A.3 Background cosmological spacetime

We consider that the spacetime is describe by a manifold ℳ with metric g μν with signature (− ,+, +,+ ). In the case of a Minkowsky spacetime g μν = η μν.

In the cosmological context, we will describe the universe by a Friedmann–Lemaître spacetime with metric

ds2 = − dt2 + a2(t)γijdxidxj (235 )
where t is the cosmic time, a the scale factor and γij the metric on the constant time hypersurfaces. The Hubble function is defined as H ≡ ˙a∕a. We also define the redshift by the relation 1 + z = a0∕a, with a0 the scale factor evaluated today.

The evolution of the scale factor is dictated by the Friedmann equation

8πG K Λ H2 = ----ρ − ---+ -, (236 ) 3 a2 3
where ρ = xiρi is the total energy density of the matter components in the universe. Assuming the species i has a constant equation of state wi = Pi∕ρi, each component evolves as ρi = ρi0(1 + z)2(1+wi). The Friedmann equation can then be rewritten as
H2 ∑ --2= Ωi (1 + z )3(1+wi) + ΩK (1 + z)2 + ΩΛ, (237 ) H0
with the density parameters defined by
Ω ≡ 8-πG-ρi0 , Ω ≡ − -K---, Ω ≡ -Λ--. (238 ) i 3H20 i 3H20 Λ 3H20
They clearly satisfy ∑ Ωi + ΩK + Ω Λ = 1.

Concerning the properties of the cosmological spacetime, I follow the notations and results of [409].

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