The cosmological interpretation of a measured microwave background power spectrum requires, to some extent, the introduction of a particular space of models. A very simple, broad, and well-motivated set of models are motivated by inflation: a universe described by a homogeneous and isotropic background with phase-coherent, power-law initial perturbations which evolve freely. This model space excludes, for example, perturbations caused by topological defects or other ``stiff'' sources, arbitrary initial power spectra, or any departures from the standard background cosmology. This set of models has the twin virtues of being relatively simple to calculate and best conforming to current power spectrum measurements. (In fact, most competing cosmological models, like those employing cosmic defects to make structure, are essentially ruled out by current microwave background and large-scale structure measurements.) This section will describe the parameters defining the model space and discuss the extent to which the parameters can be constrained through the microwave background.
The parameters defining the model space can be broken into three types: cosmological parameters describing the background space-time; parameters describing the initial conditions; and other parameters describing miscellaneous additional physical effects. Background cosmological parameters are:
Parameters describing the initial conditions are:
Other miscellaneous parameters include:
A realistic parameter analysis might include at least 8 free parameters. Given a particular microwave background measurement, deciding on a particular set of parameters and various priors on those parameters is as much art as science. For the correct model, parameter values should be insensitive to the size of the parameter space or the particular priors invoked. Several particular parameter space analyses are mentioned below in Sec. 6.5.
While the above parameters are useful and conventional for characterizing cosmological models, the features in the microwave background power spectrum depend on various physical quantities which can be expressed in terms of the parameters. Here the physical quantities are summarized, and their dependence on parameters given. This kind of analysis is important for understanding the model space of parameters as more than just a black box producing output power spectra. All of the physical dependences discussed here can be extracted from Hu and Sugiyama (1996). By comparing with numerical solutions to the evolution equations, Hu and Sugiyama demonstrated that they had accounted for all relevant physical processes.
Power-law initial conditions are determined in a straightforward way by the appropriate parameters Q, n, r, and n_{T}, if the perturbations are purely adiabatic. Additional parameters must be used to specify any departure from power law spectra, or to specify an additional admixture of isocurvature initial conditions (e.g. Bucher et al. 1999). These parameters directly express physical quantities.
On the other hand, the physical parameters determining the evolution of the initial perturbations until decoupling involve a few specific combinations of cosmological parameters. First, note that the density of radiation is fixed by the current microwave background temperature which is known from COBE, as well as the density of the neutrino backgrounds. The gravitational potentials describing scalar perturbations determine the size of the Sachs-Wolfe effect and also magnitude of the forces driving the acoustic oscillations. The potentials are determined by _{0} h^{2}, the matter density as a fraction of critical density. The baryon density, _{b} h^{2}, determines the degree to which the acoustic peak amplitudes are modulated as described above in Sec. 5.4.
The time of matter-radiation equality is obviously determined solely by the total matter density _{0} h^{2}. This quantity affects the size of the dark matter fluctuations, since dark matter starts to collapse gravitationally only after matter-radiation equality. Also, the gravitational potentials evolve in time during radiation domination and not during matter domination: the later matter-radiation equality occurs, the greater the time evolution of the potentials at decoupling, increasing the Integrated Sachs-Wolfe effect. The power spectrum also has a weak dependence on _{0} in models with _{0} significantly less than unity, because at late times the evolution of the background cosmology will be dominated not by matter, but rather by vacuum energy (for a flat universe with ) or by curvature (for an open universe). In either case, the gravitational potentials once again begin to evolve with time, giving an additional late-time Integrated Sachs-Wolfe contribution, but this tends to affect only the largest scales for which the constraints from measurements are least restrictive due to cosmic variance (see the discussion in Sec. 6.4 below).
The sound speed, which sets the sound horizon and thus affects the wavelength of the acoustic modes (cf. Eq. (28)), is completely determined by the baryon density _{b} h^{2}. The horizon size at recombination, which sets the overall scale of the acoustic oscillations, depends only on the total mass density _{0} h^{2}. The damping scale for diffusion damping depends almost solely on the baryon density _{b} h^{2}, although numerical fits give a slight dependence on _{b} alone (Hu and Sugiyama 1996). Finally, the angular diameter distance to the surface of last scattering is determined by _{0} h and h; the angular diameter sets the angular scale on the sky of the acoustic oscillations.
