Galaxies and Supermassive Black Holes at z <= 0.1: The Velocity Dispersion Function
DDraft version April 2, 2019
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GALAXIES AND SUPERMASSIVE BLACK HOLES AT z (cid:54) .
1: THE VELOCITY DISPERSION FUNCTION
Farhanul Hasan , , Alison F. Crocker Department of Physics, Reed College, Portland, OR 97202 and Department of Astronomy, New Mexico State University, Las Cruces, NM 88003
Draft version April 2, 2019
ABSTRACTWe study the distribution of central velocity dispersion, σ , for > . (cid:54) z (cid:54) .
1. We construct the velocity dispersion function (VDF) from samples complete for all σ , where galaxies with σ greater than the σ -completeness limit of the SDSS spectroscopic survey areincluded. We compare two different σ estimates; one based on SDSS spectroscopy ( σ spec ) and anotheron photometric estimates ( σ mod ). The σ spec for our sample is systematically higher than σ mod for allranges of σ , implying that rotational velocity may affect σ spec measurements. The VDFs measuredfrom these quantities are remarkably similar for lower σ values, but the σ mod VDF falls faster than the σ spec VDF at log σ (cid:38) .
35. Very few galaxies are observed to have σ (cid:38)
350 km s − . Despite differencesin sample selection and methods, our VDFs are in close agreement with previous determinations for thelocal universe, and our results confirm that complete sampling is necessary to accurately discern theshape of the VDF at all ranges. We also find that both late and early type galaxies have σ spec >σ mod ,suggesting that the rotation component of most galaxies figure significantly into σ spec measurements.Early-type galaxies dominate the population of high σ galaxies, while late-type galaxies dominatethe low σ statistic. Our results warrant a more thorough and cautious approach in using long-slitspectroscopy to derive the statistics of local galaxies. Higher quality photometric measurements willenable more accurate and less uncertain measurements of the σ mod VDF, as described here. A follow-up paper uses the final samples from this work in conjunction with the M BH - σ relation to derive the z (cid:54) . Keywords: galaxies: general – galaxies: evolution – galaxies: elliptical and lenticular, cD – galaxies:spiral – galaxies: statistics – methods: data analysis INTRODUCTIONSupermassive black holes (SMBH) exist at the cen-ters of most, if not all, massive galaxies (Richstone et al.1998; Ferrarese and Ford 2005; Kormendy and Ho 2013).The mass of a central black hole is known to correlatewell with fundamental properties of a galaxy, includingbulge mass (Kormendy and Richstone 1995; Marconi andHunt 2003), bulge luminosity (Magorrian et al. 1998;Graham 2007), and stellar velocity dispersion (Ferrareseand Merritt 2000; Gebhardt et al. 2000; McConnell andMa 2013). The tightness of these correlations have ledmany to postulate that the growth of galaxies is intrin-sically connected to the growth of their SMBHs. Conse-quently, understanding the effect of SMBHs is of vital im-portance to any successful theoretical or semi-analyticalmodel of galaxy formation and evolution (Shankar et al.2009, 2013; Heckman and Best 2014; Aversa et al. 2015).Statistical studies of galaxies can be used as robustprobes into galaxy evolution. Distribution functions havebeen used to describe the universe’s galaxy populationsin terms of fundamental galaxy properties such as lu-minosity (e.g., Blanton et al. 2001; Bernardi et al. 2003;McNaught-Roberts et al. 2014), stellar mass (e.g., P´erez-Gonz´alez et al. 2008; Kelvin et al. 2014; Weigel et al.2016), and velocity dispersion (e.g., Sheth et al. 2003;Bernardi et al. 2010; Sohn et al. 017a), helping us char-acterize the population of galaxies in both the local uni-verse (Choi et al. 2007; Baldry et al. 2012) and at highredshifts (Marchesini et al. 2007; Pozzetti et al. 2010).Distribution functions have also been extremely helpful in describing the population of SMBHs via the black holemass function (BHMF; e.g., Salucci et al. 1999; Yu andTremaine 2002; Graham et al. 2007; Li et al. 2011). Ina subsequent paper, we will use the velocity dispersiondistribution function we derive here to estimate a newlocal BHMF.A crucial link between simulations and observations isthe dark matter halo mass distribution function. Whileboth the luminosity function (LF) and stellar mass func-tion (SMF) of galaxies have been used to trace dark mat-ter halos (Yang et al. 2008, 2013), connecting the LF toDM halo mass is non-trivial and the SMF has shownlittle dependence on galaxy characteristics such as mor-phology, color, and redshift (Calvi et al. 2012, 2013).Thus, neither the LF nor the SMF may be very strongtracers of properties of the DM halo. The central stellarvelocity dispersion (velocity dispersion, or σ , hereafter),on the other hand, is a dynamical measurement which re-flects the stellar kinematics governed by the central grav-itational potential well of a galaxy, and does not sufferfrom photometric biases or systematic uncertainties inmodeling stellar evolution (Conroy et al. 2009; Bernardiet al. 2013). The central velocity dispersion has beenobserved to be strongly correlated with DM halo mass,leading some to conclude that it may be the best ob-servable parameter connecting galaxies to their DM halomass (Wake et al. 