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.
1 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. 2008; Matarrese & Verde 2008), sometimes of stochastic type (Tseliakhovich, Hirata & Slosar 2010; Baumann et al. 2013). 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 2011; Baumann et al. 2013). 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 (1974). In its simplest formulation, it identifies haloes as regions of the linearly evolved density field above some critical density δc. 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. 2008; Desjacques et al. 2011; Smith, Ferraro & LoVerde 2012; Baumann et al. 2013).
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. 1986; Cole & Kaiser 1989), and its usefulness in dealing with non-Gaussian initial conditions has been first pointed out in Dalal et al. (2008). 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 ρ2, ρ3, …, which measure the local power-spectrum amplitude, skewness, etc.
In a companion paper (Baumann et al. 2013), 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.
The outline of this paper is as follows. After defining our notation in Section 2, we introduce our main technical tool in Section 3: a series expansion for the halo field δh 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.
2 PRELIMINARIES AND NOTATION
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. 2013). We also introduce a series representation of BC, which will be our main tool to prove the equivalence to LB and PBS in Section 4.
3.1 Review of barrier crossing
3.2 Hermite polynomial expansion
We emphasize that the series representation (equation 3.17) is mathematically equivalent to the BC model, since it is obtained by simply substituting the convergent Hermite series (equation 3.15) into the BC expression (3.14) for nh. The series representation converges for all values of |${\boldsymbol {x}}$|, but its usefulness depends on how rapidly it converges, i.e. how many terms we need to get a good approximation. For example, to compute the halo field |$\delta _{\rm h}({\boldsymbol {x}})$| at a single point |${\boldsymbol {x}}$| in real space many terms are needed (of order 100) and the series representation is not useful. On the other hand, the Fourier transformed series representation |$\delta _{\rm h}({\boldsymbol {k}}) = b_{\rm g}^{{{\rm MW}}} \delta _{M}+\sum _{n=2}^\infty \alpha _n \rho _n({\boldsymbol {k}})$| converges rapidly on large scales (i.e. k ≪ knl), as shown in Fig. 1, and the series representation is very convenient. (The series converges for all k, but only converges rapidly for k ≪ knl.)
Convergence of the series representation (equation 3.17) at low k, illustrated by comparing terms in the halo–halo power spectrum |$P_{{\rm h}{\rm h}}(k) = b_{\rm g}^2(M) P_{{\rm m}{\rm m}}(k) + \sum _{n=2}^\infty \alpha _n^2(M) P_{\rho _n\rho _n}(k)$| in a Gaussian cosmology. (Note that for Gaussian initial conditions, cross power spectra |$P_{\rho _m\rho _n}(k)$| with m ≠ n are zero.) We have taken z = 0 and a mass-weighted sample of haloes with mass M ≥ 2 × 1013 h−1 M⊙.
Equations (3.17) and (3.21) 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, 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 δh (either equation 3.17 or 3.21 for an MW or narrow mass bin, respectively) and computing the necessary power spectra |$P_{\rho _m\rho _n}(k)$| on 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.
3.3.1 τNL cosmology
Our calculation of the clustering power spectra (equation 3.34) agrees with previous calculations in the literature (e.g. Tseliakhovich et al. 2010; Baumann et al. 2013), but the series representation gives some physical intuition: the large-scale non-Gaussian clustering is due to large-scale fluctuations in the field ρ2, which we interpret as long-wavelength variations in the locally measured small-scale power. If |$\tau _{{ \rm NL}}= (\frac{6}{5} f_{{\rm NL}})^2$|, then long-wavelength variations in ρ2 are 100 per cent correlated to the matter density δ on large scales, and the non-Gaussian halo bias is non-stochastic. If |$\tau _{{ \rm NL}}> (\frac{6}{5} f_{{\rm NL}})^2$|, then ρ2 and δ are not 100 per cent correlated, leading to stochastic bias.
3.3.2 gNL cosmology
These expressions for Pmh and Phh agree with previous calculations in the literature based on the Edgeworth expansion (Desjacques et al. 2011; Smith et al. 2012; Baumann et al. 2013). Our series expansion gives some physical intuition as follows. The non-Gaussian contribution to Pmh comes from the power spectrum |$P_{\delta \rho _3}$| and can therefore be interpreted as arising from long-wavelength variations in the locally measured small-scale skewness ρ3. On large scales, the non-Gaussian fluctuations in ρ3 are 100 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 |$P_{\rho _2\rho _2}$| and can be interpreted as long-wavelength variations in small-scale power which are uncorrelated to the density field.
4 PROOF OF THE CORRESPONDENCE
4.1 Local biasing
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 δM which appears in the series representation is the non-Gaussian and linearly evolved density field, smoothed on the mass scale M. 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 δM 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 δh on the long-wavelength potential Φℓ in a non-Gaussian cosmology. In some versions of LB, δh is expanded in both δℓ and Φℓ, in order to keep the relation local. In our version, the Φℓ dependence happens automatically, since δh depends on higher cumulants ρ2, ρ3, …, and these cumulants can be correlated with Φℓ in a non-Gaussian model. To see how this happens in detail, consider the fNL model. Inspection of the power spectra in equations (3.32) and (3.33) shows (taking |$\tau _{{ \rm NL}}= (\frac{6}{5} f_{{\rm NL}})^2$|) that ρ2 is 100 per cent correlated with the field |$\Phi _\ell = \alpha _{{M}}^{-1}(k) \delta _{M}$| as k → 0. More precisely, ρ2 → 4fNLΦℓ on large scales. Making this substitution in equation (4.1), we get δh = bgδℓ + fNLβfΦℓ + ⋅⋅⋅ and recover the usual result.
This example shows that including explicit Φℓ dependence in the local expansion of δh 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 fNL model, the modulation to the locally measured power ρ2 is directly proportional to Φℓ. More generally, the expansion should be in all of the non-negligible cumulants ρ2, ρ3, ….
It is also interesting to consider the τNL model in the case |$\tau _{{ \rm NL}}> (\frac{6}{5} f_{{\rm NL}})^2$|. Here, the locally measured small-scale power ρ2 has excess power on large scales which is not 100 per cent correlated with Φℓ, leading to stochastic bias (Baumann et al. 2013). This qualitative behaviour is correctly captured by an LB model of the form δh = bgδℓ + α2ρ2, but not by an LB model of the form δh = bgδℓ + b2Φℓ.
In the narrow mass bin case, our series expansion includes derivative terms of the form ∂ρn/∂ln σM. 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.
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 M; we include higher cumulants ρ2, ρ3, … in the density field, but not additional fields such as the potential Φℓ; derivative terms appear in the narrow mass bin case and we use a Hermite polynomial expansion in δM/σM 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.
4.2 Peak-background split
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 Φ. As an example, consider the τNL model from Section 3.3.1, with constituent fields ϕ and ψ. The PBS analysis of this model has been worked out in Tseliakhovich et al. (2010) and Baumann et al. (2013), and it requires keeping track of the long-wavelength parts ϕℓ and ψℓ 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 long-wavelength variations in the locally measured small-scale power, and the two are not 100 per cent correlated in the τNL model.) This raises a conceptual puzzle: how would we get stochastic bias if we were just given correlation functions of the single field Φ, rather than a description of the τNL 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 ρ2, ρ3, … 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).
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 αn(M). One can ask which prediction agrees better with N-body simulations. In Smith et al. (2012), the two predictions for α3 were compared with simulations in the context of the gNL model. It was found that the PBS prediction (4.11) is exact (within the ≈1 per cent statistical error of the simulations) if both the bias and the mass function derivative (∂ln nh/∂κ3) are evaluated numerically from the simulations. The BC prediction (4.10) 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 αn coefficients with n > 3, but we have not attempted to verify this with the simulations. (Note that no fitting function is necessary for α2, since the relation βf ≈ 2δcbg holds to ≈10 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 ρ2, ρ3, …, 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 αn(M) and βn(M)] 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 αn(M) 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 δh in the BC model: equations (3.17) and (3.21). We showed that this allows a computationally efficient way to calculate the clustering power spectra Pmh and Phh. Moreover, 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, small-scale skewness, kurtosis, etc. Mapping 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. 2011).
We thank Marilena LoVerde, Marcel Schmittfull, David Spergel and Matias Zaldarriaga for helpful discussions. SF acknowledges support from a fellowship at the Department of Astrophysical Sciences of Princeton University. KMS 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 Province of Ontario through the Ministry of Research and Innovation. The research of DG is supported by the DOE under grant number DE-FG02-90ER40542 and the Martin A. and Helen Chooljian Membership at the Institute for Advanced Study. DB gratefully acknowledges support from a Starting Grant of the European Research Council (ERC STG grant 279617).
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 M are unpopulated with tracers, whereas the expected number of tracers in a halo of mass ≥M is proportional to the halo mass.
Note that the variance of the unsmoothed linear density contrast σ2 = 〈δ2〉 is formally infinite, but cancels in the definition (3.5) of the quantity |$f_{\hat{1},n}$| which will appear in our final expressions.
We have made an approximation here: by using connected correlation functions in equations (3.22) and (3.23), we have neglected some contributions to the power spectra |$P_{\delta \rho _n}$| and |$P_{\rho _m\rho _n}$|. More precisely, we have neglected disconnected terms whose factorization contains multiple higher cumulants (i.e. κm, n with m + n ≥ 3) and also some contributions to |$P_{\rho _m\rho _n}(k)$| which approach a constant as k → 0. (Note that subleading terms in the Hermite polynomial ρn = (δM/σM)n − n(n − 1)(δM/σM)n − 2/2 + … cancel the largest disconnected contributions to the power spectra in equations 3.22 and 3.23.) The derivation in Baumann et al. (2013) of equation (3.8) contains equivalent approximations, which is why we will shortly find agreement with the results of Baumann et al. (2013). In principle, one can avoid making any approximations by including disconnected contributions when calculating power spectra |$P_{\delta \rho _n}$| and |$P_{\rho _m\rho _n}$|. However, in appendix A of Baumann et al. (2013), we showed that these approximations are always valid in the observationally relevant regime where the initial perturbations are close to Gaussian.
We have neglected contributions to Phh(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.
In Desjacques et al. (2011), 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 (2013) for further discussion.
The identity (ρn)ℓ = (κn)ℓ holds for n ≤ 5, but has non-linear corrections for n ≥ 6. For example, |$(\rho _6)_\ell = (\kappa _6)_\ell + 10 (\kappa _3)_\ell ^2$|. We have neglected these non-linear corrections since equation (4.6) is an expansion to first order anyway.
