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Homogeneous Cosmology — Examples

1 · Prerequisites

2 · Summary

This page collects worked checks of the items of gr-homogeneous-cosmology, all carried out directly in the defining equations so that they can be read without the physical reductions. The flat power-law data are differentiated and substituted in all four scalar equations of The Friedmann, acceleration and continuity equations of an FLRW model datum, verifying the Friedmann, acceleration, continuity and spatial-component equations simultaneously (Direct check of the flat power-law data against the Friedmann-system equations). The exponential datum with positive cosmological parameter is verified against the same four equations, and the square-sign obstruction shows that no flat vacuum datum satisfies the Friedmann equation when the parameter is negative (Direct check of the flat vacuum data with a cosmological parameter against the Friedmann-system equations). A counterexample separates the equations: the static flat matter datum satisfies the spatial-component and continuity equations while failing the Friedmann and acceleration equations, so the Friedmann equation is not redundant in the reduced system (A static flat matter datum satisfying the spatial-component and continuity equations but not the Friedmann equation). Finally, an explicit radial null geodesic of an exponentially expanding flat model verifies the conservation of $a V^{t}$ and the observer-energy ratio $E_{R} / E_{E} = a (t_{E}) / a (t_{R})$ in coordinates (An explicit null geodesic in an exponentially expanding flat FLRW datum: constant aV^t and the observer-energy ratio). Every item on this page is a mathematical computation on a specified FLRW model datum; no observation and no field equation is asserted.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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Direct check of the flat power-law data against the Friedmann-system equations

Example

Fix real constants $c > 0$ , $G_{N} > 0$ , $A > 0$ , $t_{*} > 0$ , $t_{0} \in R$ and $w > - 1$ , and put

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

with real powers in the sense of Real powers for positive bases, with the zero-base positive-exponent convention. Then:

The functions $a, ε, p$ are smooth on $I$ , $a (t) > 0$ and $ε (t) > 0$ for every $t \in I$ , and on $I$

$H (t) = \frac{q}{t - t_{0}}, \frac{\ddot{a} (t)}{a (t)} = \frac{q (q - 1)}{(t - t_{0})^{2}} .$

The Friedmann-system datum $(c, G_{N}, 0, I, 0, a, ε, p)$ satisfies the Friedmann equation (F), the acceleration equation (A), the continuity equation (C) and the spatial-component equation (S) of The Friedmann, acceleration and continuity equations of an FLRW model datum on $I$ . In particular $w = 0$ gives $q = \frac{2}{3}$ and $w = \frac{1}{3}$ gives $q = \frac{1}{2}$ .

Facts & Assumptions

Given: The constants $c, G_{N}, A, t_{*}, t_{0}, w$ with $c > 0$ , $G_{N} > 0$ , $A > 0$ , $t_{*} > 0$ , $w > - 1$ , the interval $I = (t_{0}, \infty)$ , the functions $a, ε, p$ of the example, and the equations (F), (A), (C), (S) of The Friedmann, acceleration and continuity equations of an FLRW model datum.

[L1]

For $x > 0$ and real $α$ the real power $x^{α} = \exp (α \log x)$ is strictly positive, continuous and differentiable on $(0, \infty)$ with $(x^{α})^{'} = α x^{α - 1}$ , and $(x^{α})^{β} = x^{α β}$ for real $β$ (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). Smoothness on an open real interval means that derivatives of every finite order exist and are continuous; it will be established locally in step 1.1.

[L2]

Smooth functions on intervals satisfy the sum, product, quotient and chain rules; in particular, if $u$ is smooth and strictly positive and $α$ is real, then $\frac{d}{d t} u^{α} = α u^{α - 1} \dot{u}$ , and the derivative of a constant function is $0$ (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, For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term).

[L3]

The equations of The Friedmann, acceleration and continuity equations of an FLRW model datum with $k = Λ = 0$ read

$H^{2} = \frac{8 π G_{N}}{3 c^{2}} ε, \frac{\ddot{a}}{a} = - \frac{4 π G_{N}}{3 c^{2}} (ε + 3 p), \dot{ε} + 3 H (ε + p) = 0, - \frac{2 a \ddot{a} + {\dot{a}}^{2}}{c^{2}} = \frac{8 π G_{N}}{c^{4}} p a^{2},$

where the first display is (F), the second (A), the third (C) and the fourth (S), each required at every point of $I$ .

