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 moreso for three different neutrino mass eigenstates. Therefore, it has the potential to significantly speed up the computation times of cosmological-perturbation calculations. This approach should also be applicable to models with other non-cold collisionless relics.
Boltzmann solvers are an important tool for the computation of cosmological observables in the linear regime. In the presence of massive neutrinos, they involve solving the Boltzmann equation followed by an integration in momentum space to arrive at the desired fluid properties, a procedure which is known to be computationally slow. In this work we introduce the so-called generalized Boltzmann hierarchy (GBH) for massive neutrinos in cosmology, an alternative to the usual Boltzmann hierarchy, where the momentum dependence is integrated out leaving us with a two-parameter infinite set of ordina…
In this paper, we formulate a generalised photon Boltzmann hierarchy that allows us to model the evolution and creation of spectral distortion anisotropies in the early Universe. We directly build on our first paper in this series, extending the thermalisation Green's function treatment to the anisotropic case. We show that the problem can be described with the common Boltzmann hierarchy for the photon field extended by new spectral parameters -- a step that reduces the complexity of the calculation by at least two orders of magnitude. Our formalism describes the effects of i) Doppler and pote…
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 …
Starting from the collisional Boltzmann equation, we derive for the first time and from first principles the Boltzmann hierarchy for neutrinos including interactions with a scalar particle. Such interactions appear, for example, in majoron-like models of neutrino mass generation. We study two limits of the scalar mass: (i) An extremely massive scalar whose only role is to mediate an effective 4-fermion neutrino-neutrino interaction, and (ii) a massless scalar that can be produced in abundance and thus demands its own Boltzmann hierarchy. In contrast to, e.g., the first-order Boltzmann hierarch…
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…
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 …
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…
Kinetic decoupling of dark matter typically happens much later than chemical freeze-out. In fact, local thermal equilibrium is an important assumption for the usual relic density calculations based on solving the Boltzmann equation (for its 0-th moment) describing the dark matter number density. But is this assumption always justified? Here we address this question and discuss the consequences of more accurate treatments. The first treatment is relying on the inclusion of higher moments of the Boltzmann equation and the second on solving the evolution of the phase-space distribution function f…
Nonzero neutrino masses are required by the existence of flavour oscillations, with values of the order of at least 50 meV. We consider the gravitational clustering of relic neutrinos within the Milky Way, and used the $N$-one-body simulation technique to compute their density enhancement factor in the neighbourhood of the Earth with respect to the average cosmic density. Compared to previous similar studies, we pushed the simulation down to smaller neutrino masses, and included an improved treatment of the baryonic and dark matter distributions in the Milky Way. Our results are important for …
We present a new flexible, fast and accurate way to implement massive neutrinos, warm dark matter and any other non-cold dark matter relics in Boltzmann codes. For whatever analytical or numerical form of the phase-space distribution function, the optimal sampling in momentum space compatible with a given level of accuracy is automatically found by comparing quadrature methods. The perturbation integration is made even faster by switching to an approximate viscous fluid description inside the Hubble radius, which differs from previous approximations discussed in the literature. When adding one…
Analysis of cosmic microwave background radiation fluctuations favors an effective number of neutrinos, $N_ν>3$. This motivates a reinvestigation of the neutrino freeze-out process. Here we characterize the dependence of $N_ν$ on the Standard Model (SM) parameters that govern neutrino freeze-out. We show that $N_ν$ depends on a combination $η$ of several natural constants characterizing the relative strength of weak interaction processes in the early Universe and on the Weinberg angle $\sin^2θ_W$. We determine numerically the dependence $N_ν(η,\sin^2θ_W)$ and discuss these results. The extensi…
The impact of dark matter-neutrino interactions on the measurement of the cosmological parameters has been investigated in the past in the context of massless neutrinos exclusively. Here we revisit the role of a neutrino-dark matter coupling in light of ongoing cosmological tensions by implementing the full Boltzmann hierarchy for three massive neutrinos. Our tightest 95% CL upper limit on the strength of the interactions, parameterized via $u_χ=\frac{σ_0}{σ_{Th}}\left(\frac{m_χ}{100 \text{GeV}}\right)^{-1}$, is $u_χ\leq3.34 \cdot 10^{-4}$, arising from a combination of Planck TTTEEE data, Pla…
In this dissertation, we study the evolution and properties of the relic (or cosmic) neutrino distribution from neutrino freeze-out at $T=O(1)$ MeV through the free-streaming era up to today, focusing on the deviation of the neutrino spectrum from equilibrium. In particular, we demonstrate the presence of chemical non-equilibrium that continues to the present day. The work naturally separates into two parts. The first focuses on aspects of the relic neutrinos that can be explored using conservation laws. The second part studies the neutrino distribution using the full general relativistic Bolt…
