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Sandbox Physics
Cosmology from expansion to inference

L3 · Chapter 7

Read a public lensing likelihood

Sandbox Physics ·

A public power spectrum becomes a parameter measurement only after a theory and an error model are attached to it. Open Qu et al., arXiv:2304.05202v3, Section IX.1 and Eq. 45, alongside the ACT DR6 lensing v1.2 release. The calculation below follows one amplitude through that Gaussian likelihood. Every other cosmological parameter stays fixed.

Start with the measured bands

Gravitational lensing slightly rearranges the microwave sky. Correlations in that rearrangement estimate a convergence field: a dimensionless measure of how much neighbouring ray directions converge. The reconstructed field has a power spectrum. ACT DR6 publishes estimates in multipole bands, together with a covariance matrix describing how their errors move together.

In the NASA LAMBDA archive ACT_dr6_likelihood_v1.2.tgz, select rows two through eleven, counting from zero, of clkk_bandpowers_act.txt. Select the same rows and columns of covmat_act_cmbmarg.txt. The ten window rows in binning_matrix_act.txt cover multipoles forty through seven hundred and sixty-three. Each window is a weighted average over multipoles; plotting its centre does not turn the measurement into a single-multipole sample.

The covariance here is the ACT-only product with primary-CMB uncertainty marginalized. Masks, beam treatment, filtering and reconstruction biases have already entered the released observations and covariance. The browser fits those products; it does not repeat the map reconstruction. Section V.11 describes the covariance simulations, and Section IX.1 explains their use in the likelihood.

Reading: Qu et al. · NASA LAMBDA

Choose the spectrum before fitting its amplitude

Use the supplied cosmo2017_10K_acc3_lenspotentialCls.dat as a fixed reference. Its sixth column stores a scaled lensing-potential spectrum. The release software converts it to convergence power by multiplying by two pi and dividing by four. Applying the published window matrix gives one reference value for each observed band. All powers in this calculation are dimensionless, with no remaining multipole or angular-area factors.

tb=∑LWbLCL,fidκκt_b=\sum_L W_{bL}C_{L,\mathrm{fid}}^{\kappa\kappa} defines the template. The model predicts mb=Aκtbm_b=A_\kappa t_b. Here AκA_\kappa is an amplitude of this fixed convergence-power spectrum. It is different from the phenomenological parameter that smooths temperature-spectrum peaks, and it does not independently vary a matter density or dark-energy equation of state.

Figure 25 of Qu et al. reports an amplitude of 1.013 with an uncertainty of 0.023 against a Planck 2018 best-fit reference. Our supplied cosmo2017 reference is different, and cosmology-dependent reconstruction corrections stay fixed. The numerical fit below therefore answers its own stated conditional question. The comparison with Figure 25 teaches which assumptions must travel with an amplitude.

Reading: Qu et al. · NASA LAMBDA · Planck 2018 results. VI. Cosmological parameters

Keep correlations in the Gaussian fit

A likelihood assigns a relative plausibility to the observed data for each chosen amplitude. Equation 45 uses a Gaussian likelihood for the bandpowers. The full covariance sets how surprising a correlated excursion across neighbouring bands is.

Let dd denote the ten observations, CC the released covariance, and P=hC−1P=hC^{-1} the corrected precision matrix. The covariance estimate used 792 simulations. The Hartlap factor h=780/791h=780/791 adjusts its inverse for the ten retained bands. Completing the square in the amplitude gives the estimate and its conditional Gaussian uncertainty.

A worked calculation with the frozen arrays gives amplitude 0.984843, standard deviation 0.0249049 and minimum chi-square 13.6865. Ten bands and one fitted amplitude leave nine residual dimensions in this fixed model. The separate Cholesky and NumPy solutions agree. Replacing the covariance by its diagonal changes the answer because the same measured deviations now receive different weights.

−2log⁡L=(d−Aκt)TP(d−Aκt)+constant,h=Ns−nb−2Ns−1=780791-2\log\mathcal L=(d-A_\kappa t)^\mathsf TP(d-A_\kappa t)+\mathrm{constant},\qquad h=\frac{N_s-n_b-2}{N_s-1}=\frac{780}{791}
The simulation count is 792 and the retained band count is ten. The constant does not depend on amplitude.
A^κ=tTPdtTPt,σA2=1tTPt\widehat A_\kappa=\frac{t^\mathsf TPd}{t^\mathsf TPt},\qquad\sigma_A^2=\frac1{t^\mathsf TPt}
The full matrix weights both the observations and the template. This uncertainty holds the spectrum shape and reconstruction response fixed.

Reading: Qu et al. · NASA LAMBDA

Integrate over an uncertain response

Suppose an additional response offset changes the predicted amplitude in exactly the same way as the signal. Give that offset a zero-mean Gaussian distribution and integrate over its possible values. This operation is marginalization: the data must now allow for a response change as well as a signal change.

For d=(Aκ+g)t+ϵd=(A_\kappa+g)t+\epsilon, choose g∼N(0,sg2)g\sim\mathcal N(0,s_g^2) and ϵ∼N(0,C/h)\epsilon\sim\mathcal N(0,C/h). Independence makes the marginalized covariance Σ=C/h+sg2ttT\Sigma=C/h+s_g^2tt^\mathsf T. Its rank-one addition points along exactly the same data-space direction as the amplitude. The fitted centre stays fixed, and the amplitude variance increases by sg2s_g^2.

For example, an extra response standard deviation of 0.08 changes the amplitude uncertainty from about 0.0249 to about 0.0838. This extra uncertainty is a teaching choice, set to zero by default. The published covariance already contains the uncertainties assigned by ACT; the added offset is useful for understanding a degeneracy, without assigning the collaboration a new calibration prior.

Reading: Qu et al. · NASA LAMBDA

Average over the prior to compare models

A uniform prior assigns equal probability density inside a chosen interval and zero outside. Multiplying it by the likelihood and normalizing gives the posterior. When the prior extends far beyond the likelihood peak, widening it hardly changes the posterior mean or interval. The evidence still changes: it averages the likelihood over the whole normalized prior.

Compare a free-amplitude model with a model fixed at unit amplitude. Widening the uniform prior half-width from 0.25 to 0.8 lowers the log evidence ratio by about 1.16, even though the concentrated posterior barely moves. Most of the newly admitted amplitudes fit poorly. A narrow prior centred away from the likelihood peak can instead cut the posterior at an edge; its resulting interval describes the combined assumptions and observations.

For the next dataset, record the spectrum convention, retained windows, full covariance, simulation correction and frozen theory. Then write the normalized prior and the precise two models being compared. Those details make the fitted number reproducible. The synthetic remapping illustration shows how lensing rearranges a source sky; the measured ACT points remain in their own data plot.

π(Aκ)=12ΔA  inside the prior,Zfree=12ΔA∫a−ΔAa+ΔAL(Aκ) dAκ,B=ZfreeL(1)\pi(A_\kappa)=\frac1{2\Delta A}\;\text{inside the prior},\qquad Z_{\mathrm{free}}=\frac1{2\Delta A}\int_{a-\Delta A}^{a+\Delta A}\mathcal L(A_\kappa)\,dA_\kappa,\qquad B=\frac{Z_{\mathrm{free}}}{\mathcal L(1)}
The prior centre is aa and its half-width is ΔA\Delta A. Both compared models use the same response-uncertainty prior and covariance.

Reading: Qu et al. · NASA LAMBDA

Try the prediction

Keep the observed spectrum fixed

  1. Compare Data set the scale with Much wider prior. Follow both the posterior interval and the log evidence ratio.
  2. Choose Uncertain response and open the model details. Increase the extra amplitude uncertainty, then compare the width with the variance-addition example.
  3. Export the calculation. Identify the observed data, template, covariance, prior and source checksums before changing any setting.

Exercises and explanations

  1. An amplitude is fitted with uncertainty 0.0249. What happens if an independent, completely degenerate response offset has standard deviation 0.08?

    Read the explanation

    The variances add, giving 0.02492+0.082≃0.0838\sqrt{0.0249^2+0.08^2}\simeq0.0838. The fitted centre stays fixed because the offset has zero mean and follows the same template direction.

  2. The half-width of a broad uniform prior grows from 0.25 to 0.8. Estimate the change in log evidence when both intervals contain essentially all of the likelihood.

    Read the explanation

    The likelihood integral stays the same and the prior density falls by a factor of 3.2. The log evidence falls by log⁡(3.2)≃1.16\log(3.2)\simeq1.16. The posterior density renormalizes to almost its previous shape.

  3. Why should the fitted 0.984843 amplitude not be plotted as a numerical reproduction of Figure 25’s Planck-reference result?

    Read the explanation

    The template is the supplied cosmo2017 spectrum and the reconstruction response is frozen. Figure 25’s Planck-reference number uses a different fixed spectrum and the collaboration’s analysis settings. An amplitude is meaningful only relative to its declared template and likelihood.

Sources and calculation scope

  1. Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209v4 (2021)

    Sections 2.2–2.3 define the data likelihoods, section 3.1 discusses the acoustic scale, and sections 6.2 and 7.3–7.5 separate lensing smoothing, curvature, dark energy and relativistic species. Published in A&A 641, A6 (2020), with the v4 corrigendum. This course does not reproduce the full Planck likelihood or chains.

  2. Qu et al., The Atacama Cosmology Telescope: A Measurement of the DR6 CMB Lensing Power Spectrum and Its Implications for Structure Growth, arXiv:2304.05202v3

    Section VIII.2 and Figure 25 discuss a fixed-template lensing amplitude; equation (45) in section IX.1 gives a Gaussian bandpower likelihood. Our fixed-template amplitude slice uses the released bandpowers and covariance, with the exact template and omissions documented beside the Lab.

  3. NASA LAMBDA, ACT DR6 lensing likelihood data

    Frozen public ACT DR6 lensing product. The data manifest records the release, original filenames, binning, template, covariance, checksums and attribution. Additional calibration uncertainty in the Lab is an explicitly chosen teaching extension.

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