ASTR 610 Theory of Galaxy Formation

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1 ASTR 610 Theory of Galaxy Formation Lecture 13: The Halo Model & Halo Occupation Statistics Frank van den Bosch Yale University, Fall 2018

2 The Halo Model & Occupation Statistics In this lecture we discuss the halo model, an analytical model to describe the (dark matter) mass distribution on small, non-linear scales. We also introduce the concept of halo occupation statistics and discuss how data can constrain these. We end with some discussion on the galaxy-dark matter connection Topics that will be covered include: The Halo Model 1-halo vs. 2-halo terms Halo Exclusion Halo Occupation Models Conditional Luminosity Function Poisson Statistics Galaxy - Dark Matter connection ASTR 610:Theory of Galaxy Formation

3 The Halo Model

The Halo Model The Halo model is an analytical model that describes dark matter density distribution in terms of its halo building blocks, under ansatz that all dark matter is partitioned over haloes.

4 The Halo Model The Halo model is an analytical model that describes dark matter density distribution in terms of its halo building blocks, under ansatz that all dark matter is partitioned over haloes. Throughout we assume that all dark matter haloes are spherical, and have a density distribution that only depends on halo mass: Here u(r M ) is the normalized density profile: ASTR 610:Theory of Galaxy Formation (r M ) = M u(r M ) d3 x u(x M ) = 1

5 y-axis The Halo Model Imagine space divided into many small volumes, none of them contain more than one halo center. V i,which are so small that N i Let be the occupation number of dark matter haloes in cell i Then we have that N i =0, 1 and therefore N i = Ni 2 = Ni 3 = This allows us to write the matter density field as a summation: (x )= i N i M i u(x x i M i ) = halo center x-axis

6 The Halo Model = Z (x )= N i M i u(x x i M i ) * + X ( x)d 3 x = ( x) = N i M i u( x x i M i ) i i ergodicity = i N i M i u(x x i M i ) halo mass function = i dm Mn(M) V i u(x x i M) = dm Mn(M) d 3 y u (x y M) = dm Mn(M) = Q.E.D.

7 The Halo Model (x )= i N i M i u(x x i M i ) Now that we can write the density field in terms of the halo building blocks, let s focus on two-point statistics: mm(r) (x ) (x + r ) = 1 2 (x ) (x + r ) 1 (x ) (x + r ) = N i M i u(x 1 x i M i ) N j M j u(x 2 x j M j ) i j x 2 = x 1 + r = i j N i N j M i M j u(x 1 x i M i )u(x 2 x j M j ) We split this in two parts: the 1-halo term (i = j), and the 2-halo term (i = j) For the 1-halo term we obtain: N 2 i = N i (x ) (x + r ) 1h = = i i N i M 2 i u(x 1 x i M i )u(x 2 x i M i ) dm M 2 n(m) V i u(x 1 x i M)u(x 2 x i M) = dm M 2 n(m) d 3 y u (x 1 y M)u(x 2 y M) 1-halo vs. 2-halo convolution integral

8 For the 2-halo term we obtain: The Halo Model (x )= i N i M i u(x x i M i ) (x ) (x + r ) 2h =? = i j=i i j=i N i N j M i M j u(x 1 x i M i ) u(x 2 x j M j ) dm 1 M 1 n(m 1 ) dm 2 M 2 n(m 2 ) V i V j [1 + hh (x i x j M 1,M 2 )] u(x 1 x i M 1 ) u(x 2 x j M 2 ) = 2 NO: dark matter haloes themselves are clustered, i.e., have a non-zero two point correlation function. This needs to be taken into account. Clustering of dark matter haloes is characterized by halo-halo correlation function: b(m) hh(r M 1,M 2 )=b(m 1 ) b(m 2 ) lin mm(r) Here is the halo bias function. Note: the above description of the halo-halo correlation function is only valid on large (linear) scales! On small scales non-linearities and halo exclusion become important (see below).

9 For the 2-halo term we obtain: The Halo Model (x )= i N i M i u(x x i M i ) (x ) (x + r ) 2h = i j=i N i N j M i M j u(x 1 x i M i ) u(x 2 x j M j ) = i j=i dm 1 M 1 n(m 1 ) dm 2 M 2 n(m 2 ) V i V j [1 + hh (x i x j M 1,M 2 )] u(x 1 x i M 1 ) u(x 2 x j M 2 ) = 2 + dm 1 M 1 n(m 1 ) dm 2 M 2 n(m 2 ) d 3 y 1 d 3 y 2 u(x 1 y 1 M 1 ) u(x 2 y 2 M 2 ) hh (y 1 y 2 M 1,M 2 ) = 2 + dm 1 M 1 b(m 1 ) n(m 1 ) dm 2 M 2 b(m 2 ) n(m 2 ) d 3 y 1 d 3 y 2 u(x 1 y 1 M 1 ) u(x 2 y 2 M 2 ) lin mm(y 1 y 2 ) convolution integral

