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The flat barotropic power-law solutions of the homogeneous cosmological field equations

Physical scope

Explicit exact solutions of the adopted field law for the flat (k=0), Lambda=0 homogeneous model with a comoving perfect fluid whose equation of state is p = w epsilon with a constant w > -1, on a open half-line: the expanding power-law scale factors and their energy densities. The statement is a conditional verification inside the adopted model; it asserts no uniqueness, no extension beyond the displayed interval, no value of the constants, and no comparison with observations.

Statement

Adopt the homogeneous cosmological model of Adopted homogeneous and isotropic cosmological model with a comoving perfect fluid with the flat standard spatial model ( $k = 0$ , Euclidean space), the field law of the model with $Λ = 0$ , conversion constant $c > 0$ and the framework's $G_{N} > 0$ , and a comoving perfect-fluid matter sector whose equation of state is $p = w ε$ for a real constant $w > - 1$ (The constant-curvature spatial model of a homogeneous cosmology clause (d), The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy, SI unit and coordinate conventions of the classical gravitational framework).

(a) Let $t_{0} \in R$ , let $t_{*} > 0$ be a reference time, let $A > 0$ be a reference length, and put

$q : = \frac{2}{3 (1 + w)}, I : = (t_{0}, \infty), a (t) : = A {(\frac{t - t_{0}}{t_{*}})}^{q}, ε (t) : = \frac{3 c^{2} q^{2}}{8 π G_{N} (t - t_{0})^{2}},$

with $p (t) : = w ε (t)$ . Then the data $(c, G_{N}, 0, I, 0, a, ε, p)$ satisfy the Friedmann equation (F) and the acceleration equation (A) on $I$ (The Friedmann, acceleration and continuity equations of an FLRW model datum), so by Reduction of the field equations of the homogeneous cosmological model to the Friedmann and acceleration equations, with derived continuity the adopted field equations hold on the spacetime $I \times Σ$ . In particular:

for $w = 0$ (dust) one has $q = \frac{2}{3}$ and $a (t) = A ((t - t_{0}) / t_{*})^{2 / 3}$ ;
for $w = \frac{1}{3}$ (radiation) one has $q = \frac{1}{2}$ and $a (t) = A ((t - t_{0}) / t_{*})^{1 / 2}$ .

(b) On $I$ the density scales as $ε (t) = C a (t)^{- 3 (1 + w)}$ with the positive constant $C = A^{3 (1 + w)} \frac{3 c^{2} q^{2}}{8 π G_{N} t_{*}^{2}}$ , and the continuity equation (C) holds. In the adopted SI convention, $w$ and $q$ are dimensionless and $C$ has units $J m^{3 w}$ , since $ε$ has units $J m^{- 3}$ and $a$ has units $m$ (SI unit and coordinate conventions of the classical gravitational framework).

The theorem asserts no uniqueness of solutions of the reduced system for the stated equation of state, no extension of the solutions beyond the displayed interval, and no value of $A, t_{*}, t_{0}$ or of the constants.

Facts & Assumptions

Given: The adopted model with the flat standard spatial model, $Λ = 0$ , the field law, a comoving perfect fluid with equation of state $p = w ε$ , a real constant $w > - 1$ , and the data $q, t_{*}, A, I, a, ε, p$ displayed in the statement.

[L1]

The spatial model with $k = 0$ is the Euclidean standard model and satisfies the curvature condition of The constant-curvature spatial model of a homogeneous cosmology; the metric, comoving congruence and Hubble parameter are those of FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter, and the model's geometry and matter content are the adopted assumptions of Adopted homogeneous and isotropic cosmological model with a comoving perfect fluid.

[L2]

For $a (t) = A ((t - t_{0}) / t_{*})^{q}$ on $I$ with $q \neq 0$ : the function is smooth and strictly positive, $H (t) = \dot{a} / a = q / (t - t_{0})$ , and $\ddot{a} (t) / a (t) = q (q - 1) / (t - t_{0})^{2}$ (Real powers for positive bases, with the zero-base positive-exponent convention, Continuity and derivatives of positive-base real powers, The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

[L3]

The equations (F) and (A) of The Friedmann, acceleration and continuity equations of an FLRW model datum read, for $k = Λ = 0$ , $H^{2} = \frac{8 π G_{N}}{3 c^{2}} ε$ and $\frac{\ddot{a}}{a} = - \frac{4 π G_{N}}{3 c^{2}} (ε + 3 p)$ , and (C) reads $\dot{ε} + 3 H (ε + p) = 0$ ; under Reduction of the field equations of the homogeneous cosmological model to the Friedmann and acceleration equations, with derived continuity, a datum satisfying (F) and (A) satisfies the adopted field law on the spacetime.

[L4]

For the displayed data, $ε (t) = \frac{3 c^{2} q^{2}}{8 π G_{N}} (t - t_{0})^{- 2}$ and $p = w ε$ , so $ε + 3 p = (1 + 3 w) ε$ and $ε + p = (1 + w) ε$ ; moreover $q \cdot 3 (1 + w) = 2$ by the definition of $q$ , which gives $3 (1 + w) q = 2$ and $\frac{2}{3 q} = 1 + w$ .

