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

378 The Cosmic Microwave Background

Figure 17.1 Black and white representation of the COBE DMR four-year data map. The typical angular scale of fluctuations is around 10◦ and the typical amplitude is around 30µ K. Picture courtesy of George Smoot and NASA.

Translated into a value of σ8(mass) using (17.4.7) with n = 1 and a standard CDM transfer function, this suggests a value of b 1, which does not seem to allow the option of a linear bias for removing discrepancies between clustering and peculiar motions, such as those we shall discuss in Chapter 18. We should say that normalising everything to the quadrupole in this way is not a very good way of using the COBE data, which actually constitute a map of nearly the whole sky with a resolution of about 10◦. The RMS temperature anisotropy obtained from the whole map is of order 1.1 × 10−5. (Both this value and the quadrupole value are changed as more data from this experiment were analysed.) The quadrupole mode is actually not as well determined as the Cl for higher l, so a better procedure is to fit to all the available data with a convolution of the expected Cl for some amplitude with the experimental beam response and then determine the best fitting amplitude for the data. The results of more sophisticated data analysis like this are, however, in rough agreement with the simpler method mentioned above. Notice also that one can in principle determine the primordial spectral index n from the data by calculating C(ϑ) and comparing this with the expected form using Equation (17.4.5) for a given P(k) kn. The results obtained from this type of analysis are rather noisy, and do di er significantly depending on the type of analysis technique used, but they do seem consistent with n = 1.

Four years’ worth of data from the DMR experiment have now been published; the experiment was turned o in 1994. An independent detection of fluctuations on a slightly smaller scale than COBE was later announced by a team working at Tenerife using a ground-based beam-switching experiment (Hancock et al. 1993). The level and form of fluctuations detected in this experiment are consistent with those found by COBE.

17.4.3 Interpretation of the COBE results

At this stage, let us return to a point we raised above: the possible contribution of tensor perturbation modes to the large-scale CMB anisotropy. Gravitational waves do involve metric fluctuations and therefore do generate a Sachs–Wolfe e ect on scales larger than the horizon. Once inside the horizon, however, they redshift away (just like relativistic particles) and play no role at all in structure formation. Gravitational waves produce an e ect similar to scalar perturbations on large angular scales but have negligible influence upon ∆T/T on scales inside the horizon at zrec. Clearly, normalising the power spectrum P(k) to the observed Cl using (17.4.5) is incorrect if the tensor signal is significant.

One can define a power spectrum of gravitational wave perturbations in an analogous fashion to that of the density perturbations. It turns out that inflationary models also generically predict a tensor spectrum of power-law form, but with a spectral index

instead of equation (13.6.10). Since H is a small parameter the tensor spectrum will be close to scale invariant. It is also possible to calculate the ratio, R, between the tensor and scalar contributions to Cl:

To get a significant value of the gravitational wave contribution to Cl one therefore generally requires a significant value of H and therefore both scalar and tensor spectra will usually be expected to be tilted away from n = 1. If R = 1, then one can reconcile the COBE detection with a CDM model having a significantly high value of b. Because one cannot use Sachs–Wolfe anisotropies alone to determine the value of R, there clearly remains some element of ambiguity in the normalisation of P(k).

The Equations (17.4.10) and (17.4.11) are true for inflationary models with a single scalar field. More contrived models with several scalar fields can allow the two spectral indices and the ratio to be given essentially independently of each other. The shape of the COBE autocovariance function suggests that n cannot be much less than unity, so the prospects for having a single-field inflationary model producing a large tensor contribution seem small. On the other hand, we have no a priori information about the value of R so it would be nice to be able to constrain

380 The Cosmic Microwave Background

it using observations. It turns out that to perform such a test requires, at the very least, observations on a di erent (i.e. smaller) angular scale. From Figure 17.1 one can see that the scalar contribution increases around degree scales, while the tensor contribution dies away completely. We shall discuss the reasons for this shortly. In principle, one can therefore estimate R by comparing observations of Cl at di erent values of l although, as we shall see, the result is rather model dependent.

We should also mention that, if the CMB fluctuations are generated by primordial density perturbations which are Gaussian (Section 13.8), then the fluctuations ∆T/T should be Gaussian also. The nonlinear Sachs–Wolfe e ect generally produces a non-Gaussian temperature pattern, as do various extrinsic anisotropy sources we shall discuss in Section 17.6. To be precise, the prediction is that individual alm should have Gaussian distributions so that the actual sky pattern will only be Gaussian if one adds a significant number of modes for the central limit theorem to come into play. In principle it is possible to use statistical properties of sky maps to test the hypothesis that the fluctuations were Gaussian, though this task will have to wait for better data than are available at present. Notice that instrumental noise is almost always Gaussian, so if there is a lot of noise superimposed on the sky signal one can have problems detecting any non-Gaussian features which may be generated by extrinsic e ects, or non-Gaussian perturbations such as cosmic strings. At the moment, all we can say is that the COBE and Tenerife results are at least consistent with Gaussian primordial fluctuations.

