Standard Candles, Standard Rulers, and Cosmography

Standard candles infer luminosity distance from flux, standard rulers infer transverse or radial distance from an imprinted scale, and cosmography expands those distances kinematically near the observer. None is “model independent” without qualification: candles need standardization and an absolute intercept, rulers need a ruler calibration and clustering model, and a finite cosmographic series needs a variable, order, and validity domain. This page compares those contracts and reproduces one covariance-sensitive DESI DR2 distance ratio. Evidence is current through 2026-08-19; later releases or calibration changes require renewed review before quantitative verdicts are updated.

Required background. Affine and Variational Geodesics supplies the Geodesics and Jacobi Fields capability underlying null distance–redshift relations.

Helpful background. The Einstein Equation supplies the Einstein Equations used by physical expansion models; Relativistic Matter and Compact Objects supplies the Relativistic Matter and Compact Objects populations used as sources and tracers; Relativistic Cosmology orients the distance conventions; and The Wave Equation and Characteristics supplies the Waves, Green Functions, and Geometric Optics treatment of light and standard-siren propagation.

Comparison axes: observable, scale, and calibration

The useful comparison axes are the measured data, the geometric combination they constrain, the absolute calibration, and the dominant nuisance model.

Method	Measured data	Geometric combination	Absolute scale or intercept	Leading nuisance structure
Standard candle	Light curves or fluxes and redshifts	$D_L(z)$ up to an intercept	Photometric zero points plus a standardized absolute magnitude tied to an external anchor	Standardization, population drift, dust, peculiar velocity, lensing, selection
Standard ruler	Angular and radial clustering	$D_M/r_d$, $D_H/r_d$, or $D_V/r_d$	Comoving sound horizon at baryon drag, $r_d$, from a declared early-time model or external calibration	Fiducial conversion, reconstruction, bias, redshift-space distortions, nonlinear modeling, covariance
Cosmography	A distance or expansion data vector	Coefficients such as $H_0$, $q_0$, and $j_0$	Inherited from the input probe	Expansion variable, truncation order, redshift domain, correlated coefficients

Bright and dark gravitational-wave standard sirens are candle-like because a waveform supplies $D_L$, but inclination, detector calibration, selection, and host-redshift association belong to Bright and Dark Standard Sirens.

Candles and the Pantheon+ likelihood

For a standardized candle,

$$m_{\rm corr} =5\log_{10}\left(\frac{D_L}{\mathrm{Mpc}}\right)+25+M,$$

where the common intercept $M$ includes the standardized absolute magnitude. Without an external anchor, changing $H_0$ rescales every model distance and is exactly degenerate with shifting $M$.

A concrete current example is the Pantheon+ release pinned at commit c447f0fea703fcd0fff57de5000947b5ca81286b. Its distance table Pantheon+SH0ES.dat contains 1,701 accepted light curves for 1,550 distinct supernovae, and Pantheon+SH0ES_STAT+SYS.cov is the ordered full covariance. The covariance file begins with N=1701, followed by exactly $N^2$ row-major scalars; reshape it before applying any selection. For the release’s supernova-only likelihood, select strictly zHD > 0.01, use zHD, zHEL, and m_b_corr, and apply the same 1,590-row ordered mask to both covariance axes. This exact cut retains ten rows with IS_CALIBRATOR=1; adding a separate calibrator-flag cut would no longer reproduce the released likelihood. The model distance is

$$D_L=(1+z_{\rm HD})(1+z_{\rm HEL})D_A(z_{\rm HD}) =(1+z_{\rm HEL})D_M(z_{\rm HD}).$$

The distinct redshifts preserve the release’s frame convention. The released m_b_corr is already Tripp-standardized and corrected for simulation bias and host dependence; the standardization, bias, and host corrections must not be applied a second time. Repeated observations are correlated, so the plotting-only fields m_b_corr_err_DIAG and MU_SH0ES_ERR_DIAG cannot replace the matrix.

