Luminosity Function of LAEs: Computing effective volumes from MUSE data cubes

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Identifying the sources of reionization

Over the past years, various type of ionizing sources have been investigated in an effort to find the primary sources of reionization.
Following the chronological order, the first candidates that were considered were the massive pop III stars. However, because of their short life-span and because they were quickly replaced by a new generation of metal poor stars, they could not have contributed to reionization during an ex-tended period of time (see e.g., the simulation results from Ricotti & Ostriker, 2004a; Paardekooper et al., 2013). The X-ray emission from hard X-ray sources such as binaries and Active Galactic Nuclei (AGN) were also considered as ionizing sources. Even though these populations are poorly constrained at high redshift, it seems that they are not numerous enough to produce a significant fraction of the ionizing flux required for reionization (see e.g. Ricotti & Ostriker, 2004b; Willott et al., 2010; McQuinn, 2012; McGreer et al., 2013).
Nowadays, these two populations have been mostly set aside in this search for the sources of reionization. It is not excluded that they contribute to a somewhat significant level to the overall ionizing flux density budget (. 10%), but a consensus is beginning to appear in the sense that none of them are driving the reionization process. More recently, the search for ionizing sources has therefore shifted towards galaxies as they are the most likely candidates remaining. And more specifically the low mass and star forming galaxies, as their high numerical density and their hard spectra produce a large amount of ionizing flux. However, a definitive proof has yet to be made that this population do indeed drive the reionization process.
We provide below a description of these high redshift and star forming galaxies and explain how to quantify the contribution of this galaxy population with the Luminosity Function (LF) in Sect. 1.2. We then present the recent progresses and developments regarding the reionization timeline and the assessment of the contribution of SFGs to reionization in Sect. 1.3 and Sect. 1.4. Finally, we explain in Sect. 1.5 the strategy used in this present work and show how it can contribute to answer to some of the remaining questions about reionization.

High redshift and star forming galaxies

The first challenge of studying high redshift galaxies, is to find and observe them. Because the rest-frame emission of these galaxies is typically dominated by the spectra of star forming regions and young stellar population, they have a bright UV continuum associated with a steep UV slope together with a characteristic break (GP trough and Lyα forest, see Sect. 1.2.2) that helps identify-ing them. In addition they can also have recombination and forbidden emission lines representative of star forming regions.
To select sources from this population, two methods are commonly used: the drop-out selection based on multi-band photometry or the Lyα selection based on either Narrow-Band (NB) pho-tometry or IFU observations. These two selection methods and the underlying physical processes traced by these methods are presented in Sect. 1.2.2 and Sect. 1.2.1 respectively. An idealized representation of such a high redshift galaxy spectrum is presented in Fig. 1.4 to illustrate the following sections.