[L4]

The defining relations $q = 2 / (3 (1 + w))$ and $w > - 1$ give $q > 0$ and $3 (1 + w) q = 2$ ; equivalently $1 + w = 2 / (3 q)$ and $3 q (1 + w) = 2$ .

Verification

1.1L1L2L4givenalgebra

(Smoothness and positivity.) Put $u (t) : = (t - t_{0}) / t_{*} > 0$ on $I$ . For every real $α$ and integer $m \geq 0$ , repeated application of [L1] and [L2] gives $\frac{d^{m}}{d t^{m}} u (t)^{α} = t_{*}^{- m} (\prod_{r = 0}^{m - 1} (α - r)) u (t)^{α - m}$ , with empty product $1$ : at $m = 0$ this is the function itself, and differentiating the formula at $m$ introduces the factor $(α - m) / t_{*}$ and lowers the exponent by one, proving the formula at $m + 1$ . Each derivative is continuous by [L1] and continuity of the affine function $u$ , so $u^{α}$ is smooth. Taking $α = q$ proves smoothness of $a = A u^{q}$ ; the same argument with the positive affine function $v (t) = t - t_{0}$ , whose derivative is $1$ , and $α = - 2$ proves smoothness of $ε = 3 c^{2} q^{2} v^{- 2} / (8 π G_{N})$ and hence of $p = w ε$ . Finally $a > 0$ by [L1] and $A > 0$ , and $ε > 0$ because $c, G_{N} > 0$ , $q > 0$ by [L4], and $v > 0$ .

2.1L1L2step 1.1algebra

(Derivatives and the Hubble parameter.) By [L2], $\dot{u} = 1 / t_{*}$ and $\dot{a} = A q u^{q - 1} \dot{u} = A q u^{q - 1} / t_{*}$ . Since $u^{q} / (t - t_{0}) = u^{q - 1} / t_{*}$ , this is $\dot{a} = q a / (t - t_{0})$ , so $H = \dot{a} / a = q / (t - t_{0})$ . Differentiating $\dot{a} = q a / (t - t_{0})$ by the product and quotient rules gives $\ddot{a} = q (\dot{a} (t - t_{0}) - a) / (t - t_{0})^{2} = q (q - 1) a / (t - t_{0})^{2}$ , hence $\ddot{a} / a = q (q - 1) / (t - t_{0})^{2}$ . Also $\dot{ε} = - 6 c^{2} q^{2} / (8 π G_{N} (t - t_{0})^{3}) = - 2 ε / (t - t_{0})$ . This proves claim 1.

3.1L3step 2.1algebra

(Friedmann equation.) By [L3], (F) with $k = 0$ is $H^{2} = 8 π G_{N} ε / (3 c^{2})$ . By step 2.1, $H^{2} = q^{2} / (t - t_{0})^{2}$ ; by the definition of $ε$ , $\frac{8 π G_{N}}{3 c^{2}} ε = \frac{8 π G_{N}}{3 c^{2}} \cdot \frac{3 c^{2} q^{2}}{8 π G_{N} (t - t_{0})^{2}} = \frac{q^{2}}{(t - t_{0})^{2}}$ . The two sides agree at every point of $I$ .

3.2L3L4step 2.1algebra

(Acceleration equation.) By [L3], (A) with $Λ = 0$ is $\ddot{a} / a = - 4 π G_{N} (ε + 3 p) / (3 c^{2})$ . By step 2.1 and $p = w ε$ , the left-hand side is $q (q - 1) / (t - t_{0})^{2}$ and the right-hand side is $- \frac{4 π G_{N}}{3 c^{2}} (1 + 3 w) \frac{3 c^{2} q^{2}}{8 π G_{N} (t - t_{0})^{2}} = - \frac{(1 + 3 w) q^{2}}{2 (t - t_{0})^{2}}$ . Since $q > 0$ by [L4], the needed identity $q (q - 1) = - \frac{1}{2} (1 + 3 w) q^{2}$ is equivalent to $2 q - 2 = - (1 + 3 w) q$ , that is, to $3 (1 + w) q = 2$ , which is [L4]. Hence (A) holds.