10 The Halo Model: Summary (part I) (r) = 1h (r)+ 2h (r) 1h (r) = 1 2 dm M 2 n(m) d 3 y u (x y M)u(x + r y M) 2h (r) = 1 2 dm 1 M 1 b(m 1 ) n(m 1 ) dm 2 M 2 b(m 2 ) n(m 2 ) d 3 y 1 d 3 y 2 u(x y 1 M 1 ) u(x + r y 2 M 2 ) lin mm(y 1 y 2 ) Halo Model Ingredients: the halo density profiles the halo mass function the halo bias function the linear correlation function of matter (r M) =Mu(r M) n(m) b(m) lin mm(r) All of these are (reasonably) well calibrated against numerical simulations.

11 The Halo Model in Fourier Space P (k) = P 1h (k)+p 2h (k) P 1h (k) = 1 2 dm M 2 n(m) ũ(k M) 2 P 2h (k) = P lin (k) 1 dm Mb(M) n(m)ũ(k M) 2 P lin (k) =P i (k) T 2 (k) =k n s T 2 (k) ũ(k M) = u(x M)e ik x d 3 x =4 0 u(r M) sin kr kr r 2 dr Since convolutions in real-space become multiplications in Fourier space, the halo model expression for the power spectrum is much easier. Therefore, in practice, one computes P (k) and then uses Fourier transformation to obtain two-point correlation function (r)

12 The Halo Model in Fourier Space Dashed line: true non-linear power spectrum Solid line: halo model Source: Cooray & Sheth (2002) large scales small scales 2 (k) = k3 P (k) Dimensionless power spectrum

13 The Halo Model: complications P 1h (k) = 1 2 dm M 2 n(m) ũ(k M) 2 P 2h (k) =P lin (k) 1 dm Mb(M) n(m)ũ(k M) 2 However, this is ONLY true under the simplifying assumption that hh(r M 1,M 2 )=b(m 1 ) b(m 2 ) lin mm(r) In reality, on small scales, in the (quasi)-linear regime, this description of the halo-halo correlation function becomes inadequate for two reasons: lin mm(r) is no longer adequate R 2 halo exclusion R 1 R halo exclusion

The Halo Model: complications R 2 Because of these complications, the 2-halo term needs to be modified to the following, much more

min [1 + hh (r M 1,M 2 )] sin kr kr r2 dr describes the fact that dark matter haloes are clustered, as described by the halo-halo correlation

has to be modified to: hh(r M 1,M 2 )=b(m 1 ) b(m 2 ) (r) mm (r) Here (r) is called the radial bias function, and which is a higher-order

14 The Halo Model: complications R 2 Because of these complications, the 2-halo term needs to be modified to the following, much more complicated form R 1 R halo exclusion P 2h (k) = 1 2 dm 1 M 1 n(m 1 )ũ(k M 1 ) dm 2 M 2 n(m 2 )ũ(k M 2 ) Q(k M 1,M 2 ) Here Q(k M 1,M 2 )=4 r min [1 + hh (r M 1,M 2 )] sin kr kr r2 dr describes the fact that dark matter haloes are clustered, as described by the halo-halo correlation function, hh(r M 1,M 2 ), and takes halo exclusion into account by having r min = R 1 + R 2 Also, the halo-halo correlation function itself has to be modified to: hh(r M 1,M 2 )=b(m 1 ) b(m 2 ) (r) mm (r) Here (r) is called the radial bias function, and which is a higher-order bias correction. No analytical model for (r) exists; use empirical calibration... Worst of all; mm(r) is the non-linear matter-matter correlation function

15 The Halo Model: Summary (part II) The simple, `linear, halo model P 1h (k) = 1 2 dm M 2 n(m) ũ(k M) 2 P 2h (k) = P lin (k) 1 dm Mb(M) n(m)ũ(k M) 2 Only accurate to ~40-50% in the 1-halo to 2-halo transition region (~1 Mpc/h) Can still be adequate for certain applications... The more accurate halo model P 1h (k) = 1 2 dm M 2 n(m) ũ(k M) 2 P 2h (k) = 1 2 dm 1 M 1 n(m 1 )ũ(k M 1 ) dm 2 M 2 n(m 2 )ũ(k M 2 ) Q(k M 1,M 2 ) Accurate to ~5% level, but requires mm(r) as input... Still, this halo model is very useful, as it can also be used to model the correlation function (or power-spectrum) of galaxies!!!! See vdbosch+13 for details

