球狀星團約在兩百年前被發現,是目前宇宙中歷史最悠久的天體之一。但這些
球狀星團是如何形成的,以及怎麼樣的環境條件能夠形成這些球狀星團仍然是
一個謎。在過去的十年裡,人們發現了星系中球狀星團的總數及質量與其所在
的暗物質暈的質量有著明確的相關性。此外,使用宇宙流體動力學模型對球狀
星團的研究中發現,大部分的球狀星團可能是在1 < z < 4時期於富含氣體星系
裡的星際介質紊流中形成。其他的球狀星團可能是在再電離時期形成於低質量
暗物質暈的核心區域。因此,探討原始球狀星團是否存在,以及他們是如何形
成是一個很重要的研究主題,透過這個研究,我們可以進一步了解星系在早期
是如何形成以及宇宙學。
本論文主要分成下列兩個部分進行探討。第一部分,我們討論建構在高紅
移區域的球狀星團模型是否跟現在觀測裡有關球狀星團以及暗物質暈的結果相
符合。我們利用其質量和紅移的資訊去了解球狀星團落在哪一個暗物質暈,進
一步延展Press-Schechter merger trees 的應用。我們的結果顯示,使用合理的參
數,這個模型預測的低質量暗物質暈(≤ 10^10 M⊙),其暗物質暈內的球狀星團數
量與觀測的預期相符合。但是這個模型的高質量暗物質暈沒有辦法呈現與觀測
相符的結果。這個結果顯示,∼ 50%的原始球狀星團會被潮汐力瓦解。另外一
個可能的解釋是,在高質量星系裡,有其他的機制讓球狀星團在近期形成。球
狀星團的數量與暗物質暈的關係,包含結果的分佈的程度,都有可能進一步提
供更多的研究資訊。近期的測量大多來自整合不同的觀測資料,不同的觀測
資料可能造成結果的不確定度增加。因此,在第二部分的論文內容裡,我們
使用DESI Legacy Survey 去進行大尺度的球狀星團統計,我們限縮我們的研究
範圍,要求球狀星團所在星系的質量落在8 < log10 L/L⊙ < 11.5 之間,且距離
在≲ 30 Mpc 以內。我們用與之前其他研究應用在單一星系裡搜尋球狀星團方法
到DESI-LS。因為DESI-LS擁有更好的靈敏度和更廣的觀測資料,我們順利得出
平均球狀星團的數量以及其分佈跟之前針對單一星系的研究相符合。當我們確
認這個方法的可行性,我們可以進一步了解球狀星團數量與其暗物質暈質量的
關係。我們發現低質量的暗物質暈有比較多的球狀星團,這個結果跟過往的研
究相符。我們也進一步討論可以在進的地方和這個方法可以應用的範圍。

Globular clusters (GC) are among the oldest known astronomical objects in the universe. They were discovered about two hundred years ago, but the initial conditions that led to their formation are still a mystery. In the last decade, a clear correlation has been found between the total number (or mass) of GCs in a galaxy and the mass of its host dark matter halo, which provides an important constraint
on models of their formation. Meanwhile, detailed hydrodynamical models of GC populations in a cosmological context have shown that a substantial fraction may have formed in the turbulent ISM of massive, gas-rich galaxies at 1 < z < 4. However, a subdominant population of primordial GCs may also have formed as nuclear clusters in low-mass dark matter halos before the epoch of reionization. The question of whether primordial GCs exist, and if so, how they formed, is important, because their cosmic abundance may be a useful probe of the earliest epoch of galaxy formation, and hence of cosmology.
This thesis tackles two related problems. In the first part, we ask whether the simplest models of primordial cluster formation at high redshift are consistent with the relationship between GC number and present-day dark matter halo mass from current observations. We apply recipes for assigning GCs to halos, based on thresholds in mass and redshift, to extended Press-Schechter merger trees. We find such models can reproduce the abundance of GCs in present-day dark
matter halos with mass ≤ 10^10 M⊙ using plausible parameters. However, they do not reproduce the observed trend of GC abundance at higher halo masses. This suggests that tidal disruption must remove ∼ 50% of primordial GCs, and/or that another, lower-redshift formation process may create GCs, particularly in massive galaxies. The shape of the cluster number – halo mass relation, and its intrinsic scatter, are potentially important constraints on those possibilities. However, current measurements are based on many different datasets, possibly
with different sampling biases, and the observed scatter appears large. Therefore, in the second part of the thesis, we use the DESI Legacy Survey to make the first large-scale uniform statistical estimates of GC abundance around host galaxies with a luminosity range of 8 < log10 L/L⊙ < 11.5 within ≲ 30 Mpc. We use similar photometric techniques to previous ground-based single-galaxy GC surveys, which
we can now extend to many more galaxies thanks to the unique combination of depth and coverage in DESI-LS. We recover average GC counts and radial profiles that are consistent with earlier work on individual galaxies. As well as validating the technique, our results provide further support for the observed relation (and scatter) between cluster number and halo mass. At lower masses, we find higher
cluster numbers than expected. We discuss possible improvements and extensions to this technique.

