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Equivalence of the Friedmann and acceleration systems, and the derivation of acceleration from Friedmann and continuity where the Hubble parameter does not vanish

Statement

Let $(c, G_{N}, Λ, I, k, a, ε, p)$ be a Friedmann-system datum, with the equations (F), (A), (C), (S) as in The Friedmann, acceleration and continuity equations of an FLRW model datum and $H = \dot{a} / a$ . Then:

(a) On the interval $I$ , the pair of equations (F) and (S) holds if and only if the pair of equations (F) and (A) holds.

(b) If (F) and (C) hold on $I$ and $t_{0} \in I$ satisfies $H (t_{0}) \neq 0$ , then equation (A) holds at $t_{0}$ . In particular, on any subinterval on which $H$ has no zero, (F) and (C) imply (A).

The proof of (b) divides by $H (t_{0})$ ; the proposition makes no claim about points with $H (t_{0}) = 0$ , and no claim that (F) and (C) alone determine the second derivative of $a$ there.

Facts & Assumptions

Given: A Friedmann-system datum $(c, G_{N}, Λ, I, k, a, ε, p)$ with all its data smooth on the open interval $I$ , the equations (F), (A), (C), (S) of The Friedmann, acceleration and continuity equations of an FLRW model datum, and the Hubble parameter $H = \dot{a} / a$ .

[L1]

The equations are the displays of The Friedmann, acceleration and continuity equations of an FLRW model datum: (F) is $H^{2} + k c^{2} / a^{2} = \frac{8 π G_{N}}{3 c^{2}} ε + \frac{Λ c^{2}}{3}$ , (S) is $- \frac{2 a \ddot{a} + {\dot{a}}^{2}}{c^{2}} - k + Λ a^{2} = \frac{8 π G_{N}}{c^{4}} p a^{2}$ , (C) is $\dot{ε} + 3 H (ε + p) = 0$ , and (A) is $\frac{\ddot{a}}{a} = - \frac{4 π G_{N}}{3 c^{2}} (ε + 3 p) + \frac{Λ c^{2}}{3}$ , each holding at every point of $I$ .

[L2]

Differentiable functions on an open interval satisfy the sum, product and quotient rules and the chain rule; in particular $\frac{d}{d t} (\frac{1}{a^{2}}) = - \frac{2 \dot{a}}{a^{3}}$ and $\frac{d}{d t} ({\dot{a}}^{2}) = 2 \dot{a} \ddot{a}$ (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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , 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).

[L3]

$H = \dot{a} / a$ with $a > 0$ on $I$ , so $H \neq 0$ exactly where $\dot{a} \neq 0$ ; differentiating $H$ at a point of $I$ gives $\dot{H} = \frac{\ddot{a}}{a} - H^{2}$ , equivalently $\frac{\ddot{a}}{a} = \dot{H} + H^{2}$ (The Friedmann, acceleration and continuity equations of an FLRW model datum, 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 $g (c) \neq 0$ ).

Proof

technique · direct algebra for (a) and differentiation with the product and chain rules for (b)

1.1L1algebra

(Part (a): equivalence of the two systems in the presence of (F).) Assume (F) holds on $I$ , and multiply it by $3$ : $\frac{3 ({\dot{a}}^{2} + k c^{2})}{a^{2}} = \frac{8 π G_{N} ε}{c^{2}} + Λ c^{2}$ ; equivalently $\frac{{\dot{a}}^{2} + k c^{2}}{c^{2}} = \frac{8 π G_{N} a^{2} ε}{3 c^{4}} + \frac{Λ a^{2}}{3}$ . Substituting this into the left-hand side of (S) gives $- \frac{2 a \ddot{a}}{c^{2}} - \frac{{\dot{a}}^{2} + k c^{2}}{c^{2}} + Λ a^{2} = - \frac{2 a \ddot{a}}{c^{2}} - \frac{8 π G_{N} a^{2} ε}{3 c^{4}} - \frac{Λ a^{2}}{3} + Λ a^{2} = - \frac{2 a \ddot{a}}{c^{2}} - \frac{8 π G_{N} a^{2} ε}{3 c^{4}} + \frac{2 Λ a^{2}}{3}$ . Hence, under (F), equation (S) is equivalent to $- \frac{2 a \ddot{a}}{c^{2}} - \frac{8 π G_{N} a^{2} ε}{3 c^{4}} + \frac{2 Λ a^{2}}{3} = \frac{8 π G_{N} a^{2} p}{c^{4}}$ ; multiplying by $\frac{c^{2}}{a^{2}}$ and dividing by $2$ gives $- \frac{\ddot{a}}{a} = \frac{4 π G_{N}}{c^{2}} (\frac{ε}{3} + p) - \frac{Λ c^{2}}{3}$ , that is, $\frac{\ddot{a}}{a} = - \frac{4 π G_{N}}{3 c^{2}} (ε + 3 p) + \frac{Λ c^{2}}{3}$ , which is (A). Each step is an equivalence (multiplication by the nonzero numbers $\frac{c^{2}}{a^{2}}$ and $2$ , and rearrangement), so (F) and (S) hold exactly when (F) and (A) hold.

