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

388 The Cosmic Microwave Background

with β 2, where Nϑ[> S(ν)] is the number of sources per unit solid angle with a measured flux at ν greater than S(ν); see Chapter 19 for some more details. If their spectrum is proportional to να, then

∆T ϑ2/β−2να−2. (17.6.5)

T ϑ

The amplitude due to these sources would depend strongly on wavelength. The wavelength dependence can therefore, in principle, be used to identify the contribution from them, but one needs to know the luminosity function of the sources well to be able to subtract them, especially at higher frequencies. Another problem is that the telescopes used for CMB studies often have considerable ‘sidelobes’, which may pick up bright objects quite a long way away from the main beam of the telescope; these are also di cult to subtract.

A cosmological background of dust may also a ect the microwave background, particularly if it is heated by some energetic source at early times. We shall discuss the e ect of this type of process upon the spectrum of the CMB radiation in Chapter 19; here it su ces to note that dust generally emits infrared radiation and this may leak into the wavelength range covered by CMB experiments. Dust is generally a signature of structure formation (it is mainly produced in regions forming massive stars). Inhomogeneities in the dust density can lead to a temperature anisotropy of the CMB. If the dust is clustered like galaxies and the distribution evolves as in a CDM model, then it can be shown that one expects anisotropy up to ∆T/T 10−5 at 400 m, rising to 10−4 at the peak of the CMB spectrum. Given the lack of observed spectral distortions, however, it seems unlikely that dust will generate a significant CMB anisotropy.

Another way in which secondary anisotropy can be generated is connected with possible reionisation of the intergalactic gas after zrec. We have already explained in Section 17.5 how this can smooth out intrinsic anisotropy. Generally, however, reionisation will lead to significant secondary anisotropy on a smaller angular scale than we considered in that section.

Reionisation or reheating may have been generated by many di erent mechanisms. Theories involving a dark-matter particle which undergoes a radiative decay can lead to wholesale reionisation. Early star formation, active galactic nuclei or quasars could also, in principle, have caused reionisation of the intergalactic medium. Cosmological explosions may heat up the intergalactic medium in a very inhomogeneous way leading to considerable anisotropy. As we shall explain in Chapter 21, we know that something reionised the Universe some time before z 4 so these apparently exotic scenarios are not completely implausible.

Whatever caused the gas to become ionised, there is expected to be an accompanying generation of anisotropy. Suppose the plasma is heated enough to ionise it, but not enough for the electrons to become highly relativistic. If the plasma is inhomogeneous, then it will generally have a velocity field associated with it and a photon travelling through the ionised medium will su er Thomson scattering o electrons with velocities oriented in di erent directions. The rate of energy loss

due to Thomson scattering is just

where ne and v are the electron number density and velocity, respectively, and σT is the Thomson scattering cross-section; nˆ is a unit vector in the direction of photon travel. Since Thomson scattering conserves photons we can write

where the integral is taken over a line of sight from the observer to trec and δ is the dimensionless density perturbation in the medium.

The net anisotropy produced by the linear terms in (17.6.7) is extremely small. The second-order term which corresponds to the interaction between the perturbation δ and the velocity can be significant, however, particularly if the inhomogeneities are evolving in the nonlinear regime. This nonlinear term is usually called the Ostriker–Vishniac e ect (Ostriker and Vishniac 1986), although it was actually first discussed by Sunyaev and Zel’dovich (1969). For a spherically symmetric homogeneous cluster moving through the CMB rest frame the e ect is particularly simple:

for a cluster of radius R moving at a velocity v.

There is one other important source of extrinsic anisotropy, called the Sunyaev– Zel’dovich e ect. We shall, however, devote the whole of Section 17.7 to this because it is important in a wider cosmological context than structure-formation theory.

17.7 The Sunyaev–Zel’dovich E ect

The physics behind the Sunyaev–Zel’dovich (SZ) e ect is that, if CMB photons enter a hot (relativistic) plasma, they will be Thomson-scattered up to higher energies, say X-ray energies. If one looks at such a cloud in the Rayleigh–Jeans (long-wavelength) part of the CMB spectrum, one therefore sees fewer microwave photons and the cloud consequently looks cooler. For a cloud with electron pressure pe the temperature ‘dip’ is

where dl = c dt is the distance along a photon path through the cloud. This e ect has been detected using radio observations of rich Abell clusters of galaxies. Such clusters contain ionised gas at a temperature of up to 108 K (the virial temperature) and are about 1 Mpc across. The e ect has been detected at a level

390 The Cosmic Microwave Background

right ascension (J2000)

Figure 17.5 A Sunyaev–Zel’dovich (SZ) map of the cluster Abell 2163. Picture courtesy of John Carlstrom.

of order 10−4 in several clusters, but a new instrument called the Ryle Telescope, recently built in Cambridge, has improved the technique and substantially reduced the observational di culties. This instrument is very di erent from most devices used to search for intrinsic CMB anisotropy because it is supposed to map only a small part of the sky around an individual cluster. (The need to cover a large part of the sky is one of the most demanding requirements on CMB anisotropy searches.) It is possible with this instrument to create detailed maps of clusters in the SZ distortion they produce; an example is shown in Figure 17.5.

A particularly interesting aspect of this technique is that, if one has X-ray observations of a cluster, its redshift and an SZ dip, one can, in principle, get the distance to the cluster in a manner independent of the redshift. This is done by combining X-ray bremsstrahlung measurements, which are proportional to n2e Te1/2 dl, the observed X-ray spectrum, which gives Te, and the Sunyaev– Zel’dovich dip. These three sets of observations allow one to determine Te and the integrals of neTe and n2e Te1/2 through the cluster. One then assumes that the physical size of the cluster along the line of sight is the same as its size in the plane of the sky. Extracting an estimate of l, the total path length through the cluster, then yields an estimate of Rc, the physical radius. Knowing its angular size, one can thus estimate a value for the proper distance. Comparing this with the cluster redshift yields a direct estimate of the Hubble constant which is independent of the usual distance ladder methods described in Section 4.3. For example, if we model the cluster as a homogeneous isothermal sphere of radius Rc, then, from

Equation (17.7.1), the dip in the centre of the cluster will be

Obviously, more sophisticated modelling than this is necessary to obtain accurate results, but the example (17.7.2) illustrates the principles of the method.

This method, when applied to individual clusters, has so far yielded estimates of the Hubble constant towards the lower end of its accepted range. One should say, however, that many clusters are significantly aspherical, so one should really apply this technique to a sample of clusters with random orientations with respect to the line of sight. An appropriate averaging can then be used to obtain an estimate of H0 for the sample which is less uncertain than that for an individual cluster.

As well as being detectable for individual clusters, there should be an integrated SZ e ect caused by all the clusters in a line of sight from the observer to the last scattering surface. This is another complicated small-scale e ect which is rather di cult to model. In principle, however, constraints on the temperature fluctuations produced by this e ect place strong limits on the evolutionary properties of clusters of galaxies. We shall discuss this and other constraints on cosmological evolution in Chapter 21.

17.8Current Status

The last 10 years have seen a tremendous revolution in CMB physics. Starting with the COBE discovery, and its confirmation at Tenerife, increasing sensitivity and resolution have driven observers forward so that all-sky maps of the temperature pattern with arcminute resolution will shortly be available. At the moment the balloon-based results from MAXIMA and Boomerang represent the state of the art. These data strongly suggest we live in a flat universe. Combined with supernova results and other measurements these results have dramatically altered our view of what the standard model of cosmology could be; ΛCDM has emerged from the pack described in Chapter 15 and now replaces SCDM as the front runner for a complete model of structure formation.

When the issue of the intermediate-scale anisotropy is finally resolved by allsky maps, a number of other questions can be addressed, connected with extrinsic (nonlinear) anisotropies, the detailed statistical properties of high-resolution sky maps and after-e ects of reionisation. Another question which will probably become important in a few years’ time is connected with the polarisation of the CMB radiation. Thomson scattering is important during the processes of decoupling and recombination and it induces a partial linear polarisation in the scattered radiation (Rybicki and Lightman 1979). It has been calculated that the level of polarisation expected in the CMB is about 10% of the anisotropy, i.e. a fractional level of around 10−6. This figure is particularly sensitive to the ionisation history and it may yield further information about possible reheating of the Universe.

392 The Cosmic Microwave Background

Measurement of CMB polarisation is, however, not practicable with the current generation of telescopes and receivers.

Bibliographic Notes on Chapter 17

The field described in this chapter is developing extremely rapidly. To see how rapidly material has become dated, it is useful to read Hogan et al. (1982), Vittorio and Silk (1984), Kaiser and Silk (1987), Partridge (1988) and even White et al. (1994). Peacock (1999) is a good up-to-date reference for this material. CMB anisotropy studies have come of age during an era dominated by the internet. Two particularly useful resources are the CMBFAST page

http://www.physics.nyu.edu/matiasz/CMBFAST/cmbfast.html

(see Seljak and Zaldarriaga 1996) and Wayne Hu’s superb compilation of CMB theory and experiment at

http://background.uchicago.edu/˜whu/

Problems

1.Verify the approximate relations (17.2.2) and (17.6.1).

2.Derive the results (17.2.13), (17.2.14) and (17.2.15).

3.Derive Equation (17.4.5).

4.Use the results of Chapter 11 to computer the evolution of the sound horizon as a function of redshift through matter–radiation equivalence until the point of recombination.

5.Derive the result (17.6.3).

6.A beam of unpolarised radiation is incident upon an electron. Show that the degree of polarisation in the light scattered at an angle θ to the incident beam is Π, where

Π= 1 − cos2 θ . 1 + cos2 θ

18

Peculiar Motions

of Galaxies

18.1 Velocity Perturbations

In our treatment of gravitational instability in Chapters 10 and 11 we focused upon the properties of the density field ρ or, equivalently, the density perturbations δ. The equations of motion do, however, contain another two variables, namely the velocity field v and the gravitational potential ϕ. These two quantities are actually quite simple to derive once the behaviour of the density has been obtained. To show this, let us write the continuity, Euler and Poisson equations again:

cf. Equations (10.2.1). As we suggested in Section 11.2, it now proves convenient to transform to comoving coordinates; here, however, we adopt a slightly di erent approach. Since we are looking for perturbations about the uniformly expanding solution with v = Hr, we introduce a peculiar velocity term V = v − Hr, where v = dr/dt, and t is the cosmological time. Let us now change the time coordinate to conformal time τ, so that dτ = dt/a(t), where a is the cosmic scale factor. This makes the handling of the comoving equations of motion rather simpler. We also use a comoving distance coordinate x = r/a. The equations of motion (18.1.1) are expressed in terms of proper distances r and proper time t; the comoving

394 Peculiar Motions of Galaxies

equations, expressed in conformal time τ and with derivatives now with respect to comoving coordinates, are

where δ, V and ϕ are the density, velocity and gravitational potential perturbations (in the latter case, within this comoving description, the mean value of ϕ vanishes so ϕ coincides with δϕ). The most important di erence between the two sets of Equations (18.1.1) and (18.1.2) is that, in the Euler Equation (18.1.2 b), there is a term in a/a˙ (remember that a˙ = da/dτ) which is due to the fact that our new system of coordinates is following the expansion and is therefore non-inertial. This term, called the ‘Hubble drag’, causes velocities to decay in comoving coordinates. There is, however, nothing strange about this: it is merely a consequence of the choice of coordinate system.

We have shown how to solve the equations of motion to obtain the behaviour of δ for various types of perturbations in Chapter 10. We shall now concentrate upon longitudinal adiabatic fluctuations (remember that transverse, or vortical, modes are generally decaying with time), and shall ignore the pressure gradient terms in the Euler Equation (18.1.2 b) because we assume k kJ. We showed in Section 10.8 that the linear solution to the density perturbation in such a situation behaves as a complicated function of the time and the value of Ω. We shall ignore the decaying mode, so that δ(x) = D(τ)δ+(x), and D is the linear growth law for the growing mode which, for Ω = 1 and matter domination, is given by D a τ2. For Ω ≠ 1 the expression for D is complicated but we do not actually need it. In fact, we only need the expression for

which has a behaviour as a function of Ω given quite accurately by the approximate form f Ω0.6. Notice that f = 1 for Ω = 1 is exact.

Now, given a solution for the density perturbation δ, one can easily derive the velocity and gravitational potential fields in these coordinates. Because the linear velocity field is irrotational, V can be expressed as the gradient of some velocity potential, ΦV , i.e.

It is helpful now to introduce the peculiar gravitational acceleration, g, which is simply

From the Poisson equation we have

2ϕ = 32 ΩH2a2δ, (18.1.6)

and, from the linearised equations of motion, it is then quite straightforward to show that

Notice that, for an Einstein–de Sitter universe, this last relation simply becomes V = gt. It is also the case that, in this model, ϕ is constant for the growing mode of linear theory. Regardless of Ω the velocity and gravitational acceleration fields are always in the same direction in linear theory.

It is also helpful to write explicitly the relationship between g (or V) and the density perturbation field δ(x) by inverting the relevant version of Poisson’s equa-

which we anticipated in Section 17.3. The expression for g can be found from (18.1.10) with the aid of (18.1.9).

Suppose now that the density field δ(x) has a known (or assumed) power spectrum P(k). From Equation (18.1.6) it follows immediately that the power spectrum

which we anticipated in Section 13.4. In linear theory the velocity field may be obtained as either the derivative of ΦV from (18.1.7) or by noting that, from the continuity equation,

Of course, V is a vector field, whereas both δ and ϕ are scalar fields. The velocity power spectrum (18.1.13) must therefore be interpreted as the power spectrum of the three components of V, each of which is a scalar function of position.

We should stress here that knowledge of P(k) is su cient to specify all the statistical properties of δ, V and ϕ only if δ is a Gaussian random field, which is the case we shall assume here.

396 Peculiar Motions of Galaxies

18.2 Velocity Correlations

In the previous section we showed how the gravitational potential and, more importantly, velocity fields are expected to behave in the gravitational instability picture. As we did in Chapter 14 with the density field, it is now necessary to explain how one might try to characterise the properties of V in a statistical manner. We shall concentrate upon generalising the covariance functions of δ we described in Section 14.9 to the case of a vector field V (Gorski 1988; Gorski et al. 1989).

The simplest possible statistical characterisation of V is the scalar velocity covariance function, defined by

where r = |x1 − x2|. One can show (we omit the details here) that this function can be expressed as

where j0(x) = (sin x)/x is the spherical Bessel function of order zero.

This is probably the simplest statistical characterisation of the velocity field but it does not contain information about directional correlations of the di erent components of V. Since velocity information is generally available only in one direction (the radial direction), the scalar correlation function (18.2.1) is of limited usefulness.

To furnish a full statistical description of the field we must define a velocity

Using the assumption of statistical homogeneity and isotropy, we can decompose the tensor Ψ into transverse and longitudinal parts in terms of scalar functions Ψ and Ψ ,

In the linear regime ×V = 0 and there is a consequent relationship between the longitudinal and transverse functions:

One can express the two functions Ψ , defined in Equations (18.2.6) and (18.2.7) in terms of the power spectrum P(k):

One can also extend this description to quantities involving the shear of the velocity field, but we shall not discuss these here.

In principle one can test a number of assumptions about the velocity field V by estimating the radial and transverse functions from a sample of peculiar velocities. For example, one can compute the expected form of the radial and transverse functions and then compare the results with estimates obtained from the data. There are, however, a number of problems with doing this kind of thing in practice. First, one needs a rather large sample of galaxy-peculiar motions. As we mentioned in Section 4.6, such a sample is di cult to obtain because it requires the independent determination of both redshifts and distances for a large number of galaxies. Moreover, such a sample would in any case only contain information about the radial component of the galaxy-peculiar motion. One can get around this in principle (see Section 18.5), but it does make it di cult to extract information about the Ψ(r) directly from the data. Results from this type of analysis are presently inconclusive, though they may become more useful when the quantity and quality of the data improve.

There is also a deeper problem. Generally one has estimates of the peculiar velocities of galaxies at a set of discrete points (galaxy positions) in space. When dealing with the density field, the assumption that ‘galaxies trace the mass’ allows one to construct a discrete set of correlation functions which are simply related to the covariance functions of the underlying density field. For the velocity field the situation is not so simple. If one has a continuous velocity field which is sampled at random positions, xi in Equation (18.2.3), then the two points may be