Cosmology. The Origin and Evolution of Cosmic Structure - Coles P., Lucchin F

208 Introduction to Jeans Theory

entropic type, and two vortical modes. To solve the Equations (10.2.2) we look for solutions in the form of plane waves

where, i = 1, 2, 3, 4, and the perturbations δui stand for δρ, δv, δϕ and δs, respectively; the δi are functions only of time. Given that the unperturbed solutions do not depend upon position, one can search for solutions of the form

let us refer to the amplitudes δ0i as D, V, Φ and Σ. In the previous equations r is a position vector, k is a (real) wavevector, and ω is a frequency which is in general complex. Substituting from (10.2.3) and (10.2.4) into (10.2.2) and putting vs2 = (∂p/∂ρ)S (vs is the sound speed, as we mentioned above), and δ0 = D/ρ0 we obtain

Let us briefly consider at the start those solutions with ω = 0, i.e. those that do not depend upon time. One such solution corresponds to Σ = Σ ≠ 0 = const. In the absence of viscosity and thermal conduction the perturbation to s is conserved in time; this is called the entropic solution. Another two solutions with ω = 0 are obtained by putting Σ = 0 and k · V = 0: these therefore have k perpendicular to V and represent vortical modes in which × v ≠ 0, which does not imply any perturbations to the density, as is evident from (10.2.5 b) and (10.2.5 c).

The time-dependent solutions of (10.2.5), i.e. those with ω ≠ 0, are more interesting. In this case (10.2.5 d) implies that Σ = 0: the perturbations are adiabatic. From (10.2.5 a) one has that k·V ≠ 0. In this case, we can resolve into components parallel and perpendicular to V. We mentioned above the consequence of having k perpendicular to V, so now let us concentrate upon the parallel component. Perturbations with k and V parallel are longitudinal in character. Equations (10.2.5) now become

This system admits a non-zero solution for δ0, V and Φ if and only if its determinant vanishes. This means that ω and k must satisfy the dispersion relation:

The solutions are of two types, according to whether the wavelength λ = 2π/k is greater than or less than

which is called the Jeans length. Notice the same dependence upon G, ρ0 and vs as the simple qualitative description given in Section 10.1.

In the case λ < λJ the angular frequency ω obtained from (10.2.7) is real:

which represent two sound waves in directions ±k, with a dispersion given by (10.2.9). The phase velocity tends to zero for λ → λJ.

When λ > λJ the frequency is imaginary:

which represents a non-propagating solution (stationary wave) of either increasing or decreasing amplitude. The characteristic timescale for the evolution of this amplitude is

It is only this type of solution that exhibits the phenomenon we referred to above as the gravitational or Jeans instability. For scales λ λJ the characteristic time τ coincides with the free-fall collapse time, τ (Gρ0)−1/2, but for λ → λJ this characteristic timescale diverges.

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10.3 Jeans Instability in Collisionless Fluids

Let us now extend our analysis of the gravitational Jeans instability to a gas of collisionless particles. In a sense, the absence of collisions implies there is no pressure, so there would appear to be no analogy with the Jeans length in this case. However, collisionless particles do have velocities and these velocities are not necessarily represented as a single unique v at each position x as we assumed in Section 10.2 for an idealised fluid. Instead there is a distribution of random velocities at each point; in what follows we assume this distribution is isotropic. It is possible for a collisionless system to be well described by a fluid with zero pressure. That occurs when the fluid is extremely cold so that the resulting flow is nearly laminar, i.e. so that the particles always travel in nearly parallel trajectories that do not cross. In such a case it is a good approximation to suppose there is a unique velocity at every point. We shall return to this when we discuss cold dark matter. For simplicity we also assume all particles have the same mass m.

In the collisionless case, the Equations (10.2.1 a) and (10.2.1 b) should be replaced by the Liouville equation

where v ≡ (∂/∂v) by analogy with ≡ (∂/∂r). The function f(r, v; t) is the phase-space distribution function for the particles; the phase space is six dimensional, and f also depends explicitly on time. The function f therefore represents the number-density of particles in a volume dr at position r and with velocity in the volume dv at v; the actual number of particles in each of these volumes is given by f(r, v; t) dr dv. In our case, of a homogeneous and isotropic timestationary background distribution, it can be shown that the distribution function is only a function of v2.

We stress that the systems (10.2.1 a)–(10.2.1 c) and Equation (10.3.1) are both approximations to a full statistical mechanical treatment using a Boltzmann equation with a collisional term on the right-hand side of (10.3.1).

Equation (10.2.1 c) does not change in the collisionless situation, so we must bear in mind the comments we made above about the existence of stationary solutions. Nevertheless, let us consider Equation (10.2.2 c):

δf is the perturbation of the distribution function and δϕ is the perturbation of the gravitational potential, related to the gravitational acceleration g = v˙ by

By analogy with what we have done in the previous paragraph, we look for a solution to Equations (10.3.2) and (10.3.5) with δf, δϕ and δρ in the form of a plane wave. Without loss of generality, we can take the wavevector k to be in the x-direction. Applying the operator to (10.3.5) and using the fact that the operators and v commute, we obtain from (10.3.2) that

This equation, after substitution in Equation (10.3.3), becomes the dispersion relation

To find the solution appropriate to k → 0 (long wavelengths) we can develop the dispersion relation as a power series in kvx/ω; keeping only the first two terms in such a series yields

The first term vanishes for reasons of symmetry, but the second can be evaluated by integration by parts (note that f(v2) tends to zero as v → ∞): one has

where ρ is obtained from a relation analogous to (10.3.3). This result shows that there is indeed a gravitational instability in this case, with characteristic timescale

identical to the previous expression (10.2.13) for λ λJ.

The Jeans length λJ can be obtained from (10.3.7) by putting ω = 0, by analogy with what we have seen above; by similar reasoning to that which led to (10.3.10) we find

212 Introduction to Jeans Theory

The velocity v replaces the velocity of sound vs in (10.2.8). In the particular case of a Maxwellian distribution

we have v = σ.

The analysis of the evolution of perturbations for λ < λJ is complicated and we shall not go into it further in this chapter. In fact, in this case, there is a rapid dissipation of fluctuations of wavelength λ in a time of order τ λ/v because of the di usion of particles, a phenomenon known as ‘free streaming’, similar to the phenomenon known in collisionless plasma theory as ‘Landau damping’ or ‘phase mixing’.

10.4 History of Jeans Theory in Cosmology

In the subsequent chapters we shall discuss how gravitational instability might take place in a cosmological context and how this theory furnishes a more-or-less complete picture of cosmic structure formation. We shall find a number of complications of the simple picture described by Jeans. For example, we shall have to take explicit account of the expansion of the Universe. We may also need to take into account how general relativity might alter the simple Newtonian analysis outlined above. We also need to understand how the relativistic and non-relativistic components of the fluid influence the growth of fluctuations, and what is the e ect of dark matter in the form of weakly interacting particles. Before going on to cover this new ground in a mathematically complete way, it is instructive to give a brief historical outline of the application of Jeans theory in cosmology. This is an introductory survey only, and we shall give the arguments in greater technical detail in Chapters 12 and 13.

The first to tackle the problem of gravitational instability within the framework of general relativity was Lifshitz (1946). He studied the evolution of small fluctuations in the density of a Friedmann model. Curiously, it was not later that the evolution of perturbations in a Friedmann model with p ρc2 was investigated in Newtonian theory by Bonnor (1957). In some ways the relativistic cosmological theory is more simple that the Newtonian analogue, which requires considerable mathematical subtlety.

These foundational studies were made at a time when the existence of the cosmic microwave background was not known. There was no generally accepted cosmological model within which to frame the problem of structure formation, and there was no way to test the gravitational instability hypothesis for the origin of structure. Nevertheless, it was clear at this time that if the Universe was evolving with time (as the Hubble expansion indicated), then it was possible, in principle, that structure may have evolved by some mechanism similar to the Jeans process. The discovery of the microwave background in the 1960s at last gave theorists a favoured model in which to study this problem: the hot Big Bang. The existence of

