Intensity mapping cross-correlations II: HI halo models including shot noise
MMNRAS , 1–15 (2017) Preprint 8 March 2018 Compiled using MNRAS L A TEX style file v3.0
Intensity mapping cross-correlations II: HI halo modelsincluding shot noise
L. Wolz, , (cid:63) S. G. Murray, , C. Blake , and J.S. Wyithe , School of Physics, University of Melbourne, Parkville, VIC 3010, Australia ARC Centre of Excellence for All-Sky Astrophysics (CAASTRO) ICRAR, Curtin Institute of Radio Astronomy, GPO Box U1987, Perth, WA 6845, Australia Centre for Astrophysics & Supercomputing, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT H i intensity mapping data traces the large-scale structure matter distribution usingthe integrated emission of neutral hydrogen gas (H i ). The cross-correlation of theintensity maps with optical galaxy surveys can mitigate foreground and systematiceffects, but has been shown to significantly depend on galaxy evolution parametersof the H i and the optical sample. Previously, we have shown that the shot noise ofthe cross-correlation scales with the H i content of the optical samples, such that theshot noise estimation infers the average H i masses of these samples. In this article,we present an adaptive framework for the cross-correlation of H i intensity maps withgalaxy samples using our implementation of the halo model formalism (Murray et al2018, in prep) which utilises the halo occupation distribution of galaxies to predict theirpower spectra. We compare two H i population models, tracing the spatial halo andthe galaxy distribution respectively, and present their auto- and cross-power spectrawith an associated galaxy sample. We find that the choice of the H i model and thedistribution of the H i within the galaxy sample have minor significance for the shape ofthe auto- and cross-correlations, but highly impact the measured shot noise amplitudeof the estimators, a finding we confirm with simulations. We demonstrate parameterestimation of the H i halo occupation models and advocate this framework for theinterpretation of future experimental data, with the prospect of determining the H i masses of optical galaxy samples via the cross-correlation shot noise. Key words: cosmology: theory – cosmology: large-scale structure of Universe – radiolines: galaxies
The cosmological evolution of our Universe can be testedvia probes of the statistics of large-scale structure. Commontechniques include measuring the Baryon Acoustic Oscilla-tions (BAOs), which act as a standard ruler for distancemeasures constraining the Cosmic acceleration (see e.g. Reidet al. 2012; Anderson et al. 2014), as well as galaxy cluster-ing which employs the positions of galaxies to measure theircosmological power spectrum (for instance Seljak et al. 2005;Percival et al. 2007). Both BAO measurements and galaxyclustering require the determination of millions of galaxypositions over large volumes in order to minimize statisti-cal uncertainties. Traditionally, optical telescopes have beenemployed for cosmological measurements as radio telescopes (cid:63)
E-mail: [email protected] are limited in sensitivity. Beyond the nearby Universe, thedetermination of redshifts at radio frequencies, in which thespectrum is close to featureless, is extremely challenging.The most notable radio spectral line, at a rest-wavelength of21cm, is caused by the spin-flip of the neutral hydrogen (H i )and is comparably weak. It has only been directly detectedup to z = . in a single object (Fernandez et al. 2016), andin the statistically averaged spectrum via H i stacking up to z ≈ . (Rhee et al. 2018). To circumvent these limitations,H i intensity mapping provides a novel technique to map thelarge-scale structure distribution as traced by neutral hy-drogen gas via low-resolution observations of the integratedand unresolved 21cm emission of multiple objects.After H i intensity mapping was proposed as a test ofcosmology more than a decade ago (see Battye et al. 2004;Wyithe et al. 2008; Chang et al. 2008), Pen et al. (2009) re-ported the first detection of structure in H i intensity maps © a r X i v : . [ a s t r o - ph . C O ] M a r L.Wolz et al. of the local Universe. Later Chang et al. (2010) reporteda detection in observations around z ≈ . . The challengesin detecting the H i intensity mapping power spectrum arisedue to the weakness of the redshifted H i signal in comparisonto the radio foregrounds in combination with the radio tele-scope systematics, see e.g. Switzer et al. (2015); Wolz et al.(2017a); Harper et al. (2017). The cross-correlation signal ofan H i map with an overlapping galaxy survey is insensitiveto many of these systematics and increases the significanceof detection. The detection of the cosmological distributionvia the power spectrum (Masui et al. 2013) was achievedby measurements of the Green Bank telescope at mediumredshift z = . in cross-correlation with the WiggleZ DarkEnergy survey (Drinkwater et al. 2010), constraining the H i energy density and the H i bias to Ω HI b HI = . + . − . × − (Switzer et al. 2013). A more recent detection has also beenmade, using the cross-correlation of the H i intensity maps ofthe Parkes telescope with the 2dF Galaxy Redshift Surveyat z ≈ . (Anderson et al. 2017). The analysis presents a5-sigma detection of the cross-power spectrum with a sig-nificant drop of the power on smaller scales, k ≈ . h / Mpc ,indicating a strong anti-correlation of H i with the red galaxysample.The future of H i intensity mapping looks very promisingas a large number of purpose-built instruments are under de-sign and construction. The instruments can be divided into3 categories: single dish telescopes similar to the pioneeringGreen Bank and Parkes telescopes as well as equipped withmulti-beam receivers (e.g. BINGO Battye et al. 2013), dishinterferometers such as HIRAX (Newburgh et al. 2016), andcylindrical dish interferometers such as CHIME (Banduraet al. 2014) or Tianlai (Chen 2012). Additionally, the SquareKilometre Array, an international radio interferometer withunprecedented scale and sensitivity, will conduct H i inten-sity mapping for wide ranges of redshifts < z < (Santoset al. 2015). Two SKA pathfinder projects, MeerKAT andthe Australian SKA Pathfinder (ASKAP), are capable ofintensity mapping observations, and will be able to exploredifferent observational techniques such as the employment ofthe array in single-dish mode (Santos et al. 2017) or phasedarray feeds, in preparation for the SKA observations to com-mence in the next decade. Forecasts predict that the futureSKA H i intensity mapping experiments will be able to mea-sure distances via BAOs to a level that is comparable toStage IV optical experiments as well as obtaining new con-straints on higher, unobserved redshifts (Bull et al. 2015).The forth coming intensity maps will also set new constraintson non-Gaussianity through measuring the ultra-large scalesof the power spectrum (Camera et al. 2014). For all men-tioned experiments, the cross-correlation of the H i intensitymapping signal with galaxy surveys will be a crucial test forsystematics, and most likely be the first observable to delivernew scientific results.In addition to cosmological parameters, the amplitudeand the clustering power of the H i intensity mapping powerspectrum depends on the distribution of the neutral hydro-gen gas with respect to the underlying matter field, andadditionally for cross-correlations on the observed opticalgalaxy sample. Recently, H i models based on available data(see e.g. Padmanabhan et al. 2017; Padmanabhan 2017) aswell as predictive theories (Chen 2012) have been proposed to deliver the theoretical framework for interpretation of theintensity mapping signal.In this work we extend existing H i models to predictthe cross-correlation of intensity maps with galaxy surveysto enhance the interpretation of existing and forth comingdata and provide a framework to include halo occupationparameters into the cosmological analysis of future measure-ments (for forecasts see Pourtsidou et al. 2015; Sarkar et al.2016; Pourtsidou et al. 2017). In Wolz et al. (2017b) wehave shown that the shot noise in the cross-power spectrum,which is caused by the discrete nature of galaxy data, scaleswith the average H i mass per optical galaxy. Hence, intensitymapping data can be employed to determine an average H i mass for any over lapping galaxy sample. This allows deter-mination of global scaling relations between star-formationactivity as traced by the optical sample and their gas con-tents, for redshifts well beyond the current limits for directgas detection. In this work, we present a theoretical frame-work which correctly determines the shot noise contributiongiven the H i parameters of the distribution, and which canbe employed to fit the H i parameters and shot noise in futureobservational data.In this paper, we first briefly introduce the halo modelframework, along with our chosen numerical implementa-tion ( halomod ) in Sec. 2. Here we also introduce the em-ployed halo occupation