Probing the Epoch of Reionization with Lyman-alpha emitters

U NIVERSITY OF G RONINGEN

B

T

Probing the Epoch of Reionization with Lyman-alpha emitters

Author:

Thijs L

Supervisors:

Dr. Anne H

Dr. Pratika D

2

Examiner:

Prof. dr. Leon K

Abstract

In this thesis, a physical model for high-redshift Lyman-alpha emitters (LAEs) at the end of the Epoch of Reionization is analyzed. The aim of the research is to find out how the clustering of LAEs, divided

in luminosity categories faint, intermediate and bright, changes as reionization proceeds. Three snapshots of a simulation box (of size 80 h⁻¹Mpc in all three spatial dimensions) at a fixed redshift of

z= 6.6 are taken at different stages of reionization ((a) 〈χHI〉 = 0.50, (b) 〈χHI〉 = 0.25 and (c)

〈χHI〉 = 0.10). A sample of Lyman Break Galaxies (LBGs) is also analyzed for comparison between LAEs and LBGs. Bright galaxies show more clustering than intermediate ones, which show more clustering than faint galaxies. The clustering of LAEs is found to be highest at〈χHI〉 = 0.50 and lowest

at〈χHI〉 = 0.10. This means that more LAEs are detected as reionization proceeds. The clustering model is compared to observational results byKashikawa et al.(2006) andOuchi et al.(2010). The

clustering of LAEs in the model is found to be higher than the results by aforementioned authors.

Reasons for the difference between the results include the fact that only one line-of-sight to the galaxy simulation box was considered, a fixed escape fraction f_esc= 0.50 for all galaxies in the model, a limit

of the ionized fraction of〈χHI〉 = 0.10, coincidental homogeneous sky observations, or the lack of confirmed LAEs in observations.

Kapteyn Astronomical Institute

Tuesday 27

November, 2018

Contents

1 Acknowledgements 2

2 Introduction 3

2.1 Formation of hydrogen to formation of stars . . . 3

2.2 The Epoch of Reionization. . . 3

2.3 Discovery of reionozation . . . 4

2.4 Detection of reionization . . . 4

3 Theoretical Background 6 3.1 Halo mass function and luminosity function. . . 6

3.2 Lyman-alpha radiation . . . 6

3.2.1 Lyman series . . . 6

3.2.2 Lyman-alpha emitters . . . 7

3.3 Probing reionization . . . 8

3.3.1 Using quasars to probe reionization . . . 8

3.3.2 Using LAEs to probe reionization . . . 8

3.4 Lyman Break Galaxies . . . 10

4 Methodology 11 4.1 Data . . . 11

4.2 Correlation function . . . 11

4.2.1 Spatial two-point correlation function . . . 11

4.2.2 Angular correlation function. . . 12

4.2.3 Estimators . . . 13

4.2.4 Computation of correlation function . . . 13

5 Results and Discussion 16 5.1 Galaxy properties . . . 16

5.2 Correlation function . . . 17

5.3 Comparison with observations . . . 19

5.3.1 Comparison with Kashikawa et al. (2006) . . . 20

5.3.2 Comparison with Ouchi et al. (2010) . . . 21

6 Conclusions 25 6.1 Future research . . . 26

Chapter 1

Acknowledgements

First off I would like to thank dr. Anne Hutter for supervising me in this bachelor project. She could consistently provide me with excellent help and always took the time to make sure I understood the problems that she would explain to me. I would also like to thank dr. Pratika Dayal for supplying the thesis and providing the necessary help. Attending the meetings of her master and PhD research group, during which I was allowed to present a few papers, improved my skills in reading and understanding scientific articles. I would like to thank the master students and PhDs in the research group for their support and helpful insights: Laurent Legrand, Olmo Piana, Jonas Bremer, Nikki Arendse and Ruslan Brilenkov. At last I want to express my gratitude towards many students at the Kapteyn Institute that were of great help to me during the time of working on this bachelor thesis: Roi Kugul, Danny Sardjan, Francis Tang, Nick Oberg, Jesper Tjoa, Casper Farret Jentink and Bas Roelenga.

Chapter 2

Introduction

After the Big Bang, the Universe underwent two major hydrogen gas phase transitions. The first phase transition was recombination. It put an end to the hot ‘plasma soup’ 380.000 years after the Big Bang, at a redshift of z≈ 1089. For the first time ever, protons and electrons became bound to form neutral hydrogen atoms. The extensive period that followed before the first stars would form was dubbed The Dark Ages. It was long understood that there must have been a period in the timeline of the Universe between recombination and ‘today’ where the neutral hydrogen in the Universe got reionized. However, only in the year 2001, observations provided us with first hints towards reionization (the second phase transition). A quasar was discovered at z= 5.73 that showed a feature in its spectrum, exactly matching expectations of a quasar in the end of the so called Epoch of Reionization. More details about this feature can be read in2.4. Further spectroscopic studies (e.g.Becker et al.(2001),Kashikawa et al.(2006) and Ouchi et al.(2010)) improved our knowledge on the Epoch of Reionization. Today, research suggests that this period took place in the redshift range 6< z < 20, approximately.

2.1 Formation of hydrogen to formation of stars

Before the Dark Ages, the matter in the Universe was a dense, hot plasma. Atoms did not exist in this high energy state of the Universe. Photons ‘trapped’ in the plasma only traveled a very short distance, due to the very short mean free path before encountering an electron and interacting with it (exchanging energy). This particle-particle interaction is called ‘scattering’. However, due to the expansion of the Universe after the Big Bang, the mean free path of photons extended. Eventually, the matter had been diffused to such a degree that it became energetically favorable to form hydrogen;

the size of the Universe was large enough for the recombination rate of protons and electrons to be higher than the hydrogen-photon scattering rate. The expansion also meant cooling of the matter and radiation in the Universe. At some point, photons were able to escape the gas in which they were trapped. Photons that would not get absorbed by other neutral hydrogen atoms could freely travel through the Universe. These photons can still be detected today as the Cosmic Microwave Background (CMB). The CMB radiation is detected as thermal emission from all parts of the sky. As there were no stars yet, CMB photons, together with photons that would get emitted by a hydrogen spin transition were the only sources of light in the Universe at this point in time. This timeframe is therefore called The Dark Ages. It would take another few million years before the hydrogen regions in the Universe became dense enough to collapse and form stars.

2.2 The Epoch of Reionization

With the emergence of stars after about 200 million years after the Big Bang, the Dark Ages slowly came to an end. However, these stars were relatively small and faint. At some point, stars became so energetic that they were able to radiate at energies higher than the binding energy of hydrogen.

This UV radiation started to ionize neutral hydrogen gas in the intergalactic medium (IGM), marking the beginning of a new era: the Epoch of Reionization. As a consequence of the ionization, the gas in the IGM was heated. This affected galaxy formation on the faint end of the luminosity function, as the increase of temperature caused less gas to be bound in halos. Affected galaxies hence had less gas available, suppressing star formation. These mechanisms affected the subsequent evolution

2.3. DISCOVERY OF REIONOZATION CHAPTER 2. INTRODUCTION

and formation of galaxies; the Epoch of Reionization indirectly impacted galaxies that we see today.