3.3L3L4step 2.1algebra

(Continuity equation.) By step 2.1, $\dot{ε} = - 2 ε / (t - t_{0})$ ; by steps 2.1 and [L4], $3 H (ε + p) = 3 \frac{q}{t - t_{0}} (1 + w) ε$ and $3 q (1 + w) = 2$ , so $3 H (ε + p) = 2 ε / (t - t_{0})$ . Hence $\dot{ε} + 3 H (ε + p) = 0$ on $I$ , which is (C).

3.4L3L4step 2.1algebra

(Spatial-component equation.) By [L3], (S) with $k = Λ = 0$ is $- (2 a \ddot{a} + {\dot{a}}^{2}) / c^{2} = 8 π G_{N} p a^{2} / c^{4}$ . By step 2.1, $2 a \ddot{a} + {\dot{a}}^{2} = (2 q (q - 1) + q^{2}) a^{2} / (t - t_{0})^{2} = (3 q^{2} - 2 q) a^{2} / (t - t_{0})^{2}$ ; the right-hand side is $\frac{8 π G_{N}}{c^{4}} w \frac{3 c^{2} q^{2}}{8 π G_{N} (t - t_{0})^{2}} a^{2} = \frac{3 w q^{2} a^{2}}{c^{2} (t - t_{0})^{2}}$ . Equality of the two displayed expressions is equivalent to $- (3 q^{2} - 2 q) = 3 w q^{2}$ , that is, to $2 q = 3 q^{2} (1 + w)$ , which holds because $3 q (1 + w) = 2$ by [L4]; hence (S) holds.

4.1step 1.1step 2.1step 3.1step 3.2step 3.3step 3.4∎

(Conclusion.) Steps 1.1 and 2.1 establish claim 1, and steps 3.1, 3.2, 3.3 and 3.4 establish (F), (A), (C) and (S) at every point of $I$ , which is claim 2; the last sentence of claim 2 is the substitution $w = 0$ , $w = \frac{1}{3}$ in $q = 2 / (3 (1 + w))$ . No equation of state beyond $p = w ε$ , no field equation and no physical premise was used.

Remarks

What is checked. The example verifies the displayed data against all four scalar equations of The Friedmann, acceleration and continuity equations of an FLRW model datum by direct differentiation; in particular it does not invoke the equivalence proposition of the A page, so it independently exhibits data satisfying (F), (A), (C) and (S) simultaneously. The physically interpreted field-equation reading of these data belongs to the physics theorem on the A page and is not asserted here.
What is not claimed. No uniqueness, no statement about the limit $t ↓ t_{0}$ , no extension beyond $I$ and no numerical value of any constant is asserted; $A$ , $t_{*}$ and $t_{0}$ are arbitrary real model data with $A > 0$ , $t_{*} > 0$ .

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Direct check of the flat vacuum data with a cosmological parameter against the Friedmann-system equations

Example

Fix real constants $c > 0$ , $G_{N} > 0$ , $A > 0$ , $t_{0} \in R$ and $Λ \in R$ . Then:

If $Λ > 0$ , put $H_{Λ} : = c \sqrt{Λ / 3}$ and define $a : R \to (0, \infty)$ by $a (t) : = A \exp (H_{Λ} (t - t_{0}))$ , together with $ε : = 0$ and $p : = 0$ on $I : = R$ . Then the Friedmann-system datum $(c, G_{N}, Λ, I, 0, a, ε, p)$ satisfies (F), (A), (C) and (S) of The Friedmann, acceleration and continuity equations of an FLRW model datum on all of $I$ .
If $Λ < 0$ , then for every nonempty open interval $I \subseteq R$ and every smooth $a : I \to (0, \infty)$ the datum $(c, G_{N}, Λ, I, 0, a, 0, 0)$ fails the Friedmann equation (F) at every point of $I$ ; in particular no such flat vacuum datum satisfies the Friedmann-system equations.

Facts & Assumptions

Given: The constants $c, G_{N}, A, t_{0}, Λ$ with $c > 0$ , $G_{N} > 0$ , $A > 0$ , the functions $a, ε, p$ of the example, and the equations (F), (A), (C), (S) of The Friedmann, acceleration and continuity equations of an FLRW model datum.

[L1]

The real exponential is strictly positive on $R$ (The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ ) and smooth with $\exp^{'} = \exp$ ; consequently for real constants $α, β$ the function $t \mapsto β \exp (α (t - t_{0}))$ is smooth with derivative $α β \exp (α (t - t_{0}))$ and second derivative $α^{2} β \exp (α (t - t_{0}))$ (The real exponential function and the number $e$ by a power series, The exponential function is smooth and $(\exp)^{'} = \exp$ , 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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , 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 $g (c) \neq 0$ ).

