Scale factor (cosmology, $a(t)$)

The scale factor $a(t)>0$ measures how spatial lengths in an FLRW metric change with cosmic time $t$. Choose $a$ to be a dimensionless quantity with $a(t_0)=1$ at a reference epoch, usually today. For two comoving observers, their fixed radial comoving distance $\chi$ and their proper distance in cosmology $d(t)$ satisfyThe proper distance in cosmology is measured along a spatial slice at one cosmic time; it is not the distance that a travelling photon covers between emission and reception. Only ratios of the scale factor have a normalization-independent meaning: replacing $a$ by $Ca$ and $\chi$ by $\chi /C$ leaves $d$ unchanged.

For example, take $\chi =100\mathrm{Mpc}$, where $\mathrm{Mpc}$ means a million parsecs. At $a=0.5,1,1.5$, the same pair of comoving observers has proper distance in cosmology $50,100,150\mathrm{Mpc}$ respectively. Every such separation changes by the same factor, although more distant pairs gain more distance. The illustration treats the marked objects as ideal comoving observers; internal motions in galaxies need not follow this background scaling.

The beauty of the scale factor is that one function describes the changing spatial ruler for every comoving observer. Spatial homogeneity gives no preferred centre of cosmic expansion: in a flat spatial patch, relative positions obey $\mathbf{r}-{\mathbf{r}}_O=a(\mathbf{x}-{\mathbf{x}}_O)$, so recentering on any comoving observer preserves Hubble's law. The evolution of the scale factor is determined by the Friedmann equations, while its ratio between emission and reception determines cosmological redshift.

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

Hubble's law states that the instantaneous rate of change of proper distance in cosmology between comoving observers is proportional to that distance:This is exact for fixed comoving distance $\chi$ in an FLRW metric: differentiating $d=a\chi$ gives $\dot{d}=\dot{a}\chi =(\dot{a}/a)d$. The dot uses cosmic time, and both distances are compared on spatial slices at the same cosmic time. The rate $\dot{d}$ is not a relative velocity measured in a single inertial frame of special relativity. Thus the instantaneous relation is not an exact observed redshift-distance formula. The associated Hubble flow describes the ideal expansion; peculiar velocities describe departures from it.

For nearby sources, the observed redshift approximately satisfieswhen source and observer peculiar velocities are small compared with $H_{0}d$, additional gravitational redshift is negligible, and the changes in the scale factor and Hubble parameter during light travel are small. Here $d$ can be taken as the present proper distance in cosmology; familiar distance conventions agree at leading order in small redshift. Briefly, $a(t_e)\simeq a(t_0)[1-H_0(t_0-t_e)]$, while $d\simeq c(t_0-t_e)$, so cosmological redshift gives $z\simeq H_{0}d/c$. At greater redshift, use an expansion history and an explicit distance convention instead.

Stipulate the illustrative Hubble constant $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ and speed of light $c=299792.458\mathrm{km}{\mathrm{s}}^{-1}$. A $\mathrm{Mpc}$ is a million parsecs. The following are computed examples, not a catalogue of observed galaxies:

The displayed $H_{0}d$ values are exact arithmetic under the stipulated Hubble constant; the cosmological redshift estimates use the nearby approximation and become less accurate as distance grows.

Hubble's law turns a nearby observed redshift into an approximate distance indicator and makes the uniformity of cosmic expansion tangible: twice the proper distance in cosmology means twice the instantaneous separation rate. Recentring the ideal Hubble flow on another comoving observer preserves the same proportionality, as illustrated under scale factor, so recession in every direction does not select a central observer.

In his 1929 PNAS paper, Edwin Hubble presented a relation between the distances and radial velocities of extragalactic objects. His fits gave coefficients of $465$ and $513\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ for 24 individual objects and nine groups respectively, and he used about $500\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ as a rounded representative value. Those historical fits used the paper's distance estimates; they are distinct from the stipulated modern-scale illustration above. ^[ref].

The Hubble parameter is the instantaneous fractional rate of cosmic expansion:where the dot is a derivative with respect to cosmic time. Thus a short interval $\Delta t$ changes any fixed-comoving proper distance in cosmology by the fraction $\Delta d/d\simeq H\Delta t$. Its dimensions are inverse time; $H>0$ describes expansion and $H<0$ contraction. The Hubble parameter identity gives $\dot{H}+H^2=\ddot{a}/a$.

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.

For the normal flow with zero shift vector, $a_{\mathrm{loc}}=(\det h{)}^{1/6}$ obeys $H_{\mathrm{loc}}=N^{-1}{\partial }_t\log a_{\mathrm{loc}}=-K/3$ in the negative extrinsic curvature of a spatial hypersurface convention. The expansion scalar is $\theta =-K=3H_{\mathrm{loc}}$.

