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For an expanding pressureless matter universe with positive curvature, zero cosmological constant and present scale factor one, set $u_0=\arccos (2/{\Omega }_0-1)\in (0,\pi )$. The closed matter-dominated Friedmann solution then gives the displayed age of an FLRW universe. Its flat limit is $2/(3H_0)$, despite the individually vanishing numerator and denominator. To see this, write $\delta ={\Omega }_0-1$; then $u_0=2\arctan \sqrt{\delta }$ and $u_0-\sin u_0=(4/3){\delta }^{3/2}+O({\delta }^{5/2})$.

For an expanding pressureless matter universe with zero cosmological constant, normalize $x=a/a_0$. Dust relation between deceleration and density and mass conservation give ${\dot{x}}^2=H_0^2(1-2q_0+2q_0/x)$. Integrating from the Big Bang yields the age integral above.

For $q_0>1/2$, set $x=q_0(1-\cos \theta )/(2q_0-1)$ to obtain $H_0{t}_0=q_0({\theta }_{\ast }-\sin {\theta }_{\ast })/(2q_0-1{)}^{3/2}$, with $\cos {\theta }_{\ast }=(1-q_0)/q_0$ on the expanding branch. For $0<q_0<1/2$, the corresponding hyperbolic expression is $q_0(\sinh {\eta }_{\ast }-{\eta }_{\ast })/(1-2q_0{)}^{3/2}$ with $\cosh {\eta }_{\ast }=(1-q_0)/q_0$. The flat-dust and empty Milne universe limits are respectively $2/3$ and $1$.

For positive pressureless matter density, positive cosmological constant, negligible radiation and zero curvature, the Friedmann equation gives $dt=\sqrt{a}da/[H_0\sqrt{{\Omega }_m+{\Omega }_{\Lambda }{a}^3}]$. Substituting $u=\sqrt{{\Omega }_{\Lambda }/{\Omega }_m}{a}^{3/2}$ gives the displayed age of an FLRW universe measured from $a=0$. Its zero-vacuum limit is $2a^{3/2}/(3H_0\sqrt{{\Omega }_m})$; the pure positive-vacuum flat patch has no finite big-bang age.

For an expanding pressureless matter universe with zero cosmological constant, fixed current Hubble parameter $H_0$ and current matter cosmological density parameter $\Omega >0$, the age of a dust universe without a cosmological constant is $H_0{t}_0={\int }_0^1\sqrt{x}/\sqrt{\Omega +(1-\Omega )x}dx$. For every $0<x<1$ the integrand decreases strictly as $\Omega$ increases. Thus open models with $0<\Omega <1$ are older than the flat model, and closed expanding models with $\Omega >1$ are younger. The flat value is $2/3$ and the empty Milne universe limit is $1$. Specifying what is held fixed matters: at fixed present scale factor and energy density, the integral $t_0={\int }_0^{a_0}da/\sqrt{C/a-k}$ instead increases with $k$, provided all three models reach $a_0$ on their expanding branches.

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 $u=2\pi$.

In a spatially flat universe containing separately conserved pressureless matter and a coasting fluid with $w=-1/3$, the ratio $\eta ={\rho }_s/{\rho }_m$ is proportional to the scale factor. If the coasting component is modeled by the barotropic fluid perturbation equations with no anisotropic stress, its contribution to the trace gravitational source vanishes because $1+3w=0$. The matter growing mode isA decaying solution is $D_{-}=\sqrt{1+\eta }/{\eta }^{3/2}$. The reduction of order formula with integrating factor ${\eta }^{3/2}\sqrt{1+\eta }$ produces the integral solution. It has $D=\eta [1-4\eta /7+O({\eta }^2)]$ during matter domination and tends to $5/2$ during coasting-fluid domination. Thus growth freezes rather than continuing in proportion to the scale factor. The background geometry is flat despite the same background expansion law as an open dust universe.

A comoving gauge in cosmology chooses hypersurfaces with zero scalar momentum density for the specified fluid or total system. For cold pressureless matter it can be implemented by taking the matter velocity potential to vanish. The choice of component must be stated in a multi-fluid system; density contrasts on this slicing can differ from those in a Newtonian or spatially flat gauge.

For pressureless matter with zero cosmological constant, the Friedmann acceleration equation gives $q=4\pi G\rho /(3H^2)={\Omega }_m/2$. Spatial curvature need not vanish. Consequently $k{c}^2/a^2=(2q-1)H^2$; this relation does not hold for a general fluid mixture.

