We present a new open-source code that calculates one-loop power auto- and cross-power spectra for matter fields and biased tracers in real and redshift space. These spectra incorporate all ingredients required for a direct application to data: non-linear bias, redshift-space distortions, infra-red resummation, counterterms, and the Alcock-Paczynski effect. Our code is based on the Boltzmann solver CLASS and inherits its advantage: user friendliness, ease of modification, high speed, and simple interface with other software. We present detailed descriptions of the theoretical model, the code structure, approximations, and accuracy tests. A typical end-to-end run for one cosmology takes $\sim 0.3$ seconds, which is sufficient for Markov Chain Monte Carlo parameter extraction. As an example, we apply the code to data from the Baryon Oscillation Spectroscopic Survey (BOSS) and infer cosmological parameters from the shape of the galaxy power spectrum.
We present a formulation of cosmological perturbation theory where the Boltzmann hierarchies that evolve the neutrino phase-space distributions are replaced by integrals that can be evaluated easily with fast Fourier transforms. The simultaneous evaluation of these integrals combined with the differential equations for the rest of the system (dark matter, photons, baryons) are then solved with an iterative scheme that converges quickly. The formulation is particularly powerful for massive neutrinos, where the effective phase space is three-dimensional rather than two-dimensional, and even more…
$\texttt{PyCosmo}$ is a Python-based framework for the fast computation of cosmological model predictions. One of its core features is the symbolic representation of the Einstein-Boltzmann system of equations. Efficient $\texttt{C/C++}$ code is generated from the $\texttt{SymPy}$ symbolic expressions making use of the $\texttt{sympy2c}$ package. This enables easy extensions of the equation system for the implementation of new cosmological models. We illustrate this with three extensions of the $\texttt{PyCosmo}$ Boltzmann solver to include a dark energy component with a constant equation of st…
Calculations of the evolution of cosmological perturbations generally involve solution of a large number of coupled differential equations to describe the evolution of the multipole moments of the distribution of photon intensities and polarization. However, this "Boltzmann hierarchy" communicates with the rest of the system of equations for the other perturbation variables only through the photon-intensity quadrupole moment. Here I develop an alternative formulation wherein this photon-intensity quadrupole is obtained via solution of two coupled integral equations -- one for the intensity qua…
The linear matter power spectrum is an essential ingredient in all theoretical models for interpreting large-scale-structure observables. Although Boltzmann codes such as CLASS or CAMB are very efficient at computing the linear spectrum, the analysis of data usually requires $10^4$-$10^6$ evaluations, which means this task can be the most computationally expensive aspect of data analysis. Here, we address this problem by building a neural network emulator that provides the linear theory (total and cold) matter power spectrum in about one millisecond with 0.2% (0.5%) accuracy over redshifts $z …
Non-gaussianity and B-mode polarization are particularly interesting features of the cosmic microwave background, as -- at least in the standard model of cosmology -- their only sources to first order in cosmological perturbation theory are primordial, possibly generated during inflation. If the primordial sources are small, the question arises how large is the non-gaussianity and B-mode background induced in second-order from the initially gaussian and scalar perturbations. In this paper we derive the Boltzmann hierarchy for the microwave background photon phase-space distributions at second …
Inhomogeneous recombination can give rise to perturbations in the electron number density which can be a factor of five larger than the perturbations in baryon density. We do a thorough analysis of the second order anisotropies generated in the cosmic microwave background (CMB) due to perturbations in the electron number density. We show that solving the second order Boltzmann equation for photons is equivalent to solving the first + second order Boltzmann equations and then taking the second order part of the solution. We find the approximate solution to the photon Boltzmann hierarchy in l mo…
Within cosmological perturbation theory, the cosmic microwave background anisotropies are usually computed from a Boltzmann hierarchy coupled to the perturbed Einstein equations. In this setup, one set of multipoles describes the temperature anisotropies, while two other sets, of electric and magnetic types, describe the polarization anisotropies. In order to reduce the number of multipoles types needed for polarization, and thus to speed up the numerical resolution, an optimal hierarchy has been proposed in the literature for Einstein-Boltzmann codes. However, it has been recently shown that …
Distortions to the cosmic microwave background~(CMB) blackbody spectrum are calculated in the framework of cosmological perturbation theory. The second order Boltzmann equation is explicitly solved, with the spectral $y$ distortion and the frequency independent second order temperature perturbation. We also solve higher order Boltzmann equations systematically and find new type spectral distortions in the low energy limit of electrons. As an example, we concretely construct a solution to the cubic order Boltzmann equation and show that it can be characterized by three parameters: a cubic order…
As wide-field surveys yield ever more precise measurements, cosmology has entered a phase of high precision requiring highly accurate and fast theoretical predictions. At the heart of most cosmological model predictions is a numerical solution of the Einstein-Boltzmann equations governing the evolution of linear perturbations in the Universe. We present PyCosmo, a new Python-based framework to solve this set of equations using a special pur- pose solver based on symbolic manipulations, automatic generation of C++ code and sparsity optimisation. The code uses a consistency relation of the field…
Within the framework of the interacting fluid formalism, we provide the numerical solution to the Boltzmann equation describing the evolution of an inflaton field coupled to radiation. We study the behavior of the system during the slow-roll regime, in the case in which an additional stochastic source term is included in the set of equations, and we recover the expression for the cosmological perturbations previously obtained in the Warm inflation scenarios.