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 physical Hubble radius is $c/H$, the distance at which the instantaneous Hubble law recession speed equals $c$. Its comoving counterpart is $c/(aH)$. Neither radius is generally the same as a particle horizon or a causal boundary: those depend on integrals of the past or future expansion history.

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 $H_0^{-1}$.

Differentiating $H=\dot{a}/a$ gives

The comoving Hubble radius is $(aH{)}^{-1}$. It grows during ordinary decelerating expansion and shrinks during sufficiently accelerated inflation, allowing modes to cross from sub-Hubble to super-Hubble scales.

With $\mathcal{H}=aH=\dot{a}$, the Friedmann acceleration equation givesThe coefficient is positive when $a>0$, $\rho >0$ and $H\ne 0$. Thus growth of the comoving Hubble radius and linear growth of departures from flatness have the same sign criterion in an expanding universe. This local criterion is distinct from a global particle horizon, which depends on the whole past history.

In a spatially flat universe containing components with constant $w_i$,

For pressureless matter and dark energy with equation of state $w(z)$,

Hubble friction is the damping term proportional to the Hubble parameter in a field or fluid equation in an expanding universe. A homogeneous scalar field, for example, has the term $3H\dot{\varphi }$ in its equation of motion.

The homogeneous expansion velocity is proportional to physical position:

The velocity-distance relation is Hubble's law; a peculiar velocity is a departure from this homogeneous flow.

A comoving coordinate remains fixed for an observer following the homogeneous cosmological expansion.

A comoving coordinate provides labels for the comoving and proper distances comparison; proper separation additionally contains the cosmological scale factor.

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.

A comoving volume has fixed boundaries in comoving coordinates. Its physical volume grows as the cube of the scale factor. In a homogeneous adiabatic gas, cosmological entropy conservation makes $sV$ constant.

On a present FLRW spatial slice, a shell has volume $R_0^3{S}^2(\chi )d\chi d\Omega$. The radial light-ray relation gives $d\chi /dz=c/(R_{0}H)$, yielding the displayed expression in terms of transverse comoving distance. At emission the proper volume is smaller by $(1+z{)}^3$. For a conserved comoving population its proper density is greater by the same factor, so either epoch gives the same count.

Comoving number density is object count per comoving volume. With scale factor normalized to unity today, proper density is $n_p=a^{-3}{n}_c$. A conserved population moving with homogeneous expansion has constant $n_c$ despite declining proper density. Formation, mergers or changing sample selection can make $n_c$ evolve.

For constant comoving number density and complete selection in a spatially flat Einstein-de Sitter universe, integrate the present-epoch shell volume up to the population's formation redshift. The radial distance is $D_M=2c[1-(1+z{)}^{-1/2}]/H_0$, so the count per unit solid angle is $n_0{D}_M^3(z_f)/3$. It tends to a finite value as $z_f$ tends to infinity because this model's present comoving particle-horizon radius is $2c/H_0$. Formation, mergers or an incomplete observed sample change this ideal count.

For proper number density $n(t)$ in an FLRW metric $d{s}^2=d{t}^2-a^2[d{r}^2/(1-k{r}^2)+r^{2}d{\Omega }^2]$, a radial shell has proper volume $4\pi a^3{r}^2|dr|/\sqrt{1-k{r}^2}$. A radial null geodesic obeys $|dr|=\sqrt{1-k{r}^2}dt/a$. Multiplication gives the displayed shell count on a local radial branch of the past light cone. Spatially uniform number density need not be time independent or have conserved comoving number density.

Peculiar velocity is motion relative to the homogeneous Hubble flow. In comoving coordinates,where $\mathbf{u}$ is physical velocity and $\mathbf{v}$ is peculiar velocity.

A freely moving massive test particle in an FLRW metric has physical momentum $p=m\gamma v$ relative to a comoving observer, where $v$ is its peculiar velocity. For a radial geodesic, the geodesic Lagrangian gives $C=a^{2}d\chi /d\tau$ constant, and the local radial momentum is $p=mad\chi /d\tau =mC/a$. The proper-time normalization gives $dt/d\tau =\sqrt{1+C^2/(a^2{c}^2)}$, soConsequently nonrelativistic peculiar velocity scales as $a^{-1}$, while the exact relativistic law keeps $|v|<c$. The momentum magnitude of a free photon also scales as $a^{-1}$, giving cosmological redshift through $E=cp$. These statements concern force-free propagation; a bound system need not follow the cosmological expansion.

With $\delta (\mathbf{x})=\int {\delta }_k{e}^{-i\mathbf{k}\cdot \mathbf{x}}{d}^{3}k/(2\pi {)}^3$ and logarithmic growing-mode rate $f=d\log D_{+}/d\log a$, the linearized cosmological continuity equation gives ${\mathbf{v}}_k=-iaHf{\delta }_k\mathbf{k}/k^2$. Apply the Rayleigh plane-wave expansion to its velocity potential and differentiate radially to obtain the displayed expression. The factor $aH$ supplies the velocity dimensions. At the present epoch $U/H_0$ has the same formula with $aH$ removed; the commonly used $f\simeq {\Omega }_m^{0.6}$ is an approximation, not an exact general growth law.

