Relativistic Effect in Galaxy Clustering

The general relativistic description of galaxy clustering provides a complete and unified treatment of all the effects in galaxy clustering such as the redshift-space distortion, gravitational lensing, Sachs-Wolfe effects, and their relativistic effects. In particular, the relativistic description resolves the gauge issues in the standard Newtonian description of galaxy clustering by providing the gauge-invariant expression for the observed galaxy number density. The relativistic effect in galaxy clustering is significant on large scales, in which dark energy models or alternative theories of modified gravity deviate from general relativity. In this paper, we review the relativistic effect in galaxy clustering by providing a pedagogical derivation of the relativistic formula and by computing the observed galaxy two-point statistics. The relativistic description of galaxy clustering is an essential tool for testing general relativity and probing the early Universe on large scales in the era of precision cosmology.Comment: 21 pages, no figures. Invited review article, accepted for publication in Classical and Quantum Gravity focus issue on "Relativistic Effects in Cosmology", edited by Kazuya Koyam

Proper-Time Hypersurface of Non-Relativistic Matter Flows: Galaxy Bias in General Relativity

We compute the second-order density fluctuation in the proper-time hypersurface of non-relativistic matter flows and relate it to the galaxy number density fluctuation in general relativity. At the linear order, it is equivalent to the density fluctuation in the comoving synchronous gauge, in which two separate gauge conditions coincide. However, at the second order, the density fluctuations in these gauge conditions differ, while both gauge conditions represent the proper-time hypersurface. Compared to the density fluctuation in the temporal comoving and the spatial C-gauge conditions, the density fluctuation in the commonly used gauge condition ($N=1$ and $N^\alpha=0$) violates the mass conservation at the second order. We provide their physical interpretations in each gauge condition by solving the geodesic equation and the nonlinear evolution equations of non-relativistic matter. We apply this finding to the second-order galaxy biasing in general relativity, which complements the second-order relativistic description of galaxy clustering in Yoo & Zaldarriaga (2014).Comment: 16 pages, no figures, accepted for publication in PR

Gauge-Transformation Properties of Cosmological Observables and its Application to the Light-Cone Average

Theoretical descriptions of observable quantities in cosmological perturbation theory should be independent of coordinate systems. This statement is often referred to as gauge-invariance of observable quantities, and the sanity of their theoretical description is verified by checking its gauge-invariance. We argue that cosmological observables are invariant scalars under diffeomorphisms and as a consequence their theoretical description is gauge-invariant, only at linear order in perturbations. Beyond linear order, they are usually not gauge-invariant, and we provide the general law for the gauge-transformation that the perturbation part of an observable does obey. We apply this finding to derive the second-order expression for the observational light-cone average in cosmology and demonstrate that our expression is indeed invariant under diffeomorphisms.Comment: 16 pages, no figures, published in JCA

Jacobi Mapping Approach for a Precise Cosmological Weak Lensing Formalism

Cosmological weak lensing has been a highly successful and rapidly developing research field since the first detection of cosmic shear in 2000. However, it has recently been pointed out in Yoo et al. that the standard weak lensing formalism yields gauge-dependent results and, hence, does not meet the level of accuracy demanded by the next generation of weak lensing surveys. Here, we show that the Jacobi mapping formalism provides a solid alternative to the standard formalism, as it accurately describes all the relativistic effects contributing to the weak lensing observables. We calculate gauge-invariant expressions for the distortion in the luminosity distance, the cosmic shear components and the lensing rotation to linear order including scalar, vector and tensor perturbations. In particular, the Jacobi mapping formalism proves that the rotation is fully vanishing to linear order. Furthermore, the cosmic shear components contain an additional term in tensor modes which is absent in the results obtained with the standard formalism. Our work provides further support and confirmation of the gauge-invariant lensing formalism needed in the era of precision cosmology.Comment: 33 pages, no figures, published in JCA

Light-Cone Observables and Gauge-Invariance in the Geodesic Light-Cone Formalism

The remarkable properties of the geodesic light-cone (GLC) coordinates allow analytic expressions for the light-cone observables, providing a new non-perturbative way for calculating the effects of inhomogeneities in our Universe. However, the gauge-invariance of these expressions in the GLC formalism has not been shown explicitly. Here we provide this missing part of the GLC formalism by proving the gauge-invariance of the GLC expressions for the light-cone observables, such as the observed redshift, the luminosity distance, and the physical area and volume of the observed sources. Our study provides a new insight on the properties of the GLC coordinates and it complements the previous work by the GLC collaboration, leading to a comprehensive description of light propagation in the GLC representation.Comment: 25 pages, no figures, published in JCA