Lyman-Alpha emitters

Description and observational features

The Lyα is a resonant emission line of hydrogen at λLyα = 1216Å and is intrinsically the brigthest line of the hydrogen emission line spectrum. It can be emitted by atom collision and recombination of ionized hydrogen and as such is often used a tracer for star forming galaxies and star-forming regions. It is also commonly used a a signature for high redshift and young galaxies (Partridge & Peebles, 1967; Malhotra & Rhoads, 2002). Galaxies observed with this Lyα emission are called Lyman-Alpha Emitters (LAEs).
Models of stellar populations predict that normal star forming galaxies cannot produce Lyα emission lines with an Equivalent Width (EW) EWLyα & 240Å (Charlot & Fall, 1993). Exceptions to that rule can be observed in case of significant absorption of the continuum by dust, extremely poor metallicity, young galaxy ages or non standard Initial Mass Function (IMF), the three latest being indicative of primitive galaxies (Schaerer, 2003; Raiter et al., 2010). The Lyα emission of galaxies with rest-frame EWLyα much greater than 240 Å is therefore likely to be powered by AGN rather than a normal stellar population and as such is tracing a completely different population.
Being a resonant emission, a Lyα photon can be absorbed by an atom of hydrogen and re-emitted at the exact same wavelength. Regarding high redshift galaxies, this has major consequences that can be observed in the spectra of LAEs.
Spatially extended emission. The Lyα emission tends to be more extended than the UV continuum emission provided that a large amount of neutral gas can be found in the Circum-Galactic Medium (CGM) (Leclercq et al., 2017; Wisotzki et al., 2018). Therefore galaxies observed in Lyα can have varying morphologies with respect to rest-frame UV observations.
Escape fraction. The complex radiative transfer of a Lyα in the surrounding CGM leads to low escape fractions (Verhamme et al., 2017). Photons can only escape the influence of the surrounding CGM once they have travelled far enough to be redshifted out of resonance. This process can be very long since the repeated scattering can lead a Lyα photon to randomly wander in neutral halos for millions of years. This long timescale means that even a very low fraction of dust in the medium is likely to absorb a Lyα photon and that we expect the escape fraction of a Lyα photon to be very different from the escape fraction of a typical UV photon.
Line profile. The kinematic of the surrounding CGM impact significantly the observed profiles of Lyα lines. Radiative transfer simulations have shown that a static shell of gas (i.e. neutral HI) around a LAE would create a double peak emission. In that situation the velocity dispersion within the cloud absorbs the photons in the line center. At each step of the radiative transfer, the re-emitted photons have their frequency shifted by a small amount depending on the angle between the velocity of the atom and the direction of re-emission of the photon. Eventually, during this process, the photon will be shifted out of resonance, either on the blue or red side of the central emission, leading to the observed double peak emission.
Following a very similar mechanisms, an outflow of gas would redshift the peak of the line and create an asymmetry with an extended wing on the red side of the lines. This asymmetry can be caused by gas outflows driven by a rapid expansion of space or by regions with intense star formation.
The exact geometry considered and kinematics of the gas can lead to a wide range of Lyα pro-files as shown in Fig. 1.5. For high redshift galaxies, the most common profiles are the one caused by outflow and the double peak emission caused by static shells (or a combination of the two). Typical profiles like these are quite common in observations and makes easier the identification of the Lyα lines when no other spectral features can be seen. For a more complete introduction to this complex subject, we refer the reader to the simulations and analysis presented in Verhamme et al. (2006) and Laursen et al. (2011).
The complex physics of Lyα emission is a blessing for observers, as it allows to learn much about galaxy properties and the CGM, but it also introduces biases in the observations that are difficult to overcome. The first one is obviously the low escape fraction that dims the Lyα emission (see e.g. Verhamme et al., 2017) and the second one is the apparent clustering of LAEs at high redshift. Both effects are described latter in Sect. 1.3.2.

Selection methods

To select LAEs, two methods are commonly used:
◦ Using NB imaging to detect an excess of flux indicating the presence of an emission line. This technique can only be used to select LAEs in narrow redshift ranges and by construction, is only efficient to select LAEs with a high enough EWLyα. When using NB imaging, a spectroscopic follow-up can be done to confirm the likely candidates (see e.g. Rhoads et al., 2000; Ouchi et al., 2010; Sobral et al., 2018).
◦ Serendipitous detections in slit spectroscopy observations (see e.g. Cassata et al., 2011).
◦ Complete blind spectroscopic selection with Integral Fields Units (IFUs). In that case, the process is simpler and leads to a higher level of completeness compared to NB imaging (see e.g. Blanc et al., 2011; Drake et al., 2017b).
Since the Lyα is at the core of the method described in this present work, these three approaches are tackled in more details in Sect. 2.3 when talking about the issue of the completeness of Lyα selection process.

Lyman-Break Galaxies

Absorption features in Lyman-break galaxie

The Lyman-Break Galaxies (LBGs) are the galaxies selected with the drop-out technique (Steidel et al., 1996, 1999) using multi-band rest frame UV photometry. As such the term « LBG” refer only to a selection method using the spectral features described in this section. Individual galaxies can be selected as both LAE and LBGs and whether there is an intrinsic physical difference between LBGs and LAEs is not fully understood yet. The LBG selection relies on the identification of integrated spectral features that can be visible on broad-band photometry. Three features (shown in Fig. 1.4), or a combination of these, can be used for this exercise:
The Lyman break. The neutral hydrogen present in large amount in the Inter-Stellar Medium (ISM) of star forming regions causes the absorption of the photons with λ ≤ 912Å. As a result, this signature appears as a break since no (or very weak) emission can reach the observers below this rest-frame wavelength. This is called the Lyman break or the Lyman limit.
The Lyman-alpha absorption forest. This feature is caused by HI clouds placed on the line of sight (LoS) causing a series of narrow absorption lines. UV photons with 912Å ≤ λ ≤ λLyα = 1216Å are continuously redshifted as they travel and at a given point some of them will be seen by their local surrounding as photons with λ = λLyα = 1216Å. At this exact location of space they enter a resonant state and if they also happen to encounter a cloud of neutral hydrogen, these photons can be scattered as described in Sect. 1.2.1. Such random scattering will cause these resonant photons to leave the LoS trajectory and to create an absorption line in the observed spectrum. These absorption lines appear as many times and at as many different wavelengths as there are clouds along the LoS (hence the name Lyman-alpha forest). The observed wavelength of these absorption lines depends on the redshift of the various clouds with respect to the SFG. The higher the redshift the denser the Lyman forest, as the amount of neutral hydrogen increases with redshift. The same stands for Lyβ . At λ < 1216Å, both the Lyα and Lyβ forests are responsible for the dimming on the spectral continuum Historically, this mechanism was first observed in the spectra of high-redshift quasars (see e.g. Lynds, 1971) as they are the brightest objects, but the same applies to normal star forming galaxies. This phenomenon was precisely modeled to evaluate its impact on the color of high redshift galaxies in Madau (1995).
The Gunn-Peterson Trough. This spectral feature is a complete suppression of all emission for rest frame wavelengths with (1 + zeor)1215Å ≤ λ ≤ 1216Å, where zeor is the redshift of the end of the reionization process. This absorption can only be observed for sources with z > zeor. It was first described in Gunn & Peterson (1965) and observed for the first time much later in Becker et al. (2001). This effect can be seen as an extreme and continuous Lyman-alpha absorption forest:
◦ All photons with λ ≤ 912Å are completely absorbed when ionizing the neutral Inter-Galactic Medium (IGM).
◦ All photons with 912Å ≤ λ ≤ 1216Å are not energetic enough to ionize the hydrogen but as they propagate, the longest wavelength are progressively redshifted into Lyα resonance. These resonant photons are progressively scattered by the neutral IGM, causing a complete and large absorption just bluewards of Lyα. Eventually, photons with high enough wavelength will travel until the end of reionization without having been redshift into resonance.however, these photons remain affected by traces of neutral hydrogen present on the LoS for z > zeor, and remain therefore subjected to the Lyα forest phenomenon.
High redshift galaxies showing this GP trough show a complete absorption for (1+zeor)1216Å ≤ λ ≤ 1216Å, Lyα absorption forest for 912Åλ ≤ (1+zeor)1216Å and a complete absorption for λ ≤ 912Å.
Each of these three phenomenon are shown and summarized on the schematic spectra in Fig. 1.4. Depending on the columns density of hydrogen, the observed spectra can be a combination of several of the effect described here. The quasars spectra published in Fan et al. (2006) showing the evolution of their observed spectral features with redshift are shown in Fig. 1.6. The spectra at z = 5.93, 6.01 and z > 6.13 can be labelled as GP troughs. The difference with respect to the other spectra is that the higher redshift lead a complete absorption between the Lyα line and a Lyα absorption forest.

