On the correspondence between barrier crossing, peak-background split and local biasing

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1 NRAS 435, Advance Access publication 03 August 30 doi:0.093/mnras/stt7 On the correspondence between barrier crossing, peak-background split and local biasing Simone Ferraro, Kendrick. Smith,, Daniel Green 3,4,5 and Daniel Baumann 6 Princeton University Observatory, Peyton Hall, Ivy Lane, Princeton, NJ 08544, USA Perimeter Institute for Theoretical Physics, Waterloo, ON NL Y5, Canada 3 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA 4 Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94306, USA 5 Kavli Institute for Particle Astrophysics and Cosmology, Stanford, CA 9405, USA 6 D.A..T.P., Cambridge University, Cambridge CB3 0WA, UK Accepted 03 July 8. Received 03 July 7; in original form 0 December INTRODUCTION The large-scale clustering of dark matter haloes has become an important probe of primordial cosmology. In particular, non- Gaussianity in the initial conditions would leave an imprint in the scale dependence of the halo bias Dalal et al. 008; atarrese & Verde 008, sometimes of stochastic type Tseliakhovich, Hirata & Slosar 00; Baumann et al. 03. Several, apparently distinct, methods are commonly used to compute these effects. So far, these methods have been considered to be independent, even though they give the same results when applied to specific examples Desjacques, Jeong & Schmidt 0; Baumann et al. 03. In this paper, we will show that the barrier crossing BC model, the peak-background split PBS method and the local biasing LB approach are, in fact, mathematically closely related. Barrier crossing is the classic model of structure formation dating back to the pioneering work of Press & Schechter 974. In its simplest formulation, it identifies haloes as regions of the linearly evolved density field above some critical density δ c. The clustering properties of haloes can then be calculated as an Edgeworth expansion in the cumulants of the probability density of the primordial density fluctuations, which in turn can be expressed in terms of N-point functions of the potential LoVerde et al. 008; Desjacques et al. 0; Smith, Ferraro & LoVerde 0; Baumann et al. 03. Peak-background split is a method for calculating the influence of long-wavelength fluctuations larger than the halo size on the locally measured statistical properties. It has been widely used in cosmology Bardeen et al. 986; Cole & Kaiser 989, and its usefulness in dealing with non-gaussian initial conditions has been first dbaumann@damtp.cam.ac.uk ABSTRACT Several, apparently distinct, formalisms exist in the literature for predicting the clustering of dark matter haloes. It has been noticed on a case-by-case basis that the predictions of these different methods agree in specific examples, but the precise correspondence remains unclear. In this paper, we provide a simple mathematical relationship between barrier crossing, peak-background split and local biasing. Key words: cosmology: theory large-scale structure of Universe. pointed out in Dalal et al In the most common implementation, the non-gaussian field is defined as a non-linear function of auxiliary Gaussian fields, which are split into short-wavelength and long-wavelength components. By modulating the statistics of the short modes, the long modes affect the clustering statistics. In this paper, we will generalize the PBS approach so that it can be applied to arbitrary non-gaussian initial conditions, parametrized by arbitrary N-point functions of the primordial potential. This will require introducing additional fields ρ, ρ 3,..., which measure the local power-spectrum amplitude, skewness, etc. Local biasing Fry & Gaztanaga 993; Sefusatti 009; Giannantonio & Porciani 00; Baldauf, Seljak & Senatore 0; Scoccimarro et al. 0 refers to the idea of expressing the halo density field δ h as a function the local dark matter density smoothed on some scale and expanding in powers of the density contrast δ, δ h x = b δx + b δ x + b 3 δ 3 x +.. Correlation functions can then be computed straightforwardly in terms of the coefficients in the expansion. Several variations of this formalism exist in literature for example, some use an expansion in the non-linear dark matter density, while others use the linearly evolved density. In this work, we will demonstrate the equivalence between BC and a particular variant of LB, in which the expansion is in the linearly evolved and non-gaussian dark matter density contrast. In a companion paper Baumann et al. 03, we derived the clustering statistics for specific non-gaussian models, both in the PBS formalism and in the BC model. We showed for each example that both approaches give consistent results. The goal of this paper is to prove that this agreement is not accidental, but follows from a mathematical relationship between both methods. C 03 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