Reionization, and thereby the epoch named after it, finished at z≈ 6 when all hydrogen was ionized.

2.3 Discovery of reionozation

When one observes the sky at redshifts below z ≈ 6, basically everything is visible unless dust or other particles are blocking the view between an object and Earth. Neutral hydrogen absorbs radiation at wavelengths λ ≤ 1216 Å (the energy difference between the ground state and first excited state of hydrogen). Photons with wavelengths below that point are able to excite (or ionize ifλ ≤ 912 Å) hydrogen, blocking the line of sight. Because spectra of nearby objects only show very sharp absorption lines at hydrogen exciting or ionizing wavelengths, astronomers before the 90’s knew that close to all of the hydrogen in the Universe was ionized in the nearby Universe. This meant that, even before the advent of telescopes that were able to look at high redshifts (past z≈ 5), astronomers knew that at some point between the formation of hydrogen and ‘today’, the hydrogen in the Universe got reionized. They did not know when the reionization had taken place. Nonetheless, it was clear that neutral hydrogen should cause a drop of flux (at hydrogen exciting wavelengths) in the spectra of objects. It was only a matter of time before the first spectrum with such a feature was detected.

2.4 Detection of reionization

In the late 1990’s, the field of distant galaxies was evolving very rapidly: the Hubble Space Telescope (HST) made it possible to detect galaxies at redshifts of z≈ 6. In 2000, the Sloan Digital Sky Survey (SDSS) started analyzing the sky with a ground based telescope. This included spectroscopic images of quasars at redshifts above z= 5. This enabled astronomers to analyze these spectra, in hopes of finding hints towards reionization. In 2001, Djorgovski et al.(2001) saw a significant drop in the flux at wavelengths λ > 7550 Å (corresponding to energies of the radiation high enough to excite hydrogen atoms at that redshift), which the researchers claimed was a signature of the trailing edge of the Epoch of Reionization. Not only absorption regions, but also transmission spikes were visible in the spectrum. According the authors, this behavior was caused by ionized hydrogen bubbles along the line of sight to the quasar. The researches also expected this, as the models of the Epoch of Reionization predicted a ‘patchy’ reionization process. This meant that the reionization was not a gradual process from neutral to ionized hydrogen across all regions of space, but more like an irregular process where ionized bubbles first surrounded only bright galaxies. These bubbles would grow and more bubbles would be created with the formation of more galaxies. Figure2.1can aid in the visualization on this effect. Understanding how the reionization proceeded is extremely important for the research into cosmic structure formation and evolution, as the Epoch of Reionization is the bridge between the birth of the first stars and the Universe as we know it today. In this thesis, the reionization process is further examined by analyzing a galaxy clustering model at different stages of reionzation. The underlying mechanics are explained in further detail in Chapter3.

2.4. DETECTION OF REIONIZATION CHAPTER 2. INTRODUCTION

Figure 2.1: Image byDjorgovski et al.(2001) & Caltech, explaining how the analysis of the spectrum of quasar SDSS 10440125 led to the discovery of a patchy reionization process.

Chapter 3

Theoretical Background

In this Chapter, two methods for probing the Epoch of Reionization will be discussed. First, different functions that describe the halo masses and luminosities of a set of galaxies will be reviewed. The mechanics behind Lyman-alpha radiation will be explained. After that, the aforementioned method of using quasars to probe reionization is treated. Then follows a description of how to use Lyman-alpha radiation coming from distant galaxies to examine the Epoch of Reionization.

3.1 Halo mass function and luminosity function

An important component of galaxies for studying cosmic structure formation is the dark matter halo.

The halo encloses the disk of a galaxy (but extends far beyond the visible edge of a galaxy) and accounts for the majority of the galaxy’s mass. A dark matter halo is not made up of ordinary baryonic matter, but theoretical dark matter. The presence of dark matter in these halos has been derived from the rotational curves of spiral galaxies. Since dark matter itself has not been directly observed, it seems to not interact with radiation or the rest of a galaxy’s matter. Despite its undetectability, dark matter and dark matter halos in particular, prove to be useful probes for cosmic structure formation. The halo mass function (HMF) is a measure for the distribution of dark matter halos: it shows the number density of dark matter halos per mass interval. The HMF can be analyzed and linked to another property of galaxies:

luminosity. One of the most fundamental properties of a galaxy is its luminosity (in some waveband).

Hence, an important statistic in galaxy distribution is the luminosity function (LF). It describes the number (density) of galaxies with luminosities in the range L± dL /2 (Mo et al.,2010). In the modern understanding of galaxy formation, all galaxies form within a dark matter halo. The assumption that luminosity scales linearly with halo mass is often made:

L∝ Mh. (3.1)

By linking the HMF and LF and using this relation, more can be understood about how many galaxies eventually end up in halos.

3.2 Lyman-alpha radiation

3.2.1 Lyman series

Observing radiation from objects in the Universe is just a matter of pointing a telescope and collecting photons (not that telescopes are simple machines). However, the path between these objects and Earth is not simply empty space, causing all kinds of effects to come into play. One of these effects is the inter- action between radiation and matter. The region between stars inside a galaxy is called the interstellar medium (ISM). Zooming out of one galaxy, we encounter the region between multiple galaxies: the intergalactic medium (IGM). The most abundant element in the Universe, and thus in these regions, is hydrogen. When photons travel through a medium, they can be scattered by the particles that make up the medium. Consider a photon with an energy of at least the energy difference between two energy states of an atom in the ISM. If this photon hits the atom, the photon is absorbed, making the atom go into a higher energy state (excitation). If the energy of the photon is even higher, e.g. an energy of at least the binding energy of an atom, the photon with this energy can separate an electron from

3.2. LYMAN-ALPHA RADIATION CHAPTER 3. THEORETICAL BACKGROUND

the nucleus of the atom (ionization). From this point onward, ‘excitation’ and ‘ionization’ should be interpreted as the excitation and ionization of the hydrogen atom, respectively. The energy difference between the ground state (n= 1) and the first excited state (n = 2) of the hydrogen atom is:

E(n = 2) − E(n = 1)  =

 − 13.6eV − (−3.4eV)

 = 10.2eV,

which corresponds to a wavelength ofλ ≈ 1216 Å. The energy of light is related to its wavelength, according to the Planck-Einstein relation:

E= hν =hc

λ, (3.2)

where E is the energy of the light, h is Planck’s constant,ν is the frequency of the light and λ is the wavelength of the light. A larger wavelength results in a lower energy. The wavelength corresponding to the binding energy of the hydrogen atom isλbind.= 912 Å. Radiation with wavelengths shorter than λbind.can therefore separate the proton from the electron: the ionization of the hydrogen atom. Hence, exciting radiation has wavelengthλexciteand ionizing radiation has wavelengthλionize:

912 Å<λexcite≤ 1216 Å, λionize≤ 912 Å.

Whenever an atom is excited, it can also de-excite, emitting a photon with the energy difference of the initial state〈i〉 and that of the final state 〈f 〉:

λ = hc E_i− Ef

. (3.3)

The Lyman series is the spectral series of the transitions in a hydrogen atom from the n≥ 2 to the n = 1 state. The transition from n= 2 to n = 1 is called Lyman-alpha, the transition from n = 3 to n = 1 is called Lyman-beta and the sequence continues with the letters of the Greek alphabet. The limit to this series is called the Lyman limit, often denoted by LyC. Radiation with wavelengths shorter than the Lyman limitλLyC= 912 Å does not excite the atom, but ionizes it.

