The isotropic and homogenous Universe

The simplest model that one can consider for the Universe is obtained using the homogeneity and isotropy assumptions of the matter³ distribution. According to Einstein’s field equations the geometric property of the Universe is defined based on the matter distribution. As a result the 3 dimensional space should also be homogenous and isotropic, in other words any geometrical quantity on the constant time manifolds, for example the 3D Ricci scalar and extrinsic cur-vature should be independent of the position and direction. However, this of course does not necessarily hold for the 4D geometrical objects like the 4D Ricci scalar as they can be time dependent (for a detailed discussion see e.g. Amen-dola & Tsujikawa [2010], Dodelson [2003], Durrer [2008], Padmanabhan [1996], Padmanabhan [2000], Padmanabhan [2002, 2010], Weinberg [2008])

The assumption of 3D isotropy and homogeneity defines a set of equivalent observers, the special observer who sees the Universe homogenous and isotropic.

3dark matter, dark energy and radiation.

1.1. The isotropic and homogenous Universe

Other observers that have a relative velocity with respect to these observers or using a peculiar coordinate to do measurements, they see an anisotropic and inhomogeneous Universe. We, for example, are among these "bad" observers that see the Universe anisotropic, because according to our relative velocity we see the cosmic microwave background radiation anisotropic (we see a dipole in the CMB which is detected by Plank, WMAP and COBE satellite).

Considering the coordinate (t, x^α) for the special observers (who sees the Universe isotropic and homogenous) we can write a general space-time interval as following,

ds² =g_µνdx^µdx^ν =−g₀₀dt²−2g_0idtdxⁱ+σ_ijdxⁱdx^j, (1.1) where σ_ij is the spatial part of the line element. We are going to simplify the line element according to the symmetries we have. The isotropy of the Universe implies thatg_0i = 0 since a non-zero value of these elements means that the mea-sured quantity, and as a result, the geometrical objects depend on the direction of measurement. Moreover if we use the proper time of clocks carried by these observers we haveg₀₀= 1. The line element then is written as,

ds² =−dt²+σ_ijdxⁱdx^j. (1.2) At any time the three-metric σ_ij is homogenous and isotropic. According to the isotropic assumption which is equivalent to the spherical symmetry in the three-metric we can write,

σ_ijdxⁱdx^j =a(t)^2hλ²(r)dr²+r²dθ²+ sin²θdφ²ⁱ. (1.3) Now we can use the three dimensional Ricci scalar and the symmetries to simplify the previous expression. The 3D Ricci scalar, or the scalar curvature, reads

3R=σ^ijR_ij =σ^ij

The homogeneity implies that ³R as a three dimensional geometrical object should be independent of the position (here r), i. e. ³R should be a constant.

3

4According to our convention the Greek letters indices span the space-time coordinates (0,...,3) and the latin letters indices span the space coordinates (1,. . . , 3)

Integrating over r results in,

where a(t) scales the spatial metric and is called the scale factor. The previ-ous metric is called the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric which is an exact solution of Einstein’s field equations and it describes a spa-tially maximally symmetric (isotropic and homogeneous) universe. The spatial hypersurfaces of the FLRW metric have positive, zero and negative curvature for respectively k= +1, k = 0 andk =−1.

1.1.1 The geometrical properties of the FLRW metric

In order to have a geometrical intuition of the FLRW metric it is better to use the following coordinate for different values of the spatial curvature k

χ=

moreover, we change the time coordinate from the real time to the conformal time defined by dτ =dt/a(t). Then the metric reads,

ds² =a(τ)^2h−dτ²+dχ²+f²(χ)dθ² + sin²θdφ²ⁱ, (1.11)

1.1. The isotropic and homogenous Universe

The metric in (1.11) is especially useful since the curvature has gone into the angular part of the metric, so we can more easily study the distances. Let us now study the geometrical properties of this metric for different values ofk.

For k = 0 the spatial part of the metric represents the flat Euclidian three-dimensional space. For k = 1, the spatial part of the metric describes a three sphere of radius a embedded in a flat four dimensional Euclidian space. We can show such a 3-Sphere as,

x²₁+x²₂+x²₃+x²₄ =a², (1.13) wherex_i are the cartesian coordinates of the four dimensional space. The metric on the 3-Sphere embedded in the higher dimension is written as

dL² =dx²₁+dx²₂+dx²₃+dx²₄. (1.14) We can express the 3-Sphere by three angular coordinates (γ, θ, φ) instead, de-fined by

x₁ =acosγsinθsinφ, x₂ =acosγsinθcosφ, x₃ =acosγcosθ, x₄ =asinγ.

(1.15)

Substituting the new coordinate in the metric (1.14) we obtain

dL² =a^2hdγ² + sin²γdθ²+ sin²θdφ²ⁱ, (1.16) which is the same expression as (1.9) for k = 1. In fact the whole space of a positive curvature model is covered by the angle ranges 0 ≤ γ ≤ π; 0 ≤ θ ≤ whereσ is the determinant of the 3D metric.

In the case of k =−1 we have a hyperboloid geometry embedded in a four-dimensional Lorentzian space. The line element in Lorentzian space is

dL² =dx²₁+dx²₂+dx²₃ −dx²₄. (1.18) On the other hand a three-dimensional hyperboloid embedded in a 4D Lorentzian space is represented by the relation

x²₄−x²₁−x²₂−x²₃ =a². (1.19)

Like the 3-Sphere we can represent this hyperboloid with 3 angles (γ, θ, φ) defined by

x₁ =asinhχsinθsinφ, x₂ =asinhχsinθcosφ , x₃ =asinhχcosθ, x₄ =acoshχ ,

(1.20)

which by substituting to the line element we can see that this metric corresponds to the FLRW metric withk =−1. An important point about this space is thatγ is not bounded anymore and the angles ranges are 0≤φ ≤2π; 0 ≤θ ≤π; 0≤ γ ≤ ∞. This space, unlike the positive curvature space, has an infinite volume.

From now on during this thesis we alway assume k = 0 which is consistent with the cosmological observations.