A longitudinal gauge degree of freedom and the Pais Uhlenbeck field

We show that a longitudinal gauge degree of freedom for a vector field is equivalent to a Pais-Uhlenbeck scalar field. With the help of this equivalence, we can determine natural interactions of this field with scalars and fermions. Since the theory has a global U(1) symmetry, we have the usual conserved current of the charged fields, thanks to which the dynamics of the scalar field is not modified by the interactions. We use this fact to consistently quantize the theory even in the presence of interactions. We argue that such a degree of freedom can only be excited by gravitational effects like the inflationary era of the early universe and may play the role of dark energy in the form of an effective cosmological constant whose value is linked to the inflation scale.Comment: 20 pages, no figures. Minor changes and comments added to match the accepted version in JHE

The observed galaxy power spectrum in General Relativity

Measurements of the clustering of galaxies in Fourier space, and at low wavenumbers, offer a window into the early Universe via the possible presence of scale dependent bias generated by Primordial Non Gaussianites. On such large scales a Newtonian treatment of density perturbations might not be sufficient to describe the measurements, and a fully relativistic calculation should be employed. The interpretation of the data is thus further complicated by the fact that relativistic effects break statistical homogeneity and isotropy and are potentially divergent in the Infra-Red (IR). In this work we compute for the first time the ensemble average of the most used Fourier space estimator in spectroscopic surveys, including all general relativistic (GR) effects, and allowing for an arbitrary choice of angular and radial selection functions. We show that any observable is free of IR sensitivity once all the GR terms, individually divergent, are taken into account, and that this cancellation is a consequence of the presence of the Weinberg adiabatic mode as a solution to Einstein's equations. We then study the importance of GR effects, including lensing magnification, in the interpretation of the galaxy power spectrum multipoles, finding that they are in general a small, less than ten percent level, correction to the leading redshift space distortions term. This work represents the baseline for future investigations of the interplay between Primordial Non Gaussianities and GR effects on large scales and in Fourier space.Comment: 44 pages, 10 figure

The relativistic galaxy number counts in the weak field approximation

We present a novel approach to compute systematically the relativistic projection effects at any order in perturbation theory within the weak field approximation. In this derivation the galaxy number counts is written completely in terms of the redshift perturbation. The relativistic effects break the symmetry along the line-of-sight and they source, contrarily to the standard perturbation theory, the odd multipoles of the matter power spectrum or 2-point correlation function, providing a unique signature for their detection in Large Scale Structure surveys. We show that our approach agrees with previous derivations (up to third order) of relativistic effects and, for the first time, we derive a model for the transverse Doppler effect. Moreover, we proof that in the Newtonian limit this approach is consistent with standard perturbation theory at any order.Comment: 16 page

Modeling relativistic contributions to the halo power spectrum dipole

We study the power spectrum dipole of an N-body simulation which includes relativistic effects through ray-tracing and covers the low redshift Universe up to $z_{\rm max} = 0.465$ (RayGalGroup simulation). We model relativistic corrections as well as wide-angle, evolution, window and lightcone effects. Our model includes all relativistic corrections up to third-order including third-order bias expansion. We consider all terms which depend linearly on $\mathcal{H}/k$ (weak field approximation). We also study the impact of 1-loop corrections to the matter power spectrum for the gravitational redshift and transverse Doppler effect. We found wide-angle and window function effects to significantly contribute to the dipole signal. When accounting for all contributions, our dipole model can accurately capture the gravitational redshift and Doppler terms up to the smallest scales included in our comparison ($k=0.48\,h{\rm Mpc}^{-1}$), while our model for the transverse Doppler term is less accurate. We find the Doppler term to be the dominant signal for this low redshift sample. We use Fisher matrix forecasts to study the potential for the future Dark Energy Spectroscopic Instrument (DESI) to detect relativistic contributions to the power spectrum dipole. A conservative estimate suggests that the DESI-BGS sample should be able to have a detection of at least $4.4\sigma$, while more optimistic estimates find detections of up to $10\sigma$. Detecting these effects in the galaxy distribution allows new tests of gravity on the largest scales, providing an interesting additional science case for galaxy survey experiments.Comment: 45 pages, 10 figure

CMB-lensing beyond the leading order: temperature and polarization anisotropies

We investigate the weak lensing corrections to the CMB temperature and polarization anisotropies. We consider all the effects beyond the leading order: post-Born corrections, LSS corrections and, for the polarization anisotropies, the correction due to the rotation of the polarization direction between the emission at the source and the detection at the observer. We show that the full next-to-leading order correction to the B-mode polarization is not negligible on small scales and is dominated by the contribution from the rotation, this is a new effect not taken in account in previous works. Considering vanishing primordial gravitational waves, the B-mode correction due to rotation is comparable to cosmic variance for $\ell \gtrsim 3500$, in contrast to all other spectra where the corrections are always below that threshold for a single multipole. Moreover, the sum of all the effects is larger than cosmic variance at high multipoles, showing that higher-order lensing corrections to B-mode polarization are in principle detectable.Comment: 32 pages, 6 figures. New results about the signal-to-noise amplitude for next-to-leading order corrections, further clarifications about the polarization rotation and references added. Version accepted for publication in Physical Review