Suppose a homogeneous scalar field oscillates much faster than the Hubble parameter in a monomial potential $V(\varphi )=\lambda |\varphi {|}^n$. The virial theorem averaged over one oscillation gives $⟨{\dot{\varphi }}^2⟩=n⟨V⟩$, and henceA quadratic minimum therefore behaves like pressureless matter, whereas a quartic minimum behaves like radiation in cosmology.

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 $a=(9C/4{)}^{1/3}{t}^{2/3}$.

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.

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 pressureless matter in Newtonian gauge in cosmology with a common metric potential $\Phi$, let $\theta =\nabla \cdot \mathbf{v}$. The linear equations ${\delta }_m^{\prime }=-\theta +3{\Phi }^{\prime }$ and ${\theta }^{\prime }=-\mathcal{H}\theta -{\nabla }^2\Phi$, where primes denote conformal time, yieldThis follows by differentiating the first equation and eliminating $\theta$. On deeply subhorizon scales with a slowly varying matter potential, the metric time derivatives are small compared with its spatial Laplacian.

For separately conserved pressureless matter, ${\rho }_m\propto a^{-3}$. The deceleration parameter identity $\dot{H}/H^2=-(1+q)$ and ${\Omega }_m\propto {\rho }_m/H^2$ give ${\dot{\Omega }}_m=H{\Omega }_m(2q-1)$. A dust-plus-vacuum universe has $q={\Omega }_m/2-{\Omega }_{\Lambda }$; conversion to conformal time yields the displayed flow. The familiar matter-only equation requires negligible cosmological constant. In a spatially flat dust-plus-vacuum universe, ${\Omega }_m+{\Omega }_{\Lambda }=1$ and ${\Omega }_m^{\prime }=-3\mathcal{H}{\Omega }_m{\Omega }_{\Lambda }$, so flatness does not require ${\Omega }_m=1$.

In a flat universe containing pressureless matter and a positive cosmological constant, normalize the scale factor to one today. The Friedmann equation and $d\tau =dt/a$ giveThe change of variable is $a=({\Omega }_m/{\Omega }_{\Lambda }{)}^{1/3}y$. Including radiation in cosmology makes this an approximation, with its omitted early contribution of order $H_0^{-1}\sqrt{{\Omega }_r}/{\Omega }_m$. When the upper limit is large, the integral approaches $I=\Gamma (1/6)\Gamma (1/3)/(3\sqrt{\pi })$; this follows by setting $z=y^3$ and using the beta function integral.

In a flat universe containing pressureless matter and smooth radiation in cosmology, put $y=a/a_{\mathrm{eq}}$ and $C=H_0^2{\Omega }_{m,0}^2/{\Omega }_{r,0}$. The Friedmann equation gives ${\mathcal{H}}^2=C(1+y)/y^2$, so $y^{\prime 2}=C(1+y)$ and $y^{\prime \prime }=C/2$. The subhorizon cosmological Poisson equation gives ${\delta }^{\prime \prime }+\mathcal{H}{\delta }^{\prime }=3C\delta /(2y)$. The chain rule produces the displayed equation. Smoothing radiation neglects its rapid acoustic forcing after horizon entry; it is not an exact treatment of all radiation perturbations.

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.

The comoving dust ball $0\le \chi \le {\chi }_0<\pi /2$ in the Oppenheimer-Snyder model is an Oppenheimer-Snyder cloud. Its pressureless matter has uniform density on each constant-proper time slice and reaches a simultaneous final singularity in the ideal homogeneous model.

The Oppenheimer-Snyder model describes a homogeneous ball of pressureless matter collapsing from rest, with a closed FLRW metric inside and a Schwarzschild metric outside. In geometric units its interior is $d{s}^2=-d{\tau }^2+a^2(d{\chi }^2+{\sin }^2\chi d{\Omega }^2)$, with $a=a_{\max }(1+\cos \eta )/2$ and $\tau =a_{\max }(\eta +\sin \eta )/2$. The boundary is a fixed comoving angle. Density conservation gives $\rho a^3$ constant, and the Friedmann equation at the initial turning point gives $8\pi {\rho }_0{a}_{\max }^2/3=1$.

Neglecting curvature and dark energy, integration of the Friedmann equation for radiation in cosmology and pressureless matter gives the displayed comoving particle horizon. At matter-radiation equality, $a_{\mathrm{eq}}={\Omega }_{r,0}/{\Omega }_{m,0}$, this is $2(\sqrt{2}-1)$ times the radiation-only extrapolation $c{a}_{\mathrm{eq}}/(H_0\sqrt{{\Omega }_{r,0}})$. Its associated inverse length differs by an order-one factor from the matter-radiation equality scale $a_{\mathrm{eq}}{H}_{\mathrm{eq}}/c$.