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Most analysis of microwave background data and predictions about its ability to constrain cosmology have been based on the cosmological parameter space described in Sec. 6.1 above. This space is motivated by inflationary cosmological scenarios, which generically predict power-law adiabatic perturbations evolving only via gravitational instability. Considering that this space of models is broad and appears to fit all current data far better than any other proposed models, such an assumed model space is not very restrictive. In particular, proposed extensions tend to be rather ad hoc, adding extra elements to the model without providing any compelling underlying motivation for them. Examples which have been discussed in the literature include multiple types of dark matter with various properties, nonstandard recombination, small admixtures of topological defects, production of excess entropy, or arbitrary initial power spectra. None of these possibilities are attractive from an aesthetic point of view: all add significant complexity and freedom to the models without any corresponding restrictions on the original parameter space. The principle of Occam's Razor should cause us to be skeptical about any such additions to the space of models.

On the other hand, it is possible that some element is missing from the model space, or that the actual cosmological model is radically different in some respect. The microwave background is the probe of cosmology most tightly connected to the fundamental properties of the universe and least influenced by astrophysical complications, and thus the most capable data source for deciding whether the universe actually is well described by some model in the usual model space. An interesting question is the extent to which the microwave background can determine various properties of the universe independent from particular models. While any cosmological interpretation of temperature fluctuations in the microwave sky requires some kind of minimal assumptions, all of the conclusions outlined below can be drawn without invoking a detailed model of initial conditions or structure formation. These conclusions are in contrast to precision determination of cosmological parameters, which does require the assumption of a particular space of models and which can vary significantly depending on the space.

7.1. Flatness

The Friedmann-Robertson-Walker spacetime describing homogeneous and isotropic cosmology comes in three flavors of spatial curvature: positive, negative, and flat, corresponding to Omega > 1, Omega < 1, and Omega = 1 respectively. One of the most fundamental questions of cosmology, dating to the original relativistic cosmological models, is the curvature of the background spacetime. The fate of the universe quite literally depends on the answer: in a cosmology with only matter and radiation, a positively-curved universe will eventually recollapse in a fiery ``Big Crunch'' while flat and negatively-curved universes will expand forever, meeting a frigid demise. Note these fates are at least 40 billion years in the future. (A cosmological constant or other energy density component with an unusual equation of state can alter these outcomes, causing a closed universe eventually to enter an inflationary stage.)

The microwave background provides the cleanest and most powerful probe of the geometry of the universe (Kamionkowski et al. 1994). The surface of last scattering is at a high enough redshift that photon geodesics between the last scattering surface and the Earth are significantly curved if the geometry of the universe is appreciably different than flat. In a positively-curved space, two geodesics will bend towards each other, subtending a larger angle at the observer than in the flat case; likewise, in a negatively-curved space two geodesics bend away from each other, resulting in a smaller observed angle between the two. The operative quantity is the angular diameter distance; Weinberg (2000) gives a pedagogical discussion of its dependence on Omega. In a flat universe, the horizon length at the time of last scattering subtends an angle on the sky of around two degrees. For a low-density universe with Omega = 0.3, this angle becomes smaller by half, roughly.

A change in angular scale of this magnitude will change the apparent scale of all physical scales in the microwave background. A model-independent determination of Omega thus requires a physical scale of known size to be imprinted on the primordial plasma at last scattering; this physical scale can then be compared with its apparent observed scale to obtain a measurement of Omega. The microwave background fluctuations actually depend on two basic physical scales. The first is the sound horizon at last scattering, rs (cf. Eq. (29). If coherent acoustic oscillations are visible, this scale sets their characteristic wavelengths. Even if coherent acoustic oscillations are not present, the sound horizon represents the largest scale on which any causal physical process can influence the primordial plasma. Roughly, if primordial perturbations appear on all scales, the resulting microwave background fluctuations appear as a featureless power law at large scales, while the scale at which they begin to depart from this assumed primordial behavior corresponds to the sound horizon. This is precisely the behavior observed by current measurements, which show a prominent power spectrum peak at an angular scale of a degree (l = 200), arguing strongly for a flat universe. Of course, it is logically possible that the primordial power spectrum has power on scales only significantly smaller than the horizon at last scattering. In this case, the largest scale perturbations would appear at smaller angular scales for a given geometry. But then the observed power-law perturbations at large angular scales must be reproduced by the Integrated Sachs-Wolfe effect, and resulting models are contrived. If the microwave background power spectrum exhibites acoustic oscillations, then the spacing of the acoustic peaks depends only on the sound horizon independent of the phase of the oscillations; this provides a more general and precise probe of flatness than the first peak position.

The second physical scale provides another test: the Silk damping scale is determined solely by the thickness of the surface of last scattering, which in turn depends only on the baryon density Omegab h2, the expansion rate of the universe and standard thermodynamics. Observation of an exponential suppression of power at small scales gives an estimate of the angular scale corresponding to the damping scale. Note that the effects of reionization and gravitational lensing must both be accounted for in the small-scale dependence of the fluctuations. If the reionization redshift can be accurately estimated from microwave background polarization (see below) and the baryon density is known from primordial nucleosynthesis or from the alternating peak heights signature (Sec. 5.4), only a radical modification of the standard cosmology altering the time dependence of the scale factor or modifying thermodynamic recombination can change the physical damping scale. If the estimates of Omega based on the sound horizon and damping scales are consistent, this is a strong indication that the inferred geometry of the universe is correct.

