Physlib.Cosmology.FLRW.Basic

The type `SpatialGeometry` represents the possible spatial geometries for the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. It is an inductive type with three constructors: - $\text{Spherical}(k)$, representing a spherical spatial geometry with a curvature parameter $k \in \mathbb{R}$. - $\text{Flat}$, representing a flat (Euclidean) spatial geometry. - $\text{Saddle}(k)$, representing a hyperbolic (saddle-shaped) spatial geometry with a curvature parameter $k \in \mathbb{R}$.

For a given spatial geometry $s$ and a radial coordinate $r \in \mathbb{R}$, the function $S(s, r)$ is defined based on the type of geometry: - If $s$ is a spherical geometry with curvature parameter $k$, then $S(s, r) = k \sin(r/k)$. - If $s$ is a flat geometry, then $S(s, r) = r$. - If $s$ is a saddle (hyperbolic) geometry with curvature parameter $k$, then $S(s, r) = k \sinh(r/k)$. This function typically appears in the Friedmann-Lemaître-Robertson-Walker (FLRW) metric to describe the radial part of the spatial line element.

For any real numbers $r$ and $k$, the identity $k \sinh\left(\frac{r}{k}\right) = \frac{\sinh\left(\frac{r}{k}\right)}{1/k}$ holds.

For any real number $r$, the limit of the function $\frac{\sinh(rx)}{x}$ as $x$ approaches $0$ is equal to $r$. In mathematical notation, this is expressed as: $$\lim_{x \to 0} \frac{\sinh(rx)}{x} = r$$ where $\sinh$ denotes the hyperbolic sine function.

For any real number $r$, the limit of the function $f(k) = k \sinh\left(\frac{r}{k}\right)$ as $k$ approaches infinity is $r$. That is, $$\lim_{k \to \infty} k \sinh\left(\frac{r}{k}\right) = r.$$

For any real numbers $r$ and $k$, the product $k \sin(r/k)$ is equal to the quotient $\frac{\sin(r/k)}{1/k}$.

For any real number $r$, the limit of the function $\frac{\sin(rx)}{x}$ as $x$ approaches $0$ is $r$. In mathematical notation, this is expressed as: $$\lim_{x \to 0} \frac{\sin(rx)}{x} = r$$

For any real number $r$, the limit of the expression $k \sin\left(\frac{r}{k}\right)$ as $k$ approaches infinity is $r$. Mathematically, this is expressed as: $$\lim_{k \to \infty} k \sin\left(\frac{r}{k}\right) = r$$

The structure `FLRW` encapsulates the physical parameters required to define the Friedmann–Lemaître–Robertson–Walker metric for a homogeneous and isotropic universe. It consists of a scale factor $a(t)$, which is a function of time $t$, and an element representing the `SpatialGeometry` of the universe.

The first-order Friedmann equation relates the expansion rate of the universe to its energy density and geometry. For a scale factor $a: \mathbb{R} \to \mathbb{R}$, a total energy density $\rho: \mathbb{R} \to \mathbb{R}$, a spatial curvature parameter $k \in \mathbb{R}$, a cosmological constant $\Lambda \in \mathbb{R}$, Newton's gravitational constant $G \in \mathbb{R}$, and the speed of light $c \in \mathbb{R}$, the equation at cosmic time $t$ is given by: $$\left( \frac{a'(t)}{a(t)} \right)^2 = \frac{8\pi G}{3} \rho(t) - \frac{k c^2}{a(t)^2} + \frac{\Lambda c^2}{3}$$ where $a'(t)$ is the derivative of the scale factor with respect to time.

The second-order Friedmann equation (sometimes referred to as the Raychaudhuri equation) describes the acceleration of the expansion of the universe within the Friedmann-Lemaître-Robertson-Walker (FLRW) model. Given the scale factor $a(t)$, the total energy density $\rho(t)$, the pressure $p(t)$, Newton's gravitational constant $G$, the cosmological constant $\Lambda$, and the speed of light $c$, the equation at cosmic time $t$ is: $$\frac{a''(t)}{a(t)} = -\frac{4\pi G}{3} \left( \rho(t) + \frac{3p(t)}{c^2} \right) + \frac{\Lambda c^2}{3}$$ where $a''(t)$ denotes the second derivative of the scale factor $a$ with respect to time $t$.

For a given scale factor $a: \mathbb{R} \to \mathbb{R}$ representing the expansion of the universe over time, the Hubble constant $H$ at time $t$ is defined as the ratio of the time derivative of the scale factor to the scale factor itself: $$H(t) = \frac{a'(t)}{a(t)}$$ where $a'(t)$ denotes the derivative of $a$ with respect to $t$.

Given a scale factor $a: \mathbb{R} \to \mathbb{R}$ as a function of time $t$, the deceleration parameter $q$ at time $t$ is defined as: $$q(t) = - \frac{\ddot{a}(t) a(t)}{\dot{a}(t)^2}$$ where $\dot{a}(t)$ and $\ddot{a}(t)$ denote the first and second derivatives of the scale factor with respect to time, respectively.

For a given Hubble parameter $H$ and its time derivative $\dot{H} = \frac{dH}{dt}$, the deceleration parameter $q$ is defined by the relationship: $$q = - \left( 1 + \frac{\dot{H}}{H^2} \right)$$ This formula expresses the deceleration parameter in terms of the rate of change of the Hubble constant.

The time derivative of the Hubble parameter $H$ with respect to time $t$, denoted as $\dot{H}$ or $\frac{dH}{dt}$, is given by the expression: $$\dot{H} = - H^2 (1 + q)$$ where $q$ is the deceleration parameter.

There exists a time $t$ such that the Hubble constant $H$ is decreasing, which corresponds to the condition that its time derivative $\dot{H} = \frac{dH}{dt}$ is less than zero, if and only if there exists a time $t$ such that the deceleration parameter $q$ satisfies $q < -1$.