暗能量是當今天文學上未解的謎團之一。藉由宇宙微波背景輻射（CMB），完全薩克斯-瓦福（ISW）效應是一個研究暗能量的方法：當來自CMB的光子經過重力位能井時，會在掉入位能井時先藍移，之後在離開位能井時紅移；當宇宙的主要組成是物質時，重力位能井並不會隨時間改變，所以光子所經歷的藍移和紅移大小相同、相互抵消，光子並不會有任何改變；然而，當宇宙的主要組成是暗能量時，重力位能井會因為暗能量造成的宇宙加速膨脹、而隨著時間衰減，於是光子離開位能井時受到的紅移減少，不再能抵消其所受到的藍移，因此從CMB而來的光子便因為殘留在其上的藍移、而溫度升高。

我們可從CMB溫度微擾的分佈與大尺度結構間的互相關探測ISW效應。在ISW相關文獻中，大部分的研究量測威爾金森微波微擾探測器（WMAP）的CMB溫度微擾分佈與來自各種不同觀測的星系分佈之間的互相關，一些比較顯著研究探測到ISW效應達2-3σ。另一方面，在量測互相關時所使用的星系數據中，廣域紅外線全天探測衛星（WISE）是一個極為合適的資料來源：全天域，並且有足夠深的極限能見度。

普朗克衛星（Planck）為WMAP的後繼者，具有更好的靈敏度和更高的解析力，儘管如此，卻少有研究同時使用Planck與WISE作ISW效應的研究。在為數不多同時利用Planck與WISE的ISW研究中，關於星系樣本的選取、前景干擾的屏蔽、統計方法的應用對ISW訊號的影響則只有簡略的探討；然而，不同的星系樣本的選取、前景干擾的屏蔽或是統計方法的應用，都是造成各種ISW文獻中訊號大小不一致的原因。

因此，在這篇論文中，我們使用WISE的星系資料和Planck的CMB溫度微擾分佈、詳細的研究了其中的ISW效應：我們發展了一個定量的方法選取最合適的星系樣本，並且以這個樣本量到其與Planck CMB溫度微擾分佈互相關的訊號後，進一步探討不同的Planck CMB微擾分佈、不同的屏蔽前景方法和不同的互相關頻譜組合方式對ISW訊號所造成的影響。在各種Planck CMB溫度微擾分佈中，我們量測到1.2-2.5σ的ISW效應，而且，我們所量測到的ISW效應的大小與ΛCDM模型（標準宇宙模型）所預測的大小相符。這篇論文以不同於其他暗能量研究的獨立方法，證實了暗能量所造成的宇宙加速膨脹。

Dark energy is one of the biggest puzzles in modern astronomy, and its nature is still unknown. The Integrated Sachs-Wolfe (ISW) effect provides us with a direct approach to dark energy through the temperature anisotropy in the Cosmic Microwave Background (CMB). When passing through the gravitational wells, the CMB photons first become blueshifted while falling in the wells and then become redshifted while climbing out the wells. In a matter-dominated universe, the blueshifts and redshifts compensate each other, and there is no net change on the CMB photons. However, in a dark-energy-dominated universe, the gravitational wells would decay with time due to the accelerated expansion of the Universe caused by dark energy, and hence the redshifts could not cancel the blueshifts anymore, which makes the CMB photons eventually become hotter due to the remained blueshifts.

The ISW effect can be detected through the cross-correlation between the temperature anisotropies of the CMB and the large-scale structures, and in the literature, most significant detections are at $2-3\sigma$ via the cross-correlation of the CMB temperature maps from the Wilkinson Microwave Anisotropy Probe (WMAP) or Planck with various survey of galaxies. The Wide-field Infrared Survey Explorer (WISE) is an ideal data-set of galaxies for the ISW studies, because WISE has a survey area of the whole sky and is deeper than other all-sky surveys. In spite of WISE's great advantage and Planck's improvement over WMAP, few works use WISE with Planck to study the ISW effect. Most previous works, including the one using both WISE and Planck, perform simple analysis on the selection of the galaxy samples, the treatment of foreground contamination, and the statistical inference, while the detection level of the ISW effect varies among papers because of those aspects.

Therefore, using the galaxies from WISE and the CMB temperature maps from Planck, we present a comprehensive analysis of the ISW effect with a careful examination on the galaxy selection, the foreground treatment, and the statistical inference. We optimise the galaxy sample by developing a quantitative method to evaluate the uniformness of a galaxy sample. With the selected WISE galaxy sample, which is more uniform than those in the previous studies, we obtain a positive ISW signal. We analyse the cross-correlation of WISE galaxies with different foreground-cleaned CMB maps in Planck. We further investigate the influence on the ISW signal from different treatments of foreground contamination. We also check the influence on the detection level of the ISW effect from various utilisation of the cross-power spectra used in previous studies. Among the different foreground-cleaned CMB maps in Planck, we obtain $1.2-2.5\sigma$ detections of the ISW effect with amplitudes consistent with the $\Lambda$CDM prediction, providing us with an alternative evidence for the accelerated expansion driven by dark energy.

Acknowledgements ............................... iii Abstract..................................... v TableofContents ................................viii ListofIllustrations ............................... ix ListofTables .................................. xi ChapterI:Introduction ............................. 1 ChapterII:Theory ............................... 4
2.1 TheISWEffect............................. 4
2.2 Cross-correlation............................ 4 ChapterIII:Data ................................ 9 3.1CMB.................................. 9
3.2 WISEGalaxies............................. 10
3.2.1 SelectionandMasking .................... 10
3.2.2 RedshiftDistribution ..................... 15 ChapterIV:AnalysisandResults........................ 16
4.1 GalaxyBias .............................. 16
4.2 WISE-PlanckCross-powerSpectra .................. 18 ChapterV:SignificanceTestandSystematicEffects . . . . . . . . . . . . . . 21
5.1 Statistics ................................ 21
5.2 BinnedPowers............................. 22
5.3 UnbinnedPowers............................ 26 ChapterVI:Discussion ............................. 30
6.1 U73v.s.KQ75y9 ........................... 30
6.2 Uncertainty in the Redshift Distribution and the Galaxy Bias . . . . . 31
6.3 Binnedv.s.Unbinned ......................... 32
6.3.1 Comparison with Previous Work Using WISE through Binned PowerSpectra......................... 33 ChapterVII:Conclusions............................ 35 AppendixA:Computationtools ........................ 36 A.1HEALPix................................ 36
A.2SpICE ................................. 36 A.3CosmoPy................................ 36 A.4CAMB................................. 36
Bibliography .................................. 38

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