10. Where do the structures in the universe come from?
Why does the universe have structures (like galaxies, galaxy clusters,
superclusters, and so on) instead of being uniform? Without such
inhomogeneity, stars would not have formed to make life possible.
This map of about 1/7 of the whole sky contains about 3 million
galaxies.
Each small bright spot is a galaxy cluster; these group
together to form superclusters.
Superclusters in turn are arranged
like filaments surrounding darker voids.
The origin of this inhomogeneity seems to be the zero-point energy
fluctuations in the vacuum, governed by the uncertainty principle of
quantum mechanics. Quantum mechanical effects normally have little
influence on large scales, but this was an exception. It was an
exception thanks to "inflation", an exponential expansion of the universe
that stretched the quantum fluctuations by many orders of magnitude.
Inflation was driven by the potential energy of what's called a
"scalar field". The excitation of a scalar field is responsible for
spin-0 particles like the Higgs boson. The scalar field particularly
responsible for inflation is called "inflaton", which has a negative
pressure. There were quantum fluctuations in the inflaton field. Right
after the Big Bang, the total energy density of the universe was dominated
by the inflaton potential energy. Thus, fluctuations in the inflaton
field gave rise to fluctuations in the energy density.
Potential energy of the scalar field drove rapid inflation of the
universe between the cosmic times t = 10-35s and
t = 10-33s. During this short period , the causal
horizon size (red line) did not have a chance to increase much at
all, whereas the fabric of space and any comoving scales within it (blue
line) expanded by tens of orders of magnitude. The expanding space
stretched each Fourier modes of the quantum fluctuations past the causal
horizon at the first purple point. After exiting the horizon (when
a(t) = ck/H), the fluctuations no longer evolved quantum
mechanically because their scales increased beyond causal connection. The
fluctuations became frozen in to become super-horizon metric
perturbations (became a classical quantity after leaving the horizon).
Much later, the horizon (the Hubble radius) expanded enough (at the second
purple point) for the perturbations to begin growing.
These density perturbations, through gravitational effects, eventually
resulted in the large-scale structures of the universe.
According to the observations of the cosmic microwave background (CMB)
radiation, the perturbations seem to be ideally Gaussian (their Fourier
modes are uncorrelated). This suggests that the perturbation has a
quantum mechanical origin: each Fourier mode evolves independently of
others like a quanta governed by the commutation relations. Also
according to the CMB observations, the power spectrum seems to be ideally
scale-invariant (the fluctuations per decade in wavenumber is constant).
Physically this is because the horizon size c/H was nearly
constant as the Fourier components exited the horizon during inflation.
Soon, the spectral index n [defined as in P(k) ~ kn-1
] could be measured with a precision of Δn ~ 0.01.
The scale-invariance imply fractal-like structures, as might be observed
in the large-scale structures of the universe.
Finally, the observed distribution of galaxies and isotropy of the CMB
suggest that the relative amplitude of the density perturbations should be
10-5 ~ 10-4.
How inflation produced scale-independent
fluctuations
References
2000: A. R. Liddle, D. H. Lyth. Cosmological Inflation and Large-Scale Structure.
2000: P. T. P. Viana.
Constraints on Inflation.
1999: J. A. Peacock. Cosmological Physics.
1999: A. Liddle,
An Introducation to Cosmological Inflation.
1999: A. Albrecht.
Cosmic Inflation.
1999: M. Kamionkowski, A. Kosowsky.
The Cosmic Microwave Background and Particle Physics.
1998: E. W. Kolb.
Particle Physics in the Early Universe.
1997: A. Liddle,
The Early Universe.
1993: P. J. E. Peebles. Principles of Physical Cosmology.
1993: T. Padmanabhan. Structure Formation in the Universe.
1991: J. V. Narlikar, T. Padmanabhan.
Inflation for Astronomers.
1990: E. W. Kolb, M. S. Turner. The Early Universe.
The "explosion" at the Big Bang does NOT mean that a clump of matter
exploded and expanded in some pre-existing space. The space itself was
probably created at the Big Bang and has been expanding in volume since
then, just like the surface area of an inflating balloon. In
this analogy, we must imagine that the 3-dimensional volume of space is
represented by the surface of the balloon and it's expanding (stretching)
with time, not into some pre-existing space, but along some other
dimension orthogonal (perpendicular) to the 3 dimensions of space. The
"membrane" or the "fabric" of space is stretching, but that doesn't mean
that you or any other matter is also stretching in size (just as a bug or
a dust particle on the balloon doesn't get stretched as you inflate the
balloon!).
, where Mr =
(4/3)πr3ρ is the mass inside the radius r.
With v=dr/dt ≡ (r dot), we can see how the expansion rate
changes with time: 
, related to how
the density ρ(t) changes as the universe expands.
, proportional to the square root
of the density. So, as the universe expanded and became less dense, the
expansion became slower and slower. The expansion rate is related to how
the density varied over time.
Integrating this from the Beginning (when the scale was pretty much zero)
to the present (when the scale factor is just 1), the cumulutive time
since the Big Bang is: 
.
