Part 1: Observations of Global Properties

Part 2: Homogeneity and Isotropy; Many Distances; Scale Factor

Part 3: Spatial Curvature; Flatness-Oldness; Horizon

Part 4: Inflation; Anisotropy and Inhomogeneity

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The "inflationary scenario", developed by Starobinsky and by Guth,
offers a solution to the flatness-oldness problem and the
horizon problem. The
inflationary scenario invokes a
vacuum
energy density. We normally think of the vacuum as empty and
massless, and we can determine that the density of the vacuum is
less than 10^{-29} gm/cc now. But in quantum field theory, the vacuum
is not empty, but rather filled with virtual particles:

The space-time diagram above shows virtual particle-antiparticle pairs forming out of nothing and then annihilating back into nothing. For particles of mass m, one expects about one virtual particle in each cubical volume with sides given by the Compton wavelength of the particle, h/mc, where h is Planck's constant. Thus the expected density of the vacuum is rho = m

The inflationary scenario proposes that the vacuum energy was
very large during a brief period early in the history of the
Universe. When the Universe is dominated by a vacuum energy
density the scale factor grows exponentially,
a(t) = exp(H(t-t_{o})). The Hubble constant really is constant
during this epoch so it doesn't need the "naught".
If the inflationary epoch lasts long enough the exponential
function gets very large. This makes a(t) very large,
and thus makes the radius of curvature of the Universe
very large. The diagram below shows our
horizon superimposed
on a very large radius sphere on top, or a smaller sphere on
the bottom. Since we can only see as far as our horizon,
for the inflationary case on top the large radius sphere
looks almost flat to us.

This solves the flatness-oldness problem as long as the exponential growth during the inflationary epoch continues for at least 100 doublings. Inflation also solves the horizon problem, because the future lightcone of an event that happens before inflation is expanded to a huge region by the growth during inflation.

This space-time diagram shows the inflationary epoch tinted green, and the future lightcones of two events in red. The early event has a future lightcone that covers a huge area, that can easily encompass all of our horizon. Thus we can explain why the temperature of the microwave background is so uniform across the sky.

Of course the Universe is not really homogeneous, since it contains
dense regions like galaxies and people. These dense regions should
affect the temperature of the microwave background.
Sachs and Wolfe (1967, ApJ, 147, 73) derived the effect of the
gravitational potential perturbations on the CMB. The
gravitational potential,
phi = -GM/r, will be negative in dense lumps, and positive in
less dense regions. Photons lose energy when they climb out of
the gravitational potential wells of the lumps:

This conformal space-time diagram above shows lumps as gray vertical bars, the epoch before recombination as the hatched region, and the gravitational potential as the color-coded curve phi(x). Where our past lightcone intersects the surface of recombination, we see a temperature perturbed by dT/T = phi/(3*c

The map above is from COBE and is much better than Conklin's 2 standard deviation detection. The red part of the sky is hotter by (v/c)*T

The map above shows cosmic anisotropy (and detector noise) after the dipole pattern and the radiation from the Milky Way have been subtracted out. The anisotropy in this map has an RMS value of 30 microK, and if it is converted into a gravitational potential using Sachs and Wolfe's result and that potential is then expressed as a height assuming a constant acceleration of gravity equal to the gravity on the Earth, we get a height of twice the distance from the Earth to the Sun. The "mountains and valleys" of the Universe are really quite large.

Inflation predicts a certain statistical pattern in the anisotropy.
The quantum fluctuations normally affect very small regions of space,
but the huge exponential expansion during the inflationary epoch
makes these tiny regions observable.

The space-time diagram on the left above shows the future lightcones of quantum fluctuation events. The top of this diagram is really a volume which intersects our past lightcone making the sky. The future lightcones of events become circles on the sky. Events early in the inflationary epoch make large circles on the sky, as shown in the bottom map on the right. Later events make smaller circles as shown in the middle map, but there are more of them so the sky coverage is the same as before. Even later events make many small circles which again give the same sky coverage as seen on the top map.

An animated GIF file showing the spatial part of the above space-time diagram
as a function of time is available
here [1.2 MB].

The pattern formed by adding all of the effects from events of all ages is known as "equal power on all scales", and it agrees with the COBE data.

Having found that the observed pattern of anisotropy is consistent
with inflation, we can also ask whether the amplitude implies
gravitational forces large enough to produce the observed clustering
of galaxies.

The conformal space-time diagram above shows the phi(x) at recombination determined by COBE's dT data, and the worldlines of galaxies which are perturbed by the gravitational forces produced by the gradient of the potential. Matter flows "downhill" away from peaks of the potential (red spots on the COBE map), producing voids in the current distribution of galaxies, while valleys in the potential (blue spots) are where the clusters of galaxies form.

COBE was not able to see spots as small as clusters or even superclusters of galaxies, but if we use "equal power on all scales" to extrapolate the COBE data to smaller scales, we find that the gravitational forces are large enough to produce the observed clustering, but only if these forces are not opposed by other forces. If the all the matter in the Universe is made out of the ordinary chemical elements, then there was a very effective opposing force before recombination, because the free electrons which are now bound into atoms were very effective at scattering the photons of the cosmic background. We can therefore conclude that most of the matter in the Universe is "dark matter" that does not emit, absorb or scatter light. Furthermore, observations of distant supernovae have shown that most of the energy density of the Universe is a vacuum energy density (a "dark energy") like Einstein's cosmological constant that causes an accelerating expansion of the Universe. These strange conclusions have been greatly strengthened by temperature anisotropy data at smaller angular scales which was provided by the Wilkinson Microwave Anisotropy Probe (WMAP) in 2003.

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© 1996-2004 Edward L. Wright. Last modified 14 Sep 2004