Linear theory

To understand the evolution of the density fluctuations in the expanding universe, we need to solve the perturbed Friedmann equations. However, when the scale of the perturbation is small compared with the horizon (i C ct0) and the flow is nonrelativistic (v C c), a Newtonian derivation can be used. Furthermore, when the perturbations are still small, as happens in the early stages of structure formation, the equations that govern their evolution can be linearized.

We begin with the equation of mass conservation

the Euler equation

the Poisson equation

and an equation of state P = P (p). If the flow is homogeneous, the solution to the above equations is

Now we perturb this solution as follows:

and define the comoving coordinate x = r/a(t) and the proper time dr = dt/a. We obtain the following equations, where all the time and spatial derivatives are with respect to r and x:


If the perturbations are small, we can linearize the equations, so that the first equation becomes S + V- v = 0 and solve for S. Note that from the condition of adiabaticity (no heat exchange between fluid elements), it follows that

where cs = yMP/dp is the speed of sound and the last equality is the result of linearization.

The velocity field can be decomposed as follows: v = v|| + vi, where V x V||= 0 and V - vi = 0. The rotational component, equivalent to a vorticity, satisfies

and therefore decays in the expanding universe. The irrotational component satisfies

Peculiar velocities associated with this component arise owing to density fluctuations.

Finally, combining (9.19)-(9.21) and moving back to physical units, we obtain the following equation for the overdensity field:

where the second term on the left-hand side is the damping term due to the expansion of the universe, the first term on the right-hand side is the gravitational driving term, and the second term on the right-hand side represents the pressure support.

Now let us decompose S into plain-wave modes S(x, t) = SSke*x k, where A = 2na/k is the physical wavelength. The perturbation is unstable if its scale exceeds the Jeans scale, defined here as

In the case of weakly interacting dark matter, the matter pressure vanishes, and (9.25) reduces to

During the epoch of matter domination, pm > pY and all the wavelengths are unstable. A lower bound arises as a result of the free streaming of the dark matter particles, and corresponds to the cosmologically irrelevant scale of ~10-6 Earth mass for a 100 GeV dark matter particle. In fact, there are interesting variations in this scale that arise from considerations of kinetic as opposed to thermal decoupling (Chen et al., 2001) and from the possible dominance of warm dark matter (Boehm and Schaeffer, 2005).

In an Einstein-de Sitter background (Qm = 1), the scale factor grows as a ж t2/3 while p = pm ж a-3, and there are two solutions to (9.27): S ж t2/3 and S ж t-1. At late epochs, the constant energy density associated with the cosmological constant dominates over matter and the solution is S « const.

Recall that in the general case the background universe is governed by the equation

One solution to the perturbation equations is therefore

In other words, the overdensity is nearly constant during radiation domination (in fact, it grows logarithmically), and grows as t2/3 in the matter-dominated epoch.

These results can be reformulated statistically by Fourier transforming these equations and looking at the power spectrum of the density fluctuations:

where P(k) = Akn. For large scales, n ~ 1 (for example from measurements of the cosmic microwave background anisotropies; (Planck Collaboration 2013a), while on small scales, growth is suppressed during radiation domination, which results in a characteristic peak in P(k) on a scale corresponding to matter-radiation equality. It follows from (9.30) that Sp/p ж k(n+3)/2 or Sp/p ж M-(n+3)/6. Asymptotically for large M, Sp/p ж M-2/3. More precisely, the power spectrum today can be described by Pobs(k) ж knT2(k), where T(k) is a transfer function that represents the modifications to the primordial power spectrum due to the transition from radiation to matter domination. The power spectrum observed on different scales is shown schematically in Fig. 9.7. This figure nicely illustrates the complementarity of diverse observations that either directly (CMB) or indirectly (clusters, galaxies, integalactic medium (IGM)) sample the linear density fluctuation spectrum.

Power spectrum measurements (dots) and the theoretical prediction (curve). Image credit

Fig. 9.7 Power spectrum measurements (dots) and the theoretical prediction (curve). Image credit: M. Tegmark (SDSS). For the figure in color, please see the online version of the lectures.

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