models for galaxies and H i mod-els and present theoretical euqtaions for the H i auto-powerspectra and their respective cross-power spectra. In Sec. 3we describe our method of producing lognormal realisationsof joint optical and H i samples, which we will use to verifyour theoretical formalism. In Sec. 4, we review the currentunderstanding of shot noise on power spectra and discuss itsimplementation in halomod . We present and examine thecomparison of theory with lognormal simulations in Sec. 5.In the following Sec. 6, we demonstrate how halomod canbe used to constrain H i parameters via MCMC parameterestimation. We discuss our findings and present the conclu-sions in Sec. 7. The halo model (Peacock & Smith 2000; Cooray & Sheth2002) is a highly successful description of the cosmologicaldensity field that uses empirical models of the internal prop-erties of dark matter halos to access non-linear scales. It hasbeen employed, along with a prescription for the abundanceof galaxy tracers within halos termed the halo occupationdistribution (HOD), to predict the spatial statistics of vari-ous galaxy populations, typically in order to constrain var-ious properties of the selected sample (Zheng et al. 2005;Zehavi et al. 2011; Beutler et al. 2013). It has recently beenextended to the domain of H i abundance by Padmanabhan& Refregier (2017); Padmanabhan et al. (2017).The essence of the halo model consists of the assumptionthat all material is sequestered into discrete halos, which arein turn self-similar objects that scale exclusively as a func-tion of their mass. Consequently, knowledge of the spatialarrangement of the halo centres combined with a knowledgeof their internal profiles, how these scale with the halo’smass, and the abundance of halos at any given mass, yieldsa full statistical description of the matter field down to ar- MNRAS , 1–15 (2017) ntensity mapping cross-correlations II: HI halo models including shot noise bitrarily small scales in real space. Likewise, assuming thatany given tracer inhabits halos with an abundance exclu-sively as a function of their mass, the statistics of the tracerfield may also be determined.In summary, to describe the two-point statistics of atracer field (or the cross-correlation of tracers), one requiresthe following ingredients:(i) The non-linear matter power spectrum (Smith et al.2003).(ii) The radial profile of the tracer within the halo, ρ ( r ) ;we typically employ the standard NFW profile (Navarroet al. 1997), but also check the modified, or “cored” NFWemployed by Padmanabhan & Refregier (2017).(iii) The mass function of halos, n ( m ) ; we use the fittingformula of Tinker et al. (2008).(iv) The abundance and distribution of tracers within ha-los, N ( m ) ; we describe our choices for this component furtherin § c ( m ) , which defineshow the profile scales with halo mass; we use the fit of Duffyet al. (2008).(vi) The bias of halos of a given mass, b ( m ) ; we use thefunction determined by Tinker et al. (2010).Additionally, the effects of halo exclusion can be modeled,such that pairs of the tracer that are very close are proba-bilistically assigned to the same halo and excluded from thecounts between different halos, to avoid double-counting. Weomit this modeling for this introductory work, but note thatits inclusion is trivial within the halomod package that weuse. All halo model calculations performed in this work use the halomod Python library (Murray et al., 2018, in prep. ).This library is built on the hmf package (Murray et al.2013), which handles the cosmology, linear power spectra,and mass functions. The halomod code provides manymodels for halo profiles, halo bias, concentration-mass re-lations, HODs and halo exclusion, along with the necessaryframework to combine these to produce spatial statistics.A key feature of the hmf framework which is extendedto halomod is the simplicity of defining new componentmodels and “plugging” them into the calculations. Thus forinstance it is simple to define a new HOD model from a stan-dard galaxy HOD, and is instantly usable within the frame-work without having to modify the source code. We notethat versions of both hmf and halomod that calculate theresults of this paper can be obtained via the feature/HIHOD branch of each. In our study we require several HOD models: one whichdescribes the full galaxy count population, another whichdescribes a particular optically-selected sample count, anda model which describes the H i occupation. We use variants Source code at https://github.com/steven-murray/halomod . Available at https://github.com/steven-murray/hmf of the simple HOD parameterisation of Zehavi et al. (2005)(Z05) in all cases. This model depends on three parameters:the minimum halo mass to be occupied by a galaxy M min ,the characteristic halo mass M which marks the turn-overof the broken power-law, and the power-law coefficient α ofthe satellite HOD. We extend the parametrisation by addingthe maximum (cut-off) halo mass M max as a parameter.In general the HOD can be split into two separateclasses of objects; central galaxies located at the centre ofthe halo, and satellite galaxies that trace the halo’s densityprofile. The Z05 model assigns the following parameterisa-tions to each component: (cid:104) N cen ( m )(cid:105) = (cid:40) M min < m < M max otherwise (1) (cid:104) N sat ( m )(cid:105) = (cid:40) ( m / M ) α M min < m < M max otherwise (2)When stating that a sample may be described by a sep-aration of central and satellite galaxies, we furthermore as-sume (in this paper) that this separation is due to the cen-tral having a much higher probability of existence withinthe sample than its associated satellites. This may be under-stood easily in terms of optical samples, in which the centralgalaxy is typically much brighter than the satellites. To ap-proximate the effect of this a priori knowledge, our halomod algorithms assert in such cases that a central galaxy must be present before any satellites. In this case, the average to-tal occupation is accurately given by the following definition,which ensures that the total occupation is zero whenever thecentral occupation is zero, but otherwise yields the expectedsum of central and satellite: (cid:104) N ( m )(cid:105) = (cid:104) N cen ( m )(cid:105)( + (cid:104) N sat ( m )(cid:105)) . (3) halomod does not limit the form or parametrisation ofthe HODs and more complex models can be assumed.In this article, we adopt two fiducial galaxy HODs, inthis study, referred to as sample and field , where we as-sume that the galaxy field model is a description of alloptically-detectable and H i emitting galaxies and sample isan optically-detected sub-sample of the field . The HOD pa-rameters of all galaxy and H i models can be found in Tab. 1,where we choose representative values for all parameters inour toy models.In the following, we postulate two variations of the H i HOD model in the framework of the halo model. For demon-stration purposes, we base the parametrisation of the H i HODs on Z05. More physically-motivated and data-drivenmodels, such as Padmanabhan & Refregier (2017) and Paulet al. (2017), can be easily implemented in halomod andstudied with our methods.
Continuous HI distribution.
In this scenario we assumethat the H i continuously traces the dark matter halo fol-lowing an independent density profile, for example a coredNFW profile as in Padmanabhan et al. (2017). This impliesthat the H i is not associated with galaxies and there are nocentral or satellite contributions to the density. This modelis best suited to describe the cold gas distribution at theearly stages of galaxy formation at the end of the epochof reionisation and resembles semi-numerical approaches to MNRAS000
In this scenario we assumethat the H i continuously traces the dark matter halo fol-lowing an independent density profile, for example a coredNFW profile as in Padmanabhan et al. (2017). This impliesthat the H i is not associated with galaxies and there are nocentral or satellite contributions to the density. This modelis best suited to describe the cold gas distribution at theearly stages of galaxy formation at the end of the epochof reionisation and resembles semi-numerical approaches to MNRAS000 , 1–15 (2017)
L.Wolz et al.
Table 1.
HOD parameters for all models considered in this work.All masses are given as log and in units of M (cid:12) / h .Model M min M max α M log A HI Galaxy field sample i continuous 11.0 17.0 0.7 11.0 11.0H i discrete 11.0 17.0 0.7 11.0 11.0 cosmological simulations of intensity maps (e.g. Alonso et al.2014).We alter the Z05 HOD to describe the H i mass dis-tribution, by adding an extra normalisation A HI in units of M (cid:12) / h to scale the distribution to produce typical H i masses,increasing the number of parameters to five. (cid:104) M HI ( m )(cid:105) = (cid:40) A HI (( m / M ) α + ) M min < m < M max otherwise (4)The additional term + in the HOD is introduced to simplifycomparison with the our second H i model. Discrete HI distribution.
In this model, we assume thatthe H i is on average following the underlying dark matterhalo density profile throughout the halo, but specify that theH i mass is co-located with the underlying galaxy field . Thusthe H i in any given halo is discretely located. This modeldescribes a stage of galaxy evolution in which most H i isconfined within galaxies and inter-galactic cold gas is negli-gible in the intensity maps. The approach predicts a similardistribution to semi-analytic simulations which model thecold gas abundances within star-forming regions (e.g. Lagoset al. 2014; Kim et al. 2017).We model this case in a similar fashion to galaxies, inwhich we split the HOD contributions into central and satel-lite components. (cid:104) M cenHI ( m )(cid:105) = (cid:40) A HI M min < m < M max otherwise (5)and the satellite part by (cid:104) M satHI ( m )(cid:105) = (cid:40) A HI [( m / M ) α ] M min < m < M max otherwise (6)We note that this H i model has a dependence on themodel defining the underlying galaxy field with which it isco-located. While the HOD itself as defined above requiresno knowledge of the underlying field HOD, and thus theauto-power spectrum of the H i is fully self-defined, its cross-correlation with an optical sample relies on the actual dis-tribution of H i within the halo, which we have describedas being dependent on the field . We will see that this in-formation will be necessary in order to define theoreticalcross-correlations, Poisson noise, and also to self-consistentlyproduce joint simulations. The power spectrum is divided into its 2-halo and 1-halocontribution, P ( k ) = P h ( k ) + P h ( k ) . The 2-halo term closelyfollows the linear matter power spectrum P lin ( k ) and, in its most general form applicable to cross-correlation on largescales, the 2-halo term is expressed as P ij h ( k ) = b i ( k ) b j ( k ) ∗ P lin ( k ) , (7)where b i is the effective bias of the i th probe, given as b i ( k ) = n g ∫ d m n ( m ) b ( m )(cid:104) N i ( m )(cid:105) u i ( k | m ) . (8)Here b ( m ) is the halo bias, u ( k | m ) is the Fourier transform ofthe halo mass profile following the NFW model, and ¯ n g isgiven by the number density of the galaxies, computed as ¯ n g = ∫ d m n ( m )(cid:104) N ( m )(cid:105) . (9) n ( m ) is the halo mass function, for more details on the im-plementation please refer to Murray et al. (2013).The 1-halo term is given by the clustering within thehalos and depends on the number of centrals and satellitegalaxies. For the auto-correlation of one probe, this resultsin P h ( k ) = n g ∫ d m n ( m ) (cid:20) (cid:104) N cen N sat (cid:105) u ( k | m ) + (cid:104) N sat ( N sat − )(cid:105) u ( k | m ) (cid:21) . (10)The first term depends on the expectation value of the num-ber of central-satellite pairs per halo multiplied by the halomass profile and the second term on the expectation valueof the number of satellite-satellite pairs per halo mass mul-tiplied by the self-convolved mass profile. Since in our modelthere can only ever be either zero or one central galaxy ina halo, and under the assumption that the central galaxyis always the first of the halo to be included in a sample,we have (cid:104) N cen N sat (cid:105) = (cid:104) N cen (cid:105)(cid:104) N sat (cid:105) . Furthermore, for Poisson-distributed X , (cid:104) X ( X − )(cid:105) ≡ (cid:104) X (cid:105) , which means (assuming thesatellite occupation is Poisson-distributed) that P h ( k ) = n g ∫ d m n ( m ) (cid:20) (cid:104) N cen (cid:105)(cid:104) N sat (cid:105) u ( k | m ) + (cid:104) N sat (cid:105) u ( k | m ) (cid:21) , (11)This form is convenient, as it only depends on the meanoccupation functions which we have defined above.For the cross-correlation of two different galaxy sam-ples which follow different HODs and density profiles, theanalogue of Eq. 10 is P ij h ( k ) = n i ¯ n j ∫ d m n ( m ) (cid:104) (cid:104) N i cen N j sat (cid:105) u j ( k | m ) + (cid:104) N j cen N i sat (cid:105) u i ( k | m ) + (cid:104) N i sat N j sat (cid:105) u i ( k | m ) u j ( k | m ) (cid:105) . (12)In general we cannot further reduce this equation, be-cause it is not guaranteed that the absence of a centralgalaxy in one sample necessitates the absence of satellites (aswell as central) in a different sample. However, if the centralHOD happens to be a step-function, so that at any masseither all or none of the haloes have centrals, the central-satellite term decomposes as before. We note that this is MNRAS , 1–15 (2017) ntensity mapping cross-correlations II: HI halo models including shot noise an extra condition , which was not required for Eq. 11. Thisallows us to re-write the equation as follows: P ij h ( k ) = n i ¯ n j ∫ d m n ( m ) (cid:104) (cid:104) N i cen (cid:105)(cid:104) N j sat (cid:105) u j ( k | m ) + (cid:104) N j cen (cid:105)(cid:104) N i sat (cid:105) u i ( k | m ) + (cid:16) (cid:104) N i sat (cid:105)(cid:104) N j sat (cid:105) + σ i σ j R ij − Q (cid:17) u i ( k | m ) u j ( k | m ) (cid:105) , (13)where R ij is the correlation of the satellite occupation be-tween the probes, and σ i the standard deviation of the satel-lite occupation, which for a Poisson occupation is simply (cid:112) (cid:104) N sat (cid:105) . Q is equal to the expected number of shared pointsbetween the samples unless either tracer is continuously spa-tially distributed which results in Q = .In general, R ij is constrained to be within (− , ) anddepends on the complicated physical interactions of the twotracer populations. However, for the toy models we employin this paper, it is possible to provide a better descriptionwhich we present in detail in Appendix A1 and A2. Following the same arguments as the previous section, wemay derive the power spectrum of H i density fluctuationsfor both cases presented in Sec. 2.2. The 2-halo term of theH i power spectra for both models is similar to Eqs. 7 and8 with the galaxy HOD substituted by the H i occupation (cid:104) M HI ( m )(cid:105) of the respective model, such that b HI ( k ) = C HI ∫ d m n ( m ) b ( m )(cid:104) M HI ( m )(cid:105) u HI ( k | m ) . (14)where the coefficient C HI is described below. The H i halodensity profile u HI ( k | m ) is commonly defined as a modified(or cored) NFW profile (Padmanabhan et al. 2017) which inreal space reads as ρ HI ( r ) = ρ r s ( r + . r s )( r + r s ) (15)where r s is the scale radius of the dark matter halo whichis defined as r s = r vir / c ( m ) and r vir is the virial radius ofthe halo. We refrain from the use of a H i specific parametri-sation of r s and adopt the concentration-mass relation fitfrom Duffy et al. (2008). In halomod we employ the ana-lytic expression of the Fourier transform u HI ( k ) of this profile(Padmanabhan et al. 2017).H i intensity maps are measured in brightness temper-ature T HI . To follow this convention, we convert all powerspectra into temperature units, using a conversion C HI , givenby C HI = A h P c ( + z ) π m H k B ν H ( z ) (16)with h P the Planck constant, k B the Boltzmann constant, m H the mass of the hydrogen atom, A the emission coefficientof the 21cm line transmission and ν the rest frequency ofthe 21cm emission. H ( z ) is the Hubble parameter at redshift z . All presented studies are for redshift z ≈ . The plottedH i power spectra are given in units of K ( Mpc / h ) and cross-power spectra as K ( Mpc / h ) if not stated otherwise. The predicted mean brightness temperature for each H i model can be determined via T HI = C HI ∫ d m n ( m )(cid:104) M HI ( m )(cid:105) (17)The mean H i brightness temperature is directly proportionalto the H i energy density Ω HI which makes it a desired observ-able when conducting H i intensity mapping experiments. Continuous HI distribution.
The 1-halo term of theauto-power spectrum in this case, with lack of satellite com-ponents, can be written as P HI , cont1 h ( k ) = C ∫ d m n ( m )(cid:104) M HI ( m )(cid:105) u HI ( k | m ) , (18)while the cross-correlation with a galaxy sample g is P g HI , cont1 h ( k ) = C HI ¯ n g ∫ d m n ( m ) u HI ( k | m )× (cid:104) ( u g ( k | m ) (cid:104) N g sat ( m )(cid:105)(cid:104) M HI ( m )(cid:105) + R g HI ) + (cid:104) N g cen ( m )(cid:105)(cid:104) M HI ( m )(cid:105) (cid:3) (19)where R g HI is a galaxy-H i correlation coefficient. As thereis no central-satellite split in the H i HOD, the clustering issimplified into two terms - one in which the satellite galax-ies pair with the H i profile, and another in which the single(possible) central galaxy pairs with the H i profile. We fidu-cially consider a value of R = for this work, which impliesthat the H i mass is uncorrelated with the galaxy occupa-tion. The more detailed derivation of the correlation factor R and an example for a correlated toy model can be foundin Appendix A1. Discrete HI distribution.
The 1-halo power spectrum ofthe discrete H i model can be written similarly to Eq. 11,assuming that the positions of the satellite occupation arePoisson-distributed: P HI , dsc1 h ( k ) = C ∫ d m n ( m ) (cid:20) u HI ( k | m )(cid:104) M satHI ( m )(cid:105)(cid:104) M cenHI ( m )(cid:105) + (cid:104) M satHI ( m )(cid:105) u HI ( k | m ) (cid:21) . (20)The 1-halo term of the H i cross-correlation with a galaxysample reads as P g HI , dsc1 h ( k ) = C HI ¯ n g ∫ d m n ( m ) (cid:104) (cid:16) (cid:104) N g cen (cid:105)(cid:104) M satHI (cid:105) (cid:17) u HI ( k | m ) + (cid:0) (cid:104) M cenHI (cid:105)(cid:104) N g sat (cid:105) (cid:1) u g ( k | m ) + (cid:16) (cid:104) N g sat (cid:105)(cid:104) M satHI (cid:105) (cid:17) u g ( k | m ) u HI ( k | m ) (cid:105) . (21)We note the absence of the correlation term, R . This is dueto exact co-location of the H i with the optical galaxies, asexplained in detail in Appendix A2. Briefly, in this model,H i abundance depends only on the properties of the galaxyin which it is situated, and this galaxy, by construction, hasno correlation with other galaxies. Therefore all correlationsare expressed at a separation of zero, and do not affect the MNRAS000
The 1-halo power spectrum ofthe discrete H i model can be written similarly to Eq. 11,assuming that the positions of the satellite occupation arePoisson-distributed: P HI , dsc1 h ( k ) = C ∫ d m n ( m ) (cid:20) u HI ( k | m )(cid:104) M satHI ( m )(cid:105)(cid:104) M cenHI ( m )(cid:105) + (cid:104) M satHI ( m )(cid:105) u HI ( k | m ) (cid:21) . (20)The 1-halo term of the H i cross-correlation with a galaxysample reads as P g HI , dsc1 h ( k ) = C HI ¯ n g ∫ d m n ( m ) (cid:104) (cid:16) (cid:104) N g cen (cid:105)(cid:104) M satHI (cid:105) (cid:17) u HI ( k | m ) + (cid:0) (cid:104) M cenHI (cid:105)(cid:104) N g sat (cid:105) (cid:1) u g ( k | m ) + (cid:16) (cid:104) N g sat (cid:105)(cid:104) M satHI (cid:105) (cid:17) u g ( k | m ) u HI ( k | m ) (cid:105) . (21)We note the absence of the correlation term, R . This is dueto exact co-location of the H i with the optical galaxies, asexplained in detail in Appendix A2. Briefly, in this model,H i abundance depends only on the properties of the galaxyin which it is situated, and this galaxy, by construction, hasno correlation with other galaxies. Therefore all correlationsare expressed at a separation of zero, and do not affect the MNRAS000 , 1–15 (2017)
L.Wolz et al. -1 k [Mpc − h] P ( k ) [ M p c h − ] Galaxy field modelGalaxy sample modelHI discrete modelHI continuous modelHI continuous model, cored NFW
Figure 1.
