Cosmological Parameter Estimation

Cosmological parameter estimation is the machinery that turns raw astronomical measurements — CMB maps, galaxy catalogues, supernova light curves, weak-lensing shear fields — into quantitative statements about the constituents and history of the Universe. Given a parametrized model, chiefly the ΛCDM Concordance Model, one predicts observables as a function of a parameter vector $\theta$ , builds a likelihood comparing prediction to data, and characterizes the resulting probability distribution over $\theta$ . The output is the familiar table of numbers — baryon and cold-dark-matter densities, the expansion rate, the amplitude and tilt of primordial fluctuations — each with an uncertainty and a full web of correlations. The dominant paradigm is Bayesian: parameters are random variables whose posterior encodes our knowledge after seeing the data, fusing the likelihood (how well a set fits) with a prior (what we assumed before). Because the posterior is high-dimensional and rarely Gaussian, one samples it numerically — Markov Chain Monte Carlo produces a cloud of points whose density traces it — while the Fisher information matrix gives a fast Gaussian approximation for forecasting surveys and exposing degeneracies. The predictive engine underneath is a Boltzmann code (CAMB, CLASS) integrating the Einstein–Boltzmann equations for the CMB spectra, the matter power spectrum, and background quantities like the sound horizon. A single dataset typically constrains only combinations of parameters — the CMB alone leaves a geometric degeneracy that BAO and supernovae break, while clustering and lensing carve the $\sigma_8$ – $\Omega_m$ plane. Combining independent probes is the whole art, and modern estimation has reached the sub-percent regime — sharp enough to turn two persistent discrepancies, the Hubble and $S_8$ tensions, into the central open questions of the field.

Key takeaways

Cosmological parameter estimation infers the parameters of a model (usually flat ΛCDM) by comparing theory to data through a likelihood, almost always in a Bayesian framework where $P(\theta|D)\propto\mathcal{L}(D|\theta)\,\pi(\theta)$ .
The base six-parameter set $\{\omega_b,\ \omega_c,\ 100\theta_*,\ \tau,\ \ln(10^{10}A_s),\ n_s\}$ predicts the CMB spectra, the matter power spectrum, and the expansion history from which every observable follows.
For Gaussian data $-2\ln\mathcal{L}=(d-t)^{\mathsf T}C^{-1}(d-t)+\text{const}$ , a $\chi^2$ ; the CMB likelihood is nearly Gaussian in $C_\ell$ at high $\ell$ but demands an exact non-Gaussian form at low $\ell$ .
Posteriors are mapped by Markov Chain Monte Carlo; the Fisher matrix $F_{ij}=-\langle\partial_i\partial_j\ln\mathcal{L}\rangle$ gives fast forecasts and the Cramér–Rao floor $\sigma_i\ge\sqrt{(F^{-1})_{ii}}$ .
Combining CMB, BAO, Type Ia supernovae and lensing breaks degeneracies — the geometric $\Omega_m$ – $H_0$ locus and the $\sigma_8$ – $\Omega_m$ banana — and tightens everything.
Planck 2018 gives $H_0=67.36\pm0.54\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , $\Omega_m=0.3153$ , $\sigma_8=0.8111$ ; the ~ $5\sigma$ Hubble and ~ $2.7\sigma$ $S_8$ tensions flag possible new physics or systematics.
Key idea: a parametrized cosmological model plus a statistical likelihood turns diverse datasets into a joint posterior over parameters; combining independent probes breaks degeneracies, and the Fisher matrix quantifies how much information each dataset carries.

Where this lives today

Parameter estimation is the daily engine room of observational cosmology. Every Planck, ACT, and SPT analysis runs a Boltzmann code inside a Metropolis–Hastings or nested sampler; the public likelihoods (plik, CamSpec) and samplers (CosmoMC, Cobaya) are shared infrastructure. Galaxy surveys — BOSS, eBOSS, now DESI — feed BAO distances and full-shape spectra into the same framework, Pantheon+ supplies the low-redshift distance ladder, and DES, KiDS and HSC contribute weak-lensing $S_8$ . The frontier is the two tensions: the ~ $5\sigma$ Hubble discrepancy and the ~ $2.7\sigma$ $S_8$ offset have spawned a zoo of extended models — early dark energy, evolving $w_0$ – $w_a$ , extra relativistic species $N_{\rm eff}$ , interacting dark sectors — each tested by Bayes factors against base ΛCDM. Looking forward, Fisher forecasts are designing the next decade: CMB-S4, the Simons Observatory, Euclid, Rubin's LSST, and DESI are optimized around figures of merit like the dark-energy $\mathrm{FoM}\propto1/\sqrt{\det\mathrm{Cov}(w_0,w_a)}$ , tuning area, depth and redshift range to shrink the parameter ellipse before data arrive. The open question these tools were built to adjudicate is stark: are the tensions systematics, or the first cracks in the standard model of cosmology?