In summary, the physical dependence of the temperature perturbations at last scattering depends on _{0} h^{2}, _{b} h^{2}, _{0} h, and h instead of the individual cosmological parameters _{0}, _{b}, h, and . When analyzing constraints on cosmological models from microwave background power spectra, it may be more meaningful and powerful to constrain these physical parameters rather than the cosmological ones.
6.3. Power spectrum degeneracies
As might be expected from the above discussion, not all of the parameters considered here are independent. In fact, one nearly exact degeneracy exists if _{0}, _{b}, h, and are taken as independent parameters. To see this, consider a shift in _{0}. In isolation, such a shift will produce a corresponding stretching of the power spectrum in l-space. But this effect can be compensated by first shifting h to keep _{0} h^{2} constant, then shifting _{b} to keep _{b} h^{2} constant, and finally shifting to keep the angular diameter distance constant. This set of shifted parameters will, in linear perturbation theory, produce almost exactly the same microwave background power spectra as the original set of parameters. The universe with shifted parameters will generally not be flat, but the resulting late-time Integrated Sachs-Wolfe effect only weakly break the degeneracy. Likewise, gravitational lensing has only a very weak effect on the degeneracy.
But all is not lost. The required shift in is generally something like 8 times larger than the original shift in _{0}, so although the degeneracy is nearly exact, most of the degenerate models represent rather extreme cosmologies. Good taste requires either that = 0 or that = 1, in other words that we disfavor models which have both a cosmological constant and are not flat. If such models are disallowed, the degeneracy disappears. Finally, other observables not associated with the microwave background break the degeneracy: the acceleration parameter q_{0} = _{0} / 2 - , for example, is measured directly by the high-redshift supernova experiments. So in practice, this fundamental degeneracy in the microwave background power spectrum between and is not likely to have a great impact on our ability to constrain cosmological parameters.
Other approximate degeneracies in the temperature power spectrum exist between Q and r, and between z_{r} and n. The first is illusory: the amplitudes of the scalar and tensor power spectra can be used in place of their sum and ratio, which eliminates the degeneracy. The power spectrum of large-scale structure will lift the latter degeneracy if bias is understood well enough, as will polarization measurements and small-scale second-order temperature fluctuations (the Ostriker-Vishniac effect, see Jaffe and Gnedin 2000) which are both sensitive to z_{r}.
Finally, many claims have been made about the ability of the microwave background to constrain the effective number of neutrino species or neutrino masses. The effective number of massless degrees of freedom at decoupling can be expressed in terms of the effective number of neutrino species N_{} (which does not need to be an integer). This is a convenient way of parameterizing ignorance about fundamental particle constituents of nature. Contributors to N_{} could include, for example, an extra sterile neutrino sometimes invoked in neutrino oscillation models, or the thermal background of gravitons which would exist if inflation did not occur. This parameter can also include the effects of entropy increases due to decaying or annihilating particles; see Chapter 3 of Kolb and Turner (1990) for a detailed discussion. As far as the microwave background is concerned, N_{} determines the radiation energy density of the universe and thus modifies the time of matter-radiation equality. It can in principle be distinguished from a change in _{0} h^{2} because it affects other physical parameters like the baryon density or the angular diameter distance differently than a shift in either _{0} or h.
Neutrino masses cannot provide the bulk of the dark matter, because their free streaming greatly suppresses fluctuation power on galaxy scales, leading to a drastic mismatch with observed large-scale structure. But models with some small fraction of dark matter as neutrinos have been advocated to improve the agreement between the predicted and observed large-scale structure power spectrum. Massive neutrinos have several small effects on the microwave background, which have been studied systematically by Dodelson et al. (1996). They can slightly increase the sound horizon at decoupling due to their transition from relativistic to non-relativistic behavior as the universe expands. More importantly, free streaming of massive neutrinos around the time of last scattering leads to a faster decay of the gravitational potentials, which in turn means more forcing of the acoustic oscillations and a resulting increase in the monopole perturbations. Finally, since matter-radiation equality is slightly delayed for neutrinos with cosmologically interesting masses of a few eV, the gravitational potentials are less constant and a larger Integrated Sachs-Wolfe effect is induced. The change in sound horizon and shift in matter-radiation equality due to massive neutrinos cannot be distinguished from changes in _{b} h^{2} and _{0} h^{2}, but the alteration of the gravitational potential's time dependence due to neutrino free streaming cannot be mimicked by some other change in parameters. In principle the effect of neutrino masses can be extracted from the microwave background, although the effects are very small.
