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5 Cosmology and General Relativity |
which is the standard conversion formula from polar coordinates to a set of orthogonal Cartesian coordinates in the standard Euclidian space R3.
In terms of the coordinates {η, xi } the background cosmological metric takes the form:
ds02 = a(η)2 |
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(5.9.3) |
i=1
Using the same coordinates also for the scalar field, the coupled system of Einstein and Klein-Gordon equations, displayed in (5.8.7), (5.8.8) and (5.8.10), is turned into the following one:
H 2 = |
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π G ϕ 2 + 2a2V |
(5.9.4) |
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π G − ϕ 2 + 4a2V |
(5.9.5) |
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(5.9.6) |
where the prime denotes the derivative with respect to η, and where we have introduced the conformal Hubble function:
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H = a |
(5.9.7) |
From these background equations we can also derive the following identity:
H − H 2 = −2π G ϕ 2 |
(5.9.8) |
which is quite useful in further manipulations.
It is also convenient to inspect the form taken by the de Sitter solution in the conformal frame. This latter is characterized by a constant scalar field ϕ = ϕ0 sitting at an extremum of the potential, as described in (5.8.11). Under these conditions the solution of the differential system is:
a(η) = − 2 H0η
(5.9.9)
H = − 1
η
where H0 was defined in (5.8.14).
5.9.2 Deriving the Equations for the Perturbation
As a next step we parameterize the perturbations of the metric and of the scalar field around a homogeneous isotropic solution, namely around a solution of the differen-
5.9 Primordial Perturbations of the Cosmological Metric and of the Inflaton |
189 |
tial system (5.9.4), (5.9.5), (5.9.6). Since the scalar field has spin zero it follows that the relevant perturbations of the metric which are driven by scalar perturbations have also spin zero. Modulo redefinitions induced by linearized diffeomorphisms, the scalar perturbations of the metric can be encoded in two scalar functions Φ(η, xi ) and Ψ (η, xi ) that deform the line-element (5.9.3) in the following way:
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(5.9.10) |
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2Φ(η, xi ) dη2 |
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2Ψ (η, xi ) |
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where: |
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1 & Φ(η, xi ); |
1 & Ψ (η, xi ) |
(5.9.11) |
Similarly the perturbation of the scalar field is parameterized as follows: |
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ϕ(η, xi ) = ϕ(η) + δϕ(η, xi ) |
(5.9.12) |
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where we have: |
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ϕ(η) & δϕ(η, xi ) |
(5.9.13) |
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We are interested in expanding the Klein-Gordon and Einstein equations to first order in the perturbations in order to derive the linearized equations for these latter in the background of the considered isotropic homogeneous solution. It turns out to be convenient to write Einstein equations in the following way:
Gμν = 4π GT μν |
(5.9.14) |
where the first index of both the Einstein tensor and of the stress-energy tensor have been raised by means of the inverse metric gμν . Relying on this form of the equation we write:
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= G(μ0)ν + δGμν |
(5.9.15) |
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= T(μ0)ν + δT μν |
(5.9.16) |
and we obtain the linearized Einstein equations in the form:
δGμν = 4π GδT μν |
(5.9.17) |
In a similar way we have to expand to first order in the perturbation fields the KleinGordon equation. The results are obtained by means of a straightforward, although lengthy calculation. Let us spell it. For the perturbation of the Einstein tensor we obtain
δG0 |
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2Ψ − 3H Ψ − 3H 2Φ |
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∂i Ψ + H Φ |
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δG0 i = |
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2 ∂i ∂j (Φ − Ψ ) |
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5 Cosmology and General Relativity |
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δGi j |
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(5.9.20) |
where: |
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denotes the standard Laplacian operator in flat three-space.
On the other hand, the first order perturbation of the scalar field stress-energy tensor is the following one:
δT 00 |
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ϕ δϕ − ϕ 2 + a2 dϕ |
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dV |
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Finally the first order perturbation of the Klein-Gordon equation takes the following form:
δϕ − 2δϕ − ϕ |
(3Ψ + Φ) + 2H ϕ + 2a2 |
d2V |
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ϕδϕ = 0 |
(5.9.25) |
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The unknown functions are three Φ, Ψ , δϕ, but the field equations are much more, although not all independent, since Einstein equations are constrained by the Bianchi identities, as we extensively discussed in Volume 1. The net balance of these arguments is that in the perturbed Einstein-Klein-Gordon system there is just one independent scalar degree of freedom u(η, xi ), which obeys the following propagation equation:
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u − 2u − θ u = 0 |
(5.9.26) |
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where: |
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H (η) |
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(5.9.27) |
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All the other functions Φ, Ψ , δϕ can be expressed in terms of u(η, xi ).
5.9 Primordial Perturbations of the Cosmological Metric and of the Inflaton |
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5.9.2.1 Meaning of the Propagation Equation
Before deriving it, let us briefly discuss the interpretation of (5.9.26). Consider a free scalar field ξ living in a conformally flat Minkowskian manifold with a metric of the following form:
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dsλ2 = λ2(η)2dη2 − dxi2 |
(5.9.28) |
i=1
in such a background, the action of the free field ξ , takes the following explicit form:
Aξ = − Det g |
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∂μξ ∂ν ξgμν d4x |
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By means of the field redefinition: |
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after an integration by parts, the action (5.9.29) becomes:
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A [u, λ] = |
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and its variation yields the general field equation:
λu ≡ u − 2u − |
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(5.9.32) |
which is formally the same as the field equation of a free scalar field in a Minkowski space with an effective time-dependent mass:
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(η) ≡ |
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(5.9.33) |
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In particular a free scalar field propagating in the background cosmological metric corresponds to this class of actions with λ(η) = a(η), which, for de Sitter space is given by (5.9.9). Hence the effective degree of freedom encoding the combined gravitational and scalar field perturbation is just a free field propagating in an effective background manifold which has just the same structure as the physical universe but with an effective scale factor θ (η), as defined in (5.9.27), which replaces the actual scale factor a(η).
For the pure de Sitter case the time evolving effective mass is:
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(η) = |
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mdS |
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(5.9.34) |
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