Figure 3.1: Schematic view of electron (blue) transitioning from the n= 2 to the n = 1 state, emitting Lyman-alpha radiation in the process.

3.2.2 Lyman-alpha emitters

Lyman-alpha emitters (LAEs) are high-redshift galaxies with ongoing star-formation. The exciting radi- ation coming from the hot stars in these galaxies is absorbed by neutral hydrogen present in the ISM of these galaxies and is re-emitted as Lyman-alpha radiation. This absorption and re-emission process can occur several times before the radiation escapes into the IGM. The Lyman-alpha radiation is detected as a (broad) emission line in the spectrum of the galaxy. In order to account for all kinds of effects (atmospherical effects, resolution limitations, etc.) that may cause small distortions to the spectrum, a galaxy has to meet certain criteria in order to be classified as an LAE. The equivalent width EW (or W_λ) of the Lyman-alpha line has to be at least 20 Å. The equivalent width is a measure of the area of an absorption or emission line in a plot of flux versus wavelength.Peterson(1997) used the following definition: "The equivalent width of an emission line provides an estimate of how large a continuum range one would need to integrate over to obtain the same energy flux as is in the emission line." The value of EW ≥ 20 Å was used byHutter et al. (2014), who built a physical model for high-redshift

3.3. PROBING REIONIZATION CHAPTER 3. THEORETICAL BACKGROUND

LAEs. The data of LAEs analyzed in this thesis is supplied byHutter et al.(2014). Other researchers might use a different value, depending on the focus of their research. Next to this criterion, LAEs were selected on their observed absolute UV magnitude in the research byHutter et al.(2014), and thus in this thesis. A galaxy has to have an absolute UV magnitude of M_UV< −17. The value of MUV< −17 is in accordance with current observational criteria at redshift z≈ 6 − 8.

3.3 Probing reionization

3.3.1 Using quasars to probe reionization

As mentioned before, the Epoch of Reionization is believed to end around z≈ 6. Studying the Universe at such large distances imposes problems that have to be overcome. Sources have to be extremely bright to be detected on Earth. The Earth’s atmosphere absorbs UV radiation, so light at UV wavelengths cannot be observed from Earth. However, the expansion of the Universe entails an effect that proves useful for the detection of UV radiation. According to general relativity, light gets stretched when traveling large distances through the expanding Universe. UV radiation from far-away objects are redshifted to larger wavelengths that are not absorbed by the atmosphere and can thus be observed.

The Lyman-alpha wavelength isλLyα= 1216 Å: radiation at this wavelength falls in the UV part of the electromagnetic spectrum, but the objects in the Epoch of Reionization are at such large redshifts (z > 6) that this wavelength is stretched to a larger wavelength that is not absorbed by the Earth’s atmosphere. This can be seen from Equation3.4:

λobs= (1 + z)λemit, (3.4)

whereλobsis the observed wavelength andλemitthe emitted wavelength from the source. As mentioned before, very distant quasars can be used to study reionization, because they belong to the brightest ob- jects in the universe and their radiation can be detected from as far back as the Epoch of Reionization.

Back in 1965, Gunn and Peterson already predicted that Lyman-alpha radiation from distant quasars should be absorbed by neutral hydrogen and that a decrease of flux should be visible in their spectra (Gunn and Peterson,1965). At that time, technology was not advanced enough to allow for quantative studies of high-redshift objects. Almost 40 years later, the effect had finally been observed in the spec- tra of quasars observed when analyzing spectra obtained by SDSS (Djorgovski et al.(2001) andBecker et al.(2001)). Quasars can radiate at wavelengthsλ ≤ 1216 Å, meaning at energies high enough to excite or ionize hydrogen. The neutral hydrogen present in the IGM can absorb this radiation along the line of sight to the quasar. When this radiation is absorbed, the hydrogen is excited or ionized. This results in a drop of flux at those wavelengths: the Gunn-Peterson trough. Due to the low density of hy- drogen in the IGM, atoms do not easily recombine, meaning that the hydrogen stays ionized. Quasars only contribute to the reionization very minimally. As is discussed in Section3.3.2, there is another type of source which accounted for a large amount of the ionizing power. Figure3.2displays the spectrum of a quasar in which the Gunn-Peterson trough is clearly visible. Note that the observed Lyman-alpha wavelength at that redshift is larger thanλ_Lyα= 1216 Å, as can be seen from Equation (3.4). Not all radiation between Lyα and LyC is absorbed: the current theory of reionization tells us that the process happened in a patchy way. The quasar is located at z= 6.13, which is almost at the end of the Epoch of Reionization. Most of the patches or bubbles of ionized hydrogen surrounding high-energy sources had already been expanded so vastly that almost the entire Universe was reionized.

3.3.2 Using LAEs to probe reionization

Next to the aforementioned distant quasars, there is another type of object that radiates in UV in the Epoch of Reionization that is a good source for studying the topology of reionization: a Lyman-alpha emitter. At the start of the Reionization Epoch, all hydrogen in the IGM is neutral. Lyman-alpha ra- diation that escapes into the IGM is absorbed by that neutral hydrogen (due to the large scattering cross-section at Lyman wavelengths) and cannot be observed. However, due to the ionizing radiation from stars, more and more hydrogen surrounding galaxies starts to get ionized in the Reionization Epoch. Lyman-alpha radiation from LAEs is not absorbed by the regions of ionized hydrogen, because once hydrogen is ionized, exciting radiation cannot interact with it and passes right through. Thus, Lyman-alpha radiation can freely travel through regions of ionized hydrogen. In Figure3.3, two galax-

3.3. PROBING REIONIZATION CHAPTER 3. THEORETICAL BACKGROUND

Figure 3.2: A high signal-to-noise spectrum of the quasar ULAS J1319+0959 at z = 6.13 fromBecker et al.(2015), obtained with the X-Shooter spectrograph on the Very Large Telescope (VLT). The figure indicates the region where the flux is virtually zero: the Gunn-Peterson trough.

Figure 3.3: Schematic view of LAEs in the Epoch of Reionization, showing that larger HII regions allow the Lyman-alpha radiation to escape the ISM and traverse HI regions in the IGM.

ies are depicted alongside their surrounding ionized hydrogen bubbles. The Lyman-alpha radiation from the upper galaxy (at higher redshift than the lower galaxy) cannot be observed, because it travels through the ionized region (HII), but once it hits the edge of the region, it immediately gets absorbed by the neutral hydrogen (HI). The lower galaxy has a larger HII region surrounding it. Lyman-alpha radiation traveling through this region traverses such a great distance that, once the radiation hits the edge of the HII region, it has already been redshifted to a wavelength that does not correspond to an energy high enough to excite hydrogen. These photons can thus travel through the HI region without being absorbed, meaning that observation of these photons is possible.

3.4. LYMAN BREAK GALAXIES CHAPTER 3. THEORETICAL BACKGROUND

Figure 3.4: Diagram that shows ionization process with decreasing redshift. More LAEs are visible as reionization proceeds.

