Cosmological distance measure

A cosmological distance measure converts an observed redshift, angle, flux, or travel time into a distance in an expanding spacetime. Different measures answer different operational questions and therefore need not agree numerically.

The look-back time is the cosmic-time interval between emission at redshift $z$ and reception today. Differentiating $1+z=R_0/R(t)$ gives $dt/dz=-1/[(1+z)H]$ and hence the displayed integral. The radial comoving-distance integral instead has denominator $H$ alone; elapsed time and present spatial separation answer different operational questions.

A source-frame transverse physical speed $v$ produces an observed angular rate $\dot{\theta }$. cosmological time dilation converts source time to observer time, so this operational distance equals $(1+z)$ times the angular diameter distance. In an FLRW geometry it is the transverse comoving distance $R_{0}S(\chi )$. It should be derived with the time convention stated, rather than interpreting the source speed and angular rate as measurements made by the same clock.

In metric light propagation with photon conservation and the usual reciprocity assumptions, luminosity distance and angular diameter distance satisfy $D_L=(1+z{)}^2{D}_A$. Comparing received flux with apparent angular area then gives the bolometric cosmological surface-brightness dimming factor $(1+z{)}^{-4}$. Absorption, photon conversion or an inconsistent definition of emitted/observed bands must be treated separately.

The distance that converts a small angular separation into a transverse comoving distance at a specified cosmological redshift. It equals the radial comoving distance in spatially flat geometry; cosmological spatial curvature changes that equality. A comoving acoustic ruler has angular size $r_d/D_M$.

Geometrical distances to nearby calibrators determine the luminosities of indicators that can be observed farther away. Those indicators calibrate still more luminous indicators in turn. A ladder may connect parallax or maser distances to Cepheid variables and then to Type Ia supernovae. Each step propagates its calibration uncertainty to the final distance scale and the inferred Hubble constant.

An orbit model for a resolved maser disk combines angular positions, Doppler velocities, accelerations and sometimes proper motions to infer physical orbital dimensions and distance. For an ideal matched circular radius, $R=v^2/a_{\mathrm{orb}}$ and $d=R/\theta$; velocities and accelerations of unrelated maser spots cannot simply be inserted into that expression. Warps, inclination, eccentricity and feature identification enter real disk fits.^[ref]

The termination of the first-ascent red giant luminosity function near helium ignition provides a calibrated population-II distance indicator. In suitable passbands the tip luminosity varies relatively weakly over a useful old, metal-poor population range. Colour/composition corrections, foreground extinction, photometric completeness and contamination by other evolved stars must be controlled; an arbitrary brightest red star is not a reliable tip measurement.

Compare the extinction-corrected apparent magnitudes of a cluster's main sequence with calibrated absolute magnitudes at matching colour or stellar parameters. The magnitude displacement estimates its distance modulus. Age, metallicity, helium abundance, reddening and unresolved binaries can move or broaden the sequence, so a colour-magnitude displacement is not uniquely a distance change.

The angular diameter distance is defined by $d_A=x/\theta$ for a small transverse physical size $x$ subtending angle $\theta$. In a spatially flat Friedmann-Lemaître-Robertson-Walker metric,where $\chi$ is the comoving radial distance.

Normalize the present scale factor to one and assume dust plus curvature with no cosmological constant. Then $H^2=H_0^2[{\Omega }_M{a}^{-3}+(1-{\Omega }_M)a^{-2}]$ and $D_C=(c/H_0){\int }_{1/(1+z)}^{1}da/[\sqrt{a}\sqrt{{\Omega }_M+(1-{\Omega }_M)a}]$. For positive ${\Omega }_K=1-{\Omega }_M$, the transverse comoving distance is $S_k(D_C)=c\sinh (H_0\sqrt{{\Omega }_K}{D}_C/c)/(H_0\sqrt{{\Omega }_K})$. At fixed radial distance it exceeds the flat transverse distance. At fixed redshift the radial distance must also be calculated from the expansion history. The empty limit is $D_C=(c/H_0)\ln (1+z)$, while the flat dust limit is $2c[1-(1+z{)}^{-1/2}]/H_0$.