The physical picture

Think of the model as a machine with six dials. Turn them and it prints a predicted CMB spectrum and a predicted galaxy power spectrum. Estimation is the inverse problem: find the dial settings for which the printouts match the real sky, and quantify how sharply that match pins each dial. The likelihood is a scorecard measuring the mismatch; the posterior is the smeared-out region of dial space the data leave standing — a lumpy high-dimensional blob, and almost everything technical in the field is about mapping its shape honestly. A single experiment is never enough because most datasets measure combinations, not individual parameters. Different probes are like flashlights shining from different angles onto the same object. The CMB illuminates a long thin sliver in the $\Omega_m$ – $H_0$ plane — the geometric degeneracy — because the acoustic peaks fix only the angular scale $\theta_*=r_s/D_M$ , and many $(\Omega_m,H_0)$ pairs keep that ratio constant. BAO shines from another angle and lights a crossing band. Where the beams overlap is the constrained region, far smaller than either beam alone — which is why the headline numbers come from CMB + BAO + SNe and never one probe.

Figure

Two flashlights crossing the same plane: the CMB alone gives a long degenerate sliver in the Ω_m–H_0 plane, BAO gives a band that crosses it, and their overlap pins the concordance values Ω_m ≈ 0.315 and H_0 ≈ 67.4 km s⁻¹ Mpc⁻¹

The Fisher matrix is the curvature of the log-likelihood at its peak, and it makes the flashlight picture quantitative: a sharply curved direction is well measured, a flat valley is a degeneracy, and the eigenvectors of $F$ are the natural combinations the data determine — the CMB measures $\theta_*$ to $0.03\%$ , but resolving it into $\Omega_m$ and $H_0$ needs outside help. Marginalization — integrating the posterior over the parameters you are not reporting — is what produces honest error bars: it inflates a parameter's uncertainty to cover our ignorance of everything it correlates with, which is why marginalized errors are always at least as large as the errors from freezing every other parameter.

Figure

Marginalizing always widens the error bar: projecting the tilted 68% posterior ellipse onto the θ_1 axis gives σ_marg, wider than the conditional σ_cond from a slice at fixed θ_2 — the ratio is 1/√(1−ρ²) = 2.29 at correlation ρ = 0.9

The core equations

Everything begins with Bayes' theorem, which is not an approximation but the definition of the posterior. It states how the prior belief $\pi(\theta)$ is updated by data $D$ through the likelihood, with the evidence $\mathcal{Z}$ as the normalizing constant that also drives model comparison.

P(\theta\,|\,D)=\frac{\mathcal{L}(D\,|\,\theta)\,\pi(\theta)}{\mathcal{Z}},\qquad \mathcal{Z}=\int \mathcal{L}(D\,|\,\theta)\,\pi(\theta)\,\mathrm{d}\theta \qquad (1)

where: $P(\theta|D)$ — posterior probability of parameters $\theta$ given data $D$ ; $\mathcal{L}(D|\theta)$ — likelihood, the probability of the data at fixed parameters; $\pi(\theta)$ — prior over parameters; $\mathcal{Z}$ — Bayesian evidence (marginal likelihood). For parameter estimation $\mathcal{Z}$ is just a constant and can be ignored; for choosing between models it is the whole story. When the data residuals are Gaussian with covariance $C$ , the log-likelihood is a quadratic form — the celebrated $\chi^2$ plus a determinant term — the workhorse for supernova distances, BAO, and high- $\ell$ CMB bandpowers.

-2\ln\mathcal{L}(\theta)=\big[d-t(\theta)\big]^{\mathsf T} C^{-1}\big[d-t(\theta)\big]+\ln\det(2\pi C) \qquad (2)

where: $d$ — data vector (measured bandpowers, distances, correlation functions); $t(\theta)$ — theory prediction from the Boltzmann code; $C$ — data covariance matrix. When $C$ is independent of $\theta$ the second term is constant and maximizing $\mathcal{L}$ is exactly minimizing $\chi^2=(d-t)^{\mathsf T}C^{-1}(d-t)$ . The model that supplies $t(\theta)$ is, in its base form, remarkably economical: six numbers. The choice of which six matters — these are the combinations the CMB measures most cleanly, minimizing correlations in the fit.