Wide Angle Effects in Future Galaxy Surveys

Current and future galaxy surveys cover a large fraction of the entire sky with a significant redshift range, and the recent theoretical development shows that general relativistic effects are present in galaxy clustering on very large scales. This trend has renewed interest in the wide angle effect in galaxy clustering measurements, in which the distant-observer approximation is often adopted. Using the full wide-angle formula for computing the redshift-space correlation function, we show that compared to the sample variance, the deviation in the redshift-space correlation function from the simple Kaiser formula with the distant-observer approximation is negligible in galaxy surveys such as the SDSS, Euclid and the BigBOSS, if the theoretical prediction from the Kaiser formula is properly averaged over the survey volume. We also find corrections to the wide-angle formula and clarify the confusion in literature between the wide angle effect and the velocity contribution in galaxy clustering. However, when the FKP method is applied, substantial deviations can be present in the power spectrum analysis in future surveys, due to the non-uniform distribution of galaxy pairs.Comment: 17 pages, 11 figures, accepted for publication in MNRA

Exact Analytic Solution for Non-Linear Density Fluctuation in a LCDM Universe

We derive the exact third-order analytic solution of the matter density fluctuation in the proper-time hypersurface in a $\Lambda$CDM universe, accounting for the explicit time-dependence and clarifying the relation to the initial condition. Furthermore, we compare our analytic solution to the previous calculation in the comoving gauge, and to the standard Newtonian perturbation theory by providing Fourier kernels for the relativistic effects. Our results provide an essential ingredient for a complete description of galaxy bias in the relativistic context.Comment: 20 pages, no figures, published in JCA

Maximum cosmological information from type Ia supernova observations

Type Ia supernova observations yield estimates of the luminosity distance, which includes not only the background luminosity distance, but also the fluctuation due to inhomogeneities in the Universe. These fluctuations are spatially correlated, hence limiting the cosmological information. In particular, the spatial correlation of the supernova host galaxies is a dominant source of the fluctuation in the luminosity distance measurements. Utilizing the recent theoretical framework that accurately quantifies the information contents accounting for the three-dimensional correlation of the observables on the past light cone, we compute the maximum cosmological information obtainable from idealized supernova surveys, where an infinite number of observations are made over the full sky without any systematic errors up to a maximum redshift zm. Here we consider two cosmological parameters Ωm and w0 and show that the cosmological information contents are a lot more reduced than previously computed in literature. We discuss how these fundamental limits set by cosmic variance can be overcome

Incompatibility of standard galaxy bias models in general relativity

The standard model for galaxy bias is built in a Newtonian framework, and several attempts have been made in the past to put it in a relativistic framework. The focus of past works was, however, to use the same Newtonian formulation, but to provide its interpretation in a relativistic framework by either fixing a gauge condition or transforming to a local coordinate system. Here we demonstrate that these reverse-engineered approaches do not respect the diffeomorphism symmetry in general relativity, and we need to develop a covariant model of galaxy bias that is diffeomorphism compatible. We consider a simple toy model for galaxy bias and discuss the impact for measuring the primordial non-Gaussianity

Relativistic effects in galaxy clustering in a parametrized post-Friedmann universe

We explore the signatures of quintessence and modified gravity theories in the relativistic description of galaxy clustering within a parametrized post-Friedmann framework. For this purpose, we develop a calibration method to consistently account for horizon-scale effects in the linear parametrized Post-Friedmann perturbations of minimally and nonminimally coupled scalar-tensor theories and test it against the full model-specific fluctuations. We further study the relativistic effects in galaxy clustering for the normal and self-accelerating branches of the Dvali-Gabadadze-Porrati braneworld model as well as for phenomenological modifications of gravity. We quantify the impact of modified gravity and dark energy models on galaxy clustering by computing the velocity-to-matter density ratio F, the velocity contribution R, and the potential contribution P and give an estimate of their detectability in future galaxy surveys. Our results show that, in general, the relativistic correction contains additional information on gravity and dark energy, which needs to be taken into account in consistent horizon-scale tests of departures from LCDM using the galaxy-density field.Comment: 24 pages, 7 figures, 1 table; v2 matches published versio