For a relative Hubble diagram, define

$$r_i=m_{{\rm corr},i} -5\log_{10}\left(\frac{D_{L,i}}{\mathrm{Mpc}}\right)-25.$$

Profiling the intercept in a Gaussian likelihood gives

$$\widehat M =\frac{\boldsymbol 1^TC^{-1}\boldsymbol r} {\boldsymbol 1^TC^{-1}\boldsymbol 1},$$

and

$$\chi^2_{\rm prof} =\boldsymbol r^TC^{-1}\boldsymbol r -\frac{(\boldsymbol 1^TC^{-1}\boldsymbol r)^2} {\boldsymbol 1^TC^{-1}\boldsymbol 1}.$$

Cholesky solves should replace an explicit matrix inverse in an actual calculation. The Cepheid-anchored MU_SH0ES column is not a calibration-neutral substitute. In the anchored likelihood of Brout et al., Cepheid hosts use residuals $\mu_i-\mathrm{CEPH\_DIST}_i$, other selected rows use $\mu_i-\mu_{\rm model}(z_i)$, and the supernova and Cepheid-host covariance contributions are summed. Merely fixing MU_SH0ES does not implement that contract. Brout et al. specify the standardization, total covariance, likelihood, and intercept degeneracy in Brout et al. 2022, printed pp. 3–5, eqs. (1) and (5)–(15), and Appendix C, p. 22 (Open PDF). Their calibration model joins 25 photometric systems and 105 passbands through a $105\times105$ zero-point covariance Brout et al. 2022, §3.2.1, printed p. 7 (Open PDF). Sample cuts and release fields are documented in Scolnic et al. 2022, §2.2 and Table 2, printed p. 6, and Appendices A–B, pp. 10–14 (Open PDF).

For this concrete release, the 1,701 rows are the intersection of the Scolnic et al. Table 2 cuts: SALT2 fit probability and temporal coverage, $-0.3\lt c\lt0.3$, $-3\lt x_1\lt3$, uncertainty cuts on $x_1$ and peak date, Milky-Way extinction, high-redshift rest-frame-$U$ sensitivity, outliers, valid bias correction, and survival under the systematic variants. The strict redshift cut above is then applied for the 1,590-row likelihood. Analysis choices were fixed under two layers of blinding—shifted Hubble residuals and shifted cosmology chains—before the final public result was unblinded. Peculiar-velocity, selection/Malmquist, dust, population, lensing, host, calibration, and SALT2-retraining assumptions are propagated through the released vector and covariance; “full covariance” does not make them assumption-free.

Rulers and a released DESI DR2 ratio

Here $r_d$ denotes the comoving sound horizon at the baryon-drag epoch. For this comoving ruler, anisotropic baryon acoustic oscillations constrain

$$\frac{D_M(z)}{r_d}, \qquad \frac{D_H(z)}{r_d}, \qquad D_H(z)=\frac{c}{H(z)}.$$

An isotropic compression often uses

$$D_V(z) =\left[zD_M^2(z)D_H(z)\right]^{1/3}.$$

The ruler scale cancels from the Alcock–Paczyński ratio $D_M/D_H$, but not from either absolute distance. DESI DR2 specifies tracer cuts and redshift bins, catalog blinding, reconstruction, the fiducial coordinate conversion, BAO templates, covariance, and systematic budget in Abdul Karim et al. 2025, printed pp. 4–5, eqs. (2)–(5), and §§II–III, printed pp. 7–16, especially eqs. (11)–(12) and Table IV, p. 14 (Open PDF).

The following reproduction uses the public 13-element DR2 v1.1/BAO Gaussian likelihood: desi_bao_dr2/desi_gaussian_bao_ALL_GCcomb_mean.txt and its matching _cov.txt at CobayaSampler/bao_data commit bb0c1c9009dc76d1391300e169e8df38fd1096db, retrieved 2026-08-19. The underlying analysis used IterativeFFT reconstruction in pyrecon with the DR1 fiducial settings and a RascalC covariance; its analysis choices were frozen before unblinding, and the released covariance includes the stated systematic errors. At $z_{\rm eff}=0.510$, the released vector and its matching covariance block are

$$\boldsymbol x =\begin{pmatrix} D_M/r_d\\ D_H/r_d \end{pmatrix} =\begin{pmatrix} 13.58758434\\ 21.86294686 \end{pmatrix},$$ $$C= \begin{pmatrix} 0.0283473742 & -0.0326062007\\ -0.0326062007 & 0.183928040 \end{pmatrix}.$$