7.2. Coherent acoustic oscillations

If a series of peaks equally spaced in l is observed in the microwave background temperature power spectrum, it strongly suggests we are seeing the effects of coherent acoustic oscillations at the time of last scattering. Microwave background polarization provides a method for confirming this hypothesis. As explained in Sec. 4.2, polarization anisotropies couple primarily to velocity perturbations, while temperature anisotropies couple primarily to density perturbations. Now coherent acoustic oscillations produce temperature power spectrum peaks at scales where a mode of that wavelength has either maximum or minimum compression in potential wells at the time of last scattering. The fluid velocity for the mode at these times will be zero, as the oscillation is turing around from expansion to contraction (envision a mass on a spring.) At scales intermediate between the peaks, the oscillating mode has zero density contrast but a maximum velocity perturbation. Since the polarization power spectrum is dominated by the velocity perturbations, its peaks will be at scales interleaved with the temperature power spectrum peaks. This alternation of temperature and polarization peaks as the angular scale changes is characteristic of acoustic oscillations (see Kosowsky (1999) for a more detailed discussion). Indeed, it is almost like seeing the oscillations directly: it is difficult to imagine any other explanation for density and velocity extrema on alternating scales. The temperature-polarization cross-correlation must also have peaks with corresponding phases. This test will be very useful if a series of peaks is detected in a temperature power spectrum which is not a good fit to the standard space of cosmological models. If the peaks turn out to reflect coherent oscillations, we must then modify some aspect of the underlying cosmology, while if the peaks are not coherent oscillations, we must modify the process by which perturbations evolve.

If coherent oscillations are detected, any cosmological model must include a mechanism for enforcing coherence. Perturbations on all scales, in particular on scales outside the horizon, provide the only natural mechanism: the phase of the oscillations is determined by the time when the wavelength of the perturbation becomes smaller than the horizon, and this will clearly be the same for all perturbations of a given wavelength. For any source of perturbations inside the horizon, the source itself must be coherent over a given scale to produce phase-coherent perturbations on that scale. This cannot occur without artificial fine-tuning.

7.3. Adiabatic primordial perturbations

If the microwave background temperature and polarization power spectra reveal coherent acoustic oscillations and the geometry of the universe can also be determined with some precision, then the phases of the acoustic oscillations can be used to determine whether the primordial perturbations are adiabatic or isocurvature. Quite generally, Eq. (28) shows that adiabatic and isocurvature power spectra must have peaks which are out of phase. While current measurements of the microwave background and large-scale structure rule out models based entirely on isocurvature perturbations, some relatively small admixture of isocurvature modes with dominant adiabatic modes is possible. Such mixtures arise naturally in inflationary models with more than one dynamical field during inflation (see, e.g., Mukhanov and Steinhardt 1998).

7.4. Gaussian primordial perturbations

If the temperature perturbations are well approximated as a gaussian random field, as microwave background maps so far suggest, then the power spectrum Cl contains all statistical information about the temperature distribution. Departures from gaussianity take myriad different forms; the business of providing general but useful statistical descriptions is a complicated one (see, e.g., Ferreira et al. 1997). Tiny amounts of nongaussianity will arise inevitably from non-linear evolution of fluctuations, and larger nongaussian contributions can be a feature of the primordial perturbations or can be induced by ``stiff'' stress-energy perturbations such as topological defects. As explained below, defect theories of structure formation seem to be ruled out by current microwave background and large-scale structure measurements, so interest in nongaussianity has waned. But the extent to which the temperature fluctuations are actually gaussian is experimentally answerable, and as observations improve this will become an important test of inflationary cosmological models.

7.5. Tensor or vector perturbations

As described in Sec. 4.3, the tensor field describing microwave background polarization can be decomposed into two components corresponding to the gradient-curl decomposition of a vector field. This decomposition has the same physical meaning as that for a vector field. In particular, any gradient-type tensor field, composed of the G-harmonics, has no curl, and thus may not have any handedness associated with it (meaning the field is even under parity reversal), while the curl-type tensor field, composed of the C-harmonics, does have a handedness (odd under parity reversal).

This geometric interpretation leads to an important physical conclusion. Consider a universe containing only scalar perturbations, and imagine a single Fourier mode of the perturbations. The mode has only one direction associated with it, defined by the Fourier vector k; since the perturbation is scalar, it must be rotationally symmetric around this axis. (If it were not, the gradient of the perturbation would define an independent physical direction, which would violate the assumption of a scalar perturbation.) Such a mode can have no physical handedness associated with it, and as a result, the polarization pattern it induces in the microwave background couples only to the G harmonics. Another way of stating this conclusion is that primordial density perturbations produce no C-type polarization as long as the perturbations evolve linearly. On the other hand, primordial tensor or vector perturbations produce both G-type and C-type polarization of the microwave background (provided that the tensor or vector perturbations themselves have no intrinsic net polarization associated with them).

