From three-dimensional waves to an angular spectrum

Specify the spectrum and its units

Write $\mathcal R$ for primordial curvature perturbations. Their dimensionless power $\mathcal P_{\mathcal R}(k)$ gives variance per logarithmic wavenumber interval. The angular power $C_\ell^{\Theta\Theta}$ of fractional temperature $\Theta=\delta T/T_0$ is dimensionless too. Multiplying by the squared mean temperature restores temperature-squared units.

Temperature plots often use $D_\ell$ in square microkelvin. The multipole weighting emphasizes power near each logarithmic angular scale. Before reading a public file, establish which spectrum and temperature units its columns contain.

D_\ell^{TT}=\frac{\ell(\ell+1)}{2\pi}\,T_0^2 C_\ell^{\Theta\Theta}

Using microkelvin for the mean temperature gives square microkelvin here. The CAMB grid stores this rescaled temperature spectrum.

Carry one initial mode to the observer

The transfer function $\Delta_\ell^T(k)$ calculates how an initial wavenumber contributes to a sky multipole. It includes gravitational driving, photon-baryon motion, recombination and propagation. A spherical Bessel function projects the three-dimensional wave onto angular structure.

In flat conformal coordinates with light speed one, the source $S_T$ collects temperature contributions over time. Weight its projected signal by the initial power and integrate over wavenumber to obtain the theoretical angular spectrum. This averages over random initial phases; a particular sky will scatter around it.

The transfer integral below gives the linear unlensed temperature spectrum. Subsequent gravitational deflections remap the sky, mix angular scales and smooth the acoustic peaks. CAMB separately includes this lensing remapping in the spectrum stored by our grid.

\Delta_\ell^T(k)=\int_0^{\eta_0}d\eta\,S_T(k,\eta)j_\ell[k(\eta_0-\eta)]

The source normalization matches the transfer convention. Last scattering has finite width, and late gravitational redshift supplies additional contributions.

C_\ell^{\Theta\Theta}=4\pi\int d\ln k\;\mathcal P_{\mathcal R}(k)\,|\Delta_\ell^T(k)|^2

This expression assumes scalar primordial curvature perturbations and statistical isotropy. Polarization and cross-spectra require their corresponding transfers.

Reading: David Tong · CAMB source and documentation

Compensating distances can align the peaks

Changing the Hubble constant together with the constant dark-energy equation-of-state parameter $w$ can approximately preserve distance to last scattering. With early physical densities fixed, the sound horizon changes less. Similar ratios produce similar peak positions. Different parameter combinations with similar predictions are called a degeneracy.

The compensation is incomplete. Late potential evolution changes large-angle temperature; structure growth changes lensing. Polarization and other distance measurements add different information. Compare differences across the full temperature spectrum, beyond its first peak.

Flatness, fixed cold-dark-matter physical density and fixed initial-spectrum settings are conditions of this experiment. Allowing curvature, extra radiation or a flexible initial spectrum changes the degeneracy. A precise angle becomes an inference about late ingredients only after these assumptions are specified.

Reading: Planck 2018 results. VI. Cosmological parameters

Work a pair of nearly matching backgrounds

Hold physical baryon density at $\omega_b=0.0224$ and retain the other grid settings. Compare the reference $H_0=67.4\;\mathrm{km\,s^{-1}\,Mpc^{-1}}$ , $w=-1$ with $H_0=73.5\;\mathrm{km\,s^{-1}\,Mpc^{-1}}$ , $w=-1.2$ . The expansion parameters differ substantially, yet the ratio of sound horizon to transverse distance remains close.

The frozen grid gives an acoustic-angle difference of about 0.00724 percent. Both sampled spectra place the first peak at multipole 222. Their ages remain different, about $13.79\;\mathrm{Gyr}$ and $13.69\;\mathrm{Gyr}$ . Nearly aligned peaks can therefore accommodate different late expansion histories.

The largest spectral difference above multipole one hundred is about 0.355 percent, below the largest measured interpolation difference discussed next. At this resolution, the robust comparison is the near-overlap of the peak sequences; resolving those smaller residuals requires a finer grid.

Locate the numerical approximation

The experiment freezes seventy-five CAMB 1.6.6 nodes: three baryon-density values, five Hubble constants and five constant dark-energy equations of state. It uses flat geometry and PPF dark-energy treatment. The generation script saves all other settings. The browser interpolates lensed scalar temperature spectra inside the grid and refuses extrapolation.

Six interior points were recomputed at higher numerical accuracy. For multipoles at least thirty, the largest pointwise relative difference is about 0.804 percent and the largest RMS relative difference about 0.355 percent. These six points assess interpolation together with changed numerical settings. They do not establish a uniform error bound across the grid.

For a worked verification, first choose a saved node and reproduce its stored spectrum. Then choose an interior point and compare interpolation against the independently calculated reference. The first check validates reading; the second tests the approximation. Precision likelihood use would require comparing this numerical error with the accuracy demanded by the data and refining the grid accordingly.

Reading: CAMB source and documentation

From three-dimensional waves to an angular spectrum

Specify the spectrum and its units

Carry one initial mode to the observer

Compensating distances can align the peaks

Work a pair of nearly matching backgrounds

Locate the numerical approximation

Try the prediction

Find two models with nearly aligned peaks

Exercises and explanations

Sources and calculation scope