2 On the correspondence between biasing models 935 The outline of this paper is as follows. After defining our notation in Section, we introduce our main technical tool in Section 3: a series expansion for the halo field δ h in the BC model. We review some examples of non-gaussian models and show how the series expansion is used for efficiently calculating halo power spectra. In Section 4, we use the series expansion to provide a mathematical relationship among the BC model, the PBS method and the LB formalism. We conclude with brief comments in Section 5. An appendix collects some elementary properties of Hermite polynomials. PRELIINARIES AND NOTATION Non-Gaussian initial conditions can be parametrized by the connected N-point functions ξ N of the primordial gravitational potential. In Fourier space, these are defined as k k kn c = π 3 δ D k,,...,n ξ N k, k,...,k N,. where k,,...,n k + k + +k N. The primordial potential is related to the linearly evolved matter density contrast via Poisson s equation δ k z = αk, z k,. where αk, z k T kdz 3 m H0..3 Here, Tk is the matter transfer function normalized such that Tk ask 0andDz is the linear growth factor as a function of redshift z, normalized so that Dz = + z in matter domination. For notational simplicity, we will from now on suppress the redshift argument from all quantities. The field δ x denotes the linear density contrast smoothed with a top-hat filter of radius R = 3 /4π ρ m /3. In Fourier space, δ k = W kδ k,.4 where W k is the Fourier transform of the top-hat filter, W k 3 sinkr kr coskr..5 kr 3 We also define δ / and α k W kαk. The main quantity of interest, in this paper, is the halo density contrast in Lagrangian space δ h x n hx n h,.6 n h where n h x is the halo number density. To lowest order, δ h is related to the halo overdensity in Eulerian space via δh E = δ h + δ. We will determine the large-scale behaviour of the matter halo and halo halo power spectra P mh k δδ h k andp hh k δ h δ h k. We define P hh k to be the halo power spectrum after the shot noise contribution /n h has been subtracted, where n h is the halo number density. Analogously, we define P mh k to be the matter halo power spectrum after subtracting the one-halo term in practice, this term is usually negligibly small. We define the Lagrangian halo bias as bk P mhk P mm k..7 This is related to the Eulerian bias via b E = b +. A stochastic form of halo bias arises whenever the density of haloes is not 00 per cent correlated with the dark matter density Baumann et al. 03. In that case, the bias inferred from P hh will be different from the bias inferred from P mh,i.e. P hh k P mm k Pmh k P mm k..8 3 A SERIES REPRESENTATION OF BARRIER CROSSING In this section, we introduce the BC formalism and quote results from our companion paper Baumann et al. 03. We also introduce a series representation of BC, which will be our main tool to prove the equivalence to LB and PBS in Section Review of barrier crossing In the simplest version of the BC model Press & Schechter 974, haloes of mass are modelled as regions of space in which the smoothed density field δ exceeds the collapse threshold δ c,i.e.the halo number density n h xisgivenby n W h x δ x δ c, 3. where is the Heaviside step function. Equation 3. models the abundance of a mass-weighted sample of haloes whose mass exceeds some minimum value. We will also consider the case of a halo sample defined by a narrow mass bin, which is obtained from the mass-weighted case by differentiating with respect to, i.e. n N h x δ x δ c. 3. Throughout this paper, we will refer to these two types of halo samples as mass-weighted samples W and narrow samples N. The BC model allows us to compute the statistics of halo halo and halo matter correlations. To discuss correlations between quantities at two points x and x,itisusefultodefineδ = δ x, δ = δ x andr = x x. The joint cumulants of the density fields are then κ ˆm,n r, δm δ σ m σ n n c, 3.3 n c κ m,n r,, δ m δ σ mσ n. 3.4 The hat on κ ˆm,n denotes the use of the unsmoothed density field δ. In the limit k 0, we find κˆ,k P mm k/σ and κ, k P mm k/ σ. This motivates the following definitions fˆ,nk, κˆ,nk, for n, 3.5 κˆ,k, This type of sample is often assumed when fitting models to observations of luminous tracers such as galaxies or quasars. In the absence of detailed knowledge of the halo occupation distribution, a simple choice is to assume that haloes below some minimum mass are unpopulated with tracers, whereas the expected number of tracers in a halo of mass is proportional to the halo mass. Note that the variance of the unsmoothed linear density contrast σ = δ is formally infinite, but cancels in the definition 3.5 of the quantity fˆ,n which will appear in our final expressions.

