The EFT Likelihood for Large-Scale Structure in Redshift Space

We study the EFT likelihood for biased tracers in redshift space, for which the bias expansion of the galaxy velocity field $\mathbf{v}_g$ plays a fundamental role. The equivalence principle forbids stochastic contributions to $\mathbf{v}_g$ to survive at small $k$. Therefore, at leading order in derivatives the form of the likelihood ${\cal P}[\tilde{\delta}_g|\delta,\!\mathbf{v}]$ to observe a redshift-space galaxy overdensity $\tilde{\delta}_g(\tilde{\mathbf{x}})$ given a rest-frame matter and velocity fields $\delta(\mathbf{x})$, $\mathbf{v}(\mathbf{x})$ is fixed by the rest-frame noise. If this noise is Gaussian with constant power spectrum, ${\cal P}[\tilde{\delta}_g|\delta,\!\mathbf{v}]$ is also a Gaussian in the difference between $\tilde{\delta}_g(\tilde{\mathbf{x}})$ and its bias expansion: redshift-space distortions only make the covariance depend on $\delta(\mathbf{x})$ and $\mathbf{v}(\mathbf{x})$. We then show how to match this result to perturbation theory, and that one can consistently neglect the field-dependent covariance if the bias expansion is stopped at second order in perturbations. We discuss qualitatively how this affects numerical implementations of the EFT-based forward modeling, and how the picture changes when the survey window function is taken into account.Comment: 33 pages (29+4), 1 figure, matches published versio

How Gaussian can our Universe be?

Gravity is a non-linear theory, and hence, barring cancellations, the initial super-horizon perturbations produced by inflation must contain some minimum amount of mode coupling, or primordial non-Gaussianity. In single-field slow-roll models, where this lower bound is saturated, non-Gaussianity is controlled by two observables: the tensor-to-scalar ratio, which is uncertain by more than fifty orders of magnitude; and the scalar spectral index, or tilt, which is relatively well measured. It is well known that to leading and next-to-leading order in derivatives, the contributions proportional to the tilt disappear from any local observable, and suspicion has been raised that this might happen to all orders, allowing for an arbitrarily low amount of primordial non-Gaussianity. Employing Conformal Fermi Coordinates, we show explicitly that this is not the case. Instead, a contribution of order the tilt appears in local observables. In summary, the floor of physical primordial non-Gaussianity in our Universe has a squeezed-limit scaling of $k_\ell^2/k_s^2$, similar to equilateral and orthogonal shapes, and a dimensionless amplitude of order $0.1\times(n_\mathrm{s}-1)$.Comment: 26 + 18 pages, 2 figures. References added and minor typos corrected. Matches published versio

The EFT Likelihood for Large-Scale Structure

We derive, using functional methods and the bias expansion, the conditional likelihood for observing a specific tracer field given an underlying matter field. This likelihood is necessary for Bayesian-inference methods. If we neglect all stochastic terms apart from the ones appearing in the auto two-point function of tracers, we recover the result of Schmidt et al., 2018. We then rigorously derive the corrections to this result, such as those coming from a non-Gaussian stochasticity (which include the stochastic corrections to the tracer bispectrum) and higher-derivative terms. We discuss how these corrections can affect current applications of Bayesian inference. We comment on possible extensions to our result, with a particular eye towards primordial non-Gaussianity. This work puts on solid theoretical grounds the EFT-based approach to Bayesian forward modeling.Comment: 53 pages (36+17), 4 tables. v2: matches JCAP version. Added section to compare with Schmidt et al., 2018; added plot to show relative importance of different contributions to log-likelihoo

A new scale in the bias expansion

The fact that the spatial nonlocality of galaxy formation is controlled by some short length scale like the Lagrangian radius is the cornerstone of the bias expansion for large-scale-structure tracers. However, the first sources of ionizing radiation between $z\approx 15$ and $z\approx 6$ are expected to have significant effects on the formation of galaxies we observe at lower redshift, at least on low-mass galaxies. These radiative-transfer effects introduce a new scale in the clustering of galaxies, i.e. the finite distance which ionizing radiation travels until it reaches a given galaxy. This mean free path can be very large, of order $100\,h^{-1}\,{\rm Mpc}$. Consequently, higher-derivative terms in the bias expansion could turn out to be non-negligible even on these scales: treating them perturbatively would lead to a massive loss in predictivity and, for example, could spoil the determination of the BAO feature or constraints on the neutrino mass. Here, we investigate under what assumptions an explicit non-perturbative model of radiative-transfer effects can maintain the robustness of large-scale galaxy clustering as a cosmological probe.Comment: 33 pages, 5 figures, 2 tables. Added discussion on time dependence of galaxy response. Extended conclusions with discussion on velocity bias and redshift-space distortions. Main results unchange

