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General Relativistic Effect in Galaxy Clustering as a Novel Probe of Inflationary Cosmology

Periodic Reporting for period 3 - GREinGC (General Relativistic Effect in Galaxy Clustering as a Novel Probe of Inflationary Cosmology)

Reporting period: 2019-03-01 to 2020-08-31

Substantial advances in cosmology over the past decades have firmly established the standard model of cosmology. However, the physical nature of the early Universe and dark energy (or inflationary cosmology) remains poorly understood. To resolve these issues, a large number of galaxy surveys are planned to measure millions of galaxies in the sky, promising precision measurements of galaxy clustering with enormous statistical power. Despite these advances in observation, the standard theoretical description of galaxy clustering is based on the Newtonian description, inadequate for measuring the relativistic effects from the early Universe and the deviations of modified gravity from general relativity. In recent years, the applicant, for the first time, developed the linear-order general relativistic description of galaxy clustering and showed that the relativistic effect in galaxy clustering is already measurable at a few-sigma level in current surveys like the Sloan survey and significant detections (>10 sigma) are possible in upcoming surveys.

This research proposal aims to use the subtle relativistic effect in galaxy clustering to develop novel probes of inflationary cosmology. In particular, the applicant will 1) formulate the higher-order relativistic description of galaxy clustering, an essential tool for computing the bispectrum, and 2) investigate the unique relativistic signatures (linear-order and higher-order) in galaxy clustering from the early Universe and dark energy to develop novel probes of isolating those signatures and to quantify their detectabilities in future galaxy surveys. Biases in cosmological parameter estimation, if the standard Newtonian description is used, will be quantified. A comprehensive understanding of inflationary cosmology will have far-reaching consequences, shedding light on new physics beyond the standard model.

The specific goal over the first 30 months is to develop the higher-order general relativistic description of cosmological observables such as galaxy clustering and establish its theoretical foundation. On large scales, the metric perturbations along the photon path affect the photon propagation. Furthermore, the photon propagation is described in the FRW frame, while the observables and the physical quantities are defined in the observer and the source rest frames. These subtle relativistic effects need to be properly taken into account in considering the relation of the observable quantities in galaxy clustering such as the observed redshift and the angular position of source galaxies to the physical quantities of the source galaxies. In the past few years, the linear-order relativistic effect in galaxy clustering has been computed, and it was shown that these subtle relativistic effects can be detected at more than 10-σ level and can be used to test general relativity and probe the early universe in upcoming galaxy surveys. Drawing on the past work, the higher-order relativistic effects in galaxy clustering was computed with main focus on the second-order relativistic description of the observed galaxy number density and the luminosity distance, essential ingredients in the era of precision cosmology.

Twelve papers were published in the Journal of Cosmology and Astroparticle Physics, in conjunction with the Annex 1 of the Grant Agreement. The measurements of the luminosity distance from the distant supernovae provided the convincing evidence that the Universe is accelerating. However, its theoretical calculation is often based on the background Universe ignoring the inhomogeneities in the Universe, and most calculations accounting for the perturbations do not agree, as the second-order calculations are inevitably lengthy and complicated. In the publications, our team performed the detailed calculations of the luminosity distance and showed that the infrared divergences in the luminosity distance are due to the gauge artefact of the incomplete theoretical calculations. We provided the gauge-transformation properties that any physical observables including the luminosity distance and galaxy clustering should satisfy, which can be used to check the sanity of the second-order calculations. Albeit lengthy, it is the most straightforward and sound method. We held a mini workshop to establish further verification and comparison of the second-order calculations among the different research groups. Recently, we quantified the systematic errors in galaxy correlation function measurements. The general relativistic effects missing in the standard prediction are at two percent level just beyond the baryonic acoustic oscillation scale and they become larger as the scale increases.