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Selection method and photometric redshifts

Two examples of rest frame UV photometric observations with HST and leading to LBG selec-tion are shown in Fig. 1.7. Both sources are not detected in the filters centred on the shortest wavelengths. This is explained by the fact that these filters are sampling rest frame wavelengths bluewards of Lyα emission (see transmission bands in Fig. 1.4), where either of the three absorp-tion phenomena mentioned above play a role to suppress the continuum. For the galaxy shown in the top panel, the break is observed at lower wavelengths (between F435w and F606w) than for the bottom panel galaxy (between F814w and F105w), indicating that its redshift is lower.
The position of the integrated breaks can be used to give more reliable constraints on the redshift through color-color selection (as done originally in Steidel et al. (1996) or more recently in Bouwens et al. (2015b)), or with SED-fitting techniques to compute more precise photometric redshift (see e.g. HyperZ presented in Bolzonella et al. (2000) and used in e.g., Pelló et al. (2018)).
The color-color selection is a simple and fast method relying on the direct comparison of mag-nitudes in different filters, in order to identify a break combined with a steep UV slope (i.e. a rest frame “very blue” galaxy). A set of rules and conditions can be given to assess in a consistent way the colors of galaxies and sort them in different pre-defined redshift ranges.
The SED-fitting (Spectral Energy Distribution) techniques are a bit more complex and heavy to implement. From photometric observations and measurements, the procedure fits the observed SED using a library of template spectra spanning a parameter space given by e.g. star formation type and age (with or without e-line contribution), metallicity, IMF, Lyman forest prescriptions, with redshift being one of the parameters.
Photometric redshift measured are obviously less precise than spectroscopic measurements but
the agreement with true redshifts is of the order of σ(zphot − ztrue) ∼ 0.05(1 +ztrue) for a deep broad band survey. In addition to the best photometric redshift, SED-fitting procedures also provide the best fit across the parameter space. An example of SED-fitting made with HyperZ is provided in Fig. 1.8. In this figure, the photometric observations are best fitted by HyperZ when using a photometric redshift of zphot = 4.1.
Color-color diagram and photometric redshifts used to select drop-out galaxies are prone to make errors on the redshift determination. The only way to avoid this is to use narrower and more numerous photometry filters to break some of the degeneracies between low and high red-shift solutions. This is of course not feasible in most cases because of the difficulty to obtain telescope time and for other technical limitations (e.g. the S/N is degraded when using narrow filters and longer exposures are need to reach equivalent depth). A comparison between narrow and broad band photometry results within the same field can be found in Arrabal Haro et al. (2018).
Figure 1.7: Example of two drop-out selected galaxies using HST filters in the A2744 frontiers field
Lotz et al. (2017). The galaxy in the top panel has a photometric redshift at zphiot ∼ 4.4, and the one in the bottom panel has a photometric redshift around zphot ∼ 7.0.
Figure 1.8: Example of SED fitting results with HyperZ using the Hubble Space Telescope (HST) photometry displayed in the top panel of Fig. 1.7 and additional photometry from Hawk-I K band and Irac 1 and 2 bands for the longest wavelength filters. The best photometric redshift as computed by HyperZ is zphot = 4.1. The red points are the photometric measurements, the black line is the best galaxy spectra template and the green points are photometric points measured from the template (synthetic photometry) using the same filter transmissions as for the original photometry. The best fit spectra shows both a Lyman-break and a Lyman-alpha absorption forest bluewards of λLyα .

