# Gravity and Large-Scale Non-local Bias

###### Abstract

For Gaussian primordial fluctuations the relationship between galaxy and matter overdensities, bias, is most often assumed to be local at the time of observation in the large-scale limit. This hypothesis is however unstable under time evolution, we provide proofs under several (increasingly more realistic) sets of assumptions. In the simplest toy model galaxies are created locally and linearly biased at a single formation time, and subsequently move with the dark matter (no velocity bias) conserving their comoving number density (no merging). We show that, after this formation time, the bias becomes unavoidably non-local and non-linear at large scales. We identify the non-local gravitationally induced fields in which the galaxy overdensity can be expanded, showing that they can be constructed out of the invariants of the deformation tensor (Galileons), the main signature of which is a quadrupole field in second-order perturbation theory. In addition, we show that this result persists if we include an arbitrary evolution of the comoving number density of tracers. We then include velocity bias, and show that new contributions appear; these are related to the breaking of Galilean invariance of the bias relation, a dipole field being the signature at second order.

We test these predictions by studying the dependence of halo overdensities in cells of fixed dark matter density: measurements in simulations show that departures from the mean bias relation are strongly correlated with the non-local gravitationally induced fields identified by our formalism, suggesting that the halo distribution at the present time is indeed more closely related to the mass distribution at an earlier rather than present time. However, the non-locality seen in the simulations is not fully captured by assuming local bias in Lagrangian space. The effects on non-local bias seen in the simulations are most important for the most biased halos, as expected from our predictions.

Accounting for these effects when modeling galaxy bias is essential for correctly describing the dependence on triangle shape of the galaxy bispectrum, and hence constraining cosmological parameters and primordial non-Gaussianity. We show that using our formalism we remove an important systematic in the determination of bias parameters from the galaxy bispectrum, particularly for luminous galaxies.

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^{†}preprint: Version 1.10

## I Introduction

Galaxies are one of the main probes of modern cosmology. However, the galaxy clustering amplitude depends on galaxy type, so not all types can be unbiased tracers of the dark matter Kaiser (1984). Therefore, understanding and accounting for this bias is important. It is common to assume that this bias is a local and deterministic function of the dark matter density field (e.g. other properties of the field than the local overdensity, such as the tidal field, are assumed to produce negligible effects on the galaxy distribution), so the galaxy density contrast at any given time can be written as a Taylor series in the dark matter density at that time Fry and Gaztanaga (1993). One of the main goals of this paper is to show that, if there is any time at which this is a good approximation, then it is not good at any other time. A related goal is to argue is that this should be a better model at early than at later times, in a sense that will be made more precise later in this paper.

In the galaxy distribution, one expects departures from the local deterministic bias model on scales where nonlinear baryon physics matters. Nonetheless, on the scales larger than those associated with galaxy formation processes, the deterministic local bias is expected to be accurate, except for a possible constant shot-noise-type contribution Scherrer and Weinberg (1998); Matsubara (1999).
This has motivated the use of the deterministic local biasing prescription for interpreting clustering measurements in galaxy surveys, in particular this model has been heavily used in interpreting measurements of three-point functions and other measures of non-Gaussianity Fry and Gaztanaga (1993); Frieman and Gaztanaga (1994); Fry (1994); Gaztanaga and Frieman (1994); Scoccimarro *et al.* (2001a); Feldman *et al.* (2001); Verde *et al.* (2002); Baugh *et al.* (2004); Croton *et al.* (2004); Jing and Börner (2004); Kayo *et al.* (2004); Pan and Szapudi (2005); Gaztañaga *et al.* (2005); Nishimichi *et al.* (2007); Marín *et al.* (2008); McBride *et al.* (2010a, b); Marín (2011).
However, it is also common to assume that galaxies are closely associated with dark matter halos Cooray and Sheth (2002). So it is natural to ask if halo bias is a deterministic function of the local dark matter overdensity. Numerical simulations indicate that, on scales of order 20 Mpc and less, halo abundance is not a deterministic function of the dark mass Mo and White (1996); Sheth and Lemson (1999); Casas-Miranda *et al.* (2002). This manifests as stochasticity in the relation between the galaxy and dark matter density fields at the present time Dekel and Lahav (1999); Somerville *et al.* (2001). If we distinguish between the stochasticity associated with some initial or formation time, and that due to evolution from this time to the time of observation – then the question arises as to which matters more on the large scales which the next generation of galaxy surveys will probe.

