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An accelerating universe has $\ddot{a}>0$, equivalently $q<0$ while expansion continues.

Here $a$ is the scale factor and $q=-a\ddot{a}/{\dot{a}}^2$ the deceleration parameter. Accelerating expansion means $\dot{a}>0$ and $\ddot{a}>0$; the Hubble parameter $H=\dot{a}/a$ can still decrease. The observational challenge is to reconstruct that history from a past light cone, rather than watch a distant galaxy speed up over the observing campaign.

Match flat Friedmann equations histories at scale factor $R_1$ and Hubble parameter $H_1$, and give them identical subsequent evolution. Below $R_1$, constant vacuum energy gives $H=H_1$, while a radiation-dominated universe gives $H=H_1(R_1/R{)}^2$. From $ϵR_1$ to $R_1$ the former takes longer by $H_1^{-1}{\int }_{ϵ}^1(1/x-x)dx>0$. Its duration diverges as $ϵ\to 0$; the radiation duration tends to $1/(2H_1)$. A finite period of cosmic inflation requires a finite initial state and a specified matching before such age comparisons are meaningful.

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.

Baryon-number washout is erasure of a baryon asymmetry by reactions that violate baryon number. Inverse decays and scattering can drive the plasma toward its zero-asymmetry thermal equilibrium. The relevant comparison is their rate against the Hubble parameter, not just the decay width that initially produced the asymmetry.

For constant positive Hubble parameter $\mu$ and a four-thirds polytropic equation of state, a linear cosmological density perturbation obeys $\ddot{\delta }+2\mu \dot{\delta }+{\eta }^2{e}^{-3\mu t}\delta =0$. Put $\delta =e^{-\mu t}y$ and $x=2|\eta |e^{-3\mu t/2}/(3\mu )$. The resulting Bessel differential equation has order $2/3$; use ordinary Bessel functions for ${\eta }^2>0$ and modified Bessel functions for ${\eta }^2<0$. Their small-$x$ expansions give one constant limiting mode and one $e^{-2\mu t}$ mode. The constant limit need not exceed the initial amplitude without an initial-velocity condition.

A Big Rip is a finite-future-time cosmological singularity at which the scale factor, Hubble parameter, and phantom energy density diverge.

A comoving observer has constant comoving coordinates in a homogeneous expanding universe. Its physical separation from another comoving observer grows in proportion to the scale factor, giving relative recession governed by the Hubble parameter. Light exchanged between these observers undergoes cosmological redshift.

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.

For conformal time $d\tau =dt/a$, the conformal Hubble parameter is $\mathcal{H}=a^{\prime }/a=aH=\dot{a}$, where primes and dots denote conformal and cosmic time derivatives respectively. Its inverse, in units $c=1$, is the comoving Hubble radius. It differs from the ordinary Hubble parameter by one factor of the scale factor.

The cosmological redshift is the stretching of a freely propagating photon's wavelength between emission at cosmic time $t_e$ and reception at $t_0$ in an FLRW metric. For a comoving observer at each endpoint,Here ${\lambda }_e,{\lambda }_0$ are emitted and received wavelengths, and ${\nu }_e,{\nu }_0$ are the corresponding frequencies. A positive cosmological redshift means that the scale factor grew while the photon travelled. The name describes a shift toward longer wavelengths; it applies to light beyond the visible range as well.

For a computed example, let the scale factor double from $a(t_e)=0.5$ to $a(t_0)=1$. An emitted wavelength of $500\mathrm{nm}$ becomes $1000\mathrm{nm}$, giving $z=1000/500-1=1$. The received frequency is half the emitted frequency. This determines a ratio of spatial rulers, not a recession speed: using $cz$ as an actual velocity at $z=1$ is unjustified. The wavelength panel in the illustration under Hubble's law shows this doubling.

A short derivation follows successive wave crests along a radial null geodesic in FLRW spacetime. Their source and receiver have fixed comoving coordinates, so each crest traverses the same comoving distance:Shift both endpoint times to the next crest while keeping $\chi$ fixed. To first order in the very short periods $\delta t_e,\delta t_0$, this gives $\delta t_0/a(t_0)=\delta t_e/a(t_e)$. Locally, wavelength equals the speed of light times the crest period, yielding the displayed cosmological redshift relation. These periods are much shorter than the timescale on which the scale factor changes.

