CMB Power Spectrum Correction

• CMB Power Spectrum Correction is a suite of analytic, algorithmic, and simulation-based techniques that mitigate biases from recombination physics, lensing, and instrumental effects to recover true cosmological signals.
• It addresses observer motion effects, including aberration and Doppler shifts, ensuring that partial-sky surveys yield accurate CMB spectra.
• Advanced inversion and delensing methods, along with rigorous corrections for foreground and systematic errors, achieve the sub-percent precision required in modern cosmological experiments.

The Cosmic Microwave Background (CMB) power spectrum encodes crucial information about cosmological parameters and early Universe physics. Achieving sub-percent accuracy in CMB power spectrum measurements and derived cosmological constraints requires rigorous correction for all relevant physical and instrumental effects. "CMB Power Spectrum Correction" refers to the suite of analytic, algorithmic, and simulation-based procedures developed to mitigate biases and recover true sky statistics from real-world observations, taking into account lensing, observer motion, finite recombination width, instrumental systematics, beam and filtering effects, foregrounds, and sky cuts.

1. Sub-Percent Corrections from Recombination Physics

Standard Boltzmann solvers (CAMB, CLASS) model the CMB last scattering surface as an infinitely thin screen at a fixed recombination redshift $z_*\sim1090$. However, due to Silk damping, small-scale temperature anisotropies are generated over a range of earlier times, introducing a scale-dependent effective last-scattering redshift $z_*(k)$. The comoving diffusion damping scale $k_D(\eta)$ governs the exponential cutoff of small-scale modes: $$k_D^{-2}(\eta)=\int_0^{\eta}\frac{d\eta'}{6\,(1+R(\eta'))\,n_e(\eta')\,\sigma_T\,a(\eta')}[R^2(\eta')+\frac{16}{15}(1+R(\eta'))]$$ where $R=3\rho_b/4\rho_\gamma$, $n_e$ is free-electron density.

The mode-dependent visibility function is

$g(k,\eta)=D(k,\eta)\,v(\eta)$

where $v(\eta)$ is the standard visibility function and $D(k,\eta)$ the Silk damping factor. The function $g(k,\eta)$ peaks at a $k$-dependent time, $\eta_*(k)$, corresponding to a redshift $z_*(k)$. For instance, $k=0.01\,{\rm Mpc}^{-1}$ leads to $z_*\approx1090$, while $k=0.3\,{\rm Mpc}^{-1}$ gives $z_*\approx1130$, i.e., a $\sim40$ redshift shift at small scales.

Inserting $z_*(k)$ into the lensing kernel,

$$W^\kappa(z,k)=\frac{3}{2}\Omega_m H_0^2\frac{1+z}{H(z)}\frac{\chi(z)}{c}\frac{\chi_*(k)-\chi(z)}{\chi_*(k)}$$

modifies $C_\ell^{\kappa\kappa}$ and the lensed CMB power spectrum. The dominant effect is a $\Delta W^\kappa/W^\kappa\sim0.5\%$ correction at small scales ($\ell\sim4000$), with a net shift $\Delta C_\ell^{TT}/C_\ell^{TT}\sim4\times10^{-5}$ for $2500\lesssim\ell\lesssim4000$, corresponding to a $\sim0.1\sigma$ bias in a cosmic-variance-limited experiment (Hadzhiyska et al., 2017).

2. Corrections for Observer Motion: Aberration and Doppler Effects

Our motion with respect to the CMB frame ($\beta\approx1.23\times10^{-3}$) induces angular distortions ("aberration") and frequency shifts ("Doppler") of the observed field. In harmonic space, this yields a linear mixing,

$$a'_{\ell m} = \sum_{\ell'} K_{\ell\ell'm}(\beta) a_{\ell'm}$$

resulting in off-diagonal coupling of power. For full-sky power spectra, the leading effect is $\mathcal{O}(\beta^2)$, negligible for cosmological inference, but for partial skies or asymmetric masks, first-order $\mathcal{O}(\beta)$ corrections become relevant.

The fractional bias is well-approximated by

$$\frac{\Delta C_\ell}{C_\ell}\simeq -\beta\left\langle\cos\theta\right\rangle\frac{d\ln C_\ell}{d\ln\ell}$$

where $\langle\cos\theta\rangle$ is averaged over the observed footprint (Jeong et al., 2013). On deep or highly anisotropic patches, aberration can bias the power spectrum by up to $\sim1\%$ ($\ell\sim3000$). The correction is especially relevant for small-sky surveys (ACT, SPT) and for Planck analyses with $f_{\rm sky}\sim0.4$. To correct, de-boosting can be performed via inverse harmonic kernels, bias subtraction templates, or direct likelihood modeling (Notari et al., 2013, Catena et al., 2012, Pereira et al., 2010).