[L2]

For $Λ > 0$ the real power $\sqrt{Λ / 3} = (Λ / 3)^{1 / 2}$ is a strictly positive real number with $(Λ / 3)^{1 / 2} \cdot (Λ / 3)^{1 / 2} = (Λ / 3)^{1} = Λ / 3$ , and real powers satisfy $(x^{r})^{s} = x^{r s}$ and $(x y)^{r} = x^{r} y^{r}$ for $x, y > 0$ (Real powers for positive bases, with the zero-base positive-exponent convention, The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents). Hence $H_{Λ} = c (Λ / 3)^{1 / 2}$ satisfies $H_{Λ} > 0$ and $H_{Λ}^{2} = c^{2} Λ / 3$ . For $Λ < 0$ the number $Λ c^{2} / 3$ is negative.

[L3]

The equations of The Friedmann, acceleration and continuity equations of an FLRW model datum with $k = 0$ and $ε = p = 0$ read

$H^{2} = \frac{Λ c^{2}}{3}, \frac{\ddot{a}}{a} = \frac{Λ c^{2}}{3}, \dot{ε} + 3 H (ε + p) = 0, - \frac{2 a \ddot{a} + {\dot{a}}^{2}}{c^{2}} + Λ a^{2} = 0,$

where the four displays are (F), (A), (C) and (S) at every point of $I$ ; the third holds identically for $ε = p = 0$ .

[L4]

For every real number $x$ one has $x^{2} \geq 0$ : if $x = 0$ this is clear, and if $x \neq 0$ then $x^{2} > 0$ (Squares of nonzero elements are positive). Since $c > 0$ gives $c^{2} > 0$ , for $Λ < 0$ the product $Λ c^{2} / 3$ is negative (Order on the reals).

Verification

1.1L1givenalgebra

(The exponential datum is admissible.) The function $a (t) = A \exp (H_{Λ} (t - t_{0}))$ is smooth on $R$ and strictly positive because $A > 0$ and $\exp > 0$ ; by [L1] its logarithmic derivative is $H (t) = \dot{a} / a = H_{Λ}$ and $\ddot{a} / a = H_{Λ}^{2}$ . The zero functions $ε, p$ are smooth, and $(c, G_{N}, Λ, I, 0, a, ε, p)$ with $I = R$ is a Friedmann-system datum.

1.2L2L3L4algebra

( $Λ < 0$ obstruction.) Suppose $Λ < 0$ , let $I \subseteq R$ be a nonempty open interval and let $a : I \to (0, \infty)$ be smooth, and suppose the datum $(c, G_{N}, Λ, I, 0, a, 0, 0)$ satisfies (F). By [L3] the Friedmann equation reads $H (t)^{2} = Λ c^{2} / 3$ at every $t \in I$ ; by [L4] the right-hand side is negative while $H (t)^{2} \geq 0$ for the real number $H (t)$ , a contradiction. Hence no such datum satisfies (F), proving claim 2.

2.1L2L3step 1.1algebra

(Friedmann equation for $Λ > 0$ .) By [L3] and step 1.1, (F) reads $H^{2} = Λ c^{2} / 3$ ; the left-hand side is $H_{Λ}^{2}$ and $H_{Λ}^{2} = c^{2} Λ / 3 = Λ c^{2} / 3$ by [L2]. Hence (F) holds on $I$ .

2.2L2L3step 1.1algebra

(Acceleration equation for $Λ > 0$ .) By [L3] and step 1.1, (A) reads $\ddot{a} / a = Λ c^{2} / 3$ , and $\ddot{a} / a = H_{Λ}^{2} = Λ c^{2} / 3$ by [L2]. Hence (A) holds on $I$ .

2.3L2L3step 1.1algebra

(Continuity and spatial-component equations for $Λ > 0$ .) For $ε = p = 0$ every term of (C) vanishes, so (C) holds identically. For (S), step 1.1 gives $2 a \ddot{a} + {\dot{a}}^{2} = 2 a \cdot H_{Λ}^{2} a + (H_{Λ} a)^{2} = 3 H_{Λ}^{2} a^{2}$ , hence $- (2 a \ddot{a} + {\dot{a}}^{2}) / c^{2} + Λ a^{2} = - 3 H_{Λ}^{2} a^{2} / c^{2} + Λ a^{2} = - 3 (c^{2} Λ / 3) a^{2} / c^{2} + Λ a^{2} = 0$ , which is (S) with zero pressure. This completes claim 1.

