Search

We study the $S$-matrix of Goldstones in the renormalizable theory of a $U(1)$ complex scalar at finite charge, i.e. in a state that breaks Lorentz invariance. The theory is weakly coupled so that this $S$-matrix exists at all energies. Unlike the Lorentz invariant case, the resulting $S$-matrix is not analytic in the exchanged (complexified) four-momentum. The non-analyticities stem from the LSZ reduction formula, as a consequence of the energy-dependent mixing between the radial and Goldstone modes., Comment: 18+11 pages

We derive the kernels and the Effective Field Theory of Large-Scale Structure counterterms for the one-loop bispectrum of dark matter and of biased tracers in real and redshift space. This requires the expansion of biased tracers up to fourth order in fluctuations. In the process, we encounter several subtleties related to renormalization. One is the fact that, in renormalizing the momentum, a local counterterm contributes non-locally. A second subtlety is related to the renormalization of local products of the velocity fields, which need to be expressed in terms of the renormalized velocity in order to preserve Galilean symmetry. We check that the counterterms we identify are necessary and sufficient to renormalize the one-loop bispectrum at leading and subleading order in the derivative expansion. The kernels that we originally present here have already been used for the first analyses of the one-loop bispectrum in BOSS data [1, 2]., Comment: 39 + 28 pages, typos corrected, some expanded comments, ancillary Mathematica file in "Other formats"

We study the onset of inflation in 3+1 dimensional cosmologies with an inflationary potential $U$ satisfying $0 < \Lambda_1 \leq U \leq \Lambda_2$, matter satisfying the dominant and strong energy conditions, and with spatial slices that can be foliated by 2-dimensional surfaces that are orbits under an isometry group. Assuming an initial Cauchy slice with positive mean curvature everywhere, we show, via mean curvature flow, that there exists a family of spatial slices parameterized by $\lambda$, whose volume grows between the flat slicings in de Sitter spaces with cosmological constants $\Lambda_1$ and $\Lambda_2$. In particular, inflationary expansion indeed occurs in this setting with inhomogeneous initial conditions. Finally, we apply this "inflationary time coordinate" $\lambda$ to study asymptotics of the variation in the metric, the average stress-energy tensor, and the dynamics of an inflaton field on a spatial slice., Comment: 29 pages, 2 figures

The disagreement between direct late-time measurements of the Hubble constant from the SH0ES collaboration, and early-universe measurements based on the $\Lambda$CDM model from the Planck collaboration might, at least in principle, be explained by new physics in the early universe. Recently, the application of the Effective Field Theory of Large-Scale Structure to the full shape of the power spectrum of the SDSS/BOSS data has revealed a new, rather powerful, way to measure the Hubble constant and the other cosmological parameters from Large-Scale Structure surveys. In light of this, we analyze two models for early universe physics, Early Dark Energy and Rock 'n' Roll, that were designed to significantly ameliorate the Hubble tension. Upon including the information from the full shape to the Planck, BAO, and Supernovae measurements, we find that the degeneracies in the cosmological parameters that were introduced by these models are well broken by the data, so that these two models do not significantly ameliorate the tension., Comment: 15+12 pages, 7 figures. v2: Minor corrections, references added. v3: Extended appendices

We study, using Mean Curvature Flow methods, 3+1 dimensional cosmologies with a positive cosmological constant, matter satisfying the dominant and the strong energy conditions, and with spatial slices that can be foliated by 2-dimensional surfaces that are the closed orbits of a symmetry group. If these surfaces have non-positive Euler characteristic (or in the case of 2-spheres, if the initial 2-spheres are large enough) and also if the initial spatial slice is expanding everywhere, then we prove that asymptotically the spacetime becomes physically indistinguishable from de Sitter space on arbitrarily large regions of spacetime. This holds true notwithstanding the presence of initial arbitrarily-large density fluctuations., Comment: 42 pages, 3 figures

We apply the Effective Field Theory of Large-Scale Structure to analyze the $w$CDM cosmological model. By using the full shape of the power spectrum and the BAO post-reconstruction measurements from BOSS, the Supernovae from Pantheon, and a prior from BBN, we set the competitive CMB-independent limit $w=-1.046_{-0.052}^{+0.055}$ at $68\%$ C.L.. After adding the Planck CMB data, we find $w=-1.023_{-0.030}^{+0.033}$ at $68\%$ C.L.. Our results are obtained using PyBird, a new, fast Python-based code which we make publicly available., Comment: 27 pages, 6 tables, 8 figures. v2: 33 pages, 8 tables, 9 figures, matches JCAP published version. The PyBird code is available at https://github.com/pierrexyz/pybird

Gravitational-wave observations of coalescing binary systems allow for novel tests of the strong-field regime of gravity. Using data from the Gravitational Wave Open Science Center (GWOSC) of the LIGO and Virgo detectors, we place the first constraints on an effective field-theory based extension of General Relativity in which only higher-order curvature terms are added to the Einstein-Hilbert action. We construct gravitational-wave templates describing the quasi-circular, adiabatic inspiral phase of binary black holes in this extended theory of gravity. Then, after explaining how to properly take into account the region of validity of the effective field theory when performing tests of General Relativity, we perform Bayesian model selection using the two lowest-mass binary black-hole events reported to date by LIGO and Virgo -- GW151226 and GW170608 -- and constrain this theory with respect to General Relativity. We find that these data disfavors the appearance of new physics on distance scales around $\sim 150$ km. Finally, we describe a general strategy for improving constraints as more observations will become available with future detectors on the ground and in space., Comment: 13 pages, 6 figures. v2: matches published version

We resolve the issue of infrared divergences present in theories of light scalar fields on de Sitter space., Comment: 95 pages, 12 figures

The precision of the cosmological data allows us to accurately approximate the predictions for cosmological observables by Taylor expanding up to a low order the dependence on the cosmological parameters around a reference cosmology. By applying this observation to the redshift-space one-loop galaxy power spectrum of the Effective Field Theory of Large-Scale Structure, we analyze the BOSS DR12 data by scanning over all the parameters of $\Lambda$CDM cosmology with massive neutrinos. We impose several sets of priors, the widest of which is just a Big Bang Nucleosynthesis prior on the current fractional energy density of baryons, $\Omega_b h^2$, and a bound on the sum of neutrino masses to be less than 0.9 eV. In this case we measure the primordial amplitude of the power spectrum, $A_s$, the abundance of matter, $\Omega_m$, the Hubble parameter, $H_0$, and the tilt of the primordial power spectrum, $n_s$, to about $19\%$, $5.7\%$, $2.2\%$ and $7.3\%$ respectively, obtaining $\ln ( 10^{10} A_s) =2.91\pm 0.19$, $\Omega_m=0.314\pm 0.018$, $H_0=68.7\pm 1.5$ km/(s Mpc) and $n_s=0.979\pm 0.071$ at $68\%$ confidence level. A public code is released with this preprint., Comment: 15 + 10 pages, 15 figures. Code publicly available. Removed prior on the tilt, improved presentation and accuracy