For $R=A{t}^p$, $0<p<1$, integrate the radial photon path from emission to reception and multiply by the scale factor at emission. This gives the displayed angular diameter distance. The arbitrary normalization $A$ cancels, and the distance is not obtained by multiplying light speed by elapsed cosmic time.

In an Einstein-de Sitter universe, $H(z)=H_0(1+z{)}^{3/2}$ and the radial comoving distance is $2H_0^{-1}[1-(1+z{)}^{-1/2}]$. A physical Hubble radius $H(z{)}^{-1}$ has comoving size $(1+z)/H(z)$. Dividing by the radial distance gives the displayed small angle, in radians. A Hubble diameter has twice this angle; the matter-era particle-horizon radius also equals twice the Hubble radius. These distinctions matter when comparing approximate CMB angular scales.

In a standard expanding cosmology, $d_A(z)$ initially increases from zero but eventually decreases because the factor $(1+z{)}^{-1}$ dominates the bounded comoving distance. Beyond its maximum, a fixed physical object subtends a larger angle at a larger redshift.

For a flat universe with a single constant equation-of-state parameter $w>-1$, $H(z)=H_0(1+z{)}^{\alpha }$. Its angular diameter distance is $H_0^{-1}[(1+z{)}^{-\alpha }-(1+z{)}^{-1}]/(1-\alpha )$, with logarithmic limit $H_0^{-1}\log (1+z)/(1+z)$ at $\alpha =1$. Differentiation gives the displayed unique maximum, interpreted as $z_m=e-1$ at $\alpha =1$. The distance vanishes at both endpoints $z=0$ and $z\to \infty$. As $w\downarrow -1$, the maximum moves to infinite cosmological redshift.

A cosmological standard ruler is a feature with a fixed known or redshift-independent physical or comoving size. Comparing its observed angular and radial extents constrains the expansion history even when its absolute size is treated as a nuisance parameter.

Baryon acoustic oscillations are the relic spatial scale of sound waves in the pre-recombination photon-baryon fluid. The scale appears both as a preferred separation in galaxy clustering and as oscillatory structure in the cosmological density power spectrum.

The epoch at which momentum exchange with photons ceases to drag baryons efficiently. The sound horizon at this epoch calibrates the late-time baryon acoustic oscillation ruler. It is related to, but not identical with, photon last scattering.

For a comoving BAO scale ${\chi }_d$ perpendicular to the line of sight in a spatially flat universe,Its redshift dependence constrains the shape of the distance--redshift relation.

For a small comoving BAO separation parallel to the line of sight,in units with $c=1$. Combining this with a transverse measurement eliminates the unknown ruler size.

The matter power spectrum contains low-amplitude oscillatory features because baryons preserve their pre-recombination acoustic phase and later transfer part of it gravitationally to the combined matter distribution.

The Alcock-Paczyński test compares radial and transverse distance conversion using a feature that is statistically isotropic in comoving space.

The Alcock-Paczyński parameter compares radial and transverse conversions between observed separations and comoving distances at one cosmological redshift. For a small intrinsically isotropic feature of comoving size $r$, the radial and transverse relations areHere $D_M$ is the transverse comoving distance, $H$ the Hubble parameter, and the angle $\Delta \theta$ is in radians. Eliminating the unknown ruler gives the dimensionless combinationwhere $D_A$ is the angular diameter distance. In spatially flat geometry $D_M=\chi$; the earlier expression $H\chi$ additionally uses units with $c=1$. For nonzero cosmological spatial curvature, the radial comoving distance cannot simply replace $D_M$.^[ref]

This ratio is useful because an absolute ruler calibration cancels. Its inference still assumes statistical isotropy of the selected feature. Peculiar velocities produce redshift-space distortions, so an observed anisotropy must be modeled jointly with these effects before being interpreted as an incorrect distance conversion.