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Adiabatic initial conditions perturb all relevant local physical quantities along the same background time trajectory. For quantities with nonzero background time derivatives, their perturbations satisfy $\delta X/{\bar{X}}^{\prime }=\delta Y/{\bar{Y}}^{\prime }$. For separately conserved barotropic fluids, cosmological energy conservation then givesFor example, pressureless matter and radiation in cosmology obey ${\delta }_m=3{\delta }_{\gamma }/4$. Their relative particle-number ratio is unperturbed at leading order.

The density condition tests relative entropy between components. In a general nonbarotropic component it does not by itself eliminate intrinsic entropy, measured by $\delta P_i-({\bar{P}}_i^{\prime }/{\bar{\rho }}_i^{\prime })\delta {\rho }_i$. The stronger statement that every physical variable follows the same time shift includes the pressure and any independent field-velocity data. A single scalar species alone does not make the density-ratio test sufficient, since that test is then vacuous.

For vanishing total non-adiabatic pressure perturbation and negligible regular gradient terms, uniform-density curvature is conserved on superhorizon scales.^[ref] Relating this to conserved comoving curvature perturbation also requires the appropriate regular adiabatic mode; a non-attractor phase such as ultra-slow-roll inflation needs separate analysis. The ordinary growing adiabatic photon mode selects coherent acoustic phases, explaining the adiabatic acoustic-peak spacing.

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 scalar perturbations in cosmic time with $d{s}^2=-d{t}^2+a^2({\delta }_{ij}+h_{ij})d{x}^{i}d{x}^j$, define the covariant fluid velocity potential $u$ by $\delta u_i={\partial }_{i}u$. The linear momentum density is $T^0{}_i=(\bar{\rho }+\bar{P}){\partial }_{i}u$. Normalize scalar anisotropic stress bywhere derivatives inside these parentheses are Cartesian comoving derivatives. Then ${\partial }_j{\pi }^j{}_i=2{\partial }_i{\nabla }^2\Pi /(3a^2)$. Spatial stress-energy conservation gives, for nonzero scalar Fourier modes,Writing ${\pi }^S=2\Pi /(3a^2)$ reproduces the form with ${\nabla }^2{\pi }^S$; the coefficient depends on that stress normalization. The time derivative acts on the entire momentum-density product, not only on $u$.

For pressureless matter with zero scalar stress, background $\dot{\bar{\rho }}+3H\bar{\rho }=0$ reduces the equation to $\dot{u}=0$. The physical peculiar velocity is $\mathbf{v}=\nabla u/a$ and therefore decays as $a^{-1}$. This checks the difference between covariant velocity potential and physical velocity and the expansion factor in the conservative equation.

Neglect radiation in cosmology and take pressureless matter plus a separately conserved dark energy component with $p_{\mathrm{DE}}=w{\rho }_{\mathrm{DE}}{c}^2$, where $\rho$ here denotes mass-equivalent energy density. The Friedmann acceleration equation givesThus $w<-1/3$ for a positive-density component is necessary for that component to accelerate expansion, but the total must also overcome pressureless matter. For a cosmological constant, $w=-1$ and $q_0={\Omega }_{m,0}/2-{\Omega }_{\Lambda ,0}$, so acceleration requires ${\Omega }_{\Lambda ,0}>{\Omega }_{m,0}/2$. Spatial curvature of an FLRW universe enters the distance relation but has no separate term in this acceleration criterion. A Hubble diagram alone does not identify a microscopic dark energy mechanism.

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

For pressureless matter, a positive cosmological constant, flat spatial geometry and negligible radiation, the continuity equation gives ${\rho }_m\propto a^{-3}$. Substitution in the Friedmann equation yields the displayed Hubble parameter, with ${\Omega }_{m0}+{\Omega }_{\Lambda 0}=1$. radiation must be restored at early epochs; the formula does not require matter to be the largest present-day density component.

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.

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 $c\eta$.

With nonnegative conserved pressureless matter and radiation in cosmology, at least one of which has positive density, 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.

The Friedmann equations, named for Alexander Friedmann, apply the Einstein field equations to an FLRW metric. Using natural units with $c=1$, energy density $\rho$, pressure $p$ and cosmological constant $\Lambda$, they readHere $a$ is the scale factor, $H=\dot{a}/a$ the Hubble parameter, and $k$ specifies spatial curvature. The cosmological continuity equation completes the fluid description. For a flat universe with $\Lambda =0$, pressureless matter gives $a\propto t^{2/3}$ and radiation in cosmology gives $a\propto t^{1/2}$.^[ref]

Assume spatial flatness, negligible radiation, separately conserved pressureless matter and dark energy, and $a_0=1$. Let $w(z)=P_{\mathrm{DE}}/{\rho }_{\mathrm{DE}}$ be the dark-energy equation-of-state parameter. The cosmological continuity equation givesCombining this with ${\rho }_m={\rho }_{m,0}(1+z{)}^3$ in the Friedmann equation yieldswith ${\Omega }_{m,0}+{\Omega }_{\mathrm{DE},0}=1$. For $w=-1$, the dark-energy factor is constant and this reduces to the matter-plus-cosmological constant model. Nonzero cosmological spatial curvature, radiation or energy exchange between the components requires additional terms or modified continuity equations.

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.