Clustering Fossils from the Early Universe

Many inflationary theories introduce new scalar, vector, or tensor degrees of freedom that may then affect the generation of primordial density perturbations. Here we show how to search a galaxy (or 21-cm) survey for the imprint of primordial scalar, vector, and tensor fields. These new fields induce local departures to an otherwise statistically isotropic two-point correlation function, or equivalently, nontrivial four-point correlation functions (or trispectra, in Fourier space), that can be decomposed into scalar, vector, and tensor components. We write down the optimal estimators for these various components and show how the sensitivity to these modes depends on the galaxy-survey parameters. New probes of parity-violating early-Universe physics are also presented.Comment: 4 pages, 2 figures, submitted to PR

Cosmic Rulers

We derive general covariant expressions for the six independent observable modes of distortion of ideal standard rulers in a perturbed Friedmann-Robertson-Walker spacetime. Our expressions are gauge-invariant and valid on the full sky. These six modes are most naturally classified in terms of their rotational properties on the sphere, yielding two scalars, two vector (spin-1), and two tensor (spin-2) components. One scalar corresponds to the magnification, while the spin-2 components correspond to the shear. The vector components allow for a polar/axial decomposition analogous to the E/B-decomposition for the shear. Scalar modes do not contribute to the axial (B-)vector, opening a new avenue to probing tensor modes. Our results apply, but are not limited to, the distortion of correlation functions (of the CMB, 21cm emission, or galaxies) as well as to weak lensing shear and magnification, all of which can be seen as methods relying on "standard rulers".Comment: 31 pages, 3 figures (v3); v2: minor changes reflecting PRD published version; v3: evolving ruler case relegated to arXiv:1305.1299, added proper time condition for observer, additional test cases for magnificatio

Cosmic Clocks

In a perturbed Universe, comoving tracers on a two-dimensional surface of constant observed redshift are at different proper time since the Big Bang. For tracers whose age is known independently, one can measure these perturbations of the proper time. Examples of such sources include cosmic events which only happen during a short period of cosmic history, as well as evolving standard candles and standard rulers. In this paper we derive a general gauge-invariant linear expression for this perturbation in terms of space-time perturbations. As an example, we show that the observed temperature perturbations of the cosmic microwave background (CMB) on large scales are exactly given by these proper time perturbations. Together with the six ruler perturbations derived in Schmidt and Jeong (2012), this completes the set of independent observables which can be measured with standard rulers and candles.Comment: 10 pages, 2 figure

Large-Scale Structure Observables in General Relativity

We review recent studies that rigorously define several key observables of the large-scale structure of the Universe in a general relativistic context. Specifically, we consider i) redshift perturbation of cosmic clock events; ii) distortion of cosmic rulers, including weak lensing shear and magnification; iii) observed number density of tracers of the large-scale structure. We provide covariant and gauge-invariant expressions of these observables. Our expressions are given for a linearly perturbed flat Friedmann-Robertson-Walker metric including scalar, vector, and tensor metric perturbations. While we restrict ourselves to linear order in perturbation theory, the approach can be straightforwardly generalized to higher order.Comment: 24 pages, 3 figures. A review article submitted to CQG focus issue "Relativistic Effects in Cosmology". arXiv admin note: substantial text overlap with arXiv:1204.3625, v2: correct one missing referenc

Large-Scale Structure with Gravitational Waves II: Shear

The B-(curl-)mode of the correlation of galaxy ellipticities (shear) can be used to detect a stochastic gravitational wave background, such as that predicted by inflation. In this paper, we derive the tensor mode contributions to shear from both gravitational lensing and intrinsic alignments, using the gauge-invariant, full-sky results of arXiv:1204.3625. We find that the intrinsic alignment contribution, calculated using the linear alignment model, is larger than the lensing contribution by an order of magnitude or more, if the alignment strength for tensor modes is of the same order as for scalar modes. This contribution also extends to higher multipoles. These results make the prospects for probing tensor modes using galaxy surveys less pessimistic than previously thought, though still very challenging.Comment: 21 pages, 7 figures; v2: reflects PRD version, added section on second-order scalar B modes; v3: fixed typo in Eq. (A44