3 936 S. Ferraro et al. f,n k,, κ,nk,, κ, k,, σ for n, 3.6 κ m,n k,, f m,n k,, for m, n. 3.7 κ, k,, σ Using the function αk, z defined in equation., it is straightforward to relate the above cumulants to the primordial correlation functions ξ N defined in equation.. In Baumann et al. 0, we showed how the matter halo and halo halo power spectra are computed in the BC model using the Edgeworth expansion for the joint probability density function pδ,δ. We refer the reader to that paper for detailed derivations and further discussion. The result can be expressed in terms of the cumulants fˆ,n and f m, n. Taking the limit k 0 for the case of an mass-weighted sample with =, wefind P mh k, = P mm k bg W + α n fˆ,nk,, n 3.8a P hh k, = P mm k bg W + bg W α n f,n n k,, + α m α n f m,n k,, m,n, 3.8b where the coefficients α n not to be confused with the α of equation. are defined in terms of Hermite polynomials see Appendix A, e ν c / α n H n ν c, with ν c δ c. π erfc ν c n! 3.9 We also defined the Gaussian bias as bg W α. 3.0 Note that bg W is the Press Schechter bias for the massweighted halo sample. In writing equation 3.8, we have dropped non-linear terms in the Edgeworth expansion, i.e. terms involving products κ m n,κ m n...κ mpnp with p >. Similarly, for the case of a halo sample defined by a narrow mass bin, we have P mh k, = P mm k bg N + D n fˆ,nk,, 3.a n P hh k, = P mm k bg N + bg N D n f,n n k,, = + D m D n f m,n k,,, m,n = 3.b where we have defined the differential operator D n β n + β n, 3. as well as the functions bg N νc, β n H nν c ν c n! and β n H n ν c. 3.3 n! ν c Note that bg N is the Press Schechter bias of a halo sample defined by a narrow mass bin. In equation 3.b for P hh, we have assumed = for simplicity, but the variables and should be treated as independent for the purpose of taking derivatives. 3. Hermite polynomial expansion In this section, we will develop an alternative to the Edgeworth expansion algebraic framework for analysing clustering in the BC model. First, consider the case of a mass-weighted halo sample, where the halo field is modelled as a step function n W h x νx ν c, where νx δ x. 3.4 Since the Hermite polynomials H n ν are a complete basis, any function of ν can be written as a linear combination of Hermite polynomials. In particular, we can write the Heaviside step function ν ν c as ν ν c = a n ν c H n ν, 3.5 where n=0 a n ν c = dν ν ν c e ν / n! π erfc ν c n = 0 H n ν = π e ν c / H n ν c n n!. 3.6 Plugging this series expansion into equation 3.4, and normalizing the halo field to the fractional overdensity δ h,weget δ h x = a n ν c a n 0 ν c H δ x n = bg W δ x + α n ρ n x n mass-weighted sample, 3.7 where α n andbg W were introduced in equations 3.9 and 3.0, respectively. The fields ρ n are defined as δ x ρ n x H n. 3.8 On large scales, the field ρ = δ / tracks long-wavelength variations in the locally measured small-scale power, and for non-gaussian initial conditions the power spectrum P ρ ρ k may acquire extra large-scale contributions. Analogously, the field ρ 3 = δ 3 / 3 3δ / tracks long-wavelength variations in the locally measured small-scale skewness, and so on for higher ρ n. We emphasize that the series representation equation 3.7 is mathematically equivalent to the BC model, since it is obtained by simply substituting the convergent Hermite series equation 3.5 into the BC expression 3.4 for n h. The series representation converges for all values of x, but its usefulness depends on how rapidly

4 On the correspondence between biasing models 937 As a check on our formalism, we can verify that the matter halo and halo halo power spectra obtained from the series equation 3.7 agree with the results obtained previously in Baumann et al. 03 using the Edgeworth expansion. We first write the power spectrum P δρn k in terms of the correlation function., P δρn k = αk σ n α q i α k + q q i n i= ξ n+ k, q,...,q n, k q, 3. Figure. Convergence of the series representation equation 3.7 at low k, illustrated by comparing terms in the halo halo power spectrum P hh k = b g P mmk + n= α n P ρ nρ n k in a Gaussian cosmology. Note that for Gaussian initial conditions, cross power spectra P ρmρ n k with m n are zero. We have taken z = 0 and a mass-weighted sample of haloes with mass 0 3 h. it converges, i.e. how many terms we need to get a good approximation. For example, to compute the halo field δ h x at a single point x in real space many terms are needed of order 00 and the series representation is not useful. On the other hand, the Fourier transformed series representation δ h k = bg W δ + n= α nρ n k converges rapidly on large scales i.e. k k nl, as shown in Fig., and the series representation is very convenient. The series converges for all k, but only converges rapidly for k k nl. The preceding expressions have all applied to the case of a massweighted halo sample. For the case of a halo sample defined by a narrow mass bin, the halo field is modelled as n N h x δ x ν c n 0 δ x = a n ν c H n = n + ν c a n+ ν c + a n ν c ln σ n 0 δ x H n. 3.9 Normalizing n h to the fractional halo overdensity δ h,weget δ h x = n + a n+ν c + a nν c δ x H n a n ν c ν c a ν c ν = c δ x + δ x ν c ν c + n! H nν c + H n ν c n! ν n c δ x H n. 3.0 We drop the term containing δ /, since this term vanishes on large scales, k R, and write the result using the notation b g,β n and β n defined in equation 3.3: δ h x = bg N δ x + β n + β n ρ n x n= narrow sample. 3. where we have defined q d 3 q i andq n i π 3 i= q i. Similarly, we can express P ρmρn k as 3 P ρmρn k = m n σ mσ n α q i α q j α q q j i= q i j= α q ξ m+n q,...,q m, q + k, q,..., q n, q k, 3.3 where q m i= q i. Using the notation fˆ,n and f m, n defined in equations 3.5 and 3.7, and taking the limit k 0, we find P δρn k = fˆ,nk, P mm k, 3.4 P ρmρ n k = f m,n k,, P mm k. 3.5 For the case of the mass-weighted halo sample, the series representation equation 3.7 therefore gives the following matter halo and halo halo power spectra, P mh k, = P mm k α n fˆ,nk,, 3.6a n P hh k,, = P mm k α m α n f m,n k,,, m,n 3.6b in agreement with the Edgeworth calculation equation 3.8. The case of the narrow mass bin can be verified similarly. Equations 3.7 and 3. are the main results of this section and give a series representation for the halo field in the BC model, for the cases of a mass-weighted halo sample and a narrow mass bin, respectively. Using the series representation, we will give a simple, 3 We have made an approximation here: by using connected correlation functions in equations 3. and 3.3, we have neglected some contributions to the power spectra P δρn and P ρmρn. ore precisely, we have neglected disconnected terms whose factorization contains multiple higher cumulants i.e. κ m, n with m + n 3 and also some contributions to P ρmρn k which approach a constant as k 0. Note that subleading terms in the Hermite polynomial ρ n = δ / n nn δ / n / +... cancel the largest disconnected contributions to the power spectra in equations 3. and 3.3. The derivation in Baumann et al. 03 of equation 3.8 contains equivalent approximations, which is why we will shortly find agreement with the results of Baumann et al. 03. In principle, one can avoid making any approximations by including disconnected contributions when calculating power spectra P δρn and P ρmρn. However, in appendix A of Baumann et al. 03, we showed that these approximations are always valid in the observationally relevant regime where the initial perturbations are close to Gaussian.

