馬驄

(中國科學(xué)院紫金山天文臺(tái) 南京 210034)

本文中,我們系統(tǒng)地表述一種自洽的宇宙學(xué)模型分析及其參數(shù)估計(jì)方法,并將其用于Ia型超新星(SNIa)光度距離作為宇宙學(xué)標(biāo)準(zhǔn)燭光的研究.我們利用貝葉斯圖論方法及其與隨機(jī)變量間因果性、信息流和概率獨(dú)立性的對應(yīng)關(guān)系構(gòu)建我們的分析,并通過實(shí)例展示如何在宇宙學(xué)數(shù)據(jù)分析中恰當(dāng)?shù)剡\(yùn)用聯(lián)合光變曲線分析(JLA)SNIa數(shù)據(jù)集合中依賴于參數(shù)的協(xié)方差矩陣.我們的方法從概念到產(chǎn)出的結(jié)果均不同于傳統(tǒng)的分析方法.可以證明,通常形如χ2最小化的傳統(tǒng)分析等效于施加在SNIa標(biāo)準(zhǔn)化模型變量先驗(yàn)分布上的隱含失真.通過顯性地指明這一失真的形式,我們指出它無論從天體物理還是統(tǒng)計(jì)方法角度理解均缺乏合理性.與之形成對比的是,我們的貝葉斯統(tǒng)計(jì)模型明確地包含各個(gè)參數(shù)的先驗(yàn)分布.與先前研究對照,關(guān)于SNIa光度基于光變曲線時(shí)間延展和極大時(shí)期色指數(shù)的標(biāo)準(zhǔn)化改正,本文的新結(jié)果顯著支持更小的改正量.關(guān)于受到宿主星系恒星質(zhì)量控制的SNIa光度改正,我們得出了更低的統(tǒng)計(jì)顯著性.對wCDM (cold dark matter)和ΛCDM宇宙學(xué)模型,我們得出了修正的后驗(yàn)概率估計(jì),并分別得出了對w?1和更高?M估計(jì)值的支持.

這一統(tǒng)計(jì)分析方法問題在SNIa數(shù)據(jù)集合壓縮的工作中再次得到體現(xiàn).為此,我們將數(shù)據(jù)壓縮表述為統(tǒng)計(jì)推斷問題,并得出其最優(yōu)化、最大熵解及其高效而準(zhǔn)確的高斯近似.通過多種統(tǒng)計(jì)和信息學(xué)檢驗(yàn),這一近似的準(zhǔn)確性和穩(wěn)健性得到了確認(rèn).與JLA原先壓縮對比,我們得出的距離模數(shù)數(shù)值及其協(xié)方差矩陣的結(jié)構(gòu)均呈現(xiàn)出差異.使用這些新的壓縮數(shù)據(jù),我們進(jìn)行SNIa距離階梯的內(nèi)部交叉核實(shí)檢驗(yàn),并對表觀上的色改正系數(shù)演化趨勢給出了新的結(jié)果.

在此基礎(chǔ)上,我們論證了SNIa壓縮數(shù)據(jù)用于檢驗(yàn)距離對偶關(guān)系(DDR)的優(yōu)越性,即子樣本更良好的代表性和統(tǒng)計(jì)不確定度更準(zhǔn)確的建模.與輔助性的觀測信息結(jié)合后,壓縮的SNIa光度距離可與重子聲振蕩(BAO)巡天所得的角直徑距離數(shù)據(jù)對應(yīng)并直接比較.我們使用蒙特卡羅分析得出DDR判別量η的分布性質(zhì).這一計(jì)算結(jié)果與DDR在統(tǒng)計(jì)不確定度限度內(nèi)一致,以標(biāo)準(zhǔn)差表示的不確定度約為8%.假設(shè)DDR表觀背離的機(jī)制來源于對波長選擇性低的星系際塵埃吸收造成SNIa亮度系統(tǒng)性偏低,我們通過結(jié)合Planck宇宙學(xué)后驗(yàn)分布得到更緊密的限制,并將其表達(dá)為星系際消光和光深的紅移較差.其計(jì)算結(jié)果在改進(jìn)的不確定度限度內(nèi)與無消光一致.我們根據(jù)未來巡天觀測的性能指標(biāo)估算得出η可達(dá)到的統(tǒng)計(jì)不確定度約2%.這一結(jié)果顯示了系統(tǒng)不確定度對未來相關(guān)研究的影響以及針對這些影響采取自洽的統(tǒng)計(jì)解決方法的重要性.