\{\,\omega_b\equiv\Omega_b h^2,\ \omega_c\equiv\Omega_c h^2,\ 100\,\theta_*,\ \tau,\ \ln(10^{10}A_s),\ n_s\,\} \qquad (3)

where: $\omega_b,\omega_c$ — physical baryon and cold-dark-matter densities (peak ratios and heights fix them); $\theta_*$ — angular size of the sound horizon at last scattering (peak positions); $\tau$ — optical depth to reionization; $A_s,n_s$ — amplitude and tilt of the primordial power spectrum. Everything else — $H_0$ , $\Omega_m$ , $\sigma_8$ , the age of the Universe — is a derived function of these six. To forecast a survey, or read off the local shape of the posterior, one uses the Fisher information matrix — the expected curvature of the log-likelihood at the peak, whose inverse bounds the achievable errors through the Cramér–Rao inequality.

F_{ij}=-\Big\langle\frac{\partial^2\ln\mathcal{L}}{\partial\theta_i\,\partial\theta_j}\Big\rangle,\qquad \sigma_i\ge\sqrt{(F^{-1})_{ii}} \qquad (4)

where: $F_{ij}$ — Fisher information matrix element; $\theta_i$ — the $i$ -th parameter; $(F^{-1})_{ii}$ — marginalized variance forecast for parameter $i$ . The marginalized error $\sqrt{(F^{-1})_{ii}}$ is always $\ge$ the conditional error $1/\sqrt{F_{ii}}$ ; equality holds only when parameter $i$ is uncorrelated with all the rest. The reason the CMB cannot fix $H_0$ by itself is captured in one geometric relation: the acoustic peaks measure the ratio of a physical length to a distance, and that single number is what recombination stamps on the sky.

\theta_*=\frac{r_s(z_*)}{D_M(z_*)},\qquad D_M(z)=\int_0^z\frac{c\,\mathrm{d}z'}{H(z')} \qquad (5)

where: $r_s(z_*)$ — comoving sound horizon at recombination, $z_*\approx1090$ ; $D_M(z_*)$ — comoving angular-diameter distance to last scattering; $H(z)$ — Hubble parameter, set by the Friedmann equation $H^2(z)=H_0^2[\Omega_m(1+z)^3+\Omega_r(1+z)^4+\Omega_\Lambda+\Omega_k(1+z)^2]$ . Because $\theta_*$ is a ratio, a whole curve of $(\Omega_m,H_0)$ values reproduces it — the geometric degeneracy, and fitting BAO and supernova distances to this integral is what breaks it and constrains $\Omega_k$ and the dark-energy equation of state. Late-time structure probes report a different combination — clustering amplitude weighted by how much matter there is to cluster. Weak lensing is sensitive not to $\sigma_8$ alone but to this product.

S_8\equiv\sigma_8\sqrt{\Omega_m/0.3} \qquad (6)

where: $\sigma_8$ — rms linear matter fluctuation in $8h^{-1}\mathrm{Mpc}$ spheres; $\Omega_m$ — present matter density. $S_8$ runs along the $\sigma_8$ – $\Omega_m$ degeneracy that lensing leaves, so it is the quantity a lensing survey measures well. Finally, deciding whether an extended model is warranted is a question of evidence, not best fit — the Bayes factor is the ratio of evidences, with an automatic Occam penalty.

R=\frac{\mathcal{Z}_1}{\mathcal{Z}_0}=\frac{\int\mathcal{L}_1\pi_1\,\mathrm{d}\theta}{\int\mathcal{L}_0\pi_0\,\mathrm{d}\theta} \qquad (7)

where: $\mathcal{Z}_1,\mathcal{Z}_0$ — evidences of models 1 and 0 (e.g. $w$ CDM vs ΛCDM); $R$ — the odds ratio, read on the Jeffreys scale. A model with an extra parameter must fit enough better to overcome the prior volume it wastes, or $R$ will favour the simpler model.

Derivation AFrom Gaussian data to a chi-squared likelihood

The goal is to show precisely why parameter estimation so often reduces to minimizing a $\chi^2$ , and where that shortcut is legitimate. We assume the data are drawn from a multivariate Gaussian centred on the theory prediction, and track what happens to the normalization when the covariance does — or does not — depend on the parameters. This matters because a covariance that secretly depends on $\theta$ (as the cosmic-variance part of the CMB does) adds a determinant term that a naive $\chi^2$ silently drops.

Step 1. Write the residual $d-t(\theta)$ as a multivariate Gaussian, $\mathcal{L}\propto|C|^{-1/2}\exp[-\tfrac12(d-t)^{\mathsf T}C^{-1}(d-t)]$ . Why: the central limit theorem makes binned data approximately Gaussian, and $C$ carries every correlation — noise, cosmic variance, survey geometry.

Step 2. Take $-2\ln$ of both sides: $-2\ln\mathcal{L}=(d-t)^{\mathsf T}C^{-1}(d-t)+\ln\det(2\pi C)$ . Why: the log turns the exponential into a quadratic form we can differentiate, and $|C|^{-1/2}$ becomes an additive $\tfrac12\ln\det C$ .

