= Solution
The <Friedmann-Lemaître-Robertson-Walker metric> describes homogeneous, isotropic spatial slices. Here $t$ is <cosmic time>, equal to the <proper time> of a <comoving observer>; $R(t)$ is the length-valued <scale factor>; and $\chi,\theta,\phi$ are dimensionless comoving radial, polar and azimuthal coordinates. The <speed of light> is $c$, and
$$
S(\chi)=\begin{cases}\sin\chi&k=+1,\\\chi&k=0,\\\sinh\chi&k=-1,\end{cases}
$$
for positive, zero and negative <cosmological spatial curvature>. The convention used makes $ds^2$ a squared time interval, with signature $(+---)$; multiplying the metric by $c^2$ gives the more usual squared-length convention. At fixed $t$, radial proper length is $R(t)d\chi$, while a sphere has area $4\pi R^2(t)S^2(\chi)$.
<luminosity distance> is defined by $F=L/(4\pi d_L^2)$ for bolometric <radiative flux> $F$ and intrinsic <luminosity> $L$. Radial null propagation gives $d\chi=\pm c\,dt/R(t)$. Applying the same path integral to two successive wave crests gives $\Delta t_0/R_0=\Delta t_e/R_e$, hence $1+z=R_0/R_e$. Each <photon> loses an <energy> factor $1+z$, and their arrival rate loses another. For transparent propagation and isotropic emission,
$$
F=\frac{L}{4\pi R_0^2S^2(\chi)(1+z)^2},\qquad
\boxed{d_L=R_0S(\chi)(1+z)}.
$$
The two <redshift> factors have distinct physical origins; geometrical dilution alone is insufficient.
With $H=\dot R/R$, differentiating $1+z=R_0/R$ yields $dt/dz=-[(1+z)H(z)]^{-1}$. Therefore the radial <comoving coordinate> is
$$
\chi=\int_{t_e}^{t_0}\frac{c\,dt}{R(t)}
=\boxed{\frac c{R_0}\int_0^z\frac{dz'}{H(z')}}.
$$
This is independent of any particular matter model; the <transverse comoving distance> is $R_0S(\chi)$, which equals $R_0\chi$ only for flat space.
For <pressureless matter>, the <continuity equation> $\dot\rho_m+3H\rho_m=0$ integrates to $\rho_m=\rho_{m0}(R_0/R)^3=\rho_{m0}(1+z)^3$. The <cosmological constant> is kept separately from this matter density. In the flat <Friedmann equation>, define
$$
\Omega_{\Lambda0}=\frac{\Lambda}{3H_0^2},\qquad
\Omega_{m0}=\frac{8\pi G\rho_{m0}}{3H_0^2}=1-\Omega_{\Lambda0}.
$$
The <flat dust-vacuum Hubble parameter> is therefore
$$
\boxed{H(z)=H_0\sqrt{(1-\Omega_{\Lambda0})(1+z)^3+\Omega_{\Lambda0}}}.
$$
In the equation's convention $\rho_m$ is mass-equivalent <energy density> and $\Lambda$ has units $\mathrm{s^{-2}}$; the geometrical <cosmological constant> convention instead writes $c^2\Lambda_{\rm geom}$. The model contains dust and <vacuum energy> with negligible <radiation> over the relevant late epochs: “matter-dominated” here does not require matter to exceed <vacuum energy> today.
The <cosmic microwave background> supplies a standard ruler through its acoustic scale: the angular positions of the <Cosmic microwave background acoustic peaks> constrain <cosmological spatial curvature>, while their relative heights constrain matter and <baryon> densities. Together with independent distance information, these favour a nearly flat <Lambda-CDM model> containing substantial <cold dark matter> and <vacuum energy>. The <cosmic microwave background> alone has a geometrical degeneracy between parameters affecting distance to last scattering; its analysis also includes early <radiation>, which the late-time formula above omits. The nine-year <WMAP> results were already available as a December 2012 preprint.<Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results by C. L. Bennett and others>[Version 1 abstract: combined-data curvature constraint and consistency with the six-parameter Lambda-CDM model]{ref}
A standardized <Type Ia supernova> at appreciable <redshift> is fainter, hence farther in <luminosity distance>, than in a matter-only <Einstein-de Sitter universe> with the same local expansion scale. Fitting the supernova <Hubble diagram> supports late acceleration; in flat dust plus vacuum, $q_0=\Omega_{m0}/2-\Omega_{\Lambda0}$ is negative for $\Omega_{\Lambda0}>1/3$. Light-curve and colour corrections, extinction, calibration, selection bias and possible population evolution must be controlled. The supernova evidence and <cosmic microwave background> constraints are complementary; together they favour this model without proving that every alternative dark-energy model is impossible.<Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant by Adam Riess and others>[Abstract and cosmological fits of the supernova distance-redshift relation]{ref}<Measurements of Omega and Lambda from 42 High-Redshift Supernovae by Saul Perlmutter and others>[Abstract and joint matter-density/cosmological-constant fits]{ref}
For the final comparison, assume $z>0$, a common $H_0$ and $0<\Omega_{\Lambda0}<1$ in model (a). Then
$$
H_c=H_0,\qquad H_a=H_0\sqrt{(1-\Omega_{\Lambda0})(1+z)^3+\Omega_{\Lambda0}},\qquad
H_b=H_0(1+z)^{3/2}.
$$
Thus $H_c<H_a<H_b$ and the integrands obey $1/H_b<1/H_a<1/H_c$. Because all three models are flat, compare the normalization-independent distance $R_0\chi=c\int dz'/H(z')$, rather than separately normalized coordinates. The <luminosity-distance ordering of flat dust-vacuum universes> is $d_{L,b}<d_{L,a}<d_{L,c}$, hence
$$
\boxed{F_b>F_a>F_c}.
$$
Model (b) gives the largest received flux, model (c) the smallest. Equalities replace the strict ordering at the endpoint values of the vacuum fraction; all have the same leading local <Hubble law> as $z\to0$.
Back to article page