5 938 S. Ferraro et al. conceptual proof of the close correspondence of the barrier model, the PBS and LB in Section 4. However, it is useful to first build an intuition by considering a few example non-gaussian models. 3.3 Examples For a given non-gaussian model, one can analyse large-scale clustering by keeping a small set of terms in the series expansion of δ h either equation 3.7 or 3. for an W or narrow mass bin, respectively and computing the necessary power spectra P ρmρn kon large scales. This is a computationally convenient way to compute the non-gaussian clustering signal and allows the signal to be interpreted physically as arising from large-scale variations in locally measured quantities such as small-scale power and skewness, as we will see in the context of some example models τ NL cosmology Consider a non-gaussian model in which the initial Newtonian potential is given by x = φx + f NL φ x φ, 3.7 where φ is a Gaussian field. We will refer to this as the f NL model or local model. This type of non-gaussianity arises somewhat generically in multifield models of the early universe, e.g. modulated reheating models Zaldarriaga 004, curvaton models Linde & ukhanov 997; Lyth & Wands 00 or multifield ekpyrotic scenarios Buchbinder, Khoury & Ovrut 007; Lehners et al In this section, we will study a generalization of the f NL model which we will call the τ NL model. This type of non-gaussianity arises in multisource models, i.e. models in which quantum mechanical perturbations in multiple fields determine the initial adiabatic curvature perturbation Chen & Wang 00; Tseliakhovich et al. 00; Assassi, Baumann & Green 0; Baumann & Green 0. The non-gaussian potential is given in terms of two uncorrelated Gaussian fields φ and ψ, with power spectra that are proportional to each other: x = φx + ψx + f NL + ψ x ψ, 3.8 where f NL and = P φ k/p ψ k are free parameters. It is easy to compute the three- and four-point functions, [ = f NL P P + 5perms. + O fnl 3, 3.9 ξ 3 5 [ ξ 4 = τ NL P P P perms. + O τ 6 NL, 3.30 wherewehavedefinedτ NL = 6 5 f NL +, P i P k i and P ij P k i + k j. It is conventional to parametrize this model with variables {f NL,τ NL }, which correspond to the amplitudes of the three-point and four-point functions, rather than the variables {f NL, }.Thef NL model with = 0sothatψ contributes but not φ corresponds to the special case τ NL = 6 5 f NL. To compute halo clustering in the τ NL model, we keep the first two terms in the series expansion for δ h equations 3.7 and 3., obtaining { b W g δ + α ρ mass-weighted sample, δ h = bg Nδ + β + β ρ narrow sample. 3.3 Using equations 3.9 and 3.30 in equations 3. and 3.3, we obtain the following power spectra in the k 0 limit: P mm k P δρ k = 4f NL, 3.3 αk 5 P mm k P ρ ρ k = 6 τ NL α k Putting everything together, we find β f P mh k = b g + f NL P mm k, 3.34a αk P hh k = bg + b β f 5 gf NL αk + βf τ NL P 6 α mm k, k 3.34b where we have defined the non-gaussian bias parameter { 4α mass-weighted sample, β f = β narrow sample. In both the mass-weighted and narrow mass bin cases, the non- Gaussian and Gaussian parts of the bias are related by β f = δ c b g. Note that in the narrow mass bin case, there is a derivative term in δ h the term ρ / in equation 3.3, but this ends up giving zero contribution to the power spectra P mh and P hh, since the power spectra P δρ and P ρ ρ are independent of in the τ NL model. Our calculation of the clustering power spectra equation 3.34 agrees with previous calculations in the literature e.g. Tseliakhovich et al. 00; Baumann et al. 03, but the series representation gives some physical intuition: the large-scale non-gaussian clustering is due to large-scale fluctuations in the field ρ,which we interpret as long-wavelength variations in the locally measured small-scale power. If τ NL = 6 5 f NL, then long-wavelength variations in ρ are 00 per cent correlated to the matter density δ on large scales, and the non-gaussian halo bias is non-stochastic. If τ NL > 6 5 f NL,thenρ and δ are not 00 per cent correlated, leading to stochastic bias g NL cosmology The g NL model is a non-gaussian model in which the initial potential is given in terms of a single Gaussian field φ by x = φx + g NL φ 3 x 3 φ φx We keep the first three terms in the series expansion for δ h, obtaining bg W δ + α ρ + α 3 ρ 3 mass-weighted sample, δ h = bg Nδ + β + β ρ + β 3 + β 3 ρ 3 narrow sample To compute power spectra we will need the following cumulants in the g NL model: ξ 4[tree = g NL [P P P perms. ξ 4[loop + OgNL, 3.38 [ = 9 gnl P P P φ k 3 + perms., 3.39

6 [ ξ 6 = 36 g NL P P P 3 P 4 P perms Here, we have defined P ij k = P φ k i + k j + k k and P φ k P φ qp φ k q 4 φ lnklp φk, 3.4 q where φ k3 /π P φ k, and we have regulated the infrared divergence by putting the field in a finite box of size L. Note that the power spectra P δρ and P ρ ρ 3 are zero since there is a symmetry. The remaining power spectra can be calculated by substituting equations 3.38, 3.39 and 3.40 into equations 3. and 3.3. In the limit k 0, this gives P mm k P δρ3 k = 3 g NL κ f NL= 3, 3.4 αk P ρ ρ k = 4 g NL σ P ρ3 ρ 3 k = 9 gnl P mm k α k α qp φ q + 36 gnl P φk, 3.43 q κ f NL= Here, κ f NL= 3 denotes the dimensionless skewness parameter κ 3 = δ 3 x c/σ 3 in the local model with f NL =. Note that we when computing P δρ3, but use both the tree-level cumulant and the one-loop cumulant ξ 4[loop when computing P ρ ρ. Although the OgNL one-loop cumulant is use the tree-level cumulant ξ 4[tree generally smaller than the Og NL tree-level cumulant, the one-loop cumulant dominates in the k + k 0 limit which is relevant for P ρ ρ. Putting the above calculations together, we find: 4 β g P mh k = b g + g NL αk P mm k, 3.45a β g P hh k = b g + g NL P mmk + 9 αk 4 β f g NL P φk, 3.45b where β f was defined in equation 3.35 and we have defined 3 α 3 κ f NL= 3 mass-weighted sample, β g = 3 β + β κ f NL= 3 narrow sample Note that in the narrow mass bin case, there are derivative terms in δ h equation 3.37, and their contributions to P mh and P hh are non-zero unlike the previously considered τ NL model, because the power spectra P ρmρn in equations depend on halo mass via the mass-dependent quantity κ f NL= 3. These expressions for P mh and P hh agree with previous calculations in the literature based on the Edgeworth expansion Desjacques et al. 0; Smith et al. 0; Baumann et al. 03. Our series expansion gives some physical intuition as follows. The non-gaussian contribution to P mh comes from the power spectrum P δρ3 and can therefore be interpreted as arising from longwavelength variations in the locally measured small-scale skewness ρ 3. On large scales, the non-gaussian fluctuations in ρ 3 are 4 We have neglected contributions to P hh k which approach a constant as k 0; such contributions are unobservable in practice since they are degenerate with other contributions such as second-order halo bias. On the correspondence between biasing models per cent correlated to the density field, and therefore the associated halo bias is non-stochastic. The leading contribution to stochastic bias comes from the power spectrum P ρ ρ and can be interpreted as long-wavelength variations in small-scale power which are uncorrelated to the density field. 4 PROOF OF THE CORRESPONDENCE In the previous section, we showed that the BC model can be formulated as a series representation: bg W δ x + n α nρ n x mass-weighted sample, δ h x = bg Nδ x + n β n + β n ρ n x narrow sample. 4. In this section, we will use this result to prove that BC is mathematically related to LB Section 4. and PBS Section 4., and that in particular they give equivalent results. 4. Local biasing LB refers to any model of halo clustering in which the halo field is represented as a local function of the dark matter density, e.g. a power series δ h x = b δx + b δ x + b 3 δ 3 x Several versions of LB exist in the literature e.g. Fry & Gaztanaga 993; Sefusatti 009; Giannantonio & Porciani 00; Baldauf et al. 0. We notice that the series on the right-hand side of equation 4. is a type of LB expansion, since the ρ n fields are local functions of the smoothed density field δ. Therefore, our series representation proves that the BC model is mathematically equivalent to a specific version of the LB formalism. 5 In this section, we would like to elaborate on the connection between our series representation and the usual way of thinking about LB and comment on the differences with other versions of the formalism. First, the density field δ which appears in the series representation is the non-gaussian and linearly evolved density field, smoothed on the mass scale. In particular, there is no need to introduce a new smoothing scale which is distinct from the halo scale, as done in some versions of LB. We do not include non-linear evolution in δ since the standard BC model is based on thresholding the linear density field. Secondly, we do not need to introduce explicit dependence of the halo overdensity δ h on the long-wavelength potential l in a non-gaussian cosmology. In some versions of LB, δ h is expanded in both δ l and l, in order to keep the relation local. In our version, the l dependence happens automatically, since δ h depends 5 In Desjacques et al. 0, the authors have argued that the BC model can be written as a local expansion in terms of renormalized bias parameters see section III.B and III.C. We note that a local Hermite expansion is well defined and will automatically generate the correct bias coefficients in the Gaussian or weakly non-gaussian case, without the need for renormalization. See section II.D of Schmidt, Jeong & Desjacques 03 for further discussion.

