Physlib.Cosmology.FLRW.Solutions

Let $\gamma: \mathbb{R} \to \mathbb{R}$ be a function and $t \in \text{Time}$ be a point in time, where $t_{\text{val}} \in \mathbb{R}$ represents its underlying real value. If $\gamma$ has a derivative $v$ at the point $t_{\text{val}}$ (i.e., $\gamma'(t_{\text{val}}) = v$), then the time derivative $\partial_t$ of the function $s \mapsto \gamma(s_{\text{val}})$ evaluated at $t$ is equal to $v$.

For real parameters $a_0, \sigma, \Lambda,$ and $c$, the de Sitter scale factor is the function $a: \text{Time} \to \mathbb{R}$ defined by $$a(t) = a_0 \exp\left(\sigma \sqrt{\frac{\Lambda}{3}} c t\right)$$ where $a_0$ is the initial scale factor, $\Lambda$ is the cosmological constant, $c$ is the speed of light, and $\sigma$ is a sign factor (typically $\pm 1$, where $\sigma = 1$ corresponds to the expanding branch).

For any real numbers $a_0, K,$ and $x$, the function $y \mapsto a_0 e^{Ky}$ has a derivative at the point $x$ equal to $a_0 K e^{Kx}$.

For real parameters $a_0, \sigma, \Lambda,$ and $c$, let $a(t)$ be the de Sitter scale factor defined by $a(t) = a_0 \exp\left(\sigma \sqrt{\frac{\Lambda}{3}} c t\right)$. The time derivative of this scale factor, denoted $\partial_t a$, is given by $$\frac{d a(t)}{dt} = a_0 \left( \sigma \sqrt{\frac{\Lambda}{3}} c \right) \exp\left( \sigma \sqrt{\frac{\Lambda}{3}} c t\right)$$ where $a_0$ is the initial scale factor, $\Lambda$ is the cosmological constant, $c$ is the speed of light, and $\sigma$ is a sign factor.

For any real parameters $a_0, \sigma, \Lambda,$ and $c$, the second time derivative of the de Sitter scale factor $a(t) = a_0 \exp\left(\sigma \sqrt{\frac{\Lambda}{3}} c t\right)$ is given by $$\frac{d^2 a}{dt^2}(t) = a_0 \left( \sigma \sqrt{\frac{\Lambda}{3}} c \right)^2 \exp\left(\sigma \sqrt{\frac{\Lambda}{3}} c t\right)$$ where $a_0$ is the initial scale factor, $\Lambda$ is the cosmological constant, $c$ is the speed of light, and $\sigma$ is a sign factor.

For any real numbers $\sigma, \Lambda,$ and $c$ such that $\Lambda \geq 0$ and $\sigma = 1$ or $\sigma = -1$, the following equality holds: $$\left( \sigma \sqrt{\frac{\Lambda}{3}} c \right)^2 = \frac{\Lambda c^2}{3}$$ where $\Lambda$ represents the cosmological constant and $c$ is the speed of light.

For real parameters $a_0, \sigma, \Lambda, G,$ and $c$ such that $\Lambda > 0, a_0 \neq 0$, and $\sigma \in \{1, -1\}$, the de Sitter scale factor $a(t) = a_0 \exp\left(\sigma \sqrt{\frac{\Lambda}{3}} c t\right)$ satisfies the first-order Friedmann equation at any time $t$ for a universe with vanishing energy density ($\rho = 0$) and zero spatial curvature ($k = 0$): $$\left( \frac{a'(t)}{a(t)} \right)^2 = \frac{8\pi G}{3} \rho(t) - \frac{k c^2}{a(t)^2} + \frac{\Lambda c^2}{3}$$ where $a'(t)$ is the time derivative of the scale factor, $G$ is Newton's gravitational constant, and $c$ is the speed of light.

Let $a_0, \sigma, \Lambda, G,$ and $c$ be real numbers such that $\Lambda > 0$, $a_0 \neq 0$, and $\sigma = 1$ or $\sigma = -1$. Let $a(t)$ be the de Sitter scale factor defined by $$a(t) = a_0 \exp\left(\sigma \sqrt{\frac{\Lambda}{3}} c t\right)$$ where $a_0$ is the initial scale factor, $\Lambda$ is the cosmological constant, $c$ is the speed of light, and $\sigma$ is a sign factor. Then $a(t)$ satisfies the second-order Friedmann equation for a vacuum universe (where energy density $\rho(t) = 0$ and pressure $p(t) = 0$): $$\frac{\ddot{a}(t)}{a(t)} = -\frac{4\pi G}{3} \left( \rho(t) + \frac{3p(t)}{c^2} \right) + \frac{\Lambda c^2}{3}$$ where $\ddot{a}(t)$ is the second derivative of the scale factor with respect to time $t$.