In a time dependent background like de Sitter space, Feynman-Dyson perturbation theory breaks down due to infra-red divergences. We investigate an interacting scalar field theory in Schwinger-Keldysh formalism. We derive a Boltzmann equation from a Schwinger-Dyson equation inside the cosmological horizon. Our solution shows that the particle production is compensated by the reduction of the on-shell states due to unitarity. Although the degrees of freedom inside the horizon leads to a small and diminishing screening effect of the cosmological constant, there is a growing screening effect from …
The linear Einstein-Boltzmann equations describe the evolution of perturbations in the universe and its numerical solutions play a central role in cosmology. We revisit this system of differential equations and present a detailed investigation of its mathematical properties. For this purpose, we focus on a simplified set of equations aimed at describing the broad features of the matter power spectrum. We first perform an eigenvalue analysis and study the onset of oscillations in the system signaled by the transition from real to complex eigenvalues. We then provide a stability criterion of dif…
General Relativistic Entropic Acceleration at the perturbation level: a CLASS implementation and first Boltzmann-code constraints
Institute of Space Sciences (ICE, CSIC)（空间科学研究所）；School of Mathematical and Physical Sciences, University of Sheffield（谢菲尔德大学数学与物理科学学院）；Instituto de Física Teórica UAM-CSIC, Universidad Autonóma de Madrid（马德里自治大学-西班牙国家科学研究委员会理论物理研究所）
In this paper the non-linear effect of massive neutrinos on cosmological structures is studied in a conceptually new way. We have solved the non-linear continuity and Euler equations for the neutrinos on a grid in real space in $N$-body simulations, and closed the Boltzmann hierarchy at the non-linear Euler equation using the stress and pressure perturbations from linear theory. By comparing with state-of-the art cosmological neutrino simulations, we are able to simulate the non-linear neutrino power spectrum very accurately. This translates into a negligible error in the matter power spectrum…
We develop the transport operator formalism, a new line-of-sight integration framework to calculate the anisotropies of the Cosmic Microwave Background (CMB) at the linear and non-linear level. This formalism utilises a transformation operator that removes all inhomogeneous propagation effects acting on the photon distribution function, thus achieving a split between perturbative collisional effects at recombination and non-perturbative line-of-sight effects at later times. The former can be computed in the framework of standard cosmological perturbation theory with a second-order Boltzmann co…
We study linear cosmological perturbations in the ``healthy extension'' of Horava-Lifshitz gravity which has recently been analyzed \cite{BPS2}. We find that there are two degrees of freedom for scalar metric fluctuations, but that one of them decouples in the infrared limit. Also, for appropriate choices of the parameters defining the Lagrangian, the extra mode can be made well-behaved even in the ultraviolet.
In Horava's theory of gravity, Lorentz symmetry is broken in exchange for renormalizability, but the original theory has been argued to be plagued with problems associated with a new scalar mode stemming from the very breaking of Lorentz symmetry. Recently, Blas, Pujolas, and Sibiryakov have proposed a healthy extension of Horava gravity, in which the behavior of the scalar mode is improved. In this paper, we study scalar modes of cosmological perturbations in extended Horava gravity. The evolution of metric and density perturbations is addressed analytically and numerically. It is shown that …
