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We study the Sunyaev-Zeldovich effect for clusters of galaxies. We explore the relativistic corrections to the Kompaneets equation in terms of two different expansion approximation schemes, namely, the Fokker-Planck expansion approximation and delta function expansion approximation. We show that two expansion approximation formalisms are equivalent under the Thomson approximation, which is extremely good approximation for the CMB photon energies. This will clarify the situation for existing theoretical methods to analyse observation data.
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High-precision constraints on primordial non-Gaussianity (PNG) will significantly improve our understanding of the physics of the early universe. Among all the subtleties in using large scale structure observables to constrain PNG, accounting for relativistic corrections to the clustering statistics is particularly important for the upcoming galaxy surveys covering progressively larger fraction of the sky. We focus on relativistic projection effects due to the fact that we observe the galaxies through the light that reaches the telescope on perturbed geodesics. These projection effects can give rise to an effective $f_{rm NL}$ that can be misinterpreted as the primordial non-Gaussianity signal and hence is a systematic to be carefully computed and accounted for in modelling of the bispectrum. We develop the technique to properly account for relativistic effects in terms of purely observable quantities, namely angles and redshifts. We give some examples by applying this approach to a subset of the contributions to the tree-level bispectrum of the observed galaxy number counts calculated within perturbation theory and estimate the corresponding non-Gaussianity parameter, $f_{rm NL}$, for the local, equilateral and orthogonal shapes. For the local shape, we also compute the local non-Gaussianity resulting from terms obtained using the consistency relation for observed number counts. Our goal here is not to give a precise estimate of $f_{rm NL}$ for each shape but rather we aim to provide a scheme to compute the non-Gaussian contamination due to relativistic projection effects. For the terms considered in this work, we obtain contamination of $f_{rm NL}^{rm loc} sim {mathcal O}(1)$.
We derive a `Kompaneets equation for neutrinos, which describes how the distribution function of neutrinos interacting with matter deviates from a Fermi-Dirac distribution with zero chemical potential. To this end, we expand the collision integral in the Boltzmann equation of neutrinos up to the second order in energy transfer between matter and neutrinos. The distortion of the neutrino distribution function changes the rate at which neutrinos heat matter, as the rate is proportional to the mean square energy of neutrinos, $E_ u^2$. For electron-type neutrinos the enhancement in $E_ u^2$ over its thermal value is given approximately by $E_ u^2/E_{ u,rm thermal}^2=1+0.086(V/0.1)^2$ where $V$ is the bulk velocity of nucleons, while for the other neutrino species the enhancement is $(1+delta_v)^3$, where $delta_v=mV^2/3k_BT$ is the kinetic energy of nucleons divided by the thermal energy. This enhancement has a significant implication for supernova explosions, as it would aid neutrino-driven explosions.
Relativistic corrections are estimated for classical Cepheids and the Tip of the Red Giant Branch (TRGB stars), to enable future unbiased 1% measurements of Hubbles constant, $H_0$. We considered four effects: $K-$corrections, time-dilation, the apparent change of host dust extinction due to non-comoving reference frames, and the change of observed color due to redshift. Extinction-dependent $K-$corrections were computed using stellar atmosphere models applicable to giant stars for $0.005 < z < 0.03$ in HST, JWST, and 2MASS filters. The optical-NIR Wesenheit function advantageously combines filters with oppositely signed $K-$corrections and avoids complications due to host extinction. For TRGB stars, the JWST/NIRCAM F277W filter combines insensitivity to reddening with $K-$corrections $<1$% at Coma cluster distances. Missing corrections for host extinction due to circumgalactic or circumstellar material are discussed as potential systematics for TRGB distances although their impacts are insufficient to explain differences between $H_0$ based on Cepheid or TRGB supernova calibrations. All stellar standard candles require relativistic corrections to achieve an unbiased 1% $H_0$ measurement in the future. The combined relativistic correction involving $K$, redshift-Leavitt bias, and the redshift-dependence of the Wesenheit function yield an increase of the Cepheid-based $H_0$ by $0.45 pm 0.05$ km/s/Mpc to $73.65 pm 1.30$ km/s/Mpc and raises the tension with the {it Planck} value from $4.2sigma$ to $4.4sigma$. For TRGB stars, we estimate a $sim 0.5%$ increase of $H_0$ reported by Freedman et al. (to $70.2pm1.7$km/s/Mpc) and a small decrease by $-0.15%$ for $H_0$ reported by Anand et al. (to $71.4 pm 1.8$km/s/Mpc). The opposite sign of these corrections is due to different reddening systematics and reduces the difference between the studies by $sim 0.46$km/s/Mpc.[abridged]
The thermal Sunyaev-Zeldovich (tSZ) effect is produced by the interaction of cosmic microwave background (CMB) photons with the hot (a few keV) and diffuse gas of electrons inside galaxy clusters integrated along the line of sight. This effect produces a distortion of CMB blackbody emission law. This distortion law depends on the electronic temperature of the intra-cluster hot gas, $T_{e}$, through the so-called tSZ relativistic corrections. In this work, we have performed a statistical analysis of the tSZ spectral distortion on large galaxy cluster samples. We performed a stacking analysis for several electronic temperature bins, using both spectroscopic measurements of X-ray temperatures and a scaling relation between X-ray luminosities and electronic temperatures. We report the first high significance detection of the relativistic tSZ at a significance of 5.3 $sigma$. We also demonstrate that the observed tSZ relativistic corrections are consistent with X-ray deduced temperatures. This measurement of the tSZ spectral law demonstrates that tSZ effect spectral distorsion can be used as a probe to measure galaxy cluster temperatures.
In low-density or high-temperature plasmas, Compton scattering is the dominant process responsible for energy transport. Kompaneets in 1957 derived a non-linear degenerate parabolic equation for the photon energy distribution. In this paper we consider a simplified model obtained by neglecting diffusion of the photon number density in a particular way. We obtain a non-linear hyperbolic PDE with a position-dependent flux, which permits a one-parameter family of stationary entropy solutions to exist. We completely describe the long-time dynamics of each non-zero solution, showing that it approaches some non-zero stationary solution. While the total number of photons is formally conserved, if initially large enough it necessarily decreases after finite time through an out-flux of photons with zero energy. This corresponds to formation of a Bose-Einstein condensate, whose mass we show can only increase with time.
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