Remarkably, the microwave background power spectrum contains enough information to constrain numerous parameters simultaneously (Jungman et al. 1996). We would like to estimate quantitatively just how well the space of parameters described above can be constrained by ideal measurements of the microwave background. The question has been studied in some detail; this section outlines the basic methods and results, and discusses how good various approximations are. For simplicity, only temperature fluctuations are considered in this section; the corresponding formalism for the polarization power spectra is developed in Kamionkowski et al. (1997).
Given a pixelized map of the microwave sky, we need to determine the contribution of pixelization noise, detector noise, and beam width to the multipole moments and power spectrum. Consider a temperature map of the sky T^{map}() which is divided into N_{pix} equal-area pixels. The observed temperature in pixel j is due to a cosmological signal plus noise, T^{map}_{j} = T_{j} + T^{noise}_{j}. The multipole coefficients of the map can be constructed as
(31) |
where _{j} is the direction vector to pixel j. The map moments are written as d_{lm} to distinguish them from the moments of the cosmological signal a_{lm}; the former include the effects of noise. The extent to which the second line in Eqs. (31) is only an approximate equality is the pixelization noise. Most current experiments oversample the sky with respect to their beam, so the pixelization noise is negligible. Now assume that the noise is uncorrelated between pixels and is well-represented by a normal distribution. Also, assume that the map is created with a gaussian beam with width _{b}. Then it is straightforward to show that the variance of the temperature moments is given by (Knox 1995)
(32) |
where _{b} = 0.00742(_{b} / 1°) and
(33) |
is the inverse statistical weight per unit solid angle, a measure of experimental sensitivity independent of the pixel size.
Now the power spectrum can be estimated via Eq. (32) as
(34) |
where
(35) |
The individual coefficients d^{T}_{lm} are gaussian random variables. This means that C_{l}^{T} is a random variable with a ^{2}_{2l+1} distribution, and its variance is (Knox 1995)
(36) |
Note that even for w^{-1} = 0, corresponding to zero noise, the variance is non-zero. This is the cosmic variance, arising from the fact that we have only one sky to observe: the estimator in Eq. (35) is the sum of 2l + 1 random variables, so it has a fundamental fractional variance of (2l + 1)^{-1/2} simply due to Poisson statistics. This variance provides a benchmark for experiments: if the goal is to determine a power spectrum, it makes no sense to improve resolution or sensitivity beyond the level at which cosmic variance is the dominant source of error.
Equation (36) is extremely useful: it gives an estimate of how well the power spectrum can be determined by an experiment with a given beam size and detector noise. If only a portion of the sky is covered, the variance estimate should be divided by the fraction of the total sky covered. With these variances in hand, standard statistical techniques can be employed to estimate how well a given measurement can recover a given set s of cosmological parameters. Approximate the dependence of C_{l}^{T} on a given parameter as linear in the parameter; this will always be true for some sufficiently small range of parameter values. Then the parameter space curvature matrix (also known as the Fisher information matrix) is specified by
(37) |
The variance in the determination of the parameter s_{i} from a set of C_{l}^{T} with variances C_{l}^{T} after marginalizing over all other parameters is given by the diagonal element i of the matrix ^{-1}.