When scanning the Universe for LAEs during the beginning of reionization, we expect only the most massive and bright galaxies to be detected, as only these galaxies are surrounded by ionized bubbles large enough to transmit Lyman-alpha radiation. This effect is caused by the fact that there are simply more stars in massive (high-luminosity) galaxies that emit H-ionizing radiation. This leads to larger HII regions surrounding those galaxies through which Lyman-alpha radiation can travel and get redshifted to non-ionizing wavelengths. However, we should expect more fainter galaxies to start appearing as reionization proceeds. In Figure3.4, this effect is depicted. The HII regions expand, and once they expand to sizes large enough to encapsulate the fainter galaxies, the ‘barrier’ of HI between those galaxies and Earth is removed, increasing the amount of detected LAEs. With the advance of reionization also comes the formation of more stars and galaxies. These galaxies contribute to the reionization (if they contain high-energy stars) and this increases the amount of HII bubbles in the Universe, also resulting in more detected LAEs.

3.4 Lyman Break Galaxies

A Lyman Break Galaxy (or LBG) is a type of galaxy with ongoing star-formation, found at high redshift (z> 3). Stars form in regions with high hydrogen density. Because these dense molecular clouds are present in the galaxy, radiation at wavelengths shorter than the Lyman limit, 912 Å, is very likely to be absorbed by these clouds. Wavelengths between Lyman-alpha and the Lyman limit are also absorbed, but not at the same degree as wavelengths below the Lyman limit. Hence, a ‘break’ past 912 Å where there is almost zero flux can be seen in the spectrum of such a galaxy. An LBG can also be an LAE.

Chapter 4

Methodology

4.1 Data

To understand the clustering of galaxies during reionization, cosmological reionization simulations of high-redshift galaxy clustering models were analyzedHutter et al.(2014). The simulation box has a size of 80 h⁻¹Mpc (comoving Mpc) and is placed at redshift z= 6.6. The escape fraction of the H-ionizing photons is f_esc = 0.50. The observed continuum luminosity L^obs_c of the galaxies in the simulation is defined using the escape fraction:

L^obs_c = L_c^int× fesc, (4.1)

where L_c^intis the intrinsic continuum luminosity. In this research, we use three snapshots at different stages of reionization. These snapshots are taken at a neutral hydrogen fraction (or ionization fraction)

〈χHI〉 of (a) 0.50, (b) 0.25 and (c) 0.10. From this point onwards, the term ‘ionization fraction’ is used. An ionization fraction of 0.10 means a 90 % ionized Universe. The effects of reionization on the clustering of galaxies can be studied by analyzing the differences between the correlation function of these snapshots. The following properties of galaxies in the simulation box were also analyzed:

the average stellar metallicity per galaxy, average stellar mass per galaxy and average stellar age per galaxy. Further specifics of the simulation can be read inHutter et al.(2014). Four subsets of the total distribution of galaxies were defined at each ionization fraction to be able to study LAEs and LBGs in particular. Properties of each subset can be seen in Table4.1. The distribution of LAEs at the lowest ionization fraction is shown in Figure4.1.

type magnitude equivalent width luminosity

Faint LAEs M_UV< −17 EW_Lyα≥ 20 10⁴¹<_{erg s}^LLyα−1 ≤ 10⁴² Intermediate LAEs M_UV< −17 EW_Lyα≥ 20 10⁴²<_{erg s}^LLyα−1 ≤ 10⁴³ Bright LAEs M_UV< −17 EW_Lyα≥ 20 _{erg s}^LLyα−1 > 10⁴³

LBGs M_UV< −17

Table 4.1: Table showing the definitions of each subset of the simulation data.

4.2 Correlation function

4.2.1 Spatial two-point correlation function

The most well-known function that describes the clustering of galaxies is the correlation function. The spatial two-point correlation function in particular, describes the probability that, given a random galaxy in a location, another galaxy will be found within a given distance (Peebles,1980). For a homogeneous

4.2. CORRELATION FUNCTION CHAPTER 4. METHODOLOGY

Figure 4.1: The position of LAEs in the simulation box at〈χHI〉 = 0.10. The distinction between faint, intermediate and bright LAEs is made.

Poisson distribution of points (galaxies) in a volume, the probability dP₁₂of finding other points (galax- ies) in two volumes dV₁and dV₂at distance r is given by:

dP₁₂= ¯n²dV₁dV₂, (4.2)

where ¯n is the mean number density. In the case of clustering, there will be an excess with respect to the Poisson distribution. To obtain the probability of finding points in two volumes dV₁ and dV₂at distance r in this case, the two-point correlation function is needed:

dP₁₂= ¯n²[1 + ξ(r)] dV1dV₂. (4.3)

Ifξ(r) = 0, Equation (4.2) returns. This would mean that there is no clustering (i.e. a homogeneous distribution) at r. The data is said to be uncorrelated. Ifξ(r) > 0, there is an excess of points at r:

clustering, and the data is said to be correlated. Data is anti-correlated ifξ(r) < 0.

In a lot of cases, the two-point correlation function can be approximated by a power-law of the form:

ξ(r) =

r r₀

‹−γ

. (4.4)

In this equation, r₀is named correlation length and it is equal to the radius r at whichξ(r) = 1. Above this scale, the clustering approaches linearity. The correlation length is a measure for the clustering amplitude. The power-law index is denoted byγ. Calculating the two-point correlation function is a powerful tool when studying the clustering of galaxies, because it gives an accurate representation of the ‘lumpiness’ of structures from small to large scales. Measuring the function for faint, intermediate and bright LAEs can give us information about the clustering strength of those three types and more importantly, if there is a difference between them. This can help in the understanding of how structure was formed. On top of this, measuring the correlation function of all these types of galaxies at different ionization fractions can improve the understanding of the evolution of structure formation in time.

4.2.2 Angular correlation function

Another possibility for probing galaxy clustering is computing the correlation as a function of the angle separating galaxies instead of the distance. In this way, the correlation function can be compared with

4.2. CORRELATION FUNCTION CHAPTER 4. METHODOLOGY

observations of a 2D patch of the sky. The function is often denoted byω(θ) in literature. In this bachelor thesis, the angular correlation function computed for the LAEs was compared with findings of observational studies.

4.2.3 Estimators

The (two-point or angular) correlation function cannot be directly measured. Several researchers have proposed estimators of the function, using a comparison between a real distribution of points and a random one. For the spatial two-point correlation function, one considers a distance interval[r, r +dr) at a given distance r, and counts the number of galaxy pairs in the distribution that are separated by a distance x within the boundaries of the interval, meaning r≤ x < r +dr. By comparing the pair counts with the counts in a random Poisson distribution, an estimate of the correlation function at distance r can be obtained. Note that the same can be done for the angular correlation function, but instead of counting pairs in certain distance intervals, one counts pairs in angular separation intervals. In the following sections about different estimators of the function, one can thus replace the distance r by the angular separationθ and obtain results for the angular correlation function.

