and Cosmology

7.7 Origin of the Density Fluctuations
calculated,
∫
u(x, t) = Ω0.6 m
4π aH(a)
d 3 y − x
y δ(y, t)
|y − x| . 3 (7.49)
Equation (7.49) shows that the velocity field can be
derived from the density field. If the density field in the
Universe were observable, one would obtain a direct
prediction for the corresponding velocity field from the
above relations. This depends on the matter density Ω m ,
so that from a comparison with the observed velocity
field, one could estimate the value for Ω m . We will come
back to this in Sect. 8.1.6.
7.7 Origin of the Density Fluctuations
We have seen in Sect. 4.5.3 that the horizon and the
flatness problem in the normal Friedmann–Lemaître
evolution of the Universe can be solved by postulating
an early phase of very rapid – exponential – expansion
of the cosmos. In this inflationary phase of the Universe,
any initial curvature of space is smoothed away
by the tremendous expansion. Furthermore, the exponential
expansion enables the complete currently visible
Universe to have been in causal contact prior to the inflationary
phase. These two aspects of the inflationary
model are so attractive that today most cosmologists
consider inflation as part of the standard model, even if
the physics of inflation is as yet not understood in detail.
The inflationary model has another property that is
considered very promising. Through the huge expansion
of the Universe, microscopic scales are blown
up to macroscopic dimensions. The large-scale structure
in the current Universe corresponds to microscopic
scales prior to and during the inflationary phase. From
quantum mechanics, we know that the matter distribution
cannot be fully homogeneous, but it is subject to
quantum fluctuations, expressed, e.g., by Heisenberg’s
uncertainty relation. By inflation, these small quantum
fluctuations are expanded to large-scale density fluctuations.
For this reason, the inflationary model also
provides a natural explanation for the presence of initial
density fluctuations.
In fact, one can study these effects quantitatively and
attempt to calculate the initial power spectrum of these
fluctuations. The result of such investigations will depend
slightly on the details of the inflationary model
they are based on. However, these models agree in their
prediction that the initial power spectrum should have
a form very similar to the Harrison–Zeldovich fluctuation
spectrum, except that the spectral index n s of the
primordial power spectrum should be slightly smaller
than the Harrison–Zeldovich value of n s = 1. Thus, the
model of inflation can be directly tested by measuring
the power spectrum and, as we shall see in Chap. 8, the
power-law slope n s indeed seems to be slightly flatter
that unity, as expected from inflation.
The various inflationary models also differ in their
predictions of the relative strength of the fluctuations of
spacetime, which should be present after inflation. Such
fluctuations are not directly linked to density fluctuations,
but they are a consequence of General Relativity,
according to which spacetime itself is also a dynamical
parameter. One consequence of this is the existence
of gravitational waves. Although no gravitational waves
have been directly detected until now, the analysis of the
double pulsar PSR J1915+1606 proves the existence of
such waves. 9 Primordial gravitational waves provide
an opportunity to empirically distinguish between the
various models of inflation. These gravitational waves
leave a “footprint” in the polarization of the cosmic
microwave background that is measurable in principle.
A satellite mission to perform these measurements is
currently being discussed.
9 The double pulsar PSR J1915+1606 was discovered in 1974. From
the orbital motion of the pulsar and its companion star, gravitational
waves are emitted, according to General Relativity. Through this, the
system loses kinetic (orbital) energy, so that the size of the orbit decreases
over time. Since pulsars represent excellent clocks, and we
can measure time with extremely high precision, this change in the orbital
motion can be observed with very high accuracy and compared
with predictions from General Relativity. The fantastic agreement of
theory and observation is considered a definite proof of the existence
of gravitational waves. For the discovery of the double pulsar and
the detailed analysis of this system, Russell Hulse and Joseph Taylor
were awarded the Nobel Prize in Physics in 1993. In 2003, a double
neutron star binary was discovered where pulsed radiation from
both components can be observed. This fact, together with the small
orbital period of 2.4 h implying a small separation of the two stars,
makes this an even better laboratory for studying strong-field gravity.
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