Thermodynamics of Creation. Note Quote.

Just like the early-time cosmic acceleration associated with inflation, a negative pressure can be seen as a possible driving mechanism for the late-time accelerated expansion of the Universe as well. One of the earliest alternatives that could provide a mechanism producing such accelerating phase of the Universe is through a negative pressure produced by viscous or particle production effects. The viscous pressure contributions can be seen as small nonequilibrium contributions for the energy-momentum tensor for nonideal fluids.

Let us posit the thermodynamics of matter creation for a single fluid. To describe the thermodynamic states of a relativistic simple fluid we use the following macroscopic variables: the energy-momentum tensor T^αβ ; the particle flux vector N^α; and the entropy flux vector s^α. The energy-momentum tensor satisfies the conservation law, T^αβ_;β = 0, and here we consider situations in which it has the perfect-fluid form

T^αβ = (ρ+P)u^αu^β − P g^αβ

In the above equation ρ is the energy density, P is the isotropic dynamical pressure, g^αβ is the metric tensor and u^α is the fluid four-velocity (with normalization u^αu_α = 1).

The dynamical pressure P is decomposed as

P = p + Π

where p is the equilibrium (thermostatic) pressure and Π is a term present in scalar dissipative processes. Usually, it is associated with the so-called bulk pressure. In the cosmological context, besides this meaning, Π can also be relevant when particle number is not conserved. In this case, Π ≡ p_c is called the “creation pressure”. The bulk pressure,  can be seen as a correction to the thermostatic pressure when near to equilibrium, thus, it should be always smaller than the thermostatic pressure, |Π| < p. This restriction, however, does not apply for the creation pressure. So, when we have matter creation, the total pressure P may become negative and, in principle, drive an accelerated expansion.

The particle flux vector is assumed to have the following form

N^α = nu^α

where n is the particle number density. N^α satisfies the balance equation N^α_;α = nΓ, where Γ is the particle production rate. If Γ > 0, we have particle creation, particle destruction occurs when Γ < 0 and if Γ = 0 particle number is conserved.

The entropy flux vector is given by

s^α = nσu^α

where σ is the specific (per particle) entropy. Note that the entropy must satisfy the second law of thermodynamics s^α_;α ≥ 0. Here we consider adiabatic matter creation, that is, we analyze situations in which σ is constant. With this condition, by using the Gibbs relation, it follows that the creation pressure is related to Γ by

p_c = − (ρ+p)/3H Γ

where H = a ̇/a is the Hubble parameter, a is the scale factor of the Friedmann-Robertson-Walker (FRW) metric and the overdot means differentiation with respect to the cosmic time. If σ is constant, the second law of thermodynamics implies that Γ ≥ 0 and, as a consequence, particle destruction (Γ < 0) is thermodynamically forbidden. Since Γ ≥ 0, it follows that, in an expanding universe (H > 0), the creation pressure p_c cannot be positive.

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Cosmology: Friedmann-Lemaître Universes

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 u_a = −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:

C_ijkl := R_ijkl + 1/2(R_ikg_jl + R_jlg_ik − R_il g_jk − R_jkg_il) − 1/6R(g_ikg_jl − g_ilg_jk) = 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, u^a = δ^a₀ (a=0,1,2,3) —– (2)

where S(t) is the time-dependent scale factor, and the worldlines with tangent vector u^a = dx^a/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

dσ² = 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 T_ab necessarily takes a perfect fluid form relative to the preferred worldlines with tangent vector u^a:

T_ab = (μ + p/c²)u_au_b + (p/c²)g_ab —– (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 T_ab. The integrability conditions for the Einstein Field Equations are the energy-density conservation equation

T^ab_;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 {p_bar = 0 ⇔ w = 0} and radiation has {p_radc² = μ_rad/3 ⇔ w = 1/3,μ_rad = aT⁴_rad}, which by (5) imply

μ_bar ∝ S⁻³, μ_rad ∝ S⁻⁴, T_rad ∝ 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 μ_grav := 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 = μ_bar + μ_rad + μ_cdm + μ_ν 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 Ω_i0 := κμ_i0/3H₀² for each type of matter present (labelled by i);

• If Λ ≠ 0, either Ω_Λ0 := 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 Ω_i0 for each type of matter present, for example

Ω_m0 = Ω_bar0 + Ω_rad0 + Ω_cdm0 + Ω_ν0, —– (10)

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

Ω₀ = Ω_m0 + Ω_Λ0 —– (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 Ω_m0 − Ω_Λ0 —– (12)

This shows that a cosmological constant Λ can cause an acceleration (negative q₀); if it vanishes, the expression simplifies: Λ = 0 ⇒ q = 1 Ω_m0, 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 = H₀² (Ω₀ − 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).