The relation isolates the smooth expansion contribution. An observed redshift can also contain a Doppler effect from source or observer peculiar velocity and a gravitational redshift. See combined kinematic and cosmological redshift for the multiplicative combination of endpoint motion and expansion. Treating every observed redshift as pure cosmological redshift can therefore misassign a nearby source's distance.

The useful insight is that a received wavelength ratio records how much the scale factor changed during the photon's journey. Turning that ratio into a distance requires an expansion history, through the Hubble parameter, and a specified convention such as luminosity distance or proper distance in cosmology; the ratio alone supplies neither.

Cosmic expansion stretches an emitted or absorbed spectral feature's wavelength by $1+z$. A small observed wavelength interval at fixed transition wavelength gives a redshift interval by division by the rest wavelength. Cosmological radial distances then require the Hubble parameter and, for proper distance at that epoch, an extra factor $1/(1+z)$.

Differentiating $H=\dot{a}/a$ gives $\dot{H}=\ddot{a}/a-H^2$, which proves the displayed identity. A falling Hubble parameter does not by itself imply deceleration: linear growth $a\propto t$ has $H=1/t$ but $q=0$.

At maximum expansion of a closed matter-dominated Friedmann solution, the Hubble parameter vanishes while the physical mass density stays positive and finite. Its critical density therefore vanishes and the cosmological density parameter diverges. This is a singularity of that ratio rather than a curvature singularity. In the cycloid variable $\theta$, ${\Omega }_M={\sec }^2(\theta /2)$, diverging at $\theta =\pi$ and approaching one at the Big Bang and Big Crunch ends.

Approximately constant positive Hubble parameter gives a nearly de Sitter spacetime expansion in flat slicing. Its scale factor can be approximated by $a=-1/(H\tau )$ with negative conformal time increasing toward zero. Slow variation of the expansion rate limits the accuracy of this approximation.

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.

Compare a flat background Friedmann equation with a nearly homogeneous curved region on a slicing of equal local Hubble parameter. To first order in its curvature parameter $K$, subtraction gives $\delta \rho =3K/(8\pi G{a}^2)$, or $\delta =K/(a^2{H}^2)$. Conserved curvature therefore gives $\delta \propto a^2$ in radiation domination and $\delta \propto a$ in matter domination. This construction specifies the slicing: two patches at equal proper time generally do not have the same Hubble parameter. For growing dust perturbations, the continuity equation gives $H_s-H=-\dot{\delta }/3$ to first order. These different time comparisons can share growth powers without sharing the same curvature-to-density coefficient.

Cosmic expansion is the increase of physical distances between comoving observers as the scale factor grows. Its instantaneous fractional rate is the Hubble parameter.

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.

During radiation domination, the weak conversion rate scales as $T^5$ whereas the Hubble parameter scales as $\sqrt{G{g}_{\ast }}{T}^2$. Their crossing fixes the cosmological weak freeze-out temperature. A faster expansion raises it, leaving a larger neutron-to-proton ratio, and shortens the interval for free-neutron decay after freeze-out. Both effects increase the helium-4 mass fraction if the extra component does not independently change reaction rates or inject decay products.

The Hubble constant is the present value $H_0=H(t_0)$ of the Hubble parameter. The word "constant" fixes the epoch; it does not mean $H(t)$ stays unchanged throughout cosmic time. It sets the present instantaneous proportionality in Hubble's law and the approximate nearby observed redshift-distance relation.

For illustration, stipulate $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$; this is an example input, not a claim about a measured consensus value. A million parsecs is $3.085677581\dots \times 10^{19}\mathrm{km}$, givingHere a year means $365.25$ days of $86400\mathrm{s}$, and $\mathrm{Gyr}$ denotes a billion such years. The parsec conversion follows ^[ref]. This Hubble time is a present expansion timescale; the age of an FLRW universe also depends on the earlier evolution of the scale factor and need not equal $H_0^{-1}$.