For $R=x_M/x_H$, linear covariance propagation uses

$$\boldsymbol g =\left(\frac{1}{x_H},-\frac{x_M}{x_H^2}\right), \qquad \sigma_R^2=\boldsymbol g^TC\boldsymbol g.$$

The result is

$$\boxed{ \frac{D_M}{D_H} =0.6214892\pm0.0171091 },$$

which reproduces the rounded Table IV value $0.622\pm0.017$. Dropping the off-diagonal covariance gives $0.0144199$, underestimating the uncertainty by $15.7\%$. Two million bivariate-Gaussian draws from NumPy default_rng(20260819) give standard deviation $0.0171388$, agreeing with the delta-method result to $0.17\%$ and showing that nonlinearity of the ratio is small here.

If an adopted ruler changes from $r_d$ to $r_d'=\lambda r_d$, the inferred physical distances $d_M=r_dx_M$ and $d_H=r_dx_H$ and their physical covariance scale by $\lambda$ and $\lambda^2$. Their ratio and propagated error remain unchanged. This cancellation does not make BAO absolutely calibrated: $H_0r_d$ remains the relevant scale combination, so changing $r_d$ changes an inferred $H_0$ inversely.

The calibration branch was rerun numerically with $r_d'=1.01r_d$: constructing $d_M=r_dx_M$, $d_H=r_dx_H$, and $C_d=r_d^2C$ gives the same $R=0.6214892$ and $\sigma_R=0.0171091$ after rescaling. Every ratio and its transformed covariance is unchanged, so the two cosmographic fits below are unchanged as well, whereas an absolute-scale inference transforms as $H_0'=H_0/1.01$. The released dimensionless vector $\boldsymbol x$ itself is not rescaled by adopting a different numerical calibration for $r_d$.

Cosmography and the truncation test

Define the present kinematic parameters

$$H_0=\left.\frac{\dot a}{a}\right|_0, \qquad q_0=-\left.\frac{\ddot a}{aH^2}\right|_0, \qquad j_0=\left.\frac{\dddot a}{aH^3}\right|_0.$$

In a spatially flat geometry, the luminosity-distance series begins

$$\frac{H_0D_L}{c} =z+\frac12(1-q_0)z^2 -\frac16(1-q_0-3q_0^2+j_0)z^3+O(z^4).$$

This follows from FLRW kinematics without using an equation of state or Einstein’s equations; curvature enters at the same order as jerk if flatness is not assumed. Visser derives the distance series and its kinematic scope in Visser 2005, preprint §2, pp. 2–7, eqs. (2)–(20) (Open PDF).

The scale-factor/redshift map $a/a_0=1/(1+z)$ has a pole at $z=-1$, so a Taylor expansion of that map about zero has convergence radius one. This motivates caution with generic cosmographic $z$-series beyond $z\sim1$; the convergence radius of a particular observable still depends on its own analytic structure. For example, flat de Sitter expansion gives the polynomial $H D_L/c=z+z^2$, with no pole at $z=-1$. The alternative

$$y=\frac{z}{1+z}$$

maps the observed range $0\leq z<\infty$ into $0\leq y<1$, but improved formal convergence does not guarantee that a low-order truncation is accurate. Cattoën and Visser give the convergence and variable analysis in Cattoën and Visser 2007, §§4.1, 5, and 7, preprint pp. 8–14, eqs. (35) and (47)–(55) (Open PDF).

The DESI ratio supplies a sharp failure diagnostic. To second order,

$$\frac{D_M}{D_H} =z+\frac12(1+q_0)z^2+O(z^3) =y+\frac12(3+q_0)y^2+O(y^3).$$

Using the full transformed covariance for the three anisotropic DESI bins below $z=1$, a one-parameter generalized least-squares fit gives

$$\begin{array}{c|c|c} \text{basis} & q_0 & \chi^2/\text{degrees of freedom}\\ \hline z & -0.3052\pm0.0373 & 2.435/2\\ y & +2.9130\pm0.1239 & 18.600/2. \end{array}$$

Neither value is a cosmological measurement. Their violent basis dependence, and the rejected low-order $y$ fit, demonstrate truncation error that the quoted statistical errors omit. A defensible cosmographic analysis must state the variable, order, domain, covariance, coefficient priors, and a basis/order stability test; otherwise use a controlled physical $H(z)$ model.

This page deliberately keeps the Pantheon+ likelihood contract and the DESI covariance diagnostic separate. It makes no joint supernova–BAO parameter claim and therefore introduces no hidden common parameter prior. A joint cosmological fit must declare its physical $H(z)$ model, shared and probe-specific nuisance parameters, normalized priors, calibration choices, and every covariance block; that task belongs to the later inference chapter.

Common pitfalls

Confusing relative and absolute distances. Supernovae alone constrain the shape of the Hubble diagram while $M$ and $H_0$ remain degenerate. BAO alone measures distances relative to $r_d$.

Using diagonal errors from a correlated release. The Pantheon+ and DESI examples both require ordered full covariance. Masking or rotating a vector requires the same operation on both covariance axes.

Treating a low-order series as a physical model. Small formal parameter errors do not include truncation bias. Basis dependence is a necessary stress test, not an aesthetic choice.

Exercises

Show the candle and ruler calibration degeneracies explicitly. If $H_0\to\alpha H_0$ at fixed dimensionless expansion history, what shift in $M$ leaves the candle prediction unchanged? If $r_d\to\lambda r_d$, how must $H_0$ change to preserve ruler-scaled distances?

Solution

At fixed dimensionless history, every cosmological distance scales as $H_0^{-1}$. Thus $H_0\to\alpha H_0$ sends

$$5\log_{10}D_L \longrightarrow 5\log_{10}D_L-5\log_{10}\alpha.$$

The predicted apparent magnitude remains unchanged if

$$\boxed{M\longrightarrow M+5\log_{10}\alpha}.$$

For BAO, $D_M/r_d$ and $D_H/r_d$ scale as $(H_0r_d)^{-1}$ at fixed dimensionless history. Preserving them under $r_d\to\lambda r_d$ therefore requires

$$\boxed{H_0\longrightarrow H_0/\lambda}.$$

The ratio $D_M/D_H$ is invariant under both common rescalings, which is why the released-data calculation above tests anisotropic geometry but not the absolute ruler calibration.

References

• Abdul Karim, M., et al. (DESI Collaboration). “DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints.” Physical Review D 112 (2025): 083515. DOI; Open PDF; official data products.
• Brout, Dillon, et al. “The Pantheon+ Analysis: Cosmological Constraints.” Astrophysical Journal 938 (2022): 110. DOI; Open PDF.
• Cattoën, Céline, and Matt Visser. “The Hubble Series: Convergence Properties and Redshift Variables.” Classical and Quantum Gravity 24 (2007): 5985–5998. DOI; Open PDF.
• CobayaSampler. BAO Data, DESI DR2 likelihood files, commit bb0c1c9009dc76d1391300e169e8df38fd1096db, 26 June 2025; retrieved 19 August 2026. Commit-pinned desi_bao_dr2 directory.
• Pantheon+SH0ES Collaboration. DataRelease, commit c447f0fea703fcd0fff57de5000947b5ca81286b, 21 December 2022. Commit-pinned repository.
• Scolnic, Dan, et al. “The Pantheon+ Analysis: The Full Data Set and Light-curve Release.” Astrophysical Journal 938 (2022): 113. DOI; Open PDF.
• Visser, Matt. “Cosmography: Cosmology without the Einstein Equations.” General Relativity and Gravitation 37 (2005): 1541–1548. DOI; Open PDF.