2012; Zahid et al. 2016).Large-scale surveys of galaxies such as the Sloan Digi-tal Sky Survey (SDSS; York et al. 2000; Stoughton et al.2002; Alam et al. 2015; Abolfathi et al. 2018) have probedgalaxy populations in great detail, across cosmic time. a r X i v : . [ a s t r o - ph . GA ] M a r As a result, statistical analyses have been performed onthese large datasets to establish global distributions ofgalaxies in terms of observables. In particular, SDSSgalaxies have been used in deriving the velocity disper-sion function (VDF) of galaxies in the field population(Sheth et al. 2003; Mitchell et al. 2005; Choi et al. 2007;Sohn et al. 017a) and the cluster population (Munariet al. 2016; Sohn et al. 017b).Deriving meaningful statistical information from theVDF requires a sample that is complete and unbiased.Volume-limited samples are incomplete below a certainmagnitude limit, so distribution functions derived fromsuch samples may be biased by selection effects (P´erez-Gonz´alez et al. 2008; Weigel et al. 2016; Zahid et al.2016). SMFs have been derived from samples completein stellar mass, M (cid:63) , where the M (cid:63) completeness limit wasparametrized as a function of redshift (e.g, Fontana et al.2006; Marchesini et al. 2009; Pozzetti et al. 2010; Weigelet al. 2016). Recently, Sohn et al. (017a) outlined anapproach to empirically determine a redshift-dependent σ -completeness limit in order to derive a σ -complete sam-ple for a robust measurement of the VDF from local qui-escent SDSS galaxies. Here, we follow their approachto generate a σ -complete sample from ∼ z (cid:54) . σ values may include both the true σ and an additional contribution from the rotational veloc-ity, which is worse if the galaxy is more rotation domi-nated, the aperture extends further into the galaxy, orthe galaxy is more inclined. This inherent bias in the ve-locity dispersion measurements may lead to inaccuraciesin the derived VDF. Thus, in this paper, we also use ve-locity dispersions estimated by the approach of Bezansonet al. (2011) to infer the true velocity dispersion based onthe virial theorem and photometric estimates. Compari-son of the VDF generated from this method is importantto characterize the systematic uncertainties still presentin two of our best ways of experimentally determiningthe velocity dispersion for nearby galaxies.We use our σ -complete sample to construct a VDFfor (cid:38) . (cid:54) z (cid:54) . /V max methodology to determine the black hole massfunction from our σ -complete sample (Hasan & Crocker,in prep).This paper is organized as follows: In Section 2, wepresent the data used in this work, including spectro-scopic and photometric data from SDSS and morpholog-ical galaxy classifications from the Galaxy Zoo project(Lintott et al. 2008, 2011). Here, we also derive twodifferent estimates of velocity dispersion. We accountfor selection effects by constructing a σ -complete sam-ple from z (cid:54) . H = 70 km s − Mpc − , Ω m = 0 . Λ = 0 . DATAIn order to construct a VDF of local galaxies, weuse velocity dispersion measurements from SDSS spec-troscopy, as well as photometric data. SDSS, whichbegan routine operations from 2000 (York et al. 2000;Stoughton et al. 2002; Strauss et al. 2002), has collectedspectra of over a million galaxies and 100000 quasars,and imaged about a third of the sky ( ∼ Photometric data
We use the SDSS main galaxy sample (Strauss et al.2002), which is a magnitude-limited sample with r -bandPetrosian (1976) magnitude, r p < .
77 mags, limited to z (cid:46) .
3. The main galaxy sample is taken from SDSSData Release 12 (Alam et al. 2015). We restrict oursample of local galaxies to z (cid:54) .
1, which brings oursample size from ∼ ∼ D n r -band magnitudes. We use the z = 0 K-correction fromthe NYU Value Added Galaxy Catalog (Blanton et al.2005). Hereafter, the K-corrected absolute r -band mag-nitude is referred to as M r .2.2. Velocity Dispersions
SDSS selects ∼ .
9% of r p < .
77 objects from itsimages as spectroscopic targets. SDSS spectroscopy is ∼
95% complete, and produces spectra which cover thewavelength range of 3500 − R ∼ . ThePortsmouth templates are capped at σ = 420 km s − , sothat is the upper limit of reliable measurements for oursample. Furthermore, measurements where σ (cid:46)
70 kms − have low signal-to-noise (S/N) ratio, so they are un-reliable too. The σ measurements from the Portsmouthgroup all have S/N >
10 and the median uncertainty is 7km s − .The SDSS velocity dispersions are based on spectra ob-tained from a 1 . (cid:48)(cid:48) fiber. These dispersions are afterwardscorrected to an aperture of R e /8 ( R e = r -band effectiveradius) by adopting the calibration of Cappellari et al.(2006): (cid:18) σ SDSS σ e (cid:19) = (cid:18) R SDSS R e / (cid:19) − . , (1)where R SDSS = 1 . (cid:48)(cid:48) is the SDSS aperture radius and σ e is the velocity dispersion measured within a radius R e /8 from the center. The aperture corrections are smallenough that they do not have a major impact on ourresults, and our qualitative conclusions don’t change if we . . . . log σ mod [km s − ] . . . . . . . . . l og σ s p e c [ k m s − ] Figure 1.
Observed spectroscopic σ ( σ spec ) vs. modeled σ basedon photometric estimates ( σ mod ) for our final sampleThe dashedblue line indicates 1:1 match. Many log σ mod values are below ∼ .