3.1step 2.1step 2.2step 2.3step 1.2∎

(Conclusion.) Steps 2.1, 2.2 and 2.3 verify (F), (A), (C) and (S) for the exponential datum of claim 1 on all of $R$ , and step 1.2 proves the failure of (F) for every flat vacuum datum with $Λ < 0$ , which is claim 2. The verification used only differentiation of the exponential and the sign of a square; no field equation, no reference to the A page and no physical premise was used.

Remarks

What is checked. The example verifies the two sign regimes directly in the defining equations of The Friedmann, acceleration and continuity equations of an FLRW model datum: for $Λ > 0$ the exponential datum satisfies all four scalar equations, and for $Λ < 0$ the Friedmann equation alone is already unsatisfiable because a square cannot equal a negative number. The physical reading of the datum as a vacuum solution of the field law belongs to the A page and is not asserted here.
What is not claimed. No uniqueness of the solutions for $Λ > 0$ , no statement about the flat $Λ = 0$ case, no numerical value of $Λ$ or $H_{Λ}$ , and no physical claim about an expanding universe is made.

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A static flat matter datum satisfying the spatial-component and continuity equations but not the Friedmann equation

Statement refuted

Statement refuted: "Every Friedmann-system datum $(c, G_{N}, Λ, I, k, a, ε, p)$ of The Friedmann, acceleration and continuity equations of an FLRW model datum with $k = 0$ , $Λ = 0$ , constant scale factor $a \equiv A$ for some $A > 0$ , constant density $ε \equiv ε_{0}$ for some $ε_{0} > 0$ and pressure $p \equiv 0$ satisfies the Friedmann equation (F)."

Facts & Assumptions

Given: Real constants $c > 0$ , $G_{N} > 0$ , $A > 0$ , $ε_{0} > 0$ ; the interval $I : = R$ ; the constant functions $a \equiv A$ , $ε \equiv ε_{0}$ , $p \equiv 0$ ; and the equations (F), (A), (C), (S) of The Friedmann, acceleration and continuity equations of an FLRW model datum with $k = Λ = 0$ .

[L1]

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

[L2]

The derivative of a constant function is $0$ , and the sum, product and quotient rules for differentiable functions on an open interval hold (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 $g (c) \neq 0$ ).

[L3]

The real numbers form an ordered field; consequently the product and quotient of strictly positive real numbers is strictly positive, so $8 π G_{N} ε_{0} / (3 c^{2}) > 0$ and $4 π G_{N} ε_{0} / (3 c^{2}) > 0$ (Order on the reals).

Counterexample

1.1L1L2algebra

(The datum and its derivatives.) Each of $a \equiv A$ , $ε \equiv ε_{0}$ , $p \equiv 0$ is a constant function on $I = R$ , hence smooth, with $a$ taking values in $(0, \infty)$ ; thus $(c, G_{N}, 0, I, 0, a, ε, p)$ is a Friedmann-system datum of the kind named in the refuted statement. By [L2], $\dot{a} \equiv 0$ , $\ddot{a} \equiv 0$ , $\dot{ε} \equiv 0$ , so $H = \dot{a} / a \equiv 0$ .

2.1L1L3step 1.1algebra

(The Friedmann equation fails.) By [L1] and step 1.1, the left-hand side of (F) is $H^{2} + k c^{2} / a^{2} = 0$ , while the right-hand side is $\frac{8 π G_{N}}{3 c^{2}} ε_{0} + \frac{Λ c^{2}}{3} = \frac{8 π G_{N} ε_{0}}{3 c^{2}}$ , which is strictly positive by [L3]; hence $0 = \frac{8 π G_{N} ε_{0}}{3 c^{2}}$ is false and (F) fails at every point of $I$ . This refutes the displayed statement.