2.1L1L2L3step 1.1algebra∎

(Part (b): differentiation where $H \neq 0$ .) Assume (F) and (C) on $I$ , fix $t_{0} \in I$ with $H (t_{0}) \neq 0$ . The functions on both sides of (F) are smooth on $I$ , so differentiating the identity and using [L2] on the terms $\frac{k c^{2}}{a^{2}}$ and $H^{2} = \frac{{\dot{a}}^{2}}{a^{2}}$ gives $2 H \dot{H} - \frac{2 k c^{2} \dot{a}}{a^{3}} = \frac{8 π G_{N}}{3 c^{2}} \dot{ε}$ at every point of $I$ ; since $\frac{\dot{a}}{a} = H$ this is $2 H (\dot{H} - \frac{k c^{2}}{a^{2}}) = \frac{8 π G_{N}}{3 c^{2}} \dot{ε}$ . Substituting the continuity equation (C), $\dot{ε} = - 3 H (ε + p)$ , and dividing by the nonzero number $2 H (t_{0})$ yields $\dot{H} - \frac{k c^{2}}{a^{2}} = - \frac{4 π G_{N}}{c^{2}} (ε + p)$ at $t_{0}$ . By [L3], $\frac{\ddot{a}}{a} = \dot{H} + H^{2}$ at $t_{0}$ , so $\frac{\ddot{a}}{a} = \frac{k c^{2}}{a^{2}} + H^{2} - \frac{4 π G_{N}}{c^{2}} (ε + p)$ ; substituting (F) for the sum $\frac{k c^{2}}{a^{2}} + H^{2} = \frac{8 π G_{N} ε}{3 c^{2}} + \frac{Λ c^{2}}{3}$ gives $\frac{\ddot{a}}{a} = \frac{8 π G_{N} ε}{3 c^{2}} - \frac{4 π G_{N}}{c^{2}} (ε + p) + \frac{Λ c^{2}}{3} = - \frac{4 π G_{N}}{3 c^{2}} (ε + 3 p) + \frac{Λ c^{2}}{3}$ , which is (A) at $t_{0}$ . This proves part (b) at every point where $H$ does not vanish; the division by $H (t_{0})$ is the only use of the hypothesis, and no conclusion is drawn at its zeros.

Remarks

Reading of part (a). Part (a) is the exact sense in which the reduced field equations of the cosmological model can be presented either by the spatial-component equation (S) or by the acceleration equation (A): with the Friedmann equation (F) held fixed, the two presentations carry the same content. It is a statement about the four displayed equations and asserts nothing about the existence of solutions with given data.
The role of the acceleration equation. Part (b) is the usual derivation of the acceleration equation from the Friedmann equation and the continuity equation; it proceeds by division by $H$ and therefore holds only where $H \neq 0$ . On regions where $H$ vanishes identically, the derivation gives no information and the acceleration equation is retained as an independent member of the system; this proposition makes no claim about that case.
No equation of state. None of (F), (A), (C), (S) involves an equation of state; a model that adopts $p = P (ε)$ adds a relation among the data and is treated separately on this page.

Equivalence of the Friedmann and acceleration systems, and the derivation of acceleration from Friedmann and continuity where the Hubble parameter does not vanish

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources