The Friedmann, acceleration and continuity equations of an FLRW model datum

(a) Data. A Friedmann-system datum is a tuple

$(c,G_N,\mathrm{\Lambda },I,k,a,\epsilon ,p)$

consisting of positive real constants $c>0$ and $G_N>0$, a real constant $\mathrm{\Lambda }\in \mathbb{R}$, a nonempty open interval $I\subseteq \mathbb{R}$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), a real number $k$ (the curvature parameter of the datum), and smooth functions $a:I\to (0,\mathrm{\infty })$, $\epsilon :I\to \mathbb{R}$ and $p:I\to \mathbb{R}$, with values in the open interval of strictly positive reals in the case of $a$ ($C^r$ and smooth maps between smooth manifolds). We write $\dot{a},\ddot{a},\dot{\epsilon }$ for the derivatives of these functions of one variable and

$H:=\frac{\dot{a}}{a}:I\to \mathbb{R}$

for the Hubble parameter of the datum (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set). No relation between $k$ and the three signs $\{-1,0,1\}$ is imposed by this definition; the spatial model data of FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter restrict to those signs, and the real parameter is used so that the equations can also be examined after a rescaling of the spatial metric.

(b) The equations. For a Friedmann-system datum define the following three scalar equations on $I$:

(F) The Friedmann equation.

$H(t{)}^2+\frac{k{c}^2}{a(t{)}^2}=\frac{8\pi G_N}{3c^2}\epsilon (t)+\frac{\mathrm{\Lambda }{c}^2}{3}\text{for all }t\in I;$

(A) The acceleration equation.

$\frac{\ddot{a}(t)}{a(t)}=-\frac{4\pi G_N}{3c^2}(\epsilon (t)+3p(t))+\frac{\mathrm{\Lambda }{c}^2}{3}\text{for all }t\in I;$

(C) The continuity equation.

$\dot{\epsilon }(t)+3H(t)(\epsilon (t)+p(t))=0\text{for all }t\in I\mathrm{.}$

We also record the spatial-component equation

(S)

$-\frac{2a(t)\ddot{a}(t)+\dot{a}(t{)}^2}{c^2}-k+\mathrm{\Lambda }a(t{)}^2=\frac{8\pi G_N}{c^4}p(t)a(t{)}^2\text{for all }t\in I\mathrm{.}$

Each of (F), (A), (C), (S) is an equality of real numbers at each point of $I$; the equations involve the data of clause (a) alone. Their relation to the curvature components of an FLRW metric, and their logical relations among one another, are the content of the results on this page and are not asserted by this definition. In particular, this definition adopts no equation of state: a relation $p=P(\epsilon )$, when used, is an additional datum.

• Origin of the equations. In the physical model the three equations are the reductions of the field law and of the matter conservation identity for a homogeneous and isotropic metric with a comoving fluid; the stress-energy input supplies $\epsilon$ and $p$, and the cosmological parameter enters through $\mathrm{\Lambda }$. The equations as displayed are purely mathematical relations among the numerical data; the physical derivation and the meaning of $c,G_N,\mathrm{\Lambda },\epsilon ,p$ belong to the postulates and to the physics theorems of this page.

• Units. With $t$ in seconds, $a$ in metres, $c$ in metres per second, $G_N$ in ${\mathrm{m}}^3{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ and $\epsilon ,p$ in $\mathrm{J}{\mathrm{m}}^{-3}$, equation (F) is dimensionally homogeneous: $H^2$, $k{c}^2\mathrm{/}{a}^2$, $8\pi G_N\epsilon \mathrm{/}(3c^2)$ and $\mathrm{\Lambda }{c}^2\mathrm{/}3$ all carry ${\mathrm{s}}^{-2}$, and (A) and (C) carry ${\mathrm{s}}^{-2}$ and $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$ respectively. These unit readings belong to the framework's convention item and are not part of the mathematical definition.