164 Cosmic Inflation
Perturbations. Density perturbations in the cosmological fluid of wavelength훬shift
with the scale푎(푡)of the expansion, or what is the same, the comoving wavelength
is a constant. This is true during matter domination and radiation domination. The
comoving horizon grows proportionally to the conformal time휏so that wavelengths
훬which are outside the horizon early on will later move in.
The situation was different during inflation [3]. The initial singularity is a surface
of constant negative conformal time휏<0. After the Big Bang when the universe was
of zero spatial size it became spatially infinite in an infinitesimal duration of time, so
the Big Bang happened simultaneously everywhere at infinite speed. During inflation
휏is negative, becoming less negative until휏=0 when the expansion with푎(푡)com-
mences. Thus perturbations of wavelengths훬which were inside the horizon in an
infinitesimal duration of time after the initial singularity will move out and disappear
at superhorizon scales. The shortest wavelength perturbations are those which exited
the horizon at푁=0 and longer ones excited earlier. Perturbations about the same
size as our horizon today exited inflation at aboutN=60.
When the perturbations enter the horizon they behave as a classical field, creating
the inhomogeneities we observe in the CMB and the large-scale distribution of matter.
A problem is that the separation into a homogeneous background, that depends only
on time, and spatially dependent perturbations, is not unique, it depends on the
choice ofgauge.
Since the introduction of metrics in Chapter 2 we have been using the coordinates
푡,푥,푦,푧or푥^0 ,푥^1 ,푥^2 ,푥^3. The spacelike hyper-surfaces of constant푡define theslicingto
the four-dimensional space-time, while the timelike worldlines of constantxdefine the
threading. Each spacelike threading corresponds to a homogeneous universe while the
slicing is orthogonal to these universes. In the homogeneous and isotropic universe
we have studied so far our choice of coordinates was natural and we did not need to
consider alternatives.
However, in a perturbed spacetime the definition of slicing and threading is not
unique. Consider replacing the time coordinate푡by a perturbed time slicẽ푡=푡+
훿푡(푡,x ̊). A spatially homogeneous and isotropic function is only a function of time푡,
as for example the energy density휌(푡). On the new time-slicẽ푡the energy density will
not be homogeneous,휌̃̃푡x ̊=휌(푡)(̃푡x ̊). In general relativity we need both the matter
field perturbations and the metric perturbations, so we can use the freedom of gauge
transformation to trade one for the other. In the present example we can choose the
hyper-surface of constant time to coincide with the hypersurface of constant energy
density so that the real perturbations vanish,훿̃휌=0.
One simple choice is to fix a gauge where the nonrelativistic limit of the full per-
turbed Einstein equation can be recast as a Poisson equation with a Newtonian grav-
itational potential,훷. The induced metric can then be written
푑푠^2 =푎^2 (휏)[( 1 + 2 훷)d휏^2 −( 1 − 2 훹)훿ikd푥푖d푥푘]. (7.43)In the presence of Einstein gravity and when the spatial part of the energy-momentum
tensor is diagonal one has훷=훹. Other useful gauges can be defined.
In single-field inflation we define perturbations around the homogeneous back-
ground solutions for the inflaton휙and the metric푔휇휈(푡)by
휙(푡,x ̊)=휙(푡)+훿휙(푡,x ̊),푔휇휈(푡,x ̊)=푔휇휈(푡)+훿푔휇휈(푡,x ̊). (7.44)