Step 3. If $C$ is independent of $\theta$ , the determinant is constant and drops out of any $\theta$ -derivative, so maximizing $\mathcal{L}$ equals minimizing $\chi^2=(d-t)^{\mathsf T}C^{-1}(d-t)$ . Why: this is the license behind every "we minimize $\chi^2$ " — exact only when the errors do not depend on the model, true for SNe and BAO.

Step 4. When $C=C(\theta)$ , keep the determinant: the correct estimator balances fit against model-dependent noise. Why: this is why the low- $\ell$ CMB needs an exact likelihood — the variance of each multipole is proportional to $C_\ell$ itself. *Verified: reduces to $\chi^2$ for parameter-independent $C$ ; the $\ln\det$ term is retained for cosmic-variance-limited data. Standard (Dodelson, Modern Cosmology Ch. 14; Verde 2010).*

Go deeper · Derivation B · 1 minThe Fisher matrix and the Cramer-Rao forecast

Derivation BThe Fisher matrix and the Cramer-Rao forecast

The aim is to derive the fastest, most widely used tool for predicting how well a future experiment will measure parameters before a single photon is collected. We expand the log-likelihood to second order about its maximum, identify the Gaussian's covariance with the inverse curvature, and average over data realizations to get the expected information. The payoff is a recipe — build $F$ from derivatives of the theory spectra, invert, read off forecasted errors — plus the Cramér–Rao bound that no unbiased method can beat.

Step 1. Taylor expand about the maximum-likelihood point $\hat\theta$ : $\ln\mathcal{L}(\theta)\approx\ln\mathcal{L}(\hat\theta)+\tfrac12(\theta-\hat\theta)_i\,\partial_i\partial_j\ln\mathcal{L}\,(\theta-\hat\theta)_j$ , with no linear term. Why: at a maximum the gradient $\partial_i\ln\mathcal{L}$ vanishes, so the leading behaviour is the quadratic curvature — the posterior looks locally Gaussian no matter how non-Gaussian it is globally.

Step 2. Read off a Gaussian posterior $P(\theta)\propto\exp[-\tfrac12(\theta-\hat\theta)_iH_{ij}(\theta-\hat\theta)_j]$ with precision matrix $H_{ij}=-\partial_i\partial_j\ln\mathcal{L}$ , whose covariance is $H^{-1}$ . Why: a Gaussian's inverse-covariance is precisely its curvature, so the Hessian of the log-likelihood already contains the full error ellipse, including all parameter correlations.

Step 3. Average the Hessian over data realizations, $F_{ij}=\langle H_{ij}\rangle=-\langle\partial_i\partial_j\ln\mathcal{L}\rangle$ , to get the Fisher information — the expected curvature, which needs only a fiducial model, not real data. Why: this is what makes forecasting possible; you assume a truth, compute how sharply an idealized dataset would peak around it, and you have predicted the survey's power in advance.

Step 4. State the Cramér–Rao bound $\mathrm{Cov}(\theta_i,\theta_j)\ge(F^{-1})_{ij}$ , so the best marginalized error is $\sigma_i=\sqrt{(F^{-1})_{ii}}$ , always $\ge 1/\sqrt{F_{ii}}$ . Why: inverting the full matrix and taking the diagonal marginalizes over the other parameters, and the inequality $\sqrt{(F^{-1})_{ii}}\ge 1/\sqrt{F_{ii}}$ is the mathematical form of "correlations inflate error bars" seen in the second figure. *Verified: standard result; Dodelson Ch. 14, Tegmark, Taylor and Heavens (1997). For a Gaussian likelihood, $F_{ij}=t_{,i}^{\mathsf T}C^{-1}t_{,j}+\tfrac12\mathrm{Tr}[C^{-1}C_{,i}C^{-1}C_{,j}]$ , which for CMB spectra becomes a sum over $\ell$ weighted by $(2\ell+1)f_{\rm sky}/2$ .*

Go deeper · Derivation C · 2 minThe geometric degeneracy and why BAO breaks it

Derivation CThe geometric degeneracy and why BAO breaks it

This explains the single most important reason the field combines datasets: the CMB, for all its precision, measures only one geometric number well and cannot separate $\Omega_m$ from $H_0$ on its own. We fix the early-Universe physics that sets the sound horizon, show the distance to last scattering can be traded against $H_0$ at constant $\theta_*$ , then introduce the low-redshift standard ruler that measures the same scale from a different vantage, crossing the degenerate line and pinning both parameters.