總結(jié)本文的主要發(fā)現(xiàn),我們指出宇宙學(xué)模型統(tǒng)計(jì)分析中貝葉斯方法的優(yōu)勢.關(guān)于未來可期待的觀測宇宙學(xué)數(shù)據(jù),我們強(qiáng)調(diào)更深入研究的目標(biāo)在于提出新的統(tǒng)計(jì)方法,以更完好地保持這些結(jié)構(gòu)更復(fù)雜、樣本量更大的數(shù)據(jù)集合中蘊(yùn)含的信息.在本文工作的基礎(chǔ)上,我們試探性地提出了未來相關(guān)研究計(jì)劃.

In this dissertation,we set out to formalize a self-consistent method of testing cosmological models and to obtain the parameter estimations with the luminosity distance and redshift data from the Type Ia supernova (SNIa)standard candles.Using the Bayesian graphical models and their correspondence with the causality,information-flow,and independence properties among the random variables,we construct a method for properly utilizing the parameter-dependent covariance matrix in cosmological model analyses and demonstrate its application to the Joint Light-curve Analysis (JLA)SNIa distance data set.Our statistical method differs,in both concept and effect,from those used in conventional analyses.It is found that the conventional analysis in the form ofχ2minimization equivalently incurrs an implicit distortion on the prior probability of SNIa standardization parameters.The distortion,once revealed explicitly,cannot find convincing justification from statistical or astrophysical considerations.In contrast,our Bayesian method explicitly incorporates prior probabilities of all the model parameters.Our new analysis favors a lower amount of stretchand color-based corrections to the SNIa magnitude compared with previous studies,and we report a reduced statistical significance of the step correction based on the host galaxy’s stellar mass.ForwCDM (cold dark matter)and ΛCDM cosmologies,we obtain the revised posterior estimates,showing an increased favor forw?1 and higher estimates of ?M,respectively.

The same statistical issue is revisited in the task of compressing the SNIa data set and controlling the loss of cosmological information.To this end,we formalize the data compression as a statistical inference problem,and achieve an efficient and accurate Gaussian approximation to the optimal,maximal-entropy solution.Multiple tests based on statistical and information theories are applied to our proposed solution,and they validate its accuracy and robustness.Discrepancies from the JLA original compression exist in the distance modulus estimates and in the structures of their covariance matrix.We use our compressed data set to perform an internal cross-validation of the SNIa distance ladder,and find a nominal color evolution trend that is different from previous results.

Furthermore,we establish the strengths of compressed SNIa data in the distance-duality relation(DDR)tests in the form of more representative data subsamples and more accurate modelling of statistical uncertainties.In combination with auxiliary observational information,the compressed SNIa luminosity distance moduli provide a subsample of measurements matched to and comparable with the angular-diameter distance data from the baryon acoustic oscillation (BAO)surveys.We perform Monte Carlo analyses to evaluate the distributional properties of the DDR discriminant parameterη.The results are consistent with the DDR,and uncertainty bounds are estimated to be about 8% (standard deviation).Under the additional hypothesis of a grey dust component that may systematically diminish SNIa brightness and induce an apparent deviation from DDR,we improve the constraints by incorporatingPlanckcosmological posteriors,and obtain improved bounds on the intergalactic extinction and optical-depth differential.The results are consistent with zero intergalactic extinction with improved bounds.The statistical uncertainty ofηbased on future survey specifications is estimated to be ～2%,highlighting the importance of systematics and the role of consistent statistical methods in response to them.

In the light of our key discoveries,we advocate the methods of statistical analysis of cosmological models based on the Bayesian formalisms,and call for the further study into their efficacy in preserving the information content of more complex cosmological observational data sets expected in the future.Some research topics are tentatively proposed in relation to the present study.