A direct and robust method to observationally constrain the halo mass function via the submillimeter magnification bias: Proof of concept
AAstronomy & Astrophysics manuscript no. main © ESO 2021February 9, 2021
A direct and robust method to observationally constrain the halomass function via the submillimeter magnification bias:Proof of concept
Cueli M. M. , , Bonavera L. , , González-Nuevo J. , , Lapi A. , Departamento de Fisica, Universidad de Oviedo, C. Federico Garcia Lorca 18, 33007 Oviedo, Spain Instituto Universitario de Ciencias y Tecnologías Espaciales de Asturias (ICTEA), C. Independencia 13, 33004 Oviedo, Spain International School for Advanced Studies (SISSA), via Bonomea 265, I-34136 Trieste, Italy Institute for Fundamental Physics of the Universe (IFPU), Via Beirut 2, I-34014 Trieste, ItalyReceived xxx, xxxx; accepted xxx, xxxx
ABSTRACT
Aims.
The main purpose of this work is to provide a proof-of-concept method to derive tabulated observational constraints on thehalo mass function (HMF) by studying the magnification bias e ff ect on high-redshift submillimeter galaxies. Under the assumption ofuniversality, we parametrize the HMF according to two traditional models, namely the Sheth and Tormen (ST) and Tinker fits, deriveposterior distributions for their parameters, and assess their performance in explaining the measured data within the Λ cold darkmatter ( Λ CDM) model. We also study the potential influence of the halo occupation distribution (HOD) parameters in this analysisand discuss two aspects regarding the HMF parametrization, namely its normalization and the possibility of allowing negative valuesfor the parameters.
Methods.
We measure the cross-correlation function between a foreground sample of GAMA galaxies with spectroscopic redshiftsin the range 0 . < z < . . < z < . Results.
Under the assumption that all HMF parameters are positive, the ST fit only seems to fully explain the measurements byforcing the mean number of satellite galaxies in a halo to increase substantially from its prior mean value. The Tinker fit, on the otherhand, provides a robust description of the data without relevant changes in the HOD parameters, but with some dependence on theprior range of two of its parameters. When the normalization condition for the HMF is dropped and we allow negative values of the p parameter in the ST fit, all the involved parameters are better determined, unlike the previous models, thus deriving the most generalHMF constraints. While all the aforementioned cases are in agreement with the traditional fits within the uncertainties, the last onehints at a slightly higher number of halos at intermediate and high masses, raising the important point of the allowed parameter range. Key words.
Galaxies: halos – Submillimeter: galaxies – Gravitational lensing: weak
1. Introduction
Within the Λ cold dark matter ( Λ CDM) model, the hierarchi-cal growth of dark matter perturbations in the early Universeis an essential assumption needed to account for galaxy forma-tion. Due to its high temperature, baryonic matter could not haveformed gravitationally self-bound objects so early had they notbeen subject to gravitational interactions of some other naturethat could overcome thermal energy. The very early freeze-out ofdark matter allowed it to start clustering long before big bang nu-cleosynthesis could take place, providing the necessary potentialwells for baryons to fall into. As a consequence, the relevanceof dark matter halos for the probing of large-scale structure isunquestionable and has motivated the search for a quantitativeunderstanding of their mass distribution.The first attempt at estimating this quantity dates back over40 years. The Press-Schechter formalism (Press & Schechter1974) provided an analytic form for the halo mass function(HMF) based on spherical collapse and initial Gaussian fluctu-ations which laid the groundwork for ever-increasing e ff orts todetermine this quantity as accurately as possible. An alternativederivation of the Press-Schechter HMF was carried out by Bond et al. within the so-called excursion set approach (Bond et al.1991).Up until the end of the 1990s, the Press-Schechter mass func-tion agreed reasonably well with most numerical simulations.However, as their resolutions improved, important deviations be-gan to manifest themselves for halos below and above the so-called characteristic mass scale M ∗ , overestimating the formerand underestimating the latter (Sheth & Tormen 1999). The dy-namics of ellipsoidal collapse were successfully applied to theexcursion set formalism (Sheth et al. 2001) and resulted in thewidely used Sheth and Tormen (ST) parametrization of the HMF,which provides a very good fit when tested against N-body sim-ulations. For instance, using high-resolution simulations for dif-ferent cosmologies, Jenkins et al. (2001) showed that the HMF isfairly well described by the ST fit in the mass range from galax-ies to clusters and from redshift 0 to 5. They suggested an alter-native fit that provides some improvement at the high-mass tailbut cannot be extended beyond said mass range. Moreover, theyshowed that the mass function could be expressed in a universalform when appropriately rescaled, meaning that the same ana-lytical form and parameters could be used for di ff erent redshiftsand cosmologies. Article number, page 1 of 19 a r X i v : . [ a s t r o - ph . C O ] F e b & A proofs: manuscript no. main
Subsequently, a variety of fits to the HMF based on N-bodysimulations for di ff erent mass and redshift ranges were pro-posed, some of them confirming universality within a few per-cent (Reed et al. 2003, 2007; Warren et al. 2006), others quanti-fying small departures from it (Tinker et al. 2008; Crocce et al.2010; Courtin et al. 2011; Watson et al. 2013). The questionof universality is indeed a lenghty matter to discuss. However,as shown by Despali et al. (2016), departures from universalitycould be associated with the way halos are defined (see Knebeet al. 2013, for a summary of di ff erent halo finding methodologyin simulations).In essence, two common ways to obtain a halo catalog froman N-body simulation are friends-of-friends (FoF) algorithms(Davis et al. 1985) and spherical overdensity (SO) algorithms(Lacey & Cole 1994). Since there is not a universal definition ofa dark matter halo, both methods have benefits and drawbacksand departures from a universal behavior have been found for thetwo kinds of algorithms. However, Despali et al. (2016) showedthat, if SO-defined halos are defined using the virial overdensity(as opposed to other common criteria) and the mass function isexpressed in terms of a parameter accounting for it, universalitycan then be retrieved to within a few percent. Their results werein agreement with those of Courtin et al. (2011), who concludedthat deviations from universality could be accounted for if oneincorporates the redshift and cosmology dependence of the lin-ear collapse threshold and the virialization overdensity.Moreover, physical processes associated with baryons suchas radiative cooling, star formation or feedback from supernovaeand active galactic nuclei (AGN) have been shown to producenon-negligible modifications in the HMF, the e ff ects being how-ever sensitive to the modeling of the baryonic component. In-deed, Cui et al. (2012b) compared a dark-matter-only simula-tion with hydrodynamical counterparts without feedback fromAGN, obtaining an increase in the number density of high-massobjects. However, the addition of AGN feedback by Cui et al.