Landy & Szalay

The estimator used in the analysis of this project is very often found in researches that focus on the spatial distribution of galaxies. It was first introduced in 1993 byLandy and Szalay(1993), ten years after the introduction of the estimator byDavis and Peebles(1983), which is sometimes considered to be the standard estimator, but is less accurate for small deviations (Kerscher et al.,2000). The function compares a distribution (of real datapoints) with a random distribution and it is defined as:

ξˆLS(r) = N_r(Nr− 1) N(N − 1)

DD(r)

RR(r) − 2 N_r− 1 N

DR(r)

RR(r)+ 1, (4.5)

where N_r is the number of galaxies (datapoints) in the random distribution and N is the number of galaxies real distribution. Determining DD(r) requires that for every galaxy in the real distribution, the distances to all other galaxies in this distribution are known. This means that if the distribution has N galaxies, there are N(N −1) galaxy pairs for which the distance between the two galaxies that make up the pair must be known. Figure4.2is a visualization of this process. DD(r) is then the total number of pairs (in the real-real catalog) that are separated by any distance x where r≤ x < r + dr. For DR(r), the distances for every galaxy in the real distribution to all galaxies in the random distribution must be known. In this case, there are N· Nr pairs. DR(r) is calculated in the same way as DD(r), but looks at values in the real-random catalog. RR(r) is also calculated in the same way as DD(r) and DR(r) but only looks at galaxies in the random-random catalog. As can be seen from Equation (4.5), ˆξLS(r) is already normalized, as DD(r), DR(r) and RR(r) are divided by the total number of pairs in their respective catalogs:

DD(r) N(N−1)

RR(r) Nr(Nr−1)

− 2

DR(r) N·Nr

RR(r) Nr(Nr−1)

+ 1 =N_r(Nr− 1) N(N − 1)

DD(r)

RR(r) − 2 N_r− 1 N

DR(r)

RR(r)+ 1. (4.6)

4.2.4 Computation of correlation function

Two-point correlation function

As explained in Subsection4.2.3, knowing the distance between all galaxies in each sample is required for both estimators of the correlation function. The distance s between two galaxies in a box with dimensions x, y, z is given by:

s=Æ

(x2− x1)²+ (y2− y1)²+ (z2− z1)², (4.7) where x₂is the x-location of galaxy 2, x₁is the x-location of galaxy 1, etc. A module in Python called NumPy was imported in order to use its histogram function. To compute DD(r), a new function was written that, for each galaxy in the sample, computed the distance to all other galaxies and made a histogram of the number of galaxy pairs per distance interval. The values of all histograms were added

4.2. CORRELATION FUNCTION CHAPTER 4. METHODOLOGY

Figure 4.2: Figure showing the process of counting pairs and calculating the distance between the galaxies in each pair, resulting in a total of N(N − 1) pairs.

to obtain DD(r). The number of distance intervals or bins was not the same for all subsets (bright LAEs are less numerous than faint LAEs). A similar approach was used for DR(r) and RR(r), with the only difference being that for DR(r), the distance between the galaxies in the real and galaxies in the random sample was computed and for RR(r), the distance between all galaxies in the random sample was computed. The random distribution of points was made using another built-in function in the NumPy Python module. A uniform, random distribution was generated with 10 times as many points as the real distribution.

Angular correlation function

When measuring the angular correlation function for a distribution of galaxies in a simulation box, one has to find a way to convert the distance between galaxies to an angle subtended on the sky.

Using simple geometry to calculate this angle does not suffice. In the currently acceptedΛCDM model, objects at redshifts greater than z≈ 1.5 appear to subtend a larger angle on the sky than they would in Euclidean space. The angleθ in that case would be given by:

tan(θ) = s

d ≈ θ, (4.8)

where s is the distance between two objects and d the distance to one of the objects. The small-angle approximation is also used here. However, the aforementioned effect in theΛCDM model makes this approach invalid. Using the small-angle approximation, the angular diameter distance d_A does take this effect into account and relates the angleθ and the distance s between two objects at high redshift:

θ = s

d_A. (4.9)

The cosmological calculator byWright(2006) was used to obtain the angular diameter distance at a red- shift of z= 6.6, using the same values for ΩΛandΩMas used in the simulations byHutter et al.(2014).

“This calculator allows one to input user-selected values of the Hubble constant, Omega(matter), Omega(vacuum) and the redshift z, and returns the current age of the Universe, the age, the co-moving radial distance (and volume) and the angular-size distance at the specified redshift, as well as the scale (kpc/arcsec) and the luminosity distance." The calculator expresses the angular diameter distance in physical size. Since the units of the box are expressed in comoving size, the angular diameter distance was also converted to comoving size:

size_comoving= (1 + z) sizephysical. (4.10)

For all distances s between the galaxies, the approximation that all galaxies had a y-position equal to 0 was used, where y is the axis of the box in the direction parallel to the line of sight to the box. This approximation is valid since d_A≫ y. In this way, all points in the box can considered to be ‘squished’

into a 2-dimensional plane with only x and z directions. Figure4.3demonstrates this. The distance s between two galaxies is then given by:

s=Æ

(x2− x1)²+ (z2− z1)². (4.11)

4.2. CORRELATION FUNCTION CHAPTER 4. METHODOLOGY

The computation of the angular correlation function was done in the same way as the two-point corre- lation function, but instead of calculating the distance between each galaxy, the angle was calculated.

Histograms were computed per angular separation interval.

Figure 4.3: Visualization of all galaxies in box projected onto a 2-dimensional plane.

Chapter 5

Results and Discussion

5.1 Galaxy properties

Figures5.1, 5.2and5.3show the HMF, Lyman-alpha LF and UV LF of the distribution, respectively.

Each function contains 15 bins. The HMF clearly indicates a decrease in dark matter halo density as mass increases. At low masses below∼ 10^9.75M_⊙, the halo mass density decreases again, but this is due to the limits of the simulation. The trend simply indicates that there are less dark matter halos of high mass than of low mass. When looking at the UV LF, one can see that there is a peak in galaxy density at M_UV≈ −17. Note that increasing magnitude (less negative) corresponds to a decreasing luminosity.

Due to limitations of the simulation, the LF drops off at higher magnitudes than M_UV≈ −17. For the Lyman-alpha LF, this is at luminosities lower than L_Lyα≈ 10^41.5erg s⁻¹. We see a general trend up to L_Lyα≈ 10⁴³erg s⁻¹: at the lowest ionization fraction (〈χHI〉 = 0.50), the number density of galaxies is higher than at〈χHI〉 = 0.25, which is again higher than the number density at 〈χHI〉 = 0.10. As the Universe gets more ionized, more Lyman-alpha radiation is transmitted, increasing the observed Lyman- alpha luminosities of galaxies. This raises the number density of the detected galaxies. At luminosities higher than L_Lyα≈ 10⁴³erg s⁻¹however, we see an increase of the differences between the ionization fractions. This is due to the fact that there are very few galaxies with luminosities in this regime; an increase of the amount of ionized hydrogen could mean the shift to a higher luminosity bin for one galaxy. As there are few galaxies in the large luminosity bins, this shift makes a significant difference.