Step 1. The pre-recombination physics fixes the comoving sound horizon $r_s(z_*)=\int_{z_*}^{\infty}c_s(z)/H(z)\,\mathrm{d}z$ , which depends only on $\omega_b$ and $\omega_m$ . Why: the baryon-photon sound speed $c_s$ and the expansion rate before recombination are set by the physical densities, which the CMB damping tail and peak-height ratios nail down independently of the late-time expansion.

Step 2. The observed angle is $\theta_*=r_s/D_M(z_*)$ with $D_M(z_*)=\int_0^{z_*}c\,\mathrm{d}z/H(z)$ ; holding $\theta_*$ and $r_s$ fixed forces $D_M$ constant, which a whole family of $(\Omega_m,H_0)$ pairs achieves. Why: with $\omega_m$ fixed, raising $H_0$ lowers $\Omega_m=\omega_m/h^2$ and raises $\Omega_\Lambda$ in a coordinated way that leaves the distance integral unchanged — this is the degenerate direction, the long amber sliver in the first figure.

Step 3. Along that curve the entire CMB spectrum is nearly invariant, so the CMB alone cannot split $\Omega_m$ from $H_0$ ; in flat ΛCDM the late-time ISW effect and lensing partly lift it, but curvature or dynamical dark energy reopen it fully. Why: this is why a "CMB-only $H_0$ " quietly assumes flatness and a cosmological constant — relax those and the constraint balloons.

Step 4. BAO measures $D_V(z)/r_d$ at $z\sim0.1$ – $0.7$ , the same sound-horizon scale imprinted in galaxy clustering, giving a second constraint in the $(\Omega_m,H_0)$ plane that crosses the CMB line. Why: because BAO reads $r_d\approx r_s$ at low redshift where the distance–density relation is different, its band cuts across the degeneracy — the blue band in the first figure — and their intersection fixes both parameters at once. *Verified: consistent with Planck 2018 (A&A 641, A6) degeneracy-breaking analysis; the acoustic-scale numbers are reproduced in Worked Example 2.*

Limiting cases

Different data regimes call for genuinely different statistical treatments — the same Bayes theorem, but a different likelihood shape.

Case	Result	Meaning
High- $\ell$ CMB / many modes	$-2\ln\mathcal{L}\simeq(\hat C_\ell-C_\ell)^{\mathsf T}M^{-1}(\hat C_\ell-C_\ell)$	Many modes per band; the central limit theorem makes the bandpower likelihood approximately Gaussian, with $M$ the bandpower covariance.
Low- $\ell$ CMB / few modes	$\mathcal{L}(C_\ell)\propto\prod_\ell (\hat C_\ell)^{(2\ell-1)/2}\,C_\ell^{-(2\ell+1)/2}\,e^{-(2\ell+1)\hat C_\ell/2C_\ell}$	Only $2\ell+1$ modes exist; cosmic variance dominates and the exact inverse-Wishart form is mandatory, not a Gaussian in $C_\ell$ .
Gaussian posterior (strong data)	$P(\theta)\propto\exp[-\tfrac12(\theta-\theta_0)^{\mathsf T}F(\theta-\theta_0)]$	Near a sharp peak the Fisher matrix captures the full posterior; errors saturate the Cramér–Rao bound and forecasts are reliable.
Weak / prior-dominated	$P(\theta\|D)\approx\pi(\theta)$	When data barely constrain a parameter (e.g. $\tau$ without low- $\ell$ polarization), the posterior tracks the prior and reported bounds reflect assumptions, not measurement.

Common trapA tight posterior means the answer is right

It does not. A narrow, beautiful contour only means the data constrain the parameters within the assumed model. Feed a wrong or too-restrictive model — or an unmodelled systematic — into the same machinery and it returns a confidently wrong answer with tiny error bars; the likelihood cannot know the model is incomplete. Three traps follow. First, marginalized 1D error bars hide correlations, so quoting individual $\sigma$ 's without the covariance conceals the degeneracy directions that matter. Second, a good fit ( $\chi^2/\mathrm{dof}\approx1$ ) does not validate a model — it only says the model is not badly falsified by this dataset at this precision. Third, when data are weak the prior leaks in, so a bound can be a statement about the analyst rather than the Universe. The field's tensions are interesting precisely because independent, well-understood datasets give tight and mutually inconsistent answers.

Worked examples

1 · How correlation inflates a marginalized error (fully worked)

Two parameters $\theta_1,\theta_2$ have a Fisher matrix $F=\begin{pmatrix}100 & 90\\ 90 & 100\end{pmatrix}$ (in units where the diagonal is $1/\sigma^2$ ). Find the conditional error on $\theta_1$ (all else fixed), the marginalized error, and the inflation factor.