Two papers out of twelve were published, in which we investigated the nonlinear matter power spectrum and fully nonlinear application to the FRW metric. The computation of the nonlinear matter power spectrum beyond the linear order is essential in modelling the nonlinear clustering of galaxies and computing the bispectrum or the loop corrections in the power spectrum, both of which are now well measured in the current surveys. In most part, the beyond-the-linear-order calculation of the relativistic effects in galaxy clustering is to compute the 4D volume distortion and relativistic corrections of the observed galaxy number density, due to the mismatch between the physical quantities of the source galaxies and the observable quantities such as the observed redshift and angular position of the source galaxy. However, one critical step remains to be addressed in interpreting the observed galaxy number density, namely, galaxy bias --- the relation between the physical galaxy number density and the underlying matter distribution. At the linear order, a simple argument can be made that galaxy formation is a local process and it only depends on local curvature, providing justification for the use of the matter density fluctuation at the hypersurface of proper time, which is the comoving and synchronous gauge. However, beyond the linear order in perturbations, this simple relation is lost, and two gauge conditions give different matter density fluctuations. Irrespective of nonlinear biasing schemes, computation of the higher-order matter density fluctuation is necessary, even for the simplest linear biasing, as we need the second-order matter density fluctuation for the tree-level bispectrum and the third-order matter density fluctuation for one-loop corrections to the power spectrum. Our team showed that even in the popular comoving and the synchronous gauges there exits subtle effects in the matter power spectrum that may artificially appear as the primordial non-Gaussianity. Furthermore, our recent computation of the fully nonlinear formalism will provide a great new opportunity to explore cosmological applications beyond the perturbation theory.

Two papers were published, addressing the issues in weak gravitational lensing. The standard weak lensing formalism is relativistic, but not fully relativistic, because it relies on the unlensed source angular position, which is un-observable quantity. We developed fully gauge-invariant weak lensing formalism and found several missing velocity contributions as well as gravitational wave contributions. As opposed to the standard prediction, in which tensor modes rotate the lensing images, we found that the rotation induced by tensor modes is a coordinate artefact, as the local basis rotates against the coordinate when parallel transported. We plan to compute the impact of the missing relativistic effects on current and future surveys.
Theoretical foundation of the higher-order general relativistic effects:
1) Gauge-Invariance and Infrared Divergences in the Luminosity Distance
Measurements of the luminosity distance have played a key role in discovering the late-time cosmic acceleration. However, when accounting for inhomogeneities in the Universe, its interpretation has been plagued with infrared divergences in its theoretical predictions, which are in some cases used to explain the cosmic acceleration without dark energy. The infrared divergences in most calculations are artificially removed by imposing an infrared cut-off scale. We show that a gauge-invariant calculation of the luminosity distance is devoid of such divergences and consistent with the equivalence principle, eliminating the need to impose a cut-off scale. We present proper numerical calculations of the luminosity distance using the gauge-invariant expression and demonstrate that the numerical results with an ad hoc cut-off scale in previous calculations have negligible systematic errors as long as the cut-off scale is larger than the horizon scale. We discuss the origin of infrared divergences and their cancellation in the luminosity distance.

2) Unified Treatment of the Luminosity Distance in Cosmology
Comparing the luminosity distance measurements to its theoretical predictions is one of the cornerstones in establishing the modern cosmology. However, as shown in Biern & Yoo, its theoretical predictions in literature are often plagued with infrared divergences and gauge-dependences. This trend calls into question the sanity of the methods used to derive the luminosity distance. Here we critically investigate four different methods --- the geometric approach, the Sachs approach, the Jacobi mapping approach, and the geodesic light cone (GLC) approach to modelling the luminosity distance, and we present a unified treatment of such methods, facilitating the comparison among the methods and checking their sanity. All of these four methods, if exercised properly, can be used to reproduce the correct description of the luminosity distance.

3) Correlation function of the luminosity distances
We present the correlation function of the luminosity distances in a flat LCDM universe. Decomposing the luminosity distance fluctuation into the velocity, the gravitational potential, and the lensing contributions in linear perturbation theory, we study their individual contributions to the correlation function. The lensing contribution is important at large redshift (z>0.5) but only for small angular separation, while the velocity contribution dominates over the other contributions at low redshift or at larger separation. However, the gravitational potential contribution is always subdominant at all scale, if the correct gauge-invariant expression is used. The correlation function of the luminosity distances depends significantly on the matter content, especially for the lensing contribution, thus providing a novel tool of estimating cosmological parameters.

