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

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

On Graviton non-Gaussianities in the Effective Field Theory of Inflation

We derive parity-even graviton bispectra in the Effective Field Theory of Inflation (EFToI) to all orders in derivatives. Working in perturbation theory, we construct all cubic interactions that can contribute to tree-level graviton bispectra, showing that they all come from EFToI operators containing two or three powers of the extrinsic curvature and its covariant derivatives: all other operators can be removed by field redefinitions or start at higher-order in perturbations. For operators cubic in the extrinsic curvature, where the single-clock consistency relations are satisfied without a correction to the graviton two-point function, we use the Manifestly Local Test (MLT) to efficiently extract the effects of evolving graviton fluctuations to the end of inflation. Despite the somewhat complicated nature of the bulk interactions, the final boundary correlators take a very compact form. For operators quadratic in the extrinsic curvature, the leading order bispectra are a sum of contact and single exchange diagrams, which are tied together by spatial diffeomorphisms, and to all orders in derivatives we derive these bispectra by computing the necessary bulk time integrals. For single exchange diagrams we exploit factorisation properties of the bulk-bulk propagator for massless gravitons and write the result as a finite sum over residues. Perhaps surprisingly, we show these single exchange contributions have only total-energy poles and also satisfy the MLT