As suggested above, observations of the location of the first peak strongly
point to a flat universe. This is encouraging news
for adherents of inflation, a theory which initially predicted at a time when few astronomers would sign on to
such a high value (see [Liddle & Lyth, 1993] for a review). However,
the argument for inflation goes beyond the confirmation of flatness. In particular,
the discussion of the last subsection begs the question: whence
, the initial conditions of the
temperature fluctuations? The answer requires the inclusion of gravity and considerations
of causality which point to inflation as the origin of structure in the Universe.
The calculations of the typical angular scale of the acoustic oscillations
in the last section are familiar in another context: the horizon
problem. Because the sound speed is near the speed of light, the degree
scale also marks the extent of a causally connected region or particle horizon
at recombination. For the picture in the last section to hold, the perturbations
must have been laid down while the scales in question were still far outside
the particle horizon.
The recent observational verification of this basic peak structure presents
a problem potentially more serious than the original horizon problem of approximate
isotropy: the mechanism which smooths fluctuations in the Universe must also
regenerate them with superhorizon sized correlations
at the
level. Inflation is an idea that
solves both problems simultaneously.
The inflationary paradigm postulates that an early phase of near exponential expansion of the Universe was driven by a form of energy with negative pressure. In most models, this energy is usually provided by the potential energy of a scalar field. The inflationary era brings the observable universe to a nearly smooth and spatially flat state. Nonetheless, quantum fluctuations in the scalar field are unavoidable and also carried to large physical scales by the expansion. Because an exponential expansion is self-similar in time, the fluctuations are scale-invariant, i.e. in each logarithmic interval in scale the contribution to the variance of the fluctuations is equal. Since the scalar field carries the energy density of the Universe during inflation, its fluctuations induce variations in the spatial curvature [Guth & Pi, 1985,Hawking, 1982,Bardeen et al, 1983]. Instead of perfect flatness, inflation predicts that each scale will resemble a very slightly open or closed universe. This fluctuation in the geometry of the Universe is essentially frozen in while the perturbation is outside the horizon [Bardeen, 1980].
Formally, curvature fluctuations are perturbations to the space-space piece
of the metric. In a Newtonian coordinate system,
or gauge, where the metric is diagonal, the spatial curvature fluctuation is
called (see e.g. [Ma & Bertschinger, 1995]). The more familiar
Newtonian potential is the time-time fluctuation
and is approximately
. Approximate scale invariance then says that
where
is the power spectrum of
and the tilt
.
Now let us relate the inflationary prediction of scale-invariant curvature
fluctuations to the initial temperature fluctuations. Newtonian intuition based
on the Poisson equation tells us that on large scales (small
) density and hence temperature fluctuations should be negligible compared
with Newtonian potential. General relativity says otherwise because the Newtonian
potential is also a time-time fluctuation in the metric. It corresponds to a
temporal shift of
. The CMB temperature varies as the inverse of the
scale factor, which in turn depends on time as
. Therefore, the fractional change in the
CMB temperature
![]() |
(13) |
Thus, a temporal shift produces a temperature perturbation of in the radiation dominated era (when
and
in the matter dominated epoch (
) ([Peacock, 1991]; [White & Hu, 1997]). The initial temperature
perturbation is therefore inextricably linked with the initial gravitational potential
perturbation. Inflation predicts scale-invariant initial fluctuations in both
the CMB temperature and the spatial curvature in the Newtonian gauge.
Alternate models which seek to obey the causality can generate curvature fluctuations
only inside the particle horizon. Because the perturbations are then not generated
at the same epoch independent of scale, there is no longer a unique relationship
between the phase of the oscillators. That is, the argument of the cosine in
Equation (10) becomes , where
is a phase which can in principle be different for different wavevectors,
even those with the same magnitude
. This can lead to temporal incoherence
in the oscillations and hence a washing out of the acoustic peaks [Albrecht et al, 1996], most notably
in cosmological defect models [Allen et al, 1997,Seljak et al, 1997]. Complete incoherence
is not a strict requirement of causality since there are other ways to synch
up the oscillations. For example, many isocurvature models,
where the initial spatial curvature is unperturbed, are coherent since their
oscillations begin with the generation of curvature fluctuations at horizon
crossing [Hu & White, 1996]. Still they typically have
(c.f. [Turok, 1996]). Independent of the angular diameter
distance
, the ratio of the peak locations gives
the phase:
for
. Likewise independent of a constant phase, the
spacing of the peaks
gives a measure of the angular diameter
distance [Hu & White, 1996]. The observations, which
indicate coherent oscillations with
, therefore have provided a non-trivial test of the inflationary
paradigm and supplied a substantially more stringent version of the horizon
problem for contenders to solve.