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

368 The Cosmic Microwave Background

which reduces the expected temperature fluctuation still further. It was an interesting experience to those who had been working in this field for many years to see this trend change sign abruptly in 1992. The ∆T/T fluctuations seen by COBE were actually larger than predicted by the standard version of the CDM model. This must have been the first time a theory had been rejected because it did not produce high enough temperature fluctuations!

Searches for CMB anisotropy would be (and have been), on their own, enough subject matter for a whole book. In one chapter we must therefore limit our scope quite considerably. Moreover, COBE marked the start, rather than the finish, of this aspect of cosmology and it would have been pointless to produce a definitive review of all the ongoing experiments and implications of the various upper limits and half-detections for specific theories, when it is possible that the whole picture will change within a year or two. Therefore, we shall mainly concentrate on trying to explain the physics responsible for various forms of temperature anisotropy. We shall not discuss any specific models in detail, except as illustrative examples, and our treatment of the experimental side of this subject will be brief and nontechnical. Finally, we shall be extremely conservative when it comes to drawing conclusions. As we shall explain, the situation with respect to CMB anisotropy as a function of angular scale is still very confused and we feel the wisest course is to wait until observations are firmly established before drawing definite conclusions.

17.2 The Angular Power Spectrum

Let us first describe how one provides a statistical characterisation of fluctuations in the temperature of the CMB radiation from point to point on the celestial sphere.

The usual procedure is to expand the distribution of T on the sky as a sum over spherical harmonics

where θ and φ are the usual spherical angles; ∆T/T is defined by Equation (4.8.1). The l = 0 term is a monopole correction which essentially just alters the mean temperature on a particular observer’s sky with respect to the global mean over an ensemble of all possible such skies. We shall ignore this term from now on because it is not measurable. The l = 1 term is a dipole term which, as we shall see in Section 17.3, is attributable to our motion through space. Since this anisotropy is presumably generated locally by matter fluctuations, one tends to remove the l = 1 mode and treat it separately. The remaining modes, from the quadrupole (l = 2) upwards, are usually attributed to intrinsic anisotropy produced by e ects either at trec or between trec and t0. For these e ects the sum in Equation (17.2.1) is generally taken over l 2. Higher l modes correspond to fluctuations on smaller angular scales ϑ according to the approximate relation

The expansion of ∆T/T in spherical harmonics is entirely analogous to the planewave Fourier expansion of the density perturbations δ; the Ylm are a complete orthonormal set of functions on the surface of a sphere, just as the plane-wave modes are a complete orthonormal set in a flat three-dimensional space. The alm are generally complex, and satisfy the conditions

where δij is the Kronecker symbol and the average is taken over an ensemble of realisations. The quantity Cl is the angular power spectrum,

which is analogous to the power spectrum P(k) defined by Equation (14.2.5). It is also useful to define an autocovariance function for the temperature fluctuations,

and the nˆi are unit vectors pointing to arbitrary directions on the sky. The expectation values in (17.2.3) and (17.2.5) are taken over an ensemble of all possible skies. One can try to estimate Cl or C(ϑ) from an individual sky using an ergodic hypothesis: an average over the probability ensemble is the same as an average over all spatial positions within a given realisation. This only works on small angular scales when it is possible to average over many di erent pairs of directions with the same ϑ, or many di erent modes with the same l. On larger scales, however, it is extremely di cult to estimate the true C(ϑ) because there are so few independent directions at large ϑ or, equivalently, so few independent l modes at small l. Large-angle statistics are therefore dominated by the e ect of cosmic variance: we inhabit one realisation and there is no reason why this should possess exactly the ensemble average values of the relevant statistics.

As was the case with the spatial power spectrum and covariance functions, there is a simple relationship between the angular power spectrum and covariance

function:

∞

C(ϑ) = 1 (2l + 1)ClPl(cos ϑ), (17.2.7) 4π l=2

where Pl(x) is a Legendre polynomial. We have written the sum explicitly to omit the monopole and dipole contributions from (17.2.1).

