引力波作为一种新信使，在宇宙学研究中有着重要地位。目前使用引力波研究宇宙学的方法主要有标准汽笛法和强引力透镜时间延迟法。其中，标准汽笛法受益于双星并合引力波波形的精确建模，使得引力波可以作为一种新的标准距离指示器；引力波强引力透镜时间延迟法受益于时间延迟的精确测量以及低光污染，拥有较小的系统误差和较好的透镜重构结果。这些优势使得引力波有望在不久的将来为解决现代宇宙学中的“哈勃危机”问题作出贡献。

然而，引力波在传播过程中不可避免地会受到宇宙大尺度结构上弱引力透镜的影响，在其光度距离测量上产生系统误差，从而影响标准汽笛法的精度。系统评估弱引力透镜效应引入的系统误差，可以为未来引力波大样本时代标准汽笛法提供理论指导，为“去透镜”方法的寻找提供启示。
受限于引力波的定位精度，目前强透镜引力波事例的认证方法普遍存在效率低、误报率高等问题，限制了引力波强透镜方法的发展。除此之外，透镜星系中恒星等小质量天体产生的微引力透镜效应会在引力波波形上产生影响，在引力波中引入系统偏差。系统评估透镜星系中的微引力透镜效应、找到高效且低误报率的强透镜引力波事例认证方法将对引力波宇宙学应用产生深远影响。

基于以上几点问题，本人博士课题研究了不同尺度下引力透镜对引力波的影响，具体分为弱透镜系统误差、强透镜认证和微透镜波动光学效应。

本文第一章将简要介绍目前宇宙学研究的主流手段、存在的问题并介绍引力透镜和引力波的基础理论。第二至六章为博士期间五个研究课题。

第二章的研究首次使用多球面光线追踪算法探究了宇宙大尺度结构弱引力透镜效应对第二和第三代引力波探测器光度距离参数估计的影响。光度距离是引力波标准汽笛方法中最为重要的一个物理量，此研究工作为标准汽笛方法的系统误差提供了理论指导。

第三和第四章提出了一套算法用来探究强引力透镜星系中微引力透镜（主要包括恒星和恒星产物）对强透镜引力波的影响。由于引力波的长波性质，微引力透镜会在强透镜引力波波形上留下干涉的印记，此效应在强透镜引力波事例数据分析中不可忽略。然而，强透镜星系中微引力透镜数目十分庞大，使得干涉效应极难计算。所以，发展一套完整的衍射积分数值算法将有助于后续的科学研究。其中，第三章解决了算法收敛性和傅立叶变换频谱泄漏问题，第四章解决了数值分辨率、计算精度和速度问题。

第五章重点关注强透镜引力波事例认证。目前强透镜引力波事例认证的主流方法普遍存在假阳性率高、计算耗时长的问题。为解决这两个问题，本工作首次提出可以通过搜寻强透镜引力波中所特有的微透镜干涉印记来认证强透镜引力波事例。此方法可以有效地应用于未来第三代引力波探测器，为强透镜引力波信号以及后续宿主星系认证提供了新方法。

第六章基于第三和第四章提出的衍射积分数值算法，首次系统评估了微引力透镜效应对强透镜引力波参数估计及认证所产生的系统误差。为后续强透镜引力波的数据分析以及去透镜方法的寻找给出了启示。

第七章为全文的总结与后续工作展望。
 

Gravitational waves, as a new messenger, hold an important position in cosmological research. The current methods of studying cosmology using gravitational waves mainly include the standard siren method and the strong gravitational lensing time delay method. The standard siren method benefits from the precise modeling of binary merger gravitational wave waveforms, enabling gravitational waves to serve as a new standard distance indicator. The strong gravitational lensing time delay method benefits from the precise measurement of time delays and low light pollution to host galaxy (compared with strong lensing quasars), offering smaller systematic errors and better lens reconstruction results. These advantages make gravitational waves promising in resolving the "Hubble tension" issue in modern cosmology in the near future.

However, gravitational waves inevitably suffer from the influence of weak gravitational lensing from the large-scale structure of the universe during their propagation, leading to biases in their luminosity distance estimation and thus affecting the accuracy of the standard siren method. A full assessment of the systematic introduced by weak gravitational lensing can provide theoretical guidance for the standard siren method in the era of large gravitational wave samples and offer insights for the search of de-lensing method.

Limited by the localization accuracy of gravitational waves, the current identification methods for strong lensing gravitational wave generally suffer from low efficiency and high false positive rates, which limit the development of the strong lensing gravitational wave studies. In addition, microlensing effects caused by low-mass discrete objects such as stars in lensing galaxies can affect the gravitational wave waveform, introducing systematic biases. Systematically evaluating the microlensing effects in lensing galaxies, and developing methods that are both efficient and have a low false positive rate for the identification of strong lensing gravitational waves, will have a profound impact on the application of gravitational wave cosmology.

Based on the above issues, my doctoral research investigates the impact of gravitational lensing on gravitational waves at different scales, specifically focusing on weak lensing systematic errors, strong lensing identification, and microlensing wave effects.

The first chapter of this thesis will briefly introduce the mainstream methods of current cosmological research, the existing problems, and the basic theories of gravitational lensing and gravitational waves. 
Chapters two to six will cover five research topics during the doctoral period.

In the second chapter, we use a multi-sphere ray-tracing method to explore the impact of weak lensing effects from the large-scale structure of the universe on the luminosity distance estimation of the second and third generation gravitational wave detectors. Luminosity distance is one of the most important physical quantities in the standard siren method of gravitational waves, and this research provides theoretical guidance for the systematic errors of the standard siren method.

Chapters three and four propose a set of algorithms to investigate the impact of microlensing (mainly including stars and remnants) in strong lensing galaxies on strong lensing gravitational waves. Due to the long wavelength nature of gravitational waves, microlensing effect can leave an interference imprint on strong lensing gravitational wave waveform. This effect cannot be ignored in the data analysis of strong lensing gravitational wave events. However, the number of microlenses in strong lensing galaxies is enormous, making the interference effect extremely difficult to calculate. To this end, we propose a complete diffraction integral numerical algorithms for the first time. The algorithm in the third chapter solves the convergence of the algorithm and the problem of Fourier transform spectrum leakage, while the forth chapter addresses the issues of computational resolution, accuracy and speed.

Chapter five focuses on the identification of strong lensing gravitational wave events. The current methods for strong lensing gravitational wave identification suffer from high false positive rates and long computational times. To address these two issues, this work proposes for the first time that strong lensing gravitational wave events can be identified by searching for the unique microlensing interference imprints in strong lensing gravitational waves. This method can be effectively applied to future third-generation gravitational wave detectors, providing a new method for the identification of strong lensing gravitational wave signals and its host galaxy.

Chapter six, based on the diffraction integral numerical algorithms proposed in chapters three and four, systematically evaluates the systematic errors introduced by microlensing effects on the parameter estimation and the identification of strong lensing gravitational waves. It provides theoretical guidance for the subsequent data analysis of strong lensing gravitational waves and inspires the development of de-biasing methods.

Chapter seven is the summary and discussion of the entire thesis.