228 Cosmic Structures
But this difference need not be the same in′. For instance, even if an observable
behaves as a scalar under coordinate transformations in, its perturbation will not be
invariant under gauge transformations if it is time dependent in the background. Non-
Newtonian density perturbations inon superhorizon scales may have an entirely dif-
ferent time dependence in′, and the choice of gauge transformation→′is quite
arbitrary.
But arbitrariness does not imply that one gauge is correct and all others wrong.
Rather, it imposes on physicists the requirement to agree on a convention, otherwise
there will be problems in the interpretation of results. The formalism we chose in
Chapter 3, which led to the Einstein Equation (3.29) and to Friedmann’s Equations
(5.4) and (5.5), implicitly used a conventional gauge. Alternatively one could have
used gauge-invariant variables, but at the cost of a very heavy mathematical appa-
ratus. Another example concerns the electroweak theory, in which particle states are
represented by gauge fields that are locally gauged.
10.2 Structure Formation
As we have seen in the FLRW model, the force of gravity makes a homogeneous mat-
ter distribution unstable: it either expands or contracts. This is true for matter on all
scales, whether we are considering the whole Universe or a tiny localized region. But
the FLRW expansion of the Universe as a whole is not exponential and therefore it
is too slow to produce our Universe in the available time. This requires a different
mechanism to give the necessary exponential growth: cosmic inflation. Only after the
graceful exit from inflation does the Universe enter the regime of Friedmann expan-
sion, during which the Hubble radius gradually overtakes the inflated regions.
In Figure 10.1 we show schematically the fate of fluctuations as a funtion of time
or the cosmic scale푎. Inflationary fluctuations crossed the post-inflationary Hubble
radius at scale푎 1 and came back into vision recently, at≈푎 0 , with a wavelength휆hor
corresponding to the size the Hubble radius at that moment. Some galaxies will cross
the post-inflationary Hubble radius at scale푎 2 and come back into vision with a wave-
length휆gallater at the cosmic scale>푎Hwith a comoving length scale>퐻−^1.
In noninflationary cosmology, a given scale crosses the horizon but once, while in
the inflationary cosmology all scales begin subhorizon sized, cross outside the Hubble
radius during inflation, and re-enter during the post-inflationary epoch. The largest
scales cross outside the Hubble radius first and re-enter last. Causal microphysics
operates only on scales less than퐻−^1 below the strong black line. During inflation
퐻−^1 is a constant, and in the post-inflation era it is proportional to푎^1 ∕푛,where푛= 2
during radiation domination, and푛=^32 during matter domination.
Jeans Mass. Primordial density fluctuations expand linearly at a rate slower than
the Universe is expanding in the mean, until eventually they reach a maximum size
and collapse nonlinearly. If the density fluctuates locally, also the cosmic scale factor
will be a fluctuating function푎(r,푡)of position and time. In overdense regions where
the gravitational forces dominate over pressure forces, causing matter to contract