强引力透镜现象作为爱因斯坦广义相对论的预言之一，在宇宙学研究中扮演着重要的角色。该效应可以显著放大背景天体，而且具有干净的物理背景，因此在宇宙学参数限制、星系的形成和演化、以及星系和星系团的结构等方面具有广泛的应用前景。未来第四代巡天项目如LSST和CSST等有望探测到几十万个星系尺度强引力透镜，这将为宇宙学参数测量、强引力透镜统计等研究带来革命性机会。本论文主要关注星系尺度强引力透镜的宇宙学应用。红移是强引力透镜在天体物理和宇宙学应用中必需的物理量，而由于光谱观测的高耗时性，预计只有少数透镜系统会有光谱红移，大部分透镜系统仍需通过多波段测光数据得出透镜星系和源的测光红移。因此本文的研究内容主要包括两个部分：(1)测量星系尺度强引力透镜系统的测光红移；(2)强透镜红移概率分布统计学的研究。
       针对测光红移的测量，目前国际上只有测量单个星系测光红移的程序，尚无测量强引力透镜系统测光红移的程序。因此，本文将在解决从图像中将透镜星系和源进行分光的难题的基础上，测试出一套能够有效且快速地测量星系尺度强引力透镜系统测光红移的方案。首先我们提出了两套分光策略，即快速光建模和快速质量建模，随后基于模板匹配法测量了100个星系尺度强引力透镜模拟系统的背景源的测光红移。结果显示，在快速光建模方案下，样本的离群数比快速质量建模方案多了近一倍，但后者花费的时间是前者的5倍。通过分析两种方案的优缺点，我们提出了将快速光建模和快速质量建模联合起来进行分类建模的方案，该方案有望提高强引力透镜测光红移的准确性，并且在大规模样本的处理上具有更高的效率。
       基于不同类型的强引力透镜数据，例如透镜化像的角分离分布、透镜红移分布等，强引力透镜统计学方法已经广泛应用到宇宙学研究当中。鉴于透镜化像的角分离分布存在一定的选择效应，本文第二个研究选择应用透镜红移概率分布作为统计量来进行宇宙学分析。我们利用了更新的星系尺度强引力透镜样本，并考虑了三种经典的宇宙学模型（ΛCDM、ωCDM和ω0ωaCDM模型）以及透镜星系速度弥散分布函数（VDF）的三种经典的演化形式（非演化形式、幂律形式演化和指数形式演化）来探讨透镜星系演化模型对宇宙学参数的影响。结果表明，宇宙学参数和VDF参数之间的简并会导致参数估计的偏移。强引力透镜红移分布统计学对Ωm0的限制结果与用Pantheon+ Ia型超新星（SN Ia）样本限制的结果在68.3%的置信水平上是一致的，然而前者Ωm0的不确定性比后者大3到8倍。与非演化VDF模型的结果相比，Ωm0的中心值在幂律形式VDF模型中偏大，在指数形式VDF模型中偏小。在ωCDM模型中，暗能量状态方程参数ω0的限制结果与Pantheon+样本的限制结果在1σ范围内是一致的，但前者约束得到的ω0的中心值明显小于后者。在ω0ωaCDM模型中，与ωCDM模型相比，ω0的不确定性显著增大；此外，ωa的马尔可夫链在三种VDF演化模型下都不收敛。总的来说，强引力透镜红移概率分布统计学对Ωm0的约束比对暗能量状态方程参数的约束更有效。

Strong gravitational lensing, one of the consequences of Einstein’s General Theory of Relativity, plays an important role in the study of cosmology. This effect not only significantly magnifies background celestial bodies, but also has a clean physical background, making it widely applicable in cosmological parameter constraints, galaxy formation and evolution, as well as the structure of galaxies and galaxy clusters. Future fourth-generation surveys such as LSST and CSST are expected to discover hundreds of thousands of galaxy-scale strong gravitational lensing systems, which will bring revolutionary opportunities for cosmological parameter measurement or strong lensing statistics. This paper mainly focuses on the cosmological applications of galaxy-scale strong gravitational lensing, where redshift is a necessary physical quantity in astrophysical and cosmological applications. However, given the high time-consuming nature of spectroscopic observations of lens samples, it is expected that only a few lens systems will have spectroscopic redshifts, and most systems will require photometric redshifts of lens and source galaxies based on multi-band photometric data. Therefore, the research content of this paper mainly includes two parts: (1) photometric redshifts of galaxy-scale strong gravitational lening; (2) the statistics of strong lening redshift probability distribution.
Currently, there are only some programs for measuring photometric redshifts of individual galaxies, while there is no program for measuring photometric redshifts of strong gravitational lening systems. Therefore, the first research of this paper is to propose an effective and fast method for measuring photometric redshifts of galaxy-scale strong gravitational lening systems based on solving the problem of separating the light of lens galaxies and sources from images. Firstly, we propose two sets of deblending schemes: fast light modeling and fast mass modeling, and then measure the photometric redshifts of background sources of 100 galaxy-scale strong gravitational lening simulation systems based on the template matching method. The results show that under the fast light modeling scheme, the number of outliers in the sample is nearly twice as many as that under the fast mass modeling scheme, but the latter takes five times longer. By analyzing the advantages and disadvantages of this two schemes, we propose a scheme that combines fast light modeling and fast mass modeling for classification modeling, which is expected to improve the accuracy of photometric redshifts and has better efficiency in processing large-scale samples.
Statistical methods based on different types of strong gravitational lensing data such as image angular separation distribution and lens redshift distribution, have been widely applied in cosmological research. Given the observation bias in the image angular separation distribution, the second research of this paper will use the probability distribution of lens redshifts as a statistical quantity for cosmological analysis. We conduct the cosmological analysis by using the lens-redshift distribution test with the updated galaxy-scale strong lensing sample, where the considered scenarios involve three typical cosmological models (i.e., ΛCDM, ωCDM and ω0ωaCDM models) and three typical choices (i.e., non-evolving, power-law and exponential forms) for the velocity-dispersion distribution function (VDF) of lens galaxies. It shows that degeneracies between cosmological and VDF parameters lead to the shifts of estimates on the parameters. The limits on Ωm0 from the lens-redshift distribution are consistent with those from the Pantheon+ Type Ia supernova (SN Ia) sample at 68.3% confidence level, though the uncertainties on Ωm0 from the former are about 3 to 8 times larger than those from the latter. The mean values of Ωm0 shift to the larger values in the power-law VDF case and to the lower values in the exponential VDF case, compared with those obtained in the non-evolving VDF case. In the ωCDM model, the limits on ω0, i.e. the dark energy equation of state (EoS), are consistent with those from the Pantheon+ sample at 68.3% confidence level, but the mean values of ω0 from the former are significantly smaller than those from the latter. In the ω0ωaCDM model, the uncertainties on ω0 are dramatically enlarged compared with those obtained in the ωCDM model; moreover, the Markov chains of ωa, i.e. the time-varying slope of EoS, do not achieve convergence in the three VDF cases. Overall, the lens-redshift distribution test is more effective on constraining Ωm0 than on the dark energy EoS.