The galaxy luminosity function

In the previous section it was explained how to observe and select high-redshift star forming galax-ies. In this section we explain how to study these galaxies as a population and how this can be used for the study of reionization.
One of the main strategy to quantify the abundance of galaxies in a given volume is to measure their Luminosity Function (LF). The LF is simply a number density of galaxies per (co-)volume units and magnitude (or luminosity) bins. Since it aims at giving a complete assessment of a given population but observations are limited by their depth and volume, the determination of the LF is therefore limited to a restricted range of magnitude (Luminosity). Three steps are needed to derive a luminosity function:
◦ Select a galaxy population. The method used for selection has to be consistent and reliable (in the sense of « reproducible ») to properly assess the completeness of the sample. This means that galaxies selected as a result of visual inspection or irregular selection procedures are not suited for the derivation of the LF. For galaxies in the redshift domain of interest, the selection criteria commonly used are are the LBG or LAE selections (as explained in Sect. 1.2.2 or Sect. 1.2.1, respectively).
◦ Do a completeness correction, or alternatively, cut the sample to the magnitude range where the sample is 100% complete. The goal of this completeness correction is to account for the sources missed in the detection process in the final statistic of the LF. Rigurously speaking, the sample used for LF computation should be complete both in terms of luminosity and volume.
◦ Determine the distribution of galaxies from the completeness corrected sample. This can be done by deriving the most likely parametric form given the data set or simply using a magnitude (luminosity) binned representation.
When the LF is expressed as a numerical density per magnitude (luminosity) bin, it is called the differential LF. This term is used in opposition to the cumulative LF which gives a numerical density of galaxies up to a given absolute magnitude (or luminosity). Both forms of the LF can be derived from each other and give the same information. In the rest of this work, only the differential LF is considered, unless specified otherwise.
Many different methods can be used to compute a LF but will not be detailed in this work. A short overview of the main methods and a comparison of the results obtained with some of them can be found in Herenz et al. (2019). Most methods vary in their use of the selection function and in the underlying hypothesis regarding the distribution of galaxies. The most commonly used methods are variations of a maximum likelihood estimator or variations of the the 1/Vmax methods. For more details on these methods and the use of the selection function, see the introduction of Sect. 4. In the following sections, we introduce a convenient parameterization of the LF to facilitate the discussion and the comparison with previous studies (Sect. 1.2.3.1), and explain what can be learned from the study of the galaxy luminosity function (Sect. 1.2.3.2).

Schechter parameterization

A widely used parameterization for the study of the LF is the Schechter function (Schechter, 1976)
which writes as:
Because the second form is easier to read and is the one used in this work for LAEs, only this last one is commented, but the same can be transposed to the parameterization with magnitudes instead of luminosity. The term Φ(L)dL is the expected density of galaxy within the luminosity range [L, L + dL], Φ∗ is a normalization parameter, L∗ regulates the position of the transition between the exponential law at higher luminosity and the power law at low luminosity (often called the knee of the LF). The parameter α ≤ 0 is the slope of the faint end: as the Schechter function is almost always plotted in Log space, the power law of the faint end transposes as a line of slope α. An example of Schechter functions is shown in Fig. 1.9 where the impact of the value of the three parameters are also shown. Observed LFs with the best Schechter parameterization are shown in Fig. 6.5.
The motivation for such a function is that its form derives from the Press-Schechter formalism for the halo mass function developed in Press & Schechter (1974) and that it provides a good approximation for the observed LF. This formalism uses a linear perturbation theory to describe the growth of structures at the larger scales only. As specified in Press & Schechter (1974), using only the larger scales is needed to avoid as much as possible the non linear effects that are inherent to the N-body problems. The authors also do not exclude the idea that the non linear perturbations could transfer to larger scales and produce effects non predicted by their formalism.
There is therefore no strong physical evidence or guarantee that this function should correctly describe the galaxy LF at all luminosity ranges. The main reason for the use of this function is that up to now, it has been proven to work reasonably well (once again this is clearly stated by the author in Schechter (1976)) to describe the observations. In addition, it provides an easy way to compare the results obtained in different studies, and the three parameters α, L∗ and Φ∗ make it easier to discuss possible evolution of the LF shape. And finally, using a parametric form can be useful to extrapolate results to luminosity ranges that are out of reach of present observations.
The use of the Schechter function is therefore quite convenient, the main interrogation remaining is whether it can continue to accurately describe the LF as we reach lower and lower luminosity regimes, or if there is some effective turnover luminosity under which the Schechter function is no longer a good representation of the data.