The Effective Field Theory of Large-Scale Structure (EFTofLSS) is a formalism that allows us to predict the clustering of Cosmological Large-Scale Structure in the mildly non-linear regime in an accurate and reliable way. After validating our technique against several sets of numerical simulations, we perform the analysis for the cosmological parameters of the DR12 BOSS data. We assume $\Lambda$CDM, a fixed value of the baryon/dark-matter ratio, $\Omega_b/\Omega_c$, and of the tilt of the primordial power spectrum, $n_s$, and no significant input from numerical simulations. By using the one-loop power spectrum multipoles, we measure the primordial amplitude of the power spectrum, $A_s$, the abundance of matter, $\Omega_m$, and the Hubble parameter, $H_0$, to about $13\%$, $3.2\%$ and $3.2\%$ respectively, obtaining $\ln(10^{10}As)=2.72\pm 0.13$, $\Omega_m=0.309\pm 0.010$, $H_0=68.5\pm 2.2$ km/(s Mpc) at 68\% confidence level. If we then add a CMB prior on the sound horizon, the error bar on $H_0$ is reduced to $1.6\%$. These results are a substantial qualitative and quantitative improvement with respect to former analyses, and suggest that the EFTofLSS is a powerful instrument to extract cosmological information from Large-Scale Structure., Comment: 78 pages, 27 figures

We study, using Mean Curvature Flow methods, 2+1 dimensional cosmologies with a positive cosmological constant and matter satisfying the dominant and the strong energy conditions. If the spatial slices are compact with non-positive Euler characteristic and are initially expanding everywhere, then we prove that the spatial slices reach infinite volume, asymptotically converge on average to de Sitter and they become, almost everywhere, physically indistinguishable from de Sitter. This holds true notwithstanding the presence of initial arbitrarily-large density fluctuations and the formation of black holes., Comment: 18 pages, 3 figures; v2: minor corrections, CMP published version

We develop an analytic method for implementing the IR-resummation of arXiv:1404.5954, which allows one to correctly and consistently describe the imprint of baryon acoustic oscillations (BAO) on statistical observables in large-scale structure. We show that the final IR-resummed correlation function can be computed analytically without relying on numerical integration, thus allowing for an efficient and accurate use of these predictions on real data in cosmological parameter fitting. In this work we focus on the one-loop correlation function and the BAO peak. We show that, compared with the standard numerical integration method of IR-resummation, the new method is accurate to better than 0.2 %, and is quite easily improvable. We also give an approximate resummation scheme which is based on using the linear displacements of a fixed fiducial cosmology, which when combined with the method described above, is about six times faster than the standard numerical integration. Finally, we show that this analytic method is generalizable to higher loop computations., Comment: 39 pages, 11 figures, code available at http://web.stanford.edu/~senatore/

Effective field theory methods suggest that some rather-general extensions of General Relativity include, or are mimicked by, certain higher-order curvature corrections, with coupling constants expected to be small but otherwise arbitrary. Thus, the tantalizing prospect to test the fundamental nature of gravity with gravitational-wave observations, in a systematic way, emerges naturally. Here, we build black hole solutions in such a framework and study their main properties. Once rotation is included, we find the first purely gravitational example of geometries without $\mathbb{Z}_2$-symmetry. Despite the higher-order operators of the theory, we show that linearized fluctuations of such geometries obey second-order differential equations. We find nonzero tidal Love numbers. We study and compute the quasinormal modes of such geometries. These results are of interest to gravitational-wave science but also potentially relevant for electromagnetic observations of the galactic center or $X$-ray binaries., Comment: 8 pages, RevTex4. v2: Minor edits. v3: Typo in Eq. (16) corrected. Published in Physical Review Letters

The existence of light particles with spin during inflation is prohibited by the Higuchi bound. This conclusion can be evaded if one considers states with a sizeable coupling with the inflaton foliation, since this breaks the de Sitter isometries. The action for these states can be constructed within the Effective Field Theory of Inflation, or using a CCWZ procedure. Light particles with spin have prescribed couplings with soft inflaton perturbations, which are encoded in consistency relations. We study the phenomenology of light states with spin 2. These mix with the graviton changing the tensor power spectrum and can lead to sizeable tensor non-Gaussianities. They also give rise to a scalar bispectrum and trispectrum with a characteristic angle-dependent non-Gaussianity., Comment: 40 pages, 12 figures, matches JCAP version

The recent direct observation of gravitational waves (GW) from merging black holes opens up the possibility of exploring the theory of gravity in the strong regime at an unprecedented level. It is therefore interesting to explore which extensions to General Relativity (GR) could be detected. We construct an Effective Field Theory (EFT) satisfying the following requirements. It is testable with GW observations; it is consistent with other experiments, including short distance tests of GR; it agrees with widely accepted principles of physics, such as locality, causality and unitarity; and it does not involve new light degrees of freedom. The most general theory satisfying these requirements corresponds to adding to the GR Lagrangian operators constructed out of powers of the Riemann tensor, suppressed by a scale comparable to the curvature of the observed merging binaries. The presence of these operators modifies the gravitational potential between the compact objects, as well as their effective mass and current quadrupoles, ultimately correcting the waveform of the emitted GW., Comment: v1: 43+16 pages, 11 figures, 2 tables; v2: minor corrections; v3: minor corrections, JHEP published version