7 940 S. Ferraro et al. on higher cumulants ρ, ρ 3,..., and these cumulants can be correlated with l in a non-gaussian model. To see how this happens in detail, consider the f NL model. Inspection of the power spectra in equations 3.3 and 3.33 shows taking τ NL = 6 5 f NL thatρ is 00 per cent correlated with the field l = α kδ as k 0. ore precisely, ρ 4f NL l on large scales. aking this substitution in equation 4., we get δ h = b g δ l + f NL β f l + and recover the usual result. This example shows that including explicit l dependence in the local expansion of δ h is not necessary in fact, including it our model would double-count the non-gaussian clustering, if higher powers of the density field are included in the expansion. In the f NL model, the modulation to the locally measured power ρ is directly proportional to l. ore generally, the expansion should be in all of the non-negligible cumulants ρ, ρ 3,... It is also interesting to consider the τ NL model in the case τ NL > 6 5 f NL. Here, the locally measured small-scale power ρ has excess power on large scales which is not 00 per cent correlated with l, leading to stochastic bias Baumann et al. 03. This qualitative behaviour is correctly captured by an LB model of the form δ h = b g δ l + α ρ, but not by an LB model of the form δ h = b g δ l + b l. In the narrow mass bin case, our series expansion includes derivative terms of the form ρ n /. To our knowledge, derivative terms have not been proposed in any version of LB which has appeared in the literature. In the BC model, derivative terms appear naturally for a narrowly selected halo sample, since this case is obtained from the mass-weighted case which does not contain derivative terms by differentiating with respect to halo mass. Finally, even in the mass-weighted case, there is a difference between the Hermite polynomial expansions δ h x = bg W δ x + δ x α n H n 4.3 σ n and a power series expansion of the form δ h x = b δ x + b δ x + b 3δ 3 x At first sight, the two may appear equivalent: if both series are truncated at the same order N, then we can rearrange coefficients to transform either series into the other since both just parametrize an arbitrary degree-n polynomial. However, when we write the power series expansion 4.4, we are assuming that the values of the low-order coefficients b, b,... are independent of the order N at which the series is truncated. This means for example that in a Gaussian cosmology, the matter halo power spectrum P mh k = b + 3 σ b σ 4 b 5 + P mm k depends on where the series is truncated. In contrast, the Hermite expansion 4.3 is more stable: P mh k is always equal to bg W P mm k, regardless of how many terms are retained in the series. Note that the BC model has a convergent Hermite polynomial expansion 3.5, but cannot be sensibly expanded as a power series in δ, since the Heaviside step function δ / δ c is not an analytic function of δ. In summary, the BC model is mathematically equivalent to a specific version of the LB formalism in which the following choices have been made: we linearly evolve the density field and smooth it at mass scale ; we include higher cumulants ρ, ρ 3,... in the density field, but not additional fields such as the potential l ; derivative terms appear in the narrow mass bin case and we use a Hermite polynomial expansion in δ / rather than the power series expansion. Other variants of the LB formalism exist in the literature, and we are not claiming that our choices are optimal in the sense of producing best agreement with simulations; the purpose of this section was simply to point out which set of choices is equivalent to the BC model. 4. Peak-background split The PBS is a formalism for modelling halo clustering on large scales, in which one relates large-scale modes of the halo density field δ h to large-scale modes of fields whose power spectra can be calculated directly. For example, the PBS formalism was applied to an f NL cosmology in Slosar et al On large scales, k R, one can argue that the halo density is related to the linear density field δ and the Newtonian potential by δ h k = b g δk + f NL β f k, 4.5 where b g is the usual Gaussian bias and β f = ln n h / ln σ 8. Using this expression, it is easy to show that the large-scale bias is given by bk = b g + f NL β f /αk and is non-stochastic. For additional examples of the PBS formalism applied to non-gaussian models, see Tseliakhovich et al. 00, Smith et al. 0 and Baumann et al. 03. In this section, we will show how the PBS formalism generalizes to an arbitrary non-gaussian model and give a simple proof that this generalization is equivalent to the BC model. We will work out in detail the case of a mass-weighted halo sample; the narrow mass bin case follows by differentiating with respect to. There is one technical point that we would like to make explicit. We want to generalize the PBS formalism so that it applies to an arbitrary non-gaussian model, parametrized by the N-point correlation functions of the initial Newtonian