Non-thermal dark matter generation is an appealing alternative to the standard paradigm of thermal WIMP dark matter. We reconsider non-thermal production mechanisms in a systematic way, and develop a numerical code for accurate computations of the dark matter relic density. We discuss in particular scenarios with long-lived massive states decaying into dark matter particles, appearing naturally in several beyond the standard model theories, such as supergravity and superstring frameworks. Since non-thermal production favors dark matter candidates with large pair annihilation rates, we analyze …
We investigate the thermal relics as hot, warm and cold dark matter in $\mathscr{L}=\varepsilon^{2-2β}R^β+{16π}m_{\text{Pl}}^{-2}\mathscr{L}_m$ gravity, where $\varepsilon$ is a constant balancing the dimension of the field equation, and $1<β<(4+\sqrt{6})/5$ for the positivity of energy density and temperature. If light neutrinos serve as hot/warm relics, the entropic number of statistical degrees of freedom $g_{*s}$ at freeze-out and thus the predicted fractional energy density $Ω_ψh^2$ are $β-$dependent, which relaxes the standard mass bound $Σm_ν$. For cold relics, by exactly solve the simp…
We study in detail how neutrino perturbations can be followed in linear theory by using only terms up to $l=2$ in the Boltzmann hierarchy. We provide a new approximation to the third moment and demonstrate that the neutrino power spectrum can be calculated to a precision of better than $\sim$ 5% for masses up to $\sim$ 1 eV. The matter and CMB power spectra can be calculated far more precisely and typically at least a factor of a few better than with existing approximations. We then proceed to study how the neutrino power spectrum can be reliably calculated even in the presence of non-linear g…
We provide a general framework for self-interacting warm dark matter (WDM) in cosmological perturbations, by deriving from first principles a Boltzmann hierarchy which retains certain independence from a particular interaction Lagrangian. We consider elastic interactions among the massive particles, and obtain a hierarchy which is more general than the ones usually obtained for non-relativistic (as for cold DM) or for ultra-relativistic (as for neutrinos) approximations. The more general momentum-dependent kernel integrals in the Boltzmann collision terms, are explicitly calculated for differe…
We solve coupled momentum-dependent Boltzmann equations for the phase space distribution of cosmic relic particles, without resorting to approximations of assuming kinetic equilibrium or neglecting backscattering or elastic interactions. Our method is amendable to precision numerical computations. To test it, we consider two benchmark models where the momentum dependence of dark matter distribution function is potentially important: a real singlet scalar extension near the Higgs resonance and a sterile neutrino dark matter model with a singlet scalar mediator. The singlet scalar example shows …
We consider a scalar field (called $ϕ$) which is very weakly coupled to thermal bath, and study the evolution of its number density. We use the Boltzmann equation derived from the Kadanoff-Baym equations, assuming that the degrees of freedom in the thermal bath are well described as "quasi-particles." When the widths of quasi-particles are negligible, the evolution of the number density of $ϕ$ is well governed by a simple Boltzmann equation, which contains production rates and distribution functions both evaluated with dispersion relations of quasi-particles with thermal masses. We pay particu…
The relic neutrinos are expected to acquire a bulk relative velocity with respect to the dark matter at low redshifts, and neutrino wakes are expected to develop downstream of the dark matter halos. We propose a method of measuring the neutrino mass based on this mechanism. This neutrino wake will cause a dipole distortion of the galaxy-galaxy lensing pattern. This effect could be detected by combining upcoming lensing surveys with a low redshift galaxy survey or a 21 cm intensity mapping survey, which can map the neutrino flow field. The data obtained with LSST and Euclid should enable us to …
The neutrino asymmetry in the early universe plasma, $n_ν- n_{\bar ν}$, is calculated both before and after the electroweak phase transition (EWPT). The leptogenesis before EWPT within the standard model is well known to be driven by the abelian anomaly in a massless hypercharge field. The generation of the neutrino asymmetry in the Higgs phase after EWPT, in its turn, has not been considered previously because of the absence of any quantum anomaly in an external electromagnetic field for such electroneutral particles as neutrino, unlike the Adler anomaly for charged left and right polarized m…
Exponential suppression or commonly known as the Boltzmann suppression in the number density of dark matter is the key ingredient for creating chemical imbalance prior to the usual thermal freeze-out. A degenerate/quasi-degenerate dark sector can experience a different exponential suppression in the number density analogous to the radioactive decay law leading to a delayed freeze-out mechanism of dark matter known as the co-decaying dark matter. In this work, we study the dynamics of a multicomponent dark matter from thermally decoupled degenerate dark sector in a hidden U$(1)_{X}$ extension o…