The auto-power spectra predicted by our model for thecase of galaxy field population, galaxy sample , H i continuum, andH i discrete model. The H i power spectra are normalised by thesquare of the mean temperature predicted by each model usingEq. 17 for presentation purposes. Note that by construction, bothH i models predict the same mean brightness temperature. -1 k [Mpc − h] -2 -1 P ( k ) [ K M p c h − ] HI discrete - galaxy fieldHI discrete - galaxy sampleHI continuous - galaxy fieldHI continuous - galaxy sampleHI continuous cored - galaxy field
Figure 2.
The cross-power spectra predicted by our model for thecase of galaxy field population, galaxy sample, H i continuum, andH i discrete model. The H i cross-power spectra are normalised bythe mean temperature predicted by each model using Eq. 17 forpresentation purposes. Note that by construction, both H i modelspredict the same mean brightness temperature. shape of the 1-halo term. This may alternatively be seen asthe exact cancellation of the correlation term with the Qterm in Eq. 13.The auto-power spectrum predictions of halomod areshown in Fig. 1, where we show the two models of galaxypower spectra, called field and sample , with HOD param-eters defined in Tab. 1. In yellow and green, we comparethe H i power spectra of the continuous and discrete mod-els, where we renormalise the H i spectra through division by T HI2 . We can see that for k < h / Mpc both models’ pre-dictions closely agree. For k > h / Mpc , the continuous H i model falls off more quickly since the 1-halo term containsno central-satellite contribution in the discrete case. In thisfigure, we also demonstrate how the cored H i profile altersthe power of the 1-halo term in comparison to the standardNFW profile. In the remainder of the article, we employ thestandard NFW profile for our computations such that thecomparison of the cases are focused on their clustering termsrather than the impact of the density profile. The cross-power spectrum prediction of both H i modelswith the two galaxy models are shown in Fig. 2. The differ-ences in the two H i models are negligible over all scales k .Furthermore, even the differences in the two different galaxymodels are very small compared to the variation in theirauto-power spectrum. The agreement of the two models isby construction as they follow the same H i HOD parametersand we implement the continuous case to be the sum of thecentral and satellite terms of the discrete model. Therefore,they correlate in a similar fashion with the galaxy sampleswhich is not to be expected in a general case.In both figures, we neglect the shot noise, also referredto as the Poisson Noise (PN) contribution P PN , and we willdiscuss its contribution in detail in the following sections. In order to test the accuracy of the analytic routines within halomod , we create a number of mock realisations of thetracer populations. As this is done explicitly to test the rou-tines, the simulations are prepared to mimic the assump-tions of the halo model formalism at the simplest level. Inthis section we describe the method used to generate thesesimulations.
We consider a cube of volume L ( Mpc / h ) with N gridcells, in which we generate a lognormal density field (Coles& Jones 1991) using the powerbox package . We choosematching input power spectra and parameters to halomod to ensure comparability of the results using the Planck15cosmological model (Planck Collaboration et al. 2016).We choose a minimum halo mass M min , h such thatall halos containing galaxies in our sample lie above thethreshold. We then draw a number density of halo masses n h = ∫ M min , h n ( m ) d m from the halo mass function distribution.These halos are placed probabilistically within the grid vol-ume, with the probability of landing in a certain cell givenby its relative density. The final positions of each halo aredrawn randomly within each cell, rendering sub-grid scaleshighly inaccurate. We note that the mass of each halo doesnot affect its placement, which effectively means that thehalo bias is unity for all masses when comparing simula-tions to theory, we therefore set the theoretical halo bias tounity in halomod .Finally, we use the resultant halo catalog, with massesand positions, as the scaffolding on which to assign the tracerpopulation. Here we will describe the methods used for pro-ducing a single tracer population, suitable for comparing toauto-spectra. We use a routine in which for each halo i weperform the following steps:(i) Sample a single number (zero or one) C i from aBernoulli distribution with mean (cid:104) N cen ( m i )(cid:105) (ii) If C i = , place a galaxy at (cid:174) x i and continue, else pro-ceed to next halo.(iii) Sample a number N i sat from a Poisson distributionwith mean (cid:104) N s ( m i )(cid:105) . Available at https://github.com/steven-murray/powerbox .MNRAS , 1–15 (2017) ntensity mapping cross-correlations II: HI halo models including shot noise (iv) If N is > , sample N i sat radii, r ij from the halo’s pro-file, ρ ( r , m ) , and sample ( θ j , φ j ) isotropically to yield 3D co-ordinates, (cid:174) x ij centred at the origin.(v) Assign N i sat galaxies to positions (cid:174) x i + (cid:174) x ij .We note that this procedure does not take into account haloexclusion – halos are allowed to overlap arbitrarily – andthus to reproduce the results analytically also requires nohalo exclusion model.In our simulations, we first apply the steps outlinedabove using the field HOD to create a galaxy cataloguewhich is assumed to contain all available galaxies. We thencreate a sub-sample of the galaxy catalogue which followsthe HOD of the galaxy sample . Similarly to steps ( i ) and ( ii ) , for each galaxy in the field catalogue we draw a singlenumber C i ∈ ( , ) from the Bernoulli distribution with mean P i = (cid:104) N sample ( m i )(cid:105)/(cid:104) N field ( m i )(cid:105) to determine if the galaxy ispart of the sample . The positions of the galaxies are keptidentical. We note that this procedure does not strictly re-tain the Poisson-distributed nature of the satellite galaxiesin the sample . Nevertheless, the mean is retained, and we donot expect the departure from Poisson to be significant. For the continuous model,we assign an H i mass to each halo produced by the lognor-mal realisations, where we draw the H i masses according tothe input H i HOD at halo mass m assuming a Gaussian dis-tribution with a standard deviation σ HI = . (cid:104) M HI ( m )(cid:105) . Inorder to mimic the continuous H i distribution throughoutthe halo, we convolve the resulting H i mass with a densityprofile using the method as follows. We note that any ar-bitrary density profile independent of the underlying halodensity profile can be used in this routine.According to the convolution theorem, the convolutionof the H i masses with any given profile is a multiplication inFourier space which is more computationally efficient. How-ever, generally the halo profile is a function of halo mass m i . In order to reduce computation, we apply a projectionalgorithm for the convolution. The H i masses are thereforebinned according to their halo mass into N bin bins. We create N bin cubes with each H i mass located at their respective halocentre position. Each cube is Fourier transformed and mul-tiplied by the Fourier-transformed profile of the mean halomass m i of the respective halo mass bin. We then sum allcubes to create the final intensity mapping cube. For the caseof the NFW profile, the algorithm converges for N bin = .We note that the continuous H i distribution is basedon the same underlying halo distribution of the lognormalrealisation but is independent of the field or sample galaxydensities and satellite positions. Discrete HI distribution.