For the UV LF, there is no distinction between the ionization fractions. This is due to the fact that the intrinsic UV luminosity of galaxies in the simulation was calculated atλ = 1500 Å (rest frame of the galaxy). Radiation at wavelengths larger thanλ_Lyα= 1216 Å are not absorbed by neutral hydrogen in the first place, which means that an increase of ionized hydrogen does not affect the observed UV luminosity (or magnitude). The linear relation between the halo mass and luminosity as described by Equation (3.1) seems to be obeyed. Faint galaxies form in small dark matter halos and bright galaxies form in large dark matter halos. In Figure5.4we see the average age of all stars per galaxy plotted against the average metallicity of all stars per galaxy. There is no clear trend visible; most galaxies are centered around Z_star= 0.001 and age = 10^7.9yr with outliers also surrounding this point. The fact that

Figure 5.1: Halo mass function of galaxies in(80 h⁻¹Mpc)³simulation box. Functions are the same for each ionization fraction.

5.2. CORRELATION FUNCTION CHAPTER 5. RESULTS AND DISCUSSION

Figure 5.2: Lyman-alpha luminosity function of galaxies in(80 h⁻¹Mpc)³simulation box at three ionization fractions.

Figure 5.3: UV luminosity function of galax- ies in(80 h⁻¹Mpc)³simulation box. Functions are the same for each ionization fraction.

the majority of galaxies are concentrated roughly between 10^7.8yr and 10^8.0yr hints towards an era of high star formation. The age of the Univese at z= 6.6 is 0.852 Gyr (Wright,2006). This means that (when subtracting the average age of the stars) this star formation era began around 0.752 Gyr after the Big Bang. When looking at the plot for the three types of LAEs, we see that the bright LAEs have the highest average stellar metallicities, followed by the intermediate ones. The bright LAEs contain the largest amount of stars, but also the brightest stars, which are able to fuse elements higher up in the periodic table. They also live shorter, thereby greatly contributing to increasing metallicity. Figures5.6 and5.7tells us something about what types of galaxies form at what time. We see that the youngest galaxies are faint (with a few intermediate ones), just like the oldest ones. Also, roughly all bright galaxies formed in the aforementioned star formation era. As galaxies get fainter, the age variation between the galaxies increases. This tells us that almost no new bright galaxies formed after a certain point, while faint galaxies were being formed throughout the whole Epoch of Reionization. Figure5.8 shows a strong correlation between the average stellar metallicity of galaxies and their average stellar mass. When we consider the different categories of LAEs in Figure5.9, the relation between mass, luminosity and metallicity becomes even more apparent. High luminosity generally means a high mass and high metallicity.

Figure 5.4: Scatterplot of the mean stellar age per galaxy against the mean stellar metallicity per galaxy. All galaxies in simulation box are taken into account.

Figure 5.5: Scatterplot of the mean stellar metallicity per galaxy against the mean stellar mass per galaxy for three types of LAEs.

5.2 Correlation function

In Figure5.10, the (spatial) two-point correlation for the different subsets of data at three ionization fractions ((a)〈χHI〉 = 0.50, (b) 〈χHI〉 = 0.25 and (c) 〈χHI〉 = 0.10), calculated using the estimator by

5.2. CORRELATION FUNCTION CHAPTER 5. RESULTS AND DISCUSSION

Figure 5.6: Scatterplot of the mean stellar age per galaxy against the mean stellar mass per galaxy. All galaxies in simulation box are taken into account.

Figure 5.7: Scatterplot of the mean stellar age per galaxy against the mean stellar mass per galaxy for three types of LAEs.

Figure 5.8: Scatterplot of the mean stellar metallicity per galaxy against the mean stel- lar mass per galaxy. All galaxies in simulation box are taken into account.

Figure 5.9: Scatterplot of the mean stellar metallicity per galaxy against the mean stellar mass per galaxy for three types of LAEs.

Landy and Szalay(1993), is displayed. All functions have been fitted by a power-law using a least- squares method by a fitting module in AstroPy. The value of the correlation function of intermediate LAEs was found to be larger at small scales than that of faint LAEs. The sample of bright LAEs was composed of only∼ 30 galaxies (different depending on ionization fraction). Therefore, the results for this sample were less reliable. Still, a higher value for the correlation function of the bright sample was found at small scales. This general trend can be expressed as:

ξbright(rsmall) > ξintermediate(rsmall) > ξfaint(rsmall). (5.1) When we factor in the LBGs as well, we see that the clustering is even less strong for these types of galaxies. A higher value for the correlation function at a certain scale means more clustering at that scale. At larger scales, all correlation functions tend towards zero. This is also what is expected, as by the Cosmological Principle, the distribution of matter in the Universe is homogeneous and isotropic when viewed on large enough scale. When zooming in to smaller scales, however, we see that high- luminosity (large) galaxies are more strongly clustered than (small) low-luminosity galaxies. This is consistent with expectations, since the simulations are based on the current understanding of structure formation, in which the matter in the Universe gets ‘lumped’ together more as time progresses. The density differences between regions become stronger with decreasing redshift. Large galaxies form within the high density peaks of the matter distribution in the Universe, whereas the smaller galaxies can form in lower density regions. As can also be seen in the HMF in Figure5.1, there are significantly more low-mass dark matter halos than high-mass ones. The low-mass halos are more homogeneously distributed: the small galaxies that form inside these low-mass halos are, as a consequence, also more

5.3. COMPARISON WITH OBSERVATIONS CHAPTER 5. RESULTS AND DISCUSSION

homogeneously spread. The sample of LBGs contains significantly more galaxies than the LAEs. Since there is no restriction for the Lyman-alpha luminosity or equivalent width, the sample consists of all types of galaxies. A certain amount of faint galaxies that did not meet the selection criteria for LAEs are included in this sample. The faint galaxies are overpopulated with respect to the bright galaxies in the LBG sample, thus resulting in the most homogeneous distribution for LBGs.

For all LAE types, we see that scenario (a) (〈χHI〉 = 0.50) has the largest measured correlation function and (c) (〈χHI〉 = 0.10) has the lowest. This means that clustering is stronger when there is more neutral hydrogen present in the IGM. We see a general relation between the completeness of reionization and the correlation function. The higher the amount of neutral hydrogen in the IGM, the more the LAEs are clustered. The only way that clustering can decrease in a simulation box where all galaxies have a fixed position is when more galaxies start to be detected as LAEs, creating a more homogeneous distribution.

This nicely matches the theory as explained in Section3.3.2and visualized in Figure3.4. Only for the bright LAE sample, this relation is not completely satisfied. This can be explained by the fact that this sample is relatively small compared to the intermediate and faint sample, so statistical variations play a bigger role. On top of this, the differences between the correlation function for the bright sample at different ionization fractions are relatively small, which is to be expected from bright LAEs. An increase of the ionized fraction is not likely to cause significantly more bright LAEs to be detected, as they are already the first galaxies with ionized bubbles and thus the first to transmit Lyman-alpha radiation. The amount of detected LBGs does not increase at all when reionization proceeds. This is because galaxies classified as LBGs are only selected on their observed UV magnitude M_UV, which is calculated at 1500 Å.

Wavelengths larger than 1216 Å do not get absorbed by neutral hydrogen, as explained in Section5.1.

Hence, reionization does not affect the amount of detected LBGs.

The angular correlation function in Figure5.11 shows really similar curves. The overall clustering strength is, however, lower in the angular two-point correlation function. This difference between the spatial and angular correlation function can only be due to the projection of the spatial distribution onto the two-dimensional plane. Consider the distance between a galaxy in the corner of the simula- tion box with(x, y, z) = (0, 0, 0) and a galaxy in the opposite corner with (x, y, z) = (80, 80, 80). When all points in the simulation box are projected onto a plane, this distance shortens. This results in more galaxy pairs within certain distance bins (and thus angular separation bins for the angular correlation function), which means a decrease of clustering strength.

