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M. B. Wise
reheats because of the inflations coupling to other fields, for example, the standard
model, Higgs field. I will not dwell on how hard it actually is to dynamically
realize this situation. The upshot is that we end up with an extremely smooth (the
exponential expansion dilutes all classical inhomogeneities) region that includes our
whole visible universe (and perhaps much more).
This scenario provides a mechanism to dynamically generate the density perturbations through quantum fluctuations in a scalar degree of freedom. In different
gauges precisely how the scalar degree of freedom is defined takes different forms.
We prefer to work in a gauge where the inflation field is spatially homogeneous.
In this gauge it is the curvature perturbation δ ζ (x, t) during the inflationary era that
is the scalar degree of freedom that characterizes the fluctuations about pure de
Sitter space time. Just after inflation the energy density perturbations δ ρ that cause
the microwave background anisotropies and the correlations of galaxy positions (on
large scales) have wavelengths well outside the horizon. They reenter the horizon
during the matter dominated era and on long wavelengths are related to the curvature
perturbations generated during the inflationary era by,
δ ρ ∝ ∇
2 δ ζ .
(2)
1.1 Symmetries of Space Time Metrics
Let us start with the metric for flat Minkowski space time, ds 2 = −dt 2 + dx i dx i .
Some symmetries of this metric are translations of time t → t + d, translations of
space x i → x i = x i + c i and rotations x i → x i = R ij x j , RR T = I . These
symmetries have implications for the equal time correlations of a scalar quantum
field φ(t, x) in a Minkowski space background. Fourier transforming,
φ(t, x) =
d 3 k
(2π) 3 e
ik·x ˜
φ(t, k)
(3)
one has that
˜
φ(t, k) ˜
φ(t, k
) = P φ (k)(2π)
3 δ(k + k
).
(4)
The Dirac delta function arises from the spatial translation invariance, and the fact
that the power spectrum P φ (k) only depends on k, the magnitude of the wave vector
k follows from rotational invariance. Time translation invariance did not play a role
since we evaluated the equal time two-point correlation.
For a de Sitter space time background the metric is no longer invariant under time
translations. Instead a combined time translation and scale transformation
t → t
= t + d, x
i
→ x
i
= λx
i , λ = e
−H d
(5)
M. B. Wise
reheats because of the inflations coupling to other fields, for example, the standard
model, Higgs field. I will not dwell on how hard it actually is to dynamically
realize this situation. The upshot is that we end up with an extremely smooth (the
exponential expansion dilutes all classical inhomogeneities) region that includes our
whole visible universe (and perhaps much more).
This scenario provides a mechanism to dynamically generate the density perturbations through quantum fluctuations in a scalar degree of freedom. In different
gauges precisely how the scalar degree of freedom is defined takes different forms.
We prefer to work in a gauge where the inflation field is spatially homogeneous.
In this gauge it is the curvature perturbation δ ζ (x, t) during the inflationary era that
is the scalar degree of freedom that characterizes the fluctuations about pure de
Sitter space time. Just after inflation the energy density perturbations δ ρ that cause
the microwave background anisotropies and the correlations of galaxy positions (on
large scales) have wavelengths well outside the horizon. They reenter the horizon
during the matter dominated era and on long wavelengths are related to the curvature
perturbations generated during the inflationary era by,
δ ρ ∝ ∇
2 δ ζ .
(2)
1.1 Symmetries of Space Time Metrics
Let us start with the metric for flat Minkowski space time, ds 2 = −dt 2 + dx i dx i .
Some symmetries of this metric are translations of time t → t + d, translations of
space x i → x i = x i + c i and rotations x i → x i = R ij x j , RR T = I . These
symmetries have implications for the equal time correlations of a scalar quantum
field φ(t, x) in a Minkowski space background. Fourier transforming,
φ(t, x) =
d 3 k
(2π) 3 e
ik·x ˜
φ(t, k)
(3)
one has that
˜
φ(t, k) ˜
φ(t, k
) = P φ (k)(2π)
3 δ(k + k
).
(4)
The Dirac delta function arises from the spatial translation invariance, and the fact
that the power spectrum P φ (k) only depends on k, the magnitude of the wave vector
k follows from rotational invariance. Time translation invariance did not play a role
since we evaluated the equal time two-point correlation.
For a de Sitter space time background the metric is no longer invariant under time
translations. Instead a combined time translation and scale transformation
t → t
= t + d, x
i
→ x
i
= λx
i , λ = e
−H d
(5)