利用非對稱星系聚集與本徵排列進行重力的宇宙學檢驗

Abstract

在宇宙尺度上檢驗引力理論對於理解宇宙加速膨脹及引力機制至關重要。本論文提出了一種利用非對稱星系聚類與內在排列（IA）統計來檢驗廣義相對論的框架。隨著當前及未來高精度星系巡天數據的提供，從這些新型探針中提取宇宙學信息的高級方法變得愈發關鍵。本論文主要針對兩個核心挑戰：在非對稱聚類統計中納入高階奇數極矩，以及在 IA 統計中選擇最佳極展開基底。

本論文的第一部分探討了高階極矩在非對稱星系聚類中的作用。Saga 等人最近的研究提出，利用不同星系樣本之間的互相關函數來檢驗愛因斯坦等效原理（EEP）的關鍵組成部分——局部位置不變性（LPI），但其分析僅限於偶極矩。本論文在其基礎上進一步研究了高階奇數極矩——八極矩在 LPI 違反參數 α 之約束能力，並預測了來自 Dark Energy Spectroscopic Instrument、Euclid Space Telescope、Subaru Prime Focus Spectrograph 及 Square Kilometre Array 等星系巡天的結果。我們發現，當分析局限於較大尺度，即選擇較大的最小分離距離 smin 時，結合八極矩與偶極矩能顯著提升約束能力。在保守設定 smin = 15Mpc/h 的情況下，相較於僅使用偶極矩，平均約束能力提升 11%。本研究結果突顯了高階極矩的重要性，提供了一種更穩健的方法來在宇宙尺度上檢驗 EEP。

論文的第二部分探討了極展開基底選擇對 IA 統計的影響。我們研究了 IA 統計的自然基底——伴隨勒讓德多項式，並將該基底所得的約束與傳統勒讓德基底所得的結果進行比較。我們發現，IA 功率譜極矩在伴隨勒讓德基底下收斂至完整 2D 功率譜的信息量較慢。在高星系數密度情況下，即便使用最高至 l = 6 的極矩，伴隨勒讓德基底下的 IA 互功率譜與自功率譜對哈勃參數的誤差，分別比完整 2D 分析大約 6% 和 10%。本研究結果表明，基底選擇雖然是任意的，但會根據樣本與統計方法影響極矩所承載的信息量。

總結來說，本論文證明了高階奇數極矩在非對稱聚類中的重要性，以及基底選擇對內在排列統計的影響，從而改進了對引力理論與標準宇宙學模型的檢驗。雖然本研究基於理論 Fisher 分析，未來的工作應將這些方法應用於模擬與觀測數據，以獲得更精確的約束並深入理解宇宙的基本性質。

Testing gravity on cosmological scales is crucial for understanding the accelerated expansion of the Universe and the mechanisms of gravity. This thesis proposes a framework for testing general relativity using asymmetric galaxy clustering and intrinsic alignment (IA) statistics. As ongoing and upcoming galaxy survey data with high precision provide, the need for advanced methods to extract the cosmological information from these novel probes becomes increasingly critical. This thesis addresses two key challenges: the incorporation of a higher-order odd multipole in asymmetric clustering statistics and the optimal choice of the basis for multipole expansion in IA statistics.

The first part explores the role of higher-order multipoles in asymmetric galaxy clus- tering. Recent work by Saga et al. proposed the validation of the local position invariance (LPI), a key aspect of the Einstein equivalence principle (EEP), using the cross-correlation function between distinct galaxy samples, but their analysis focused solely on the dipole. In this thesis, we extend their work by further analyzing a higher-order odd multipole, the octupole, in the constraints on the LPI-violating parameter, α, expected from galaxy surveys such as Dark Energy Spectroscopic Instrument, Euclid space telescope, Subaru Prime Focus Spectrograph, and Square Kilometre Array. We demonstrate that combining the octupole and dipole moments significantly improves the constraints, particularly when the analysis is restricted to larger scales, characterized by a large minimum separation smin. For a conservative setup with smin = 15Mpc/h, we find an average improvement of 11% compared to using the dipole alone. Our results highlight the importance of higher-order multipoles, providing a more robust approach to testing the EEP on cosmological scales.

In the second part, we investigate the impact of basis choice for multipole expansion in IA statistics. We examine the associated Legendre polynomials, a natural basis for IA statistics, and compare the constraints obtained from this basis with those obtained from the conventional Legendre basis. Our findings reveal that the information in the multipoles of the IA power spectra in the associated Legendre basis converges more slowly to that in the full 2D power spectra than in the Legendre basis. In this high number density case, we show that the errors on the Hubble parameter obtained from the multipoles of the IA cross- and auto-power spectra in the associated Legendre basis are respectively about 6% and 10% larger than the full 2D case even when we use multipoles up to l = 6. Our results demonstrate that the choice of basis is arbitrary but changes the information content encoded in multipoles depending on the sample and statistics under consideration.

In conclusion, this thesis demonstrates the importance of higher-order odd multipoles in asymmetric clustering and the choice of basis in IA statistics, improving tests of gravity and the standard cosmological model. While based on theoretical Fisher analysis, future work should apply these methods to simulations and observational data, leading to more precise constraints and deeper insights into the fundamental nature of the Universe.