Step 1. The conditional error freezes $\theta_2$ and uses only the diagonal curvature: $\sigma_1^{\rm cond}=1/\sqrt{F_{11}}=1/\sqrt{100}=\mathbf{0.10}$ . This is the width of a vertical slice through the peak — the narrow red bar in the second figure.

Step 2. The marginalized error requires inverting $F$ . The determinant is $\det F=100\times100-90\times90=10000-8100=1900$ , so $(F^{-1})_{11}=F_{22}/\det F=100/1900=0.05263$ and $\sigma_1^{\rm marg}=\sqrt{0.05263}=\mathbf{0.229}$ .

Step 3. The inflation factor is $\sigma_1^{\rm marg}/\sigma_1^{\rm cond}=0.229/0.10=\mathbf{2.29}$ . Cross-check against the correlation: the covariance correlation coefficient is $\rho=-F_{12}/\sqrt{F_{11}F_{22}}=-90/100=-0.9$ , and $1/\sqrt{1-\rho^2}=1/\sqrt{1-0.81}=1/\sqrt{0.19}=2.29$ . The two routes agree exactly.

Step 4. Interpretation: a $90\%$ correlation between two parameters more than doubles the honest error bar on either one. This is why breaking a degeneracy with a second probe, which drives $\rho$ toward zero, is worth more than collecting more of the same data. σ_cond = 0.10, σ_marg = 0.229, inflation = 2.29 at ρ = 0.9.

2 · The acoustic scale, from length and distance to a peak position (fully worked)

Using the Planck comoving sound horizon $r_s(z_*)=144.4\ \mathrm{Mpc}$ and comoving distance to last scattering $D_M(z_*)=13\,870\ \mathrm{Mpc}$ , compute $\theta_*$ , the reported $100\,\theta_*$ , and the acoustic multipole $\ell_A=\pi/\theta_*$ . Compare to Planck's $100\,\theta_*=1.04109$ .

Step 1. The angle is a pure ratio: $\theta_*=r_s/D_M=144.4/13\,870=\mathbf{0.010411\ rad}$ . In more familiar units that is $0.010411\times(180/\pi)\times60=\mathbf{35.8\ arcmin}$ — the physical size of the sound horizon subtends about a third of a degree on the sky.

Step 2. The quantity actually reported is $100\,\theta_*=100\times0.010411=\mathbf{1.0411}$ , matching the Planck value $1.04109$ to four significant figures — a measurement good to about $0.03\%$ , one of the most precise numbers in cosmology.

Step 3. The acoustic scale in multipole space is $\ell_A=\pi/\theta_*=3.14159/0.010411=\mathbf{301.8}$ . This sets the harmonic spacing of the CMB peaks; the first peak sits lower, near $\ell\approx220$ , because gravitational driving and the early ISW effect shift it, but the comb spacing is governed by $\ell_A$ .

Step 4. The degeneracy made numerical: this exquisite number constrains only the ratio $r_s/D_M$ . Hold it fixed and slide $H_0$ from $67.4$ toward $73$ — with $\omega_m$ pinned, $\Omega_m=\omega_m/h^2$ falls from $0.314$ to $0.267$ while $D_M$ readjusts to keep $\theta_*$ fixed. The CMB cannot tell these universes apart; BAO, measuring the same $r_s$ at $z\sim0.5$ , can. 100θ_* = 1.0411 (Planck 1.04109), ℓ_A = 301.8.

3 · Quantifying the S_8 tension (fully worked)

Planck 2018 gives $\sigma_8=0.8111$ and $\Omega_m=0.3153$ . Compute $S_8$ . Given a weak-lensing value $S_8=0.759\pm0.024$ (KiDS-1000-like) and a Planck value $S_8=0.832\pm0.013$ , quantify the tension.

Step 1. Evaluate the definition: $S_8=\sigma_8\sqrt{\Omega_m/0.3}=0.8111\times\sqrt{0.3153/0.3}=0.8111\times1.0252=\mathbf{0.832}$ . The combination is designed so that lensing surveys, which trail along the $\sigma_8$ – $\Omega_m$ banana, report $S_8$ with a much smaller error than either factor alone.

Step 2. The discrepancy is $\Delta S_8=0.832-0.759=0.073$ ; treating the measurements as independent, $\sigma=\sqrt{0.013^2+0.024^2}=\sqrt{0.000745}=0.0273$ , so the tension is $\Delta S_8/\sigma=0.073/0.0273=\mathbf{2.7\sigma}$ — the milder of the two, with late-time surveys seeing less clustering than the CMB extrapolates.