For real parameters $a_0, \sigma, \Lambda,$ and $c$ where $a_0 \neq 0$, let $a(t) = a_0 \exp\left(\sigma \sqrt{\frac{\Lambda}{3}} c t\right)$ be the de Sitter scale factor. Then the Hubble parameter $H(t) = \frac{a'(t)}{a(t)}$ at any time $t$ is given by: $$H(t) = \sigma \sqrt{\frac{\Lambda}{3}} c$$

For the de Sitter scale factor $a(t) = a_0 \exp\left(\sigma \sqrt{\frac{\Lambda}{3}} c t\right)$ with parameters $a_0, \sigma, \Lambda, c \in \mathbb{R}$ satisfying $\Lambda > 0$, $c > 0$, $a_0 \neq 0$, and $\sigma \in \{1, -1\}$, the deceleration parameter defined by $$q(t) = - \frac{\ddot{a}(t) a(t)}{\dot{a}(t)^2}$$ is equal to $-1$ for all time $t$.

For a reference time $t_0 > 0$ and a power-law index $n \in \mathbb{R}$, let the scale factor be defined as $a(t) = \left(\frac{t}{t_0}\right)^n$. For any time $t > 0$, the Hubble parameter $H(t) = \frac{a'(t)}{a(t)}$ is given by: $$H(t) = \frac{n}{t}$$

Consider a power-law scale factor $a(s) = \left( \frac{s}{t_0} \right)^n$ for some reference time $t_0 > 0$ and index $n \in \mathbb{R}$ with $n \neq 0$. For any time $t > 0$, the deceleration parameter $q(t)$, defined as $$q(t) = - \frac{\ddot{a}(t) a(t)}{\dot{a}(t)^2},$$ is constant and equal to $\frac{1 - n}{n}$.

Consider a cosmological model with Newton's gravitational constant $G > 0$, speed of light $c$, and a power-law index $n \in \mathbb{R}$. For a reference time $t_0 > 0$, let the scale factor be $a(s) = (s/t_0)^n$ and the energy density be $\rho(s) = \frac{3n^2}{8\pi G s^2}$. For any cosmic time $t > 0$, these functions satisfy the first-order Friedmann equation with spatial curvature $k = 0$ and cosmological constant $\Lambda = 0$: $$\left( \frac{a'(t)}{a(t)} \right)^2 = \frac{8\pi G}{3} \rho(t) - \frac{k c^2}{a(t)^2} + \frac{\Lambda c^2}{3}$$ where $a'(t)$ is the derivative of the scale factor with respect to time.

For a reference time $t_0 > 0$, gravitational constant $G > 0$, speed of light $c > 0$, and a non-zero constant $n \neq 0$, let the scale factor be $a(t) = (t/t_0)^n$, the energy density be $\rho(t) = \frac{3n^2}{8\pi G t^2}$, and the pressure be $p(t) = w \rho(t) c^2$. If the equation of state parameter $w$ satisfies the relation $1 + 3w = \frac{2(1-n)}{n}$, then for any cosmic time $t > 0$, these functions satisfy the second-order Friedmann equation with a cosmological constant $\Lambda = 0$: $$\frac{\ddot{a}(t)}{a(t)} = -\frac{4\pi G}{3} \left( \rho(t) + \frac{3p(t)}{c^2} \right)$$ where $\ddot{a}(t)$ denotes the second derivative of the scale factor with respect to time.

The radiation-dominated scale factor $a(t)$ is a function that maps a time $t$ to a real number, given a reference time $t_0$. It is defined by the power law $a(t) = \left( \frac{t}{t_0} \right)^{1/2}$, which is normalized such that $a(t_0) = 1$. In this cosmological model, $t = 0$ corresponds to the Big Bang, and the definition is physically valid for $t > 0$.

