Friedmann equations

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

For a spatially flat FLRW metric containing conserved pressureless matter and radiation in cosmology, with $a_0=1$ and conformal time zero at the Big Bang, the Friedmann equation gives $a^{\prime 2}=H_0^2({\Omega }_{M,0}a+{\Omega }_{R,0})$. Separating the positive square root gives the displayed scale factor. At matter-radiation equality, $a_{\mathrm{eq}}={\Omega }_{R,0}/{\Omega }_{M,0}$ and ${\eta }_{\mathrm{eq}}=2(\sqrt{2}-1)\sqrt{{\Omega }_{R,0}}/(H_0{\Omega }_{M,0})$. Here the prime denotes differentiation with respect to $\eta$, not $c\eta$.

A smooth cosmological bounce is a positive minimum of the scale factor: contraction changes to expansion, with $a(t_b)>0$, $\dot{a}(t_b)=0$ and $\ddot{a}(t_b)>0$ in the nondegenerate case. The closed slicing of de Sitter spacetime gives $a=H_{\Lambda }^{-1}\cosh [H_{\Lambda }(t-t_b)]$, whose minimum is $H_{\Lambda }^{-1}$. The Friedmann acceleration equation with zero separate cosmological constant requires $\varrho +3p<0$ there.

The equation of state $p=-\varrho$ makes the cosmological continuity equation imply constant energy density ${\varrho }_0$. For ${\varrho }_0>0$, put $H_{\Lambda }^2=8\pi G{\varrho }_0/3$. Solving both the Friedmann equation and Friedmann acceleration equation givesUp to shifts of cosmic time and time reversal, the positive scale factors are $A{e}^{H_{\Lambda }t}$ for $k=0$, $H_{\Lambda }^{-1}\cosh (H_{\Lambda }t)$ for $k=1$, and $H_{\Lambda }^{-1}\sinh (H_{\Lambda }t)$ on $t>0$ for $k=-1$. They are the flat, closed and open slicings of de Sitter spacetime. The open zero is a coordinate boundary with finite Ricci scalar $12H_{\Lambda }^2$ and finite Kretschmann scalar $24H_{\Lambda }^4$.

For the expanding flat branch, the cosmological event horizon has comoving radial distance ${\int }_t^{\infty }ds/a(s)=1/[H_{\Lambda }a(t)]$ and proper distance in cosmology $H_{\Lambda }^{-1}$. An event on the horizon reaches the central comoving observer only at infinite reception time; events beyond it never do.

For positive constant cosmological constant, zero cosmological spatial curvature and a separate positive-density fluid with constant equation-of-state parameter $w>-1$, set $H_{\Lambda }=\sqrt{\Lambda /3}$ in the time-unit convention. Eliminating the fluid density gives $\dot{H}=-3(1+w)(H^2-H_{\Lambda }^2)/2$. On the expanding Big Bang branch,The origin is the Big Bang. The fluid's early-time power-law expansion approaches exponential vacuum domination at late times.

For a constant-$w$ material fluid and positive vacuum energy, the Friedmann acceleration equation gives $\ddot{a}/(a{H}_0^2)={\Omega }_{\Lambda 0}-(1+3w){\Omega }_{m0}(1+z{)}^{3(1+w)}/2$. If $w>-1/3$, its zero determines the displayed transition. A past transition additionally requires the right-hand side of the displayed equation to exceed one. The equation does not imply the same transition for a fluid that already has sufficiently negative pressure.

The flat barotropic matter-vacuum solution has $H_{\Lambda }/H_0=\sqrt{{\Omega }_{\Lambda 0}}$ and therefore the displayed age, measured from its Big Bang. At fixed $H_0$ and density fractions it is $1/(1+w)$ times the dust-plus-vacuum age. Its $w=0$ case is the age of a flat matter-Lambda universe.