It is quite straightforward to calculate the cosmic variance corresponding to an

ˆ

estimate obtained from observations of a single sky, C(ϑ), of the ‘true’ autocovariance function, C(ϑ):

370 The Cosmic Microwave Background

where the aˆlm are obtained from a single realisation on the sky. The statistical procedure for estimating these quantities is by no means trivial, but we shall not describe the various possible approaches here: we refer the reader to the bibliography for more details. In fact the variance of the estimated aˆlm across an

We have again explicitly omitted the monopole and dipole terms from the sums in (17.2.8) and (17.2.9).

In Sections 17.4–17.6 we shall discuss the various physical processes that produce anisotropy with a given form of Cl (we mentioned these briefly in Section 4.8); the dipole is discussed in Section 17.3. Generally the form of Cl must be computed numerically, at least on small and intermediate scales, by solving the transport equations for the matter–radiation fluid through decoupling in the manner discussed in Chapter 13. We shall make some remarks on how this is done later in this chapter. As we shall see, the comparison of a theoretical Cl against an observed

ˆ ˆ

Cl or C(ϑ) in principle provides a powerful test of theories of galaxy formation. Before discussing the physics, however, it is worth making a few remarks about observations of the CMB anisotropy.

The fluctuations one is looking for generally have an amplitude of order 10−5. One is therefore looking for a signal of amplitude around 30 K in a background temperature of around 3 K. One’s observational apparatus, even with the aid of sophisticated cooling equipment, will generally have a temperature much higher than 3 K and one must therefore look for a tiny variation in temperature on the sky against a much higher thermal noise background in the instrument. From the ground, one also has the problem that the sky is a source of thermal emission at

microwave frequencies. Noise of these two kinds is usually dealt with by integrat-

√

ing for a very long time (thermal noise decreases as t, where t is the integration time) and using some kind of beam-switching design in which one measures not ∆T at individual places but temperature di erences at a fixed angular separation (double beam switching) or alternate di erences between a central point and two adjacent points (triple beam switching). Recovering the ∆T at any individual point (i.e. to produce a map of the sky) from these types of observations is therefore not trivial. Moreover, any radio telescope capable of observing the microwave sky will have a finite beamwidth and will therefore not observe the temperature point by point, but would instead produce a picture of the sky convolved with some

It is generally more convenient to work in terms of l than in terms of ϑ so we shall express the response of the instrument as Fl; the relationship between Fl

372 The Cosmic Microwave Background

our Galaxy with respect to a cosmologically comoving frame in which the CMB is isotropic. The angle θ is the angle between the observation and the direction of motion of the observer. The e ect is not a simple Doppler e ect. The actual level of anisotropy is of order β = v/c 10−3, so for the derivation of the result we shall ignore relativistic corrections. The point is that the Doppler e ect will increase the energy of photons seen in the direction of motion relative to that of a static observer in an isotropic background. However, the interval of frequencies dν is also increased by the same factor of (1 + β cos θ). Since the temperature is defined in terms of energy per unit frequency, the net Doppler e ect on the temperature is zero. There are, however, two other e ects. The first is that the moving observer actually sweeps up more photons. In a direction θ the observer collects (c dt+v cos θ dt)/c dt more photons than an observer at rest, which gives rise to an increase in the temperature by a factor of (1 + β cos θ). The second e ect is aberration: the solid angle for a moving observer gets smaller by a factor (1 + β cos θ)−2, so the flux goes up by the reciprocal of this factor. Hence the spectral intensity seen by a moving observer is

Inserting all the factors in (9.5.1) gives the Planck spectrum with T(θ) = T0(1 + β cos θ). Including all the relativistic e ects, to leading order in β, gives

cf. Equations (4.8.2) and (11.7.3). The reason why this is accepted to be due to our motion is that the quadrupole moment (variation on 90◦ scale; l = 2) is much less: if it were generated by intrinsic anisotropy, one should expect these two scales to contribute roughly the same order of magnitude to ∆T/T. By making a map of T(θ, φ) on the sky, one can determine the velocity vector that explains the dipole. The measured velocity is 390 ± 30 km s−1. After subtracting the Earth’s motion around the Sun, the Sun’s motion around the Galactic centre and the velocity of our Galaxy with respect to the centroid of the Local Group, this dipole anisotropy tells us the speed and direction of the Local Group through the cosmic reference frame. The result is a velocity of about 600 km s−1 in the direction of HydraCentaurus (l = 268◦, b = 27◦) (Rowan-Robinson et al. 1990).