(2014) causes the opposite e ff ect, a trend that has been con-firmed using higher-resolution simulations, where a general de-crease in the HMF is reported, more noticeable at low massesand redshifts (Sawala et al. 2013; Bocquet et al. 2015; Castroet al. 2020). Lastly, there could be physics beyond the StandardModel with a non-negligible e ff ect on structure formation. In-deed, some authors have studied the inclusion of massive neutri-nos (Costanzi et al. 2013) or the e ff ect of an interaction betweendark energy and cold dark matter (Cui et al. 2012a). An e ff ort to-ward observational constraints on the HMF could therefore pro-vide some insight into these questions in addition to a validationof the results from N-body simulations.Although some recent studies have provided observationalmethods to determine the HMF (Castro et al. 2016; Sonnenfeldet al. 2019; Li et al. 2019), all of them su ff er from the uncertain-ties that arise when observational properties of cosmic structuresare linked to the underlying halo mass. Our goal is not to assignhalo masses to galaxies (or any of their observational properties)and empirically construct the HMF from there. In other words,we do not make use of a mass-richness relation, nor do we aim atobtaining one. We propose instead the use of an observable that,given its direct dependence on the halo mass and clustering of theforeground lenses, provides a robust measurement of the HMF.This physical quantity is the foreground-background galaxy an-gular cross-correlation function, together with background sam-ples of submillimeter galaxies, which we argue to be promisingcandidates for cosmological analysis through the magnificationbias e ff ect (González-Nuevo et al. 2017; Bonavera et al. 2019; Bonavera, L. et al. 2020; Gonzalez-Nuevo et al. 2020). We termthis observable the submillimeter galaxy magnification bias.The aim of this paper is therefore to study two di ff erent HMFuniversal fits (namely the ST and Tinker models) with the aimof constraining their parameters and providing bounds to theHMF itself. This will be done by computing the angular cross-correlation function between two source samples with nonover-lapping redshift distributions and fitting the result through aMarkov chain Monte Carlo (MCMC) algorithm to its theoret-ical prediction within the halo model formalism. Although theconstrained HMF is in principle only representative of the galax-ies producing the lensing e ff ect, the comparison of the auto- andcross-correlation results by Bonavera, L. et al. (2020) shows thatthe lens properties are indistinguishable from the galaxy parentpopulation.The paper has been structured as follows. Section 2 providesa theoretical description of the physical situation. The usual for-malism describing the HMF is presented, as well as a descriptionof the chosen parametrizations. We also discuss the halo modelprediction for our observable, the foreground-background angu-lar cross-correlation function. Section 3 describes the method-ology followed in our work process. We describe in detail thebackground and foreground galaxy samples as well as the cross-correlation measurement method. The MCMC algorithm usedto fit the data to the model is presented, as well as the di ff erentruns we perform. Section 4 provides a discussion of the mainresults we obtained for the ST and Tinker fits and Section 5 de-tails some further studies on the non-normalization of the HMFand the non-positivity of its parameters. The values for the z =
2. Theoretical basis
The common strategy when studying the statistical propertiesof mass fluctuations is to consider the overdensity field linearlyextrapolated to the present, δ ( x ), and smooth it with a filter ofscale R , that is, δ R ( x ) ≡ (cid:90) d x (cid:48) δ ( x (cid:48) ) W ( x + x (cid:48) ; R ) = (cid:90) d k ˆ W ( k R ) δ , k e i k · x , where ˆ W ( kR ) is the Fourier transform of the filter function W ( x ; R ), which, for the case of a top-hat in real space is givenbyˆ W ( kR ) = kR − kR cos kR ]( kR ) . If we associate a mass M with a comoving scale R via M = π R ρ , where ρ is the mean matter density of the Universe at presenttime, we can interchangeably characterize a filter by its mass orlength scale. The mass variance of the filtered linear overdensityfield is thus σ ( M ) ≡ (cid:104) [ δ R ( x )] (cid:105) = π (cid:90) ∞ k P ( k ) ˆ W ( kR ) dk , where P ( k ) is the linear matter power spectrum at redshift z = Article number, page 2 of 19ueli M. M. et al.: Halo Mass Function measurement with Magnification Bias
Although its physical definition is clear, the mathematicalparametrization of the HMF varies widely in the literature, socare must be taken when comparing results and di ff erent models.The (di ff erential) HMF n ( M , z ) is the comoving number densityof halos at a given redshift per unit mass, that is, n ( M , z ) dM is the comoving number density of halos of mass in the range[ M , M + dM ] at redshift z .One common way to parametrize it, which arises naturallyfrom the excurstion set formalism, is n ( M , z ) = ρ M f ( ν, z ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ ln ν ( M , z ) ∂ ln M (cid:12)(cid:12)(cid:12)(cid:12) , (1)where ρ is the comoving mean matter density of the Universeand ν ( M , z ) ≡ (cid:20) ˆ δ c ( z ) σ ( M , z ) (cid:21) , with σ ( M , z ) ≡ D ( z ) σ ( M ), where D ( z ) is the linear growthfactor for a Λ CDM universe, and ˆ δ c ( z ) is the linear critical over-density at redshift z for a region to collapse into a halo at thatsame redshift according to the spherical collapse model . It isclear that ν depends on redshift and cosmology. However, if thefunction f ( ν, z ) is the same for all redshifts and cosmologies, thatis, if f ( ν, z ) ≡ f ( ν ) for all cosmologies, the mass function is saidto be universal.For instance, the ST and Tinker z = f ST ( ν ) = A S (cid:114) a S ν π (cid:20) + (cid:18) a S ν (cid:19) p S (cid:21) e − a S ν/ (2) f T ( ν, z = = A T (cid:20) + (cid:18) B T √ ν (cid:19) p T (cid:21) e − C T ν , (3)where A S = . a S = . p S = . , and A T = . B T = . δ (0) C T = . δ (0) p T = . . It should however be noted that some authors parametrize theHMF solely in terms of σ ( M , z ), and care should be taken whenrelating the parameters from each definition.Furthermore, Sheth & Tormen (Sheth & Tormen 1999) im-posed a normalization condition, which in our parametrizationreads (cid:90) ∞ f ( ν ) M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ log ν∂ log M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dM = a S and p S ), since the normalization param-eter ( A S ) was fixed by (4), yielding A S ( p ) = (cid:104) + − p √ π Γ (1 / − p ) (cid:105) − It should be noted that other authors define ν ( M , z ) without thesquare. The redshift dependence of ˆ δ c ( z ) is weak and usually neglected, thatis, ˆ δ c ( z ) ≈ .
686 for all z . However, we have taken it into account viathe fitting function from Kitayama & Suto (1996). and the condition p < /
2. Most authors have fit these models ortheir own to numerical simulations without imposing condition(4), thus having an extra parameter. Although we find it morecoherent with our halo model description to employ it in thiswork, we will only do so for the ST fit, since the Tinker fit asshown in equation (3) cannot be normalized in this manner .Given the fact that they are the most important and mostwidely used models, our analysis focuses on these two univer-sal fits for the HMF: namely, a two-parameter ST fit, f ( ν ; a , p ) = A ( p ) (cid:114) a ν π (cid:20) + (cid:18) a ν (cid:19) p (cid:21) e − a ν/ , (5)and a four-parameter Tinker-like fit, f ( ν ; A , B , C , p ) = A (cid:20) + (cid:18) B √ ν (cid:19) p (cid:21) e − C ν . (6) The standard halo model considers that the matter density fieldat a point in space can be thought of as a sum over the densityprofiles of halos. In this context, the galaxy-dark matter cross-power spectrum can be parametrized by P g-dm ( k , z ) = P ( k , z ) + P ( k , z ) , where P is the so-called 1-halo term, accounting for contri-butions within the same halo and P is the so-called 2-haloterm, accounting for contributions among di ff erent halos.These two quantities can be further expressed (Cooray &Sheth 2002) as P ( k , z ) = (cid:90) ∞ dM M n ( M , z )¯ ρ ( z ) (cid:104) N g (cid:105) M ¯ n g ( z ) | u dm ( k , z | M ) || u g ( k , z | M ) | p − (7) P ( k , z ) = P ( k , z ) (cid:104) (cid:90) ∞ dM M n ( M , z )¯ ρ ( z ) b ( M , z ) u dm ( k , z | M ) (cid:105) ·· (cid:104) (cid:90) ∞ dM n ( M , z ) b ( M , z ) (cid:104) N g (cid:105) M ¯ n g ( z ) u g ( k , z | M ) (cid:105) , (8)where b ( M , z ) is the linear deterministic halo bias, ¯ ρ ( z ) is themean matter density of the Universe, ¯ n g ( z ) is the mean numberdensity of galaxies, (cid:104) N (cid:105) M is the mean number of galaxies in ahalo of mass M and u ( k , z | M ) is the normalized Fourier trans-form of the matter distribution (be it dark matter or galaxies).Some comments should be made concerning (7) and (8). Firstly,it is a reasonable approximation (Sheth & Diaferio 2001) to setthe Fourier transform of the galaxy distribution to that of darkmatter. Secondly, the mean number of galaxies within a halo ofmass M is split into a contribution from central galaxies anda contribution from satellite galaxies, parametrizing it in termsof the halo occupation distribution (HOD) parameters α , M min and M , following Zehavi et al. (2005) and Zheng et al. (2005).Lastly, the exponent p should be set to 1 for central galaxies andto 2 for satellites (Cooray & Sheth 2002). More detailed infor-mation concerning the computation of all these quantities can befound in Appendix A. The results one would obtain using the lesser-known normalizableTinker fit are qualitatively similar. Article number, page 3 of 19 & A proofs: manuscript no. main
This cross-correlation between galaxies and dark matter canbe probed via the weak lensing tangential shear-galaxy corre-lation (Bartelmann & Schneider 2001) or via the foreground-background source correlation function. This work exploits thelatter method, which is based on the fact that foreground sourcestrace the mass density field a ff ecting the number counts of back-ground sources.Indeed, in the presence of lensing, number counts observedin direction θ and exceeding a flux S are modified according to(Bartelmann & Schneider 2001) N S ( θ ) = N S µ β − ( θ ) , where N s denotes the intrinsic source number counts exceedingflux S , β is their logarithmic slope and µ ( θ ) is the magnificationfactor in direction θ . In the weak-lensing limit, µ ( θ ) ≈ + κ ( θ ),where κ ( θ ) is the convergence. As a consequence, the fluctua-tions in the background number counts, which are due to magni-fication bias, can be written as δ N b ( θ ) ≡ N b ( θ )¯ N b − = µ β − ( θ ) − ≈ β − κ ( θ ) . Concerning the foreground sources, since they are supposed totrace the density field, the fluctuations in their number counts aredue to pure clustering, that is, δ N f ( θ ) ≡ (cid:90) χ H d χ g f ( χ ) δ g ( θ , χ ) , where χ H denotes the comoving radial distance to the horizonand g f ( χ ) is the radial distribution of foreground sources.The angular cross-correlation between the foreground andbackground sources is then given by (Cooray & Sheth 2002) w f b ( θ ) ≡ (cid:104) δ N f ( ϕ ) δ N b ( ϕ + θ ) (cid:105) == β − (cid:90) χ H d χ g f ( χ ) ˆ W lens ( χ ) ·· (cid:90) ∞ dk k π P g-dm ( k , z ) J ( kd A θ ) , (9)where θ = | θ | , d A ( χ ) is the comoving angular diameter distance, J is the zeroth-order Bessel function of the first kind andˆ W lens ( χ ) = H c a ( χ ) E ( χ ) (cid:90) χ H χ d χ (cid:48) d A ( χ ) d A ( χ (cid:48) − χ ) d A ( χ (cid:48) ) g b ( χ (cid:48) ) . In terms of redshift, (9) becomes w f b ( θ ) = β − (cid:90) ∞ dz χ ( z ) n f ( z ) W lens ( z ) ·· (cid:90) ∞ dl l π P g-dm ( l /χ ( z ) , z ) J ( l θ ) , (10)where we have defined l ≡ kd A ( z ), W lens ( z ) = H c (cid:20) E ( z )1 + z (cid:21) (cid:90) ∞ z dz (cid:48) χ ( z ) χ ( z (cid:48) − z ) χ ( z (cid:48) ) n b ( z (cid:48) ) , and n b ( z ) ( n f ( z )) is the unit-normalized redshift distribution ofthe background (foreground) sources. β is the logarithmic slopeof the background source number counts and it is commonlyfixed to 3 for submillimeter galaxies (Lapi et al. 2011, 2012; Caiet al. 2013; Bianchini et al. 2015, 2016; González-Nuevo et al.2017; Bonavera et al. 2019). In this model, β provides a generalnormalization whose possible changes are almost fully balanced by variations of M min (e.g., a ≈
15% increase in β corresponds toa log M min reduction of ≈ P g-dm . Therefore, we used this observable to con-strain such parameters. Moreover, aside from the HMF parame-ters, the cross-correlation function depends on both the cosmol-ogy and the HOD parameters. Throughout our analysis, whichassumes universality of the mass function, we keep the cosmol-ogy fixed to Planck’s (Planck Collaboration et al. 2020) but aimto discuss the role of the HOD parameters by also including themin the MCMC analysis in some cases, as will be described inSection 3.3.
3. Work methodology
The background and foreground samples have been selectedas described in detail in González-Nuevo et al. (2017) andBonavera et al. (2019). The foreground sources consist of a sam-ple of the GAMA II (Driver et al. 2011; Baldry et al. 2010, 2014;Liske et al. 2015) spectroscopic survey, with 0 . < z < . ∼ z med = . ∼
147 deg ,and the part of the South Galactic Pole (SGP) that overlapswith the foreground sample ( ∼
60 deg ). To ensure no overlapin the redshift distributions of lenses and background sources,we selected only background sources with photometric redshift1 . < z < .
0. The redshift estimation is described in González-Nuevo et al. (2017) and Bonavera et al. (2019). After performingsuch a selection, we end up with 57930 galaxies, approximately24% of the initial sample.It should be stressed that both the H-ATLAS and the GAMAII surveys were carried out to maximize the common area cov-erage. Both surveys covered the three equatorial regions at 9,12, and 14.5 h (referred to as G09, G12 and G15, respectively)and the H-ATLAS SGP was also partially observed by GAMAII. Thus, the resulting common area is of about ∼ , sur-veyed down to a limit of r (cid:39) . p ( z | W ) of thegalaxies selected by our window function and takes into ac-count the e ff ect of random errors in photometric redshifts, as inGonzález-Nuevo et al. (2017); Bonavera, L. et al. (2020). The H-ATLAS survey is divided into five di ff erent fields: threeGAMA fields in the ecliptic (9h, 12h, 15h) and two in the Northand South Galactic Poles (NGP and SGP). The H-ATLAS scan-ning strategy produced a characteristic repeated diamond shapein most of their fields that was named "Tiles." The area of eachtile is ∼
16 deg . In order to maintain a regular shape for the tiles,a small overlap among such regions is needed, typically lowerthan 20% of their area. Considering the common area betweenforeground and background surveys, we have 16 di ff erent tiles, Article number, page 4 of 19ueli M. M. et al.: Halo Mass Function measurement with Magnification Bias
Fig. 1.
Description of the surveyed areas and tiling scheme. Top panel:Mollweide view of the sky distribution of the G09, G12, G15 and SGPregions in equatorial coordinates. Bottom panel: Representation of theTiles scheme for G09, the pattern being similar for the other regions. redshift0.00.20.40.60.81.0 N o r m a li z e d nu m b e r o f G a l a x i e s BackgroundForeground
Fig. 2.
Normalized redshift distribution of the background H-ATLASsample (red) and the foreground GAMA one (blue). which helps diminish the e ff ects of cosmic variance. In particu-lar, Figure 1 (bottom panel) illustrates the diamond-shaped Tilesscheme in the G09 region. The other considered regions have ananalogous pattern.In this work, we use the angular cross-correlation functionmeasured by Gonzalez-Nuevo et al. (2020) using the Tiles areafor the same spectroscopic sample. We chose this particular set of measurements based on the analysis performed by Gonzalez-Nuevo et al. (2020), which studied the large-scale biases for dif-ferent samples and tiling schemes. The measurements from thespectroscopic sample are only a ff ected by the so called integralconstraint (IC; Roche & Eales 1999), but the correction for thechosen tiling scheme is almost negligible (IC = · − ). It a ff ectsonly marginally the measurements at the largest angular scales.For completeness, we summarize here the pipeline used toestimate the measured cross-correlation function (black circlesin Figures 3 and 4). As described in detail in González-Nuevoet al. (2017), we used a modified version of the Landy & Szalay(1993) estimator (Herranz 2001):˜ w ( θ ) = D f D b ( θ ) − D f R b ( θ ) − D b R f ( θ ) + R f R b ( θ )R f R b ( θ ) , (11)where D f D b , D f R b , D b R f and R f R b are the normal-ized foreground-background, foreground-random, background-random and random-random pair counts for a given separation θ . The cross-correlation is computed for each tile and its statis-tical error is obtained by averaging over 10 di ff erent realizations(using di ff erent random catalogs each time). The final cross-correlation measurement for a given angular separation bin cor-responds to the mean value of the cross-correlation functions es-timated for every tile. The associated uncertainty is the standarderror of the mean, that is, σ µ = σ/ √ n , with σ the standard de-viation of the population and n the number of independent areas(each selected region can be assumed to be statistically indepen-dent due to the small overlap between the tiles). The estimation of the HMF parameters will be carried outthrough an MCMC method using the open source emcee soft-ware package (Foreman-Mackey et al. 2013), a Python imple-mentation of the Goodman & Weare a ffi ne invariant MCMC en-semble sampler (Goodman & Weare 2010).As described in Section 2.1, we will adopt two di ff erent fitsfor the HMF. Assuming Gaussian errors, the log-likelihood func-tion takes the formlog L ( θ , . . . , θ n ; { p j } j ) = − n (cid:88) i = (cid:20) log 2 πσ i ++ [ w ( θ i ; { p j } j ) − ˜ w ( θ i )] σ i (cid:21) , where { p j } j is the set of HMF parameters, σ i is the error in the i th measurement and w ( θ i ) and ˜ w ( θ i ) are the theoretical and mea-sured value of the cross-correlation at angular scale θ i .With regard to the choice of priors, we consider it a delicateissue. We opted for uniform distributions for all HMF parame-ters, but the range of these intervals is not obvious at first sight.Furthermore, while some parameters are mathematically forcedto be nonnegative ( a in the ST fit and A , B and C in Tin-ker’s), others could a priori be allowed to be negative ( p in theST fit and p in Tinker’s). Traditional methods to determine theHMF imply using an optimizer to find the single tuple of param-eter values that best fits the simulations through a χ analysisand provide no information about whether negative values wereallowed in the search. In fact, we have found no mention whatso-ever to the potential non-positivity of any of the parameters. As aconsequence, for example, while previous simulation-based fitshave yielded a value of p ≈ . Article number, page 5 of 19 & A proofs: manuscript no. main there is a physically motivated reason to exclude negative valuesfrom its priors. As a consequence, even though the main caseswe have performed assume all HMF parameters are positive, wealso decided to consider the non-negativity of p , as we will dis-cuss in Section 5 together with the possibility of not applying thenormalization condition (4) to the ST fit.Concerning the HOD parameters, for the runs in which wekeep them fixed, we selected the following values based on theBonavera, L. et al. (2020) results: α = . M min = . M = . , where M min and M are expressed in M (cid:12) / h , while the Gaussiandistributions for the runs in which we include them are extractedfrom recent literature, as described in Bonavera, L. et al. (2020).In particular, they are based on Sifón et al. (2015) (making useof the recipe by Pantoni et al. (2019) to switch from stellar mass M (cid:63) to halo mass M h ) for M min and M (in agreement with Aversaet al. (2015) for M min ), and on Viola et al. (2015) for α .Therefore, the main MCMC runs, along with their respectiveprior distributions are the following: Run 1 analyzes the two-parameter ST fit with uniform priors, U , on a and p and fixedHOD parameters, that is, a ∼U [0 , p ∼U [0 , . . Run 2 studies the two-parameter ST fit with uniform priors on a and p and Gaussian, N , priors on the HOD parameters: a ∼U [0 , p ∼U [0 , . α ∼N (0 . , . M min ∼N (12 . , .
1) log M ∼N (13 . , . . Run 3 analyzes the four-parameter Tinker-like fit with uniformpriors on A , B , C and p and fixed HOD parameters: A ∼U [0 , B ∼U [0 , C ∼U [0 , p ∼U [0 , . Run 4 studies the four-parameter Tinker-like fit with uniformpriors on A , B , C and p and Gaussian priors on the HODparameters: A ∼U [0 , B ∼U [0 , C ∼U [0 , p ∼U [0 , α ∼N (0 . , .
15) log M min ∼N (12 . , . M ∼N (13 . , . p and not normalizingthe ST fit. We will describe them in detail in Section 5.
4. Main results
Table 1 shows the results from the first run of the MCMC al-gorithm, namely the peaks, means and narrowest 68% and 95%credible intervals of the marginalized one-dimensional distribu-tions. Figure B.3 (in blue) shows the corner plot with the one-dimensional and two-dimensional posterior distributions of bothparameters. While a presents a constraining marginalized pos-terior with a clear peak at a = . p can only be assignedupper bounds, namely p < .
17 and p < .
31 at 68% and 95%credibility, respectively. For our fixed choice of HOD parame-ters, the traditional parameter values of the ST fit are compat-ible given the wide uncertainties in the posterior distributions,although the marginal mean value of p hints at smaller values. The upper-left panel of Figure 3 shows the resulting cross-correlation function when the full posterior distribution is sam-pled (solid red lines), along with the lines corresponding tothe traditional ST fit (dotted black) and the "mimic" marginalpeak values (dashed light red), corresponding to a = .
88 and p = .
20. Since the marginalized posterior of p does not dis-play a peak, the latter line has been chosen so that it providesa reasonable fit and serves only as a visual aid, hence the wordmimic. As can be seen from its comparison to the measured data(black circles), there is more probability density toward smallercross-correlation values at angular scales θ > z = M > . M (cid:12) / h ). At low masses,the HMF is well-constrained, whereas our treatment provides in-teresting upper bounds for the HMF at the aforementioned largescale.As expected, if we now introduce the HOD parameters in theMCMC analysis (with Gaussian priors as discussed in Section3.3), the results, which we present in Table 2 and Figure B.3 (inred) vary quantitatively. With respect to the fixed HOD case, the a and p marginalized distributions present some di ff erences.In particular, both the peak and the mean of the a distributionare displaced to the right to values of 1.58 and 1.88, respectively.Moreover, the p distribution, while still right-skewed, becomesmainly concave with a mode of p = .
