Cosmic microwave background anisotropies:

tantalizingly close to expectations

Sean Carroll, University of Chicago
carroll@theory.uchicago.edu, http://pancake.uchicago.edu/~carroll/
Ever since the detection of temperature anisotropies in the cosmic microwave background (CMB) by the COBE satellite in 1992, cosmologists have anticipated that information about the amplitude of these fluctuations across a range of angular scales could be an extraordinarily powerful constraint on cosmological models (see for example [1]). Now a series of new experiments -- the TOCO98 run of the MAT ground-based telescope in Chile [2], the balloon-borne Boomerang experiment flown both in Texas [3] and Antarctica [4], and the balloon-borne Maxima experiment flown in Texas [5] -- have turned these expectations into reality. The figure shows the new results combined with previous experiments, presented as amplitude of fluctuation vs. multipole moment l in a spherical harmonic decomposition.
  
Figure 1: Amplitude of CMB temperature anisotropies, as a function of multipole moment l (so that angular scale decreases from left to right). The data points are averaged from all of the experiments performed as of Summer 2000. The curve is a theoretical model with scale-free adiabatic scalar perturbations in a flat universe dominated by a cosmological constant, with a slightly higher baryon density than implied by big-bang nucleosynthesis. Courtesy of Lloyd Knox.
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Angular size decreases from left to right in the figure, as multipole moment is related to angular scale roughly by $\theta \sim 180^\circ /l$. This plot manifests three crucial features:
1.
A well-defined, narrow ( $\Delta l/l \sim 1$) peak in the power spectrum. This is strong evidence in favor of ``inflationary'' primordial perturbations.
2.
Location of the peak at $l\sim 200$. This is strong evidence in favor of a nearly flat spatial geometry ( $\Omega_{\rm tot} = 1$).
3.
A secondary peak ($l\sim 500$) which is rather small, if one is indicated at all. Previous best-fit models predicted a noticeable peak at this location; this might be evidence of tilt in the perturbation spectrum, a higher-than-expected baryon density, or more profound physics.
Let's consider each of these features in turn. The adjective ``inflationary'' refers to adiabatic perturbations that have been imprinted with (nearly) equal amplitudes on all scales (both greater and less than the Hubble radius H-1) before recombination. ``Adiabatic'' means that fluctuations in each species are correlated, so that the number density ratios of photons/baryons/dark matter are spatially constant. (Here, ``baryons'' is cosmology-speak for ``charged particles.'') These are the kinds of perturbations predicted by inflationary cosmology; it is entirely possible that a mechanism other than inflation could generate perturbations of this type, although no theories which do so have thus far been proposed. When we observe temperature fluctuations in the CMB, on scales which are larger than the Hubble radius at recombination the dominant effect is the gravitational redshift/blueshift as photons move through potential wells (the Sachs-Wolfe effect), while on smaller scales the intrinsic temperature anisotropy is dominant. An adiabatic mode of wavelength $\lambda$ (which grows along with the cosmic scale factor) will remain approximately constant in amplitude while $\lambda > H^{-1}$, after which it will begin to evolve under the competing effects of self-gravity (which works to increase the density contrast) and radiation pressure (which works to smooth it out). The result is an acoustic wave which oscillates during the period between when the mode becomes sub-Hubble-sized and recombination (when radiation pressure effectively ends). As the wave evolves it is also damped as photons dissipate from overdense to underdense regions. We therefore expect to see a series of peaks in the CMB spectrum, with the largest peak corresponding to a physical length scale equal to that of the Hubble radius at recombination. A crucial point is that the sharpness of this peak is evidence for the temporal coherence of the waves -- the evolution of a wave at any one wavelength is related in a simple way to that at other wavelengths, which enables the spectral features to be well-defined (see [6] for a discussion). In models where the perturbations are continually generated at all times (such as with topological defects), or