In the next few years, we are going to probe the low-redshift universe with unprecedented accuracy. Among the various fruits that this will bear, it will greatly improve our knowledge of the dynamics of dark energy, though for this there is a strong theoretical preference for a cosmological constant. We assume that dark energy is described by the so-called Effective Field Theory of Dark Energy, which assumes that dark energy is the Goldstone boson of time translations. Such a formalism makes it easy to ensure that our signatures are consistent with well-established principles of physics. Since most of the information resides at high wavenumbers, it is important to be able to make predictions at the highest wavenumber that is possible. The Effective Field Theory of Large-Scale Structure (EFTofLSS) is a theoretical framework that has allowed us to make accurate predictions in the mildly non-linear regime. In this paper, we derive the non-linear equations that extend the EFTofLSS to include the effect of dark energy both on the matter fields and on the biased tracers. For the specific case of clustering quintessence, we then perturbatively solve to cubic order the resulting non-linear equations and construct the one-loop power spectrum of the total density contrast., Comment: 53 pages, 6 figures, code available at http://web.stanford.edu/~senatore/; minor corrections, version published in JCAP

The Effective Field Theory of Large-Scale Structure (EFTofLSS) provides a novel formalism that is able to accurately predict the clustering of large-scale structure (LSS) in the mildly non-linear regime. Here we provide the first computation of the power spectrum of biased tracers in redshift space at one loop order, and we make the associated code publicly available. We compare the multipoles $\ell=0,2$ of the redshift-space halo power spectrum, together with the real-space matter and halo power spectra, with data from numerical simulations at $z=0.67$. For the samples we compare to, which have a number density of $\bar n=3.8 \cdot 10^{-2}(h \ {\rm Mpc}^{-1})^3$ and $\bar n=3.9 \cdot 10^{-4}(h \ {\rm Mpc}^{-1})^3$, we find that the calculation at one-loop order matches numerical measurements to within a few percent up to $k\simeq 0.43 \ h \ {\rm Mpc}^{-1}$, a significant improvement with respect to former techniques. By performing the so-called IR-resummation, we find that the Baryon Acoustic Oscillation peak is accurately reproduced. Based on the results presented here, long-wavelength statistics that are routinely observed in LSS surveys can be finally computed in the EFTofLSS. This formalism thus is ready to start to be compared directly to observational data., Comment: 38 pages, 8 figures. Code available at http://web.stanford.edu/~senatore/

We construct the Supersymmetric Effective Field Theory of Inflation, that is the most general theory of inflationary fluctuations when time-translations and supersymmetry are spontaneously broken. The non-linear realization of these invariances allows us to define a complete SUGRA multiplet containing the graviton, the gravitino, the Goldstone of time translations and the Goldstino, with no auxiliary fields. Going to a unitary gauge where only the graviton and the gravitino are present, we write the most general Lagrangian built out of the fluctuations of these fields, invariant under time-dependent spatial diffeomorphisms, but softly-breaking time diffeomorphisms and gauged SUSY. With a suitable St\"uckelberg transformation, we introduce the Goldstone boson of time translation and the Goldstino of SUSY. No additional dynamical light field is needed. In the high energy limit, larger than the inflationary Hubble scale for the Goldstino, these fields decouple from the graviton and the gravitino, greatly simplifying the analysis in this regime. We study the phenomenology of this Lagrangian. The Goldstino can have a non-relativistic dispersion relation. Gravitino and Goldstino affect the primordial curvature perturbations at loop level. The UV modes running in the loops generate three-point functions which are degenerate with the ones coming from operators already present in the absence of supersymmetry. Their size is potentially as large as corresponding to $f_{\rm NL}^{\rm equil.,\,orthog.}\sim 1$ or, for particular operators, even $\gg 1$. The non-degenerate contribution from modes of order $H$ is estimated to be very small., Comment: 55 pages, 2 figures

Planning to explore the beginning of the Universe? A lightweight introductory guide to the theory of Inflation., Comment: 87 pages, 29 figures

In homogeneous and isotropic Friedmann-Robertson-Walker cosmology, the topology of the universe determines its ultimate fate. If the Weak Energy Condition is satisfied, open and flat universes must expand forever, while closed cosmologies can recollapse to a Big Crunch. A similar statement holds for homogeneous but anisotropic (Bianchi) universes. Here, we prove that $arbitrarily$ inhomogeneous and anisotropic cosmologies with "flat" (including toroidal) and "open" (including compact hyperbolic) spatial topology that are initially expanding must continue to expand forever at least in some region at a rate bounded from below by a positive number, despite the presence of arbitrarily large density fluctuations and/or the formation of black holes. Because the set of 3-manifold topologies is countable, a single integer determines the ultimate fate of the universe, and, in a specific sense, most 3-manifolds are "flat" or "open". Our result has important implications for inflation: if there is a positive cosmological constant (or suitable inflationary potential) and initial conditions for the inflaton, cosmologies with "flat" or "open" topology must expand forever in some region at least as fast as de Sitter space, and are therefore very likely to begin inflationary expansion eventually, regardless of the scale of the inflationary energy or the spectrum and amplitude of initial inhomogeneities and gravitational waves. Our result is also significant for numerical general relativity, which often makes use of periodic (toroidal) boundary conditions., Comment: 6 pages, 2 figures. v2: Important reference and minor clarifications added, v3: added appendix with extended explanations, JCAP published version

We further develop the description of redshift space distortions within the Effective Field Theory of Large Scale Structures. First, we generalize the counterterms to include the effect of baryonic physics and primordial non-Gaussianity. Second, we evaluate the IR-resummation of the dark matter power spectrum in redshift space. This requires us to identify a controlled approximation that makes the numerical evaluation straightforward and efficient. Third, we compare the predictions of the theory at one loop with the power spectrum from numerical simulations up to $\ell=6$. We find that the IR-resummation allows us to correctly reproduce the BAO peak. The $k$-reach, or equivalently the precision for a given $k$, depends on additional counterterms that need to be matched to simulations. Since the non-linear scale for the velocity is expected to be longer than the one for the overdensity, we consider a minimal and a non-minimal set of counterterms. The quality of our numerical data makes it hard to firmly establish the performance of the theory at high wavenumbers. Within this limitation, we find that the theory at redshift $z=0.56$ and up to $\ell=2$ matches the data to percent level approximately up to $k \sim 0.13 \, h { \rm Mpc^{-1}}$ or $k \sim 0.18 \, h { \rm Mpc^{-1}}$, depending on the number of counterterms used, with potentially large improvement over former analytical techniques., Comment: 34 pages, 10 figures; v2: minor typos fixed; v3: 35 pages, minor corrections, PRD version