07, as opposed to the firstrun. Concerning the HOD parameters, whereas the marginalizedposterior distributions of α and log M min hardly deviate fromtheir priors (with peaks at 0.94 and 12.48, respectively), that oflog M does substantially, with a clear peak at 12.74, more than3 σ away from its prior mean.The lower-left and lower-right panels of Figure 3 show thecorresponding posterior-sampled cross-correlation and z = M > . M (cid:12) / h .In summary, when the HOD parameters are fixed, the two-parameter ST fit is not able to fully explain the cross-correlationsignal at angular scales θ > a would help in this direction (as a parameter sensitivity anal-ysis shows), this would provide a poorer general fit to the databecause it would cause the small-scale cross-correlation, whichis better constrained by observations, to decrease. It should benoted that the role of p is not as significant in this argumentgiven the little room for manoeuvre (prior-wise) at its disposal.However, the situation di ff ers for the case in which the HODparameters are introduced in the MCMC analysis. As describedin Bonavera, L. et al. (2020), a decrease in parameter M mainlycauses an increase in the cross-correlation function, this e ff ectbeing more noticeable at angular scales between 1 and 4 arcminand almost negligible at larger scales. As a consequence, a cannow be increased in order to accommodate the data without im-poverishing the fit by demanding that M be decreased, that is,that there be more satellite galaxies. The sampling of the fullposterior (lower-left panel of Figure 3) reflects this situation Article number, page 6 of 19ueli M. M. et al.: Halo Mass Function measurement with Magnification Bias
Table 1.
Parameter priors and marginalized posterior peaks, means, 68%, and 95% credible intervals for run 1 of the MCMC algorithm, that is, atwo-parameter ST fit with positive p and fixed HOD values. Parameter Prior Peak Mean 68% CI 95% CI a U [0,10] 0 .
88 1 .
29 [0 . , .
53] [0 . , . p U [0,0.50] − .
13 [ − , .
17] [ − , . Table 2.
Parameter priors and marginalized posterior peaks, means, 68%, and 95% credible intervals for run 2 of the MCMC algorithm, that is, atwo-parameter ST fit with positive p and Gaussian priors on the HOD parameters. Parameters M min and M are expressed in M (cid:12) / h . Parameter Prior Peak Mean 68% CI 95% CI a U [0,10] 1 .
58 1 .
88 [0 . , .
42] [0 . , . p U [0 , .
50] 0 .
07 0 .
15 [ − , .
20] [ − , . α N [0 . , .
15] 0 .
94 0 .
95 [0 . , .
09] [0 . , . M min N [12 . , .
10] 12 .
48 12 .
46 [12 . , .
57] [12 . , . M N [13.95,0.3] 12 .
74 13 .
03 [12 . , .
26] [12 . , . Fig. 3.
Full posterior sampling (solid lines) and mimic marginal mode values (dashed lines) for runs 1 (in red) and 2 (in blue) of the MCMCalgorithm, that is, a two-parameter ST fit with fixed and Gaussian priors on the HOD values, respectively. The left panels show the cross-correlationfunction (the black filled circles being our measurements), while the right panels display the z = clearly. It should also be mentioned that larger values of M min have an increasing e ff ect on all scales, again to the detriment ofsmaller-scale values and thus diminishing its influence.Although the posterior distribution for M is physically rea-sonable, it di ff ers substantially from those obtained by Bonavera,L. et al. (2020) or Gonzalez-Nuevo et al. (2020) using the tradi-tional ST fit, which should serve as additional motivation for theanalysis in Section 5. In any event, as compared to the traditionalone, the ST fit as described in this section hints at a smaller num- ber of halos, especially for the largest masses, an e ff ect that ismainly driven by the cross-correlation measurements at θ > Table 3 and Figure B.4 (in blue) show the corresponding re-sults for the third run of the MCMC algorithm: a four-parameterTinker fit with fixed HOD values. Whereas A and C show Article number, page 7 of 19 & A proofs: manuscript no. main
Table 3.
Parameter priors and marginalized posterior peaks, means, 68%, and 95% credible intervals for run 3 of the MCMC algorithm, that is, afour-parameter Tinker fit and fixed HOD values.
Parameter Prior Peak Mean 68% CI 95% CI A U [0,5] 0.15 0 .
20 [0 . , .
29] [0 . , . B U [0,5] 0 .
82 1 .
66 [ − , .
96] [ − , − ] C U [0,5] 0.56 0 .
78 [0 . , .
00] [0 . , . p U [0,5] − − [ − , − ] [ − , − ] Table 4.
Parameter priors and marginalized posterior peaks, means, 68%, and 95% credible intervals for run 4 of the MCMC algorithm, that is, afour-parameter Tinker fit with Gaussian priors on the HOD parameters. Parameters M min and M are expressed in M (cid:12) / h . Parameter Prior Peak Mean 68% CI 95% CI A U [0,5] 0 .
16 0 .
20 [0 . , .
28] [ − , . B U [0,5] 0 .
91 1 .
71 [ − , .
04] [ − , − ] C U [0,5] 0 .
63 0 .
85 [0.27,1.11] [0.01,1.80] p U [0,5] − − [ − , − ] [ − , − ] α N [0.92,0.15] 0 .
89 0 .
91 [0.77,1.06] [0.62,1.21]log M min N [12.40,0.10] 12 .
43 12 .
42 [12.32,12.42] [12.23,12.62]log M N [13.95,0.30] 12 .
91 13 .
56 [12.58,14.20] [12.48,14.97]
Fig. 4.
Full posterior sampling (solid lines) and mimic marginal mode values (dashed lines) for runs 3 (in red) and 4 (in blue) of the MCMCalgorithm, that is, a four-parameter Tinker fit with fixed and Gaussian priors on the HOD values, respectively. The left panels show the cross-correlation function (the black filled circles being our measurements), while the right panels display the z = z = constraining marginalized posterior distributions with peaks at A = .
15 and C = . B and p remain unconstrained, theformer hinting toward a peak value of 0.91 and the latter be-ing completely prior-dominated. This issue is not resolved bywidening the priors (even considering negative values for p ) and therefore compromises the reliability of the statistical con-clusions, since the credible intervals on the HMF will eventuallydepend on the prior range of B and p . However, we suspectthat the derived HMF is not too sensitive to this issue, althoughwe consider it delicate and have not gone further into a quanti- Article number, page 8 of 19ueli M. M. et al.: Halo Mass Function measurement with Magnification Bias tative analysis. It should also be added that parameters A and C are very robust to the widening or narrowing of said priordistributions.The red lines in Figure 4 show the posterior-sampled cross-correlation function (upper-left panel) and z = A = . B = . C = .
56 and p = .
50. As opposed to the ST fit, the vast majority of the sam-pled cross-correlation lines seem to properly explain the large-scale data, while the traditional Tinker fit underestimates themeasurements above 1 arcmin. As can be seen from the upper-right panel of Figure 4, the z = ff at high masses, although less pronounced.However, the recovered HMF shows a wider spread for low andintermediate masses, M < . M (cid:12) / h , as compared to the pre-vious subsection.We now turn to analyzing the introduction of the HOD pa-rameters in the MCMC analysis. Table 4 and Figure B.4 (in red)show the corresponding results. In this case, the marginalizeddistributions of the HMF parameters practically show no di ff er-ence when compared to the fixed HOD case (Table 3 and FigureB.4 in blue). Only the peak and the mean of the C distributionare visibly displaced to the right to values of 0 .
63 and 0 .
85, re-spectively. Regarding the HOD parameters, the situation resem-bles that of the previous case up to a certain point; the marginal-ized posterior distributions of α and log M min hardly move awayfrom their priors (with peaks at 0.89 and 12.43, respectively),while that of log M does (to the left), but in this case appearsto maintain a high probability region toward values around theprior.The lower panels of Figure 4 show the posterior-sampled(solid blue lines) cross-correlation function (lower-left panel)and z = A = . B = . C = .
63 and p = .
50. Since the data was al-ready properly explained by the fixed HOD case, we only ob-serve an expected increase in the spread of the HMF, mainly inthe form of higher upper bounds at every mass, and especiallyfor M > . M (cid:12) / h .In summary, while the Tinker-like fit shows a more robustbehavior with respect to the HOD parameters than the ST fit (dueto the fact that, unlike the latter, it can properly explain the cross-correlation signal without changes in them), the statistical resultsconcerning the HMF depend on the prior range of two of itsparameters ( B and p ), which cannot be bounded. Although wedo not suspect this is a major issue, it should nevertheless beclarified that the credible intervals derived for the HMF in runs3 and 4 have assumed the prior ranges described in section 3.3.
5. Further discussion
As discussed in Section 2.1, the normalization condition im-posed on the ST fit assumes all mass in the Universe is boundup in halos. Although the present work has incorporated this as-sumption on the grounds of coherence with the underlying halomodel, it is of interest to analyze the situation when the A pa-rameter is left free in the MCMC analysis. In this scenario, p could, in principle, take values that are larger than 0.5 (or evennegative; see next subsection) but, for the sake of comparison,we will keep the priors on a and p the same as in Section 4.1.The results are displayed in Table B.1 and Figure B.1 (inred). Parameter A shows a well-constrained marginalized dis-tribution with a peak at A = .
59, while that of a is narrowerthan that of run 1 and barely displaced to the right, with a peakat a = .
93. It should be noted that parameter p is now uncon-strained on both sides, hinting again at a preference for negativevalues. Figure 5 further compares the posterior sampling of run5 with that of run 1. It permits us to conclude that, while keep-ing p positive and smaller than 0.5, the introduction of A asa free parameter in the ST fit allows the cross-correlation func-tion to take larger values for θ > p to be [0 , . ff erences with respectto run 1 and to serve as a link between section 4 and the nextsubsection. Another point regarding the HMF parameters was raised in Sec-tion 3.3. We find no mathematical reason why parameters p and p cannot take negative values. As to a possible physical expla-nation, an analysis of the excursion set formalism or of otherworks that derive a HMF template purely from physical argu-ments still yields no reason why this cannot be the case. Wewould like to emphasize that the usual methods consist in find-ing the single tuple of parameters that provides the best fit tothe simulation in question, but we have found no further detailsabout the range of parameter values that is used in said searches(are negative values explored?). Since prior distributions are ofparamount importance in Bayesian statistics, we deem this a del-icate issue. As a consequence, we decided to analyze the pos-sibility of allowing parameter p in the ST fit to take negativevalues, both in the case where the normalization condition is ap-plied (two-parameter fit) and in the case where it is not (three-parameter fit). The results for both cases are shown in FigureB.2 and the statistical results are summarized in Tables B.2 and5, respectively.In the two-parameter case (Figure B.2 in blue), we now ob-serve clear peaks in both parameters, at values a = .
46 and p = − .
43, and a strong degeneracy direction that produces theappearance of long tails in both one-dimensional marginalizeddistributions. Figure 6 shows the posterior-sampled cross corre-lation function (left panel) and z = p helps to accountfor the high correlation at large angular scales ( θ > ffi cient as varying the HOD parameters orthe normalization parameter A in the MCMC analysis, as showsthe fainter blue line density in the cross-correlation sampling.Moreover, the large degeneracy between a and p translates toa much wider spread in the HMF, which is clearly visible at thesmallest and largest mass values.On the other hand, the three-parameter case (Table 5 and Fig-ure B.2 in red) presents a very symmetric marginalized posteriordistribution for p with a clear peak at p = − .
25. This pa-rameter shows again a degeneracy with a , although this doesnot originate one-sided tails in this case. Parameter A peaks at Article number, page 9 of 19 & A proofs: manuscript no. main
Fig. 5.
Full posterior sampling (solid lines) and (mimic) marginal mode values (dashed lines) from runs 1 (in red) and 5 (in blue) of the MCMCalgorithm, that is, a two-parameter fixed HOD and a three-parameter fixed HOD ST fit, respectively. Parameter p is assumed to be in the range[0 , . z = Fig. 6.
Full posterior sampling (solid lines) and (mimic) marginal mode values (dashed lines) from runs 1 (in red) and 6 (in blue) of the MCMCalgorithm, that is, a two-parameter fixed HOD ST fit with p > p allowed to be negative, respectively. The left panel shows the cross-correlation function (the black filled circles being our measurements), while the right one displays the z = Table 5.
Parameter priors, marginalized posterior peaks, means, 68%, and 95% credible intervals for run 7 of the MCMC algorithm, that is, athree-parameter ST fit, p allowed to be negative and fixed HOD values. Parameter Prior Peak Mean 68% CI 95% CI A U [0,5] 0 .
66 0 .
55 [0 . , .
87] [ − , − ] a U [0,10] 1 .
29 1 .
30 [0 . , .
55] [0 . , . p U [-10,10] − . − .
15 [ − . , .
11] [ − . , . A = . a does at 1 . z = p values can clearly explain the cross-correlationdata and, as it can be seen in the right panel of Figure 7, thederived HMF appears to hint at a larger number of halos whencompared to run 1, notably in the range 10 < M < M (cid:12) / h .Comparing the two results of the three-parameter case, weobserve that the peaks for A are almost the same in both scenar-ios (there is only a slight increase in the credible intervals for thenonpositivity case). However, the a peak value increases from0.93 to 1.29, as do the mean and the upper credible interval. This di ff erence clearly arises from the fact that p appears to be drivenby the data to take negative values and, in turn, a has to increasein order to counteract this e ff ect. Unlike the first run, p now hasa wide enough range within which it can move, hence the con-straining posterior distributions. In summary, introducing A asa free parameter along with allowing p to take negative valuesallows us to bypass the two problematic aspects that we have en-countered in this paper: the long one-sided tails in the a and p marginalized posterior distributions and the lack of generality inthe choice of prior range. Article number, page 10 of 19ueli M. M. et al.: Halo Mass Function measurement with Magnification Bias
Fig. 7.
Full posterior sampling (solid lines) and (mimic) marginal mode values (dashed lines) from runs 1 (in red) and 7 (in blue) of the MCMCalgorithm, that is, a two-parameter fixed HOD ST fit with p > p allowed to be negative,respectively. The left panel shows the cross-correlation function (the black filled circles being our measurements), while the right one displays the z = With a view to constraining the HMF itself at any redshift (ir-respective of its parameters), we now make use of one of themain advantages of performing a Bayesian analysis and studythe spread of the full posterior distribution so as to obtain cred-ible intervals for the value of the z = to10 . M (cid:12) / h for each case. The associated numerical values aretabulated in Tables B.3 and B.4.Figure 8 shows the median, 68%, and 95% credible inter-vals for the z = ff erent mass values for the two-parameter ST fit with p > p > p is allowed to be negative and greater than 0.5 (run7, in blue). There is good agreement with the traditional ST fit(black dotted line), with a tendency toward fewer massive halosat mass values larger than M (cid:38) M (cid:12) / h in the first two cases.The three-parameter ST fit shows the previously mentioned ten-dency toward a larger number of halos at intermediate masses,between 10 . and 10 M (cid:12) / h , although still compatible with thetraditional ST fit within the uncertainties.Figure 9 shows the corresponding results for the four-parameter Tinker fit with p > p > p > M < . M (cid:12) / h ) whencompared to the ST fits.
