivalent circular velocities are 7.5 km s-1, 88 km s-1, and 250 kmI>s-1. At z = 10, the 1 - sigma, 2 - sigma, and 3 - sigma fluctuations correspond to halo masses of 1.3 × 103 Msun, 5.7 × 107 Msun, and 4.8 × 109 Msun, respectively. The corresponding virial temperatures are 6.2 K, 7.9 × 103 K, and 1.5 × 105 K. The equivalent circular velocities are 0.41 km s-1, 15 km s-1, and 65 km s-1. Atomic cooling is efficient at Tvir gtapprox 104 K, or a circular velocity Vc gtapprox 17 km s-1. This corresponds to a 1.2 - sigma fluctuation and a halo mass of 2.1 × 108 Msun at z = 5, and a 2.1 - sigma fluctuation and a halo mass of 8.3 × 107 Msun at z = 10. Molecular hydrogen provides efficient cooling down to Tvir ~ 300 K, or a circular velocity Vc ~ 2.0 km s-1. This corresponds to a 0.76 - sigma fluctuation and a halo mass of 3.5 × 105 Msun at z = 5, and a 1.3 - sigma fluctuation and a halo mass of 1.4 × 105 Msun at z = 10.

In Figure 10 we show the halo mass function dn / d ln(M) at several different redshifts: z = 0 (solid curve), z = 5 (dotted curve), z = 10 (short-dashed curve), z = 20 (long-dashed curve), and z = 30 (dot-dashed curve). Note that the mass function does not decrease monotonically with redshift at all masses. At the lowest masses, the abundance of halos is higher at z > 0 than at z = 0.

Figure 10

Figure 10. Halo mass function at several redshifts: z = 0 (solid curve), z = 5 (dotted curve), z = 10 (short-dashed curve), z = 20 (long-dashed curve), and z = 30 (dot-dashed curve).

3 The coefficient of 1/2 in equation (27) would be exact for a singular isothermal sphere, rho(r) propto 1 / r2. Back.

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