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.

The Friedmann-Robertson-Walker spacetime describing homogeneous and isotropic cosmology comes in three flavors of spatial curvature: positive, negative, and flat, corresponding to > 1, < 1, and = 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
. 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
= 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
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
. The microwave
background fluctuations actually depend on two basic physical scales.
The first is the sound horizon at last scattering, *r*_{s}
(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
_{b}
*h*^{2}, 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
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 *C*_{l} 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 *n*_{T} of the tensor
perturbations.

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 *z*_{r} independent of any cosmological model.

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.