17.5 Intermediate Scales

As we have already explained, the large-scale features of the microwave sky are expected to be primordial in origin. Smaller scales are closer to the size of the Hubble horizon at zrec so the density fluctuations present there may have been modified by various damping and dissipation processes. Moreover, there are physical mechanisms other than the Sachs–Wolfe e ect which are capable of generating anisotropy in the CMB on these smaller scales. We shall concentrate upon intrinsic sources of anisotropy in this section, i.e. those connected with processes occurring around trec; we mention some extrinsic (line-of-sight) sources of anisotropy in Section 17.6.

Let us begin with some naive estimates. For a start, if the density perturbations are adiabatic, then one should expect fluctuations in the photon temperature of the same order. Using ρr T4 and the adiabatic condition, 4δm = 3δr, we find that

which is also stated implicitly in Section 12.2. Another mechanism, first discussed by Zel’dovich and Sunyaev, is simply a Doppler e ect. Density perturbations at trec will, by the continuity equation, induce streaming motions in the plasma. This generates a temperature anisotropy because some electrons are moving towards

the observer when they last scatter the radiation and some are moving away. It turns out that the magnitude of this e ect for perturbations on a scale λ at time t is

where ct is of order the horizon scale at t.

The actual behaviour of the background radiation spectrum is, however, much more complicated than these simple arguments might suggest. The detailed computation of fluctuations originating on these scales is consequently much less straightforward than was the case for the Sachs–Wolfe e ect. In general one therefore resorts to a full numerical solution of the Boltzmann equation for the photons through recombination, taking into account the e ect of Thomson scattering, as described briefly in Section 11.10. The usual approach is to expand the distribution function of the radiation in spherical harmonics thereby generating a coupled set of equations for di erent l-modes of the distribution function; in Section 12.10 we used the brightness function, δ(r), to represent the perturbation to the radiation and wrote down a set of equations (11.9.7) for the l-modes, σl, defined by

δ(kr)(µ, t) = (2l + 1)Pl(µ)σl(k, t); (17.5.3)

l

µ = cos ϑ is the cosine of the angle between the photon momentum and the wave vector k. The solution of (11.9.7) is a fairly demanding numerical task. Given a set of σl, however, it is straightforward to show that the autocovariance function C(ϑ) of the sky at the present time is just

where the integral takes the distribution from Fourier space back to real space and the factor of 4 is due to the fact that δr = 4∆T/T. Fortunately, it is now possible to perform computations of both the transfer functions we described in Chapter 15 and the predicted temperature fluctuations rapidly and accurately using an approach that bypasses the complex hierarchy we described above. The code that does this, CMBFAST (Seljak and Zaldarriaga 1996), is available freely on the web so that anyone interested in computing the predicted pattern of fluctuations for their favourite model may download it.

As mentioned above, one can also allow for the e ect of di erent beam profiles and experimental configurations. For example, a double-beam experiment of the form (17.2.14) would have

× {1 + 13 J0[2αkr0(1 − µ2)1/2] − 43 J0[αkr0(1 − µ2)1/2]}

× exp[−k2σ2r02(1 − µ2)1/2] dk dµ, (17.5.5)

when pressure forces become negligible, these waves are left with phases which depend on their wavelength. Both the photon temperature fluctuations (17.5.1) and the velocity perturbations (17.5.2) are therefore functions of wavelength (both contribute to ∆T/T in this regime) and this manifests itself as an almost periodic behaviour of Cl. The use of the term ‘Doppler peak’ to describe only the first maximum of these oscillations is misleading because it is actually just the first (and largest-amplitude) Sakharov oscillation. Although velocities are undoubtedly important in the generation of this feature, it is wrong to suggest that the physical origin of the first peak in the angular power spectrum is qualitatively di erent from the others.

The power spectrum of the matter fluctuations is also expected to display oscillations relating to this phase e ect but with a much lower contrast. The reason for this is that most of the matter in standard models is neither baryonic nor collisional. Consequently it neither interacts by scattering with radiation nor produces restoring forces to support induced oscillations. Essentially the CMB anisotropy is influenced by the baryonic component only so the oscillations are dominant, while the power spectrum of the dark matter is smooth with only small baryonic oscillations superimposed upon it.