1 What make us study globular cluster (GC)....1
1.1 The role of GCs in astronomy . . . . . . . . . . . . . . . . . .2
1.2 Difficulties in the study of GC formation .. . . .. . . . . . . .3
1.3 The role of this thesis . . . . . . . . . . . . . . . . . . . .4
2 What is a GC...8
2.1 Globular cluster: a city of stars . . . . . . . . . . . . . . . 9
2.2 Globular cluster luminosity function . . . . . . . . . . . . . 11
2.3 Globular cluster classification . . . . . . . . . . . . . . . . 13
2.3.1 Classification based on observational properties . . . . . . 13
2.3.2 Classification based on formation theories . . . . . . .. . . 14
2.4 Primordial globular cluster formation: an analog to galaxy formation ...15
3 Observations and simulations on GC ..... 18
3.1 Observational catalogs . . . . . .. . . . . . . . . . . . . . . 19
3.1.1 Harris’ GC system catalog . . . . .. . . . . . . . . . . . . 19
3.1.2 Catalogs of GC systems in low-mass galaxies . . . . . . . . 20
3.2 The EMOSAICS simulations . . . . . . . . . . . . . . . . . . 22
3.3 Modeling GC – halo mass relation . . . . . . . . . . . . . . . 23
3.3.1 The number of GCs – galaxy z-band magnitude relation . . 24
3.3.2 Yang et al. (2021) group catalog . . . .. . . . . . . . . . 28
3.3.3 Apply NGC − Mz relation to galaxy catalog . . . . . . . . . 28
4 Semi-analytical study on GC formation ... 30
4.1 Semi-analytical works in literature . . . . . . . . . . . . . . 31
4.2 The extended Press-Schechter method ... 31
4.2.1 Our semi-analytical method . . . . . . . . . . . . . . . . . 33
4.2.2 Comparing our semi-analytical method to observations . . . 34
5 GC surface density profile ...37
5.1 Primary galaxies selection . . .. . . . . . . . . . . . . . . 39
5.1.1 Siena Galaxy Atlas (SGA) . . . .. . . . . . . . . . . . . . 39
5.1.2 Extragalactic Distance Database (EDD) . . . . . . . . . . . 39
5.1.3 Criteria for primary galaxies selection . . . . . . . . . . 40
5.2 Identifying globular cluster candidates . . . . . . . . . . . 43
5.2.1 The DESI Legacy Surveys . . . . . . . . . . . . . . . . . . . 43
5.2.2 Gaia catalog . . . . . . . . . . . . . . . . . . . . . . . 44
5.2.3 Criteria to distinguish GC candidates from contamination . 44
5.3 Globular cluster projected density profile . . . . . . . . . . 48
5.3.1 Correction for the background subtraction . . . . . . . . . . 49
5.3.2 Correction for the density profile from GCLF . . . . . . . . 49
5.4 Elliptical versus spiral primaries: any difference in their GC profiles?... 53
5.5 Number of globular clusters as a function of host halo mass . . 57
5.5.1 Stellar mass – halo mass relation . . . . . . . . . . . . . 58
5.5.2 Number of GCs – host DM halo mass relation . . . . . . . . 58
6 Discussion & conclusion ... 62
Reference ... 69
Appendix .... 76
A Appendix .... 76
A.1 Semi-analytical method . . . . . . . . . . . . . . . . . . . 76
A.1.1 Compare to blue GCs . . . . . . . . . . . . . 76
A.1.2 Different DM models on GC formation . . . . . . . . . . . . 77
A.2 Color selection of GC candidates . . . . . . . . . . . . . . 79
A.3 Error estimation. . . . . . . . . . . . . . . . . . . 80

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