In the Excursion set model of halo formation, abundance and clustering, it is the initial fluctuation field which is fundamental Bond *et al.* (1991); Lacey and Cole (1993); Sheth (1998). In these models, the origin of the first source of stochasticity is relatively straightforward to understand: the initial random fluctuation field is expected to have structure on arbitrary small scales, so the substructure within large patches of the same large scale overdensity may differ from one patch to another. Whether or not a small patch forms a halo is known to be closely related to the initial overdensity of the patch. If the initial overdensity is the only parameter which matters (e.g., in spherical evolution models, where tidal fields etc. play no role) then the fact that the small scale density is correlated with the density on larger scales produces stochasticity in halo abundances within large spheres of fixed initial overdensity. Much of this scatter is just a sort of shot-noise which decreases as the cell size is increased Sheth and Lemson (1999). So, on large scales, a deterministic model for the bias can be quite accurate. If halo formation depends on quantities other than local density Bond and Myers (1996); Sheth *et al.* (2001), then this may contribute to the stochasticity seen in the initial conditions. But if these other quantities are correlated over shorter scales than is the density, then their effects will be subdominant on large scales, and so they may be neglected in studies of sufficiently large scale bias. In what follows, we will assume this is the case.

That is to say, the main goal of this paper is to study departures from the local deterministic bias model which may appear on scales larger than those associated with galaxy or halo formation (i.e., above a few tens of Mpc). We will show that, even if the bias is local and deterministic at some given time (which we will usually call the formation time), then subsequent nonlinear gravitational evolution will generate non-local bias. In this respect, our results serve as a well-motivated model for non-local bias. Other works on non-local bias have provided models Matsubara (1995, 1999); McDonald and Roy (2009) based on statistical (as opposed to dynamical) considerations. The virtue of our approach is that it gives a concrete form of non-local bias that must be present even if formation bias is truly local, and we demonstrate for the first time their presence in numerical simulations. In addition, we show that our non-local bias model solves a systematic effect in the determination of the linear bias from bispectrum measurements for biased tracers.

Since evolution plays an important role in the discussion, we devote a substantial part of this paper to the study of the evolution of bias and how it generates non-local bias. The evolution of halo bias, under the assumption that the number of halos was conserved and their motions were not biased relative to the mass, was first studied by Mo and White (1996). They showed that the predictions for this evolution, based on a spherical collapse model for the dynamics, provided a good description of how halo bias evolves. At linear order (linear theory evolution of the linear bias factor), this calculation agrees with that from combining the continuity equation with perturbation theory, again assuming no velocity bias Fry (1996). At linear order, the perturbation theory approach can be generalized to include stochasticity and galaxy formation as a source Pen (1998); Tegmark and Peebles (1998). However, going beyond linear order, either in evolution or in bias, is less straightforward.

Evolution of the higher order bias factors was investigated in Mo *et al.* (1997); Scoccimarro *et al.* (2001b), but these works approximated the nonlinear gravitational evolution using the spherical collapse model. This simplification leads to the inaccurate conclusion that a local bias at formation stays local. That gravitational evolution generates non-local bias can be seen from the results of Fry (1996) in second-order in perturbation theory, although this particular point was not noted in that work. The best known example is the limit of this result when the “formation time” is taken to be at the far past, the so-called local Lagrangian bias, and was first emphasized in Catelan *et al.* (1998) and further explored in Catelan *et al.* (2000). In this paper we develop a formalism that contains all these results in particular limits. Moreover, it extends them
i) to higher-order in perturbation theory,
ii) to include non-conservation of tracers
(arbitrary formation rate and merging),
iii) to consider biased tracers that do not flow with the dark matter
(velocity bias).
Non-local bias is particularly interesting in view of the fact that the local biasing prescription does not seem to agree well with simulations Roth and Porciani (2011); Manera and Gaztañaga (2011). Our model of the non-locality generated by evolution gives a well-motivated model for non-local bias.

This paper is organized as follows. In Section II, we develop a formalism to generalize previous work on bias evolution to include velocity bias. We show that gravitational evolution induces a quadrupole, and hence non-locality of bias, on large scales. If velocity bias is present, then a dipole is also induced. We illustrate these effects for the case of the evolution of initially scale independent local bias.

Section III shows that, when there is no velocity bias, then the same results can be obtained from a Lagrangian formalism, provided the initial conditions are treated self-consistently. In so doing, we demonstrate that Eulerian and Lagrangian treatments do, in fact, yield the same bispectrum; we discuss this in the context of what appear to be contradictory statements in the literature. Section IV studies bias evolution when comoving number densities are not conserved, either because of merging, or because of the formation of new objects. In this case also, no dipole contribution is generated if there is no velocity bias. In Section V, we extend our calculation to third order (for the case of no velocity bias), and show that the structure of the non-local bias generated is most easily described by Galileon fields, with a dipole arising from breaking of Galilean invariance of the bias relation when there is velocity bias. Appendix A makes the connection between the conserved and non-conserved non-local bias in the most general terms.

A comparison with simulations is done in Section VI, where we use the results of previous sections to motivate a search for correlations between the halo overdensity at fixed matter overdensity with the different non-local fields that our calculations singled out, finding signatures of non-local bias and its dependence on halo mass. In Section VII we discuss the impact of non-local bias on the bispectrum, and quantify the magnitude of non-local bias in simulations. A final section summarizes our conclusions.

Where necessary, we assume a flat CDM cosmology with and . In this paper we use galaxies, halos, and biased tracers interchangeably. Those readers interested in skipping the technical details and focussing on the main results, the detection of non-local bias in the simulations and their implications, can jump directly to Section VI, where the main results derived previously are summarized.

## Ii Non-local bias generation with conserved tracers

### ii.1 Conserved Tracers with Velocity Bias

We start by generalizing previous results Fry (1996); Tegmark and Peebles (1998) on the evolution of a tracer density perturbation (galaxies or halos), under the assumption that they form at a single instant in time with local bias, and thereafter evolve conserving their comoving number density (we relax this assumption in Section IV). In particular, we include the possibility that these tracers do not flow with the dark matter, and therefore have their own velocity field. To fully specify the evolution of their velocity field however one needs to make some assumptions, here we will assume that the tracers are massless so we can ignore their contribution to the gravitational potential which is only sourced by the dark matter. This is a reasonable approximation for galaxies, since only about 20% of the matter density is in baryons and an even smaller fraction of baryons is in galaxies Fukugita and Peebles (2004). At the large-scales of most interest, we can neglect dynamical friction and any pressure contribution, so we effectively treat the tracers as a pressure-less ideal fluid moving under the gravitational force generated by matter perturbations. In many respects, our approach is very similar to the perturbation theory treatment of two-fluids in Somogyi and Smith (2010), a connection we will make more explicitly below (see also Elia *et al.* (2010); Bernardeau *et al.* (2011)). In section VI we will apply our results to dark matter halos in simulations. In this case we are effectively assuming that halos may be treated as test particles (represented by their center of mass) whic move in the gravitational field due to the full matter distribution (i.e. all other halos).

In what follows, we will make heavy use of results from perturbation theory (PT, see Bernardeau *et al.* (2002) for a review). See section V for a simpler approach (in real instead of Fourier space) that neglects velocity bias, but which goes to third-order in PT instead of the second-order calculations we do here. We assume that our tracers (which we will henceforth denote as galaxies) are formed at a single instant, with a spatial distribution that is a local function of overdensity , and a velocity bias that is linear. We thus have two density and two velocity fields, one each for matter and tracers, and equations of motion that follow from imposing conservation of mass and tracers (we go beyond conserved tracers in section IV) and momentum conservation describing motion under the gravitational potential that is sourced by matter perturbations.

For a single-component fluid, mass and momentum conservation can be combined into a single equation for a two-component “vector” which simplifies obtaining the evolution of density and velocity fields at once Scoccimarro (1998, 2001). In what follows, we generalize this to a four-component vector equation for our two-component model. That is, we consider

(1) | |||||

(2) | |||||

(3) | |||||

(4) |

where and are the density contrast and velocity divergence of dark matter and and are the corresponding quantities for galaxies. is conformal time, , and denotes . The mode-coupling kernels and are defined as

(5) |

We then introduce as the time variable, where is the linear growth factor for the matter perturbations satisfying

(6) |

Since , with , is a very good approximation throughout the evolution Scoccimarro *et al.* (1998), the equations of motion Eqs. (1-4) can be written in compact form by defining a four-component “vector” as

(7) |

which yields

(8) |

where integration over and is implied and the entries of are zero except for

(9) | |||||

(10) |

and the matrix reads

(11) |

In this section we assume that galaxies are formed at a single epoch with linear density bias and linear velocity bias . Our choice of means that we can restore the more general time dependence by replacing in all formulas below. The initial conditions can be handled conveniently by Laplace transforms. Taking the Laplace transform with respect to , Eq. (8) becomes

(12) |

where represents the Laplace transform of , is the initial condition and

(13) |

for some in the region of convergence of . Collecting the terms linear in , we have

(14) |

with . Finally, we go back to the -space by the taking the inverse Laplace transform

(15) |

where , called the linear propagator Scoccimarro (1998, 2001), is given by

(16) |

where is a real number larger than the real parts of the poles of . We then have,

(17) |

We note that the block in the upper left corner is the same as the linear propagator for dark matter derived in Scoccimarro (1998, 2001). The linear propagator satisfies the relation

(18) |

which is the expected law for linear evolution generalized for arbitrary mixing of growing and decaying modes. The linear propagator has the usual (matter only) growing and decaying modes,

(19) |

i.e. and . In addition, there is an iso-density decaying mode and an iso-density-velocity decaying mode ,

(20) |

where here we restored (following Somogyi and Smith (2010)) temporarily a contribution to the overall mass density fraction of for matter and for galaxies (). The first eigenmode here satisfies , corresponding to a constant mode with zero total density perturbation, while the second eigenmode satisfies and corresponds to a vanishing total density and total velocity divergence perturbation. Our assumption of tracers as test particles (massless) means we have set and in our approximation. In the more general case, the same results we find here will apply with small corrections proportional to (see Somogyi and Smith (2010)). Note that the standard eigenmodes are of course independent of as they correspond to in-phase motion of the two fluids as if they were one.

We have transformed the equations of motion Eq. (8) into an integral equation Eq. (15) with explicit dependence on initial conditions that can be solved perturbatively. To specify the initial conditions we assume that they can be expanded as follows,

(21) |

where is the initial dark matter density contrast, and the vector can be similarly expanded as

(22) |

Putting Eqs. (21) and (22) into Eq. (15), and collecting terms of the same order in , we get a recursion relation for the kernels,

Note that the kernels obtained from Eq. (II.1) are not symmetric in the arguments , but they can be symmetrized afterwards. Only the symmetric part contributes to .

### ii.2 Generation of Non-Local Bias

#### ii.2.1 “Initial Conditions” at Formation

We now explore the results of Eq. (II.1) to study the generation of non-local bias by gravitational evolution from local-bias initial conditions. That is, we assume that biased tracers at formation time (corresponding to growth factor and ) can be written as a local function of matter density that is then Taylor expanded,

(24) |

where we assumed that the tracers have velocities that are only linearly biased with respect to matter. In the examples below we assume but our results in this section also apply if the velocity bias is -dependent. Small-scale velocity bias has been seen in simulations Colín *et al.* (2000); Berlind *et al.* (2003); Diemand *et al.* (2004), at the level. At large scales is predicted by peak theory Desjacques and Sheth (2010); Desjacques *et al.* (2010) although in a statistical sense, i.e. peaks move locally with the dark matter but their statistics can be thought of as if there is a velocity bias that is -dependent and goes to unity at very large scales as . But the situation for baryons, and therefore galaxies (as opposed to halos), can be somewhat different, e.g. at early times the relative velocity between the dark matter and baryons is typically supersonic Tseliakhovich and Hirata (2010); Dalal *et al.* (2010), and there might be
a non-trivial component to the relation between dark matter and baryons on large scales due to isocurvature modes, see e.g. Grin *et al.* (2011).

We assume that the matter is in the growing mode, so, to linear order, the initial conditions kernel at formation time is given by,

(25) | |||||

Here we have expanded the initial conditions in terms of the eigenmodes of the linear propagator (recall that and in Eq. 20). This makes clear that density bias excites the iso-density decaying mode and that velocity bias excites, in addition, the iso-density-velocity decaying mode. At second-order we have,

(26) |

where and are the second-order kernels that describe the density and velocity divergence to quadratic order in the linear matter fluctuations (see Eqs. 34-36 below for the multipole expansion of ). They are generated by the second term in Eq. (II.1) during time evolution up to for matter fluctuations (which satisfies the same equations for as Eq. (II.1) restricted to ). They are functions of the wavectors through because gravitational evolution is nonlocal.

