A Line of Sight Approach to Cosmic Microwave Background Anisotropies

TL;DR

The paper introduces an exact line-of-sight integration method to compute linear CMB anisotropy spectra by integrating sources along the photon past light cone, decoupling a model-independent geometric kernel from a model-dependent source term. The multipole evolution is recast so that the observed fluctuations follow $Δ^{(S)}_{T,l}(k, au_0)=\int_0^{\tau_0} S^{(S)}_{T}(k,\tau) j_l[k(\tau_0-\tau)] d\tau$ (and a tensor counterpart with $χ^l_k$), enabling precomputation of the geometry and evaluation of the source at a small number of points. This reduces the number of required differential equations to roughly 35 and allows sparse sampling in $k$ and $l$, yielding about a 100× speedup with accuracy better than ~0.1–0.2% up to $l\sim 1500$ over a broad range of cosmological models. The method delivers the same predictive power as traditional Boltzmann codes while enabling rapid exploration of large parameter spaces, making it particularly valuable for upcoming precision CMB data and parameter estimation; extensions to non-flat geometries are discussed for future work.

Abstract

We present a new method for calculating linear cosmic microwave background (CMB) anisotropy spectra based on integration over sources along the photon past light cone. In this approach the temperature anisotropy is written as a time integral over the product of a geometrical term and a source term. The geometrical term is given by radial eigenfunctions which do not depend on the particular cosmological model. The source term can be expressed in terms of photon, baryon and metric perturbations, all of which can be calculated using a small number of differential equations. This split clearly separates between the dynamical and geometrical effects on the CMB anisotropies. More importantly, it allows to significantly reduce the computational time compared to standard methods. This is achieved because the source term, which depends on the model and is generally the most time consuming part of calculation, is a slowly varying function of wavelength and needs to be evaluated only in a small number of points. The geometrical term, which oscillates much more rapidly than the source term, does not depend on the particular model and can be precomputed in advance. Standard methods that do not separate the two terms and require a much higher number of evaluations. The new method leads to about two orders of magnitude reduction in CPU time when compared to standard methods and typically requires a few minutes on a workstation for a single model. The method should be especially useful for accurate determinations of cosmological parameters from CMB anisotropy and polarization measurements that will become possible with the next generation of experiments. A programm implementing this method can be obtained from the authors.

Figures (5)

• Figure 1: CMB spectra produced by varying the number of evolved photon multipole moments, together with the relative error (in %) compared to the exact case. While using $l_\gamma=5$ produces up to 2% error, using $l_\gamma=7$ gives results almost identical to the exact case.
• Figure 2: Relative error between the exact and interpolated spectrum, where every 20th, 50th or 70th multipole is calculated. The maximal error for the three approximations is less than 0.2, 0.4 and 1.2%, respectively. The rms deviation from the exact spectrum is further improved by finer sampling, because the interpolated spectra are exact in the sampled points. For the sampling in every 50th multipole the rms error is 0.1%.
• Figure 3: Error in the spectrum caused by insufficient temporal sampling of the source term. Inaccurate sampling of the source during recombination leads to numerical errors, which can reach the level of 1% if the source is sampled in only 10 points across the recombination epoch. Finer sampling in time gives much smaller errors for this model. Comparisons with other models indicate that sampling in 40 points is needed for accurate integration.
• Figure 4: In (a) $\Delta_{T,150}^{(S)}(k)$ is plotted as a function of wavevector $k$. In (b) $\Delta_{T,150}^{(S)}(k)$ is decomposed into the source term $S_T^{(S)}$ integrated over time and the spherical Bessel function $j_{150}(k\tau_0)$. The high frequency oscillations of $\Delta_{T,150}^{(S)}(k)$ are caused by oscillations of the spherical Bessel function $j_{150}(k\tau_0)$, whereas the source term varies much more slowly. This allows one to reduce the number of $k$ evaluations in the line of sight integration method, because only the source term needs to be sampled.
• Figure 5: Error in the spectrum caused by insufficient $k$-mode sampling of the source term. Sampling the source with 40 points up to $k = 2l_{\rm max}$ leads to 1% errors, while with 60 or 80 points the maximal error decreases to 0.2%. Comparisons with other models indicate that sampling in 60 points is sufficient for accurate integration.