Estimates of this kind were first made by Jungman et al. (1996) and subsequently refined by Zaldarriaga et al. (1997) and Bond et al. (1997), among others. The basic result is that a map with pixels of a few arcminutes in size and a signal-to-noise ratio of around 1 per pixel can determine , _{b} h^{2}, _{m} h^{2}, h^{2}, Q, n, and z_{r} at the few percent level simultaneously, up to the one degeneracy mentioned above (see the table in Bond et al. 1997). Significant constraints will also be placed on r and N_{}. This prospect has been the primary reason that the microwave background has generated such excitement. Note that , h, _{b}, and are the classical cosmological parameters. Decades of painstaking astronomical observations have been devoted to determining the values of these parameters. The microwave background offers a completely independent method of determining them with comparable or significantly greater accuracy, and with fewer astrophysical systematic effects to worry about. The microwave background is also the only source of precise information about the spectrum and character of the primordial perturbations from which we arose. Of course, these exciting possibilities hold only if the universe is accurately represented by a model in the assumed model space. The model space is, however, quite broad. Model-independent constraints which the microwave background provides are discussed in Sec. 7.
The estimates of parameter variances based on the curvature matrix would be exact if the power spectrum always varied linearly with each parameter. This, of course, is not true in general. Given a set of power spectrum data, we want to know two pieces of information about the cosmological parameters: (1) What parameter values provide the best-fit model? (2) What are the error bars on these parameters, or more precisely, what is the region of parameter space which defines a given confidence level? The first question can be answered easily using standard methods of searching parameter space; generally such a search requires evaluating the power spectrum for fewer than 100 different models. This shows that the parameter space is generally without complicated structure or many false minima. The second question is more difficult. Anything beyond the curvature matrix analysis requires looking around in parameter space near the best-fit model. A specific Monte Carlo technique employing a Metropolis algorithm has recently been advocated (Christensen and Meyer 2000); such techniques will certainly prove more flexible and efficient than recent brute-force grid searches (Tegmark and Zaldarriaga 2000). As upcoming data sets contain more information and consequently have greater power to constrain parameters, efficient techniques of parameter space exploration will become increasingly important.
To this point, the discussion has assumed that the microwave background power spectrum is perfectly described by linear perturbation theory. Since the temperature fluctuations are so small, parts in a hundred thousand, linear theory is a very good approximation. However, on small scales, non-linear effects become important and can dominate over the linear contributions. The most important non-linear effects are the Ostriker-Vishniac effect coupling velocity and density perturbations (Jaffe and Kamionkowski 1998, Hu 2000), gravitational lensing by large-scale structure (Seljak 1996), the Sunyaev-Zeldovich effect which gives spectral distortions when the microwave background radiation passes through hot ionized regions (Birkinshaw 1999), and the kinetic Sunyaev-Zeldovich effect which doppler shifts radiation passing through plasma with bulk velocity (Gnedin and Jaffe 2000). All three effects are measurable and give important additional constraints on cosmology, but more detailed descriptions are outside the scope of these lectures.
Finally, no discussion of parameter determination would be complete without mention of galactic foreground sources of microwave emission. Dust radiates significantly at microwave frequencies, as do free-free and synchrotron emission; point source microwave emission is also a potential problem. Dust emission generally has a spectrum which rises with frequency, while free-free and synchrotron emission have falling frequency spectra. The emission is not uniform on the sky, but rather concentrated in the galactic plane, with fainter but pervasive diffuse emission in other parts of the sky. The dust and synchrotron/free-free emission spectra cross each other at a frequency of around 90 GHz. Fortunately for cosmologists, the amplitude of the foreground emission at this frequency is low enough to create a frequency window in which the cosmological temperature fluctuations dominate the foreground temperature fluctuations. At other frequencies, the foreground contribution can be effectively separated from the cosmological blackbody signal by measuring in several different frequencies and projecting out the portion of the signal with a flat frequency spectrum. The foreground situation for polarization is less clear, both in amplitude and spectral index, and could potentially be a serious systematic limit to the quality of cosmological polarization data. On the other hand, it may be no greater problem for polarization fluctuations than for temperature fluctuations. For an overview of issues surrounding foreground emission, see Bouchet and Gispert 1999 or the WOMBAT web site, http://astro.berkeley.edu/wombat.