potential. As an example, consider the τ NL model from Section 3.3., with constituent fields φ and ψ. The PBS analysis of this model has been worked out in Tseliakhovich et al. 00 and Baumann et al. 03, and it requires keeping track of the long-wavelength parts φ l and ψ l of both fields, in order to correctly predict non-gaussian stochastic bias on large scales. Intuitively, multiple fields are needed because we need to keep track of long-wavelength density fluctuations and longwavelength variations in the locally measured small-scale power, and the two are not 00 per cent correlated in the τ NL model. This raises a conceptual puzzle: how would we get stochastic bias if we were just given correlation functions of the single field, rather than a description of the τ NL model involving multiple constituent fields? As we will now see, we must extend the PBS formalism by introducing additional fields which correspond to the locally measured small-scale power, small-scale skewness, kurtosis, etc. These fields are precisely the quantities ρ, ρ 3,... which appeared earlier in our series expansion in Section 3. This will allow us to connect the PBS formalism with the BC model and in fact prove that the two are mathematically equivalent. Consider a large subvolume of the Universe containing many haloes, but over which the long mode is reasonably constant, and let l denote a spatial average over the subvolume. Let us assume that the halo number density n h l in the subvolume is a function of the one-point PDF of the underlying dark matter field δ when linearly evolved and smoothed on the halo scale. For weakly non- Gaussian fields, the one-point PDF in each subvolume can be characterized completely by its mean δ l,varianceσ l and higher cumulants κ n l = δ n c/σ n l for n 3. Therefore, we can write n h l n h δ l, σ l, {κ n l }. Taylor expanding to first order in

8 On the correspondence between biasing models 94 these parameters, we get n h l = n h + ln n h δ l + ln n h δ l σ σ σ l l ln n h + κ n l. 4.6 κ n l n=3 Here, we have used the notation σ l to denote the variance of δ restricted to the subvolume, and σ to denote the global variance. To make contact with our previous notation, note that σ l σ = σ ρ l and κ n l = ρ n l. 6 aking these substitutions in equation 4.6, we get δ h l = ln n h δ l + σ ln n h δ l ln n h ρ σ l + ρ n l. κ l n=3 n l 4.7 Since this equation applies when taking the subvolume average l over any large subvolume, it also applies to any large-scale Fourier mode: δ h k k 0 ln n h δ k + σ ln n h δ σ ρ k + n=3 ln n h κ n ρ n k. 4.8 Let us compare this expression with our series representation of δ h in the BC model: δ h k = bg W δ k + α n ρ n k. 4.9 n The form of the two series representations is the same, but the coefficients appear to be different. In the BC model, we have the following explicit formula for the coefficient α n ofthenth term in the series: α n = π e ν c / H n ν c, 4.0 erfc ν c n! whereas in the PBS derivation, α n is given by a suitable derivative of the halo mass function: α = σ ln n h σ and α n = ln n h κ n for n If we assume a Press Schechter mass function, then one can evaluate the mass function derivatives in the above equation using the machinery from LoVerde et al The result agrees precisely with the explicit formula 4.0. Therefore, the BC model and the generalized PBS formalism with fields ρ, ρ 3,... are formally equivalent, but only under the assumption of a Press Schechter mass function note that this assumption is built in to the BC model. If we relax the assumption of a Press Schechter mass function, then the BC model and the generalized PBS formalism can both be written as series expansions with the same general form, but make different predictions for the coefficients α n. One can ask which prediction agrees better with N-body simulations. In Smith et al. 0, the two predictions for α 3 were compared with simulations 6 The identity ρ n l = κ n l holds for n 5, but has non-linear corrections for n 6. For example, ρ 6 l = κ 6 l + 0κ 3 l. We have neglected these non-linear corrections since equation 4.6 is an expansion to first order anyway. in the context of the g NL model. It was found that the PBS prediction 4. is exact within the per cent statistical error of the simulations if both the bias and the mass function derivative ln n h /κ 3 are evaluated numerically from the simulations. The BC prediction 4.0 is an approximation: although it is based on an exact calculation within the BC model, this model is an approximation to the true dynamics of an N-body simulation. The approximation works reasonably well for large halo mass but breaks down for low masses, motivating the use of fitting functions for practical data analysis. It is natural to conjecture that the same qualitative statements will be true for the α n coefficients with n > 3, but we have not attempted to verify this with the simulations. Note that no fitting function is necessary for α, since the relation β f δ c b g holds to 0 per cent accuracy in N-body simulations. In summary, the BC model is closely related to the PBS formalism, appropriately generalized to an arbitrary non-gaussian cosmology by introducing additional fields ρ, ρ 3,..., plus the additional assumption of a Press Schechter mass function. The BC model is analytically tractable [e.g. one can derive closed-form expressions for the coefficients α n andβ n and usually a reasonable approximation, making it very useful for analytic studies or forecasts. However, for data analysis, it may be necessary to go beyond the Press Schechter approximation by replacing the closed-form expressions for coefficients such as α n with their PBS counterparts measured from simulations. 