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2.4. Primeval Gravitational Energy

In the LambdaCDM cosmology the gravitational binding energy of the present mass distribution has two contributions. The first, which we term primeval, is a result of the purely gravitational growth of mass fluctuations out of the small adiabatic departures from a homogeneous mass distribution present in the initial conditions for the Friedmann-Lemaître cosmology. The second, to be discussed in the next subsection, is the result of dissipative settling of baryons that produced the baryon-dominated luminous parts of the galaxies along with stars and star remnants. We can find sensible approximations to the primeval and dissipative components because, as we will discuss, the characteristic length scales are well separated.

The primeval gravitational energy is defined by imagining a universe with initial conditions identical to ours in all respects except that the baryonic matter in our universe is replaced by an equal mass of CDM in the reference model. At the present world time this reference model contains a clustered distribution of massive halos with gravitational binding energy density that we term the primeval component. 5 This component is estimated as follows.

The Layzer (1963) - Irvine (1961) equation for the evolution of the kinetic and gravitational energies of nonrelativistic matter, such as CDM, that interacts only by gravity is

Equation 57 (57)

The kinetic energy per unit mass is

Equation 58 (58)

where vector{v} is the peculiar velocity of a particle with mass m. The gravitational potential energy per unit mass is

Equation 59 (59)

where rhom is the mean mass density and xi is the reduced mass autocorrelation function.

We can use a simple limiting case of the Layzer-Irvine equation, because in the LambdaCDM cosmology the universe has now entered Lambda-dominated expansion, which has caused a significant suppression of the growth of large-scale departures from homogeneity. This means that the first term in equation (57) has become small compared to the second term. Thus it is a reasonable approximation to take 2K = - W, the usual virial equilibrium relation. Then the gravitational binding energy per unit mass is U = K + W = W / 2, and the cosmic mean primeval gravitational binding energy is the product of U with the mean mass density Omegam in matter (eq. [8]). With the normalization P(k) = integ d3r xi(r) eivector{k} . vector{r} for the mass fluctuation power spectrum, the expression for the primeval gravitational binding energy in this approximation is

Equation 60 (60)

For a numerical value we use the mass fluctuation power spectrum P(k) in Figure 37 of Tegmark et al. (2004b) at k < 0.1h Mpc-1. At smaller scales the Tegmark et al. spectrum decreases too rapidly with increasing wavenumber k to be a good approximation to the present mass distribution (Davis & Peebles 1983). We approximate the spectrum as

Equation 61 (61)

at k > 0.1h Mpc-1. This is based on the Fourier transform of the pure power law model for the galaxy autocorrelation function (eq. [5]),

Equation 62 (62)

which gives P = 5200h-3 Mpc3 at k = 0.1h Mpc-1. We choose the somewhat larger normalization in equation (61) to fit the SDSS measurement. The numerical result is

Equation 63 (63)

The integral of the Tegmark et al. (2004b) power spectrum over all wavenumbers is 2700h-2 Mpc, which is 0.55 times the value in equation (63). Our estimate of the integral is based on measurements of the actual power spectrum, not the spectrum that would have obtained if there were no dissipative settling of baryons, but the error is small because the integral is dominated by lengths large compared to the scale of separation of baryons from the dark matter.

Equations (60) and (63) yield the entry in line 4 of Table 1,

Equation 64 (64)

This is the density parameter of the gravitational binding energy of the present departure from a homogeneous mass distribution, ignoring the effects of the dissipative settling of baryons. It will be useful to note that a measure of the length scale of the gravitational energy is the half-point of the integral in equation (63), at wavenumber k1/2 = 0.27h Mpc-1, or half wavelength

Equation 65 (65)

The kinetic energy per unit mass belonging to equation (64) is

Equation 66 (66)

The velocity in parentheses is the one-dimensional single-particle line-of sight rms peculiar velocity.

The gravitational binding energy is not equal to a sum over the contributions from individual objects, but we can write useful approximations to the decomposition into the three components - the virialized parts of the massive halos of L ~ L* galaxies, the rich clusters, and large-scale clustering - shown in category 4 in the inventory.

Since galaxy rotation curves tend to be close to flat we write the binding energy of the virialized parts of the dark matter halos of the galaxies as

Equation 67 (67)

where Omegam is the matter density parameter (eq. [8]), sigma is the characteristic velocity dispersion in equation (10) and the last factor is the virialized mass fraction (eq. 13). This is about 10% of the total gravitational binding energy. The dissipative settling that produced the baryon-dominated luminous parts of the galaxies would have perturbed the massive halos, but the disturbance to the primeval gravitational energy is small because the baryon mass fraction is small.

Our estimate of the binding energy of the dark halos of the galaxies may be compared to the density parameter given by equation (60) when the integral over P(k) is restricted to small scales, k > pi / rv, where rv is the virial radius in in equations (10) and (67). The result, with equation (61) for P(k), is twice the value in equation (67). The difference is an indication of the ambiguity of separating gravitational energy into components. For entry 4.1 we adopt the value in equation (67) as the more directly interpretable.

Equation (67) is a reasonable approximation for most of the mass in galaxy-size dark halos in luminous field galaxies such as the Milky Way, but in rich clusters the galaxies tend to share a dark halo that is close to smoothly distributed across the cluster. Our estimate in entry 4.2 for the primeval gravitational binding energy belonging to rich clusters follows equation (67), with sigma = 800 km s-1 and Omegacl from equation (46). Rich clusters share about 15% of the total gravitational binding energy.

Entry 4.3 is the difference between the total in entry 4 and the sum of entries 4.1 and 4.2. The Layzer-Irvine equation indicates that the binding energy is dominated by a length scale (eq. [65]) that is much larger than galaxy virial radii. Consistent with this, entry 4.3 is larger than entry 4.1 for the binding energy of the dark halos of galaxies. The difference is not large, however, because at small scales the integral over the power spectrum converges slowly, as k-0.23.

5 One surely would say that in this model universe the virialized dark matter halos have gravitational binding energy. Since there was no energy transfer to some other form, one might also want to say that this binding energy must have been present in the initial conditions. Furthermore, one can assign to a linear mass density fluctuation with contrast delta(t) > 0, and comoving radius x (physical radius xa(t)), a gravitational energy per unit mass, W' ~ - G <rho> delta(ax)2, which is constant in linear perturbation theory and comparable to the binding energy of the final virialized halo. Perhaps one can use this as a guide to a definition of the primeval energy belonging to the density fluctuation, despite the problem that the mean of W' vanishes, and the fact that in general relativity theory there is no general definition of the global energy density of a statistically homogeneous system. We have not been able to find a useful approach along these lines. We might add that if our universe had been Einstein-de Sitter then we would have defined the primeval energy density by a numerical solution of equation (57). Back.

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