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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - GREinGC (General Relativistic Effect in Galaxy Clustering as a Novel Probe of Inflationary Cosmology)

Teaser

Summary

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-Ïƒ 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

Work performed

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 a

Final results

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.