In the discrete model, the H i HOD is associated with an underlying galaxy field
HOD (cid:104) N field (cid:105) which describes the distribution of all H i emit-ting objects. In our algorithm, the galaxy field is drawnfrom (cid:104) N field (cid:105) as described in Sec. 3.1. We then assignthe H i mass of each galaxy from a Gaussian distributionwith mean (cid:104) M HI , field (cid:105) = (cid:104) M HI ( m i )(cid:105)/(cid:104) N field ( m i )(cid:105) and stan-dard deviation σ HI = . (cid:104) M HI , field ( m i )(cid:105) for satellites and centrals respectively. The assumption that (cid:104) M HI , field (cid:105) = (cid:104) M HI ( m i )(cid:105)/(cid:104) N field ( m i )(cid:105) is only true if the probability of select-ing a galaxy is independent of H i mass, which precludes theuse of this algorithm for creating correlated samples. Thismodel allows for the galaxy field HOD and the H i HODto follow independent models and parametrisations withinthe limitation that M min and M max of the H i sample can-not be outside the defined galaxy mass range. In our study,we choose M min , HI = M min , field and M max , HI = M max , field forsimplicity. Additionally, the H i mass can be scaled by anindependent H i density profile, similarly to the continuouscase. In our study, we set the H i profile equal to the NFWprofile of the underlying galaxy field . The procedures described above produce galaxy cataloguesand H i intensity maps useful for determining their 1-haloclustering and Poisson noise. It is non-trivial to populate aphysically-motivated model for correlated galaxy-H i samplesin the framework of halomod . Commonly, the H i mass ofgalaxies is associated with their star-formation activity andother more complex mechanisms depending on the galaxy’sevolutionary state, which is beyond the scope of our work.As previously stated, in the continuous H i case the cor-relation factor R is determined through the dependence ofthe galaxy numbers on the H i mass per halo, or vice versa (a basic example of this is set out in Appendix A1), and thisimpacts the 1-halo contribution of the cross-power spectrum.For the discrete H i case, we demonstrated that a correla-tion between the H i distribution and the galaxy abundanceshas no impact on the 1-halo term. However, if H i masses andthe galaxy abundances are correlated, the averaged H i massper galaxy over the sample is modulated and hence the am-plitude of the cross-shot noise is changed, as we detail in thefollowing section. As one of our primary concerns is investi-gating the cross-shot noise, we demonstrate this effect withcorrelated simulations through the following procedure.In order to create a H i correlation, we either up- ordown-weight the H i masses of the galaxies in the sample bydrawing for each galaxy i a Gaussian variable δ M i , HI withzero mean and multiplying the absolute value | δ M i , HI | witha weighting factor w = { + , − } . When assigning H i masses,we then add w × | δ M i , HI | to the mean H i mass at m given bythe H i HOD (cid:104) M HI ( m )(cid:105) . This implies that all galaxies in the sample either have higher or lower H i mass than the mean ofthe Gaussian. Galaxies which are not part of the sample arenot affected by the weighting and their H i masses fluctuatearound the mean. This process slightly alters the measuredbrightness temperature of the H i intensity maps. However,if the galaxy sample is a small enough sub-sample of thewhole galaxy field , this effect will be minor. The additive shot noise contribution to the power spectrum,also referred to as Poisson noise in the literature, is due tothe finite number of data points used to probe a continu-ous field. In galaxy surveys, the shot noise is caused by the
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L.Wolz et al. finite number of galaxies in the sample employed to tracethe matter field. The resulting Poisson noise on the powerspectrum is scale-independent with the amplitude equal tothe inverse of the galaxy density: P PN = n g = N g / V . (22)The total power measured from galaxy survey data is P ( k ) = P h ( k ) + P h ( k ) + P PN .The shot noise of a galaxy distribution is not strictlyPoissonian. The deviations from the Poisson limit were ex-amined by e.g. (Hamaus et al. 2011; Baldauf et al. 2013;Paech et al. 2017). The deviations are caused by halo ex-clusion, non-linear clustering on small scales and satellitegalaxy distributions, where the fraction of satellite galaxiescan determine if the noise is sub- or super-Poissonian (Bal-dauf et al. 2013).In the halo model context, the shot noise of the halopower spectrum may be determined by the k → limit ofthe 1-halo term of the power spectrum, which results in thePoisson limit. This approach is correct when treating trac-ers without sub-sampling the halo with satellite populations.For galaxy populations including a central / satellite split,the k → limit of the 1-halo term does not result in the Pois-son limit and over-estimates the shot noise. Ginzburg et al.(2017) investigate the shot noise expression for dark matter,halos and tracers in the halo model framework, consideringgalaxy populations with satellites. They derive the correc-tion terms to the 1-halo term to accurately determine thedeviations from the Poissonian noise on scales k (cid:28) h / Mpc .For our scales of interest where the shot noise dominates theoverall power for k (cid:29) h / Mpc , the shot noise must convergetowards the Poisson limit of the 1-halo term neglecting thesatellite correlations (Ginzburg et al. 2017). For the remain-der of this study, we will only consider the Poisson limit ofthe shot noise and use the terms Poisson noise and shot noiseinterchangeably.We derive the Poisson limit of the shot noise through P g PN = (cid:169)(cid:173)(cid:171)∫ d m n ( m ) (cid:213) i = sat , cen (cid:104) N i ( m )(cid:105) (cid:170)(cid:174)(cid:172) − . (23)In intensity mapping, the nature of the shot noise de-pends on the H i model used. In general, the shot noise isgiven by the standard deviation (or second moment) of theobserved field (see Kovetz et al. 2017 and Breysse et al.2017), in this case the H i mass distribution, such that P HIPN = C ∫ d m n ( m )(cid:104) M HI ( m )(cid:105) . (24)In the halo model context this is equal to taking the k → limit of the 1-halo term neglecting the existence of satel-lite distributions, similar to in Eq. 18, see also Castorina &Villaescusa-Navarro (2017) for a similar result.In our specific case of the continuous model, the H i masses are sampled per halo, which means that the numberof samples is equivalent to the number of halos. However, theH i is not discretely populated, but convolved with the haloprofile which results in a continuous map of the H i in voxelspace. From a strict definition of Poisson noise originatingfrom discrete sampling and resulting in a scale-independentnoise, this means that the H i continuous power spectrum does not contain a Poisson noise contribution. The absenceof H i shot noise in the continuous case is due to the strictsmoothness of the H i distribution tracing the halo. Alter-natively, one could think of the 1-halo term as the Poissoncontribution which is convolved by the halo profile.For our H i discrete model, we assume that the H i massesare sampled per galaxy, rather than halo, so we need todetermine the second moment of the H i distribution pergalaxy where the H i per galaxy is given as (cid:104) M i HI , field ( m )(cid:105) = (cid:104) M i HI ( m )(cid:105)/(cid:104) N i field ( m )(cid:105) with i = { cen , sat } , (again we note thatthis is strictly only correct if H i and galaxy abundances areuncorrelated). The resulting Poisson noise of this model is P HI , dscPN = C ∫ d m n ( m ) (cid:213) i = sat , cen (cid:104) M i HI , field ( m )(cid:105) (cid:104) N i field (cid:105) . (25)We do not usually know the HOD of the underlying galaxy field , as well as the H i HOD, in order to determine the H i pergalaxy as a function of halo mass. In practise, the Poissonnoise can be modelled as a single additive number and fit toobservations. The shot noise in the cross-power spectrum of two galaxysamples, is determined by the galaxy density of the overlapof the two samples (see e.g. Smith 2009). If the two galaxysamples are mutually-exclusive, the amplitude of the Poissonnoise in the power spectrum is zero.As outlined in the previous paragraph, the continuousH i power spectrum does not contain a scale-independentPoisson noise contribution. Similarly, there is no Poissonnoise generated in the cross-correlation of a continuum anda discrete galaxy sample, as the H i distribution is assumedto be completely smooth and hence no additional samplingnoise can correlate with the sampling noise of the galaxies.Again, alternatively, one could think of the sampling noisebeing incorporated in the 1-halo term as the Poisson contri-bution is convolved by the smooth H i profile.The discrete H i model can be approached similarly tothe case of two galaxy samples where shot noise is deter-mined by the cross-section. In intensity mapping, it is as-sumed that each object emits H i and contributes to the H i maps. The cross-section of the H i maps and the sample ishence the number density of the sample and the Poissonnoise is inversely proportional to the galaxy number den-sity. The H i contribution to the Poisson noise is determinedby the average H i emission of the galaxies in the sample .This general expression for the cross-shot noise can also bederived considering the k → limit of the H i -galaxy 1-haloterm in absence of satellite populations. We derive the Pois-son noise of the cross-correlation of the discrete case as P g HI , dscPN = C HI (cid:169)(cid:173)(cid:171)∫ d m n ( m ) (cid:213) i = sat , cen (cid:104) M i HI , field ( m )(cid:105)(cid:104) N i sample (cid:105) (cid:170)(cid:174)(cid:172) × (cid:18)∫ d mn ( m )(cid:104) N ( m )(cid:105) (cid:19) − (26)This equation agrees with the derivation in Wolz et al.(2017b), where it was shown that the Poisson noise is di-rectly proportional to the averaged H i mass per galaxy in MNRAS , 1–15 (2017) ntensity mapping cross-correlations II: HI halo models including shot noise -1 k [Mpc − h] P ( k ) [ M p c h − ] Galaxy field modelGalaxy field model + PoissonGalaxy sample modelGalaxy sample model + PoissonGalaxy field lognormalsGalaxy sample lognormals
Figure 3.