CMB anisotropies and spectral distortions: constraining inflation at small scales

Anisotropies in the angular power spectra of the Cosmic Microwave Background (CMB) temperature and polarization are sourced by inflationary perturbations on scales from 10^1 Mpc to 10^4 Mpc. Deviations of the CMB frequency spectrum from a black-body, instead, can probe inflationary perturbations on scales from 10^−4 Mpc to 10^−2 Mpc. These length scales are inaccessible to CMB and large-scale structure measurements. CMB spectral distortions, averaged over the whole sky, constrain the two-point function of primordial perturbations. Correlation of temperature and spectral distortion anisotropies, instead, can constrain their three-point function (making them a probe of primordial non-Gaussianity). In the first part of this thesis I study what is the level of sensitivity needed, by an experiment measuring the CMB frequency spectrum, to detect the running of the spectral index of inflationary perturbations. I then investigate what is the minimal contribution to the correlation function between temperature and spectral distortion anisotropies that is expected in standard inflationary scenarios. Finally, I discuss what are the secondary contributions (arising from late-time gravitational evolution) to such angular correlation, and how they could bias constraints on primordial non-Gaussianity

The Likelihood for LSS: Stochasticity of Bias Coefficients at All Orders

In the EFT of biased tracers the noise field $\varepsilon_g$ is not exactly uncorrelated with the nonlinear matter field $\delta$. Its correlation with $\delta$ is effectively captured by adding stochasticities to each bias coefficient. We show that if these stochastic fields are Gaussian (the impact of their non-Gaussianity being subleading on quasi-linear scales anyway) it is possible to resum exactly their effect on the conditional likelihood ${\cal P}[\delta_g|\delta]$ to observe a galaxy field $\delta_g$ given an underlying $\delta$. This resummation allows to take them into account in EFT-based approaches to Bayesian forward modeling. We stress that the resulting corrections to a purely Gaussian conditional likelihood with white-noise covariance are the most relevant on scales where the EFT is under control: they are more important than any non-Gaussianity of the noise $\varepsilon_g$.Comment: 15 pages. v2: very minor change in Eqs. (2.6), (2.8), (2.17), (2.19), (3.16), (3.18), (3.19) to show factorization in real space of the normalization of the likelihood, updated acknowledgements. v3: added section on renormalization, matches published versio

Constraints on the early and late integrated Sachs-Wolfe effects from the Planck 2015 cosmic microwave background anisotropies in the angular power spectra

The Integrated Sachs-Wolfe (ISW) effect predicts additional anisotropies in the Cosmic MicrowaveBackground due to time variation of the gravitational potential when the expansion of the universeis not matter dominated. The ISW effect is therefore expected in the early universe, due to thepresence of relativistic particles at recombination, and in the late universe, when dark energy startsto dominate the expansion. Deviations from the standard picture can be parameterized byAeISWandAlISW, which rescale the overall amplitude of the early and late ISW effects. Analyzing themost recent CMB temperature spectra from the Planck 2015 release, we detect the presence of theearly ISW at high significance withAeISW= 1.06±0.04 at 68% CL and an upper limit for thelate ISW ofAlISW<1.1 at 95% CL. The inclusion of the recent polarization data from the Planckexperiment erases such 1.5σhint forAeISW6= 1. When considering the recent detections of the lateISW coming from correlations between CMB temperature anisotropies and weak lensing, a value ofAlISW= 0.85±0.21 is predicted at 68% CL, showing a 4σevidence. We discuss the stability of ourresult in the case of an extra relativistic energy component parametrized by the effective neutrinonumberNeffand of a CMB lensing amplitudeA