5 CONCLUSIONS In this paper, we have demonstrated the precise mathematical relationship between BC, PBS and LB. We first introduced a Hermite polynomial expansion of the halo density contrast δ h in the BC model: equations 3.7 and 3.. We showed that this allows a computationally efficient way to calculate the clustering power spectra P mh and P hh. oreover, the series expansion makes the formal equivalence of the various halo modelling formalisms very transparent. First, it automatically takes the form of an LB model, in which the non-gaussian and linearly evolved density contrast is expanded in Hermite polynomials. Secondly, it provides a very natural connection between BC and PBS. To make this relationship manifest, we generalized the PBS formalism so that it can be applied to the most general set of non-gaussian initial conditions, parametrized by the N-point functions of the primordial potential. This extension of PBS involves additional fields which correspond to the locally measured small-scale power, smallscale skewness, kurtosis, etc. apping those fields to fields in the Hermite polynomial expansion of the BC model, we showed the close relationship between PBS and BC. Finally, although in this paper we have concentrated on computing power spectra, our series expansion should also be useful for analysing the effects of primordial non-gaussianity on other clustering statistics, such as the halo bispectrum Baldauf et al. 0. ACKNOWLEDGEENTS We thank arilena LoVerde, arcel Schmittfull, David Spergel and atias Zaldarriaga for helpful discussions. SF acknowledges support from a fellowship at the Department of Astrophysical Sciences of Princeton University. KS was supported by a Lyman Spitzer fellowship in the Department of Astrophysical Sciences at Princeton University. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the

9 94 S. Ferraro et al. Province of Ontario through the inistry of Research and Innovation. The research of DG is supported by the DOE under grant number DE-FG0-90ER4054 and the artin A. and Helen Chooljian embership at the Institute for Advanced Study. DB gratefully acknowledges support from a Starting Grant of the European Research Council ERC STG grant REFERENCES Assassi V., Baumann D., Green D., 0, J. Cosmol. Astropart. Phys.,, 047 Baldauf T., Seljak U., Senatore L., 0, J. Cosmol. Astropart. Phys., 04, 006 Bardeen J.., Bond J., Kaiser N., Szalay A., 986, ApJ, 304, 5 Baumann D., Green D., 0, Phys. Rev. D, 85, 0350 Baumann D., Ferraro S., Green D., Smith K.., 03, J. Cosmol. Astropart. Phys., 305, 00 Buchbinder E. I., Khoury J., Ovrut B. A., 007, Phys. Rev. D, 76, 3503 Chen X., Wang Y., 00, J. Cosmol. Astropart. Phys., 004, 07 Cole S., Kaiser N., 989, NRAS, 37, 7 Dalal N., Dore O., Huterer D., Shirokov A., 008, Phys. Rev. D, 77, 354 Desjacques V., Jeong D., Schmidt F., 0, Phys. Rev. D, 84, 0635 Fry J. N., Gaztanaga E., 993, ApJ, 43, 447 Giannantonio T., Porciani C., 00, Phys. Rev. D, 8, Lehners J.-L., cfadden P., Turok N., Steinhardt P. J., 007, Phys. Rev. D, 76, 0350 Linde A. D., ukhanov V. F., 997, Phys. Rev. D, 56, 535 LoVerde., iller A., Shandera S., Verde L., 008, J. Cosmol. Astropart. Phys., 0804, 04 Lyth D. H., Wands D., 00, Phys. Lett. B, 54, 5 atarrese S., Verde L., 008, ApJ, 677, L77 Press W. H., Schechter P., 974, ApJ, 87, 45 Schmidt F., Jeong D., Desjacques V., 03, Phys. Rev. D, 88, 0355 Scoccimarro R., Hui L., anera., Chan K. C., 0, Phys. Rev. D, 85, Sefusatti E., 009, Phys. Rev. D, 80, 300 Slosar A., Hirata C., Seljak U., Ho S., Padmanabhan N., 008, J. Cosmol. Astropart. Phys., 0808, 03 Smith K.., Ferraro S., LoVerde., 0, J. Cosmol. Astropart. Phys., 03, 03 Tseliakhovich D., Hirata C., Slosar A., 00, Phys. Rev. D, 8, Zaldarriaga., 004, Phys. Rev. D, 69, APPENDIX A: HERITE POLYNOIALS In this paper, we have used the probabilists definition of Hermite polynomials H n ν = n e ν / dn dν n e ν /, satisfying the recursion relation H n+ ν = νh n ν H n ν and the orthogonality condition dν π e ν / H m νh n ν = m! δ mn. A. A. A.3 For reference, we list some of the low-order Hermite polynomials H 0 ν =, H ν = ν, H ν = ν, H 3 ν = ν 3 3ν, H 4 ν = ν 4 6ν + 3. A.4 A.5 A.6 A.7 A.8 We have made use of the following integral dν e ν / erfc ν c n = 0 H n ν =. n! ν c π n! π e ν c / H n ν c n A.9 This paper has been typeset from a TEX/LATEX file prepared by the author.