Consider a cosmological model with a reference time $t_0 > 0$ and gravitational constant $G > 0$. For any cosmic time $t > 0$, the radiation-dominated scale factor $a(t) = (t/t_0)^{1/2}$ and the energy density $\rho(t) = \frac{3}{32 \pi G t^2}$ satisfy the first-order Friedmann equation: $$\left( \frac{a'(t)}{a(t)} \right)^2 = \frac{8\pi G}{3} \rho(t) - \frac{k c^2}{a(t)^2} + \frac{\Lambda c^2}{3}$$ evaluated with spatial curvature $k = 0$ and cosmological constant $\Lambda = 0$.

For a reference time $t_0 > 0$, gravitational constant $G > 0$, and speed of light $c > 0$, the radiation-dominated scale factor $a(t) = (t/t_0)^{1/2}$ satisfies the second-order Friedmann equation: $$\frac{\ddot{a}(t)}{a(t)} = -\frac{4\pi G}{3} \left( \rho(t) + \frac{3p(t)}{c^2} \right) + \frac{\Lambda c^2}{3}$$ where the energy density is $\rho(t) = \frac{3}{32 \pi G t^2}$, the pressure is $p(t) = \frac{\rho(t) c^2}{3}$, and the cosmological constant is $\Lambda = 0$, for all cosmic times $t > 0$.

For a radiation-dominated cosmological model with a scale factor $a(t) = (t/t_0)^{1/2}$ defined for a reference time $t_0 > 0$ and any time $t > 0$, the deceleration parameter $q(t) = - \frac{\ddot{a}(t) a(t)}{\dot{a}(t)^2}$ is equal to 1.

In a radiation-dominated cosmological model with a scale factor defined by $a(t) = \left( \frac{t}{t_0} \right)^{1/2}$ for a reference time $t_0 > 0$, the Hubble constant $H(t) = \frac{a'(t)}{a(t)}$ evaluated at $t = t_0$ is given by: $$H(t_0) = \frac{1}{2t_0}$$

For a given reference time $t_0$, the Einstein-de Sitter scale factor is a function of time $t$ defined as $a(t) = \left( \frac{t}{t_0} \right)^{2/3}$. This model describes a flat, dust-dominated universe where the scale factor is normalized such that $a(t_0) = 1$ and the Big Bang singularity occurs at $t = 0$.

Let $t_0 > 0$ be a reference time, $G > 0$ be Newton's gravitational constant, and $c$ be the speed of light. For any time $t > 0$, the Einstein-de Sitter scale factor $a(t) = (t/t_0)^{2/3}$ and the energy density $\rho(t) = \frac{1}{6\pi G t^2}$ satisfy the first-order Friedmann equation with spatial curvature $k = 0$ and cosmological constant $\Lambda = 0$: $$\left( \frac{a'(t)}{a(t)} \right)^2 = \frac{8\pi G}{3} \rho(t) - \frac{k c^2}{a(t)^2} + \frac{\Lambda c^2}{3}$$

Let $t_0$ be a positive reference time, $G$ be the gravitational constant, and $c$ be the speed of light. For any time $t > 0$, the Einstein-de Sitter scale factor $a(t) = (t/t_0)^{2/3}$ satisfies the second-order Friedmann equation $$\frac{a''(t)}{a(t)} = -\frac{4\pi G}{3} \left( \rho(t) + \frac{3p(t)}{c^2} \right) + \frac{\Lambda c^2}{3}$$ with energy density $\rho(t) = \frac{1}{6 \pi G t^2}$, pressure $p(t) = 0$, and cosmological constant $\Lambda = 0$.

For a reference time $t_0 > 0$ and any time $t > 0$, the deceleration parameter $q(t)$ of the Einstein-de Sitter universe, characterized by the scale factor $a(t) = \left( \frac{t}{t_0} \right)^{2/3}$, is equal to $1/2$. Here, the deceleration parameter is defined as $q(t) = - \frac{\ddot{a}(t) a(t)}{\dot{a}(t)^2}$, where $\dot{a}(t)$ and $\ddot{a}(t)$ are the first and second time derivatives of the scale factor, respectively.

In the Einstein-de Sitter model of the universe with a reference time $t_0 > 0$, where the scale factor is defined as $a(t) = (t/t_0)^{2/3}$, the Hubble constant $H$ evaluated at time $t_0$ is given by $$H(t_0) = \frac{2}{3 t_0}$$ This implies that the reference time can be expressed in terms of the Hubble constant at that time as $t_0 = \frac{2}{3 H_0}$.

