Analytic spectrum of relic gravitational waves modified by neutrino free streaming and dark energy

We include the effect of neutrino free streaming into the spectrum of relic gravitational waves (RGWs) in the currently accelerating universe. For the realistic case of a varying fractional neutrino energy density and a non-vanishing derivative of mode function at the neutrino decoupling, the integro-differential equation of RGWs is solved by a perturbation method for the period from the neutrino decoupling to the matter-dominant stage. Incorporating it to the analytic solution of the whole history of expansion of the universe, the analytic solution of GRWs is obtained, evolving from the inflation up to the current acceleration. The resulting spectrum of GRWs covers the whole range of frequency $(10^{-19}\sim 10^{10})$Hz, and improves the previous results. It is found that the neutrino free-streaming causes a reduction of the spectral amplitude by $\sim 20$ in the range $(10^{-16}\sim 10^{-10})$ Hz, and leaves the other portion of the spectrum almost unchanged. This agrees with the earlier numerical calculations. Examination is made on the difference between the accelerating and non-accelerating models, and our analysis shows that the ratio of the spectral amplitude in accelerating $\Lambda$CDM model over that in CDM model is $\sim 0.7$, and within the various accelerating models of $\Omega_{\Lambda}> \Omega_m$ the spectral amplitude is proportional to $\Omega_m/\Omega_{\Lambda}$ for the whole range of frequency. Comparison with LIGO S5 Runs Sensitivity shows that RGWs are not yet detectable by the present LIGO, and in the future LISA may be able to detect RGWs in some inflationary models.Comment: 22 pages,12 figures, accepeted by PR

Patterns of Scalable Bayesian Inference

Datasets are growing not just in size but in complexity, creating a demand for rich models and quantification of uncertainty. Bayesian methods are an excellent fit for this demand, but scaling Bayesian inference is a challenge. In response to this challenge, there has been considerable recent work based on varying assumptions about model structure, underlying computational resources, and the importance of asymptotic correctness. As a result, there is a zoo of ideas with few clear overarching principles. In this paper, we seek to identify unifying principles, patterns, and intuitions for scaling Bayesian inference. We review existing work on utilizing modern computing resources with both MCMC and variational approximation techniques. From this taxonomy of ideas, we characterize the general principles that have proven successful for designing scalable inference procedures and comment on the path forward

Effects of cold dark matter decoupling and pair annihilation on cosmological perturbations

Weakly interacting massive particles are part of the lepton-photon plasma in the early universe until kinetic decoupling, after which time the particles behave like a collisionless gas with nonzero temperature. The Boltzmann equation for WIMP-lepton collisions is reduced to a Fokker-Planck equation for the evolution of the WIMP distribution including scalar density perturbations. This equation and the Einstein and fluid equations for the plasma are solved numerically including the acoustic oscillations of the plasma before and during kinetic decoupling, the frictional damping occurring during kinetic decoupling, and the free-streaming damping occurring afterwards and throughout the radiation-dominated era. An excellent approximation reduces the solution to quadratures for the cold dark matter density and velocity perturbations. The subsequent evolution is followed through electron pair annihilation and the radiation-matter transition; analytic solutions are provided for both large and small scales. For a 100 GeV WIMP with bino-type interactions, kinetic decoupling occurs at a temperature $T_d=23$ MeV. The transfer function in the matter-dominated era leads to an abundance of small cold dark matter halos; with a smooth window function the Press-Schechter mass distribution is $dn/d\ln M\propto M^{-1/3}$ for $M<10^{-4} (T_d/$10 MeV)$^{-3}$ M$_\odot$.Comment: 18 pages, 12 figures; corrected error in bino decoupling temperature, figures update

Dilepton production from a viscous QGP

This work calculates the first correction to the leading order q\={q} dilepton production rates due to shear viscosity in an expanding gas. The modified rates are integrated over the space-time history of a viscous hydrodynamic simulation of RHIC collisions. The net result is a {\em hardening} of $q_\perp$ spectrum with the magnitude of the correction increasing with invariant mass. We argue that a thermal description is reliable for invariant masses less than $M_{max}\approx(2\tau_0 T_0^2)/(\eta/s)$. For reasonable values of the shear viscosity and thermalization time $M_{max}\approx 4.5$ GeV. Finally, the early emission from a viscous medium is compared to emission from a longitudinally free streaming plasma. Qualitative differences in $q_\perp$ spectrum are seen which could be used to extract information on the thermalization time, viscosity to entropy ratio and possibly the thermalization mechanism in heavy-ion collisions.Comment: 17 pages, 8 figure

Small scale aspects of warm dark matter : power spectra and acoustic oscillations

We provide a semi-analytic study of the small scale aspects of the power spectra of warm dark matter (WDM) candidates that decoupled while relativistic with arbitrary distribution functions. These are characterized by two widely different scales $k_{eq} \sim 0.01\,(\mathrm{Mpc})^{-1}$ and k_{fs}= \sqrt{3}\,k_{eq}/2\,^{1/2} with $^{1/2} \ll 1$ the velocity dispersion at matter radiation equality. Density perturbations evolve through three stages: radiation domination when the particle is relativistic and non-relativistic and matter domination. An early ISW effect during the first stage leads to an enhancement of density perturbations and a plateau in the transfer function for $k \lesssim k_{fs}$. An effective fluid description emerges at small scales which includes the effects of free streaming in initial conditions and inhomogeneities. The transfer function features \emph{WDM-acoustic oscillations} at scales $k \gtrsim 2 \,k_{fs}$. We study the power spectra for two models of sterile neutrinos with $m \sim \,\mathrm{keV}$ produced non-resonantly, at the QCD and EW scales respectively. The latter case yields acoustic oscillations on mass scales $\sim 10^{8}\,M_{\odot}$. Our results reveal a \emph{quasi-degeneracy} between the mass, distribution function and decoupling temperature suggesting caveats on the constraints on the mass of a sterile neutrino from current WDM N-body simulations and Lyman-$\alpha$ forest data. A simple analytic interpolation of the power spectra between large and small scales and its numerical implementation is given.Comment: 47 pages, 17 figures, section with comparison with Boltzmann code