If time-translations are spontaneously broken, so are boosts. This symmetry breaking pattern can be non-linearly realized by either just the Goldstone boson of time translations, or by four Goldstone bosons associated with time translations and boosts. In this paper we extend the Effective Field Theory of Multifield Inflation to consider the case in which the additional Goldstone bosons associated with boosts are light and coupled to the Goldstone boson of time translations. The symmetry breaking pattern forces a coupling to curvature so that the mass of the additional Goldstone bosons is predicted to be equal to $\sqrt{2}H$ in the vast majority of the parameter space where they are light. This pattern therefore offers a natural way of generating self-interacting particles with Hubble mass during inflation. After constructing the general effective Lagrangian, we study how these particles mix and interact with the curvature fluctuations, generating potentially detectable non-Gaussian signals., Comment: 30 pages, 2 figures. Extended explanations, version published in JCAP

Using numerical solutions of the full Einstein field equations coupled to a scalar inflaton field in 3+1 dimensions, we study the conditions under which a universe that is initially expanding, highly inhomogeneous and dominated by gradient energy can transition to an inflationary period. If the initial scalar field variations are contained within a sufficiently flat region of the inflaton potential, and the universe is spatially flat or open on average, inflation will occur following the dilution of the gradient and kinetic energy due to expansion. This is the case even when the scale of the inhomogeneities is comparable to the initial Hubble length, and overdense regions collapse and form black holes, because underdense regions continue expanding, allowing inflation to eventually begin. This establishes that inflation can arise from highly inhomogeneous initial conditions and solve the horizon and flatness problems, at least as long as the variations in the scalar field do not include values that exceed the inflationary plateau., Comment: 14 pages, 6 figures; version accepted by JCAP

We study the local response to long wavelength fluctuations in cosmological $N$-body simulations, focusing on the matter and halo power spectra, halo abundance and non-linear transformations of the density field. The long wavelength mode is implemented using an effective curved cosmology and a mapping of time and distances. The method provides an alternative, most probably more precise, way to measure the isotropic halo biases. Limiting ourselves to the linear case, we find generally good agreement between the biases obtained from the curvature method and the traditional power spectrum method at the level of a few percent. We also study the response of halo counts to changes in the variance of the field and find that the slope of the relation between the responses to density and variance differs from the naive derivation assuming a universal mass function by 18%. This has implications for measurements of the amplitude of local non-Gaussianity using scale dependent bias. We also analyze the halo power spectrum and halo-dark matter cross-spectrum response to long wavelength fluctuations and derive second order halo bias from it, as well as the super-sample variance contribution to the galaxy power spectrum covariance matrix., Comment: 29 pages, 14 figures

We study the prediction of the dark matter power spectrum at two-loop order in the Effective Field Theory of Large Scale Structures (EFTofLSS) using high precision numerical simulations. In our universe, short distance non-linear perturbations, not under perturbative control, affect long distance fluctuations through an effective stress tensor that needs to be parametrized in terms of counterterms that are functions of the long distance fluctuating fields. We find that at two-loop order it is necessary to include three counterterms: a linear term in the over density, $\delta$, a quadratic term, $\delta^2$, and a higher derivative term, $\partial^2\delta$. After the inclusion of these three terms, the EFTofLSS at two-loop order matches simulation data up to $k\simeq 0.34 \,h\, {\rm Mpc}^{-1}$ at redshift $z=0$, up to $k\simeq 0.55\,h\, {\rm Mpc}^{-1}$ at $z=1$, and up to $k\simeq 1.1\,h\, {\rm Mpc}^{-1}$ at $z=2$. At these wavenumbers, the cosmic variance of the simulation is at least as small as $10^{-3}$, providing a high precision comparison between theory and data. The actual reach of the theory is affected by theoretical uncertainties associated to not having included higher order terms in perturbation theory, for which we provide an estimate, and by potentially overfitting the data, which we also try to address. Since in the EFTofLSS the coupling constants associated with the counterterms are unknown functions of time, we show how a simple parametrization gives a sensible description of their time-dependence. Overall, the $k$-reach of the EFTofLSS is much larger than previous analytical techniques, showing that the amount of cosmological information amenable to high-precision analytical control might be much larger than previously believed., Comment: v2: JCAP published version, added comments and explanations

With the completion of the Planck mission, in order to continue to gather cosmological information it has become crucial to understand the Large Scale Structures (LSS) of the universe to percent accuracy. The Effective Field Theory of LSS (EFTofLSS) is a novel theoretical framework that aims to develop an analytic understanding of LSS at long distances, where inhomogeneities are small. We further develop the description of biased tracers in the EFTofLSS to account for the effect of baryonic physics and primordial non-Gaussianities, finding that new bias coefficients are required. Then, restricting to dark matter with Gaussian initial conditions, we describe the prediction of the EFTofLSS for the one-loop halo-halo and halo-matter two-point functions, and for the tree-level halo-halo-halo, matter-halo-halo and matter-matter-halo three-point functions. Several new bias coefficients are needed in the EFTofLSS, even though their contribution at a given order can be degenerate and the same parameters contribute to multiple observables. We develop a method to reduce the number of biases to an irreducible basis, and find that, at the order at which we work, seven bias parameters are enough to describe this extremely rich set of statistics. We then compare with the output of $N$-body simulations. For the lowest mass bin, we find percent level agreement up to $k\simeq 0.3\,h\,{\rm Mpc}^{-1}$ for the one-loop two-point functions, and up to $k\simeq 0.15\,h\,{\rm Mpc}^{-1}$ for the tree-level three-point functions, with the $k$-reach decreasing with higher mass bins. This is consistent with the theoretical estimates, and suggests that the cosmological information in LSS amenable to analytical control is much more than previously believed., Comment: 54 pages, 16 figures, v2: added references and explanations, corrected typos