2.2L1L3step 1.1algebra

(The acceleration equation fails and the other equations hold.) By [L1] and step 1.1, the left-hand side of (A) is $\ddot{a} / a = 0$ while the right-hand side is $- \frac{4 π G_{N}}{3 c^{2}} ε_{0} < 0$ by [L3], so (A) also fails. In contrast, (C) reads $\dot{ε} + 3 H (ε + p) = 0 + 0 = 0$ and (S) reads $- 0 / c^{2} = 0 = 8 π G_{N} \cdot 0 \cdot a^{2} / c^{4}$ , so both hold identically. Thus the spatial-component equation (S) and the continuity equation (C) do not by themselves imply the Friedmann equation, and (F) is not redundant in the pair (F), (S).

3.1step 1.1step 2.1step 2.2∎

(Conclusion.) The displayed universal statement is false: the flat static matter datum of step 1.1 has constant positive density and vanishing pressure and scale-factor velocity, and it satisfies (C) and (S) while failing (F) and (A) at every point.

Remarks

Consistency with the equivalence proposition. The example does not contradict Equivalence of the Friedmann and acceleration systems, and the derivation of acceleration from Friedmann and continuity where the Hubble parameter does not vanish clause (a), which assumes (F) and compares the pairs (F), (S) and (F), (A); here (F) fails, so that proposition does not apply. It shows that the hypothesis (F) in that equivalence cannot be dropped.
Scope. The datum is a formal Friedmann-system example, not a claim about a physical configuration. The conclusions concern only the four scalar equations defined in The Friedmann, acceleration and continuity equations of an FLRW model datum; no field law, observation, or physical model is asserted here.

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An explicit null geodesic in an exponentially expanding flat FLRW datum: constant aV^t and the observer-energy ratio

Example

Fix real constants $c > 0$ , $A > 0$ , $H > 0$ , $t_{0} \in R$ , $r_{0} \in R$ . Let $(I, Σ, γ, k, a)$ be the FLRW model datum with $I = R$ , $Σ = R^{3}$ carrying the standard Euclidean metric $γ$ in the standard coordinates $x = (x^{1}, x^{2}, x^{3})$ , $k = 0$ and $a (t) : = A \exp (H (t - t_{0}))$ (FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). On the spacetime $M = I \times Σ$ define the curve $η : (0, \infty) \to M$ ,

$η (s) : = (t (s), (r_{0} + \frac{c}{A H^{2} s}, 0, 0)), t (s) : = t_{0} + \frac{1}{H} \log (H s),$

and let $V : = η^{'}$ with components $V^{t} = d t / d s$ , $V^{1}, V^{2}, V^{3}$ . Then:

The curve $η$ is smooth and its tangent $V$ is a future-directed null vector, $g (V, V) = 0$ and $V^{t} > 0$ , at every point of $(0, \infty)$ ; moreover $η$ is a geodesic of the metric-selected connection with affine parameter $s$ , in the sense of Geodesic of an affine connection.
Along $η$ the product $a (t (s)) V^{t} (s)$ is the constant $A$ ; in particular $a V^{t}$ is preserved by the flow of the null geodesic.
For $0 < s_{E} < s_{R}$ let $t_{E} = t (s_{E})$ , $t_{R} = t (s_{R})$ , let $n = c^{- 1} \partial_{t}$ be the comoving congruence of the datum and let $E (s) : = - c g_{η (s)} (V (s), n (η (s)))$ be the observer energy of Observer-relative energy, spatial momentum and rest mass of a tangent vector clause (a). Then $E (s) = c^{2} / (H s) > 0$ and

$\frac{E (s_{R})}{E (s_{E})} = \frac{a (t_{E})}{a (t_{R})} = \frac{s_{E}}{s_{R}} .$

Facts & Assumptions

Given: The constants $c, A, H, t_{0}, r_{0}$ with $c > 0$ , $A > 0$ , $H > 0$ , the FLRW model datum of the example, its spacetime $M = R \times R^{3}$ , metric $g$ , comoving congruence $n = c^{- 1} \partial_{t}$ and metric-selected connection $\nabla$ , and the curve $η$ with tangent $V$ .