In the gravitational instability picture this velocity can be explained as being due to the net gravitational pull on the Local Group generated by the inhomogeneous distribution of matter around it. In fact the net gravitational acceleration is just

g = G ρ(r)r dV, (17.3.3) r3

where the integral should formally be taken to infinity. As we shall see in Section 18.1, the linear theory of gravitational instability predicts that this gravitational acceleration is just proportional to, and in the same direction as, the net velocity. Moreover, the constant of proportionality depends on f Ω0.6. If one

can measure ρ from a su ciently large sample of galaxies, then one can in principle determine Ω. Of course, the ubiquitous bias factor intrudes again, so that one can only determine f/b, and that only as long as b is constant.

The technique is simple. Suppose we have a sample of galaxies with some welldefined selection criterion so that the selection function, the probability that a galaxy at distance r from the observer is included in the catalogue, proportional to the function ψ in Section 16.3, has some known form φ(r). Then the acceleration vector g at the origin of the coordinates can be approximated by

where the ri are the galaxy positions, M is a normalisation factor with the dimension of mass to take into account the masses of the galaxies at ri, and the factor 1/φ(ri) allows for the galaxies not included in the survey. The sum in Equation (17.3.4) is taken over all the galaxies in the sample. The dipole vector D can be computed from the catalogue and, as long as it is aligned with the observed CMB dipole anisotropy, one can estimate Ω00.6. It must be emphasised that this method measures only the inhomogeneous component of the gravitational field: it will not detect a mass component that is uniform over the scale probed by the sample. This technique has been very popular over the last few years, mainly because the various IRAS galaxy catalogues are very suitable for this type of analysis. There are, however, a number of di culties which need to be resolved before the method can be said to yield an accurate determination of Ω.

First, and probably most importantly, is the problem of convergence. Suppose one has a catalogue that samples a small sphere around the Local Group, but that this sphere is itself moving in roughly the same direction. For this to happen, the Universe must be significantly inhomogeneous on scales larger than the catalogue can probe. In this circumstance, the actual velocity explained by the dipole of the catalogue is not the whole CMB dipole velocity but only a part of it. It follows then that one would overestimate the Ω0.6 factor by attributing all of the observed velocity to the observed local dipole D when, in reality, this dipole is only responsible for part of this velocity. One must be sure, therefore, that the sample is deep enough to sample all contributions to the Local Group motion if one is to determine Ω with any accuracy. Analyses of the dipole properties of the IRAS catalogues seem to indicate a rather high value of f/b, consistent with Ω = 1. On the other hand, catalogues of rich clusters, which have a selection function φ(r) that falls less steeply on large scales than that of IRAS galaxies, seem to indicate Ω 0.3 (Plionis et al. 1993).

Another problem is that, because of the weighting in Equation (17.3.4), one must ensure that the selection function is known very accurately, especially at large r. This essentially means knowing the luminosity function extremely well, particularly for the brightest objects (the ones that will be seen at great distances). There is also the problem that galaxy properties may be evolving with time so the luminosity function for distant galaxies may be di erent from that of nearby ones. There is also the problem of bias. We have assumed a linear bias throughout the

374 The Cosmic Microwave Background

above discussion. The ramifications of nonlinear and/or non-local biases have yet to be worked out in any detail.

Finally, we should mention the e ect of redshift-space distortions, cf. Section 18.5. On the scales needed to probe large-scale structure, it is not practicable to obtain distances for all the objects, so one uses redshifts to estimate distances. One might expect this to be a good approximation at large r, but working in redshift space rather than real space introduces alarming distortions into the analysis. One can illustrate some of the problems with the following toy example. Suppose an observer sits in a rocket and flies through a uniform distribution of galaxies. If he looks at the distribution in redshift space, even if the galaxies have no peculiar motions, he will actually see a dipole anisotropy caused by his motion. He may, if he is unwise, thus determine Ω from his own velocity and this observed dipole: the answer would, of course, be entirely spurious and would have nothing whatsoever to do with the mean density of the Universe.

The combination of redshift-space e ects, bias and lack of convergence is di - cult to unravel. We therefore suggest that determinations of Ω by this method be treated with caution. For the latest developments in dipole analysis, see RowanRobinson et al. (2000).

