Cosmology: Friedmann-Lemaître Universes

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Cosmology starts by assuming that the large-scale evolution of spacetime can be determined by applying Einstein’s field equations of Gravitation everywhere: global evolution will follow from local physics. The standard models of cosmology are based on the assumption that once one has averaged over a large enough physical scale, isotropy is observed by all fundamental observers (the preferred family of observers associated with the average motion of matter in the universe). When this isotropy is exact, the universe is spatially homogeneous as well as isotropic. The matter motion is then along irrotational and shearfree geodesic curves with tangent vector ua, implying the existence of a canonical time-variable t obeying ua = −t,a. The Robertson-Walker (‘RW’) geometries used to describe the large-scale structure of the universe embody these symmetries exactly. Consequently they are conformally flat, that is, the Weyl tensor is zero:

Cijkl := Rijkl + 1/2(Rikgjl + Rjlgik − Ril gjk − Rjkgil) − 1/6R(gikgjl − gilgjk) = 0 —– (1)

this tensor represents the free gravitational field, enabling non-local effects such as tidal forces and gravitational waves which do not occur in the exact RW geometries.

Comoving coordinates can be chosen so that the metric takes the form:

ds2 = −dt2 + S2(t)dσ2, ua = δa0 (a=0,1,2,3) —– (2)

where S(t) is the time-dependent scale factor, and the worldlines with tangent vector ua = dxa/dt represent the histories of fundamental observers. The space sections {t = const} are surfaces of homogeneity and have maximal symmetry: they are 3-spaces of constant curvature K = k/S2(t) where k is the sign of K. The normalized metric dσ2 characterizes a 3-space of normalized constant curvature k; coordinates (r, θ, φ) can be chosen such that

2 = dr2 + f2(r) dθ2 + sin2θdφ2 —– (3)

where f (r) = {sin r, r, sinh r} if k = {+1, 0, −1} respectively. The rate of expansion at any time t is characterized by the Hubble parameter H(t) = S ̇/S.

To determine the metric’s evolution in time, one applies the Einstein Field Equations, showing the effect of matter on space-time curvature, to the metric (2,3). Because of local isotropy, the matter tensor Tab necessarily takes a perfect fluid form relative to the preferred worldlines with tangent vector ua:

Tab = (μ + p/c2)uaub + (p/c2)gab —– (4)

, c is the speed of light. The energy density μ(t) and pressure term p(t)/c2 are the timelike and spacelike eigenvalues of Tab. The integrability conditions for the Einstein Field Equations are the energy-density conservation equation

Tab;b = 0 ⇔ μ ̇ + (μ + p/c2)3S ̇/S = 0 —– (5)

This becomes determinate when a suitable equation of state function w := pc2/μ relates the pressure p to the energy density μ and temperature T : p = w(μ,T)μ/c2 (w may or may not be constant). Baryons have {pbar = 0 ⇔ w = 0} and radiation has {pradc2 = μrad/3 ⇔ w = 1/3,μrad = aT4rad}, which by (5) imply

μbar ∝ S−3, μrad ∝ S−4, Trad ∝ S−1 —– (6)

The scale factor S(t) obeys the Raychaudhuri equation

3S ̈/S = -1/2 κ(μ + 3p/c2) + Λ —– (7)

, where κ is the gravitational constant and Λ is the cosmological constant. A cosmological constant can also be regarded as a fluid with pressure p related to the energy density μ by {p = −μc2 ⇔ w = −1}. This shows that the active gravitational mass density of the matter and fields present is μgrav := μ + 3p/c2. For ordinary matter this will be positive:

μ + 3p/c2 > 0 ⇔ w > −1/3 —– (8)

(the ‘Strong Energy Condition’), so ordinary matter will tend to cause the universe to decelerate (S ̈ < 0). It is also apparent that a positive cosmological constant on its own will cause an accelerating expansion (S ̈ > 0). When matter and a cosmological constant are both present, either result may occur depending on which effect is dominant. The first integral of equations (5, 7) when S ̇≠ 0 is the Friedmann equation

S ̇2/S2 = κμ/3 + Λ/3 – k/S2 —– (9)

This is just the Gauss equation relating the 3-space curvature to the 4-space curvature, showing how matter directly causes a curvature of 3-spaces. Because of the spacetime symmetries, the ten Einstein Filed Equations are equivalent to the two equations (7, 9). Models of this kind, that is with a Robertson-Walker (‘RW’) geometry with metric (2, 3) and dynamics governed by equations (5, 7, 9), are called Friedmann-Lemaître universes (‘FL’). The Friedmann equation (9) controls the expansion of the universe, and the conservation equation (5) controls the density of matter as the universe expands; when S ̇≠ 0 , equation (7) will necessarily hold if (5, 9) are both satisfied. Given a determinate matter description (specifying the equation of state w = w(μ, T) explicitly or implicitly) for each matter component, the existence and uniqueness of solutions follows both for a single matter component and for a combination of different kinds of matter, for example μ = μbar + μrad + μcdm + μν where we include cold dark matter (cdm) and neutrinos (ν). Initial data for such solutions at an arbitrary time t0 (eg. today) consists of,

• The Hubble constant H0 := (S ̇/S)0 = 100h km/sec/Mpc;

• A dimensionless density parameter Ωi0 := κμi0/3H02 for each type of matter present (labelled by i);

• If Λ ≠ 0, either ΩΛ0 := Λ/3H20, or the dimensionless deceleration parameter q := −(S ̈/S) H−20.

Given the equations of state for the matter, this data then determines a unique solution {S(t), μ(t)}, i.e. a unique corresponding universe history. The total matter density is the sum of the terms Ωi0 for each type of matter present, for example

Ωm0 = Ωbar0 + Ωrad0 + Ωcdm0 + Ων0, —– (10)

and the total density parameter Ω0 is the sum of that for matter and for the cosmological constant:

Ω0 = Ωm0 + ΩΛ0 —– (11)

Evaluating the Raychaudhuri equation (7) at the present time gives an important relation between these parameters: when the pressure term p/c2 can be ignored relative to the matter term μ (as is plausible at the present time, and assuming we represent ‘dark energy’ as a cosmological constant.),

q0 = 1/2 Ωm0 − ΩΛ0 —– (12)

This shows that a cosmological constant Λ can cause an acceleration (negative q0); if it vanishes, the expression simplifies: Λ = 0 ⇒ q = 1 Ωm0, showing how matter causes a deceleration of the universe. Evaluating the Friedmann equation (9) at the time t0, the spatial curvature is
K0:= k/S02 = H020 − 1) —– (13)
The value Ω0 = 1 corresponds to spatially flat universes (K0 = 0), separating models with positive spatial curvature (Ω0 > 1 ⇔ K0 > 0) from those with negative spatial curvature (Ω0 < 1 ⇔ K0 < 0).
The FL models are the standard models of modern cosmology, surprisingly effective in view of their extreme geometrical simplicity. One of their great strengths is their explanatory role in terms of making explicit the way the local gravitational effect of matter and radiation determines the evolution of the universe as a whole, this in turn forming the dynamic background for local physics (including the evolution of the matter and radiation).
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