6. Summary, conclusions and future prospects
This paper has explored the submillimeter galaxy magnificationbias as a cosmological observable to provide a proof-of-conceptmethod to extract information about the HMF. By means of a
Fig. 8.
Credible intervals (68% in bold and 95% in faint colors) for the z = ff erent mass values when the full posterior distributionis sampled for the ST fit in the two-parameter fixed HOD case (red),the two-parameter Gaussian HOD case (green), and the three-parametercase (blue). The plots for each case are slightly displaced in the horizon-tal direction just for visual purposes. halo model interpretation of the foreground-background cross-correlation function between samples of GAMA II (with spec-troscopic redshift between 0 . < z < . z med = .
28) andH-ATLAS galaxies (with photometric redshift between 1 . < z < . z med = . ff erent universal HMF models with the aim of study-ing which of them provides a better fit to the data and derivingobservation-based credible intervals for the number density ofthe dark matter halos associated with the lenses at certain massvalues. We have also studied the potential influence of the HODparameters in our conclusions.We have begun our analysis with the apparently common as-sumption that all HMF parameters should be positive. In thisscenario, we have found that the two-parameter ST fit can onlyproperly explain the cross-correlation signal at angular scaleslarger than 3 arcmin when the HOD parameters are introducedin the MCMC analysis and thus allowed to vary. Indeed, the two-parameter ST fit is shown to be sensitive to the variation of the Article number, page 11 of 19 & A proofs: manuscript no. main
Fig. 9.
Credible intervals (68% in bold and 95% in faint colors) for the z = ff erent mass values when the full posterior distributionis sampled for the four-parameter Tinker fit in the fixed HOD case (darkorange) and in the Gaussian HOD case (purple). The two-parameterST fit with fixed HOD is also shown for comparison (red). The plotsfor each case are slightly displaced in the horizontal direction just forvisual purposes. HOD and a decrease in M (which substantially deviates fromits prior distributions) with a corresponding increase in a al-lows it to properly reproduce the data. On the other hand, thefour-parameter Tinker fit is quite robust to changes in the HODparameters and easily accommodates the large-scale data, buttwo of its parameters cannot be constrained. In fact, the extent oftheir posterior distributions depends on the corresponding rangeof their priors and this is a delicate issue when trying to derivestatistical results. In other words, care should be taken when in-terpreting our statistics of the Tinker fit, since they rely on ourspecific assumption of prior ranges, although we do not suspectmajor di ff erences would appear if they were modified.Both cases have nonetheless yielded credible intervals forthe z = M < . M (cid:12) / h ).We next analyzed the possibility of relaxing the normal-ization assumption for the ST fit while keeping parameter p within the range [0 , .
5) for the sake of comparison with the two-parameter case. We found that, under these conditions, adding A as a free parameter in the analysis allows the ST model toproperly explain the cross-correlation data at the largest scaleswithout resorting to changes in the HOD parameters. Parameter p , however, now becomes unconstrained on both ends, whichserves as a hint that other values should be explored.Indeed, motivated by the large relevance of prior distribu-tions in Bayesian inference and by the impossibility of constrain-ing parameter p on both sides with the previous studies, we de-cided to consider the case of a wide enough prior range for it,since we believe there is no physical reason against p takingnegative values. We analyzed both the two-parameter and thethree-parameter case. The former presents a strong degeneracydirection in the a - p plane with the presence of long one-sidedtails that reduce the constraining power with respect to the HMF.The three-parameter case, on the other hand, provides a robustconstraint on all the involved parameters. In our opinion this is the most general fit, with fewer assumptions on the prior infor-mation of the parameters, and the one to be used in future works.In fact, it hints at a slightly di ff erent behavior of the HMF at in-termediate and high masses with respect the traditional ST fit(but still compatible within the uncertainty range).In this respect, we strongly emphasize that future analyses ofthe HMF from N-body simulations should provide the range ofallowed or explored parameter values used to derived the best-fitbecause it is an important piece of information. Moreover, basedon our conclusions, we would like to recommend the allowanceof negative values for p in their best-fit calculations.Lastly, we provided a tabulated form of the constrained z = Euclid mission (Laureijs et al. 2011) will certainly provide addi-tional lenses at z > . / near-infrared James Webb SpaceTelescope (JWST, Gardner et al. 2006) will certainly increasethe area and / or the density of the background sources. Acknowledgements.
MMC, LB and JGN acknowledge the PGC 2018 projectPGC2018-101948-B-I00 (MICINN / FEDER). AL acknowledges support fromPRIN MIUR 2017 prot. 20173ML3WW002, ‘Opening the ALMA window on
Article number, page 12 of 19ueli M. M. et al.: Halo Mass Function measurement with Magnification Bias the cosmic evolution of gas, stars and supermassive black holes’, the MIURgrant ‘Finanziamento annuale individuale attivitá base di ricerca’, and the EUH2020-MSCA-ITN-2019 Project 860744 ‘BiD4BEST: Big Data applications forBlack hole Evolution STudies’. We deeply acknowledge the CINECA award un-der the ISCRA initiative, for the availability of high performance computing re-sources and support. In particular the project “SIS20_lapi” in the framework“Convenzione triennale SISSA-CINECA”. In this work, we made extensive useof
GetDist (Lewis 2019), a Python package for analysing and plotting MC sam-ples. In addition, this research has made use of the python packages ipython (Pérez & Granger 2007), matplotlib (Hunter 2007) and
Scipy (Jones et al.2001)
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Article number, page 14 of 19ueli M. M. et al.: Halo Mass Function measurement with Magnification Bias
Appendix A: The ingredients of the model
The dark matter transfer function has been computed usingEisenstein and Hu’s fitting formula (Eisenstein & Hu 1998),which takes baryonic e ff ects into account in a Λ CDM model.Having chosen an analytical computation of the power spec-trum over the traditional numerical one using CAMB (Lewiset al. 2000) is mainly due to computation time. The galaxy-darkmatter cross-power spectrum has been computed through equa-tions (7) and (8), where the linear dark matter power spectrum isevolved to redshift z via the linear growth factor approximationof Carroll et al. (1992).The HMF has of course been parametrized according to (1)for the two di ff erent fits we have described in Section 2.1. Thedeterministic bias associated with each model has been derivedusing the peak background split as in Sheth & Tormen (1999).Furthermore, we have expressed the mean number of galax-ies in a halo of mass M as (cid:104) N g (cid:105) M = (cid:104) N c g (cid:105) M + (cid:104) N s g (cid:105) M , where (cid:104) N c g (cid:105) M and (cid:104) N s g (cid:105) M are the mean number of central andsatellite galaxies in a halo of mass M , respectively, expressed interms of three HOD parameters ( α, log M , log M min ) as (cid:104) N c g (cid:105) M = Θ ( M − M min )and (cid:104) N s g (cid:105) M = (cid:16) MM (cid:17) α Θ ( M − M min )following Zehavi et al. (2005) and Zheng et al. (2005). Inessence, M min is the minimum mean halo mass required to host a(central) galaxy and M > M min is the mean halo mass at whichexactly one satellite galaxy is hosted. The mean number densityof galaxies at redshift z is then given by¯ n g ( z ) = (cid:90) ∞ dM n ( M , z ) (cid:104) N g (cid:105) M . The halo density profile has been assumed to match aNavarro-Frenk-White (NFW) profile (Navarro et al. 1997). Thenormalized Fourier transform of the dark matter distributionwithin a halo of mass M is then given by (Cooray & Sheth 2002) u ( k , z | M ) = πρ s r s ( M , z ) M (cid:104) sin kr s (cid:2) Si([1 + c ] kr s ) − Si( kr s ) (cid:3) −− sin ckr s [1 + c ] kr s + cos kr s (cid:2) Ci([1 + c ] kr s ) − Ci( kr s ) (cid:3)(cid:105) , where r s ( M , z ) ≡ R vir c ( M , z ) (A.1)and ρ s are a scale radius and density that parametrize the profile,concentration parameter of a halo of mass M at redshift z , whichsatisfies M = πρ s r s (cid:104) ln [1 + c ( M , z )] − c ( M , z )1 + c ( M , z ) (cid:105) (A.2)for an NFW profile. The virial radius R vir has been computedthrough the virial overdensity at redshift z , using the fit by Wein-berg & Kamionkowski (2003). It should be noted that we havenot defined halos at a certain redshift as overdense regions of aconstant factor (say 200) times the background or critical den-sity, but using the virial overdensity instead, which depends onredshift. In practice, for a halo of mass M , we have adopted theconcentration parameter by Bullock et al. (2001), computed r s through (A.1) and, subsequently, calculated ρ s using (A.2) Article number, page 15 of 19 & A proofs: manuscript no. main
Appendix B: Additional tables and figures
Table B.1.
Parameter priors, marginalized posterior peaks, means, 68%, and 95% credible intervals for run 5 of the MCMC algorithm, that is, athree-parameter ST fit with 0 < p < . Parameter Prior Peak Mean 68% CI 95% CI A U [0,1] 0.59 0 .
60 [0 . , .
73] [0 . , . a U [0,10] 0 .
93 1 .
12 [0 . , .
32] [0 . , . p U [0,0.50] − − − − Fig. B.1.
One- and two-dimensional (contour) posterior distributions from run 5 (in red) and run 1 (in blue), that is, a three-parameter and atwo-parameter ST fit with 0 < p < .
5, respectively.
Table B.2.
Parameter priors, marginalized posterior peaks, means, 68%, and 95% credible intervals for run 6 of the MCMC algorithm, that is, atwo-parameter ST fit, p allowed to be negative and fixed HOD values. Parameter Prior Peak Mean 68% CI 95% CI a U [0,10] 1 .
46 2 .
86 [0 . , .
42] [ − , . p U [-10,0.50] − . − .
24 [ − . , .
31] [ − . , − ] Fig. B.2.
One- and two-dimensional (contour) posterior distributions from run 6 (in blue) and run 7 (in red), that is, a two-parameter ST fit with − < p < . − < p <
10 and fixed HOD values, respectively.Article number, page 16 of 19ueli M. M. et al.: Halo Mass Function measurement with Magnification Bias
Fig. B.3.
One- and two-dimensional (contour) posterior distributions from run 1 (in blue) and run 2 (in red), that is, a two-parameter ST fit withfixed values and with Gaussian priors for the HOD parameters, respectively. The p parameter is assumed to be positive. Table B.3.
Tabulation of the z = M (cid:12) / h and the median, lowerand upper bounds of the credible intervals are expressed in h Mpc − M − (cid:12) . two-parameter ST fit: fixed HOD two-parameter ST fit: Gaussian HODlog M log Med log
68% CI log
95% CI log Med log
68% CI log
95% CI10.0 -10.54 [ − . , − .
47] [ − . , − .
38] -10.48 [ − . , − .
41] [ − . , − . − . , − .
40] [ − . , − .
33] -11.42 [ − . , − .
35] [ − . , − . − . , − .
33] [ − . , − .
29] -12.37 [ − . , − .
30] [ − . , − . − . , − .
27] [ − . , − .
25] -13.33 [ − . , − .
25] [ − . , − . − . , − .
22] [ − . , − .
22] -14.30 [ − . , − .
22] [ − . , − . − . , − .
18] [ − . , − .
18] -15.30 [ − . , − .
18] [ − . , − . − . , − .
14] [ − . , − .
14] -16.35 [ − . , − .
17] [ − . , − . − . , − .
11] [ − . , − .
10] -17.49 [ − . , − .
15] [ − . , − . − . , − .
12] [ − . , − .
10] -18.83 [ − . , − .
29] [ − . , − . − . , − .
25] [ − . , − .
19] -20.54 [ − . , − .
59] [ − . , − . − . , − .
59] [ − . , − .
49] -23.05 [ − . , − .
23] [ − . , − . − . , − .
48] [ − . , − .
22] -27.38 [ − . , − .
74] [ − . , − . Article number, page 17 of 19 & A proofs: manuscript no. main
Fig. B.4.
One- and two-dimensional (contour) posterior distributions from run 3 (in blue) and run 4 (in red), that is, a four-parameter Tinker fitwith fixed values and with Gaussian priors for the HOD parameters, respectively. The p parameter is assumed to be positive. Table B.4.
Tabulation of the z = M (cid:12) / h and themedian, lower and upper bounds of the credible intervals are expressed in h Mpc − M − (cid:12) . four-parameter Tinker fit: fixed HOD four-parameter Tinker fit: Gaussian HODlog M log Med log
68% CI log
95% CI log Med log
68% CI log
95% CI10.0 -10.36 [ − . , − .
25] [ − . , − .
18] -10.39 [ − . , − .
24] [ − . , − . − . , − .
21] [ − . , − .
15] -11.35 [ − . , − .
21] [ − . , − . − . , − .
18] [ − . , − .
10] -12.30 [ − . , − .
15] [ − . , − . − . , − .
12] [ − . , − .
03] -13.24 [ − . , − .
08] [ − . , − . − . , − .
04] [ − . , − .
89] -14.20 [ − . , − .
00] [ − . , − . − . , − .
96] [ − . , − .
73] -15.17 [ − . , − .
91] [ − . , − . − . , − .
90] [ − . , − .
57] -16.17 [ − . , − .
82] [ − . , − . − . , − .
88] [ − . , − .
45] -17.23 [ − . , − .
79] [ − . , − . − . , − .
95] [ − . , − .
53] -18.40 [ − . , − .
83] [ − . , − . − . , − .
23] [ − . , − .
81] -19.86 [ − . , − .
90] [ − . , − . − . , − .
85] [ − . , − .
56] -21.98 [ − . , − .
10] [ − . , − . − . , − .
94] [ − . , − .
28] -25.67 [ − . , − .
27] [ − . , − . Article number, page 18 of 19ueli M. M. et al.: Halo Mass Function measurement with Magnification Bias
Table B.5.
Tabulation of the z = − < p <
10 andfixed HOD. For convenience, we have tabulated the base-10 logarithm of all quantities; the masses are expressed in M (cid:12) / h and the median, lowerand upper bounds of the credible intervals are expressed in h Mpc − M − (cid:12) . log M log Med log
68% CI log
95% CI10.0 -10.50 [ − . , − .
31] [ − . , − . − . , − .
23] [ − . , − . − . , − .
15] [ − . , − . − . , − .
06] [ − . , − . − . , − .
92] [ − . , − . − . , − .
69] [ − . , − . − . , − .
48] [ − . , − . − . , − .
38] [ − . , − . − . , − .
46] [ − . , − . − . , − .
85] [ − . , − . − . , − .
70] [ − . , − . − . , − .
01] [ − . , − .40]