For a given constant $c \in \mathbb{R}$, the Milne scale factor is the function $a: \text{Time} \to \mathbb{R}$ defined by $a(t) = c t$. In this cosmological model, the Big Bang occurs at $t = 0$, and the model is physically defined for times $t > 0$.

For any real constant $c$, the time derivative of the Milne scale factor $a(t) = ct$ is the constant function $a'(t) = c$.

For any constant $c \in \mathbb{R}$, let $a(t) = ct$ be the Milne scale factor. The second time derivative of the Milne scale factor is zero for all $t$, which can be expressed as: $$\frac{d^2 a(t)}{dt^2} = 0$$ where $a(t)$ is the function mapping time $t$ to $ct$.

For any gravitational constant $G \in \mathbb{R}$ and speed of light $c > 0$, and for any time $t > 0$, the Milne scale factor $a(t) = ct$ satisfies the first-order Friedmann equation: $$\left( \frac{a'(t)}{a(t)} \right)^2 = \frac{8\pi G}{3} \rho(t) - \frac{k c^2}{a(t)^2} + \frac{\Lambda c^2}{3}$$ with vanishing energy density $\rho(t) = 0$, spatial curvature $k = -1$, and vanishing cosmological constant $\Lambda = 0$.

For any gravitational constant $G \in \mathbb{R}$, speed of light $c \in \mathbb{R}$, and time $t$, the Milne scale factor $a(t) = ct$ satisfies the second-order Friedmann equation $$\frac{a''(t)}{a(t)} = -\frac{4\pi G}{3} \left( \rho(t) + \frac{3p(t)}{c^2} \right) + \frac{\Lambda c^2}{3}$$ with energy density $\rho(t) = 0$, pressure $p(t) = 0$, and cosmological constant $\Lambda = 0$.

For the Milne cosmological model with scale factor $a(t) = ct$, where $c \in \mathbb{R}$ is a constant and $t$ represents time, the deceleration parameter $q(t)$ defined by $$q(t) = - \frac{\ddot{a}(t) a(t)}{\dot{a}(t)^2}$$ is equal to $0$ for all $t$.

In the Einstein static universe model, consider a scale factor $a(t)$ and an energy density $\rho(t)$ at cosmic time $t$. Assume that Newton's gravitational constant $G > 0$ and the second time derivative of the scale factor vanishes, $\ddot{a}(t) = 0$. If the second-order Friedmann equation holds at time $t$ for a dust-filled universe (where the pressure $p$ is $0$), i.e., $$\frac{\ddot{a}(t)}{a(t)} = -\frac{4\pi G}{3} \rho(t) + \frac{\Lambda c^2}{3}$$ then the energy density $\rho(t)$ is given by: $$\rho(t) = \frac{\Lambda c^2}{4 \pi G}$$ where $\Lambda$ is the cosmological constant and $c$ is the speed of light.

Consider a cosmological model defined by the scale factor $a: \text{Time} \to \mathbb{R}$ and energy density $\rho: \text{Time} \to \mathbb{R}$. Let $k$ be the spatial curvature parameter, $\Lambda$ the cosmological constant, $G$ Newton's gravitational constant, and $c$ the speed of light. At a given cosmic time $t$, if the universe is static—meaning the first and second time derivatives of the scale factor vanish, $a'(t) = 0$ and $a''(t) = 0$— and satisfies both the first-order Friedmann equation and the second-order Friedmann equation (for a pressureless "dust" universe, where $p=0$), then the following relationship holds: $$\frac{k c^2}{a(t)^2} = 4 \pi G \rho(t)$$

Let $a(t)$ be the scale factor and $\rho(t)$ be the energy density of a Friedmann-Lemaître-Robertson-Walker universe at cosmic time $t$. Let $k \in \mathbb{R}$ be the spatial curvature parameter, $G$ be Newton's gravitational constant, $c$ be the speed of light, and $\Lambda$ be the cosmological constant. Suppose the universe is static at time $t$, such that the first and second time derivatives of the scale factor vanish ($a'(t) = 0$ and $a''(t) = 0$). If the universe satisfies the first-order Friedmann equation and the second-order Friedmann equation (with zero pressure), and assuming $G > 0$, $c > 0$, $a(t) \neq 0$, and a positive energy density $\rho(t) > 0$, then the spatial curvature parameter $k$ must be positive ($k > 0$).