在論文的第一部分,我們簡扼介紹了二次型重力理論並論證了純二次型重力理論不滿足一些物理要求。因此,在第二章節我們開始著手研究耦合於純量場並且共形對稱破缺的二次型重力理論的解析性質,並在宇宙學背景下,我們發現了這個理論的運動方程式有著部分可積性,同時我們也給出了一些簡單的宇宙學解析解。在第三章裡,我們發現了一個可能可以解決暗能量問題非平凡真空解析解。利用該真空解析解所提供的最佳擬合參數,我們找出了能夠與超新星數據良好擬合併能夠外插至宇宙學復合時期紅移量的宇宙學數值解。在第四章節中,我們發現了一族純二次型重力理論的球對稱解析解,但這些解析解的事件視界卻是由真實且可察覺的裸奇點所構成的超曲面,這再次反映了純二次型重力理論的非物理的性質。在最後一個章節裡,我們證明了在平行傳播時空裡,假如本模型中的純量場在平行傳播波的波前上是常數且非奇異的,那麼在每個 x-y 平面上此純量場都必須是常數,而最後我們也給出了一般解。

In the first chapter of this thesis, we give a brief introduction to the quadratic gravities and suggest the impossibility of pure quadratic gravities. Subsequently, in the second chapter, we study the analytic perspective of quadratic gravity theories coupled to a scalar field with broken conformal symmetry. In the cosmological setting, we find that the equation of motion is partially integrable, and some trivial exact solutions are given. In the third chapter, exact nontrivial solutions to this model are found, which could solve the dark energy problem. Around the parameters suggested by this exact vacuum solution, we find numerical solutions with the presence of radiation and cold dust which fit the Hubble parameter with the supernova data well and are extensible to the recombination redshift. In the fourth chapter, we give a family of exact spherically symmetric solutions of the pure quadratic gravities whose event horizons are hypersurfaces of true, detectable and naked singularities, which again suggests that the pure quadratic gravities are unphysical. In the last chapter, we prove that the scalar field must be constant on each x-y plane provided that the scalar field is constant on each wave-front of the pp-wave and non-singular.

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