At time (or redshift ) will be, precisely because of in the second term in Eq. (II.1), a non-local function of . We therefore are interested in separating out the non-local contribution to the galaxy density perturbations,

(27) |

where is a local function of and thus can be expressed in terms of local bias parameters (the evolved version of the ’s). We will in fact construct and order by order, e.g. by first substracting linear bias and analyzing local and non-local contributions at second order, then substracting local quadratic bias and analyzing what happens at third order, and so on.

In this section we take the first step in this analysis: to quantify the non-local contributions to second order. To do so, it is convenient to define the field

(28) |

which neglects all terms in of Eq. (27). We can then study at second order and decompose it in Legendre polynomials ,

(29) | |||||

where . A local contribution to should be proportional to , and therefore corresponds to a monopole () contribution with independent of . Any piece cannot be written as local functions of and thus will be entirely the result of non-local contributions. As we shall see, a quadrupole contribution () is inevitable for biased tracers, and velocity bias generates in addition a dipole (). We shall not go beyond second order here, see Section V for results to third order (see also Appendix A), and the next step in the construction of Eq. (27).

#### ii.2.2 Evolution of Linear Density and Velocity Bias

We now turn to the solution of Eq. (II.1). To linear order, only its first term contributes. From the decomposition into eigenmodes of the propagator, Eq. (25), we can read off the evolution of each field, which is precisely of the local form given in the initial conditions but with a prescribed time dependence. For matter density and velocity fields we have linear growing-mode evolution,

(30) |

while, for density and velocity bias we have, respectively,

(31) |

and

(32) |

Note that when there is no velocity bias, , we recover from Eq. (31) the well-known result Mo and White (1996); Fry (1996)

(33) |

that density bias asymptotes to unity in the long-time limit if the comoving number density of tracers is conserved. On the other hand, our generalization to does not agree with recent assumptions about the evolution of peaks in the initial density field Desjacques *et al.* (2010), which do not show the presence of the iso-density-velocity mode contribution that gives the decay in Eqs. (31-32). This disagreement results from different assumptions.
Peaks move locally with the dark matter but their velocity statistics can be thought of as if they had a statistical velocity bias that remains constant with evolution. Because of this difference in treatment, the peaks calculation cannot be directly compared to what we do here, although it is important to clarify which treatment is a more accurate description of velocity statistics of tracers. We hope to report on this in the near future.

Figure 1 shows and as a function of the scale factor . We have set and , and 0.9. Note that slows down the relaxation of the density bias slightly while speeds it up. Velocity bias also relaxes to unity eventually.

#### ii.2.3 Quadratic Order: Emergence of Non-Local Bias

Because the vertex is quadratic in , only multipole moments will be present. For the matter density field, the multipole expansion of reads,

(34) | |||||

(35) | |||||

(36) |

These correspond to the multipole expansion of . The monopole represents the second-order nonlinear growth in the spherical collapse dynamics, the dipole the transport of matter by the velocity field, and the quadrupole describes tidal gravitational effects.

The multipole moments for the galaxy density perturbation to second-order, , are given by

(37) | |||||

(38) | |||||

(39) |

where and are proportional to density and velocity bias, respectively:

(40) |

(41) |

and vanish for fully unbiased tracers. They are also defined to be zero at formation time (), leaving only the prescribed monopole from local bias. In the long-time limit () they asymptote to

(42) |

Thus, the effects proportional to dominate, but they are suppressed by another factor of compared to the usual second-order effects. Finally, note that when there is no velocity bias () Eqs. (37-39) reduce to,

(43) |

with . In this case there is no dipole, and the induced quadrupole and monopole are opposite in sign. The induced structure when there is no velocity bias for conserved tracers is further explored in Section V to third-order in PT, and in Appendix A for the non-conserved case.

Figure 2 shows the evolution of the multipoles (normalized by their dark matter counterparts, Eqs. 34-36) as a function of for three different choices of velocity bias, (solid), 1 (dashed) and 0.9 (dotted). We see that even though the bias at formation () is local (only a monopole is present), higher-order multipoles get generated. If there is no velocity bias then only a quadrupole gets generated; if then a dipole is also generated (with sign determined by ). All of these normalized multipoles eventually relax to zero because the galaxy multipoles grow more slowly than those of the dark matter. If , the relaxation of the monopole is slowed down, whereas the quadrupole relaxes faster; the opposite holds for . We see that even for a significant velocity bias of , the generated dipole is only of that in the dark matter. A dipole contribution in galaxy bias can enhance the shift of the BAO peak in the correlation function Crocce and Scoccimarro (2008), but since the dark matter dipole effect is at the percent level, velocity bias is unlikely to change this in any significant way except possibly for the very highly biased tracers. See Section VI for more discussion on the effects of such dipole term from numerical simulations.

Thus, we see that non-local bias is inevitably induced by gravitational evolution, and that the locality assumption cannot be self-consistent. In practice, because galaxy formation happens in a continuous fashion, we don’t expect locality to be valid at any time, even if the formation bias were local. We explore this in Section IV. In addition, there is no reason to expect the bias at formation to be purely local, even for dark matter halos, since the barrier for collapse is known to depend on quantities other than the overdensity (e.g. Sheth *et al.* (2001)).

## Iii Comparison with Local Lagrangian Bias

In this section, we would like to compare our results with those known from the literature on local Lagrangian bias, which can be thought of as a particular limit of our results when formation time is at the far past () and there is no velocity bias. While such calculations are usually done in Lagrangian PT (see e.g. Catelan *et al.* (1998, 2000)), clearly one should obtain the same results if done in Eulerian PT as we have used so far. It is however instructive to redo this calculation in a Lagrangian description and compare.

Since there is no velocity bias and tracers are conserved, the continuity equations Eq. (1) and Eq. (3) can be used to relate to through the matter velocity divergence field Catelan *et al.* (1998),

(44) |

where we used the Lagrangian or total derivative following the motion of a fluid element,

(45) |

Upon integration of Eq. (44), we get

(46) |

where is a function depending on initial fields at the Lagrangian coordinate related to the Eulerian through the displacement field

(47) |

In terms of the initial condition is clearly given by

(48) |

where the Lagrangian fields are evaluated at the initial time .
This is the same result as that given in Catelan *et al.* (1998, 2000); Desjacques *et al.* (2010) except for the denominator , which was implicitly dropped in those works (it is however included in Bartolo *et al.* (2010)). However, to reproduce the decaying modes found in the previous section, this denominator is required.

For comparison with the results in the previous section, we now assume local Lagrangian bias in Eq. (48) to quadratic order,

(49) |

where the Lagrangian bias parameters are the equivalent to the parameters in the previous section. To linear order, we can assume in this equation, but to go to second order we need to include the displacement field to first order, i.e. in the Zel’dovich approximation (hereafter ZA, Zel’Dovich (1970)). This is given by

(50) |

where we have used the fact that, in the ZA, the decaying mode is constant Scoccimarro (1998). Note that at formation time , as it should. We emphasize again that this decaying mode, which is often neglected in the literature, must be included if one wishes to fully reproduce the results in the previous section to second order.

To linear order Eq. (49) reads , where all fields have the same argument . Therefore we deduce the Eulerian linear bias

(51) |

which is Eq. (33), with . This seems different a priori from the often quoted relationship between linear Eulerian and Lagrangian bias . The reason is twofold: first, it is customary to define the Lagrangian bias with respect to the extrapolated linear density field rather than the Lagrangian density field as we have done here, so the more standard definition is instead

(52) |

and second, if we neglect the third term in Eq. (51) coming from the denominator in Eq. (48), then we recover the familiar . This second step is justified for objects that are not arbitrarily close to unbiased in which case as and , increases without bound for objects with fixed , so that in this limit. Although this step is unjustified for unbiased objects for which , keeping only this term does no harm as its contribution to Eq. (51) vanishes as .

(53) |

which upon Fourier transform, after using Eq. (50), we can write as the quadratic kernel for galaxies

(54) |

where we have used the second-order matter results Eqs. (34-36), and . It is easy to check that this equation agrees with Eq. (43) with , after using that with given by Eq. (51) and .

We can now take the limit in Eq. (54) as for the linear result above and compare with the results in the literature. Now we need to redefine the quadratic Lagrangian bias in terms of the present density fluctuations, as done for the linear bias in Eq. (55),

(55) |

and assuming as before we then get for Eq. (54),

(56) |

which agrees with Eq. (8) in Catelan *et al.* (2000). Note however that in Catelan *et al.* (2000) it is argued that the dipole term (proportional to in Eq. 56) is a new feature of local Lagrangian bias as opposed to local Eulerian bias. This interpretation is not correct: in local Eulerian bias the second-order galaxy kernel is, apart from local contributions of quadratic bias, that of the matter multiplied by linear bias
,
so the precise amplitude of the dipole agrees with that in Eq. (56). Subtracting this local Eulerian piece to construct
,
shows that the new contributions are indeed of the form given by Eq. (43). That is, the new qualitative contribution is a quadrupole term, not a dipole. As we showed in the previous section, an additional dipole will only appear if there is velocity bias. The physical reason for this (breaking of the Galilean invariance of the bias relation) is discussed in Section V.

Thus, we have shown that, in the appropriate limit, we reproduce known local Lagrangian bias results. However, our approach in the previous section is more flexible as it does not require “formation” to be in the distant past, and it also allows for velocity bias. On the other hand, we have, so far, assumed that the comoving density of tracers is conserved. We now discuss how to go beyond this assumption.

## Iv Non-local bias generation with non-conserved tracers

Galaxies form at a range of redshifts and merge. So it is important to extend the previous results to the more realistic case when the comoving number density of galaxies changes with redshift due to some arbitrary source field , which effectively includes the effects of galaxy formation and merging. Our description here is similar to Tegmark and Peebles (1998) (see also Matarrese *et al.* (1997); Moscardini *et al.* (1998)), but we shall extend the analysis to higher order in PT. For simplicity here we assume that the bias at formation is local (as we have done so far), Appendix A discusses what happens in the most general case (see also Eq. 114 below).
The evolution equation for the physical galaxy number density now becomes

(57) |

Note that we factorized the source term into two components, and , where roughly parametrizes the epoch of galaxy formation (e.g. following star formation history) and describes the effects of dark matter on the formation and merging of galaxies. Nonetheless, we stress that our main results are independent of the detailed functional form of and , and we use the assumed functional forms only to make the plots shown in Fig. 3. For example, can be a log-normal profile

(58) |

where and are free parameters. For we take a simple quadratic form

(59) |

where and are model parameters, and is the average matter density today. Appendix A considers the implications of depending on non-local functions of , or other fields.

The second term in Eq. (57) can be eliminated if we use comoving rather than physical number densities, , so we have

(60) |

where . We then write,

(61) |

and solve Eq. (60) by perturbation theory. We will also assume that there is no velocity bias, so that galaxies and matter share the same velocity field, which is known from solving the evolution of matter. We then expand the source term on the right hand side of Eq. (60) to second order

(62) |

This invites us to interpret and as the instantaneous formation bias and of the galaxies formed (or destroyed) at time

(63) |

For example, the form of in Eq. (59) gives

(64) |

which imply a simple relation . This relation only holds at the formation time, as we will see evolution inevitably generates non-locality and breaks this. Note that in Eq. (64) the bias parameters have a pole when is negative. This means that quadratic approximation is no longer valid and the higher order terms in the expansion are important.

Figure 3 shows in the top left and center panels and for three sets of values of and , corresponding to (solid), (dashed), and (dot-dashed). As we mentioned before, these choices are just illustrative with no special physical significance, but serves to show a range of possibilities for the local biases at formation.

### iv.1 Background Solution

We now look for the evolution of the galaxy comoving number density by solving the background equation,

(65) |

where is the mean matter density. The solution is

(66) |

where we have assumed that there are no galaxies at . The top right panel in Fig. 3 shows the resulting for three sets of parameters and in three different backgound profiles . As expected the background number density is predominantly determined by the profile .

### iv.2 First-Order: The Evolution of Linear Bias

To first order in PT we write in Eq. (60), and using the background evolution in Eq. (65), we get

(67) |

which using linear theory evolution for matter with growth factor can be rewritten as

(68) |

where , where is the linear growth factor for galaxies, and after using Eq. (65), we arrive at . Looking for a solution of the form

(69) |

And since initially, we have