[L1]

In the product chart $(x^{0}, x^{1}, x^{2}, x^{3}) = (t, x^{1}, x^{2}, x^{3})$ the Christoffel symbols of $\nabla$ are $Γ^{0}_{i j} = (a \dot{a} / c^{2}) γ_{i j}$ , $Γ^{i}_{0 j} = Γ^{i}_{j 0} = H_{F L} δ^{i}_{j}$ with $H_{F L} = \dot{a} / a$ , $Γ^{i}_{j k} =^{(3)} Γ^{i}_{j k}$ , and all others vanish; here $γ$ is the Euclidean metric in the standard coordinates, so $^{(3)} Γ^{i}_{j k} = 0$ (Christoffel symbols of the FLRW metric in product charts, Metric compatibility, torsion freedom and the metric-selected connection class).

[L2]

In a chart a smooth curve is a geodesic with affine parameter $s$ if and only if ${\dot{V}}^{a} + \sum_{b, c} Γ^{a}_{b c} V^{b} V^{c} = 0$ for $a = 0, 1, 2, 3$ on every parameter subinterval lying in the chart (Coordinate geodesic equation, Geodesic of an affine connection).

[L3]

The comoving congruence $n = c^{- 1} \partial_{t}$ is a smooth future-directed unit timelike field, $g (n, n) = - 1$ ; a nonzero vector $V$ is future-directed exactly when $g (V, \partial_{t}) < 0$ , and null exactly when $g (V, V) = 0$ (The FLRW metric is smooth Lorentzian; the comoving field is unit timelike and cosmic time parametrizes comoving curves by proper time; the slices carry the scaled spatial metric, Timelike, null, spacelike and causal vectors of a Lorentz vector space, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L4]

The observer energy is $E_{n} (P) = - c g_{p} (P, n_{p})$ , and for the unit field $n = c^{- 1} \partial_{t}$ the pairing with a vector $V$ is $g (V, n) = - c V^{t}$ , where $V^{t} = d t (V)$ in the product chart (Observer-relative energy, spatial momentum and rest mass of a tangent vector, The FLRW metric is smooth Lorentzian; the comoving field is unit timelike and cosmic time parametrizes comoving curves by proper time; the slices carry the scaled spatial metric).

[L5]

On $(0, \infty)$ the functions $s \mapsto 1 / (H s)$ and $s \mapsto r_{0} + c / (A H^{2} s)$ are smooth with derivatives $- 1 / (H s^{2})$ and $- c / (A H^{2} s^{2})$ . The exponential is smooth with $\exp^{'} = \exp$ and the natural logarithm is its inverse on $(0, \infty)$ with $\log^{'} (x) = 1 / x$ (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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The real exponential function and the number $e$ by a power series, The exponential function is smooth and $(\exp)^{'} = \exp$ , The natural logarithm as the inverse of the exponential function, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t).

Verification

1.1L5algebra

(Tangents.) By [L5] the curve $η$ is smooth on $(0, \infty)$ and its tangent has components $V^{t} (s) = \frac{1}{H s} > 0, V^{1} (s) = - \frac{c}{A H^{2} s^{2}}, V^{2} (s) = V^{3} (s) = 0.$

2.1L3L5step 1.1algebra

(Nullness and future direction.) By [L5], $a (t (s)) = A \exp (H (t (s) - t_{0})) = A \cdot H s$ , so $a (t (s)) V^{1} (s) = A H s \cdot (- c / (A H^{2} s^{2})) = - c / (H s) = - c V^{t} (s)$ . Hence $a^{2} (V^{1})^{2} = c^{2} (V^{t})^{2}$ , and since $V^{2} = V^{3} = 0$ the metric pairing is $g (V, V) = - c^{2} (V^{t})^{2} + a^{2} (V^{1})^{2} = 0$ : $V$ is null. Since $V^{t} (s) > 0$ , [L3] gives $g (V, \partial_{t}) = - c^{2} V^{t} < 0$ , so $V$ is future-directed.