17.4 Large Angular Scales

17.4.1The Sachs–Wolfe e ect

Having dealt with the dipole, we should now look at sources of intrinsic CMB temperature anisotropy. On large scales the dominant contribution to ∆T/T is expected to be the Sachs–Wolfe e ect (Sachs and Wolfe 1967). This is a relativistic e ect due to the fact that photons travelling to an observer from the last scattering surface encounter metric perturbations which cause them to change frequency. One can understand this e ect in a Newtonian context by noting that metric perturbations correspond to perturbations in the gravitational potential, δϕ, in Newtonian theory and these, in turn, are generated by density fluctuations, δρ. Photons climbing out of such potential wells su er a gravitational redshift but also a time dilation e ect so that one e ectively sees them at a di erent time, and thus at a di erent value of a, to unperturbed photons. The first e ect gives

where λ is the scale of the perturbation. This argument is not rigorous, as the split into potential and time-delay components is not gauge invariant but does explain why (17.4.1) is not the whole e ect.

So far we have considered only adiabatic fluctuations. Since the Sachs–Wolfe e ect is generated by fluctuations in the metric, then one might expect that isocurvature fluctuations (perturbations in the entropy which leave the energy density unchanged and therefore, one might expect, produce negligible fluctuations in the metric) should produce a very small Sachs–Wolfe anisotropy. This is not the case, for two reasons. Firstly, initially isocurvature fluctuations do generate significant fluctuations in the matter component and hence in the gravitational potential, when they enter the horizon; this is due to the influence of pressure gradients. In addition, isocurvature fluctuations generate significant fluctuations in the radiation density after zeq, because the initial entropy perturbation is then transferred into the perturbation of the radiation. The total anisotropy seen is therefore the sum of the Sachs–Wolfe contribution and the intrinsic anisotropy carried by the radiation. The upshot of all this is that the net anisotropy is six times larger for isocurvature fluctuations than for adiabatic ones. This is su cient on its own to rule out most isocurvature models since the level of anisotropy detected is roughly that expected for adiabatic perturbations.

According to Equation (17.4.3), the temperature anisotropy is produced by gravitational potential fluctuations sitting on the last scattering surface. In fact this is not quite correct, and there are actually two other contributions arising from the

where the integral is taken along the path of a photon from the last scattering surface to the observer. This e ect, usually called the Rees–Sciama e ect, is due to the change in depth of a potential well as a photon crosses it. If the well does not deepen, a photon does not su er a net shift in energy from falling in and then climbing up. If the potential changes while the photon moves through it, however, there will be a net change in the frequency. In a flat universe, δϕ is actually constant in linear theory (see Section 18.1 for a proof) so one needs to have nonlinear evolution in order to produce a nonlinear Sachs–Wolfe e ect. Since the potential fluctuations are of order δϕ δ(λ/ct)2 one requires nonlinear evolution of δ on very large scales to obtain a reasonably large contribution. To calculate the e ect in detail for a background of perturbations is quite di cult because of the inherent nonlinearity involved. On the other hand, it is possible to calculate the e ect using simplified models of structure. For example, a large void region can be modelled as an isolated homogeneous underdensity (the inverse of the spherical top hat discussed in Section 14.1) which can be evolved analytically into the nonlinear regime. It turns out that, for a spherical void of the same diameter as the large void seen in Bootes, one expects to see a cold spot corresponding to ∆T/T 10−7 on an angular scale around 15◦. Large clusters or superclusters can be modelled using top-hat models, the Zel’dovich approximation or perturbative techniques. The Shapley concentration of clusters, for example, is expected

376 The Cosmic Microwave Background

to produce a hotspot with ∆T/T 10−5 on a scale around 20◦. In general these e ects are smaller than the intrinsic CMB anisotropies we have described, but may be detectable in large, sensitive sky maps: the position on the sky of these features should correspond to known features of the galaxy distribution.

The second additional contribution comes from tensor metric perturbations, i.e. gravitational waves. These do not correspond to density fluctuations and have no Newtonian analogue but they do produce redshifting as a result of the perturbations in the metric. As we shall see at the end of this section, gravitational waves capable of generating large-scale anisotropy of this kind are predicted in many inflationary models, so this is potentially an important e ect.

