= Solution
For successive radial null rays from a comoving source, $\int_{t_1}^{t_0}c\,dt/R(t)$ equals the same spatial integral. Subtracting two nearby wavecrests gives $\delta t_0/R_0=\delta t_1/R_1$, hence the <cosmological redshift>
$$
\boxed{1+z=\nu_1/\nu_0=R_0/R_1.}
$$
For transparent propagation, photon energies are reduced by $1+z$ and arrival intervals increased by $1+z$. The observer-area sphere is $4\pi(R_0r)^2$, so the observed bolometric flux is
$$
F=\frac{L}{4\pi(R_0r)^2(1+z)^2},\qquad\boxed{D_L=(1+z)R_0r.}
$$
Thus the requested $D_L=(1+z)D$ uses $D=R_0r$, the present transverse proper/area distance. In a curved metric this is not the radial proper distance $R_0\int_0^rdr'/\sqrt{1-Kr'^2}$; the two coincide for $K=0$. The source's unqualified “proper distance” needs this distinction.
<Type Ia supernovae> are standardizable candles: light-curve shape and colour calibrate their luminosity. Distant events are fainter, hence have larger inferred <luminosity distances>, than appropriate matter-only decelerating models predict. Their redshift–distance relation favours recent accelerated expansion. In general relativity, sufficiently negative pressure, $\rho+3P/c^2<0$, drives acceleration. Dust, evolution, calibration and selection are checked as possible systematics; a single faint supernova is not the argument.<Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant by Adam Riess and others>[1998 luminosity-distance evidence]{ref}
An independent complementary test combines the <cosmic microwave background> acoustic distance/near-flat geometry with matter-density and <baryon acoustic oscillation> constraints. Matter alone does not supply the required late-time expansion history; a dark-energy component fits the joint data. CMB geometry alone has curvature/expansion degeneracies and is not an assumption-free detection of a cosmological constant.<Planck 2013 results. XVI. Cosmological parameters by Planck Collaboration>[CMB,BAO and expansion-model constraints]{ref}
In natural units $c=\hbar=1$, a regulated <zero-point energy> estimate is
$$
\rho_{\rm zp}(M)=\sum_i\frac{(-1)^{F_i}g_i}{2}\int_{|\mathbf k|<M}\frac{d^3k}{(2\pi)^3}\sqrt{k^2+m_i^2},\qquad\Lambda_{\rm geom}=\Lambda_{\rm bare}+8\pi G\rho_{\rm zp}.
$$
The integral is ultraviolet divergent and requires renormalization; it is not a finite prediction from counting “virtual pairs.” An electroweak scale estimate gives $|\rho_{\rm vac}|\sim m_t^4=(174)^4\,\mathrm{GeV^4}\simeq9.2\times10^8\,\mathrm{GeV^4}$, or somewhat smaller with loop factors. Compared with $10^{-47}\,\mathrm{GeV^4}$ this is roughly \b[$10^{56}$ too large].
\b[The source's units are inconsistent:] $10^{-47}\,\mathrm{GeV^4}$ is a vacuum energy density, whereas $8\pi G\rho_{\rm vac}$ has units $\mathrm{GeV^2}$. The comparison above consistently uses densities. Bare and quantum contributions must cancel to extraordinary accuracy if this naive scale estimate is representative: the <cosmological constant problem>. Its severity is not a measured value of an individual cutoff integral.
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