2.2L1L2L5step 1.1algebra

(Geodesic equations.) By [L1] the nonvanishing symbol families in the product chart are $Γ^{0}_{i j} = a \dot{a} δ_{i j} / c^{2}$ and $Γ^{i}_{0 j} = Γ^{i}_{j 0} = H_{F L} δ^{i}_{j}$ , where $\dot{a} = H a$ gives $H_{F L} = H$ . Since $V^{2} = V^{3} = 0$ , only $Γ^{0}_{11} = H a^{2} / c^{2}$ contributes to the time-component equation, while the two symmetric mixed symbols give the factor $2$ in the spatial equations. The $t$ -equation of [L2] reads ${\dot{V}}^{t} + \frac{H a^{2}}{c^{2}} (V^{1})^{2} = 0$ : by step 1.1 and [L5], ${\dot{V}}^{t} = - 1 / (H s^{2})$ and $\frac{H a^{2}}{c^{2}} (V^{1})^{2} = \frac{H (A H s)^{2}}{c^{2}} \cdot \frac{c^{2}}{A^{2} H^{4} s^{4}} = \frac{1}{H s^{2}}$ , so the equation holds. The $x^{1}$ -equation reads ${\dot{V}}^{1} + 2 H V^{t} V^{1} = 0$ : by step 1.1, ${\dot{V}}^{1} = 2 c / (A H^{2} s^{3})$ and $2 H V^{t} V^{1} = 2 H \frac{1}{H s} (- \frac{c}{A H^{2} s^{2}}) = - \frac{2 c}{A H^{2} s^{3}}$ , so the equation holds. For $a = 2$ the equation of [L2] reads ${\dot{V}}^{2} + 2 H V^{t} V^{2} = 0$ , and it holds because $V^{2} = 0$ and ${\dot{V}}^{2} = 0$ ; the $a = 3$ equation is identical with $V^{3} = 0$ . Hence $η$ is a geodesic with affine parameter $s$ .

3.1L5step 1.1step 2.1step 2.2algebra

(The conserved product.) By step 1.1 and the identity $a (t (s)) = A H s$ of [L5], $a (t (s)) V^{t} (s) = A H s \cdot \frac{1}{H s} = A$ for every $s \in (0, \infty)$ : $a V^{t}$ is constant along the ray with value $A$ . This proves claims 1 and 2.

4.1L4step 1.1step 3.1algebra

(Observer energies and their ratio.) By [L4], $E (s) = - c g (V (s), n) = - c \cdot (- c V^{t} (s)) = c^{2} V^{t} (s) = \frac{c^{2}}{H s} > 0$ . Hence $\frac{E (s_{R})}{E (s_{E})} = \frac{s_{E}}{s_{R}}$ , and since $a (t (s)) = A H s$ we also have $\frac{a (t_{E})}{a (t_{R})} = \frac{s_{E}}{s_{R}}$ ; combining the two equalities proves the displayed energy ratio.

5.1step 1.1step 2.1step 2.2step 3.1step 4.1∎

(Conclusion.) Steps 1.1, 2.1 and 2.2 exhibit a smooth future-directed null geodesic with affine parameter $s$ , step 3.1 verifies the conservation law $a V^{t} = constant$ in this explicit family, and step 4.1 verifies the observer-energy ratio $\frac{E (s_{R})}{E (s_{E})} = \frac{a (t_{E})}{a (t_{R})}$ between comoving observers. The verification used only the metric, its Christoffel symbols and the observer-energy definition; no field equation and no matter model was used, and no physical claim about a measured wavelength is made.

Remarks

Independence from the field equations. The family $η$ exists for every exponential scale factor, whatever the matter content or the value of the cosmological parameter, because the computation uses only the metric geometry. It is a special case of the general conservation law of the A page, verified here by explicit coordinates.
Physical reading. The relation $\frac{E (s_{R})}{E (s_{E})} = \frac{a (t_{E})}{a (t_{R})}$ is a model prediction about ideal observer energies of an ideal null geodesic; identifying $E$ with an observed frequency requires the light propagation and measurement model of the framework, and no such identification or measurement is asserted here.
Choice of zero. The parameter $s$ ranges over $(0, \infty)$ and $t (s) \to \pm \infty$ as $s \to 0^{+}, \infty$ ; the ray is defined on the whole cosmic time axis. Only the ratio of energies is asserted, so the additive freedom in the affine parameter and the constant $r_{0}$ do not enter.

Homogeneous Cosmology — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Direct check of the flat power-law data against the Friedmann-system equations

Example

Facts & Assumptions

Verification

Remarks

Direct check of the flat vacuum data with a cosmological parameter against the Friedmann-system equations

Example

Facts & Assumptions

Verification

Remarks

A static flat matter datum satisfying the spatial-component and continuity equations but not the Friedmann equation

Statement refuted

Facts & Assumptions

Counterexample

Remarks

An explicit null geodesic in an exponentially expanding flat FLRW datum: constant aV^t and the observer-energy ratio

Example

Facts & Assumptions

Verification

Remarks

Sources