Wigner-Eckart theorem in cosmology: Bispectra for total-angular-momentum waves

Total-angular-momentum (TAM) waves provide a set of basis functions for scalar, vector, and tensor fields that can be used in place of plane waves and that reflect the rotational symmetry of the spherical sky. Here we discuss three-point correlation functions, or bispectra in harmonic space, for scalar, vector, and tensor fields in terms of TAM waves. The Wigner-Eckart theorem dictates that the expectation value, assuming statistical isotropy, of the product of three TAM waves is the product of a Clebsch-Gordan coefficient (or Wigner-3j symbol) times a function only of the total-angular-momentum quantum numbers. Here we show how this works, and we provide explicit expressions relating the bispectra for TAM waves in terms of the more commonly used Fourier-space bispectra. This formalism will be useful to simplify calculations of projections of three-dimensional bispectra onto the spherical sky.Comment: 19 pages, 1 figure (version as appeared in PRD

Anisotropic imprint of long-wavelength tensor perturbations on cosmic structure

Inflationary models predict a correlation between primordial density perturbations (scalar metric perturbations) and gravitational waves (tensor metric perturbations) in the form of a scalar-scalar-tensor three-point correlation, or bispectrum in Fourier space. The squeezed limit of this bispectrum implies a quadrupolar asymmetry in the observed local power spectrum for matter and galaxies. Here we show (like others before) that an infrared divergence in the amplitude of this power asymmetry predicted in single-field slow-roll models is canceled by projection effects when considering the observed power spectrum. We then further evaluate the nonzero, but finite, residual quadrupolar power asymmetry that remains after the divergences are canceled. While the quadrupolar power asymmetry is small, it is conceptually important. Our calculation moreover clarifies how the predictions for this power asymmetry may change in models with different scalar-scalar-tensor bispectra, and shows that convincing detection of the quadrupolar power asymmetry would rule out the single-field slow-roll models of inflation.Comment: 20 pages, 3 figures, revised to match the published versio

Primordial Non-Gaussianity and the Statistics of Weak Lensing and other Projected Density Fields

Estimators for weak lensing observables such as shear and convergence generally have non-linear corrections, which, in principle, make weak lensing power spectra sensitive to primordial non-Gaussianity. In this paper, we quantitatively evaluate these contributions for weak lensing auto- and cross-correlation power spectra, and show that they are strongly suppressed by projection effects. This is a consequence of the central limit theorem, which suppresses departures from Gaussianity when the projection reaches over several correlation lengths of the density field, L_P~55 [Mpc/h]. Furthermore, the typical scales that contribute to projected bispectra are generally smaller than those that contribute to projected power spectra. Both of these effects are not specific to lensing, and thus affect the statistics of non-linear tracers (e.g., peaks) of any projected density field. Thus, the clustering of biased tracers of the three-dimensional density field is generically more sensitive to non-Gaussianity than observables constructed from projected density fields.Comment: 10 pages, 6 figure

Large-Scale Galaxy Bias

This review presents a comprehensive overview of galaxy bias, that is, the statistical relation between the distribution of galaxies and matter. We focus on large scales where cosmic density fields are quasi-linear. On these scales, the clustering of galaxies can be described by a perturbative bias expansion, and the complicated physics of galaxy formation is absorbed by a finite set of coefficients of the expansion, called bias parameters. The review begins with a detailed derivation of this very important result, which forms the basis of the rigorous perturbative description of galaxy clustering, under the assumptions of General Relativity and Gaussian, adiabatic initial conditions. Key components of the bias expansion are all leading local gravitational observables, which include the matter density but also tidal fields and their time derivatives. We hence expand the definition of local bias to encompass all these contributions. This derivation is followed by a presentation of the peak-background split in its general form, which elucidates the physical meaning of the bias parameters, and a detailed description of the connection between bias parameters and galaxy statistics. We then review the excursion-set formalism and peak theory which provide predictions for the values of the bias parameters. In the remainder of the review, we consider the generalizations of galaxy bias required in the presence of various types of cosmological physics that go beyond pressureless matter with adiabatic, Gaussian initial conditions: primordial non-Gaussianity, massive neutrinos, baryon-CDM isocurvature perturbations, dark energy, and modified gravity. Finally, we discuss how the description of galaxy bias in the galaxies' rest frame is related to clustering statistics measured from the observed angular positions and redshifts in actual galaxy catalogs.Comment: 259 pages, 39 figures, 15 tables; published in Physics Reports; v2: minor corrections and clarifications, references added; v3: substantially revised and improved version; v4: minor edits and clarifications reflecting published version, corrected mistakes in Eqs. (7.57)-(7.58); v5: minor corrections [Eq. (5.5)] and updated reference