We study the prediction of the Effective Field Theory of Large Scale Structures (EFTofLSS) for the matter power spectrum at different redshifts. In previous work, we found that the two-loop prediction can match the nonlinear power spectrum measured from $N$-body simulations at redshift zero within approximately 2% up to $k\sim 0.6\,h\, {\rm Mpc}^{-1}$ after fixing a single free parameter, the so-called "speed of sound". We determine the time evolution of this parameter by matching the EFTofLSS prediction to simulation output at different redshifts, and find that it is well-described by a fitting function that only includes one additional parameter. After the two free parameters are fixed, the prediction agrees with nonlinear data within approximately 2% up to at least $k\sim 1\,h\, {\rm Mpc}^{-1}$ at $z\geq 1$, and also within approximately 5% up to $k\sim 1.2\,h\, {\rm Mpc}^{-1}$ at $z=1$ and $k\sim 2.3\,h\, {\rm Mpc}^{-1}$ at $z=3$, a major improvement with respect to other perturbative techniques. We also develop an accurate way to estimate where the EFTofLSS predictions at different loop orders should fail, based on the sizes of the next-order terms that are neglected, and find agreement with the actual comparisons to data. Finally, we use our matter power spectrum results to perform analytical calculations of lensing potential power spectra corresponding to both CMB and galaxy lensing. This opens the door to future direct applications of the EFTofLSS to observations of gravitational clustering on cosmic scales., Comment: 40 pages, 20 figures. v2: JCAP published version, improved theory-error estimates, extended explanations and minor corrections

The large scale structures of the universe will likely be the next leading source of cosmological information. It is therefore crucial to understand their behavior. The Effective Field Theory of Large Scale Structures provides a consistent way to perturbatively predict the clustering of dark matter at large distances. The fact that baryons move distances comparable to dark matter allows us to infer that baryons at large distances can be described in a similar formalism: the backreaction of short-distance non-linearities and of star-formation physics at long distances can be encapsulated in an effective stress tensor, characterized by a few parameters. The functional form of baryonic effects can therefore be predicted. In the power spectrum the leading contribution goes as $\propto k^2 P(k)$, with $P(k)$ being the linear power spectrum and with the numerical prefactor depending on the details of the star-formation physics. We also perform the resummation of the contribution of the long-wavelength displacements, allowing us to consistently predict the effect of the relative motion of baryons and dark matter. We compare our predictions with simulations that contain several implementations of baryonic physics, finding percent agreement up to relatively high wavenumbers such as $k\simeq 0.3\,h\, Mpc^{-1}$ or $k\simeq 0.6\, h\, Mpc^{-1}$, depending on the order of the calculation. Our results open a novel way to understand baryonic effects analytically, as well as to interface with simulations., Comment: 33 pages, 13 figures

We introduce a formalism, valid both for dark matter and collapsed objects, that allows us to describe redshift space distortions in the context of the Effective Field Theory of Large Scale Structures (EFTofLSS). Expressing density perturbations in redshift space corresponds to performing a change of coordinates and the resulting expressions contain products of density perturbations and velocity fields evaluated at the same location. These terms are sensitive to non-perturbative short-distance physics and in order to correctly treat them they need to be renormalized by adding suitable counterterms. Therefore more counterterms are required in redshift space expressions compared to their real space analogs. In particular in the expression for the one-loop matter power spectrum there are two new counterterms. Just as in real space, long wavelength displacements affect correlation functions in redshift space and need to be resummed. We generalize the real space formulas for IR resummation to this case: the final expressions are conceptually similar but are more challenging to compute numerically due to their reduced symmetry., Comment: 21 pages

Given the fantastic experimental effort, it is important to thoroughly explore the signature space of inflationary models. The fact that higher derivative operators do not renormalize lower derivative ones allows us to find a large class of technically natural single-clock inflationary models where, in the context of the Effective Field Theory of Inflation, the leading interactions have many derivatives. We systematically explore the 3-point function induced by these models and their overlap with the standard equilateral and orthogonal templates. We find that in order to satisfactorily cover the signature space of these models, two new additional templates need to be included. We then perform the optimal analysis of the WMAP 9-year data for the resulting four templates, finding that the overall significance of a non-zero signal is between 2--2.5$\sigma$, depending on the choice of parameter space, partially driven by the preference for nonzero $f_{NL}^{\rm orth}$ in WMAP9., Comment: 22+14 pages

We study how to describe collapsed objects, such as galaxies, in the context of the Effective Field Theory of Large Scale Structures. The overdensity of galaxies at a given location and time is determined by the initial tidal tensor, velocity gradients and spatial derivatives of the regions of dark matter that, during the evolution of the universe, ended up at that given location. Similarly to what recently done for dark matter, we show how this Lagrangian space description can be recovered by upgrading simpler Eulerian calculations. We describe the Eulerian theory. We show that it is perturbatively local in space, but non-local in time, and we explain the observational consequences of this fact. We give an argument for why to a certain degree of accuracy the theory can be considered as quasi time-local and explain what the operator structure is in this case. We describe renormalization of the bias coefficients so that, after this and after upgrading the Eulerian calculation to a Lagrangian one, the perturbative series for galaxies correlation functions results in a manifestly convergent expansion in powers of $k/k_{\rm NL}$ and $k/k_{\rm M}$, where $k$ is the wavenumber of interest, $k_{\rm NL}$ is the wavenumber associated to the non-linear scale, and $k_{\rm M}$ is the comoving wavenumber enclosing the mass of a galaxy., Comment: 45 pages; v2: added explanations, minor corrections and references

Given the importance of future large scale structure surveys for delivering new cosmological information, it is crucial to reliably predict their observables. The Effective Field Theory of Large Scale Structures (EFTofLSS) provides a manifestly convergent perturbative scheme to compute the clustering of dark matter in the weakly nonlinear regime in an expansion in $k/k_{\rm NL}$, where $k$ is the wavenumber of interest and $k_{\rm NL}$ is the wavenumber associated to the nonlinear scale. It has been recently shown that the EFTofLSS matches to $1\%$ level the dark matter power spectrum at redshift zero up to $k\simeq 0.3 h\,$Mpc$^{-1}$ and $k\simeq 0.6 h\,$Mpc$^{-1}$ at one and two loops respectively, using only one counterterm that is fit to data. Similar results have been obtained for the momentum power spectrum at one loop. This is a remarkable improvement with respect to former analytical techniques. Here we study the prediction for the equal-time dark matter bispectrum at one loop. We find that at this order it is sufficient to consider the same counterterm that was measured in the power spectrum. Without any remaining free parameter, and in a cosmology for which $k_{\rm NL}$ is smaller than in the previously considered cases ($\sigma_8=0.9$), we find that the prediction from the EFTofLSS agrees very well with $N$-body simulations up to $k\simeq 0.25 h\,$Mpc$^{-1}$, given the accuracy of the measurements, which is of order a few percent at the highest $k$'s of interest. While the fit is very good on average up to $k\simeq 0.25 h\,$Mpc$^{-1}$, the fit performs slightly worse on equilateral configurations, in agreement with expectations that for a given maximum $k$, equilateral triangles are the most nonlinear., Comment: 39 pages, 12 figures; v2: JCAP published version, improved numerical data, added explanation and clarifications