Proof

technique · verify the two reduced equations by direct substitution in the displayed data

1.1L1L2L4algebra

(Smoothness and derivatives.) The function $a$ is smooth and strictly positive on $I$ because $t - t_{0} > 0$ , $t_{*} > 0$ , $A > 0$ and a positive power of a positive number is positive; $ε$ is smooth and strictly positive on $I$ . By [L2], $H = q / (t - t_{0})$ and $\ddot{a} / a = q (q - 1) / (t - t_{0})^{2}$ ; by [L4], $p = w ε$ with $w > - 1$ , so $q = 2 / (3 (1 + w))$ is a positive real number. In particular all the data of a Friedmann-system datum are present, and $ε > 0$ on $I$ .

2.1L3step 1.1algebra

(Friedmann equation.) Since $k = 0$ , (F) is $H^{2} = \frac{8 π G_{N}}{3 c^{2}} ε$ . By step 1.1, $H^{2} = q^{2} / (t - t_{0})^{2}$ , while the right-hand side is $\frac{8 π G_{N}}{3 c^{2}} \cdot \frac{3 c^{2} q^{2}}{8 π G_{N} (t - t_{0})^{2}} = q^{2} / (t - t_{0})^{2}$ . The two sides agree identically on $I$ ; the cancellation of $G_{N}$ , $c$ and the numerical constants is exact, and no choice of $A$ or $t_{*}$ enters (F).

2.2L3L4step 1.1algebra

(Acceleration equation.) Since $Λ = 0$ , (A) is $\frac{\ddot{a}}{a} = - \frac{4 π G_{N}}{3 c^{2}} (ε + 3 p)$ . By steps 1.1 and [L4], the left-hand side is $q (q - 1) / (t - t_{0})^{2}$ and the right-hand side is $- \frac{4 π G_{N}}{3 c^{2}} (1 + 3 w) \frac{3 c^{2} q^{2}}{8 π G_{N} (t - t_{0})^{2}} = - \frac{(1 + 3 w) q^{2}}{2 (t - t_{0})^{2}}$ . The two agree exactly when $q (q - 1) = - \frac{1}{2} (1 + 3 w) q^{2}$ ; since $q > 0$ , this is equivalent to $q - 1 = - \frac{1}{2} (1 + 3 w) q$ , that is, to $2 q - 2 = - (1 + 3 w) q$ , that is, to $3 (1 + w) q = 2$ , which is exactly the defining relation for $q$ by [L4]. Hence (A) holds on $I$ .

3.1L2L3L4step 2.1step 2.2algebra∎

(Conclusion: field equations, density scaling and continuity.) By steps 2.1 and 2.2 the datum satisfies (F) and (A) on $I$ , so by [L3] and Reduction of the field equations of the homogeneous cosmological model to the Friedmann and acceleration equations, with derived continuity the adopted field equations hold on the spacetime, which is the first part of claim (a); the special values $w = 0$ and $w = \frac{1}{3}$ give $q = \frac{2}{3}$ and $q = \frac{1}{2}$ by [L4], which is the rest of claim (a). For claim (b), compute $ε (t) a (t)^{3 (1 + w)} = \frac{3 c^{2} q^{2}}{8 π G_{N}} (t - t_{0})^{- 2} A^{3 (1 + w)} (\frac{t - t_{0}}{t_{*}})^{3 (1 + w) q}$ ; by [L4], $3 (1 + w) q = 2$ , so the power of $(t - t_{0})$ is $- 2 + 2 = 0$ and the product equals the constant $C = A^{3 (1 + w)} \frac{3 c^{2} q^{2}}{8 π G_{N} t_{*}^{2}}$ , which is positive; the exponent of $t_{*}$ is $3 (1 + w) q = 2$ . Differentiating $ε = C a^{- 3 (1 + w)}$ with respect to $t$ and using $\dot{a} = H a$ gives $\dot{ε} = - 3 (1 + w) H ε = - 3 H (ε + p)$ , which is the continuity equation (C).

Remarks

What the solutions are and are not. The theorem verifies a family of exact solutions of the flat, $Λ = 0$ barotropic branch; it does not claim that these are the only solutions with the stated equation of state, and it makes no statement about the behaviour of the scale factor as $t ↓ t_{0}$ , about spacetime singularities, about the physical interpretation of the coordinates, or about any observation.
Values of the constants. $A$ , $t_{*}$ and $t_{0}$ are arbitrary model data; the density at a reference time determines the combination $C$ , and the expansion rate at a reference time determines $t - t_{0}$ there. No numerical value of $H$ , of a density or of an age is asserted; any such value would be an empirical premise requiring a source-backed measurement.

The flat barotropic power-law solutions of the homogeneous cosmological field equations

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources