# Samples from the double Schechter luminosity funcion

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I want to produce samples from a double Schechter luminosity function.

Ideally, I am looking for a python implementation of this problem but am open to other tools if necessary.

Do you know of any such package that will facilitate this?

## An elliptical galaxy luminosity function and velocity dispersion sample of relevance for gravitational lensing statistics

We have selected 42 elliptical galaxies from the literature and estimated their velocity dispersions at the effective radius (σRe) and at 0.54 effective radii (σ0.54Re). We find by a dynamical analysis that the normalized velocity dispersion of the dark halo of an elliptical galaxy σDM is roughly σRe multiplied by a constant, which is almost independent of the core radius or the anisotropy parameter of each galaxy. Our sample analysis suggests that σ DM ∗ lies in the range 178–198 km s −1 . The power law relation we find between the luminosity and the dark matter velocity dispersion measured in this way is (L/L∗)=(σ DM /σ DM ∗) γ , where γ is between 2 and 3. These results are of interest for strong gravitational lensing statistics studies. In order to determine the value of σ DM ∗ , we calculate M B T 0 ∗ in the same B T 0 band in which σ DM ∗ has been estimated. We select 131 elliptical galaxies as a complete sample set with apparent magnitudes B T 0 between 9.26 and 12.19. We find that the luminosity function is well fitted to the Schechter form, with parameters M B T 0 ∗ =−19.66+5·log10h±0.30, α=0.15±0.55, and the normalization constant φ∗=(1.34±0.30)×10 −3 h 3 Mpc −3 , with the Hubble constant Ho=100 h km s −1 Mpc −1 . This normalization implies that morphology type E galaxies make up (10.8 ± 1.2) per cent of all galaxies.

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In: Astrophysical Journal , Vol. 631, No. 1 I, 20.09.2005, p. 208-230.

Research output : Contribution to journal › Article › peer-review

T1 - The properties and luminosity function of extremely low luminosity galaxies

N2 - We examine a sample of low-redshift (10 h-1 Mpc < d < 150 h-1 Mpc) field galaxies including galaxies with luminosities as low as Mr - 5 log h ∼ -12.5, selected from the Sloan Digital Sky Survey Data Release 2 (SDSS DR2). The sample is unique in containing galaxies of extremely low luminosities in a wide range of environments, selected with uniform and well-understood criteria. We present the luminosity function, as well as the broadband properties, of low-luminosity galaxies in this sample. A Schechter function is an insufficient parameterization of the r-band luminosity function there is an upturn in the slope for Mr - 5 log h > -18. The resulting slope at low luminosities in this sample is α2 ∼ -1.3. However, we almost certainly miss a large number of galaxies at very low luminosities due to low surface brightness selection effects, and we estimate that the true low-luminosity slope may be as steep as or steeper than α2 ∼ -1.5. The results here are consistent with previous SDSS results and, in the g band, roughly consistent with the results of the Two Degree Field Galaxy Redshift Survey. Extremely low luminosity galaxies are predominantly low surface brightness, exponential disks, the majority of which are red.

AB - We examine a sample of low-redshift (10 h-1 Mpc < d < 150 h-1 Mpc) field galaxies including galaxies with luminosities as low as Mr - 5 log h ∼ -12.5, selected from the Sloan Digital Sky Survey Data Release 2 (SDSS DR2). The sample is unique in containing galaxies of extremely low luminosities in a wide range of environments, selected with uniform and well-understood criteria. We present the luminosity function, as well as the broadband properties, of low-luminosity galaxies in this sample. A Schechter function is an insufficient parameterization of the r-band luminosity function there is an upturn in the slope for Mr - 5 log h > -18. The resulting slope at low luminosities in this sample is α2 ∼ -1.3. However, we almost certainly miss a large number of galaxies at very low luminosities due to low surface brightness selection effects, and we estimate that the true low-luminosity slope may be as steep as or steeper than α2 ∼ -1.5. The results here are consistent with previous SDSS results and, in the g band, roughly consistent with the results of the Two Degree Field Galaxy Redshift Survey. Extremely low luminosity galaxies are predominantly low surface brightness, exponential disks, the majority of which are red.

## Digital [email protected]

33 fast radio bursts (FRBs) had been detected by 2018 March. Although the sample size is still limited, meaningful statistical studies can already be carried out. The normalized luminosity function places important constraints on the intrinsic power output, sheds light on the origin(s) of FRBs, and can guide future observations. In this paper, we measure the normalized luminosity function of FRBs. Using Bayesian statistics, we can naturally account for a variety of factors such as receiver noise temperature, bandwidth, and source selection criteria. We can also include astronomical systematics, such as host galaxy dispersion measure, FRB local dispersion measure, galaxy evolution, geometric projection effects, and Galactic halo contribution. Assuming a Schechter luminosity function, we show that the isotropic luminosities of FRBs have a power-law distribution that covers approximately three orders of magnitude, with a power-law index ranging from −1.8 to −1.2 and a cut off ∼2×1044ergs−1⁠. By using different galaxy models and well-established Bayesian marginalization techniques, we show that our conclusions are robust against unknowns, such as the electron densities in the Milky Way halo and the FRB environment, host galaxy morphology, and telescope beam response.

#### Keywords

Stars: Luminosity function, mass function ISM: General Galaxies: Structure Cosmology: Theory

## The luminosity function, halo masses and stellar masses of luminous Lyman-break galaxies at redshifts 5 < z < 6

We present the results of a study of a large sample of luminous (zAB < 26) Lyman-break galaxies (LBGs) in the redshift interval 4.7 < z < 6.3 , selected from a contiguous 0.63 deg 2 area covered by the UKIRT Infrared Deep Sky Survey Ultra Deep Survey and the Subaru XMM–Newton Survey. Utilizing the large area coverage and the excellent available optical+near-infrared data, we use a photometric redshift analysis to derive a new, robust, measurement of the bright end (LL ★ ) of the ultraviolet-selected luminosity function at high redshift. When combined with literature studies of the fainter LBG population, our new sample provides improved constraints on the luminosity function of redshift 5 < z < 6 LBGs over the luminosity range 0.1L ★ ≲L≲ 10L ★ . A maximum likelihood analysis returns best-fitting Schechter function parameters of M1500=−20.73 ± 0.11, φ ★ = 0.0009 ± 0.0002 Mpc −3 and α=−1.66 ± 0.06 for the luminosity function at z= 5 , and M1500=−20.04 ± 0.12, φ ★ = 0.0018 ± 0.0005 Mpc −3 and α=−1.71 ± 0.11 at z= 6 . In addition, an analysis of the angular clustering properties of our LBG sample demonstrates that luminous 5 < z < 6 LBGs are strongly clustered (r0= 8.1 +2.1 −1.5h −1 70 Mpc) , and consistent with the occupation of dark matter haloes with masses of ≃10 11.5−12 M . Moreover, by stacking the available multiwavelength imaging data for the high-redshift LBGs, it is possible to place useful constraints on their typical stellar mass. The results of this analysis suggest that luminous LBGs at 5 < z < 6 have an average stellar mass of log10(M/M) = 10.0 +0.2 −0.4 , consistent with the results of the clustering analysis assuming plausible values for the ratio of stellar to dark matter. Finally, by combining our luminosity function results with those of the stacking analysis we derive estimates of ≃1 × 10 7 and ≃4 × 10 6 M Mpc −3 for the stellar mass density at z≃ 5 and 6, respectively.

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In: Astrophysical Journal , Vol. 591, No. 2 I, 10.07.2003, p. 764-783.

Research output : Contribution to journal › Article › peer-review

T1 - Galaxy Luminosity Functions from Deep Spectroscopic Samples of Rich Clusters

N2 - Using a new spectroscopic sample and methods accounting for spectroscopic sampling fractions that vary in magnitude and surface brightness, we present .R-band galaxy luminosity functions (GLFs) for six nearby galaxy clusters with redshifts of 4000 km s-1 < cz < 20,000 km s-1 and velocity dispersions of 700 km s-1 < σ > 1250 km s -1. In the case of the nearest cluster, A1060, our sample extends to MR = -14 (7 mag below M*), making this the deepest spectroscopic determination of the cluster GLF to date. Our methods also yield composite GLFs for cluster and field galaxies to MR = -17 (M* + 4), including the GLFs of sub- samples of star-forming and quiescent galaxies. The composite GLFs are consistent with Schechter functions (M R* = -21.14-0.17+0.17, α = -1.21-0.07+0.08for the clusters, MR* = -21.15 -0.16+0.16, α = -1.28-0.11 +0.12 for the field). All six cluster samples are individually consistent with the composite GLF down to their respective absolute magnitude limits, but the GLF of the quiescent population in clusters is not universal. There are also significant variations in the GLF of quiescent galaxies between the field and clusters that can be described as a steepening of the faint-end slope. The overall GLF in clusters is consistent with that of field galaxies, except for the most luminous tip, which is enhanced in clusters versus the field. The star formation properties of giant galaxies are more strongly correlated with the environment than those of fainter galaxies.

AB - Using a new spectroscopic sample and methods accounting for spectroscopic sampling fractions that vary in magnitude and surface brightness, we present .R-band galaxy luminosity functions (GLFs) for six nearby galaxy clusters with redshifts of 4000 km s-1 < cz < 20,000 km s-1 and velocity dispersions of 700 km s-1 < σ > 1250 km s -1. In the case of the nearest cluster, A1060, our sample extends to MR = -14 (7 mag below M*), making this the deepest spectroscopic determination of the cluster GLF to date. Our methods also yield composite GLFs for cluster and field galaxies to MR = -17 (M* + 4), including the GLFs of sub- samples of star-forming and quiescent galaxies. The composite GLFs are consistent with Schechter functions (M R* = -21.14-0.17+0.17, α = -1.21-0.07+0.08for the clusters, MR* = -21.15 -0.16+0.16, α = -1.28-0.11 +0.12 for the field). All six cluster samples are individually consistent with the composite GLF down to their respective absolute magnitude limits, but the GLF of the quiescent population in clusters is not universal. There are also significant variations in the GLF of quiescent galaxies between the field and clusters that can be described as a steepening of the faint-end slope. The overall GLF in clusters is consistent with that of field galaxies, except for the most luminous tip, which is enhanced in clusters versus the field. The star formation properties of giant galaxies are more strongly correlated with the environment than those of fainter galaxies.

KW - Galaxies: Clusters: General

KW - Galaxies: Clusters: Individual (a85, a496, a754, a1060, a1631, a3266)

## 5. Summary

LAMOST is one of the most powerful telescopes in terms of accessing the spectra of celestial objects. As a key project associated with LAMOST, LaCoSSPAr, provides the most complete dataset of LEGAS up to now. In this work, we analyzed the redshift incompleteness in the LaCoSSPAr survey quantitatively, and obtained the first measurements of the galaxy LFs in the 0.1 g, 0.1 r, 0.1 i bands using LAMOST spectroscopic data.

We employed both parametric (STY) and non-parametric (SWML) maximum likelihood methods to construct LFs, and found good agreements between the results. Our LFs are comparable to previous works using SDSS data. Thanks to the deeper magnitude limit of LAMOST, compared to results based on SDSS data, we were able to extend the faint end of the LFs by ** mag. Our luminosity densities are consistent with the luminosity density evolution obtained by Loveday et al. 2012.

We divided our sample into emission line galaxies and absorption line galaxies, and derived their LFs separately. Our results show that, in every band, the SWML estimate of emission line galaxy LFs has a Schechter function profile with a steeper faint end slope than that of the total sample. The LFs of absorption line galaxies exhibit an obvious dip near ** mag in all three bands, and cannot be fitted by Schechter functions. On the other hand, double-Gaussian functions, with two characteristic absolute magnitudes and , provide excellent fits to them. This may hint at a bimodality in the population of absorption line galaxies (representing passive galaxies), with the two sub-populations having distinctively different characteristic luminosities (masses): the more massive sub-population has the luminosity of galaxies, while galaxies in the less massive sub-population are ** mag (i.e., ** ) fainter. Investigations using the group catalog of Yang et al. 2007 indicate that the former tend to be the master galaxies in groups while most of the latter are satellites.

This work is based on a small size galaxy sample within a ** 40 deg 2 survey area, which leads to large statistical uncertainties in LF estimates. In the future, we can expect a sample covering a large area when LAMOST finishes its LEGAS survey which can give us better-constrained and unbiased estimates for LFs.

## Characterizing the evolving K -band luminosity function using the UltraVISTA, CANDELS and HUDF surveys

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T1 - Characterizing the evolving K -band luminosity function using the UltraVISTA, CANDELS and HUDF surveys

AU - Mármol-Queraltó, Esther

N2 - We present the results of a new study of the K-band galaxy luminosity function (KLF) at redshifts z ≤ 3.75, based on a nested combination of the UltraVISTA, Cosmic Assembly Near-infrared Deep Legacy Extragalactic Survey and HUDF surveys. The large dynamic range in luminosity spanned by this new data set (3-4 dex over the full redshift range) is sufficient to clearly demonstrate for the first time that the faint-end slope of the KLF at z ≥ 0.25 is relatively steep (-1.3 ≤ α ≤ -1.5 for a single Schechter function), in good agreement with recent theoretical and phenomenological models. Moreover, based on our new data set, we find that a double Schechter function provides a significantly improved description of the KLF at z ≤ 2. At redshifts z ≥ 0.25, the evolution of the KLF is remarkably smooth, with little or no evolution evident at faint (MK ≥ -20.5) or bright magnitudes (MK ≤ -24.5). Instead, the KLF is seen to evolve rapidly at intermediate magnitudes, with the number density of galaxies at MK ≃-23 dropping by a factor of ≃5 over the redshift interval 0.25 ≤ z ≤ 3.75. Motivated by this, we explore a simple description of the evolving KLF based on a double Schechter function with fixed faint-end slopes (α1 = -0.5, α2 = -1.5) and a shared characteristic magnitude (MK^). According to this parametrization, the normalization of the component which dominates the faint end of the KLF remains approximately constant, with φ ^2 decreasing by only a factor of ≃2 between z ≃0 and 3.25. In contrast, the component which dominates the bright end of the KLF at low redshifts evolves dramatically, becoming essentially negligible by z ≃3. Finally, we note that within this parametrization, the observed evolution of MK^ between z ≃0 and 3.25 is entirely consistent with MK^ corresponding to a constant stellar mass of M⋆ ≃5 × 1010 M⊙ at all redshifts.

AB - We present the results of a new study of the K-band galaxy luminosity function (KLF) at redshifts z ≤ 3.75, based on a nested combination of the UltraVISTA, Cosmic Assembly Near-infrared Deep Legacy Extragalactic Survey and HUDF surveys. The large dynamic range in luminosity spanned by this new data set (3-4 dex over the full redshift range) is sufficient to clearly demonstrate for the first time that the faint-end slope of the KLF at z ≥ 0.25 is relatively steep (-1.3 ≤ α ≤ -1.5 for a single Schechter function), in good agreement with recent theoretical and phenomenological models. Moreover, based on our new data set, we find that a double Schechter function provides a significantly improved description of the KLF at z ≤ 2. At redshifts z ≥ 0.25, the evolution of the KLF is remarkably smooth, with little or no evolution evident at faint (MK ≥ -20.5) or bright magnitudes (MK ≤ -24.5). Instead, the KLF is seen to evolve rapidly at intermediate magnitudes, with the number density of galaxies at MK ≃-23 dropping by a factor of ≃5 over the redshift interval 0.25 ≤ z ≤ 3.75. Motivated by this, we explore a simple description of the evolving KLF based on a double Schechter function with fixed faint-end slopes (α1 = -0.5, α2 = -1.5) and a shared characteristic magnitude (MK^). According to this parametrization, the normalization of the component which dominates the faint end of the KLF remains approximately constant, with φ ^2 decreasing by only a factor of ≃2 between z ≃0 and 3.25. In contrast, the component which dominates the bright end of the KLF at low redshifts evolves dramatically, becoming essentially negligible by z ≃3. Finally, we note that within this parametrization, the observed evolution of MK^ between z ≃0 and 3.25 is entirely consistent with MK^ corresponding to a constant stellar mass of M⋆ ≃5 × 1010 M⊙ at all redshifts.

## Abstract

Galaxy and Mass Assembly (GAMA) is a project to study galaxy formation and evolution, combining imaging data from ultraviolet to radio with spectroscopic data from the AAOmega spectrograph on the Anglo-Australian Telescope. Using data from Phase 1 of GAMA, taken over three observing seasons, and correcting for various minor sources of incompleteness, we calculate galaxy luminosity functions (LFs) and their evolution in the ugriz passbands.

At low redshift, z < 0.1, we find that blue galaxies, defined according to a magnitude-dependent but non-evolving colour cut, are reasonably well fitted over a range of more than 10 magnitudes by simple Schechter functions in all bands. Red galaxies, and the combined blue plus red sample, require double power-law Schechter functions to fit a dip in their LF faintwards of the characteristic magnitude M* before a steepening faint end. This upturn is at least partly due to dust-reddened disc galaxies.

We measure the evolution of the galaxy LF over the redshift range 0.002 < z < 0.5 both by using a parametric fit and by measuring binned LFs in redshift slices. The characteristic luminosity L* is found to increase with redshift in all bands, with red galaxies showing stronger luminosity evolution than blue galaxies. The comoving number density of blue galaxies increases with redshift, while that of red galaxies decreases, consistent with prevailing movement from blue cloud to red sequence. As well as being more numerous at higher redshift, blue galaxies also dominate the overall luminosity density beyond redshifts z≃ 0.2. At lower redshifts, the luminosity density is dominated by red galaxies in the riz bands, and by blue galaxies in u and g.