3. Lensing Kernel and Power Spectrum Inversion Techniques

CMB lensing by large-scale structure remaps anisotropies, smoothing $C_\ell$ and mixing power. Full forward modeling with Boltzmann solvers includes lensing in the transfer function, but recent work has emphasized the importance of inverting the observed ("lensed") spectra to recover unlensed, intrinsic $C_\ell$. Matrix inversion delensing,

$$\widetilde{\mathbf{C}} = (\mathbf{I}+\boldsymbol{\delta k})\mathbf{C}$$

where $\boldsymbol{\delta k}$ is the lensing kernel, enables direct reconstruction of unlensed spectra from observed data. This approach, when calibrated against forward CAMB/CLASS runs, recovers $C_\ell$ with $<10^{-3}$ accuracy for $\ell\lesssim1000$ (Pal et al., 2013). Iterative non-linear delensing methods, such as the Non-Linear Iterative Richardson-Lucy (NIRL) algorithm, further allow recovery of the primordial power spectrum $P_R(k)$ from the lensed spectrum, including all lensing-induced non-linearities, and outperform template-based correction schemes for precision recovery of non-trivial spectral features (Chandra et al., 2021).

4. Instrumental and Analysis-Induced Corrections

Real CMB power spectrum estimation is confounded by beam asymmetries, anisotropic filtering, sky cuts, and foregrounds. Direct optimal quadratic estimators (DQML), computed from time-ordered data (TOD) and designed to explicitly correct for beam convolution and mask-induced leakage, can be implemented to yield unbiased $TT$, $EE$, $BB$, $TE$ spectra even up to $\ell=1500$, as demonstrated on Planck 70 GHz simulations (Keihänen et al., 2016).

Pseudo-$C_\ell$ (MASTER-style) pipelines approximate signal filtering biases by a one-dimensional $F_\ell$ transfer function. However, in practice, full 2D mixing arises from time-domain filtering and incomplete sky coverage. Simulation-based methods construct the full $\mathbf{J}_{\ell\ell'}$ transfer matrix by injecting single-mode realizations, running the analysis pipeline, and measuring mode mixing. This procedure corrects for all linear couplings and is signal-independent, enabling robust, unbiased recovery of the true $C_\ell$ even in the presence of strong mode-mixing (Leung et al., 2021).

For CMB lensing power spectra estimated via quadratic estimators, masking and filtering break statistical isotropy and introduce non-trivial bandpower response matrices. These effects are measured by constructing special simulation-based “response maps,” with all mode-coupling calibrated and inverted at the percent level (Carron, 2022). MAP-based lensing mass map estimators require analogous simulations to calibrate mean-fields, response biases, and realization-dependent noise, ensuring sub-percent accuracy for cosmological constraints from polarization data (Legrand et al., 2023).

5. Foreground and Point-Source Contamination Corrections

Astrophysical foregrounds (CIB, tSZ, radio point sources) can strongly bias CMB $C_\ell$ at high $\ell$. Multitracer cleaning, using external large-scale structure (LSS) catalogs and linear combinations of tracers, enables removal of correlated CIB/tSZ foregrounds without precise SED modeling. These "deCIB/de-(CIB+tSZ)" methods, when applied to multifrequency data and coupled with constrained internal linear combination (ILC) approaches, improve small-scale CMB and kSZ power spectrum SNRs by $20$–$50\%$ in currently available and future datasets (Kusiak et al., 2023).

For radio point sources, spectra can be corrected either by masking detected sources or by direct removal followed by a residual bias correction and a "shot-noise" power estimate for unresolved sources. The cleaning procedure (masking or removal) produces consistent results at the $<0.1\%$ level over $2\le\ell\le1000$ (Scodeller et al., 2012).

6. Special Corrections: Anisotropies and Topological Defects

Additional corrections arise from large-scale cosmological anisotropies or the presence of topological defects, such as cosmic strings. A direction-dependent primordial power spectrum originating from Finslerian inflation induces a quadrupolar correction to $C_\ell^{TT}$, parameterizable as

$$C_\ell^{TT}=C_\ell^{TT,(0)}+\int k^2\,dk\,P_{iso}(k)B(k)[\Delta_\ell^T(k)]^2g_\ell$$

where $g_\ell$ is a geometric factor and $B(k)$ encodes the quadrupolar amplitude. This approach partially addresses the low-$\ell$ CMB power deficit, improving the fit for $\ell=2$–$10$, but cannot simultaneously match the full range $2<\ell<40$ (Chang et al., 2018).

For B-modes, a primordial dipole anisotropy modulates the lensing kernels, but the corrections to $C_\ell^{TT}$ and $C_\ell^{EE}$ are negligible ($<10^{-3}$), while $C_\ell^{BB}$ is shifted by up to $1\%$ at low-$\ell$ ($\ell\sim30-50$), decreasing rapidly with increasing multipole (Agarwal et al., 2019).

Power spectrum corrections from topological defects (e.g., Nambu-Goto cosmic strings) are computed by extracting high-dimensional unequal time correlators (UETCs) from simulations, diagonalizing to obtain eigenmodes, and convolving them through a Boltzmann solver. The resulting spectra are used to set upper limits on the string tension $G\mu$ at the $10^{-7}$ level (Lazanu et al., 2014).

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In summary, rigorous CMB power spectrum correction is essential for both primary analysis and parameter inference as experimental sensitivity and statistical power continue to rise. Addressing all known physical and instrumental sources of bias—including recombination physics, observer motion, lensing, instrumental transfer, foreground contamination, and specific cosmological sources—enables unbiased and robust cosmological measurements at the sub-percent level demanded by next-generation CMB experiments.