Measurements of cosmological C-polarization in the microwave background are free of contributions from the dominant scalar density perturbations and thus can reveal the contribution of tensor modes in detail. For roughly scale-invariant tensor perturbations, most of the contribution comes at angular scales larger than 2° (2 < l < 100). Figure 4 displays the C and G power spectra for scale-invariant tensor perturbations contributing 10% of the COBE signal on large scales. A microwave background map with forseeable sensitivity could measure gravitational wave perturbations with amplitudes smaller than 10-3 times the amplitude of density perturbations (Kamionkowski and Kosowsky 1998). The C-polarization signal also appears to be the best hope for measuring the spectral index nT of the tensor perturbations.

Figure 4

Figure 4. Polarization power spectra from tensor perturbations: the solid line is ClG and the dashed line is ClC. The amplitude gives a 10% contribution to the COBE temperature power spectrum measurement at low l. Note that scalar perturbations give no contribution to ClC.

7.6. Reionization redshift

Reionization produces a distinctive microwave background signature. It suppresses temperature fluctuations by increasing the effective damping scale, while it also increases large-angle polarization due to additional Thomson scattering at low redshifts when the radiation quadrupole fluctuations are much larger. This enhanced polarization peak at large angles will be significant for reionization prior to z = 10 (Zaldarriaga 1997). Reionization will also greatly enhance the Ostriker-Vishniac effect, a second-order coupling between density and velocity perturbations (Jaffe and Kamionkowski 1998). The nonuniform reionization inevitable if the ionizing photons come from point sources, as seems likely, may also create an additional feature at small angular scales (Hu and Gruzinov 1998, Knox et al. 1998). Taken together, these features are clear indicators of the reionization redshift zr independent of any cosmological model.

7.7. Magnetic Fields

Primordial magnetic fields would be clearly indicated if cosmological Faraday rotation were detected in the microwave background polarization. A field with comoving field strength of 10-9 gauss would produce a signal with a few degrees of rotation at 30 GHz, which is likely just detectable with future polarization experiments (Kosowsky and Loeb 1996). Faraday rotation has the effect of mixing G-type and C-type polarization, and would be another contributor to the C-polarization signal, along with tensor perturbations. Depolarization will also result from Faraday rotation in the case of significant rotation through the last scattering surface (Harari et al. 1996) Additionally, the tensor and vector metric perturbations produced by magnetic fields result in further microwave background fluctuations. A distinctive signature of such fields is that for a range of power spectra, the polarization fluctuations from the metric perturbations is comparable to, or larger than, the corresponding temperature fluctuations (Kahniashvili et al. 2000). Since the microwave background power spectra vary as the fourth power of the magnetic field amplitude, it is unlikely that we can detect magnetic fields with comoving amplitudes significantly below 10-9 gauss. However, if such fields do exist, the microwave background provides several correlated signatures which will clearly reveal them.

7.8. The topology of the universe

Finally, one other microwave background signature of a very different character deserves mention. Most cosmological analyses make the implicit assumption that the spatial extent of the universe is infinite, or in practical terms at least much larger than our current Hubble volume so that we have no way of detecting the bounds of the universe. However, this need not be the case. The requirement that the unperturbed universe be homogeneous and isotropic determines the spacetime metric to be of the standard Friedmann-Robertson-Walker form, but this is only a local condition on the spacetime. Its global structure is still unspecified. It is possible to construct spacetimes which at every point have the usual homogeneous and isotropic metric, but which are spatially compact (have finite volumes). The most familiar example is the construction of a three-torus from a cubical piece of the flat spacetime by identifying opposite sides. Classifying the possible topological spaces which locally have the metric structure of the usual cosmological spacetimes (i.e. have the Friedmann-Robertson-Walker spacetimes as a topological covering space) has been studied extensively. The zero-curvature and positive-curvature cases have only a handful of possible topological spaces associated with them, while the negative curvature case has an infinite number with a very rich classification. See Weeks (1998) for a review.

If the topology of the universe is non-trivial and the volume of the universe is smaller than the volume contained by a sphere with radius equal to the distance to the surface of last scattering, then it is possible to detect the topology. Cornish et al. (1998) pointed out that because the last scattering surface is always a sphere in the covering space, any small topology will result in matched circles of temperature on the microwave sky. The two circles represent photons originating from the same physical location in the universe but propagating to us in two different directions. Of course, the temperatures around the circles will not match exactly, but only the contributions coming from the Sachs-Wolfe effect and the intrinsic temperature fluctuations will be the same; the velocity and Integrated Sachs-Wolfe contributions will differ and constitute a noise source. Estimates show the circles can be found efficiently via a direct search of full-sky microwave background maps. Once all matching pairs of circles have been discovered, their number and relative locations on the sky strongly overdetermine the topology of the universe in most cases. Remarkably, the microwave background essentially allows us to determine the size of the universe if it is smaller than the current horizon volume in any dimension.

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