We present a new method to resum the effect of large scale motions in the Effective Field Theory of Large Scale Structures. Because the linear power spectrum in $\Lambda$CDM is not scale free the effects of the large scale flows are enhanced. Although previous EFT calculations of the equal-time density power spectrum at one and two loops showed a remarkable agreement with numerical results, they also showed a 2% residual which appeared related to the BAO oscillations. We show that this was indeed the case, explain the physical origin and show how a Lagrangian based calculation removes this differences. We propose a simple method to upgrade existing Eulerian calculations to effectively make them Lagrangian and compare the new results with existing fits to numerical simulations. Our new two-loop results agrees with numerical results up to $k\sim 0.6 h/$Mpc to within 1% with no oscillatory residuals. We also compute power spectra involving momentum which is significantly more affected by the large scale flows. We show how keeping track of these velocities significantly enhances the UV reach of the momentum power spectrum in addition to removing the BAO related residuals. We compute predictions for the real space correlation function around the BAO scale and investigate its sensitivity to the EFT parameters and the details of the resummation technique., Comment: 37 pages, 9 figures. v2: JCAP published version, added references and extended explanations

Last year we argued that if slow-roll inflation followed the decay of a false vacuum in a large landscape, the steepening of the scalar potential between the inflationary plateau and the barrier generically leads to a potentially observable suppression of the scalar power spectrum at large distances. Here we revisit this analysis in light of the recent BICEP2 results. Assuming that both the BICEP2 B-mode signal and the Planck analysis of temperature fluctuations hold up, we find that the data now discriminate more sharply between our scenario and $\Lambda$CDM. Nonzero tensor modes exclude standard $\Lambda$CDM with notable but not yet conclusive confidence: at $\sim 3.8\,\sigma$ if $r\approx0.2$, or at $\sim 3.5\,\sigma$ if $r=0.15$. Of the two steepening models of our previous work, one is now ruled out by existing bounds on spatial curvature. The other entirely reconciles the tension between BICEP2 and Planck. Upcoming $EE$ polarization measurements have the potential to rule out unmodified $\Lambda$CDM decisively. Next generation Large Scale Structure surveys can further increase the significance. More precise measurements of $BB$ at low $\ell$ will help distinguish our scenario from other explanations. If steepening is confirmed, the prospects for detecting open curvature increase but need not be large., Comment: 16 pages, 4 figures. v2: journal version, minor corrections

We introduce a Lagrangian-space Effective Field Theory (LEFT) formalism for the study of cosmological large scale structures. Unlike the previous Eulerian-space construction, it is naturally formulated as an effective field theory of extended objects in Lagrangian space. In LEFT the resulting finite size effects are described using a multipole expansion parameterized by a set of time dependent coefficients and organized in powers of the ratio of the wavenumber of interest $k$ over the non-linear scale $k_{\rm NL}$. The multipoles encode the effects of the short distance modes on the long-wavelength universe and absorb UV divergences when present. There are no IR divergences in LEFT. Some of the parameters that control the perturbative approach are not assumed to be small and can be automatically resummed. We present an illustrative one-loop calculation for a power law universe. We describe the dynamics both at the level of the equations of motion and through an action formalism., Comment: 38+13 pages. 3 figures. Minor changes. Version to appear in JCAP

Large scale structure surveys promise to be the next leading probe of cosmological information. It is therefore crucial to reliably predict their observables. The Effective Field Theory of Large Scale Structures (EFTofLSS) provides a manifestly convergent perturbation theory for the weakly non-linear regime of dark matter, where correlation functions are computed in an expansion of the wavenumber k of a mode over the wavenumber associated with the non-linear scale k_nl. Since most of the information is contained at high wavenumbers, it is necessary to compute higher order corrections to correlation functions. After the one-loop correction to the matter power spectrum, we estimate that the next leading one is the two-loop contribution, which we compute here. At this order in k/k_nl, there is only one counterterm in the EFTofLSS that must be included, though this term contributes both at tree-level and in several one-loop diagrams. We also discuss correlation functions involving the velocity and momentum fields. We find that the EFTofLSS prediction at two loops matches to percent accuracy the non-linear matter power spectrum at redshift zero up to k~0.6 h/Mpc, requiring just one unknown coefficient that needs to be fit to observations. Given that Standard Perturbation Theory stops converging at redshift zero at k~0.1 h/Mpc, our results demonstrate the possibility of accessing a factor of order 200 more dark matter quasi-linear modes than naively expected. If the remaining observational challenges to accessing these modes can be addressed with similar success, our results show that there is tremendous potential for large scale structure surveys to explore the primordial universe., Comment: 56 pages, 10 figures. v2: JCAP published version, corrected a small algebra mistake, extended explanations, results unchanged

We assess two potential signals of the formation of our universe by the decay of a false vacuum. Negative spatial curvature is one possibility, but the window for its detection is now small. However, another possible signal is a suppression of the CMB power spectrum at large angles. This arises from the steepening of the effective potential as it interpolates between a flat inflationary plateau and the high barrier separating us from our parent vacuum. We demonstrate that these two effects can be parametrically separated in angular scale. Observationally, the steepening effect appears to be excluded at large l; but it remains consistent with the slight lack of power below l about 30 found by the WMAP and Planck collaborations. We give two simple models which improve the fit to the Planck data; one with observable curvature and one without. Despite cosmic variance, we argue that future CMB polarization and most importantly large-scale structure observations should be able to corroborate the Planck anomaly if it is real. If we further assume the specific theoretical setting of a landscape of metastable vacua, as suggested by string theory, we can estimate the probability of seeing a low-l suppression in the CMB. There are significant theoretical uncertainties in such calculations, but we argue the probability for a detectable suppression may be as large as O(1), and in general is significantly larger than the probability of seeing curvature., Comment: 34 pages plus appendices, 16 figures. v2: Corrections and clarifications to section 3.2, conclusions unchanged. References added

