## Periodic Reporting for period 4 - GREinGC (General Relativistic Effect in Galaxy Clustering as a Novel Probe of Inflationary Cosmology)

Reporting period: 2020-09-01 to 2022-02-28

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.

We have developed a proper relativistic description of galaxy clustering at the second order in perturbations, applied the theoretical predictions to modified gravity theories and inflationary models, and quantified the systematic errors in the standard theoretical predictions. A correct prediction for the primordial non-Gaussianity that can be measured in observations is at the slow-roll parameter (~0.01) with all the relativistic effects contributing order unity but cancelling each other, while the standard predictions in literature are order a few, as they miss the full relativistic contributions. A proper comparison to observations and correct interpretation of precision measurements in the upcoming surveys can only be made with theoretical descriptions as accurate as the measurement precision demands.

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.

We have developed a proper relativistic description of galaxy clustering at the second order in perturbations, applied the theoretical predictions to modified gravity theories and inflationary models, and quantified the systematic errors in the standard theoretical predictions. A correct prediction for the primordial non-Gaussianity that can be measured in observations is at the slow-roll parameter (~0.01) with all the relativistic effects contributing order unity but cancelling each other, while the standard predictions in literature are order a few, as they miss the full relativistic contributions. A proper comparison to observations and correct interpretation of precision measurements in the upcoming surveys can only be made with theoretical descriptions as accurate as the measurement precision demands.

@ 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.

@ Second-order gauge-invariant formalism for the cosmological observables: Complete verification of their gauge-invariance: Accounting for all the relativistic effects, we have developed the fully nonlinear gauge-invariant formalism for describing the cosmological observables and presented the second-order perturbative expressions associated with light propagation and observations without choosing a gauge condition. For the first time, we have performed a complete verification of the validity of our second-order expressions by comparing their gauge-transformation properties from two independent methods: one directly obtained from their expressions in terms of metric perturbations and the other expected from their nonlinear relations. The expressions for the cosmological observables such as galaxy clustering and the luminosity distance are invariant under diffeomorphism and gauge-invariant at the observed position. We compare our results to the previous work and discuss the differences in the perturbative expressions. Our second-order gauge-invariant formalism constitutes a major step forward in the era of precision cosmology and its applications in the future will play a crucial role for going beyond the power spectrum and probing the early universe.

@ Non-Gaussianity in the Squeezed Three-Point Correlation from the Relativistic Effects: Assuming a LCDM universe in a single-field inflationary scenario, we compute the three-point correlation function of the observed matter density fluctuation in the squeezed triangular configuration, accounting for all the relativistic effects at the second order in perturbations. This squeezed three-point correlation function characterizes the local-type primordial non-Gaussianity, and it has been extensively debated in literature whether there exists a prominent feature in galaxy clustering on large scales in a single-field inflationary scenario either from the primordial origin or the intrinsic nonlinearity in general relativity. We show that the relativistic effects associated with light propagation in observations cancel each other, and hence there exists no non-Gaussian contribution from the so-called projection effects.

@ Second-order gauge-invariant formalism for the cosmological observables: Complete verification of their gauge-invariance: Accounting for all the relativistic effects, we have developed the fully nonlinear gauge-invariant formalism for describing the cosmological observables and presented the second-order perturbative expressions associated with light propagation and observations without choosing a gauge condition. For the first time, we have performed a complete verification of the validity of our second-order expressions by comparing their gauge-transformation properties from two independent methods: one directly obtained from their expressions in terms of metric perturbations and the other expected from their nonlinear relations. The expressions for the cosmological observables such as galaxy clustering and the luminosity distance are invariant under diffeomorphism and gauge-invariant at the observed position. We compare our results to the previous work and discuss the differences in the perturbative expressions. Our second-order gauge-invariant formalism constitutes a major step forward in the era of precision cosmology and its applications in the future will play a crucial role for going beyond the power spectrum and probing the early universe.

@ Non-Gaussianity in the Squeezed Three-Point Correlation from the Relativistic Effects: Assuming a LCDM universe in a single-field inflationary scenario, we compute the three-point correlation function of the observed matter density fluctuation in the squeezed triangular configuration, accounting for all the relativistic effects at the second order in perturbations. This squeezed three-point correlation function characterizes the local-type primordial non-Gaussianity, and it has been extensively debated in literature whether there exists a prominent feature in galaxy clustering on large scales in a single-field inflationary scenario either from the primordial origin or the intrinsic nonlinearity in general relativity. We show that the relativistic effects associated with light propagation in observations cancel each other, and hence there exists no non-Gaussian contribution from the so-called projection effects.

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. We found several relativistic effects missing in the standard descriptions and plan to compute the systematic errors in the standard theoretical modeling. Since the second-order relativistic descriptions are intrinsically complicated, many different groups performed the calculations, but the predictions do not agree due to the difference in the theoretical descriptions. To overcome this issue, we checked the gauge-invariance of the second-order expressions for the first time. In the second half of the ERC project, we 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. With our theoretical descriptions, we showed that there exist subtle cancellations of many relativistic contributions, which are not captured in previous work. The correct predictions for the standard inflationary model were smaller by two orders of magnitude.