the microwave background at the present time implied that there was a period in which the Universe comprised a plasma of matter and radiation in thermal equilibrium. Under these physical conditions, there are a number of processes, due to viscosity and thermal conduction in the radiative plasma, which could influence the evolution of a perturbation with wavelength less than λJ. The pioneering works by Silk (1967, 1968), as well as Doroshkevich et al. (1967), Peebles and Yu (1970), Weinberg (1971), Chibisov (1972) and Field (1971), amongst many others, represented the first attempts to derive a theory of galaxy and structure formation within the framework of modern cosmology. At this time there was in fact a rival theory in which it was proposed that galaxies were formed as a result of primordial cosmic turbulence, i.e. large-scale vortical motions rather than longitudinal adiabatic perturbations. This theory, however, rapidly fell from fashion when it was realised that it should lead to large fluctuations in the temperature of the microwave background on the sky. In fact, this point about the microwave background was then and is now important in all theories of galaxy formation. If structure grows by gravitational instability, it is in principle possible to reconcile the present highly inhomogeneous Universe with a past Universe which was much smoother. The microwave background seemed to be at the same temperature in all directions to within about one part in a thousand in this period, indicating a comparable lack of inhomogeneity in the early Universe. If gravitational instability were the correct explanation for the origin of structure, however, there should be some fluctuations in the microwave background temperature. This initiated a search, which has only recently been successful, for fluctuations in the cosmic microwave background on the sky. But more of that later.

10.5The E ect of Expansion: an Approximate Analysis

The original Jeans theory of gravitational instability, formulated in a static Universe, cannot be applied to an expanding cosmological model. We also have to contend with some features in the cosmological case which do not appear in the original analysis. For example, what happens to the Jeans instability if the Universe is radiation dominated? In this chapter our goal is to translate the usual language of gravitational instability into the context of the Friedmann models. We can then go on, in the next two chapters, to examine the physics of expanding universe models in more detail.

It is useful perhaps to outline the basic results we obtain later with an approximate argument that explains the basic physics. We assume for the moment that the Universe is dominated by pressureless material. The di culty with the expanding Universe is that the density of matter varies with time according to the approximate relation

216 Introduction to Jeans Theory

of the previous chapter into the framework of the expanding universe models. For simplicity, we concentrate upon the case of a dust (zero-pressure) model, and we shall adopt a Newtonian approach as before.

The system of Equations (10.2.1) admits a solution that describes the expansion (or contraction) of a homogeneous and isotropic distribution of matter:

r is a physical coordinate, related to the comoving coordinate r0 by the relation

One defect of the solution (10.6.1) is that for r → ∞, both v and ϕ diverge. Only a relativistic treatment can remedy this problem, so we shall ignore it for the present, making some comments later, in Section 11.12, on the correct analysis.

We proceed by looking for small perturbations δρ, δv, δϕ and δp to the zeroorder solution represented by Equations (10.6.1). The equations for the perturbations can then be written

where the dots denote partial derivatives with respect to time. We now neglect the terms in r · because we make the calculations in a coordinate system where the background velocity v is zero. In fact, this trick does not always work: these terms actually correspond to terms which appear only in the Newtonian framework and they give rise to inconsistencies if there is a non-zero pressure; see Lima et al. (1997).

As we did earlier, we now look for solutions in the form of small plane-wave departures from the exact solution represented by (10.6.1):

where the variables ui, for i = 1, 2, 3, 4, are related to the quantities D, V, Φ, Σ introduced in Section 10.2; their amplitudes here, however, have to depend on time; the perturbation in the pressure is again expressed in terms of δρ and δs. The ui(t) cannot be functions of the type u0i exp(iωt), because the coe cients of the equations depend on time. We should also note that the wavevector k corresponds to a wavelength λ which varies with time according to the law (10.6.2), simply because of the expansion of the Universe:

for this reason the exponential in (10.6.4) does not depend upon time. One can obtain (after some work!) the perturbation equations corresponding to those given in (10.2.5):

This system admits a static (time-independent) solution of entropic type, in which

The vortical solutions can be obtained by putting D = Φ = Σ = 0 and the condition that V is perpendicular to k. From (10.6.6 b) we get

with V0 perpendicular to k. The Equation (10.6.9) can be obtained in another way, by applying the law of conservation of angular momentum L, due to the absence of dissipative processes,

(V is the modulus of V).

The solutions with Σ = 0 and V parallel to k are more interesting from a cos-