6.5. Current constraints and upcoming experiments
As the Como School began, results from the high-resolution balloon-born experiment MAXIMA (Hanany et al. 2000) were released,complementing the week-old data from BOOMERanG (de Bernardis et al. 2000) and creating a considerable buzz at coffee breaks. The derived power spectrum estimates are shown in Fig. 3. The data from the two measurements appear consistent up to calibration uncertainties, and for simplicity will be referred to here as ``balloon data'' and discussed as a single result. While a few experimenters and data analyzers were members of both experimental teams, the measurements and data reductions were done essentially independently. Earlier data from the previous year (Miller et al. 1999) had clearly demonstrated the existence and angular scale of the first peak in the power spectrum and produced the first maps of the microwave background at angular scales below a degree. But the new results from balloon experiments utilizing extremely sensitive bolometric detecters represent a qualitative step forward. These experiments begin to exploit the potential of the microwave background for ``precision cosmology''; their power spectra put strong constraints on several cosmological parameters simultaneously and rule out many variants of cosmological models. In fact, what is most interesting is that, at face value, these measurements put significant pressure on all of the standard models outlined above.
Figure 3. Two current measurements of the microwave background radiation temperature power spectrum. Triangles are BOOMERanG measurements multiplied by 1.21; squares are MAXIMA measurements multiplied by 0.92. The normalization factors are within the calibration uncertainties of the experiments, and were chosen by Hanany et al. (2000) to give the most consistent results between the two experiments. |
The balloon data shows two major features: first, a large peak in the power spectrum centered around l = 200 with an amplitude of approximately l^{2} C_{l} = 36000 µK^{2}, and second, a broad plateau between l = 400 and l = 700 with an amplitude of approximately l^{2} C_{l} = 10000 µK^{2}. The first peak is clearly delineated and provides good evidence that the universe is spatially flat, i.e. = 1. The issue of a second acoustic peak is much less clear. In most flat universe models with acoustic oscillations, the second peak is expected to appear at an angular scale of around l = 400. The angular resolution of the balloon experiments is certainly good enough to see such a peak, but the power spectrum data shows no evidence for one. I argue that a flat line is an excellent fit to the data past l = 300, and that any model which shows a peak in this region will be a worse fit than a flat line. This does not necessarily mean that no peak is present; the error bars are too large to rule out a peak, but the amplitude of such a peak is fairly strongly constrained to be lower than expected given the first peak.
What does this mean for cosmological models? Within the model space outlined in the previous section, there are three ways to suppress the second peak. The first would be to have a power spectrum index n substantially less than 1. This solution would force abandonment of the assumption of power law initial fluctuations, in order to match the observed amplitude of large-scale structure at smaller scales. While this is certainly possible, it represents a drastic weakening in the predictive power of the microwave background: essentially, a certain feature is reproduced by arbitrarily changing the primordial power spectrum. While no physical principle requires power law primordial perturbations, we should wait for microwave background measurements on a much wider range of scales combined with overlapping large-scale structure measurements before resorting to departures from power-law initial conditions. If the universe really did possess an initial power spectrum with a variety of features in it, most of the promise of precision cosmology is lost. Recent power spectra extracted from the IRAS Point Source Survey Redshift Catalog (Tegmark and Hamilton 2000), which show a remarkable power law behavior spanning three orders of magnitude in wave number, seem to argue against this possibility.
The second possibility is a drastic amount of reionization. It is not clear the extent to which this might be compatible with the height of the first peak and still suppress the second peak sufficiently. This possibility seems unlikely as well, but would show clear signatures in the microwave background polarization.
The most commonly discussed possibility is that the very low second peak amplitude reflects an unexpectedly large fraction of baryons relative to dark matter in the universe. The baryon signature discussed in Sec. 5.4 gives a suppression of the second peak in this case. However, primordial nucleosynthesis also constrains the baryon-photon ratio. Recent high-precision measurements of deuterium absorption in high-redshift neutral hydrogen clouds (Tytler et al. 2000) give a baryon-photon number ratio of = 5.1 ± 0.5 × 10^{10}, which translates to _{b} h^{2} = 0.019 ± 0.002 assuming that the entropy (i.e. photon number) per comoving volume remains constant between nucleosynthesis and the present. Requiring _{b} to satisfy this nucleosynthesis constraint leads to microwave background power spectra which are not particularly good fits to the data. An alternative is that the entropy per comoving volume has not remained fixed between nucleosynthesis and recombination (see, e.g., Kaplinghat and Turner 2000). This could be arranged by having a dark matter particle which decays to photons, although such a process must evade limits from the lack of microwave background spectral distortions (Hu and Silk 1993). Alternately, a large chemical potential for the neutrino background could lead to larger inferred values for the baryon-photon ratio from nucleosynthesis (Esposito et al. 2000). Either way, if both the microwave background measurements and the high-redshift deuterium abundances hold up, the discrepancy points to new physics. Of course, a final explanation for the discrepancies is simply that the balloon data has significant systematic errors.