For the moment, we shall assume that we are dealing with temperature fluctuations produced by potential fluctuations of the form (17.4.3). What is the form of Cl predicted for fluctuations generated by this e ect? This can be calculated quite straightforwardly by writing δϕ as a Fourier expansion and using the fact that the power spectrum of δϕ is proportional to k−4P(k), where P(k) is the power spectrum of the density fluctuations. Expanding the net ∆T/T in spherical harmonics and averaging over all possible observer positions yields, after some work,

where jl is a spherical Bessel function and x = 2c/H0. One can also show quite straightforwardly that, for an initial power spectrum of the form P(k) k, the quantity l(l+1)Cl is independent of the mode order l for the Sachs–Wolfe perturbations. In any case the shape of Cl for small l is determined purely by the shape of P(k), the shape of the primordial fluctuation spectrum before it is modified by the transfer function. The reason for this is easy to see: the scale of the horizon at zrec is of order

so that ϑH 2◦ for zrec 1000, which is the usual situation. Fluctuations on angular scales larger than this will retain their primordial character since they will not have been modified by any causal processes inside the horizon before zrec. One must therefore be seeing the primordial (unprocessed) spectrum. This is particularly important because observations of Cl at small l can then be used to normalise P(k) in a manner independent of the shape of the power spectrum, and therefore independent of the nature of the dark matter.

One simple way to do this is to use the quadrupole perturbation modes which have l = 2. There are five spherical harmonics with l = 2, so the quadrupole has five components a2m (m = −2, −1, 0, 1, 2) that can be determined from a map of the sky even if it is noisy. From (17.4.5), we can show that, if P(k) k, then

This connects the observed temperature pattern on the sky with the mass fluctuations δM/M = σM observed at the present epoch on a scale R.

17.4.2 The COBE DMR experiment

Such is the importance of the COBE discovery that it is worth describing the experiment in a little detail. The COBE satellite actually carried several experiments on it when it was launched in 1989. One of these (FIRAS) measured the spectrum displayed in Figure 9.1. The anisotropy experiment, called the DMR, yielded a positive detection of anisotropy after one year of observations. The advantage of going into space was to cut down on atmospheric thermal emission and also to allow coverage of as much of the sky as possible (ground-based observations are severely limited in this respect). The orbit and inclination of the satellite is controlled so as to avoid contamination by reflected radiation from the Earth and Moon. Needless to say, the instrument never points at the Sun. The DMR detector consists of two horns at an angle of 60◦; a radiometer measures the di erence in temperature between these two horns. The radiometer has two channels (called A and B) at each of three frequencies: 31.5, 53 and 90 GHz, respectively. These frequencies were chosen carefully: a true CMB signal should be thermal and therefore have the same temperature at each frequency; various sources of galactic emission, such as dust and synchrotron radiation, have an e ective antenna temperature which is frequency dependent. Combining the three frequencies therefore allows one to subtract a reasonable model of the contribution to the observed signal which is due to galactic sources. The purpose of the two channels is to allow a subtraction of the thermal noise in the DMR receiver. Assuming the sky signal and DMR instrument noise are statistically independent, the net temperature variance observed is

Adding together the input from the two channels and dividing by two gives an estimate of σobs2 ; subtracting them and dividing by two yields an estimate of σDMR2 , assuming that the two channels are independent. Taking these two together, one can therefore obtain an estimate of the RMS sky fluctuation. The first COBE announcement in 1992 gave σsky = 30 ± 5 K, after the data had been smoothed on a scale of 10◦.

In principle the set of 60◦ temperature di erences from COBE can be solved as a large set of simultaneous equations to produce a map of the sky signal. The COBE team actually produced such a map using the first year of data from the DMR experiment. It is important to stress, however, that, because the sky variance is of the same order as the DMR variance, it is not correct to claim that any features seen in the map necessarily correspond to structures on the sky. Only when the signal-to-noise ratio is much larger than unity can one pick out true sky features with any confidence. The first-year results should therefore be treated only as a statistical detection.

The value of a2lm 1/2 obtained by COBE is of order 5 × 10−6. This can also be expressed in terms of the quantity Qrms, which is defined by