With nonnegative conserved pressureless matter and radiation in cosmology, set $M=(8\pi G/3){\rho }_m{a}^3$, $Q=(8\pi G/3){\rho }_r{a}^4$, and $L=(8\pi G/3){\rho }_{\Lambda }$. The Friedmann equation becomes $(da/dt{)}^2+V(a)=-k$. For negative vacuum density the potential rises strictly from negative infinity to positive infinity, so an expanding branch has one maximum scale factor. For positive vacuum density it has one maximum: energy above that maximum gives monotonic expansion or contraction, energy below it gives disjoint recollapsing and bouncing branches, and equality admits an Einstein static universe together with asymptotic monotonic branches.

For a FRW metric with positive scale factor, nonnegative dust and radiation, and negative vacuum density, the Friedmann acceleration equation gives $a_{tt}\le -{\omega }^{2}a$, with ${\omega }^2=8\pi G|{\rho }_{\Lambda }|/3$. At maximum expansion, $a(0)=a_{\max }$ and $a_t(0)=0$. The forced-oscillator integral formula gives $a(t)\le a_{\max }\cos (\omega t)$ until the first zero, since its forcing is nonpositive and its sine kernel is nonnegative for $0\le t\le \pi /(2\omega )$. Thus each half of the expansion-recollapse history lasts at most $\pi /(2\omega )$. Nonzero dust or radiation makes the endpoint a physical density singularity; the matter-free limit instead bounds the open anti-de Sitter coordinate patch.

For a perfect fluid in general relativity with constant equation-of-state parameter $w$, use conformal time $dt=ad\tau$ and the conformal Hubble parameter $\mathcal{H}=a^{\prime }/a=aH$. The Friedmann acceleration equation gives ${\mathcal{H}}^{\prime }=a\ddot{a}=-4\pi G{a}^2(1+3w)\rho /3$. The Friedmann equation eliminates $\rho$ and gives this Riccati equation. For positive energy density, $\alpha \ne 0$ and appropriately shifted conformal time, its usual branches are $\cot (\alpha \tau )$ for $k=1$, $1/(\alpha \tau )$ for $k=0$, and $\coth (\alpha \tau )$ for $k=-1$. Their scale factors are proportional to ${\sin }^{1/\alpha }(\alpha \tau )$, $(\alpha \tau {)}^{1/\alpha }$ and ${\sinh }^{1/\alpha }(\alpha \tau )$ respectively on intervals where the bases are positive. At $\alpha =0$ the conformal Hubble parameter is constant and the separate solution is $a=A{e}^{h\tau }$.

For pressureless matter, put $C=8\pi G{\rho }_m{a}^3/3>0$ and use units $c=1$. The flat Friedmann equation is ${\dot{a}}^2=C/a+\Lambda a^2/3$. Substitution $u=a^{3/2}$ gives ${\dot{u}}^2=(9/4)(C+\Lambda u^2/3)$. With a Big Bang at $t=0$, positive cosmological constant gives $a=(3C/\Lambda {)}^{1/3}{\sinh }^{2/3}(\sqrt{3\Lambda }t/2)$. Negative cosmological constant replaces the hyperbolic sine by $\sin (\sqrt{3|\Lambda |}t/2)$ and produces recollapse at $2\pi /\sqrt{3|\Lambda |}$. Zero cosmological constant gives $a=(9C/4{)}^{1/3}{t}^{2/3}$.

For pressureless matter and a positive cosmological constant, the Friedmann equation gives ${\dot{a}}^2=H_0^2[{\Omega }_{m,0}{a}_0^3/a+{\Omega }_{k,0}{a}_0^2+{\Omega }_{\Lambda ,0}{a}^2]$. Differentiating with respect to $a$ gives the displayed unique minimum of this expression for positive matter density. It is a minimum of the actual expanding solution's speed only if that branch reaches it; sufficient positive spatial curvature can cause recollapse first. At the minimum, ${\rho }_m=2{\rho }_{\Lambda }$ and the Friedmann acceleration equation changes sign.