Step 3. For contrast, the Hubble constant — Planck $H_0=67.36\pm0.54$ against a distance-ladder $73.04\pm1.04$ — gives $\Delta H_0/\sqrt{0.54^2+1.04^2}=5.68/1.17=\mathbf{4.9\sigma}$ , the ~ $5\sigma$ Hubble tension. Parameter estimation is the arena where extended models are tested to see if they lower one tension without raising the other. S_8 = 0.832, S_8 tension ≈ 2.7σ, Hubble tension ≈ 4.9σ.

Physics of the portrait

Cosmological parameter estimation has no single inventor because it is a confluence: an eighteenth-century theorem of inverse probability, a twentieth-century theory of statistical information, a mid-century sampling trick born of the hydrogen bomb, and a turn-of-the-millennium flood of precision data. The portrait below is of the method itself, the way the refractive-index portrait was of a number seen three ways.

Origins · 1 mincosmological parameter estimation and how this was discovered

cosmological parameter estimation

from Bayes and Fisher to Planck and MCMC · Bayesian inference in precision cosmology

The method rests on a theorem published in 1763, two years after Thomas Bayes died, when his friend Richard Price read "An Essay towards solving a Problem in the Doctrine of Chances" to the Royal Society — the first statement of how evidence updates belief. Pierre-Simon Laplace independently rediscovered and vastly extended inverse probability, using it to estimate the mass of Saturn from orbital data. For over a century the machinery lay largely unused for physics until Ronald Fisher, in the 1920s, built the modern theory of estimation — the likelihood function, maximum-likelihood estimators, and the information that bears his name, whose inverse (with the 1940s Cramér–Rao bound) floors any measurement. The computational half arrived in 1953, when Metropolis, the Rosenbluths and the Tellers devised Monte Carlo sampling at Los Alamos — the algorithm that, generalized by Hastings in 1970, became the MCMC that samples cosmological posteriors today. The synthesis came in the 1990s: Tegmark, Taylor and Heavens (1997) showed how a Fisher matrix forecasts what a CMB experiment will measure, guiding the design of WMAP. When WMAP (2003) and then Planck (2018) delivered all-sky data, the same pipeline — Boltzmann code, likelihood, MCMC, marginalization — turned photographs of the infant Universe into a six-number table quoted to sub-percent precision. What Bayes began as a philosophical puzzle is now the instrument that reads the contents of the cosmos.

History

1763

— Bayes' essay on inverse probability is read to the Royal Society by Richard Price

1922

— Fisher formalizes the likelihood function and the concept of statistical information

1953

— Metropolis, Rosenbluth, Rosenbluth, Teller and Teller devise Monte Carlo sampling

1997

— Tegmark, Taylor and Heavens bring the Fisher matrix to CMB survey forecasting

2018

— Planck delivers the six base ΛCDM parameters to sub-percent precision

Going further

Texts. Dodelson, Modern Cosmology, 2nd ed., Ch. 11 (Probes of Inhomogeneities) and Ch. 14 (Data Analysis) · Ryden, Introduction to Cosmology, 2nd ed., Ch. 8–9 · Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 29–30 · Mo, van den Bosch and White, Galaxy Formation and Evolution, Ch. 6 · Verde, "Statistical methods in cosmology", Lect. Notes Phys. 800 (2010).

Provenance. Parameter values are Planck 2018 TT,TE,EE+lowE+lensing (Planck Collaboration 2020, A&A 641, A6): $\Omega_b h^2=0.02237\pm0.00015$ , $\Omega_c h^2=0.1200\pm0.0012$ , $H_0=67.36\pm0.54$ , $n_s=0.9649\pm0.0042$ , $\sigma_8=0.8111\pm0.0060$ , $\Omega_m=0.3153\pm0.0073$ , $\Omega_k=0.0007\pm0.0019$ , $w=-1.03\pm0.03$ . Bayes' theorem, the Gaussian likelihood, and the Fisher/Cramér–Rao results are textbook (Dodelson Ch. 14; Tegmark, Taylor and Heavens 1997; Verde 2010). The acoustic-scale figures ( $r_s=144.4$ Mpc, $D_M=13\,870$ Mpc, $100\,\theta_*=1.04109$ , $\ell_A=301.8$ ), $S_8=0.832$ , and the tension magnitudes (Hubble $\approx4.9\sigma$ , $S_8\approx2.7\sigma$ ) are recomputed in the worked examples from the Planck values and representative late-time datasets.

Prerequisites. The ΛCDM Concordance Model · The CMB Anisotropy Power Spectrum · Transfer Function and the Matter Power Spectrum · The Primordial Power Spectrum · The Sound Horizon and the Standard Ruler. Leads to. Baryon Acoustic Oscillations · Dark Energy and the Cosmological Constant · Cold Dark Matter.

Ask the tutor: Want to walk through how a real MCMC chain turns Planck bandpowers into the $\Omega_m$ – $H_0$ contour, step by step?