4) 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.

5) 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 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.

6) Nonlinear general relativistic effects in the observed redshift
We present the second-order expression for the observed redshift, accounting for all the relativistic effects from the light propagation and from the frame change at the observer and the source positions. We derive the generic gauge-transformation law that any observable quantities should satisfy, and we verify our second-order expression for the observed redshift by explicitly checking its gauge transformation property. This is the first time an explicit verification is made for the second-order calculations of observable quantities. We present our results in popular gauge choices for easy use and discuss the origin of disagreements in previous calculations.

7) Galaxy two-point correlation function in general relativity
We perform theoretical and numerical studies of the full relativistic two-point galaxy correlation function. Using the gauge-invariant relativistic description of galaxy clustering, we demonstrate that the complete theoretical expression is devoid of any long-mode contributions from scalar or tensor perturbations and it lacks the infrared divergences in agreement with the equivalence principle. Using the full gauge-invariant expression, we numerically compute the galaxy two-point correlation function and study the individual contributions in the conformal Newtonian gauge. Compared to the standard Newtonian theoretical predictions, the relativistic effects in galaxy clustering result in a few percent-level systematic errors beyond the scale of the baryonic acoustic oscillation. Our theoretical and numerical study provides a comprehensive understanding of the relativistic effects in the galaxy two-point correlation function.

Nonlinear computation of the matter density fluctuation:
1) 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 LCDM 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.

2) Exact non-linear equations for cosmological perturbations
We present a complete set of exact and fully non-linear equations describing all three types of cosmological perturbations -- scalar, vector and tensor perturbations. We derive the equations in a thoroughly gauge-ready manner, so that any spatial and temporal gauge conditions can be employed. The equations are completely general without any physical restriction except that we assume a flat homogeneous and isotropic universe as a background. We also comment briefly on the application of our formulation to the non-expanding Minkowski background.

General relativistic effects in weak gravitational lensing
1) Gauge-invariant formalism of cosmological weak lensing
We present the gauge-invariant formalism of cosmological weak lensing. By constructing the local tetrad bases at the observer and the source positions, we clarify the relation of the weak lensing observables such as the convergence, the shear, and the rotation to the physical size and shape defined in the source rest-frame and the observed angle and redshift measured in the observer rest-frame. Compared to the standard lensing formalism, additional relativistic effects contribute to all the lensing observables. We explicitly verify the gauge-invariance of the lensing observables and compare our results to previous work. In particular, we demonstrate that even in the presence of the vector and tensor perturbations, the physical rotation of the lensing observables vanishes at the linear order, while the tetrad basis rotates along the light propagation compared to a FRW coordinate. Though the latter is often used as a probe of primordial gravitational waves, the rotation of the tetrad basis is indeed not a physical observable. We further clarify its relation to the E-B decomposition in weak lensing. Our formalism provides a transparent and comprehensive perspective of cosmological weak lensing.

2) Jacobi mapping approach for a precise cosmological weak lensing formalism
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.
We investigate the general relativistic effects of all large-scale structure probes such as galaxy clustering, weak gravitational lensing, and CMB in the standard cosmology. The theoretical predictions in the standard cosmology are incomplete, because they often miss several relativistic effects and they are gauge dependent. We develop fully gauge-invariant theoretical descriptions of cosmological observables and check the gauge-invariance. Galaxy clustering was put in a proper general relativistic framework, and we extended the calculations to higher order perturbation theory. Furthermore, we develop for the first time fully gauge-invariant weak gravitational lensing formalism. The weak lensing formalism in the standard cosmology is on a good relativistic footing, but it is again incomplete and suffers from several gauge issues. We found several relativistic effects missing in the standard descriptions and plan to compute the systematic errors in the standard theoretical modeling. In the second half of the ERC project, we will apply the relativistic formalism to inflationary models and modified gravity theories to quantify their unique relativistic signatures and identify novel ways to distinguish them from the LCDM predictions.