For a closed Friedmann equation normalized to $k{c}^2=1$, constant equation-of-state parameter $w$, and conformal time $\tau$, let $q=(1+3w)/2$. The cosmological continuity equation and Friedmann acceleration equation give ${\mathcal{H}}^{\prime }=-q(1+{\mathcal{H}}^2)$, where $\mathcal{H}=a^{\prime }/a$. For $q\ne 0$ this integrates to the stated scale factor on any interval where the sine is positive, with ${\alpha }^{2q}=8\pi G{\rho }_0/3$. For $w=-1/3$ the separate solution is $a=A{e}^{H_0\tau }$ with $H_0^2+1=8\pi G{\rho }_0/3$; the sine-power formula is inapplicable.

For pressureless matter and a positive cosmological constant in a spatially flat universe, $H^2=H_0^2[{\Omega }_{m0}{a}^{-3}+1-{\Omega }_{m0}]$. Substituting $y=a^{3/2}$ reduces the expanding branch to $\dot{y}=(3H_0/2)\sqrt{{\Omega }_{m0}+(1-{\Omega }_{m0})y^2}$. With $a(0)=0$, integration gives $a=[{\Omega }_{m0}/(1-{\Omega }_{m0}){]}^{1/3}{\sinh }^{2/3}(3H_0\sqrt{1-{\Omega }_{m0}}t/2)$. Its matter-dominated early limit is $t^{2/3}$ and its vacuum-dominated late limit is exponential. Radiation must be restored to describe the actual earliest universe.

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.

In a dust-plus-cosmological-constant universe with an FRW metric with $a_0=1$, a nondegenerate minimum at $a_{\ast }<1$ has $H(a_{\ast })=0$ and $\ddot{a}(a_{\ast })>0$. Combining these conditions with ${\Omega }_m+{\Omega }_{\Lambda }+{\Omega }_k=1$ gives the strict version of the displayed bound; equality is the limiting stationary case. A source at cosmological redshift $z$ requires $z_{\ast }=a_{\ast }^{-1}-1\ge z$, so a high-redshift source bounds the allowed present matter density parameter even without directly observing the 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 $H\ne 0$.

In an FLRW metric with zero separate cosmological constant, $\varrho +3p\ge 0$ makes the Friedmann acceleration equation imply $\ddot{a}\le 0$. If $a_0=a(t_0)>0$ and $v_0=\dot{a}(t_0)>0$, then the scale factor is a concave function andthroughout its regular past interval. Thus a regular positive scale factor cannot persist earlier than $t_0-a_0/v_0$. If the solution continues regularly whenever $a>0$, its first past zero occurs within $a_0/v_0=H(t_0{)}^{-1}$, where $H$ is the Hubble parameter.

The continuation condition matters: without restrictions on the equation of state, an earlier curvature singularity can occur at positive $a$. For example, in fixed units $a(t)=1+\sqrt{t}$, $t>0$, with $k=0$, defines energy density $\varrho =3(\dot{a}/a{)}^2/(8\pi G)$ and pressure $p=-\varrho /3-\ddot{a}/(4\pi Ga)$. These satisfy the Friedmann equations and $\varrho +3p>0$, yet $a\to 1$ at the singular past boundary. Conversely the Milne model reaches zero without divergent curvature. A finite duration, a zero of the scale factor, and a curvature singularity are distinct conclusions.

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 matter universe with zero cosmological constant has the cycloid parametrization displayed above. In natural units, if the Friedmann equation is ${\dot{a}}^2=C/a-k$ with $C,k>0$, then $A=C/(2k)$ and $u=\sqrt{k}\tau$ for conformal time measured from the Big Bang. Substituting in $a^{\prime 2}=Ca-k{a}^2$ verifies the scale factor; integrating $dt=ad\tau$ gives cosmic time. The universe reaches maximum size at $u=\pi$ and recollapses into a Big Crunch at $u=2\pi$.

The cycloidal closed matter-dominated Friedmann solution reaches its maximum scale factor at $u=\pi$ and the Big Crunch at $u=2\pi$. Thus its total lifetime is twice its time of maximum expansion. The displayed formula uses a presently expanding normalization with $H_0>0$ and $a_0=1$; the remaining time is this lifetime minus the age of a closed dust universe.