Cosmological Parameters and Derived Quantities

A cosmological parameter is never just a number: it is a coordinate in a declared physical model, evaluated with a versioned likelihood and a prior measure. A base parameter is varied directly by the inference, whereas a derived quantity is computed from each sampled state. The distinction matters because a derived posterior inherits the base coordinates, priors, nuisance treatment, calibration, and theory implementation. This page develops that map and checks it on a public DESI DR2 compressed likelihood; it does not promote any dated best fit to universal truth.

Required background. Probability and Statistical Inference supplies conditional probability, priors, covariance, weighted samples, and posterior diagnostics.

Helpful background. Einstein Equations supplies the gravitational field equations behind the expansion map; Relativistic Matter and Compact Objects supplies stress-energy and component models; Relativistic Cosmology supplies the background and perturbation observables entering parameter maps; and Computational Relativity supplies numerical convergence and independent-benchmark discipline.

Base parameters are coordinates on a model

Let $D$ denote the selected data, $M$ a cosmological model, $\boldsymbol\theta$ its sampled cosmological parameters, and $\boldsymbol\eta$ calibration, foreground, or other nuisance parameters. The posterior density is

$$p(\boldsymbol\theta,\boldsymbol\eta\mid D,M) =\frac{p(D\mid\boldsymbol\theta,\boldsymbol\eta,M) \,\pi(\boldsymbol\theta,\boldsymbol\eta\mid M)} {p(D\mid M)}.$$

A derived quantity is a deterministic map

$$\boldsymbol\phi =g(\boldsymbol\theta,\boldsymbol\eta;M,\mathcal C),$$

where $\mathcal C$ includes numerical and convention choices. Its posterior is the pushforward of the joint posterior, not an additional measurement. Marginalizing nuisance parameters before applying $g$ is safe only if $g$ does not depend on them.

For a concrete coordinate choice, the Planck 2018 base-$\Lambda$CDM analysis sampled six combinations: $\omega_b=\Omega_bh^2$, $\omega_c=\Omega_ch^2$, an angular acoustic-scale parameter, $\tau$, $\ln(10^{10}A_s)$, and $n_s$. Quantities such as $H_0$, $\Omega_m$, $\sigma_8$, and the cosmic age were computed afterward. That list also fixed spatial flatness, a power-law primordial spectrum, and a specified neutrino sector; changing those assumptions changes the map even if a symbol such as $H_0$ is retained. Planck Collaboration 2020, §2.1 and Table 1 states the sampled priors, neutrino prescription, and base-versus-derived split.

Derived quantities inherit physics and units

Some translations are algebraic. With $H_0=100h\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$,

$$\Omega_m=\frac{\omega_b+\omega_c+\omega_\nu}{h^2}.$$

Even this formula needs a neutrino convention: $\omega_\nu$ may be fixed, sampled, or computed from a mass hierarchy. Other translations require solving the background or perturbation equations. For example,

$$D_H(z)=\frac{c}{H(z)}, \qquad D_M(z)=S_K\!\left(\int_0^z\frac{c\,dz'}{H(z')}\right),$$

and the BAO combinations $D_M/r_d$ and $D_H/r_d$ additionally inherit the drag-epoch sound horizon $r_d$ and its recombination physics. Likewise,

$$\sigma_R^2(z)=\frac{1}{2\pi^2} \int_0^\infty dk\,k^2P_{\rm L}(k,z)W^2(kR), \qquad R=8h^{-1}\,\mathrm{Mpc},$$

requires a linear transfer calculation, a growth history, a window convention, and the $h$-dependent smoothing scale.

Here $P_{\rm L}$ is the dimensional linear matter power spectrum in comoving coordinates, with units of volume. The usual $\sigma_8$ uses the spherical real-space top-hat window $W(x)=3(\sin x-x\cos x)/x^3$, with $W(0)=1$, and means $\sigma_8(z)=\sigma_R(z)$ at the stated radius. The commonly reported $S_8=\sigma_8(0)\sqrt{\Omega_m/0.3}$ is another derived combination; the reference $0.3$ fixes its definition and is not an additional measurement. These definitions explain why changing $h$ changes both the perturbation calculation and the physical smoothing radius.