An appealing theory is that our current patch of universe was born as a nucleation bubble from a phase of false vacuum eternal inflation. We search for evidence for this theory by looking for the signal imprinted on the CMB that is generated when another bubble "universe" collides with our own. We create an efficient and optimal estimator for the signal in the WMAP 7-year data. We find no detectable signal, and constrain the amplitude, a, of the initial curvature perturbation that would be generated by a collision: -4.66 \times 10^{-8} < a (\sin{\thetabubble})^{4/3} < 4.73 \times 10^{-8} [Mpc^{-1}] at 95% confidence where \thetabubble is the angular radius of the bubble signal., Comment: 5 pages, 3 figures

Studying loop corrections to inflationary perturbations, with particular emphasis on infrared factors, is important to understand the consistency of the inflationary theory, its predictivity and to establish the existence of the slow-roll eternal inflation phenomena and its recently found volume bound. In this paper we show that \zeta-correlators are time-independent at large distances at all-loop level in single clock inflation. We write the n-th order correlators of \dot\zeta\ as the time-integral of Green's functions times the correlators of local sources that are function of the lower order fluctuations. The Green's functions are such that only non-vanishing correlators of the sources at late times can lead to non-vanishing correlators for \dot\zeta\ at long distances. When the sources are connected by high wavenumber modes, the correlator is peaked at short distances, and these diagrams cannot lead to a time-dependence by simple diff. invariance arguments. When the sources are connected by long wavenumber modes one can use similar arguments once the constancy of \zeta\ at lower orders was established. Therefore the conservation of \zeta\ at a given order follows from the conservation of \zeta\ at the lower orders. Since at tree-level \zeta\ is constant, this implies constancy at all-loops by induction., Comment: 14 pages, 3 figures

Large scale structure surveys will likely become the next leading cosmological probe. In our universe, matter perturbations are large on short distances and small at long scales, i.e. strongly coupled in the UV and weakly coupled in the IR. To make precise analytical predictions on large scales, we develop an effective field theory formulated in terms of an IR effective fluid characterized by several parameters, such as speed of sound and viscosity. These parameters, determined by the UV physics described by the Boltzmann equation, are measured from N-body simulations. We find that the speed of sound of the effective fluid is c_s^2 10^(-6) and that the viscosity contributions are of the same order. The fluid describes all the relevant physics at long scales k and permits a manifestly convergent perturbative expansion in the size of the matter perturbations \delta(k) for all the observables. As an example, we calculate the correction to the power spectrum at order \delta(k)^4. The predictions of the effective field theory are found to be in much better agreement with observation than standard cosmological perturbation theory, already reaching percent precision at this order up to a relatively short scale k \sim 0.24 h/Mpc., Comment: v2: typos corrected, JHEP published version

Studying loop corrections to inflationary perturbations, with particular emphasis on infrared factors, is important to understand the consistency of the inflationary theory, its predictivity and to establish the existence of the slow-roll eternal inflation phenomena and its recently found volume bound. In this paper we prove that the zeta correlation function is time-independent at one-loop level in single clock inflation. While many of the one-loop diagrams lead to a time-dependence when considered individually, the time-dependence beautifully cancels out in the overall sum. We identify two subsets of diagrams that cancel separately due to different physical reasons. The first cancellation is related to the change of the background cosmology due to the renormalization of the stress tensor. It results in a cancellation between the non-1PI diagrams and some of the diagrams made with quartic vertices. The second subset of diagrams that cancel is made up of cubic operators, plus the remaining quartic ones. We are able to write the sum of these diagrams as the integral over a specific three-point function between two very short wavelengths and one very long one. We then apply the consistency condition for this three-point function in the squeezed limit to show that the sum of these diagrams cannot give rise to a time dependence. This second cancellation is thus a consequence of the fact that in single clock inflation the attractor nature of the solution implies that a long wavelength zeta perturbation is indistinguishable from a trivial rescaling of the background, and so results in no physical effect on short wavelength modes., Comment: 47 pages, 7 figures; v2: JHEP published version, typos and minor corrections

In single clock models of inflation the coupling between modes of very different scales does not have any significant dynamical effect during inflation. It leads to interesting projection effects. Larger and smaller modes change the relation between the scale a mode of interest will appear in the post-inflationary universe and will also change the time of horizon crossing of that mode. We argue that there are no infrared projection effects in physical questions, that there are no effects from modes of longer wavelength than the one of interest. These potential effects cancel when computing fluctuations as a function of physically measurable scales. Modes on scales smaller than the one of interest change the mapping between horizon crossing time and scale. The correction to the mapping computed in the absence of fluctuations is enhanced by a factor N_e, the number of e-folds of inflation between horizon crossing and reheating. The new mapping is stochastic in nature but its variance is not enhanced by N_e., Comment: 13 pages, 1 figure; v2: JHEP published version, added minor comments and references

We apply the Effective Field Theory of Inflation to study the case where the continuous shift symmetry of the Goldstone boson \pi is softly broken to a discrete subgroup. This case includes and generalizes recently proposed String Theory inspired models of Inflation based on Axion Monodromy. The models we study have the property that the 2-point function oscillates as a function of the wavenumber, leading to oscillations in the CMB power spectrum. The non-linear realization of time diffeomorphisms induces some self-interactions for the Goldstone boson that lead to a peculiar non-Gaussianity whose shape oscillates as a function of the wavenumber. We find that in the regime of validity of the effective theory, the oscillatory signal contained in the n-point correlation functions, with n>2, is smaller than the one contained in the 2-point function, implying that the signature of oscillations, if ever detected, will be easier to find first in the 2-point function, and only then in the higher order correlation functions. Still the signal contained in higher-order correlation functions, that we study here in generality, could be detected at a subleading level, providing a very compelling consistency check for an approximate discrete shift symmetry being realized during inflation., Comment: v2 minor revisions; 39 pages, 5 figures

It has recently been shown that in single field slow-roll inflation the total volume cannot grow by a factor larger than e^(S_dS/2) without becoming infinite. The bound is saturated exactly at the phase transition to eternal inflation where the probability to produce infinite volume becomes non zero. We show that the bound holds sharply also in any space-time dimensions, when arbitrary higher-dimensional operators are included and in the multi-field inflationary case. The relation with the entropy of de Sitter and the universality of the bound strengthen the case for a deeper holographic interpretation. As a spin-off we provide the formalism to compute the probability distribution of the volume after inflation for generic multi-field models, which might help to address questions about the population of vacua of the landscape during slow-roll inflation., Comment: 24 pages, 5 figures

