修正重力理論：宇宙學與天文物理

Abstract

修正重力理論在近代理論物理中是一門相當熱門的研究領域。一般來說，這些重力理論必須要給出廣義相對論中所有成功的宇宙學預測和天文學預測，此外，他們還預期能解決一些廣義相對論無法解釋的問題，例如各種指向暗物質暗能量的存在的觀測證據、暗示了廣義相對論不完備的時空奇異點，以及廣義相對論和量子理論的矛盾等等。由於現階段我們依舊沒有一個完整的量子重力理論，因此，從現象學的角度來看，我們或許可以將修正重力理論當成是一個基本但未知的量子重力理論的有效理論。再者，我們原則上可以藉由最新的觀測結果來測試這些重力理論，例如宇宙微波背景、重力透鏡、黑洞剪影、Ia 型超新星、重子聲學震盪、重力波、以及哈伯參數測量等等。藉由探討這些修正重力理論所蘊含的物理以及他們在觀測上的預測，我們可以更深入的了解我們的宇宙。 在這篇論文中，我們考慮一些有趣的修正重力理論以及他們在宇宙學和天文學的延伸。我們首先著重在具有 Born-Infeld 結構的理論，並且研究他們在古典宇宙學底下的解。在所謂的模仿 Born-Infeld 重力中，我們額外考慮了他的黑洞解，這是因為我們預期這個理論會讓黑洞有和廣義相對論不同的內部結構。接著我們考慮兩種不同的方法來研究 Eddington-inspired-Born-Infeld 理論 (EiBI) 的量子效應。第一個方 法是利用量子時空幾何動力論，來解出可以用來描述整個宇宙的量子行為的 Wheeler-DeWitt 方程。藉由這個方法，我們研究在 EiBI 理論中，利用量子效應來去除時空奇異點的可能性。在第二個方法中，我們研究具有 O(4) 對稱性的瞬子來描述 EiBI 理論中的量子穿隧效應。其中，我們研究奇異瞬子和常規瞬子，並且發現了一些廣義相對論中沒有的有趣結果，例如奇異極點的去除，以及羅倫茲蟲洞的生成等等。 接著，我們探討藉由黑洞微擾所發出的重力波來測試修正重力理論的可能性，這個研究方向在近年來已然成為一門顯學。在這篇論文中，我們研究一些理論中黑洞解所產生的準簡正模 (QNMs)，這些理論包含 Palatini-f(R) 重力、EiBI 重力、以及真空共形重力理論中的非奇異黑洞解。我們得出描述 QNMs 的主方程，並且利用 WKB 方法來計算這些 QNM 頻率。如同我們所預期，這些黑洞的 QNM 頻譜會與廣義相對論中的結果有所差異，這暗示了藉由未來的重力波探測來測試這些黑洞解的可能性。 原則上，我們有無窮多種可能的方法來修正廣義相對論，每一種方法都有屬於自己的理論基礎，並且分別給出了不同的物理預測。藉由未來的觀測技術發展，大部分的修正重力理論都可以被預測，甚至被否證。這將會漸漸的讓我們了解應該用什麼樣的方式來描述與理解我們的宇宙。

Modified theories of gravity have been generally regarded as a promising research direction in modern theoretical physics. Generically, they are required to reproduce all the successful cosmological and astrophysical predictions made by Einstein’s General Relativity (GR). In addition, they are motivated to resolve problems which by now cannot be well explained by GR, such as the strong observational evidences hinting towards the existence of dark energy and dark matter, the troublesome spacetime singularities which indicate the incompleteness of GR itself, and the potential conflict between GR and the quantum theory. Since there is still no firmly established quantum theory of gravity so far, modified theories of gravity can be treated as effective theories of such a fundamental yet unknown quantum theory of gravity from phenomenological perspectives. Furthermore, these theories of gravity can in principle be tested by the latest observations such as cosmic microwave background, gravitational lensing, black hole shadows, type Ia supernovae, baryon acoustic oscillations, gravitational waves, and measures of the Hubble parameter. By understanding better the physical implications of these modified theories of gravity and what their observational predictions are, it is believed that we can get more hints towards the fundamental nature of our Universe. In this thesis, we consider some interesting modified theories of gravity and study their cosmological and astrophysical implications. We first focus on the theories with Born-Infeld inspired structures and investigate the cosmological solutions of these theories at the classical level. In the so-called mimetic Born-Infeld gravity, we pay additional attention to its black hole solutions since significant differences from GR of the interior structure are expected. Then we consider two different approaches towards quantum gravitational effects of the so-called Eddington-inspired-Born-Infeld gravity (EiBI). The first approach is the quantum geometrodynamics based on solving the Wheeler-DeWitt equation which describes the quantum dynamics of the Universe as a whole. Within this approach we investigate the quantum avoidance of spacetime singularities in the EiBI gravity. In the second approach we investigate the O(4)-symmetric instantons which describe quantum tunneling processes in the EiBI gravity. The singular and regular instantons are studied and several interesting outcomes absent in GR are obtained, such as the avoidance of singular poles and the formation of Lorentzian wormholes. We then turn our consideration to the possibility of testing modified gravity theories through gravitational waves emitted by perturbed black holes. This research direction has become an interesting arena recently. In this thesis, we investigate the quasi-normal modes (QNMs) for black hole solutions in the following modified gravity theories: the Palatini-f (R) gravity, the EiBI gravity, and a class of nonsingular black holes which can be treated as the vacuum solution of a family of conformal gravity theories. The master equations describing the QNMs are derived, and the QNM frequencies are evaluated with the Wentzel-Kramers-Brillouin (WKB) method. As expected, the QNM spectra of these modified black holes would deviate from their GR counterparts, indicating the possibility of testing these black hole solutions with the help of future gravitational wave detections. In principle, there could be infinitely many ways to modify GR. Each of them stems from different theoretical grounds and gives distinct predictions. With future developments of observations, most modified theories of gravity can be tested or even falsified, revealing a much clearer picture of how our Universe should be comprehended.