What exactly is the difference between the likelihood and the posterior?

The likelihood $\mathcal{L}(D|\theta)$ is the probability of the data for fixed parameters — evaluated by running the Boltzmann code and comparing to the measurements. The posterior $P(\theta|D)$ is the probability of the parameters given the data you have, and it is what you want; Bayes' theorem connects them as posterior = likelihood × prior, normalized. The distinction matters because the likelihood is not a probability distribution over $\theta$ at all — you cannot integrate it over parameters and get one — whereas the posterior is, which is why error bars, credible intervals and marginalization are all posterior statements.

Why sample with MCMC instead of just gridding the parameter space?

Because the curse of dimensionality is brutal. A grid with 20 points per axis in the base six-parameter model already needs $20^6=6.4\times10^7$ likelihood evaluations, each running a Boltzmann code — infeasible, and hopeless for the fifteen-plus parameters of a realistic analysis with nuisances. MCMC instead spends effort only where the posterior has support: it random-walks through parameter space, accepting or rejecting steps so the density of visited points converges to the posterior. The sample count scales roughly linearly with dimension rather than exponentially, and the chain yields marginalized constraints by simply histogramming the coordinates you care about.

What is the geometric degeneracy in plain terms?

The CMB acoustic peaks are a comb whose spacing is set by one angle, $\theta_*$ — the apparent size on the sky of the sound horizon, the farthest sound could travel before recombination. That angle is a ratio: a physical length divided by the distance to last scattering. Many universes share the same ratio, because at fixed early-Universe physics you can trade the Hubble constant against the matter density and keep $\theta_*$ unchanged. So the CMB measures the combination superbly but leaves a long thin valley in the $\Omega_m$ – $H_0$ plane, which a second ruler at low redshift, like BAO, crosses to fix both.

Are the Hubble and S_8 tensions signs of new physics or just mistakes?

Nobody knows yet, which is why they are the field's central questions. Both possibilities are live. On the systematics side, the distance ladder involves Cepheid calibration and metallicity corrections, and weak lensing involves photometric redshifts and intrinsic alignments — any could bias the late-time numbers. On the new-physics side, early dark energy can raise the CMB-inferred $H_0$ by shrinking the sound horizon, and modified growth can lower $S_8$ . The trouble is that no single simple extension relieves both tensions at once without spoiling some other well-measured quantity, and Bayes factors so far do not decisively favour any over base ΛCDM. Resolving it will likely take the next generation of independent data.

Why are there exactly six base parameters, and where did the others go?

The six are a minimal spanning set: two physical densities ( $\omega_b$ , $\omega_c$ ), one geometric angle ( $\theta_*$ ), one reionization depth ( $\tau$ ), and two for the primordial spectrum ( $A_s$ , $n_s$ ), chosen because the CMB measures these combinations with the least correlation. Everything else quoted — $H_0$ , $\Omega_m$ , $\sigma_8$ , the age of the Universe — is derived: a deterministic function of the six, computed at every chain point. Parameters like curvature $\Omega_k$ , the equation of state $w$ , the neutrino mass, or $N_{\rm eff}$ are held at their ΛCDM defaults and only added to test an extension, at which point the Bayes factor judges whether the data justify the extra freedom.

When is it safe to use a Fisher matrix instead of a full MCMC?

The Fisher matrix assumes the posterior is Gaussian — a single elliptical peak — so it is trustworthy when the data constrain parameters strongly and the likelihood is smooth near its maximum, which is common for well-measured parameters and for forecasting idealized surveys. It is fast, transparent, and exposes degeneracy directions cleanly. It fails, sometimes badly, when the posterior is non-Gaussian: hard prior boundaries (a positive neutrino mass, $\Omega_k$ near zero), banana-shaped degeneracies, or multimodal likelihoods, where the Fisher error can be off by large factors and only a full MCMC gives honest contours. Use Fisher for survey design and intuition, MCMC for any number you intend to publish.

What does the Bayesian evidence add that the best-fit chi-squared does not?

The best-fit $\chi^2$ tells you how well a model can fit if tuned optimally, but it always improves when you add parameters — a more flexible model fits noise better, so raw goodness-of-fit rewards complexity. The evidence $\mathcal{Z}=\int\mathcal{L}\pi\,\mathrm{d}\theta$ instead averages the likelihood over the whole prior volume, penalizing a model for all the prior space where its extra parameter does nothing useful. This is Occam's razor made quantitative: an extension must fit enough better to repay the volume it wastes. The Bayes factor $R=\mathcal{Z}_1/\mathcal{Z}_0$ , read on the Jeffreys scale, tells you whether the data actually prefer $w$ CDM over ΛCDM, not merely whether $w$ CDM can fit.