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An expanding flat vacuum solution for positive cosmological parameter and non-existence for negative parameter

Physical scope

Exact vacuum solutions of the adopted field law for the flat (k=0) homogeneous model: the exponentially expanding solution for a positive cosmological parameter, and the non-existence of any flat vacuum solution for a negative cosmological parameter. The claims are conditional on the adopted model, field law and parameter sign; no numerical value of Lambda or of the expansion rate, no assertion about singularities, global structure or observations, and no claim that the model describes nature is made.

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 with the framework's constants $c > 0$ , $G_{N} > 0$ and a cosmological parameter $Λ \in R$ , and the vacuum matter sector, so that $T_{a b} = 0$ and $ε = p = 0$ (The constant-curvature spatial model of a homogeneous cosmology clause (d), SI unit and coordinate conventions of the classical gravitational framework). Then:

(a) If $Λ > 0$ , then for every $t_{0} \in R$ and every $A > 0$ the datum with $I : = R$ ,

$H_{Λ} : = c \sqrt{\frac{Λ}{3}}, a (t) : = A \exp (H_{Λ} (t - t_{0})), ε (t) : = 0, p (t) : = 0$

satisfies the Friedmann equation (F) and the acceleration equation (A) on $R$ (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 field equations hold on the spacetime $R \times Σ$ with the metric $g = - c^{2} d t^{2} + a (t)^{2} γ$ .

(b) If $Λ < 0$ , then there is no nonempty open interval $I$ and no smooth function $a : I \to (0, \infty)$ such that the datum $(c, G_{N}, Λ, I, 0, a, 0, 0)$ satisfies the Friedmann equation (F); consequently the flat vacuum model has no solution of the field equations for $Λ < 0$ .

No uniqueness of the solutions of (a) is asserted, no numerical value of $Λ, H_{Λ}, A$ or $t_{0}$ is asserted, and nothing is claimed about the limit $t \to - \infty$ , about singularities or about global structure.

Facts & Assumptions

Given: The adopted model with the flat standard spatial model, the field law with parameters $c, G_{N}, Λ$ , the vacuum matter sector $ε = p = 0$ , and the datum $H_{Λ}, A, t_{0}, a$ of the statement.

[L1]

The flat spatial model is the Euclidean standard model of The constant-curvature spatial model of a homogeneous cosmology, and the model's geometry, matter sector and field law are the adopted assumptions of Adopted homogeneous and isotropic cosmological model with a comoving perfect fluid used together with Reduction of the field equations of the homogeneous cosmological model to the Friedmann and acceleration equations, with derived continuity, which states: on the spacetime the field equation holds if and only if (F) and (A) hold for the model data, and the vacuum sector is the case $ε = p = 0$ with $T = 0$ .

[L2]

For every real $x$ , $\exp (x) > 0$ (The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ ); the real exponential function is $C^{\infty}$ with $\exp^{'} = \exp$ , and for constants $α, β$ the function $t \mapsto β \exp (α t)$ satisfies $\frac{d}{d t} (β \exp (α t)) = α β \exp (α t)$ and $\frac{d^{2}}{d t^{2}} (β \exp (α t)) = α^{2} β \exp (α t)$ (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 $g (c) \neq 0$ ).

[L3]

For $Λ > 0$ the real number $H_{Λ} = c \sqrt{Λ / 3}$ is positive and satisfies $H_{Λ}^{2} = c^{2} Λ / 3$ (Rational powers $a^{r}$ of a positive base); for $Λ < 0$ the real number $Λ c^{2} / 3$ is negative.

[L4]

The equations (F) and (A) of The Friedmann, acceleration and continuity equations of an FLRW model datum with $k = 0$ and $ε = p = 0$ read $H (t)^{2} = Λ c^{2} / 3$ and $\ddot{a} (t) / a (t) = Λ c^{2} / 3$ . Every real number $H (t)$ satisfies $H (t)^{2} \geq 0$ .

Proof

technique · direct verification for part (a) and a sign contradiction for part (b)

1.1L1L2L3L4algebra

(Part (a): the exponential datum satisfies the reduced equations.) The function $a (t) = A \exp (H_{Λ} (t - t_{0}))$ is smooth and strictly positive on $R$ because $A > 0$ and $\exp > 0$ ; its logarithmic derivative is $H (t) = \dot{a} / a = H_{Λ}$ by [L2], and $\ddot{a} / a = H_{Λ}^{2}$ . The Friedmann equation with $k = ε = 0$ is $H^{2} = Λ c^{2} / 3$ , and it holds because $H^{2} = H_{Λ}^{2} = c^{2} Λ / 3$ by [L2] and [L3]; the acceleration equation with $ε = p = 0$ is $\ddot{a} / a = Λ c^{2} / 3$ , and it holds for the same reason. Hence (F) and (A) hold on $R$ , and by [L1] the field equations hold on the spacetime.

1.2L1L3L4

(Part (b): negative $Λ$ is impossible.) Suppose $Λ < 0$ and suppose a nonempty open interval $I$ and a smooth $a : I \to (0, \infty)$ are given such that $(c, G_{N}, Λ, I, 0, a, 0, 0)$ satisfies (F). By [L4] the Friedmann equation with $k = 0$ and $ε = 0$ reads $H (t)^{2} = Λ c^{2} / 3$ for every $t \in I$ . By [L3] the right-hand side is a negative real number, while the left-hand side is a square of a real number and hence is nonnegative by [L4]; this is a contradiction. Therefore no such datum satisfies (F); since by [L1] any solution of the field equations in the vacuum flat sector would produce such a datum satisfying (F), no solution exists for $Λ < 0$ .

2.1step 1.1step 1.2∎

(Conclusion.) Step 1.1 proves claim (a) for every $t_{0}$ and $A$ and every $Λ > 0$ , and step 1.2 proves the non-existence claim (b) for $Λ < 0$ . The argument used only the reduction theorem, the pointwise signs of the reduced equations and the elementary properties of the exponential function; no uniqueness, no asymptotic statement and no observational input was used.

Remarks

The flat $Λ = 0$ case. For $Λ = 0$ the constant scale factor $a \equiv A$ satisfies (F) and (A) with $ε = p = 0$ trivially ( $H = \ddot{a} = 0$ ), so it is a further vacuum solution of the reduced system; the lemma makes no claim that it is isometric to Minkowski space, and no statement about which of the vacuum solutions a physical cosmology selects.
No empirical content. The parameter $Λ$ is model data; the theorem is a conditional statement about the adopted field law. In particular the observed accelerated expansion of the universe is not used, asserted or explained here, and associating $Λ$ with a measured dark-energy density requires a separate, source-backed empirical premise and analysis.

An expanding flat vacuum solution for positive cosmological parameter and non-existence for negative parameter

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources