Luminosity distance

Luminosity distance is defined by $F=L/(4\pi d_L^2)$ and includes geometric dilution, photon redshift, and arrival-rate dilation.

For one conserved constant-$w$ perfect fluid in a flat expanding FLRW metric, $H(z)=H_0(1+z{)}^{3(1+w)/2}$. Integrating the radial null geodesic giveswhen $w\ne -1/3$. Its continuous coasting limit is $D_L=c(1+z)\log (1+z)/H_0$. Expansion gives $D_L=(c/H_0)[z+(1-3w)z^2/4+O(z^3)]$, agreeing with the deceleration parameter $q_0=(1+3w)/2$. Curvature, additional independently evolving fluids or a varying $w$ change this single-fluid expression.

At fixed positive redshift, intrinsic luminosity and present Hubble constant, a flat dust-vacuum model with $0<{\Omega }_{\Lambda 0}<1$ has luminosity distance between those of an Einstein-de Sitter universe and pure positive vacuum. The Hubble parameter of the dust-only model is largest at every positive integration redshift; that of pure vacuum is smallest. Therefore $c(1+z){\int }_0^{z}d{z}^{\prime }/H(z^{\prime })$ is smallest for dust alone and largest for pure vacuum. Received radiative flux has the reverse ordering. This compares physical distances rather than arbitrarily normalized comoving coordinates.

In curvature coordinates the present sphere area is $4\pi (R_{0}r{)}^2$. Its transverse area distance is $R_{0}r$, whereas radial proper distance is $R_0{\int }_0^{r}dr/\sqrt{1-K{r}^2}$. Confusing them invalidates a flat-space distance identity in curved geometry.

In an Einstein-de Sitter universe normalized by $a_0=1$, the Friedmann equation gives $H(z)=H_0(1+z{)}^{3/2}$. Integrating the radial null geodesic gives $r(z)=c{\int }_0^{z}d{z}^{\prime }/H(z^{\prime })=2(c/H_0)[1-(1+z{)}^{-1/2}]$. The luminosity distance is $(1+z)r(z)$, yielding the displayed expression. Near the observer it is $(c/H_0)[z+z^2/4+O(z^3)]$; the limiting radial comoving distance is finite even though the luminosity distance diverges.

For an open dust-only Friedmann-Lemaître-Robertson-Walker metric with zero cosmological constant and $0<\Omega <1$, $H(z)=H_0(1+z)\sqrt{1+\Omega z}$. The present radial cosmological proper distance is $d_S={\int }_0^{z}d{z}^{\prime }/H(z^{\prime })$, and the areal radial coordinate satisfies $a_{0}r=\sinh [H_0\sqrt{1-\Omega }{d}_S]/[H_0\sqrt{1-\Omega }]$. Integrating with $v=\sqrt{1+\Omega z}$ gives the displayed relation. It has $r\sim z/(a_0{H}_0)$ near the observer and $r\to 2/(a_0{H}_0\Omega )$ as $z\to \infty$. The luminosity distance is $(1+z)a_{0}r$, not the radial cosmological proper distance.

Type Ia supernova cosmology uses standardized supernova luminosities to measure the shape of the luminosity-distance--redshift relation. In a spatially flat universe,An unknown common absolute luminosity is degenerate with the overall distance scale but does not erase the redshift dependence.

The useful observable is a relative luminosity distance at measured redshift, rather than an object's brightness alone. Spectroscopy supplies $z={\lambda }_{\mathrm{obs}}/{\lambda }_{\mathrm{rest}}-1$; repeated photometry supplies a light curve. A Type Ia supernova is a standardizable candle: a fitted relation between light curve shape and luminosity replaces the incorrect assumption that every explosion has identical peak absolute magnitude. Nearby objects establish the relation and its common intercept; distant objects test its continuation and the shape of the Hubble diagram.