Figure 5.10: Two-point correlation function of LBGs and of LAEs computed using the estimator by Landy & Szalay. LAEs are in three different luminosity categories (faint, intermediate and bright). All functions are computed at three different ionization fractions.

5.3 Comparison with observations

The correlation function of simulated LAEs analyzed in this thesis was compared to results from ob- servations. A slice of the simulation box was taken that had the same size as the observed sky regions in two other researches. In this way, an analysis was made on how well the observations match with galaxy clustering models.

5.3. COMPARISON WITH OBSERVATIONS CHAPTER 5. RESULTS AND DISCUSSION

Figure 5.11: Angular correlation function of LBGs and of LAEs computed using the estimator by Landy

& Szalay. LAEs are in three different luminosity categories (faint, intermediate and bright). All functions are computed at three different ionization fractions.

5.3.1 Comparison with Kashikawa et al. (2006)

InKashikawa et al.(2006), the angular correlation function of a sample of 58 photometric candidate LAEs at z= 6.5 was measured. 17 of these were spectroscopically confirmed LAEs. The survey region had an angular size of 876 arcmin². Their comoving survey volume was 2.17× 10⁵h⁻³₇₀Mpc³, where h₇₀is defined as:

H₀= h7070 km s⁻¹Mpc⁻¹= 70 km s⁻¹Mpc⁻¹, (5.2) meaning h₇₀= 1. The definition of h in this thesis is not equal to h70. The definition of the Hubble constant adopted in this thesis is:

H₀= h100 km s⁻¹Mpc⁻¹= 70 km s⁻¹Mpc⁻¹, (5.3) so h= 0.7. Converting the angular size of 876 arcmin²to comoving Mpc²gives an area of 5728.62 Mpc². The cosmological calculator byWright(2006) was used for this conversion. This gives sides of 52.98 h⁻¹Mpc along the x and z axes of the simulation box. These sides are the new field of view (FoV). The depth of the box (in the y-direction) was obtained by dividing the volume of 2.17× 10⁵h⁻³₇₀Mpc³by the area, resulting in a depth of 26.52 h⁻¹Mpc. The measured correlation function should be indepen- dent of FoV when viewed on large enough scale, but the restricted FoV leads to an angular correlation function that is not independent of sample variance (Hutter et al.,2015). Four slices were taken out of the simulation box in order to get an idea of the variation between different regions. The LAE density of the four slices with the aforementioned FoV and depth (with overlapping volumes) was normalized according to:

¯

n(1 + ω(r)) = 1 N

∑N i=1

n_i(1 + ωi(r)), (5.4)

where ¯n is the mean density of the whole box,ω(r) is the mean angular correlation function, N is the number of slices, n_i is the number density of the slice andωi(r) is the correlation function computed in the slice. LAEs with Lyman-alpha luminosity of L_Lyα≥ 10⁴²erg s⁻¹ were used byKashikawa et al.

(2006), so only the LAEs with this minimum luminosity were selected when taking the slice of the simulation box. One slice that was taken can be seen in Figure5.12. The correlation function of the most homogeneous slice at the three ionization fractions, alongside the values byKashikawa et al.

(2006) are displayed in Figure5.13. The analyzed sky byKashikawa et al.(2006) shows almost no correlation, while the LAEs in the slice clearly show a correlation as angular separation decreases. The authors (Kashikawa et al.) did not rule out the possibility of looking at a really homogeneous sky distribution. This may explain the difference between the correlation functions. On top of that, only 17 of the total of 58 galaxies were spectroscopically confirmed to be LAEs, which may have caused the sample to be enriched with other (homogeneously distributed) galaxies. The variations for the correlation function in the slices are substantial. This supports the possibility thatKashikawa et al.

5.3. COMPARISON WITH OBSERVATIONS CHAPTER 5. RESULTS AND DISCUSSION

(2006) looked at a homogeneous sky distribution. However, every slice does show clustering that increases with decreasing angular separation. Another possible explanation for the difference between observations and the model might be the assumed value of the escape fraction f_esc of the photons in LAEs. The value adopted in the model is f_esc= 0.50 for all LAEs. In reality, one would not expect the same escape fraction for every galaxy. The effects of dust or gas heavily influence the escape fraction.

Small mass halos (and thus small galaxies) are much more susceptible to supernova feedback. When gas gets blown out during a supernova, it increases the escape fraction of the galaxy. The smaller (fainter) LAEs might have an escape fraction higher than assumed in the model, while the larger (brighter) LAEs might have a lower value. A higher f_esc for faint galaxies would mean even more faint galaxies to be detected as LAEs, which would decrease the clustering strength. Alongside this, the decrease of f_esc for brighter galaxies would mean less detected LAEs on the bright side of the spectrum, increasing the homogeneity of the sample. Both of these effects would cause a correlation function that would be in better accordance with the observations by Kashikawa et al. (2006). The highest ionization fraction (〈χHI〉 = 0.10) shows the least amount of clustering (at small angular separation), as expected.

However, the observed patch of sky at z= 6.5 might already be completely ionized. Referring to the theory in2.4, the reionization proceeded in a patchy way. The patch of sky might be located in a large, completely ionized sphere. For an even higher ionization fraction than the highest considered here (e.g. 〈χHI〉 = 0.05 or 〈χHI〉 = 0.01), we expect less clustering. This would also result in a correlation function that would be in better accordance with the observations.

Figure 5.12: Visualization of how the slice (with same volume as sky observation byKashikawa et al.

(2006)) of the total simulation box was taken. Dots indicate all LAEs with luminosity L_Lyα≥ 10⁴²erg s⁻¹ in this slice.

5.3.2 Comparison with Ouchi et al. (2010)

InOuchi et al.(2010), the clustering properties of a set of 207 LAEs at z= 6.6 on the 1 deg²sky of the Subaru/XMM-Newton Deep Survey field are examined. Here, 1 deg of sky corresponds to ∼ 150 Mpc = 105 h⁻¹Mpc. Since the simulation box is only 80 h⁻¹Mpc, the comparison between the correlation functions is not based on the same sky size. However, the depth of the sky analyzed byOuchi et al.

(2010) is taken into account. The redshift distribution of their LAE selection is displayed in Figure 5.14. The∆z of their distribution is approximately 0.16, which corresponds to a comoving distance of 44.31 h⁻¹Mpc. The cosmological calculator byWright(2006) was again used for this calculation.

Ouchi et al.(2010) select LAEs with L_Lyα ≥ 2.5 × 10⁴²erg s⁻¹ and EW_Lyα ≥ 14 Å. Four slices (with overlapping volumes) of the box were taken with a depth of 44.31 h⁻¹Mpc including all galaxies that

5.3. COMPARISON WITH OBSERVATIONS CHAPTER 5. RESULTS AND DISCUSSION

Figure 5.13: Angular correlation functions of the slice of the simulation box at three ionization frac- tions. Errorbars show the variation between slices. Angular correlation function byKashikawa et al.

(2006) is also plotted.

meet the mentioned LAE selection criteria. Figure5.15shows the most homogeneous slice and Figure 5.16shows the results for the computation of the correlation functions for the LAEs in this slice. The correlation functions of the LAEs at the three ionization fractions in the slice are all considerably larger (by about a factor 10). The reasons for the difference between the slice and observations may, again, be due to the fact that the escape fraction of f_esc= 0.50 was assumed for all galaxies in the model, which may not be the case in reality. The other explanation mentioned in Section5.3.1(limit to ionization fraction of〈χHI〉 = 0.10) could also apply in this case. Another explanation for the differences between observations and the simulation box can be sought in the method of projecting the simulation box onto a 2D patch of sky. For every computation of the angular correlation function in this thesis, the y-axis was considered to be the axis parallel to the line-of-sight. Therefore, the positions of galaxies along this axis were all stacked on top of eachother and one dimension was lost. There is an underdense region in the box that is visible in Figure4.3. The regions that show up as underdense and overdense change depending on what side the box is viewed. When the box is viewed from a different side, the angular correlation function might give different results.

5.3. COMPARISON WITH OBSERVATIONS CHAPTER 5. RESULTS AND DISCUSSION

Figure 5.14: Figure byOuchi et al.(2010) of redshift distribution of their LAEs with a spectroscopic identification. Histogram presents LAEs confirmed by their Keck/DEIMOS observations.

Figure 5.15: Visualization of how the slice (with same depth as sky observations byOuchi et al.(2010)) of the total simulation box was taken. Dots indicate all LAEs with luminosity L_Lyα≥ 2.5 × 10⁴²erg s⁻¹ and EW_Lyα≥ 14 in this slice.

5.3. COMPARISON WITH OBSERVATIONS CHAPTER 5. RESULTS AND DISCUSSION

Figure 5.16: Angular correlation functions of the LAEs (with same selection criteria asOuchi et al.

(2010)) in slice (44.31 h⁻¹Mpc) of the simulation box at three ionization fractions. Errorbars show the variation between slices. Angular correlation function byOuchi et al.(2010) is also plotted.

Chapter 6

Conclusions

In this thesis, a physical model for high-redshift Lyman-alpha emitters (LAEs) was analyzed. The aim of the project was to find out how the clustering of LAEs (divided in luminosity categories faint, in- termediate and bright) changes as reionization proceeds. Three snapshots of a simulation box (of size 80 h⁻¹Mpc in all three spatial dimensions) were taken at different stages of reionization ((a)

〈χHI〉 = 0.50, (b) 〈χHI〉 = 0.25 and (c) 〈χHI〉 = 0.10). Selection criteria were applied to all galaxies in the model to identify LAEs and LBGs. Galaxies with a UV magnitude of M_UV< −17 were considered LBGs.

Galaxies that, in addition to this criterion, had a Lyman-alpha equivalent width of EW_Ly≥ 20 Å were considerd LAEs. They and were divided in three luminosity categories: faint (10⁴¹ < _{erg s}^LLyα−1 ≤ 10⁴²), intermediate (10⁴² < _{erg s}^LLyα−1 ≤ 10⁴³) and bright (_{erg s}^LLyα−1 > 10⁴³). The spatial and angular two-point correlation function were calculated for each of the specified galaxies at each of the three ionization stages. First it was ascertained that both the angular and spatial correlation function demonstrated that bright LAEs are clustered more than intermediate LAEs and intermediate LAEs are more clustered than faint ones. This is explained by the fact that high-mass galaxies only form in high density peaks of the matter distribution in the Universe, whereas the low-mass galaxies are also able to form in low density regions. It was found that the clustering signal of LAEs decreased with an increasing fraction of ionized hydrogen in the Universe. As more regions become ionized, the larger the space becomes through which Lyman-alpha radiation can freely travel. The growing regions result in more radiation that is redshifted to non-exciting wavelengths, causing less radiation to be absorbed. This leads to more LAEs being detected. Ultimately, this decreases the clustering signal.

Slices of the simulation box were compared to observational results byKashikawa et al.(2006) and Ouchi et al.(2010). The clustering of the identified LAEs in the model was found to be larger than that found by previously mentioned authors. This could be due to a multitude of reasons:

1. The authors happened to look at a patch of sky with noticeably high homogeneity.

2. The sample of spectroscopically confirmed LAEs in observational results was too small, which did not give an accurate representation of the actual LAE distribution. The sample might have been enriched with other homogeneously distributed galaxies.

3. The assumption adopted in the model that the escape fraction of H-ionizing photons of f_esc = 0.50 is the same for all galaxies is incorrect. Low-mass (faint) galaxies are more susceptible to supernova feedback (causing gas to be blown out, increasing the escape fraction) than high-mass (bright) galaxies. Faint galaxies should have a higher escape fraction, which results in more faint LAEs being detected and thus decreasing the clustering strength of the sample. For bright galaxies, the opposite is true, also resulting in a decrease of clustering strength.

4. The highest ionization fraction considered (〈χHI〉 = 0.10), was not representative of the actual percentage of ionization at the observed parts of the Universe. Even higher ionization fractions would result in a lower clustering strength.

5. The box is viewed from the ‘wrong’ side. When computing the angular correlation function, the assumption was made that the y-axis of the box was parallel to the line-of-sight. This caused all

6.1. FUTURE RESEARCH CHAPTER 6. CONCLUSIONS

positions of galaxies along the y-axis to be stacked on top of eachother. Overdense and under- dense regions change depending on what axis is parallel to the line-of-sight. Another line-of-sight might have given different results.

It is hard to evaluate which of these reasons is the most likely to cause the difference, but it is most probable to be a combination of multiple.

6.1 Future research

Future research based on this thesis can take into account a wider variety of ionization fractions, es- pecially fractions that tend to 0. Also, escape fractions different for each galaxy could improve the model. The mass resolution of the simulation is approximately M_h^limit= 10^9.5M_⊙. When one wants to improve the simulation, the resolution could be enhanced. This would mean that even smaller dark matter halos are included in the model. Therefore, even smaller galaxies could be accounted for, which could have an effect on the measured clustering. Next to this, future research could prove if using different lines-of-sight to the simulation box gives different results for the angular correlation func- tion. The aim of this research was to find out how the clustering of LAEs changed with reionization.

However, only one snapshot at a fixed redshift was analyzed. When the model is extended to include other redshifts, one could probe the differences between ionization fractions at different redshifts. At redshifts larger than z= 6.6, one should expect the number of galaxies to decrease, as the formation of galaxies happened throughout the Epoch of Reionization. More importantly, the ratio of faint to bright galaxies should be getting closer to 1. As can be seen in Figure5.6in Section5.1of this thesis, the last galaxies that formed were faint. As one observes regions in the Universe further back in time (and redshift), these faint galaxies would ‘disappear’, because they had not been formed yet. This would affect the correlation function: the relative increase of bright LAEs with respect to the total amount of LAEs in the sample should cause the sample to show more clustering. With the advent of even more advanced telescopes like the James Webb Space Telescope (JWST) and the European Extremely Large Telescope (EELT), the detection of even fainter galaxies and other objects at even higher redshifts will be made possible. Observations from these telescopes might make the measured correlation functions more accurate.