16 Halo Occupation Modeling

17 The Galaxy Power Spectrum P 1h (k) = 1 2 dm M 2 n(m) ũ(k M) 2 P 2h (k) = 1 2 dm 1 M 1 n(m 1 )ũ(k M 1 ) dm 2 M 2 n(m 2 )ũ(k M 2 ) Q(k M 1,M 2 ) The above equations describe the halo model predictions for the matter power spectrum The same formalism can also be used to compute the galaxy power spectrum: M hni M n g simply replace: M 2 2 hn(n n 2 g 1)i M ũ(k M) ũ g (k M) M n g u g (r M) M Here describes average number of galaxies (with certain properties) that reside in a halo of mass, is the average number density of those galaxies, and is the normalized, radial number density distribution of galaxies in haloes of mass.

18 Halo Occupation Statistics When describing halo occupation statistics, it is important to treat central and satellite galaxies separately. Central Galaxies: those galaxies that reside at the center of their dark matter (host) halo Satellite Galaxies: those galaxies that reside at the center of a dark matter sub-halo, and are orbitting inside a larger host halo. = central = satellite Central Galaxies 1X N c M = N c P (N c M) =P (N c =1 M) N c =0 1X Nc 2 M = N c =0 N 2 c P (N c M) =P (N c =1 M) = N c M Satellite Galaxies N s M = 1X N s P (N s M) N s =0 Ns 2 M = 1X Ns 2 P (N s M) N s =0 u c (r M) = D (r) u s (r M) = TBD

19 Halo Occupation Statistics Central Galaxies 1X N c M = N c P (N c M) =P (N c =1 M) N c =0 1X Nc 2 M = N c =0 N 2 c P (N c M) =P (N c =1 M) = N c M Satellite Galaxies N s M = 1X N s P (N s M) N s =0 Ns 2 M = 1X Ns 2 P (N s M) N s =0 u c (r M) = D (r) u s (r M) = TBD Calculating galaxy-galaxy correlation functions requires following halo occupation statistic ingredients: Halo occupation distribution for centrals Halo occupation distribution for satellites Radial number density profile of satellites P (N c M) P (N s M) u s (r M) P (N c,n s M) In principle, as we will see, one also requires the probability function, but it is common practice to assume that the occupation statistics of centrals and satellites are independent, i.e., that P (N c,n s M) =P (N c M) P (N s M)

20 Halo Occupation Statistics: the first moment Source: Zheng et al., 2005, ApJ, 633, 791 Consider a luminosity threshold sample; all galaxies brighter than some threshold luminosity. The halo occupation statistics for such a sample are typically parameterized as follows: hn c i M = 1 2 hn s i M = apple 1 + erf ( log M M M 1 log M log Mmin log M if M>M cut 0 if M<M cut M min hn s i M / M M min log M M cut M 1 = characteristic minimum mass of haloes that host centrals above luminosity threshold = characteristic transition width due to scatter in L-M relation of centrals = cut-off mass below which you have zero satellites above luminosity threshold = normalization of satellite occupation numbers = slope of satellite occupation numbers This particular HOD model, which is fairly popular in the literature, requires 5 parameters (M min,m 1,M cut, log M, ) to characterize the occupation statistics of a given luminosity threshold sample, and is (partially) motivated by the occupation statistics in hydro simulations of galaxy formation...

21 Halo Occupation Statistics: the first moment = d log N s /d log M Increasing the slope boosts the 1-halo term of the correlation function. It also boosts the 2-halo term, but to a lesser extent. The latter arises because a larger value of α implies that satellites, on average, reside in more massive haloes, which are more strongly biased. The 1-halo term scales with satellite occupation numbers as 2 N s M while the 2-halo term scales as Ns M. This means that the relative clustering strengths in the 1-halo and 2-halo regimes constrains the satellite fractions. Source: Berlind & Weinberg, 2002, ApJ, 575, 587

22 Halo Occupation Statistics: the first moment An alternative parameterization, which has the advantage that it describes the occupation statistics for any luminosity sample (not only threshold samples), is the conditional luminosity function. (L M) = c (L M)+ s (L M) The CLF describes the average number of galaxies of luminosity L that reside in a dark matter halo of mass M. (L) = Z 1 0 (L M) n(m)dm CLF is the direct link between the halo mass function and the galaxy luminosity function. L M = Z 1 0 (L M) L dl CLF describes link between luminosity and mass N x M = Z L2 L 1 x(l M)dL CLF describes first moments of halo occupation statistics of any luminosity sample

