Friedmann equation

$H=\dot{a}/a$, a fluid with $P=w\rho$ has $\rho \propto a^{-3(1+w)}$, and ${\rho }_{\mathrm{crit}}=3c^2{H}^2/(8\pi G)$.

With positive curvature, zero cosmological constant and radiation density ${\rho }_0{a}^{-4}$ in units $8\pi G=1$, $A=\sqrt{{\rho }_0/3}$. The scale factor is positive for $0<\tau <\pi$, reaches its maximum at $\pi /2$, and has a Big Crunch at conformal time $\pi$ or proper time $2A$. This complements the closed matter-dominated Friedmann solution.

For a perfect fluid in general relativity of constant equation of state $P=w\rho$, conservation gives $\rho ={\rho }_0{a}^{-3(1+w)}$. In units $8\pi G=1$, the Friedmann equation becomes ${\dot{a}}^2/2+V(a)=0$. Allowed scales have $V\le 0$, and the Friedmann acceleration equation is $\ddot{a}=-V^{\prime }(a)$. The curvature term shifts the potential vertically; positive cosmological constant bends it downward and negative cosmological constant bends it upward.

In a flat universe containing separately conserved radiation in cosmology and pressureless matter, the Friedmann equation becomes $(a^{\prime }{)}^2=H_0^2({\Omega }_{r,0}+{\Omega }_{m,0}a)$ in conformal time. Setting $a=0$ at $\tau =0$ gives the stated quadratic-plus-linear scale factor. The linear term controls radiation domination and the quadratic term controls matter domination. The formula presumes $a_0=1$, ${\Omega }_{r,0}+{\Omega }_{m,0}=1$ and no cosmological constant.

A flat universe with separately conserved pressureless matter and a coasting fluid satisfies $a^{\prime 2}=H_0^2[{\Omega }_{m}a+(1-{\Omega }_m)a^2]$. For both densities positive, choosing the Big Bang at zero conformal time gives $a={\Omega }_m[\cosh (\alpha \tau )-1]/[2(1-{\Omega }_m)]$, with $\alpha =H_0\sqrt{1-{\Omega }_m}$. Physical time follows by integrating $dt=ad\tau$. The hyperbolic parametrization is formally like the open dust solution, but the spatial geometry here is flat.

The conformal Friedmann equation is ${\mathcal{H}}^2=H_0^2[{\Omega }_m/a+1-{\Omega }_m]$. Differentiating and using $a^{\prime }=a\mathcal{H}$ eliminates the matter term and yields a first-order Riccati equation. The expanding Big-Bang branch for $0<{\Omega }_m<1$ has $\mathcal{H}=\alpha \coth (\alpha \tau /2)$.

A negatively curved universe with pressureless matter and no dark energy expands forever. With $a_0=1$ and $0<{\Omega }_{m,0}<1$, its scale factor is $a={\Omega }_{m,0}(\cosh \eta -1)/[2(1-{\Omega }_{m,0})]$ and its cosmic time is $t={\Omega }_{m,0}(\sinh \eta -\eta )/[2H_0(1-{\Omega }_{m,0}{)}^{3/2}]$. The parameter is conformal time scaled by $\sqrt{-K}$ and measured from the Big Bang.

For pressureless matter normalized by $\rho =3\mu /(8\pi a^3)$ in units $G=c=1$, the Friedmann equation isAllowed scale factors satisfy $V(a)\le -k$, and equality marks a turning point.

For positive spatial curvature and positive cosmological constant, the effective potential has its maximum atThe critical case $V(a_E)=-1$ hasand contains the Einstein static universe.

When the energy line $-k$ lies below the maximum of the pressureless-matter effective potential, its large-$a$ allowed region is separated from the Big-Bang region. A contracting large universe reaches the outer turning point at positive scale factor and re-expands, producing a nonsingular bounce.

For a test mass on the surface of a uniform sphere of radius $R$ and mass $M=4\pi \rho R^3/3$, zero Newtonian specific energy givesWith $R\propto a$, this becomes the spatially flat Friedmann equation $H^2=8\pi G\rho /3$.

Combining the perfect-fluid continuity equation withshows thatis constant in time. This constant is the spatial-curvature integration constant in the first Friedmann equation.

For energy density $\rho$, pressure $P$, and cosmological constant $\Lambda$,

For a homogeneous pressureless fluid in comoving coordinates, the Euler equation giveswhile the Poisson equation givesTaking the divergence of the first relation and comparing them yieldsThis local fluid derivation describes a homogeneous infinite universe without choosing a physical centre of expansion.

For positive spatial curvature, zero cosmological constant, and $\rho ,P\ge 0$, continuity makes $\rho a^3$ nonincreasing during expansion. The negative curvature term then prevents unbounded growth of $a$, while the acceleration equation forces $\dot{a}$ to reach zero in finite time.

With zero cosmological constant, the condition $\rho +3P\ge 0$ implies $\ddot{a}\le 0$ andwherever $H\ne 0$.

If $H>0$ at one time, the inequality $d(H^{-1})/dt\ge 1$ forces $H\to +\infty$ and $a\to 0$ within finite time to the past. If $H<0$, it forces $H\to -\infty$ and $a\to 0$ within finite time to the future.

In a flat radiation-plus-$\Lambda$ universe,The expansion changes from deceleration to acceleration when the radiation and cosmological-constant terms are equal, at

If the present radiation and cosmological-constant densities are respectively $\beta$ and $1$ times the present critical density, while $a_0=1$, positive curvature givesTurning points are therefore the positive roots of $x^2-\beta x+\beta$ with $x=a^2$.

For $\beta >4$, the turning polynomial has two positive roots and its smaller root lies below $2$. A universe expanding from $a=0$ reaches that root with $H=0$ and $\dot{H}<0$, then contracts to a finite-time Big Crunch.

For $\beta =4$ and $a(0)=0$, the expanding branch isIt grows as $2\sqrt{H_{0}t}$ near zero and approaches $\sqrt{2}$ exponentially at late times.

A positively curved pressureless Friedmann universe has the cycloidal parametrization $a=A(1-\cos \theta )$ and $t=B(\theta -\sin \theta )$, with $B=A/\sqrt{k}$. It expands from a Big Bang, reaches maximum size at $\theta =\pi$, and recollapses at $\theta =2\pi$.