Cosmological Perturbation Theory Using the FFTLog: Formalism and Connection to QFT Loop Integrals (1708.08130v1)

Abstract: We present a new method for calculating loops in cosmological perturbation theory. This method is based on approximating a $\Lambda$CDM-like cosmology as a finite sum of complex power-law universes. The decomposition is naturally achieved using an FFTLog algorithm. For power-law cosmologies, all loop integrals are formally equivalent to loop integrals of massless quantum field theory. These integrals have analytic solutions in terms of generalized hypergeometric functions. We provide explicit formulae for the one-loop and the two-loop power spectrum and the one-loop bispectrum. A chief advantage of our approach is that the difficult part of the calculation is cosmology independent, need be done only once, and can be recycled for any relevant predictions. Evaluation of standard loop diagrams then boils down to a simple matrix multiplication. We demonstrate the promise of this method for applications to higher multiplicity/loop correlation functions.

• The paper introduces an FFTLog-based method to decompose the linear power spectrum, enabling efficient one- and two-loop evaluations for power spectrum and bispectrum.
• It maps complex loop integrals in cosmology to analogous QFT integrals using hypergeometric functions, significantly reducing computational complexity.
• Precomputation of the cosmology-independent components allows rapid matrix multiplication for iterative analyses, enhancing precision in large-scale survey data.

Cosmological Perturbation Theory Using the FFTLog: Formalism and Connection to QFT Loop Integrals

In the paper entitled "Cosmological Perturbation Theory Using the FFTLog: Formalism and Connection to QFT Loop Integrals," the authors refine methods for calculating loop corrections to the power spectrum and bispectrum of cosmological perturbation theory. Utilizing the FFTLog algorithm, they approximate a Universe following a $\Lambda$CDM cosmology as a sum of complex power-law universes. This innovative decomposition allows the computation of loop integrals in cosmology to be directly related to loop integrals found in massless quantum field theories (QFT), allowing for the solution of such integrals via generalized hypergeometric functions.

A key outcome of this work is the formulation of explicit expressions for the one-loop and two-loop power spectra and the one-loop bispectrum. A notable advantage is that the more computationally demanding parts related to the loop calculations are cosmology independent, allowing these to be precomputed and recycled across different contexts, reducing these computations to matrix multiplications for any particular realization, thereby significantly enhancing the efficiency of cosmological perturbation analysis.

Theoretical Implications and Numerical Results

The approach laid out by the authors represents a significant simplification in the computation of loop diagrams in the cosmological perturbation theory framework. Their formalism yields several distinct advantages:

1. Efficiency and Logarithmic Decomposition: By leveraging FFTLog, the formalism efficiently decomposes the linear power spectrum into a superposition of power laws, dramatically simplifying the numerical evaluation of loop corrections.
2. Novel Integral Solutions: By connecting cosmic structures in perturbative regimes to QFT structures, the paper demonstrates that loop integrals can be analytically tackled using hypergeometric functions, allowing for rapid evaluations relative to brute-force numerical integrations.
3. Irreducibility and Reusability of Computational Complexity: The difficult parts of the calculations must be done only once and saved as a table of numbers, significantly reducing the computational burden for individual scenario evaluations. This builds an efficient path forward for Markov Chain Monte Carlo parameter estimation processes where repeated evaluations are fundamental.

The numerical results corroborate the theoretical predictions of the framework. The paper showcases the exceptional accuracy of their method when applied to the one-loop power spectrum and bispectrum, highlighting sub-percent precision efficiency, given appropriate sampling ($N\approx50$ for one-loop evaluations) without the computational overhead of conventional integration schemes.

Implications for Cosmological Research and Future Developments

The implications of this work are vast for theoretical and applied cosmology. Practically, it enables the processing of extensive datasets anticipated from ongoing and upcoming cosmic surveys, like those from DESI or CHIME, with enhanced precision. By making higher-order loop calculations more tractable, it broadens the ability to probe cosmology over large scales, crucially benefiting the search for new physics that extends beyond the $\Lambda$CDM paradigm.

Theoretically, the inspiration drawn from QFT and quantum scattering amplitudes might unearth further synergies between cosmological phenomena descriptions and particle physics computations. For future efforts, extending this method to investigate covariance matrices and potentially the one-loop trispectrum could further improve the understanding of non-linear cosmic phenomena and structure formation.

In summary, Simonovi㎝ et al. provide a robust and efficient method for loop corrections in cosmological perturbation theory using the FFTLog, reducing computational complexity and paving the way for enhanced cosmological parameter studies in current and future observational regimes.