Calculation of the Friedmann Equation  ↑

The metric for a homogeneous isotropic universe has the form
Using

Using dot for a time derivative and prime for differentiation with respect to ρ, the non-vanishing partial derivatives of the metric are:

The non-vanishing Christoffel symbols,
are:

On raising the first index,

The Ricci Curvature Tensor is

Calculate the diagonal coefficients. The time component is
Calculate the terms individually.
Substitute into R₀₀,

Calculate the radial component of the Ricci tensor.
Calculate the terms individually.
Substitute into R₁₁,
Observe that f ⁻¹f '' = −k.

Calculate the θ component of the Ricci tensor.
Calculate the terms individually,
Substitute into R₂₂,
Observe that f '' = −kf, and 1 − f '² = kf ².

Calculate the φ component of the Ricci tensor.
Calculate the terms individually,

Substitute into R₃₃,
Observe that f '' = −kf, and f '² = 1 − kf ².


Scalar curvature is
Then we find
Weyl’s postulate states that the net behaviour of galaxies is as for a perfect fluid. Then the stress energy tensor is
Einstein’s field equation,
yields
Rearranging gives Friedmann’s equation,

The other components of G do not add anything beyond local energy-momentum conservation


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Solutions of the Friedmann Equation

The Einstein-de Sitter model

The Einstein-de Sitter, Flat space, no-Λ model has the simplest solution. It has some physical interest because, in all models, for small a, expansion is dominated by the density term. In the early universe all models are approximated by the Einstein-de Sitter model. With k = 0 and Λ = 0, the Friedmann equation reduces to
On integration, and using the initial condition t = 0 at a = 0
Let
(the mass of a Euclidean sphere of density ρ₀ and radius a₀). Recall

Flat space, Λ models

With k = 0, the Friedmann equation reduces to
Substitute u, with
Then,
Substitute v with
Then,
where the constant of integration is found at t = 0, a = 0, so that u = 0, and v = π.
Using


This model collapses after a time t = 2π ⁄ √(−3Λ).


As with all models, expansion for a ≈ 0 is approximated by the Einstein-de Sitter model. As a increases accelerating expansion takes over. The concordance model has this solution.

Curved Space, No Λ models

With Λ = 0, the Friedmann equation reduces to
Substitute u, with
Substitute θ with
where the constant of integration is found at t = 0, a = 0, so that u = 0, θ = 0. We have
Let ω = 2θ, we have parametric equations for a and t
Using
For a positive curvature universe, these are the parametric equations of cycloid.



For a negative curvature universe, the substitution ω →iω yields


As with all models, the initial expansion is approximated by the Einstein-de Sitter model. For large t, a ⁄ t → 1.

Behaviour for large a

For large a, behaviour is dominated by the cosmological constant (if it is non-zero). The Friedmann equation is approximated by
For Λ > 0, expansion will assymptotically approach . If Λ < 0, this is cyclic. Expansion slows down and the universe collapses.

Einstein Static Solution

For k = 1, for a given value of M, there is a particular solution with a critical value Λ_E such that and a = a_E, constant, for which the Friedmann equation reduces to
However, if, even in a local fluctuation, a rises above a_E, the model moves into the large a regime of accelerating expansion and is thus unstable.

Other positive curvature models

At the critical value, Λ = Λ_E, if a < a_E, the mass term determines that , so the model expands assymptotically towards the Einstein static solution.

If a > a_E we have the Eddington-Lemaitre model, which approaches the Einstein static solution as time runs backwards.

The Lemaitre model is given by Λ > Λ_E, in which expansion does not stop.

If 0 < Λ < Λ_E, then either expansion stops at a < a_E, and the universe collapses, or a > a_E, in which case the model has accelerating expansion and has come from contraction in the past.

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