23 The Conditional Luminosity Function The CLF can be obtained from galaxy group catalogues. Yang, Mo & van den Bosch (2008) have shown that the CLF is well parameterized using the following functional form: c(l M)dL = s(l M)dL = 1 exp 2 c {L c,l s, c, s, s} s L s L L s s exp ln(l/l c ) 2 c 2 (L/L s ) 2 dl dl L Note: all depend on halo mass. These dependencies are typically parameterized using ~10 free parameters. Free parameters are constrained by the data, which can be galaxy group catalogs, galaxy clustering, galaxy-galaxy lensing, satellite kinematics, etc... Source: Yang, Mo & van den Bosch, 2008, ApJ, The CLFs inferred from a SDSS galaxy group catalog. Symbols are data, while the solid, black line is bestfit using the CLF parameterization indicated above...

24 Halo Occupation Statistics: the second moment The 1-halo term of the galaxy-galaxy correlation function requires the second moment hn(n 1)i M = hnc 2 i M +2hN c N s i M + hns 2 i M hn c i M hn s i M = hn s (N s 1)i M +2hN c i M hn s i M where we assumed that occupation statistics of centrals and satellites are independent Thus, we need to specify the second moment of the satellite occupation distribution: N s (N s 1) M = 1X N s =0 where we have introduced the function (M) N s (N s 1) P (N s M) (M) N s 2 If the occupation statistics of satellite galaxies follow Poisson statistics, i.e., P (N s M) = N s e N s! with = hn s i M then (M) =1. Distributions with > 1 ( < 1) are broader (narrower) than Poisson. The second moment of the halo occupation statistics is completely described by (M)

25 Halo Occupation Statistics: the second moment It is common practice to assume that satellites obey Poisson statistics. This is motivated by finding that dark matter subhaloes have occupation statistics that are (close to) Poissonian Source: Kravtsov et al., 2004, ApJ, 609, 35 hn(n 1)i 1/2 /hni N includes host halo itself simulation satellites all galaxies Source: Zheng et al., 2005, ApJ, 633, 791 Occupation statistics of dark matter subhaloes in numerical (dark-matter-only) simulations. Subhaloes follow Poisson statistics... Occupation statistics of simulated galaxies in hydrodynamical SPH simulations. Satellite galaxies follow Poisson statistics...

26 Occupation Statistics from Galaxy Group Catalog Poisson Poisson Poisson Source: Yang, Mo & van den Bosch, 2008, ApJ, Even real data shows that the occupation statistics of satellites are (close to) Poissonian.

27 Occupation Statistics from Galaxy Group Catalog Solid lines are the Poisson distribution corresponding to <Ns> Source: Yang, Mo & van den Bosch, 2008, ApJ, Even real data shows that the occupation statistics of satellites are (close to) Poissonian.

28 The Radial Number Density Profile of Satellites The radial number density profile of satellite galaxies is typically modelled as a `generalized NFW profile : u s (r M) r R r s apple 1+ r R r s Here is a parameter that controls the central cusp slope, and sets the ratio between the concentration parameter of the satellites and that of the dark matter. For = R =1satellites are an unbiased tracer of the mass distribution (within individual haloes) 3 R = c sat /c dm The radial number density profile of satellites controls the clustering on small scales (only has significant effect on 1-halo term). The two-point correlation function of galaxies, calculated using the halo model. Solid dots are data from the APM catalogue. The solid line is the model s matter correlation function, and the other lines are galaxy correlation functions in which the number density profile of satellite galaxies is varied. Source: Berlind & Weinberg, 2002, ApJ, 575, 587

29 The Radial Number Density Profile of Satellites The radial number density profile of satellite galaxies can be constrained using the clustering data itself, or by directly measuring the (projected) profiles of satellite galaxies in groups/clusters, or around isolated centrals... Source: Lin, Mohr & Stanford, 2004, ApJ, 610, 745 Source: Guo et al. 2012, arxiv: The surface density profile of satellite galaxies in clusters. Solid line is the best-fit NFW profile. The surface density profiles of satellite galaxies around isolated centrals in the SDSS. Satellites are identified in photometric catalogue using statistical background subtraction. Lines are best-fit NFW profiles.

30 The Radial Number Density Profile of Satellites Although several studies have suggested that satellite galaxies follow a radial number density profile that is well fitted by NFW profile, others find that u s (r M) has a core and is less centrally concentrated than the dark matter. This is consistent with distribution of subhaloes in dark-matter-only simulations... Source: More, vdb, et al. 2009, MNRAS, 392, 801 The surface density profile of satellite galaxies around isolated centrals. Here both centrals & satellites are obtained from the spectroscopic SDSS. Note that cored profiles are better fit than NFW profile.