We generalize the effective field theory of single clock inflation to include dissipative effects. Working in unitary gauge we couple a set of composite operators in the effective action which is constrained solely by invariance under time-dependent spatial diffeomorphisms. We restrict ourselves to situations where the degrees of freedom responsible for dissipation do no contribute to the density perturbations at late time. The dynamics of the perturbations is then modified by the appearance of `friction' and noise terms, and assuming certain locality properties for the Green's functions of these composite operators, we show that there is a regime characterized by a large friction term \gamma >> H in which the \zeta-correlators are dominated by the noise and the power spectrum can be significantly enhanced. We also compute the three point function <\zeta\zeta\zeta> for a wide class of models and discuss under which circumstances large friction leads to an increased level of non-Gaussianities. In particular, under our assumptions, we show that strong dissipation together with the required non-linear realization of the symmetries implies |f_NL| ~ \gamma/(c_s^2H) >> 1. As a paradigmatic example we work out a variation of the `trapped inflation' scenario with local response functions and perform the matching with our effective theory. A detection of the generic type of signatures that result from incorporating dissipative effects during inflation, as we describe here, would teach us about the dynamics of the early universe and also extend the parameter space of inflationary models., Comment: 52+20 pages, 15 figures. (Summary of main results in first approx 15 pages.) v2. Published version, to appear in JHEP

We point out that detectable inflationary tensor modes can be generated by particle or string sources produced during inflation, consistently with the requirements for inflation and constraints from scalar fluctuations. We show via examples that this effect can dominate over the contribution from quantum fluctuations of the metric, occurring even when the inflationary potential energy is too low to produce a comparable signal. Thus a detection of tensor modes from inflation does not automatically constitute a determination of the inflationary Hubble scale., Comment: 32 pages, 1 figure. v2: JCAP published version; some overestimates corrected; main results unchanged

Length scales probed by large scale structure surveys are becoming closer to the horizon scale. Further, it has been recently understood that non-Gaussianity in the initial conditions could show up in a scale dependence of the bias of galaxies at the largest distances. It is therefore important to include General Relativistic effects. Here we provide a General Relativistic generalization of the bias, valid both for Gaussian and non-Gaussian initial conditions. The collapse of objects happens on very small scales, while long-wavelength modes are always in the quasi linear regime. Around every collapsing region, it is therefore possible to find a reference frame that is valid for all times and where the space time is almost flat: the Fermi frame. Here the Newtonian approximation is applicable and the equations of motion are the ones of the N-body codes. The effects of long-wavelength modes are encoded in the mapping from the cosmological frame to the local frame. For the linear bias, the effect of the long-wavelength modes on the dynamics is encoded in the local curvature of the Universe, which allows us to define a General Relativistic generalization of the bias in the standard Newtonian setting. We show that the bias due to this effect goes to zero as the squared ratio of the physical wavenumber with the Hubble scale for modes longer than the horizon, as modes longer than the horizon have no dynamical effects. However, the bias due to non-Gaussianities does not need to vanish for modes longer than the Hubble scale, and for non-Gaussianities of the local kind it goes to a constant. As a further application, we show that it is not necessary to perform large N-body simulations to extract information on long-wavelength modes: N-body simulations can be done on small scales and long-wavelength modes are encoded simply by adding curvature to the simulation and rescaling the coordinates., Comment: 48 pages, 4 figures; v2: added references, JCAP published version

The effective theory of adiabatic fluctuations around arbitrary Friedmann-Robertson-Walker backgrounds - both expanding and contracting - allows for more than one way to obtain scale-invariant two-point correlations. However, as we show in this paper, it is challenging to produce scale-invariant fluctuations that are weakly coupled over the range of wavelengths accessible to cosmological observations. In particular, requiring the background to be a dynamical attractor, the curvature fluctuations are scale-invariant and weakly coupled for at least 10 e-folds only if the background is close to de Sitter space. In this case, the time-translation invariance of the background guarantees time-independent n-point functions. For non-attractor solutions, any predictions depend on assumptions about the evolution of the background even when the perturbations are outside of the horizon. For the simplest such scenario we identify the regions of the parameter space that avoid both classical and quantum mechanical strong coupling problems. Finally, we present extensions of our results to backgrounds in which higher-derivative terms play a significant role., Comment: 17 pages + appendices, 3 figures; v2: typos fixed

The universe is smooth on large scales but very inhomogeneous on small scales. Why is the spacetime on large scales modeled to a good approximation by the Friedmann equations? Are we sure that small-scale non-linearities do not induce a large backreaction? Related to this, what is the effective theory that describes the universe on large scales? In this paper we make progress in addressing these questions. We show that the effective theory for the long-wavelength universe behaves as a viscous fluid coupled to gravity: integrating out short-wavelength perturbations renormalizes the homogeneous background and introduces dissipative dynamics into the evolution of long-wavelength perturbations. The effective fluid has small perturbations and is characterized by a few parameters like an equation of state, a sound speed and a viscosity parameter. These parameters can be matched to numerical simulations or fitted from observations. We find that the backreaction of small-scale non-linearities is very small, being suppressed by the large hierarchy between the scale of non-linearities and the horizon scale. The effective pressure of the fluid is always positive and much too small to significantly affect the background evolution. Moreover, we prove that virialized scales decouple completely from the large-scale dynamics, at all orders in the post-Newtonian expansion. We propose that our effective theory be used to formulate a well-defined and controlled alternative to conventional perturbation theory, and we discuss possible observational applications. Finally, our way of reformulating results in second-order perturbation theory in terms of a long-wavelength effective fluid provides the opportunity to understand non-linear effects in a simple and physically intuitive way., Comment: 84 pages, 3 figures

Non-Gaussianities of the primordial density perturbations have emerged as a very powerful possible signal to test the dynamics that drove the period of inflation. While in general the most sensitive observable is the three-point function in this paper we show that there are technically natural inflationary models where the leading source of non-Gaussianity is the four-point function. Using the recently developed Effective Field Theory of Inflation, we are able to show that it is possible to impose an approximate parity symmetry and an approximate continuos shift symmetry on the inflaton fluctuations that allow, when the dispersion relation is of the form $\omega\sim c_s k$, for a unique quartic operator, while approximately forbidding all the cubic ones. The resulting shape for the four-point function is unique. In the models where the dispersion relation is of the form $\omega\sim k^2/M$ a similar construction can be carried out and additional shapes are possible., Comment: 13 pages, 1 figure. v2: extended discussion on near-de-Sitter models