I digress for a brief editorial comment about data analysis. Straightforward searches of the conventional cosmological model space described above for good fits to the balloon data give models with very low dark matter densities, high baryon fractions, and very large cosmological constants (see model P1 in Table 1 of Lange et al. 2000). Such models violate other observational constraints on age, which must be at least 12 billion years (see, e.g., Peacock et al. 1998), and quasar and radio source strong lensing number counts, which limit a cosmological constant to 0.7 (Falco et al. 1998). The response to this situation so far has been to invoke Bayesian prior probability distributions on various quantities like _{b} and the age. This leads to a best-fit model with a nominally acceptable (Lange et al. 2000, Tegmark et al. 2000 and others). But be wary of this procedure when the priors have a large effect on the best-fit model! The microwave background will soon provide tighter constraints on most parameters than any other source of prior information. Priors probabilities on a given parameter are useful and justified when the microwave background data has little power to constrain that parameter; in this case, the statistical quality of the model fit to the microwave background data will not be greatly affected by imposing the prior. However, something fishy is probably going on when a prior pulls a parameter multiple sigma away from its best fit value without the prior. This is what happens presently with _{b} when the nucleosynthesis prior is enforced. If your priors make a big difference, it is likely either that some of the data is incorrect, or that the model space does not include the correct model. Both the microwave background measurements and the high-redshift deuterium detections are taxing observations dominated by systematic effects, so it is certainly possible that one or both are wrong. On the other hand, MAXIMA and BOOMERanG are consistent with each other while using different instruments, different parts of the sky, and different analysis pipelines, and the deuterium measurements are consistent for several different clouds. This suggests possible missing physics ingredients, like extreme reionization or an entropy increase mentioned above, or perhaps significant contributions from cosmic defects. It has even been suggested by otherwise sober and reasonable people that the microwave background results, combined with various difficulties related to dynamics of spiral galaxies, may point towards a radical revision of the standard cosmology (Sellwood and Kosowsky 2000). We should not rest lightly until the cosmological model preferred by microwave background measurements is comfortably consistent with all relevant priors derived from other data sources of comparable precision.
The picture will come into sharper relief over the next two years. The MAP satellite (http://map.gsfc.nasa.gov), due for launch by NASA in May, 2001, will map the full microwave sky in five frequency channels with an angular resolution of around 15 arc minutes and a temperature sensitivity per pixel of a part in a million. Space missions offer unequalled sky coverage and control of systematics, and if it works as advertised, MAP will be a benchmark experiment. Prior to its launch, expect to see the first interferometric microwave data at angular scales smaller than a half degree from the CBI interferometer experiment (http://www.astro.caltech.edu/~tjp/CBI/). In this same time frame, we also may have the first detection of polarization. The most interesting power spectrum feature to focus on will be the existence and amplitude of a third acoustic peak. If a third peak appears with amplitude significantly higher than the putative second peak, this almost certainly indicates conventional acoustic oscillations with a high baryon fraction and possibly new physics to reconcile the result with the deuterium measurements. If, on the other hand, the power spectrum remains flat or falls further past the second peak region, then all bets are off. In a time frame of the next 5 to 10 years, we can reasonably expect to have a cosmic-variance limited temperature power spectrum down to scales of a few arcminutes (say, l = 4000), along with significant polarization information (though probably not cosmic-variance limited power spectra). In particular, ESA's Planck satellite mission (http://astro.estec.esa.nl/SA-general/Projects/Planck/) will map the microwave sky in nine frequency bands at significantly better resolution and sensitivity than the MAP mission. For a comprehensive listing of past and planned microwave background measurements, see Max Tegmark's experiments web page, http://www.hep.upenn.edu/~max/cmb/experiments.html.