For a fluid in an expanding universe, the peculiar-velocity equation is the Euler equations for an inviscid fluid after decomposing physical velocity into Hubble flow and peculiar velocity. It contains the Hubble-drag term $H\mathbf{v}$ and force $-\nabla \Phi /a$ from the peculiar gravitational potential.

The peculiar-velocity divergence is $\theta =\nabla \cdot \mathbf{v}$. For irrotational linear perturbations it captures the scalar part of the peculiar velocity field.

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.

A redshift estimated from multi-band fluxes or an inferred spectral energy distribution rather than directly resolved spectral-line wavelengths. calibration, templates, dust and population differences can produce biases and catastrophic outliers. In lensing or clustering analyses, the inferred source redshift distribution affects the geometric kernel as well as assignment to redshift bins.

Let the scale factor grow by $f$ from an initially thermal photon distribution. geodesic redshift gives ${\nu }_i=f\nu$, and conserving photons in corresponding comoving frequency bins gives $n_{\nu }(t)=f^{-2}{n}_{f\nu }(t_i)$. Inserting Planck's law leaves the same spectral form at temperature $T_i/f$. Total photon-number conservation alone would not prove that the entire radiation spectrum stays Planckian; frequency redshift and collisionless propagation are also required. Absorption, injection or spectral mixing can invalidate this one-temperature argument.

For a comoving emitter and receiver, photon energy falls as $1/(1+z)$ and arrival intervals grow by $1+z$. Their combination reduces bolometric flux by $(1+z{)}^2$ in addition to the area dilution.

A cosmological redshift relates a particular emission event to a particular reception event along a null geodesic. Two sources' redshifts on Earth's past light cone do not specify the light exchanged between those sources. If a photon from B passes through A before reaching Earth, and A's comparison emission lies at that passage event, then $1+z_{BA}=(1+z_B)/(1+z_A)$. Without this causal and temporal specification the ratio need not give the redshift measured in A.

Frequency ratios multiply when a photon is transformed first between a source and its local comoving frame, then propagated through an expanding universe. Thus the kinematic redshift and cosmological redshift factors multiply. Their redshifts themselves add only to first order when both are small.

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 $1/(1+z)$.

A local redshift spacing between absorbers corresponds to proper path length $c\Delta z/[H(z)(1+z)]$. Using the same rest-frame transition for all lines gives $\Delta z=\Delta \lambda /{\lambda }_{\mathrm{rest}}$. This conversion assumes cosmological redshift dominates the line spacing and uses a local Hubble-rate approximation.

Redshift drift is the change in a comoving source's cosmological redshift with observer time. It directly probes $\dot{z}=(1+z)H_0-H(z)$.

The Sandage-Loeb test seeks the spectroscopic velocity drift of stable cosmological absorption features over an observer-time baseline. Many Lyman-alpha forest lines in bright distant quasars can be combined, but wavelength calibration, instrumental stability and peculiar accelerations must be controlled below the tiny cosmological signal. The absorption-template proposal is developed in the original 1998 redshift-drift study.

For a transition with fixed rest wavelength, ${\lambda }_{\mathrm{obs}}=(1+z){\lambda }_{\mathrm{rest}}$. The apparent spectroscopic velocity shift is $dv=cd{\lambda }_{\mathrm{obs}}/{\lambda }_{\mathrm{obs}}=cdz/(1+z)$. It is a convenient wavelength-shift unit, not necessarily a change in a source's peculiar velocity. Omitting the denominator $1+z$ overestimates the signal at high redshift.

For positive matter and vacuum fractions with ${\Omega }_m+{\Omega }_{\Lambda }=1$, redshift drift vanishes when $s^2={\Omega }_m{s}^3+{\Omega }_{\Lambda }$, $s=1+z$. Factoring out the trivial observer root $s=1$ leaves ${\Omega }_m{s}^2-{\Omega }_{\Lambda }s-{\Omega }_{\Lambda }=0$, giving the displayed positive root of a polynomial. It lies at a positive redshift exactly when ${\Omega }_m<2/3$. It is independent of the Hubble constant at fixed density fractions and is not the redshift of the acceleration-to-deceleration transition.

Differentiating $1+z=a_0/a(t)$ gives

Cosmic chronometers infer the Hubble parameter from differential ages of passively evolving galaxies:They probe $H(z)$ locally in redshift rather than through a distance integral.

Observed time intervals from a comoving source are stretched by the same factor $1+z$ as photon wavelengths.

After photon decoupling in an expanding universe, every frequency scales as $\nu \propto a^{-1}$. Therefore a Planck occupation factor retains its thermal form when its temperature is assigned the same scaling,