The stretch method rescales the light curve's time axis by $s(1+z)$ and corrects peak apparent magnitude by $\alpha (s-1)$, fitting $\alpha$ jointly with cosmology.^[ref]

Here is a deliberately stipulated local model that exposes the signs. Let $t_{\mathrm{obs}}$ be observer time in days, $t_{\max }$ the observed peak time and $T(u)$ a dimensionless template with $T(0)=1$. In a single photometric passband, writeHere $u$ is rest-frame time in days; the template's argument carries those units. For illustration, assign the intrinsic relation $M_B(s)=M_{B,0}-\alpha (s-1)$ with $\alpha >0$: broader events are intrinsically brighter. Define the sign of the K correction by $m_{\mathrm{obs}}=M_B(s)+\mu +A_B+K$, where $A_B$ is interstellar extinction in magnitudes. ThenRemoving attenuation makes the magnitude smaller; standardizing a broad, intrinsically bright event makes it larger. The fitted cosmological time dilation must be removed before comparing intrinsic durations. This schematic factorization explains the bookkeeping; an actual multicolour fit must also model changing intrinsic colours and correlated uncertainties.

For two equally standardized sources, cancellation of their common absolute magnitude gives an especially clean result:A stipulated residual of $+0.20$ mag therefore means a distance ratio $10^{0.04}=1.09648$ and a radiative flux ratio $10^{-0.08}=0.83176$ relative to the comparison prediction, provided luminosity and attenuation have been correctly standardized. This is a conversion of a hypothetical residual, not a measured survey value.

The High-Z Supernova Search Team analysis fitted multicolour light curves and interstellar extinction, and compared a second decline-rate method. Similar local and distant spectroscopy supported, but did not establish, population stability. Nearly colourless attenuation remained a qualified alternative.^[ref]

The Supernova Cosmology Project compared low- and high-redshift colours, removed red outliers in its primary fit, and tested Malmquist bias with artificial-source recovery.^[ref]

The inference can be organized as a chain of independently testable failure modes. A magnitude zero point offset between nearby and distant observations imitates a distance modulus offset; a common offset instead changes the intercept. An erroneous photometric passband response changes a K correction in a redshift-dependent way because different emitted wavelengths cross the detector band. Residual host-galaxy light changes the recovered radiative flux. Color excess constrains wavelength-dependent interstellar extinction only after intrinsic colour is modeled; grey attenuation is not bounded by colour alone. Changes in stellar population or metallicity could shift standardized absolute magnitude without large changes in the chosen shape parameter. These are logical degeneracies, so increasing the sample size alone cannot remove them.

A useful control strategy therefore compares subsets with different host environments, fits consistent rest-frame photometric passbands, and checks artificial-source recovery through the same image processing and selection. A Malmquist bias tends to select brighter detectable sources, but the relative bias between the nearby and distant samples is what matters. Peculiar velocity mainly perturbs the nearby redshift anchor; weak gravitational lensing changes individual brightnesses and the selection distribution. Neither can be replaced by an arbitrary extra scatter term when its mean or selection dependence is uncertain.

A Hubble diagram plots a distance indicator, such as standardized supernova apparent magnitude or distance modulus, against redshift. The low-redshift relation sets a distance-scale intercept, while curvature of the relation at larger redshift tests the expansion history through luminosity distance. A common unknown absolute magnitude remains degenerate with the Hubble constant; the diagram's redshift-dependent shape can still constrain other cosmological parameters.

In Type Ia supernova cosmology, the predicted apparent magnitude depends on the common absolute magnitude $M_0$ and the Hubble constant only through $\beta =M_0-5{\log }_{10}(H_0/H_{\mathrm{ref}})$, after choosing a fixed reference $H_{\mathrm{ref}}>0$. A flat improper prior on $M_0$ makes the likelihood integrated over $M_0$ independent of $H_0$, by translation of the integration variable. If the other prior factors are independent and the posterior is proper, the marginal posterior of $H_0$ equals its prior. An external calibration of $M_0$ can break this lack of identifiability.

Expansion reduces bolometric flux by one factor of $1+z$ from photon energy and one from arrival rate, in addition to inverse-area dilution.

A comoving source's spectral luminosity $L_{{\nu }_e}$ specifies emitted power per unit emitted frequency. The cosmological redshift reduces photon energy and arrival rate by two factors of $1+z$, but also narrows the received frequency interval by one factor. Consequently its observed spectral flux density is $L_{{\nu }_e}/[4\pi a_0^2{r}^2(1+z)]$, with ${\nu }_e=(1+z){\nu }_0$. Substituting luminosity distance $d_L=(1+z)a_{0}r$ gives the displayed relation. Integrating over received frequency recovers the bolometric radiative flux $L/(4\pi d_L^2)$. A fixed-frequency comparison must evaluate the source spectrum at the shifted emitted frequency.