models of ``isocurvature'' fluctuations in which different species are uncorrelated, this coherence is absent, and it is very difficult to get a sharp peak. The new observations thus strongly favor primordial adiabatic perturbations. As mentioned, the location of the first peak corresponds to the Hubble radius at the last scattering surface, $H_{\rm LS}^{-1}$. In a spatially flat universe, the observed angular scale of the peak would be the ratio of $H_{\rm LS}^{-1}$ to the angular diameter distance $r_\theta$ between us and the surface of last scattering. It turns out that, in a Friedmann-Robertson-Walker cosmology with plausible values of the various cosmological parameters, both $H_{\rm LS}^{-1}$ and $r_\theta$ depend on these parameters in roughly the same way: they are each proportional to $H_0^{-1}/\sqrt{\Omega_{{\rm M}0}}$, where $\Omega_{\rm M}$ is the ratio of the matter density to the critical density and subscripts 0 refer to quantities evaluated at the present time. The ratio $H_{\rm LS}^{-1}/r_\theta$ is thus approximately independent of the cosmological parameters. The observed angular scale of the first peak therefore depends primarily on the spatial geometry through which the photons have traveled; in a positively/negatively curved space, a fixed physical size corresponds to a larger/smaller angular size. The spatial geometry can be quantified by the total density parameter $\Omega_{\rm tot}$, and the angular dependence of the peak turns out to be $l_{\rm peak}\sim 200
\Omega_{\rm tot}^{-1/2}$. Thus, the observed peak at $l\sim 200$ provides excellent evidence for a flat universe. The most recent data are sufficiently precise that sub-dominant effects become relevant, and more careful analyses are necessary [7]. The quantitative results depend somewhat on which parameters are allowed to vary and which additional data are taken into account; the CMB data alone are actually best fit by a universe with a very small positive spatial curvature, but a perfectly flat universe is within the errors, while an open matter-dominated universe with $\Omega_{\rm tot}<0.5$ is strongly ruled out. Taking existing data on the Hubble parameter and large-scale structure distribution into acount implies the need for a positive cosmological constant, thus providing some independent confirmation for the striking supernova results [8]. The most unexpected feature of the observed CMB power spectrum, from the point of view of previously favored cosmological parameters, is the absence of an easily distinguishable secondary peak. It turns out that the expected peak can be suppressed in two straightforward ways: by ``tilting'' the primordial spectrum so that there is slightly less power on small scales, or by increasing the baryon-to-photon ratio. The tilting option, while plausible, is hard to accommodate within simple inflationary models, as a sufficient tilt is necessarily accompanied by additional tensor fluctuations on large scales [9], ruining the rest of the fit. The baryon density is most conveniently expressed in terms of $\Omega_{\rm b}h^2$, where $\Omega_{\rm b}$ is the density parameter in baryons and h = H0/(100 km/sec/Mpc). The CMB data [7] imply $\Omega_{\rm b}h^2 = 0.032\pm 0.009$, while big-bang nucleosynthesis [10] implies $\Omega_{\rm b}h^2 = 0.019 \pm 0.002$ (at $95\%$ confidence), with an ``extreme upper limit'' [11] of $\Omega_{\rm b}h^2 \leq 0.025$. Hence, consistency is just barely preserved at the edges of the allowed values. It would seem at this point most likely that some combination of slight tilt and ordinary experimental error have combined to create this apparent tension, but there remains the possibility of interesting new physics. (Note that the upper limit on the baryon density provides additional support for the necessity of non-baryonic dark matter.) The new CMB data are in a sense the idea experimental result, in that they provide useful constraints within the context of a successful theory while raising questions about aspects of that theory that can only be addressed by future experiments. The near future will see a number of new balloon, ground-based and satellite measurements of the CMB power spectrum on even smaller angular scales (including the presumed location of the third peak and beyond), which should reveal whether we are seeing a spectacular confirmation of the standard cosmology or the first signs of important deviations from it.

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