Table of contents :

1 Introduction 
1.1 The early Universe
1.1.1 First structure and galaxy formation
1.1.2 Epoch of reionization
1.1.3 Identifying the sources of reionization
1.2 High redshift and star forming galaxies
1.2.1 Lyman-Alpha emitters
1.2.1.1 Description and observational features
1.2.1.2 Selection methods
1.2.2 Lyman-Break Galaxies
1.2.2.1 Absorption features in Lyman-break galaxies
1.2.2.2 Selection method and photometric redshifts
1.2.3 The galaxy luminosity function
1.2.3.1 Schechter parameterization
1.2.3.2 What can we learn from the study of the LF ?
1.3 Constraints on the timeline of reionization
1.3.1 Quasar observations
1.3.2 High redshift LAEs
1.3.3 Thomson optical depth and summary
1.4 Investigating SFGs as main drivers of reionization
1.4.1 The UV and LAE LF
1.4.2 Using strong lensing clusters
1.4.3 Total SFGs contribution to reionization
1.5 This work
2 The VLT/MUSE instrument 
2.1 General overview and technical features
2.2 Main science goals of MUSE
2.3 MUSE to study the galaxy population
2.3.1 Detection of faint emission line objects
2.3.2 Advantages of a blind spectroscopic selection for the LAE LF
2.4 Noise structure in MUSE cubes
2.4.1 Structure of the instrument
2.4.2 Sky emission lines
2.4.3 Sensitivity of the instrument and final combination
2.5 Cataloging sources in a MUSE FoV
3 Lensing clusters: methodology and MUSE observations 
3.1 Mass modelling methodology
3.1.1 Constraining the mass distribution with gravitational lensing
3.1.2 Parametric modelling with Lenstool
3.2 Observations
3.2.1 MUSE observations
3.2.2 Complementary HST observations
3.2.3 Source detection
3.3 Correcting for lensing
3.3.1 Description of the models used
3.3.2 Image plane
3.3.3 Source plane projection
4 Luminosity Function of LAEs: Computing effective volumes from MUSE data cubes 
4.1 Motivations
4.2 Source detection in MUSE cubes with Muselet
4.2.1 Definitions
4.3 Computing 2D detection masks
4.3.1 Presentation of the algorithm
4.3.2 Results, examples and tests
4.3.3 Direct application to mask 3D cubes in the source plane
4.4 Adopted method to efficiently mask 3D cubes in the source plane
4.4.1 Definition of noise levels and S/N
4.4.2 Main simplifications
4.4.3 assembling 3D masks in image and source plane
4.5 Volume integration and results
4.5.1 Effect of S/N sampling
4.5.2 Discussion on the method
5 Determination of the Luminosity Function of LAEs at 3 . z
5.1 Lyα flux computation and final selection of lensed sources
5.1.1 Source selection and flux weighted magnification
5.2 Vmax computation summary
5.3 Completeness determination
5.3.1 Source profile reconstruction
5.3.2 Source recovery experiments
5.3.3 Results and discussion
5.4 Computing the LF points
5.4.1 Tests with different luminosity binnings
6 Results: The LAE LF
6.1 Presentation of the lensed LAE sample and comparison with the MUSE-HUDF sample
6.2 LF analysis
6.2.1 Lensing sample only
6.2.2 Schechter fit of the LAE LF
6.2.2.1 Fitting method and results
6.2.2.2 Impact of luminosity binning
6.2.2.3 Discussion: Evolution of the LF with redshift
6.3 Discussion: Implication for the reionization
6.3.1 Impact of the mass model
6.3.2 Ionizing flux density
7 Intersection of the LAE and LBG populations
7.1 Source selection
7.1.1 Astrodeep catalog: filtering and cross matching with MUSE detections
7.1.2 SED fitting and photometric redshift with HyperZ
7.1.3 selection criteria
7.2 Results
7.2.1 Overview
7.2.2 Evolution with redshift
7.2.3 Evolution with luminosity and UV magnitude
7.3 Possible interpretation
8 Conclusion and future prospects

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