The galaxy power spectra predicted by halomod forthe entire galaxy field and the selected galaxy sample in compari-son with an average power spectrum of 100 lognormal realisationswith a box of length / h drawn from the respective galaxyHOD. We show halomod predictions including and excludingPoisson noise contribution. the sample . This also implies that the amplitude of the Pois-son noise is sensitive to any correlations between H i and theabundance of galaxies in the sample . In the following, weverify these expressions by comparing the halomod predic-tions to simulations and showcase how the Poisson noise canbe fit in order to determine the averaged H i masses of galaxysamples. We run a suite of lognormal simulations with different boxsizes with length L ∈ { , , , } Mpc / h and N = pixels per side to create a valid comparison for all relevantscales k . We find that the halomod prediction agrees wellwith the lognormal simulations on all scales. In order to re-solve the scales dominated by the H i and cross Poisson noise,we closely inspect simulations with volume V = ( / h ) ,which are presented in the following figures. We simulate100 realisations of each lognormal field and the error barsof the following plots are given by the standard deviation ofthese realisations.In Fig. 3, we show the comparison of the auto-powerspectra of the lognormal galaxy population models with theassociated halomod prediction. The power spectrum mea-surements from the lognormal realisations naturally containPoisson noise contributions which we add to the halomod predictions using Eq. 23. We see that the halomod predic-tion including the Poisson noise is in agreement with esti-mates from the lognormal simulations. In this plot, we showour two galaxy models, field and sample (cf. Tab. 1). Thegalaxy densities of the populations are predicted by halo-mod as n sample = . ( h / Mpc ) and n field = . ( h / Mpc ) and estimated from the realisations as n sample = . ± . ( h / Mpc ) and n field = . ± . ( h / Mpc ) .In Fig. 4, the H i power spectra of the lognormal realisa-tions using the continuous and discrete model are shown incomparison to the analytic halomod predictions in units of K ( Mpc / h ) . For the continuous case describing smooth H i -1 k [Mpc − h] -5 -4 -3 -2 P ( k ) [ K M p c h − ] HI discrete modelHI discrete model + PoissonHI continuous modelHI discrete lognormalsHI continuous lognormals
Figure 4.
The H i power spectra predicted by halomod for the H i continuous model and the H i discrete model in comparison with aaverage power spectrum of 100 lognormal realisations with a boxof / h drawn from the respective galaxy HOD. Note that theH i continuous model does not include a scale-independent Poissonnoise contribution since it is estimated from a continuum field. Weshow halomod predictions including and excluding Poisson noisecontribution. distributions within halos independent of galaxy positions,we can see that the average of the simulations and the ana-lytic prediction agree very well within the errors. In the dis-crete model, the H i distribution is co-located with the galaxypositions and hence this model includes a Poisson noise con-tribution as described in Sec. 4. We add the theoretical pre-diction of the Poisson noise using Eq. 25 to the predictionsof halomod . The combined amplitude is in agreement withthe estimates of the lognormal distributions. Both H i modelsfollow the same HOD parameter model, except the discretemodel uses two additional parameters to describe the shapeof the underlying galaxy field HOD ( α field and M , field ). Byconstruction, both H i models predict the same H i bright-ness temperature T HI = . K . The lognormal simulationsof the continuous case produce T HI = . ± . K andin the discrete model produce T HI = . ± . K . Theerrors in these measurements increase with σ HI , the scatterwith which the H i masses per object were drawn from theH i HOD.The cross-power spectra of the galaxy sample with thetwo H i models are presented in Fig. 5. Even though thetheory calculation of the two models does not predict anyvisible deviation on all considered scales, we observe that theinclusion of Poisson noise in the discrete model considerablyincreases the power in the range k (cid:38) h / Mpc . The theoret-ical prediction of the cross Poisson noise is added to halo-mod using Eq. 26. As previously discussed, the cross Poissonnoise scales with the H i content of the galaxy populationaveraged over all halo masses, in this case for the galaxy sample . For this galaxy sample , we can measure an averageH i mass of log ( M HI , sample / M (cid:12) h ) = . from the lognor-mal realisations, which is very close to the prediction of thetheoretical model with log ( M HI , sample / M (cid:12) h ) = . . Wenote that in the considered lognormal realisations with vol-ume ( / h ) , the mean number of galaxies in the sample is relatively low, with 121. MNRAS000
The H i power spectra predicted by halomod for the H i continuous model and the H i discrete model in comparison with aaverage power spectrum of 100 lognormal realisations with a boxof / h drawn from the respective galaxy HOD. Note that theH i continuous model does not include a scale-independent Poissonnoise contribution since it is estimated from a continuum field. Weshow halomod predictions including and excluding Poisson noisecontribution. distributions within halos independent of galaxy positions,we can see that the average of the simulations and the ana-lytic prediction agree very well within the errors. In the dis-crete model, the H i distribution is co-located with the galaxypositions and hence this model includes a Poisson noise con-tribution as described in Sec. 4. We add the theoretical pre-diction of the Poisson noise using Eq. 25 to the predictionsof halomod . The combined amplitude is in agreement withthe estimates of the lognormal distributions. Both H i modelsfollow the same HOD parameter model, except the discretemodel uses two additional parameters to describe the shapeof the underlying galaxy field HOD ( α field and M , field ). Byconstruction, both H i models predict the same H i bright-ness temperature T HI = . K . The lognormal simulationsof the continuous case produce T HI = . ± . K andin the discrete model produce T HI = . ± . K . Theerrors in these measurements increase with σ HI , the scatterwith which the H i masses per object were drawn from theH i HOD.The cross-power spectra of the galaxy sample with thetwo H i models are presented in Fig. 5. Even though thetheory calculation of the two models does not predict anyvisible deviation on all considered scales, we observe that theinclusion of Poisson noise in the discrete model considerablyincreases the power in the range k (cid:38) h / Mpc . The theoret-ical prediction of the cross Poisson noise is added to halo-mod using Eq. 26. As previously discussed, the cross Poissonnoise scales with the H i content of the galaxy populationaveraged over all halo masses, in this case for the galaxy sample . For this galaxy sample , we can measure an averageH i mass of log ( M HI , sample / M (cid:12) h ) = . from the lognor-mal realisations, which is very close to the prediction of thetheoretical model with log ( M HI , sample / M (cid:12) h ) = . . Wenote that in the considered lognormal realisations with vol-ume ( / h ) , the mean number of galaxies in the sample is relatively low, with 121. MNRAS000 , 1–15 (2017) L.Wolz et al. -1 k [Mpc − h] -2 -1 P ( k ) [ K M p c h − ] HI discrete - galaxy sample model HI discrete - galaxy sample model + PoissonHI continuous - galaxy sample modelHI discrete - galaxy sample lognormalsHI continuous - galaxy sample lognormals
Figure 5.
The cross-power spectra predicted by halomod forthe H i continuous model with the galaxy sample , and the H i dis-crete model with the galaxy sample , in comparison with a av-erage power spectrum of 100 lognormal realisations of a box oflength / h drawn from the respective H i and galaxy HOD.Note that the H i continuous model does not include a scale-independent Poisson noise contribution since it is estimated froma continuum field. We show halomod predictions including andexcluding Poisson noise contribution. In the above example, the distribution of the H i within thegalaxy field for each halo mass m follows a Gaussian distri-bution with standard deviation σ HI . Thus there is no depen-dence of the H i content on the galaxy occupation within the sample . In reality, the amount of H i present in the galaxydepends on its evolutionary state. In general terms, blue,star-forming galaxies are expected to be H i -rich whereas red,quiescent galaxies are H i -deficient. In this work, focusing onthe concept of Poisson noise in intensity mapping, we do notconcern ourselves with details such as luminosity functionswhich would be required to accurately model these depen-dencies.In order to mimic the effect that a correlation betweenluminosity and H i mass would impose on the Poisson noise,we assume that the galaxy sample describes the HOD of aspecific type of galaxy which is correlated or anti-correlatedwith the H i content as described in Sec. 3.3. This correla-tion, as predicted, has no effect on the H i auto-power or theshape of the cross-power, but it changes the amplitude ofthe cross-Poisson noise as the averaged H i mass per galaxyin the sample is modified. Fig. 6 presents the result of theweighting of the H i for the galaxy sample . To demonstratethe change in the Poisson noise, we added the measuredPoisson noise from the lognormal simulations with colouredhorizontal lines to Fig. 6.In general, a specific mathematical model of the cor-relation of two samples is not available, and so we cannotdetermine the Poisson noise amplitude a priori . However,the halomod theory predictions can be used as a tool to fitthe pure Poisson noise contribution as well as measure thedeviation compared to an uncorrelated sample. In this section, we demonstrate the utility of the halomod algorithms to recover the parameters of a specific H i model -1 k [Mpc − h] -2 -1 P ( k ) [ K M p c h − ] Model + PoissonLognormals, weight=0Lognormals, weight=1Lognormals, weight=-1
Figure 6.