9, while few such values are observed for log σ spec . adopt a slightly different radial dependence. Hereafter,we refer to these σ e as σ spec .The velocity dispersion estimate for many emission linegalaxies are likely compromised by rotation, in whichcase the reported σ values are overestimated. Hence, thepresence of rotation in most galaxies introduces biases ina VDF measured from σ spec . To account for this issue, weinfer another estimate of σ based on the virial theoremand photometric estimates of mass and size ( R e ). Inparticular, we follow Taylor et al. (2010) to determine amodeled σ , σ mod , for each galaxy: σ mod = (cid:115) GM (cid:63) K v ( n ) R e , (2)where M (cid:63) is the stellar mass, n the S´ersic index, and K v ( n ) a correction term accounting for the effects ofstructure on stellar dynamics, approximated by (Bertinet al. 2002): K v ( n ) (cid:39) . .
465 + ( n − . + 0 . . (3)For this analysis, we obtained M (cid:63) estimates fromthe MPA-JHU DR8 catalogues (Kauffmann et al. 2003;Brinchmann et al. 2004) and n and R e estimates fromthe NYU VAGC. The effective radii are based on fits toazimuthally averaged light profiles and are equivalent tocircularized effective radii.We compare the spectroscopically derived σ spec to thephotometrically inferred σ mod in Fig. 1. We find thatthe galaxies in our sample have systematically higher σ spec than σ mod for the entire range of velocity disper-sions (log σ ∼ . − . σ mod is in general lower than σ spec for the low end ( σ mod (cid:46) σ spec relative to σ mod is ex-actly what is expected for rotation-dominated galaxieswhich are more plentiful at lower velocity dispersions.Being noisy below σ (cid:46)
70 km s − , SDSS measurementsdon’t report many galaxies with log σ (cid:46) .
9. Seeing thata larger proportion of galaxies have log σ mod (cid:46) σ spec (cid:46) σ spec may be giving us artificially high velocitydispersions for many galaxies. This leads to an incorrectrepresentation of their true velocity dispersions.While the modeling adds a layer of complication to anotherwise directly measured quantity, we consider it avaluable complementary method of obtaining a velocitydispersion estimate. Going forward, we construct VDFsbased on both these σ estimates in section 4.1.2.3. Galaxy Zoo
Galaxy Zoo is a web-based project in which the publicis invited to visually classify galaxies from the SDSS bythe two primary morphological types: spirals (late-types)and ellipticals and lenticulars (early-types) (Lintott et al.2008). The classifications of the general public agree withthose of professional astronomers to an accuracy higherthan 10 percent, and hundreds of thousands of systemsare reliably classified by morphology at more than > σ confidence. A fundamental advantage of Galaxy Zoo isthat it obtains morphological information by direct visualcomparison instead of proxies such as color, concentra-tion index or other structural parameters. Using these asproxies for morphology may introduce various systematicbiases, leading to unreliable classifications and the needfor more direct and robust means (Lintott et al. 2008;Bamford et al. 2009).Each individual user assigns a vote for either spi-ral (late-type) or elliptical (early-type) for each galaxy.The fraction of the vote for each type for all objects isweighted as described in Lintott et al. (2008) and thendebiased in a consistent fashion (though Bamford et al.(2009) outlines complications with the debiasing). Fi-nally, a vote fraction of 80% ensures an object is classi-fied with certainty as either early or late type (the restbeing classified “uncertain”). Thus, we only selected ob-jects with either spiral ( ∼ ∼ SELECTION EFFECTS: COMPLETENESS OFTHE SAMPLEA statistically complete sample is necessary to mea-sure the statistical distribution of any galaxy property.A magnitude-limited sample such as the SDSS spec-troscopic sample can easily be made volume-limited bychoosing an appropriate absolute magnitude limit foreach redshift z , such that within the entire volume allgalaxies brighter than this magnitude limit should bedetected. However, this volume-limited sample is onlycomplete for a range in absolute magnitude, M r , not fora range in velocity dispersions (Sohn et al. 017a; Zahidand Geller 2017). Sohn et. al. (their Figure 1) show howthere is substantial scatter in σ for a fixed M r . Similarly,we find that for example, at M r = − . σ spec variesfrom ∼
50 km s − to ∼
315 km s − . Converting the abso-lute magnitude limit for a volume-limited sample, M r, lim ,to a limit at which σ is complete is therefore not a trivialexercise (Sheth et al. 2003; Sohn et al. 017a).Because of this broad distribution in σ spec and σ mod forany M r , there are many low σ galaxies which make it tothe sample by virtue of being bright enough, while somegalaxies boasting high σ are excluded for being less lu-minous. Completeness analysis helps us ensure that oursample isnt preferentially selecting, say just the brightestgalaxies. The goal in this section is to obtain the limitat which σ is ( ∼ σ spec and σ mod samples. 3.1. Constructing σ -complete samples In order to generate a σ -complete sample, we need toparametrize σ lim , the limit at which the sample is com-plete for σ > σ lim , as a function of redshift z . We takean empirical approach in doing this, which closely fol-lows the methodology of Sohn et. al. (017a). First, wedivide the volume-limited sample of 0 . (cid:54) z (cid:54) . (cid:52) z = 0 . z = 0 .
01, and ending at z = 0 .
1. For eachsubsample, we derive the 95 th percentile distribution of σ for galaxies with M r (cid:54) M r, lim to obtain σ lim , the σ -completeness limit at the maximum redshift ( z max ) of thesubsample. We repeat this process so that σ lim is foundfor z max ranging from 0.01 to 0.1. .