Running the running

We use the recent observations of Cosmic Microwave Background temperature and polarization anisotropies provided by the Planck satellite experiment to place constraints on the running $\alpha_\mathrm{s} = \mathrm{d}n_{\mathrm{s}} / \mathrm{d}\log k$ and the running of the running $\beta_{\mathrm{s}} = \mathrm{d}\alpha_{\mathrm{s}} / \mathrm{d}\log k$ of the spectral index $n_{\mathrm{s}}$ of primordial scalar fluctuations. We find $\alpha_\mathrm{s}=0.011\pm0.010$ and $\beta_\mathrm{s}=0.027\pm0.013$ at $68\%\,\mathrm{CL}$, suggesting the presence of a running of the running at the level of two standard deviations. We find no significant correlation between $\beta_{\mathrm{s}}$ and foregrounds parameters, with the exception of the point sources amplitude at $143\,\mathrm{GHz}$, $A^{PS}_{143}$, which shifts by half sigma when the running of the running is considered. We further study the cosmological implications of such preference for $\alpha_\mathrm{s},\beta_\mathrm{s}\sim0.01$ by including in the analysis the lensing amplitude $A_L$, the curvature parameter $\Omega_k$, and the sum of neutrino masses $\sum m_{\nu}$. We find that when the running of the running is considered, Planck data are more compatible with the standard expectations of $A_L = 1$ and $\Omega_k = 0$ but still hint at possible deviations. The indication for $\beta_\mathrm{s} > 0$ survives at two standard deviations when external datasets such as BAO and CFHTLenS are included in the analysis, and persists at $\sim 1.7$ standard deviations when CMB lensing is considered. We discuss the possibility of constraining $\beta_\mathrm{s}$ with current and future measurements of CMB spectral distortions, showing that an experiment like PIXIE could provide strong constraints on $\alpha_\mathrm{s}$ and $\beta_\mathrm{s}$.Comment: 10+1 pages, 9 figures, 10 tables. Matches published versio

Imprints of Oscillatory Bispectra on Galaxy Clustering

Long-short mode coupling during inflation, encoded in the squeezed bispectrum of curvature perturbations, induces a dependence of the local, small-scale power spectrum on long-wavelength perturbations, leading to a scale-dependent halo bias. While this scale dependence is absent in the large-scale limit for single-field inflation models that satisfy the consistency relation, certain models such as resonant non-Gaussianity show a peculiar behavior on intermediate scales. We reconsider the predictions for the halo bias in this model by working in Conformal Fermi Coordinates, which isolate the physical effects of long-wavelength perturbations on short-scale physics. We find that the bias oscillates with scale with an envelope similar to that of equilateral non-Gaussianity. Moreover, the bias shows a peculiar modulation with the halo mass. Unfortunately, we find that upcoming surveys will be unable to detect the signal because of its very small amplitude. We also discuss non-Gaussianity due to interactions between the inflaton and massive fields: our results for the bias agree with those in the literature.Comment: 27+15 pages, 6 figures, matches JCAP versio

Spectral distortion anisotropies from single-field inflation

Distortions of the Cosmic Microwave Background energy spectrum of the $\mu$ type are sensitive to the primordial power spectrum through the dissipation of curvature perturbations on scales $k\simeq 50$ - $10^4\,\mathrm{Mpc}^{-1}$. Their angular correlation with large-scale temperature anisotropies is then sensitive to the squeezed limit of the primordial bispectrum. For inflationary models obeying the single-field consistency relation, we show that the observed $\mu T$ angular correlation that would correspond to the local shape vanishes exactly. All leading non-primordial contributions, including all non-linear production and projection effects, are of the "equilateral shape", namely suppressed by $k^2/{\cal H}_f^2$, where ${\cal H}_f\simeq 10^{-1}\,{\rm Mpc}^{-1}$ is the Hubble radius at the end of the $\mu$-era. Therefore, these non-primordial contributions are orthogonal to a potential local primordial signal (e.g. from multi-field inflation). Moreover, they are very small in amplitude. Our results strengthen the position of $\mu$ distortions as the ultimate probe of local primordial non-Gaussianity.Comment: 34 pages (21+13), 2 figures, matches JCAP versio