The physical origin of these oscillations is interesting enough, but their importance in present and future cosmological investigations is paramount. The reason for this is that the position and relative amplitudes of the Doppler peak and its ‘harmonics’ are a sensitive diagnostic not just of the precise mix of dark matter and baryons, but also the values of the principal cosmological parameters. For instance, the position of the first peak is a direct route to the density parameter Ω0 or, rather, the global curvature k. The physical length scale at which this peak occurs corresponds to the size of the sound horizon (cstrec, where cs is the sound speed) at the surface of last scattering roughly defined by trec. This does not vary much with cosmological parameters. However, this length scale subtends an angle that depends on the geometry of the Universe. Consequently the spherical harmonic l that corresponds to the Doppler peak changes if the background curvature changes. In a flat universe the peak occurs around l 200. If the universe has positive curvature, geodesics converge towards the observer so the angle subtended by a ‘rod’ of fixed size is larger than in a flat universe. The peak therefore moves to smaller l in this case. If spatial sections are negatively curved, then the peak moves to higher l; see Figure 2.3 to see why the angle looks smaller in an open universe. This shows how important the first peak is, but the detailed shape of the power spectrum has a strong dependence on the other parameters too. An accurate measurement of these features promises to nail many of the uncertainties facing cosmology in one fell swoop. For further discussion of open universes see Kamionkowski and Spergel (1994).

There are complications, of course. One is the relatively slow rate of recombination. One e ect of this is that the optical depth to the last scattering surface can be quite large, and small-scale features can be smoothed out. For example, as we discussed in Section 9.4 in the context of the standard theory of recombination, the last scattering surface can have an e ective ‘width’ up to ∆z 400,

384 The Cosmic Microwave Background

which corresponds to a proper distance now of ∆L 40h−1 Mpc, and to an angular scale 20 arcmin. The finite thickness of the last scattering surface can mask anisotropies on scales less than ∆L in the same way that a thick piece of glass prevents one from seeing small-scale features through it. This causes a damping of the contribution at high l and thus a considerable reduction in the ∆T/T relative to the photon temperature fluctuations (17.5.1).

High angular frequency fluctuations are also quite sensitive to the possibility that the Universe might have been reionised at some epoch. As we shall see in Chapter 20, we know that the intergalactic medium is now almost completely ionised. If this happened early enough, it could smear out the fluctuations on scales less than a few degrees, rather than the few arcminutes for standard recombination, the case shown in Figure 17.2. Some non-standard cosmologies involve such a late recombination so that ∆z might be much larger. The minimum allowable redshift is, however, z 30 because an optical depth τ 1 requires enough electrons (and therefore baryons) to do the scattering; a value z < 30 would be incompatible with Ωb < 0.1; we discussed this in Chapter 9. In any case, if some physical process caused the Universe to be reheated after trec, then it might smooth out anisotropy on scales less than the horizon scale at the time when the reionisation occurred. Recall from Equation (17.4.6) that the angular scale corresponding to the particle horizon at z is of order (Ω/z)1/2, so late reionisation at z 30 could smooth out structure on scales of 10◦ or less, but not scales larger than this. We shall see in Section 17.6 that, if this indeed occurred, one might expect to see a significant anisotropy on a smaller angular scale, generated by secondary e ects.

The message one should take from these comments is that the fluctuations on these scales are much more model dependent than those on larger scales. In principle, however, they enable one to probe quite detailed aspects of the physics going on at trec and are quite sensitive to parameters which are otherwise hard to estimate. Moreover, tensor modes do not produce any Doppler motions and their contribution to Cl should therefore be small for high l. Although these oscillatory features are potentially a very sensitive diagnostic of the perturbations generating the CMB anisotropy, it is di cult to resolve them.

The problem with these experiments, which are all either balloon borne or ground based, is twofold. Firstly, they usually probe a relatively small part of the sky and the signal they see may not be representative of the whole sky, i.e. they are dominated by ‘sample variance’. The second problem is that, until recently, they generally did not have the ability to remove point sources (because of the smaller beam) or non-thermal emission (because of the smaller number of frequency channels) as e ectively as COBE. Observational programmes aimed at improving the situation have been pursued with great vigour over the last few years, as indicated by the forest of error bars in Figure 17.2.

Over the last few years the situation has changed dramatically with two longduration balloon flights bearing sensitive bolometers finally giving convincing measurements of the Doppler peak and its first one or two overtones (Hanany

Figure 17.3 The angular power spectrum of the CMB estimated by MAXIMA-1 and Boomerang. Picture courtesy of Andrew Ja e.

et al. 2000; Ja e et al. 2001); see Figure 17.3. The crucial point about this result is the position of the first peak. This tightly constrains the curvature to be very small. Taken together with the supernova results and the relatively low apparent matter density discussed in Chapter 4, this strongly suggests the existence of a cosmological constant in the Einstein field equations.

These measurements still come from relatively small patches of the sky but show how strong the constraints on cosmological models are likely to become in the near future when all-sky satellites are launched. As we write, in 2002, a US-led mission called MAP (Microwave Anisotropy Probe) is already in space collecting data from which high-resolution whole-sky maps will be constructed. In 2007 the European Space Agency’s Planck Surveyor will do a similar job at even higher resolution.