02 0 .
04 0 .
06 0 .
08 0 . z . . . . . . . l og σ [ k m s − ] σ spec , lim σ mod , lim Sohn et. al. (2017)
Figure 2. σ -completeness limit as a function of redshift for the σ spec sample (dashed purple), σ mod sample (solid dark blue), andSohn et. al.’s (017a) sample. All of these are 2 nd order polynomialfits to the σ -completeness limit in bins of ∆ z = 0 . . (cid:54) z (cid:54) .
1, and both of our fits for0 . (cid:54) z (cid:54) . For both the σ spec and σ mod samples, we fit the distri-bution of σ lim against z to a 2 nd order polynomial. Thepolynomial fits we obtain are:log σ spec , lim ( z ) = 1 .
68 + 9 . z − . z (4)log σ mod , lim ( z ) = 1 .
49 + 12 . z − . z (5)The σ -completeness limit is plotted as function of red-shift in Fig. 2, for our σ spec and σ mod samples, as well as that of Sohn et. al. (017a), whose fit was also a 2 nd orderpolynomial, but in the redshift range 0 . (cid:54) z (cid:54) .
1. Asexpected, we find that σ lim increases as z max increases.We only include galaxies with σ spec > σ spec , lim in the fi-nal σ spec sample and those with σ mod > σ mod , lim in thefinal σ mod sample.The final σ spec sample consists of ∼ galaxiesand the final σ mod sample contains ∼ galaxiesin the redshift range 0 . (cid:54) z (cid:54) .
1. These are both ∼ . D n σ -complete sam-ples with those of the magnitude-limited samples, wefind that galaxies in the σ -complete samples are brighterand have less uncertain σ measurements. The σ valuesare also larger in general – the mean σ spec and σ mod in-creases from ∼
100 km s − to ∼
165 km s − and ∼ − to ∼
160 km s − , respectively, as we go frommagnitude-limited samples to σ -complete samples. The σ -complete samples eliminate many of the low σ galax-ies which were present in the magnitude-limited sample,causing the mean σ to go up. THE VELOCITY DISPERSION FUNCTIONThe VDF is defined as the number density of galax-ies with a given σ per unit logarithmic σ . There area few ways to compute the VDF, including the 1 /V max method (Schmidt 1968), the parametric maximum like-lihood STY method (Efstathiou et al. 1988), and thenon-parametric stepwise maximum likelihood (SWML)method (Sandage et al. 1979). Several studies, includ-ing Weigel et al. (2016) review these methods and thedistribution functions resulting from them. We choosethe 1 /V max method for our computation due to its sim-plicity and because we do not have to initially assumea functional form of the VDF. One drawback of thismethod is that it may produce biased results if there areinhomogeneities on large scales. The 1 /V max and SWMLmethods have been shown to produce equivalent results(Weigel et al. 2016; Sohn et al. 017a).4.1. Constructing the VDF
The 1 /V max method takes into account the relativecontribution to the VDF of each galaxy with disper-sion σ , by volume-weighting the velocity dispersions.Each object is weighed by the maximum volume it couldbe detected in, given the redshift range and the σ -completeness of the sample. This corrects for the well-known Malmquist bias (Malmquist 1925).To generate our VDFs for both of our final samples,we first divide velocity dispersions in equal bins of width∆ log σ = 0 .
02, ranging from log σ = 1 . − tolog σ = 2 . − . Then, the number density of galaxiesin a specific σ bin j is given by the following sum:Φ j ( σ )∆ log σ = N bin (cid:88) i V max ,i , (6)where N bin is the number of galaxies in the bin and1 /V max is the maximum volume at which a galaxy i withvelocity dispersion σ i could be detected in. In a flat uni-verse, the comoving volume, 1 /V max , is given by (Hogg1999): V max ,i = 4 π survey Ω sky (cid:2) D C ( z max ,i ) − D C ( z min ,i ) (cid:3) . (7)For all galaxies, z min ,i = 0.03, since that is the lowerredshift limit of our sample. Ω sky = 41253 deg is thesolid angle of the sky and Ω survey = 9200 deg is the solidangle covered by SDSS. For z max ,i , we take the maxi-mum redshift galaxy i could have, based on σ i and theparametrized completeness limits from above (Fig. 2 andeqs. 4 and 5).The VDF, Φ( σ ), was calculated from eq. 6, for both σ spec and σ mod . The uncertainties on the VDF were esti-mated by a Monte Carlo method. We ran 10000 simula-tions of the VDF calculation, each time randomly mod-ifying σ values with the associated uncertainties, ∆ σ ,assuming a gaussian error distribution. For σ mod , wepropagated the errors on uncertainties in M (cid:63) , R e , and n . The resulting VDF, and associated uncertainties, aretabulated in Table 1.Fig. 3 shows the VDFs estimated from our σ -complete σ spec and σ mod samples. The VDFs are fairly reliablefor ∼ . (cid:54) log σ (cid:54) .
6. There are very few objectswith log σ (cid:38) .
6, resulting in uncertain VDF measure-ments for those bins. The shape of the VDFs are verysimilar throughout the range of velocity dispersions stud-ied. Both VDFs decline slowly (toward higher σ ) at lowand medium σ values (log σ (cid:46) . σ (cid:38) .
3. The σ mod VDF fallsmore rapidly than the σ spec VDF at log σ (cid:38) .
35, caus-ing it to shift to the left relative to the σ spec VDF. Athigher velocity dispersions (i.e. higher masses), we ex-pect galaxies to be less rotation-dominated, which is intension with our σ mod VDF being lower than the σ spec VDF at log σ (cid:38) .
5. If anything, we would expect the σ mod VDF to be lower than σ spec at the lowest σ s, or per-haps a systematic divergence between these throughoutthe σ range studied. So we speculate that the calculated σ mod VDF may be dubious, especially at higher σ .We refrain from making claims about the “true” VDFand which tracer – σ spec or σ mod – does a better job ofmeasuring it. Nevertheless, the σ mod VDF is a valuabletool to interpret σ spec measurements, as well as the statis-tics based on SDSS spectroscopy. We note here that theapparent discontinuity just below log σ ∼ . Schechter function
The LF, SMF, VDF and other distribution functionsof galaxies are often described by a Schechter (1976)function. Schechter’s original function to approximatethe observed LF, Φ( L ), exhibits a power-law behavior atlower values of L , followed by an exponential cutoff at acharacteristic L (cid:63) . We adopt a similar expression for ourVDF, Φ( σ ), but find that adding an additional parame-ter, β , gives a better approximation to the data. Thus,following the functional form of the BHMF from Allerand Richstone (2002), we adopt the following modifiedSchechter function: . . . . log σ [km s − ] − − − − − − l og Φ [ M p c − d e x − ] This work: σ spec This work: σ mod Figure 3.
VDFs for our σ -complete samples, based on σ spec –purple diamonds – and σ mod – blue circles. The purple dashedcurve shows the 4-parameter best-fit Schechter function (eq. 8) forour σ spec VDF, while the blue dashed curve is the best-fit func-tion for our σ mod VDF. The pink and light blue shaded regionsrepresent the 68% confidence intervals for the σ spec and σ mod fits,respectively. Φ( σ ) dσ = Φ (cid:63) (cid:18) σσ (cid:63) (cid:19) α +1 exp (cid:34) − (cid:18) − σσ (cid:63) (cid:19) β (cid:35) dσ . (8)where the characteristic truncation value of σ is σ (cid:63) and Φ( σ (cid:63) ) = Φ (cid:63) . The slope of the power law is α + 1( α = − σ < σ (cid:63) ).The σ spec and σ mod VDFs are both parametrized withthe form of eq. 8. The parameters of these fits are givenin Table 4.2. The uncertainties on the parameters werederived by using the same Monte-Carlo simulations bywhich uncertainties on the VDF were found (assuminggaussian error distributions again).4.3.
Comparison with literature
The differences in the shape of the derived VDF forlow σ , and in the characteristic σ at which the VDFtruncates, were attributed by Choi et. al. (2007) to dif-ferences in sample selection. For example, Sheth et. al.’s(2003) sample of only early-type galaxies, was not com-plete in σ , and their resulting VDF declined noticeablyfor log σ (cid:46) . − . Comparison between previousdeterminations of the VDF illuminate the fact that as-tute sample selection is critical in determining the VDF(Choi et al. 2007).In Fig. 4, we compare our VDFs derived from the σ -complete sample with previous determinations. Of these,the Bernardi et. al. (2010) and Sohn et. al. VDFs weredetermined directly from spectroscopic measurements of σ from SDSS galaxies. Chae (2010) used a combinationof Monte Carlo-realized SDSS early-type and late-typeVDF to estimate the total VDF. Of the VDFs shown,the Sheth et. al. (2003) VDF is an early-type VDF only.Furthermore, Sohn et. al.’s sample selection criteria es-sentially limited their sample to early-type galaxies sincethe 4000 ˚A break strength is strongly correlated withstellar population age, and by extension, galaxy type(Kauffmann et al. 2003; Zahid et al. 2016). Table 1
The σ spec VDF and σ mod VDF with associated upper and lower limit uncertainties log σ Φ( σ spec ) ∆Φ( σ spec ) Φ( σ mod ) ∆Φ( σ mod ) [km s − ] [Mpc − dex − ] [Mpc − dex − ] [Mpc − dex − ] [Mpc − dex − ] . × − +1 . × − − . × − . × − +2 . × − − . × − . × − +5 . × − − . × − . × − +6 . × − − . × − . × − +4 . × − − . × − . × − +4 . × − − . × − . × − +4 . × − − . × − . × − +3 . × − − . × − . × − +3 . × − − . × − . × − +2 . × − − . × − . × − +4 . × − − . × − . × − +3 . × − − . × − . × − +4 . × − − . × − . × − +3 . × − − . × − . × − +5 . × − − . × − . × − +4 . × − − . × − . × − +5 . × − − . × − . × − +8 . × − − . × − . × − +7 . × − − . × − . × − +1 . × − − . × − . × − +1 . × − − . × − . × − +4 . × − − . × − . × − +3 . × − − . × − . × − +9 . × − − . × − . × − +3 . × − − . × − . × − +4 . × − − . × − Note . — Table 1 will be published in its entirety in machine-readable format when we submit to the Astronomicalor Astrophysical Journal
Table 2
Best-fit parameters for modified Schechter function (eq. 8) fits to VDFs for our σ spec and σ mod sample Sample α β σ (cid:63) Φ (cid:63) [km s − ] [Mpc − dex − ] σ spec − . ± .
31 2 . ± .
31 172 . ±
23 (5 . ± . × − σ mod − . ± .
