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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 $\theta =2\pi$.

An isotropic cosmic string network with $\rho =C/a^2$ has equation-of-state parameter $w=-1/3$. The Friedmann equation gives ${\dot{a}}^2=8\pi GC/3-K$. On the expanding branch with positive right side, $a=B(t-t_B)$, the comoving Hubble radius is constant, and the cosmological density parameter is constant. There is no growing Flatness problem, but flatness is not an attractor. If this exact coasting solution extends all the way to $t_B$, then ${\int }_{t_B}^{t}d{t}^{\prime }/a(t^{\prime })$ diverges and there is no finite particle horizon. A finite starting time changes that global conclusion.

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 $\mathcal{H}=\alpha \coth (\alpha \tau /2)$.

The density parameter is $\Omega =\rho /{\rho }_{\mathrm{crit}}$. In a Friedmann universe, its departure from one measures the spatial-curvature term relative to the density or expansion term.

For an expanding Friedmann-Lemaitre-Robertson-Walker metric with positive energy density, put $\Omega =8\pi G\rho /(3H^2)$ and $w=P/\rho$, in units $c=1$. The Friedmann equation, Friedmann acceleration equation and cosmological perfect-fluid continuity equation giveFor constant $w$, linearizing at $\Omega =1$ gives $\Omega -1\propto a^{1+3w}$. For variable $w$, replace the power by $\exp \int (1+3w(a))d\log a$. This describes the Flatness problem for decelerating expansion, away from epochs at which $H=0$.

At Hubble parameter $H$, the critical energy density is

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.

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 a homogeneous canonical scalar field, the cosmological perfect-fluid continuity equation reduces to $\dot{\varphi }(\ddot{\varphi }+3H\dot{\varphi }+V^{\prime })=0$. It implies the inflaton equation of motion wherever $\dot{\varphi }\ne 0$, and by continuity at isolated turning points. It cannot by itself exclude a frozen field on a time interval. Indeed, choose $\varphi ={\varphi }_0$ with $V({\varphi }_0)>0$ but $V^{\prime }({\varphi }_0)\ne 0$; a de Sitter scale factor in flat slicing with $H^2=V({\varphi }_0)/(3M_{\mathrm{Pl}}^2)$ satisfies the homogeneous Friedmann equation and Friedmann acceleration equation, but not the scalar equation. Varying the inflaton action in an expanding universe supplies the missing equation also for a frozen field, requiring $V^{\prime }({\varphi }_0)=0$.

In a flat matter-dominated era with $a\propto {\tau }^2$, one has $\mathcal{H}=2/\tau$ andThe growing mode is proportional to the scale factor.

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.

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 $H^2=8\pi G\rho /3$.