As a final remark, we should stress that intrinsic CMB temperature anisotropy is expected to be Gaussian on these scales, since it is generated by linear processes from density perturbations which are themselves Gaussian. As with the Sachs– Wolfe e ect, one can in principle use the properties of ∆T/T to test the Gaussian hypothesis on these scales also. For example, in the cosmic-string scenario the dominant contribution to the CMB anisotropy is generated by cosmic strings lying between the observer and the last scattering surface which distort the photon trajectories. The detailed statistical properties of the pattern of temperature maps on intermediate and large scales in this scenario will be very di erent from those in Gaussian scenarios.

17.6Smaller Scales: Extrinsic E ects

As explained in the introduction to this chapter, one of the main motivations for studying the temperature anisotropy of the cosmic microwave background is that one can, in principle, look directly at the e ects of primordial density fluctuations and therefore probe the initial conditions from which structure is usually

386 The Cosmic Microwave Background

Figure 17.4 A simulation of the CMB sky as it might be seen by MAP or Planck.

supposed to have grown. In the previous two sections we have elucidated the physical mechanisms responsible for generating intrinsic anisotropy and shown that these do indeed involve the primordial density perturbations. The problem is that the length scales probed by these anisotropies are much larger than those of direct relevance to galaxy and cluster formation. In fact, there is a simple rule relating a given (comoving) length scale to the angle that scale subtends on the last scattering surface:

As we explained in Section 17.5, temperature anisotropies due to fluctuations on length scales up to 40h−1 Mpc will probably be smoothed out by the finite thickness of the last scattering surface. One cannot therefore probe scales of direct relevance to cluster and galaxy formation using measurements of intrinsic CMB anisotropy. COBE and related experiments can only constrain theories of structure formation if there is a continuous spectrum of density fluctuations with a welldefined shape so that a measurement of the amplitude on the scale of a thousand Mpc or so, corresponding to COBE, can be extrapolated down to smaller scales. Because these experiments do not in themselves supply a test of the shape of the power spectrum on smaller scales, theories must be constrained by combining CMB anisotropy measurements with galaxy-clustering data or peculiar velocity data; the latter will be discussed in the next chapter.

There are various ways, however, in which small-scale anisotropy measurements can yield important information on short-wavelength fluctuations due to extrinsic e ects, rather than the intrinsic e ects we have discussed so far. We shall discuss some possible mechanisms of this type in this section. Because these are highly model dependent and, in some cases, involve complicated physical pro-

cesses, we shall restrict ourselves to a qualitative discussion without many technicalities. The interested reader is referred to the bibliography for further details.

One important consideration on scales of arcminutes and less is the contribution of various kinds of extragalactic sources to the CMB anisotropy. Point sources generally have a non-thermal spectrum so they can, in principle, be accounted for using multi-frequency observations, but this is by no means straightforward in practice. The brightest point sources can be removed quite easily as they may be resolved by the experimental beam. An integrated background due to large numbers of relatively faint sources is, however, very di cult to deal with. Many of the intermediate scale measurements mentioned in Section 17.5 also su er from the di culty of point-source subtraction. Although CMB measurements may in principle place constraints on the evolution of various kinds of radio source, in practice these are usually treated as a nuisance which is to be removed. Nevertheless, it is useful to calculate the approximate contribution to ∆T/T from point sources distributed in di erent ways. Firstly, suppose the objects were actually present before zrec, which seems rather unlikely. The radiation from them would have to be thermalised by some agent, such as grains of dust, otherwise it would lead to a spectral distortion of order q, the fraction of the CMB energy density which they generate. If the sources are randomly distributed in space, then the e ective anisotropy is just due to Poisson statistics for ϑ > ϑH(zrec) = ϑ given by Equation (17.4.6):

where Nϑ is the mean number of sources in a beam of width ϑ. On angles less than ϑ the radiation would be smoothed out. For example, if we have a population of sources with (comoving) mean spacing ls at a redshift zs, it is quite easy to show that

This corresponds to two-dimensional white noise filtered on a scale ϑ .

Now consider the case of sources at 1 z < zrec. In this case there is no filtering and there will be a spectral distortion because this radiation cannot be thermalised. The resulting ∆T/T is just like (17.6.3) with ϑ = 0. As we remarked above, limits on the departure of the spectrum from a black-body form can therefore constrain the contribution from such sources.

The expression (17.6.3) must be modified considerably if one is dealing with local sources, by which we mean those with zs 1 or thereabouts. Local sources are usually referred to as ‘contamination’, which gives some idea of how astronomers regard them. The contribution from such objects is dominated by the brightest ones found in a solid angle ϑ2 and is therefore closely connected with the log N–log S relationship (the radio astronomers equivalent of the number– magnitude relation). One generally has