The cross power spectra predicted by halomod forthe H i discrete model with the galaxy sample with different H i weighting, in comparison with a average power spectrum of 100lognormal realisations with a box of length / h drawn fromthe respective H i and galaxy HOD. The dependence of the shapeof the power spectra on the H i weighting is negligible but theamplitude of the Poisson noise changes significantly. We show halomod predictions including and excluding Poisson noise con-tribution. via a Monte-Carlo Markov Chain (MCMC) maximum like-lihood fit. We use the Python package emcee (Foreman-Mackey et al. 2013) and fit the theory prediction of eachH i model to the estimated power spectra of the lognormalswith box size ( / h ) , which optimally resolves the shotnoise regime of the power spectra. We fit the averaged powerspectra of 100 realisations, as well as 10 individual realisa-tions. The covariance of each power spectrum measurementis given by the standard deviation of all realisations whichincludes Cosmic variance, fluctuations in the number densi-ties due to the small box size and variations due to the pop-ulation of H i masses. We note that due to the limited sizeof the box, which measures a minimum scale k ≈ . h / Mpc ,our parameter fitting is limited to the scales dominated bythe shot noise. The overall fitting could be improved us-ing a wider range of wavenumbers, however, some intensitymapping experiments such as the interferometer ASKAP areonly sensitive to similar scales k > . h / Mpc . For all cases,we only fit H i parameters, efficiently employing very tightpriors for parameters of the galaxy sample distribution. Wedo not attempt to fit the maximum halo mass M max , HI as forour tested box size, the abundance of these high mass halosis very low and the cut-off cannot be tested.We ran MCMCs with a total of samples and testedconvergence of the chains via the Gelman-Rubin criteriawhere all parameters passed with a threshold of R GR = . .We set Gaussian priors with the standard deviations of allHOD parameters α given as σ ( α i ) = . with i = { HI , field } and the standard deviation of all other HOD parametersas σ ( θ i ) = . . We note that our results, in particular forthe discrete case in which the model is primarily fit to theconstant amplitude of the Poisson noise, are not indepen-dent of the chosen priors. In particularly the H i amplitudeparameter, log A HI , is constrained by the prior and can notbe fit efficiently by the MCMCs unless a total temperatureconstraint is imposed. The best fit values are derived bycumulative statistics as the marginalised parameter likeli- MNRAS , 1–15 (2017) ntensity mapping cross-correlations II: HI halo models including shot noise . . . . α H I M , H I . . . . α fi e l d . . . M , fi e l d .
65 10 .
80 10 .
95 11 .
10 11 . M min , HI . . . . l og A H I . . . . α HI M , HI . . . . α field . . . M , field . . . . log A HI Figure 7.
The likelihood contours of the MCMC fit to therealisation-averaged power spectrum of the discrete H i model, us-ing the H i auto-correlation and its Poisson noise to constrain theHOD parameters of halomod . The dashed lines indicate the in-put parameter values. All masses are given as log and in unitsof M (cid:12) / h . hoods exhibit Non-Gaussian characteristics, as can be seenin Fig. 7.The resulting likelihoods of the MCMCs of the auto-power spectra of both H i models are displayed in Figs. 7and 8. In Tables 2, and 3 we present the outcomes of theparameter estimation for the auto- and cross-power spectraof both H i models. We individually fit the power spectrarather than perform a joint analysis as in many upcomingexperiments only one or the other will be available due tolimitations in the quality of data or lack of an optical galaxysample. The parameter fits can be extremely biased due tothe fluctuations in the lognormal realisations. In order to de-rive mean parameter fits and the expected variance includingCosmic variance while remaining computationally feasible,we run MCMCs on the mean power spectrum of all 100 reali-sations presented under names { Auto , Cross } in each table, inaddition to running MCMCs on 10 individual realisations,and presenting the mean and standard deviation of theirfits under { Auto (cid:205) = , Cross (cid:205) = } in the tables. Whereas theconstraints given by the MCMCs of the mean demonstratethe degeneracy within the halo model parameters, the stan-dard deviation over 10 realisation shows limitations due toCosmic variance.Table 2 and Fig. 7 presents the parameter constraintsof the discrete H i model, where the theory is primarily fitto the Poisson noise amplitude in the given k range. Fromcross-correlation, we derive the ensemble-averaged H i massof the galaxy sample from the estimated parameters which isgiven by the numerator of the cross Poisson noise. The inputparameters correspond to (cid:104) log ( M HI , g / M (cid:12) h )(cid:105) = . and the M min , HI = 10 . +0 . − . . . . α H I α HI = 0 . +0 . − . M , H I M , HI = 10 . +0 . − . .
80 10 .
88 10 . M min , HI . . . . l og A H I .
64 0 .
72 0 . α HI M , HI . . . . log A HI log A HI = 11 . ± . Figure 8.
The likelihood contours of the MCMC fit to therealisation-averaged power spectrum of the continuous H i model,using the H i -galaxy sample auto-correlation and to constrain theHOD parameters of halomod . The dashed lines indicate the in-put parameter values. All masses are given as log and in unitsof M (cid:12) / h . mean of the N = realisations gives (cid:104) log ( M HI , g / M (cid:12) h )(cid:105) N = . ± . .Table 3 and Fig. 8 display the results of the continuousH i model, where only 4 parameters need to be estimated inthe auto- and cross-correlation. The individual parameterconstraints are much tighter due both to the fewer numberof parameters, and the fact that the spectrum shape is notdominated by a single Poisson noise term. The uncertaintydue to Cosmic variance are comparable to the H i discretecase. In this study we present a new, adaptive description ofthe intensity mapping auto-power spectrum and cross-powerspectrum with galaxy surveys in the halo model framework(using halomod ). We introduce two different implementa-tions for the description of H i populations; the continuousH i model which populates H i within Dark Matter halos fol-lowing a smooth profile, and the discrete H i model whichco-locates H i masses with the positions of an underlyinggalaxy field, where H i and field can follow independent HODdescriptions. The models represent the opposite ends of thespectrum of currently used H i simulations. We inspect theimpact of the different H i models on the shapes of the auto-and cross-power spectra and find that the H i power spectraof both models only differ on scales k > h / Mpc , caused bythe additional central-satellite contributions in the 1-haloterm of the discrete model. The prediction of the 1-halo
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Table 2.
Discrete H i model: Marginalised parameter likelihoods given by the MCMC fit to the averaged auto- and cross-power spectrum,demonstrating the degeneracies within the HOD parameters, and the mean and the standard deviation of the MCMC fit to 10 realisationsmarked with (cid:205) i = , indicating the Cosmic variance. All masses are given as log and in units of M (cid:12) / h .Model M min , HI α HI M , HI α field M , field log A HI Input 11.0 0.7 11.0 0.5 11.0 11.0Auto . + . − . . + . − . . + . − . . + . − . . + . − . . ± . Auto (cid:205) i = . ± . . ± . . ± .
522 0 . ± . . ± . . ± . Cross . + . − . . ± .
12 10 . + . − . . + . − . . + . − . . + . − . Cross (cid:205) i = . ± . . ± . . ± . . ± . . ± . . ± . Table 3.
Continuous H i model: Marginalised parameter likelihoods given by the MCMC fit to the averaged auto- and cross-powerspectrum, demonstrating the degeneracies within the HOD parameters, and the mean and the standard deviation of the MCMC fit to10 realisations marked with (cid:205) i = , indicating the Cosmic variance. All masses are given as log and in units of M (cid:12) / h .Model M min , HI α HI M , HI log A HI Input 11.0 0.7 11.0 11.0Auto . + . − . . + . − . . + . − . . ± . Auto (cid:205) i = . ± . . ± . . ± . . ± . Cross . + . − . . + . − . . + . − . . ± . Cross (cid:205) i = . ± . . ± . . ± . . ± . terms of both H i models in the cross-power spectrum withgalaxies are very similar if the same H i halo profiles are used.We verified our analytic predictions with a set of lognor-mal realisations, and find that the major difference betweenthe models is the presence or absence of shot noise contri-butions. We review the current understanding of shot noisein galaxy and H i intensity mapping data and state analyticexpressions to determine the amplitude of the shot noisegiven the underlying HODs. Most notably, the shot noise onthe cross-power spectrum directly scales with the averagedH i mass of the optical galaxies, which is well-defined in thehalo model framework.We examine the shot noise properties of both H i modelsand find that the implementation of the continuous H i mod-els has no Poisson noise contribution to any power spectradue to the continuous, smooth nature of the H i density field.The shot noise of the discrete H i model is correctly predictedby halomod for the auto-correlation and cross-correlationwith a galaxy sample , given the H i content is independent ofthe galaxy sample abundances. In our examples, the Poissonnoise contributions dominate the amplitude of the overallpower spectra from scales k > h / Mpc .The cross-Poissonnoise is proportional to the averaged H i mass per galaxy inthe sample and, as such, can be used to determine the av-erage H i mass of galaxy samples without directly observingtheir H i content. Our halomod implementation is the firsttool to predict the cross-Poisson noise given H i and galaxyHOD parameters and will be useful for experimental fore-casts as well as observational interpretations.We demonstrate how the H i model parameters of the halomod predictions can be fit to the simulations usingMCMC techniques. These fits also estimate derived H i prop-erties such as the average brightness temperature, which is directly proportional to Ω HI , and the averaged H i mass pergalaxy in the cross-correlation with galaxy sample , a quan-tity of great interest in future cross-correlation experimentson small scales. This way, halomod has the potential to esti-mate the unknown parameters of the H i distribution tracedby the H i intensity maps, as well as determining the av-eraged H i masses of galaxy samples in intensity mappingcross-correlation experiments.We note that our study exclusively focuses on the im-pact of the halo occupation parameters on the power spec-tra and Poisson noise. We have not considered the degener-acy of cosmological parameters and halo occupation param-eters, but on the scales considered in this work the effectof galaxy evolution dominates. We note that non-linear ef-fects of the power spectrum and peculiar velocities alter theshape and amplitude of the 1-halo term, however, on smallenough scales, the contribution of the Poisson noise is consid-erably more dominant than the 1-halo term. In these cases,the halomod prediction of the cross-Poisson noise could beadded to a more sophisticated power spectrum model whichincludes these effects or the fits could be performed to theprojected correlation function to suppress redshift-space dis-tortions.In this project, we did not employ data-motivated H i models as we aim to demonstrate a maximally flexible frame-work for H i auto- and cross-power spectrum and their Pois-son noise predictions that can be adapted to individual ex-periments’ needs. Our model allows us to easily import anyshape and parametrisation of the H i to-halo relation and ex-amine their predictions. In future work, more data-drivenmodels will be implemented in order to compare predictionswith observations. MNRAS , 1–15 (2017) ntensity mapping cross-correlations II: HI halo models including shot noise ACKNOWLEDGEMENTS
LW is supported by an ARC Discovery Early Career Re-searcher Award (DE170100356). This research was con-ducted by the Australian Research Council Centre of Excel-lence for All-sky Astrophysics (CAASTRO), through projectnumber CE110001020. This work was performed on thegSTAR national facility at Swinburne University of Tech-nology. gSTAR is funded by Swinburne and the AustralianGovernment’s Education Investment Fund.