28 2 . ± .
37 189 . ±
27 (4 . ± . × − The shapes of both of our VDFs are very similar tothose of Sohn et. al., Bernardi et. al. and Chae et.al., despite differences in methods and sample selection.The Sohn et. al. sample is complete in σ like ours andis a reliable estimate of the quiescent population VDF.Their VDF flattens somewhat at log σ < .
1, whereasboth of ours exhibit a slow upturn towards lower σ s.The Bernardi et. al. VDF, which has a similar shapeto ours (albeit predicts much higher number densities),drew from a sample which was magnitude-limited aftercorrection for incompleteness. They used 1 /V max like us,in contrast to Sohn et. al. (who used the SWML method,but found nearly equivalent results with 1 /V max ). Chae,on the other hand, converted the LFs of early-type andlate-type galaxies (from a magnitude-limited sample) toVDFs separately and added a correction term for high σ galaxies.It is certainly interesting that the VDF from an in-ferred quantity – σ mod – reproduces results obtained fromdirect measurements of the velocity dispersion across theliterature. The approximate match between our σ mod VDF and σ spec VDFs from a variety of sample selec-tion methods is a promising sign for better uncoveringthe true velocity dispersion statistics for large samples of galaxies. Bezanson et al. (2011) derived the total VDFfor 0 . < z (cid:54) . σ mod (eq. 2) and foundthat the VDF flattens toward lower σ s with decreasingredshift, and that the number of galaxies at the highest σ bins (log σ (cid:38) .
5) change little. We find that their0 . < z (cid:54) . σ mod VDF is similar to ours, possibly im-plying little evolution from z ∼ z ∼ .
3. In any case,the agreement of our σ mod VDF with our σ spec VDF aswell as those in the literature, compels a more thoroughassessment of the differences in σ estimates from spec-troscopic and photometric measurements, especially sothat the effect of galactic rotation is taken into accountin correcting the observed σ spec and obtaining the “true”estimate of central velocity dispersion.4.4. Early and late types
Using Galaxy Zoo’s classification of ∼ σ spec and σ mod . Early type galaxies, comprising of ellipticalsand lenticulars, tend to be the most massive galaxies inthe universe. These are populated by older, redder starsand due to a lack of cold gas reservoirs, have low star for-mation activity (Jura 1977; Crocker et al. 2011). In con- . . . . log σ [km s − ] − − − − − − l og Φ [ M p c − d e x − ] This work: σ spec This work: σ mod Sheth et. al. (2003)Bernardi et. al. (2010)Chae (2010)Sohn et. al. (2017)
Figure 4.
Comparison of the VDFs from this work with with past works. Our σ spec VDF is the dashed purple curve ( ± σ region inpink), and our σ mod VDF is shown in dashed blue ( ± σ region in light blue). The past VDFs are shown as dot-dashed curves: Sheth et al.(2003) – green; Bernardi et al. (2010) – magenta; Chae (2010) – black, and individual data points: Sohn et. al. (017a) – red triangles. . . . . log σ mod [km s − ] . . . . . . . . . l og σ s p e c [ k m s − ] . . . . log σ mod [km s − ]Early-types Late-types Figure 5.
Comparison of velocity dispersion estimates for early-type galaxies (left) and late-types galaxies (right), similar to Fig.1. trast, spirals, or late type galaxies, tend to be less mas-sive, bluer and populated mostly by younger stars. Theyhave been shown to exhibit much higher rates of starformation than early type systems (Young et al. 1996;Seigar 2017).While no type-dependent separation of the VDF basedon modeled velocity dispersions can be found in the lit-erature yet, many studies have examined the distribu-tion of early and late types based on the spectroscopic σ . Bernardi et. al. (2010), for example, derived theVDF of low redshift SDSS galaxies (based on σ spec ) andfound that the VDF obtained from E+S0s (early-types)declined toward low σ , but the addition of spirals likeSas increased the number density at those ranges. In a . . . . . . log σ [km s − ] − − − − l og Φ [ M p c − d e x − ] σ spec σ mod . . . . . . log σ [km s − ] σ spec σ mod Early-types Late-types
Figure 6.
VDF for early-type galaxies (left) and late-types galax-ies (right).
Left : red = σ spec VDF, orange = σ mod VDF.
Right :purple = σ spec VDF, blue = σ mod VDF. similar way, Weigel et. al. (2016) observed a decliningSMF for lower masses for local early-types in the SDSS,while the late-type SMF showed an increasing SMF forlower masses.Fig. 5 compares the σ spec and σ mod , as in Fig. 1,for both galaxy types. While it would be expected thatspirals in general have σ spec >σ mod , we see the same istrue for early-types in our sample. In fact, the early-type σ spec values seems to be higher than σ mod , compared tolate-types. For the late-types, we find that σ mod predictsmuch lower σ values than σ spec for the lowest velocitydispersions (log σ (cid:46) σ spec may need to be corrected for rotation in notjust late-type galaxies, but early-types too. Blanton andMoustakas (2009) note that lenticulars (S0) are mostlyfast rotators, and that even most giant ellipticals mayrotate appreciably.As Fig. 6 shows, the VDFs from our early-type andlate-types galaxies do differ. As expected, late-typegalaxies have a high fraction of low σ galaxies and con-tribute the most to the low- σ distribution, while early-type galaxies dominate the high- σ distribution. Here,we observe a flattening of the early-type VDF towardlow σ mod , and a slight rise in the late-type VDF forlog σ mod (cid:46) .
15. Both the σ spec VDFs seem to flattenat the lowest σ s. For the early-types, we see that the σ spec VDFs overestimate the σ mod VDFs at log σ (cid:38) . CONCLUSIONWe measure the VDF of galaxies from the SDSS at0 . (cid:54) z (cid:54) .
1, using two different estimates of velocitydispersion: the directly measured σ spec , and the pho-tometrically inferred σ mod . We observe systematicallyhigher σ spec relative to σ mod in these galaxies, possiblydue to SDSS spectroscopy being unable to separate therotation component from the “true” velocity dispersion.We construct σ -complete samples following the approachof Sohn et. al. (017a), by selecting every galaxy with σ >σ lim ( z ), the σ -completeness limit as a function of redshiftfor an originally magnitude-limited sample at r p < . σ spec and σ mod samples consist of over 100000galaxies each and is complete for all log σ spec (cid:38) . σ mod (cid:38) .
6, respectively. These represent the largestsamples ever used in measureing the z (cid:54) . σ at low σ , followed by an exponential decline at high σ .The most uncertain number densities are observed forthe highest σ bins, since there are very few galaxies withlog σ (cid:38) . − . The VDFs derived from σ spec and σ mod agree very well, especially at log σ (cid:46) .
3, but the σ mod VDF falls faster than the σ spec VDF at log σ (cid:38) . σ mod VDF beingerroneous at the high σ range, so we caution against im-mediately drawing conclusions about which tracer is abetter descriptor of the true VDF, or about the extentto which the σ spec VDF overestimates the true VDF.However, the fact that our results do show divergencesbetween σ spec and σ mod – hypothetically the same phys-ical property – and their respective VDFs supports theargument that values of σ measured from SDSS long-slitspectroscopy may be biased in some form.We also use Galaxy Zoo’s classification of a fraction ofgalaxies in our sample as either early-type or late-type,and obtain VDFs for both subsamples separately. Wesee that σ spec is in general higher than σ mod , especiallyat log σ (cid:46)
2, for both types, again implying rotationalvelocity could be contaminating σ spec . Overall, early-type galaxies dominate the VDF at high σ (log σ (cid:38) . σ (cid:46) . σ spec VDF predicts higher number densities at log σ (cid:38) . σ mod VDF is higher at that range for the late-types.A comparison with the local VDFs derived in the liter-ature from direct spectroscopic measurements agree re-markably well with both our σ mod and σ spec VDFs, de-spite differences in sample selection and methods. Thus,we believe that the literature and our work has narroweddown on the “true” VDF for z (cid:54) . σ mod .The VDF may be an important tool in probing themass distribution of DM halos, and its accurate repro-duction by numerical simulations is paramount in tight-ening our constraints on the evolution of galaxies. Com-bining VDF estimates from large samples such as theone used here with numerical simulations would there-fore be a powerful window to understanding the large-scale structure formation and evolution of the universe.A follow-up paper will present the black hole mass func-tion derived from our σ -complete sample, using Van denBosch (2016)’s scaling of σ to SMBH mass, M BH .ACKNOWLEDGEMENTSF.H would like to thank Jubee Sohn for his helpfulcomments regarding the sample selection at the begin-ning of this research. This research has made use ofNASAs Astrophysics Data System (ADS) BibliographicServices. This publication has also made use of Astropy,a community-developed core PYTHON package for As-tronomy (Astropy Collaboration, 2013). Furthermore,we used the Tool for OPerations on Catalogues And Ta-bles (TOPCAT ) and AstroConda, a free Conda chan-nel maintained by the Space Telescope Science Institute(STScI) in Baltimore, Maryland.Galaxy Zoo is supported in part by a Jim Gray re-search grant from Microsoft, and by a grant from TheLeverhulme Trust. Galaxy Zoo was made possible bythe involvement of hundreds of thousands of volunteer“citizen scientists.”Funding for SDSS-III has been provided by the AlfredP. Sloan Foundation, the Participating Institutions, theNational Science Foundation, and the U.S. Departmentof Energy Office of Science. SDSS-III is managed bythe Astrophysical Research Consortium for the Partici-pating Institutions of the SDSS-III Collaboration. TheParticipating Institutions are the American Museum ofNatural History, Astrophysical Institute Potsdam, Uni- versity of Basel, University of Cambridge, Case WesternReserve University, University of Chicago, Drexel Uni-versity, Fermilab, the Institute for Advanced Study, theJapan Participation Group, Johns Hopkins University,the Joint Institute for Nuclear Astrophysics, the Kavli In-stitute for Particle Astrophysics and Cosmology, the Ko-rean Scientist Group, the Chinese Academy of Sciences(LAMOST), Los Alamos National Laboratory, the MaxPlanck Institute for Astronomy (MPIA), the Max PlanckInstitute for Astrophysics (MPA), New Mexico StateUniversity, Ohio State University, University of Pitts-burgh, University of Portsmouth, Princeton University,the United States Naval Observatory and the Universityof Washington. The SDSS website is .REFERENCES Abazajian, K. N. et al.: 2009,
The Astrophysical JournalSupplement Series , 543Abolfathi, B. et al.: 2018,
ApJS , 42Aihara, H. et al.: 2011,
The Astrophysical Journal SupplementSeries , 29Alam, S. et al.: 2015,
ApJS , 12Aller, M. C. and Richstone, D.: 2002, AJ , 3035Aversa, R. et al.: 2015, ApJ , 74Baldry, I. K. et al.: 2012,
MNRAS , 621Bamford, S. P. et al.: 2009,
MNRAS , 1324Bernardi, M. et al.: 2003, AJ , 1849Bernardi, M. et al.: 2010, MNRAS , 2087Bernardi, M. et al.: 2013,
MNRAS , 697Bertin, G., Ciotti, L., and Del Principe, M.: 2002,
A&A , 149Bezanson, R. et al.: 2011,
ApJ , L31Blanton, M. R. et al.: 2001, AJ , 2358Blanton, M. R. et al.: 2005, AJ , 2562Blanton, M. R. and Moustakas, J.: 2009, ARA&A , 159Brinchmann, J. et al.: 2004,
MNRAS , 1151Calvi, R. et al.: 2012,
MNRAS , L14Calvi, R. et al.: 2013,
MNRAS , 3141Cappellari, M. et al.: 2006,
MNRAS , 1126Chae, K.-H.: 2010,
MNRAS , 2031Choi, Y.-Y., Park, C., and Vogeley, M. S.: 2007,
ApJ , 884Conroy, C., Gunn, J. E., and White, M.: 2009,
ApJ , 486Crocker, A. F. et al.: 2011,
MNRAS , 1197Efstathiou, G., Ellis, R. S., and Peterson, B. A.: 1988,
MNRAS , 431Ferrarese, L. and Ford, H.: 2005,
Space Sci. Revs. , 523Ferrarese, L. and Merritt, D.: 2000,
Astrophys. J. L. , L9Fontana, A. et al.: 2006,
A&A , 745Gebhardt, K. et al.: 2000,
Astrophys. J. L. , L13Graham, A. W.: 2007,
MNRAS , 711Graham, A. W. et al.: 2007,
MNRAS , 198Heckman, T. M. and Best, P. N.: 2014,
ARA&A , 589 Hogg, D. W.: 1999, ArXiv Astrophysics e-prints
Jura, M.: 1977,
ApJ , 634Kauffmann, G. et al.: 2003,
MNRAS , 33Kelvin, L. S. et al.: 2014,
MNRAS , 1647Kormendy, J. and Ho, L. C.: 2013,
ARA&A , 511Kormendy, J. and Richstone, D.: 1995, ARA&A , 581Li, Y.-R., Ho, L. C., and Wang, J.-M.: 2011, ApJ
MNRAS , 1179Lintott, C. J. et al.: 2011,
MNRAS , 166Magorrian, J. et al.: 1998, AJ , 2285Malmquist, K. G.: 1925, Meddelanden fran Lunds AstronomiskaObservatorium Serie I , 1Marchesini, D., , et al.: 2009,
ApJ , 1765Marchesini, D. et al.: 2007,
ApJ , 42Marconi, A. and Hunt, L. K.: 2003,
ApJ , L21McConnell, N. J. and Ma, C. P.: 2013,
ApJ , 184McNaught-Roberts, T. et al.: 2014,
MNRAS , 2125Mitchell, J. L. et al.: 2005,
ApJ , 81Munari, E. et al.: 2016,
ApJ , L5Oke, J. B. and Sandage, A.: 1968,
ApJ , 21P´erez-Gonz´alez, P. G. et al.: 2008,
ApJ , 234Petrosian, V.: 1976,
ApJ , L1Pozzetti, L. et al.: 2010,
A&A , A13Richstone, D. et al.: 1998,
Nature , A14Salucci, P. et al.: 1999,
MNRAS , 637Sandage, A., Tammann, G. A., and Yahil, A.: 1979,
ApJ , 352Schechter, P.: 1976,
ApJ , 297Schmidt, M.: 1968,
ApJ , 393Seigar, M. S.: 2017, in
Spiral Structure in Galaxies , 2053-2571,pp 5–1 to 5–13, Morgan & Claypool PublishersShankar, F., Weinberg, D. H., and Miralda-Escud´e, J.: 2009,
ApJ , 20Shankar, F., Weinberg, D. H., and Miralda-Escud´e, J.: 2013,
MNRAS , 421Sheth, R. K. et al.: 2003,
ApJ , 225Sohn, J. et al.: 2017b,
ApJS , 20Sohn, J., Zahid, H. J., and Geller, M. J.: 2017a,
ApJ , 73Stoughton, C. et al.: 2002, AJ , 485Strauss, M. A. et al.: 2002, AJ , 1810Taylor, E. N. et al.: 2010, ApJ , 1Thomas, D. et al.: 2013,
MNRAS , 1383Van den Bosch, R. C. E.: 2016,
ApJ , 134van Uitert, E. et al.: 2013,
A&A , A7Wake, D. A., van Dokkum, P. G., and Franx, M.: 2012,
ApJ ,L44Weigel, A. K., Schawinski, K., and Bruderer, C.: 2016,
MNRAS , 2150Yang, X. et al.: 2013,
ApJ , 115Yang, X., Mo, H. J., and van den Bosch, F. C.: 2008,
ApJ ,248York, D. G. et al.: 2000, AJ , 1579Young, J. S. et al.: 1996, AJ , 1903Yu, Q. and Tremaine, S.: 2002, MNRAS , 965Zahid, H. J. et al.: 2016,
ApJ , 203Zahid, H. J. and Geller, M. J.: 2017,
ApJ841