31 Testing with Galaxy Mock Catalogs Source: van den Bosch, et al. 2013, MNRAS, 430, 725 One can test accuracy of halo model using mock redshift surveys. These can be constructed by populating haloes in numerical simulations with mock galaxies. Dots show mock data from simulation box, while solid lines show the model predictions from CLF+halo model.

32 Constraints on Halo Occupation Statistics Source: Zehavi et al. 2011, ApJ, 736, 59 Zehavi et al used halo occupation models to fit the projected correlation functions obtained from the SDSS for 9 different luminosity threshold samples. The left-hand panel shows data+fits (offset vertically for clarity). The right-hand panel shows first moments of best-fit halo occupation distributions.

33 The Galaxy - Dark Matter Connection Source: van den Bosch et al. 2007, MNRAS, 376, 841

34 The Galaxy - Dark Matter Connection 0.2 f bar Source: Behroozi, Wechsler & Conroy, 2012, arxiv:

35 Source: Cacciato et al., 2009, MNRAS, 394, 929 Source: Tinker et al., 2012, ApJ, 745, 16 Cosmology Dependence Both these models fit clustering data equally well

Galaxy-Galaxy Lensing The mass associated with galaxies lenses

galaxy Lensing causes correlated ellipticities, the tangential shear,

(R) crit (R) = (< R) = (R) is line-of-sight projection of

36 Galaxy-Galaxy Lensing The mass associated with galaxies lenses background galaxies background sources lensing due to foreground galaxy Lensing causes correlated ellipticities, the tangential shear, t, which is related to the excess surface density,, according to t (R) crit (R) = (< R) = (R) is line-of-sight projection of galaxy-matter cross correlation Ds (R) = [1 + g,dm (r)] d 0 ASTR 610:Theory of Galaxy Formation

37 Galaxy-Galaxy Lensing: Results Data: SDSS measurements by Mandelbaum+06 Cacciato, vdb et al. (2009) Combination NOTE: this of clustering is not a fit, & but lensing a prediction can constrain based cosmology!!! on CLF

38 Cosmological Constraints Cacciato, vdb et al. (2013) New physics beyond the vanilla LCDM cosmology or systematic errors?

39 Lecture 13 Summary

40 Summary: Key words & important facts Key words Halo model halo exclusion galaxy-galaxy lensing 1-halo & 2-halo terms Halo Occupation Distribution (HOD) Conditional Luminosity Function (CLF) The Halo model is an analytical model that describes dark matter density distribution in terms of its halo building blocks, under ansatz that all dark matter is partitioned over haloes. In combination with a halo occupation model (HOD or CLF), the halo model can be used to compute galaxy-galaxy correlation function and galaxy-matter cross-correlation function. The latter is related to the excess surface density measured with galaxy-galaxy lensing. HOD is mainly used to model clustering of luminosity threshold samples. CLF can be used to model clustering of galaxies of any luminosity (bin). It is common to assume that satellite galaxies obey Poisson statistics, such that <N s (N s -1) M> = <N s > 2, and only the first moment of P(N s M) is required. This is not exact and may cause significant errors in the predicted clustering.

41 Summary: Key equations & expressions halo model P (k) = P 1h (k)+p 2h (k) P 1h (k) = 1 2 dm M 2 n(m) ũ(k M) 2 P 2h (k) = P lin 1 (k) dm Mb(M) n(m)ũ(k M) 2 Galaxy-Galaxy lensing: tangential shear, excess surface density and galaxy-matter cross correlation t(r) crit = (R) = (< R) (R) (R) = 0 D s [1 + g,dm (r)] d CLF: the link between light and mass (L) = Z 1 0 (L M) n(m)dm L M = Z 1 0 (L M) L dl N x M = Z L2 L 1 x(l M)dL Characteristic examples of CLF and HOD for both centrals and satellites c(l M)dL = s(l M)dL = 1 exp 2 c s L s L L s s exp 2 ln(l/l c ) 2 c (L/L s ) 2 dl dl L hn c i M = 1 2 hn s i M = apple 1 + erf ( M M 1 log M log Mmin log M if M>M cut 0 if M<M cut

Cosmological Constraints from a Combined Analysis of Clustering & Galaxy-Galaxy Lensing in the SDSS. Frank van den Bosch.

Cosmological Constraints from a Combined Analysis of Clustering & Galaxy-Galaxy Lensing in the SDSS In collaboration with: Marcello Cacciato (Leiden), Surhud More (IPMU), Houjun Mo (UMass), Xiaohu Yang