Reported object	Directly varied in the stated example?	Hidden dependence that must travel with it
$\omega_b$, $\omega_c$, $n_s$	Yes, in base $\Lambda$CDM	prior ranges, pivot scale, likelihood version
$H_0$ and $\Omega_m$	No	background model, curvature, neutrino prescription
$D_M/r_d$, $D_H/r_d$	Data-vector coordinates in a compressed BAO likelihood	fiducial conversion, ruler calculation, covariance, redshift bin
$\sigma_8$ and $S_8$	Usually derived	transfer solver, nonlinear boundary, neutrino clustering, $8h^{-1}\,\mathrm{Mpc}$ convention
age or recombination redshift	Derived	expansion and recombination implementations

The 2026 Particle Data Group review emphasizes both the non-uniqueness of the parameterization and the fact that observations often constrain degenerate combinations rather than individual coordinates. Cortês, Lahav, and Liddle 2026, §25.1.4 and Table 25.1 also separates the six fitted combinations from the quantities derived under flat $\Lambda$CDM.

A prior is a measure, not a label

Under a one-to-one change of variable $\psi=f(\theta)$,

$$\pi_\psi(\psi) =\pi_\theta\!\left(\theta(\psi)\right) \left\lvert\frac{d\theta}{d\psi}\right\rvert.$$

Thus “flat” is coordinate dependent. A prior uniform in $\theta$ is generally not uniform in $\theta^2$, $\ln\theta$, or a derived density fraction. Two analyses that use the same likelihood but declare flat priors in different nonlinear bases are different statistical models.

Existing weighted samples can be transformed exactly by evaluating $\phi_s=g(\theta_s)$ while retaining their weights $w_s$. Reweighting from an old prior $\pi_0$ to a new prior $\pi_1$ uses

$$w_s'\propto w_s\frac{\pi_1(\theta_s)}{\pi_0(\theta_s)}.$$

This is valid only where the original chain has support. The effective sample size

$$N_{\rm eff}=\frac{\left(\sum_s w_s'\right)^2}{\sum_s (w_s')^2}$$

exposes weight collapse; it does not prove that an unsampled tail is absent. A change that broadens support requires a new run, not importance sampling.

This is an importance-weight diagnostic, not an autocorrelation correction. For correlated chain draws it can overstate the number of independent samples. One must also measure autocorrelation for each reported quantity, including nonlinear derived quantities and tail probabilities.

Checked transformation of a DESI DR2 likelihood product

The public DESI DR2 BAO product at repository commit bb0c1c9009dc76d1391300e169e8df38fd1096db contains a 13-component mean vector and its full covariance. The files inspected on 2026-08-19 have SHA-256 values 9ac154ab583ce759c0f7eef3c978c7c70a6ead2d18774caceadf1a350a640585 for the mean and 252a143274c8a07c78694c119617d36594f6d7965d00319ca611c6ffb886e509 for the covariance. DESI specifies the distance definitions, tracer and redshift selections, reconstruction, fiducial transformation, blinding, and systematic budget in Abdul Karim et al. 2025, Eqs. (2)–(4), §§II–III, Table IV, and §X.

At $z=0.510$, the released mean has $x=D_M/r_d=13.58758434$ and $y=D_H/r_d=21.86294686$, with covariance block

$$C= \begin{pmatrix} 0.0283473742 & -0.0326062007\\ -0.0326062007 & 0.183928040 \end{pmatrix}.$$

The isotropic combination is

$$q=\frac{D_V}{r_d}=(z x^2y)^{1/3}=12.7209963.$$

Using the Jacobian $J=(2q/3x,\,q/3y)$ gives

$$\sigma_q^2=JCJ^{\mathsf T}=0.01006760, \qquad \sigma_q=0.1003374.$$

This is a reproducible first-order uncertainty propagation for a released Gaussian compressed likelihood. Because $q(x,y)$ is nonlinear, $q$ at the released mean is not exactly the mean of the transformed distribution, and $JCJ^{\mathsf T}$ is its linearized variance. Sample-based transformation is preferable if the fractional errors are large or the result is skewed. It is not a cosmological posterior: turning these distances into $H_0$, $\Omega_m$, or dark-energy parameters requires a theory model, ruler calibration, priors, and any external likelihoods. Dropping the negative off-diagonal covariance would change the propagated uncertainty.

From an official chain to a defensible summary

A chain-level reproduction should record the release and retrieval date; model and base coordinates; neutrino, recombination, and nonlinear prescriptions; likelihood and theory-code versions; every data, scale, multipole, and redshift selection cut; external priors; chain columns, weights, burn-in or filtering; and convergence threshold. Weighted means and quantiles must be recomputed from the released weights, not from an unweighted row count. Equal-tail and highest-posterior-density intervals should be named because they differ for skewed marginals.

The transformed chain should then pass three checks. First, reproduce one collaboration-provided base and derived summary within rounding. Second, verify the transformation on selected samples directly with the pinned theory code. Third, compare posterior predictions with the observed data or held-out summaries. ACT DR6 illustrates why the likelihood identity matters: its spectra analysis binds masks, transfer functions, beams, calibration, foreground parameters, covariance, multipole cuts, and null simulations before quoting parameter chains. Louis et al. 2025, §§3–8 and Appendix H documents those dependencies and the later DESI DR2 combination.

Results and limitations through 2026-08-19

As of 2026-08-19, Planck 2018, ACT DR6.02, and DESI DR2 provide distinct versioned examples, not interchangeable columns in a timeless parameter table. Their calibration domains, selections, nuisance parameters, covariance structures, and physical model assumptions differ. A current summary therefore names the exact combination and reports statistical uncertainty separately from systematic and model dependence.

This page has reproduced a derived coordinate from the official DESI Gaussian data product. It has not loaded a complete Planck or ACT posterior chain, rerun a Boltzmann solver, or recomputed an experiment’s posterior predictive checks. No Planck or ACT chain summary is reported here; reporting one requires the stated artifacts, versions, convergence diagnostics, and independently verified weighted transformations.

Exercise: propagate a correlated transformation

Suppose $\omega_m=0.1430\pm0.0012$ and $h=0.674\pm0.005$, with correlation coefficient $\rho=-0.40$. Estimate $\Omega_m=\omega_m/h^2$ and its standard deviation to first order. Why would treating the two inputs as independent be wrong?

Solution

The central value is $\Omega_m=0.1430/0.674^2=0.3148$. The derivatives are $\partial\Omega_m/\partial\omega_m=h^{-2}=2.201$ and $\partial\Omega_m/\partial h=-2\omega_m/h^3=-0.934$. The cross-covariance is $C_{\omega h}=\rho\sigma_\omega\sigma_h=-2.4\times10^{-6}$. Therefore

$$\sigma_{\Omega_m}^2 =h^{-4}\sigma_\omega^2 +\frac{4\omega_m^2}{h^6}\sigma_h^2 -\frac{4\omega_m}{h^5}C_{\omega h},$$

which gives $\sigma_{\Omega_m}\simeq0.0062$. Because both the $h$ derivative and the covariance are negative, the cross term is positive here; setting it to zero would understate the uncertainty. This is only linear propagation. A released weighted chain is preferable when the posterior or transformation is appreciably non-Gaussian.

References

• Abdul Karim, M., et al. (DESI Collaboration). 2025. “DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints.” Physical Review D 112: 083515. arXiv:2503.14738v3. DOI. Open PDF.
• Cortês, Marina, Ofer Lahav, and Andrew R. Liddle. 2026. “Cosmological Parameters.” In Review of Particle Physics, revised October 2025. Particle Data Group. Official PDF.
• Louis, Thibaut, et al. (ACT Collaboration). 2025. “The Atacama Cosmology Telescope: DR6 Power Spectra, Likelihoods and ΛCDM Parameters.” arXiv:2503.14452v2. Open PDF. Official DR6.02 release.
• Planck Collaboration. 2020. “Planck 2018 Results. VI. Cosmological Parameters.” Astronomy & Astrophysics 641: A6. DOI. Open PDF.