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APPENDIX A: CORRELATION MODELS FORHI-GALAXY SAMPLESA1 Continuous HI Model
In this model, the gas is not co-located with observed galax-ies, forming a spatially-independent smooth profile withinthe halo. In reality, we expect that while the averaged profileof the gas is smooth, it will be lumpy on scales much smallerthan the halo radius. It is then expected that the prospectsof observing a galaxy in the sample may be dependent onthe H i density around the location of the galaxy. However,dealing with this general situation, in which spatial scaleswithin the halo are correlated according to the typical sizeof the H i “lumps” is rather difficult, and we may considertwo extreme cases in more detail. The first is that in whichthe lumps are infinitely broad, or rather that the H i profile isperfectly smooth for every halo. The second is that in whichthe “lumps” are Dirac- δ functions, but this is equivalent to MNRAS000
In this model, the gas is not co-located with observed galax-ies, forming a spatially-independent smooth profile withinthe halo. In reality, we expect that while the averaged profileof the gas is smooth, it will be lumpy on scales much smallerthan the halo radius. It is then expected that the prospectsof observing a galaxy in the sample may be dependent onthe H i density around the location of the galaxy. However,dealing with this general situation, in which spatial scaleswithin the halo are correlated according to the typical sizeof the H i “lumps” is rather difficult, and we may considertwo extreme cases in more detail. The first is that in whichthe lumps are infinitely broad, or rather that the H i profile isperfectly smooth for every halo. The second is that in whichthe “lumps” are Dirac- δ functions, but this is equivalent to MNRAS000 , 1–15 (2017) L.Wolz et al. the discrete H i model which we consider in the followingsubsection.Suppose that the H i profile of every halo is always com-pletely smooth, and is constant with the underlying halomass. Suppose also that there is a distribution of H i massesfor a given halo mass, for which the mean is (cid:104) M HI ( m )(cid:105) , andthe variance is σ (the distribution remains arbitrary, butone may like to think of it as a Gaussian or Lognormal). Ifa particular halo has an H i mass from the upper-tail of itsdistribution, then the H i density of that halo is increaseduniformly everywhere in the halo, because it is necessar-ily completely smooth. Now consider a sample of observedgalaxies. The probability of finding a galaxy at any point ina given halo may depend on the density of the H i in thatlocation (in fact, it may depend on much more than that,for example, it may depend on the H i density in nearby lo-cations, or the dynamical state of the H i rather than justits abundance, but these are considered to be minor compli-cations which we will ignore). However, since the density ofH i at any given location is determined by the density at allother locations, or rather, the density at any location is fullyspecified by the total H i mass in the halo – due to its smooth-ness – the total expected number of observed galaxies in thehalo is completely determined by its H i mass. Summarily,we have the following system: M i HI ∼ φ ( m HI , m ) , N i ∼ Poisson ( f ( M i HI )) , (A1)where f is some function which converts the actual H i massof a halo into the expected number of observed galaxies.While this function is arbitrary, a simple but flexible toymodel is such that f = n ( M i HI / A ) γ . Letting ˜ A ( m ) be theaverage amount of H i per galaxy in halos of mass m , weobtain that (cid:104) N (cid:105) = (cid:104) M HI (cid:105)/ ˜ A . Furthermore we find that ˜ A = A γ n (cid:104) M HI (cid:105)(cid:104) M γ HI (cid:105) . (A2)We focus hereafter on some special cases, γ ∈ (− , , ) , cor-responding to anti-correlated, uncorrelated and correlatedcases. For these we have ˜ A = ( An ) − (cid:104) M HI (cid:105)(cid:104) M − (cid:105) , γ = − (cid:104) M HI (cid:105) n , γ = An , γ = . (A3)We note that the distribution of N in such a setup is notnecessarily Poisson, as we generally assume it to be. Never-theless, it is not likely to be significantly different to Poisson,and in any case, this fact does not affect the rest of our cal-culations.We wish to calculate the value of (cid:104) N s M HI (cid:105) . This can beachieved by using the law of total expectation, (cid:104) N s M HI (cid:105) = ∫ dm (cid:48) m (cid:48) φ ( m (cid:48) , m ) ∞ (cid:213) k = k Pois ( f ( m (cid:48) )) = ∞ (cid:213) k = n A γ k ( k − ) ! ∫ dm (cid:48) m (cid:48) γ k + φ ( m (cid:48) , m ) e − n ( m (cid:48) / A ) γ . (A4)If we assume that φ is a Gaussian distribution, then formasses m at which the expected number of galaxies is large, since the Poisson distribution tends to a Gaussian, the resulttends to (cid:104) N s M HI (cid:105) = n A − γ √ πσ ∫ dm (cid:48) m (cid:48) + γ e −( m (cid:48) −(cid:104) M HI (cid:105)) / σ . (A5)While this is in general unsolvable, it yields solutions for ourthree cases of interest: (cid:104) N s M HI (cid:105) ≈ n A , γ = − n (cid:104) M HI (cid:105) = (cid:104) N (cid:105)(cid:104) M HI (cid:105) , γ = n (cid:104) M HI (cid:105) + σ A , γ = . (A6)Thus the correlation function is (in the large- N limit): R ( m ) = (cid:104) N s M HI (cid:105) − (cid:104) N (cid:105)(cid:104) M HI (cid:105) (cid:112) (cid:104) N (cid:105) σ HI = σ HI (cid:113) (cid:104) M HI (cid:105) ˜ A (cid:18) (cid:104) M − (cid:105) − (cid:104) M HI (cid:105) (cid:19) , γ = − , γ = σ HI √ ˜ A (cid:104) M HI (cid:105) , γ = . (A7)The question of how to calculate (cid:104) M − (cid:105) remains. Wefind that a useful empirical formula is such that (cid:104) M − (cid:105) ≈ (cid:104) M HI (cid:105) + σ (cid:104) M HI (cid:105) (A8)when M HI is a Gaussian variable, and (cid:104) M HI (cid:105)/ σ HI > . Thislatter condition must be obeyed in any case to ensure thedescription is physically appropriate, otherwise a significantpart of the probability density puts M HI < . Under thisapproximation, the correlation function becomes R γ = − ( m ) = − σ HI (cid:104) M HI (cid:105) (cid:113) M HI ˜ A (cid:104) M HI (cid:105) + σ (A9) A2 Discrete HI Model
The result in the case of the discrete model, in which the“lumps” are Dirac- δ functions, is much simpler. The typicalassumption of the galaxies obeying a Poisson distributioncarries with it the assumption that they are spatially in-dependent, and we have by construction specified that theH i components are also spatially independent. This impliesthat while the probability of observation of a galaxy at acertain point may be dependent on the H i in that location,it is entirely uncorrelated with any other point. This meansthat all correlations exist at a separation of zero, which isnot represented in the power spectrum at all. Alternativelyone may consider Eq. 13, in which the total contribution ofthe satellite-satellite term has a − Q term, which accounts forall self-pairs. After subtracting the self pairs, no other pairscontain any correlations, and so in general we have that (cid:104) N s M HI (cid:105) − Q = (cid:104) N s (cid:105)(cid:104) M HI (cid:105) . (A10)Nevertheless, while these correlations cannot change the shape of the power spectrum, they do affect the level ofshot-noise present. This is simple to conceptualise; since theshot-noise depends on the average H i mass within galaxiesof the sample , a correlation which favours observing galaxieswhich contain a higher H i mass will therefore accordinglyincrease the shot-noise, and vice versa. The net result is a MNRAS , 1–15 (2017) ntensity mapping cross-correlations II: HI halo models including shot noise constant additive factor to the observed power spectrum,and thus detailed modelling will not usually be required —the constant may just as well be fit and then interpreted.In practice, one may conceive of the correlation occur-ring in multiple ways. In any of these, if the mean H i massof the sample can be calculated, it can be used to directlyinfer the amplitude of the shot noise. This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS000