arXiv:1204.0019v1 [astro-ph.CO] 30 Mar 2012 rbto HD ucin h te prahi the is approach dis- other occupation The halo different the function. with to (HOD) populates according tribution then and of clustering halo- halos scale types the large matter is the dark from One of starts days. which these model galaxies based of bias dependent describe to to leads [7]. able eventually constraints not and [9] cosmological is samples inconsistent formula analysis among fitting differences empirical the such in the one used and as widely samples [7], expect different observed 8] among would scale-dependence [7, varies data surveys the the present scale, linear in for large The even relatively challenging. down at more breaks underlying be model the may bias and field etc.) matter galaxies dark HI Ly be- and quasars, clusters relationship (galaxies, forest, the tracers However, renormalization observable time various [6]. tween the theory (TRG) and theory resummation [5], group theory Lagrangian closure the the [4], the- [1–3], perturbation (RPT) stan- renormalized been ory the have the beyond methods including sophisticated matter proposed, Many dark approach. the dard per- of nonlinear the theory has developing it in turbation interest un- regime, of relevant in lots in role attracted data important observed a the pattern. plays derstanding clustering dynamics observed nonlinear pre- the Since of theoretically in achieve the accuracy to dicting of us level requires unprecedented it uni- dramati- surveys, an decreases early generation uncertainty next of in statistical cally the the As and modified verse. energy, masses dark probing gravity, in information mological ∗ lcrncades [email protected] address: Electronic w ieetapoce xs nmdln h scale- the modeling in exist approaches different Two cos- of wealth a provide surveys structure Large-scale u aayfrainhsoyb eprlyweighted-avera temporally rate. by trace formation history the formation of formation galaxy automatical the ous approach after evolution our As gravitational Eu evolution, by years. density both recent galaxy applying in of developed by tion field achieved matter is dark convergence of This techniques the improved. therefore e is counterparts, also quantities model matter such dark for their Lagrangi constructed ge a we further with t propagators and combining multipoint In theory, by evolution perturbation clustering. gravitational galaxy resummed quent describing of in context enough the accurate o not approach perturbative is standard informati the cosmological scales, extracting quasi-linear in crucial is tribution eateto hsc n srnm,JhsHpisUnivers Hopkins Johns , and Physics of Department h eainhpbtenosre rcr uha galaxies as such tracers observed between relationship The .INTRODUCTION I. eumdPrubto hoyo aayClustering Galaxy of Theory Perturbation Resummed i Wang Xin Dtd a 6 2018) 26, May (Dated: ∗ α n lxSzalay Alex and r o atrdsrbto 4.Te 1]as included also [13] Then the- [4]. resummation distribution similar distor- matter a redshift for on and ory based in- bias which proposed halo (LBM) was the model tion direc- bias incorporate this Lagrangian to along a tended [12], done In been tracer. biased tion. already of have clustering works expect the Several would describe pertur- to one matter theory nonstandard field, bation previous matter similar of any very dark generalize is in to model situation bias the nonlinear to the in encountered 11]. [10, tracer matter the non- of the the evolution on with gravitational consistent depends local not only is which density locally, galaxy distribution the functional the cal- of that the form assume in simply difficulties authors intrinsic most much culation, parame- too bias bring nonlocal won’t ters Furthermore, artificial some scales. of adopting range although whole the within calculation spectrum fail such large-scale would power parameters, bias the the higher-order of both on depend shape Since small-scale times. and later amplitude at and regime scales nonlinear the small dark into the ob- evolve as of down perturbations time breaks matter model the this at model Therefore, distribution servation. bias matter on dark depends Eulerian density Eulerian the number the In galaxy observed the problematic. (EBM), also is model not usually is approach sur- such large-scale of satisfactory. accuracy of Moreover, the most probe, where simulation. veys regime from quasi-linear the calibration in heav- on and parameteri- parameters, depends HOD free ily of viable lots a contains However, usually zation scale. small qualitatively at statistics careful two-point well a the With ap- describe first can parameters. the proach function, bias HOD few calibrated first and character- parameterized the and with distribution, de- it matter functional ize dark unknown the complicated general on the pending a packs into which physics baryonic model, bias perturbative ic h icliso h etraieconvergence perturbative the of difficulties the Since bias perturbative standard the hand, other the On n stelna ismdlbek onat down breaks model bias linear the As on. h olna ueinba oe (EBM) model bias Eulerian nonlinear the f ,adas losu oicueacontinu- a include to us allows also and r, yicroae h o-oaiyinduced non-locality the incorporates ly igrlvn uniiswt h galaxy the with quantities relevant ging ndsrpino aais oin The motion. galaxies’ of description an rpryo ttsisbituo these upon built statistics of property i ae,w ics uhamdlin model a such discuss we paper, his neie rmteLgaga descrip- Lagrangian the from inherited hbtepnnildmigsmlrto similar damping exponential xhibit eaiei oicroaetesubse- the incorporate to it neralize n h neligdr atrdis- matter dark underlying the and einadLgaga resummation Lagrangian and lerian t,Blioe D228 US 21218, MD , ity, 2 the halo bias in the time renormalization group methods. II. PERTURBATION THEORY OF DARK [14, 15] approached in a different way by redefining the MATTER bias parameters. As we will show in this paper, with the help of recent development in the renormalized pertur- In this section, we will briefly review both Eulerian bation theory [1–3], it is also possible to construct a re- perturbation theory (EPT) and the Lagrangian pertur- summed perturbation theory for nonlinear Eulerian bias bation theory (LPT) of the field, including model, and the perturbative expansion of statistics can both the standard approach and resummed calculation. be rewritten through the so-called Γ−expansion, first in- troduced by [3], with the building blocks known as multi- point propagators. One advantage of the Γ−expansion is A. Dynamical Equation of Motion that for Gaussian initial conditions, all contributions are positive and centralized within a limited range of scale. The gravitational dynamics of a pressureless fluid be- As we will see later, such an expansion is quite general fore shell crossing is governed by the continuity, Euler, and applicable to many circumstances as long as the mul- and Poisson equations. tipoint propagators can be well estimated. Therefore in the following, this paper will mainly concentrate on con- ∂δ(x, τ) + ∇ · 1+ δ(x, τ) u(x, τ) =0, structing the multipoint propagators for various objects. ∂τ The validity of such technique has already been verified ∂u(x, τ)    + H(τ)u(x, τ)+ u(x, τ) · ∇u(x, τ) in calculating the power spectrum of logarithmic map- ∂τ ping of the density field [16]. It have been shown that = −∇ΦN (x, τ) even for the slow converging function such as logarithm, 3 the calculation agrees with the simulation quite well. ∇2Φ (x, τ)= Ω (τ)H2(τ)δ(x, τ). (1) N 2 m We further generalize the Eulerian bias model by sepa- Here, H = d ln a(τ)/d ln τ is the Hubble expansion rate, rating the complicated nonlinear physical processes dur- a(τ) is the scale factor, Ωm(τ) is the ratio of matter den- ing the galaxy formation and the subsequent gravita- sity to critical density, and ΦN (x, τ) is the Newtonian tional evolution until been observed by a survey. As we potential. will show, this can be achieved by combining the Eu- Following [1, 2], the equation of motion in Fourier space lerian bias model and Lagrangian description of galaxy can be written in a compact form by defining the two- motion introduced in [12]. Like EBM, we assume the component variable galaxy distribution created at some time is characterized by a general unknown functional, and the galaxy number Ξa(k, η)= δ(k, η), − θ(k, η)/H , (2) density observed at a Eulerian position later can be nat- where θ denotes the divergence of peculiar velocity ∇ · u. urally described by the Lagrangian version of continuity The index a ∈ {1, 2} stands for matter density or ve- equation. Consequently, this approach also enables us to locity variables, respectively. In the following, we will incorporate continuous galaxy formation by temporally interchangeably use Ξ , Ξ and δ for matter overden- averaging the distribution of newborn galaxies weighted 1 m m sity, while using both Ξ and δ for galaxy perturbation. with the galaxy formation rate. In such a framework, g g The equation of motion then reads while the complex galaxy formation physics is simplified by a few bias parameters, we are able to capture the fol- ∂ηΞa(k, η) + ΩabΞb(k, η) lowing gravitational evolution after its creation, which k k k Ξ k Ξ k would in principal induce non-locality and can not be ac- = γabc( , 1, 2) b( 1, η) c( 2, η), (3) curately described by a local Eulerian bias model. As in- with the convention that repeated Fourier arguments are herited from the Lagrangian bias model and our resumed integrated over. The time η = ln a(τ) in a Einstein-de calculation of Eulerian bias model, this model can also Sitter (EdS) . Here the constant matrix be expressed in a resummed manner. 0 −1 Ω = , (4) In section II, we briefly review both Eulerian and La- ab −3/2 1/2 grangian perturbation theory of dark matter field, includ-   ing both standard approach and resummed theory. We derived for an EdS universe, is still applicable in other will also introduce the concept of the multipoint propa- with negligible corrections to the coefficients, gator in the context of Lagrangian perturbation theory, using η = ln D(τ) (with D the linear growth factor). and give a explicit calculation in Appendix A. In section The symmetrized vertex matrix γabc is given by III, we first review the standard Eulerian and Lagrangian |k + k |2(k · k ) bias model, and then introduce our continuous galaxy 1 2 1 2 γ222(k, k1, k2) = δD(k − k1 − k2) 2 2 formation model. We construct the resummed Eulerian 2k1k2 bias model in Section IV, and the generalize it in Section (k1 + k2) · k1 γ121(k, k1, k2) = δD(k − k1 − k2) 2 (5) V. Finally, we conclude in Section VI. 2k1 3

γ112(k, k1, k2)= γ121(k, k2, k1), and γ = 0 otherwise. be obtained in terms of gab and γabc recursively. Then the formal integral solution to Eq. (3) can be (n) derived as Fa (k1, ··· , kn; η)δD(k − k1···n)=

η n η dη′g (η − η′) γ (k, k , k ) Ξ (k, η) = g (η)φ (k)+ dη′ g (η − η′) m=1 0 ab bcd 1···m m+1···n a ab b ab  Z0 P R ′ ′ (m) ′ (n−m) ′ ×γbcd(k, k1, k2)Ξc(k1, η )Ξd(k2, η ) (6) × Fc (k1···m; η ) Fc (km+1···n; η ) (11) symmetrized where φa(k) denotes the initial condition φa(k) ≡ (1) Ξa(k, η = 0), and the linear propagator gab(η) is given For n = 1, Fa (η)= gab(η)ub. It should be noted that in by this formalism, the kernel depends on the time η, since it includes subleading terms in eη [1]. If one only considers eη 3 2 e−3η/2 −2 2 the fastest-growing mode, F (n)(η) equals the well-known g (η)= − . (7) a ab 5 3 2 5 3 −3 PT kernel Dn(η){F (n), G(n)}, where D(η) is the linear     growth factor. In the following, we adopt growing-mode initial condi- tions φa(k)= δ0(k)ua, with ua = (1, 1). In Lagrangian perspective, the dynamic of a fluid ele- F (n) ment at initial Lagrangian position q is entirely described by the final Eulerian position x, or equivalently the dis- placement field Ψ(q)= x − q, which is governed by the equation of motion,

d2 d Ψ + H(τ) Ψ = −∇Φ (8) dτ 2 dτ N FIG. 1: Diagrammatic elements of Eulerian perturbation the- ory. left: perturbative expansion (Eq. 10); right: initial power where ∇ is still the derivative with respect to Eulerian spectrum P0(k). coordinates x, and the gravitational potential ΦN is de- termined by Poisson equation as shown in Eq. (1). The The power spectrum Pm(k, η) of the matter density density contrast is then related to Ψ(q) via mass conser- perturbation is defined as vation ′ ′ Pm(k, η)δD(k + k )= hΞm(k, η) Ξm(k , η)i (12) −1 −1 1+ δm(x)= J = det [δij +Ψi,j (q)] (9) For Gaussian initial conditions, all the statistical infor-   mation is encoded in the initial power spectrum where J is the Jacobian of the transformation between ′ ′ hφa(k)φb(k )i = δD(k + k )Pab(k), (13) Eulerian and Lagrangian space, and Ψi,j = ∂Ψi/∂qj. To linear order, Eq. (8) is then solved by the Zel’dovich ap- ′ (1) where Pab(k) = uaubP0(k), with P0(k) = hδ0(k)δ0(k )i. proximation ∇q ·Ψ (q)= −D(τ)δ0(q), with Lagrangian derivative ∇ . Diagrammatically, as shown in the right-hand diagram q of Fig. (1), the ensemble average is obtained by gluing two open circles together to form the symbol ⊗, which represents the initial power spectrum Pab(k). B. Eulerian Perturbation Theory As the density fluctuations evolve into the non-linear regime at later times, the validity of standard perturba- A perturbative solution to Eq. (6) could be obtained tion theory breaks down, loop contributions become ill- by expanding in terms of initial fields behaved and the convergence of perturbation series gets out of control. As one of several different approaches be- ∞ yond SPT been proposed recently, we will briefly review (n) Ξa(k, η) = Ξa (k, η) the renormalized perturbation theory (RPT) introduced n=1 X by [1–3], and then the Lagrangian resummation theory (n) 3 (n) [4] in the next subsection. Ξ (k, η) = d q1···n δD(k − q1···n)F (q1, ··· , a ab1···bn The crucial step of renormalized perturbation theory Z is to define the generalized growth factor, known as the ··· , qn; η)φb1 (q1) ··· φbn (qn) (10) (1) propagator Γm ab(k) 3 3 3 where d q1···n is short for d q1 ··· d qn, and q1···n de- (n) notes q +···+q . The kernels F are fully symmetric (1) δΞa(k, η) 1 n Γ (k, η)δ (k − k′) ≡ , (14) functions of the wave vectors. As shown in [2], they can m ab D δφ (k′)  b  4

= ++ + · · ·

(1) FIG. 2: The nonlinear propagator Γm (k, η) has an infinite number of loop contributions. which effectively describes the time evolution of individ- known as the Γ−expansion ual Fourier modes when non-linear mode-coupling is in- cluded. Here δ denotes the functional derivative. The 3 Pm ab(k, η) = n! d q1···n δD(k − q1···n) propagator measures the dependence of a non-linearly n≥1 Z evolved Fourier mode Ξa(k, η) on its initial state φb(k) X (n) q q (n) q q on average. Intuitively, one should expect Gab to decay Γm a( 1, ··· , n; η) Γm b( 1, ··· , n; η) to zero at small scales since non-linear mode-coupling has P0(q1) ··· P0(qn). (18) erased all the information from the initial state at that k. Indeed, with the help of the Feynman diagrams intro- Note that Eq. (18) describes nothing but an alter- duced in [1], the dominant contribution can be summed native way of taking ensemble averages, or diagram- up explicitly in the large-k limit, and giving the exponen- matic speaking, gluing initial states. Instead of gluing 2 2 2 two density fields order by order, one can first construct tial decay Gab(k) ≈ exp(−k D σΨ/2). In this paper, the non-linear propagator is diagrammatically presented as a the objects by gluing initial states of individual density grey circle with an incoming branch. It is a summation fields with n incoming branches, known as an (n + 1)- of infinite number of loop contributions, as illustrated in point propagator, and the final non-linear power spec- Fig. (2). trum is then obtained by gluing two propagators to- Given above, [1] was able to rewrite the nonlinear mat- gether. Therefore, this Γ−expansion is quite general and ter density power spectrum as a sum of two contributions applicable to the ensemble average of any two arbitrary statistical fields hx x i, which are functionals of some 2 1 2 Pm(k, η)= G (k, η)P0(k)+ PMC (k, η), (15) other field x0 where G is the density propagator G(k, η)= G1bub, and k 3k (n) k k PMC (k, η) is the mode-coupling term. Therefore, the xi[x0( n), λ] = d 1···n Xi ( 1, ··· , n; λ) n non-linear power spectrum at any k is composed of two X Z parts. One is proportional to the initial power spectrum × [x0(k1) ··· x0(kn)], (19) at the same k; the other comes from mode-coupling of other k′. As G(k) decays at small scales, more and more as long as the building blocks are well estimated. Here (n) power comes from the mode-coupling contribution. [3] λ symbolize some parameters, and X are symmetric showed that these complicated mode-coupling contribu- on their arguments. In the case that x0 is Gaussian, tions can be expressed as a summation of multi-point the Wick’s theorem ensures that the ensemble average propagators, defined as of an odd number of fields vanishes hx0(1) ··· x0(2n + 1)i = 0 and the average of an even number of fields (n) k k k k hx0(1) ··· x0(2n)i can be decomposed as summation of Γm ab1···bn ( 1, ··· , n; η)δD( − 1···n) two-point correlations 1 δnΞ (k, η) = a , (16) n! δφb1 (k1) ··· δφb (kn)  n  hx0(1) ··· x0(2n)i = hx0(i)x0(j)i. (20) ij which is nothing but a generalization of two-point prop- allX pairs Y agator Gab. Similarly, the dominant part of multi-point propagators can also be summed and decay into the non- Following the idea of [3], there are three types of two- point correlations hx0(i)x0(j)i involved in calculating linear regime at the same rate as two-point propagator. ′ [3] showed that a simple approximation which general- hx1(λ)x2(λ )i: pairs within x1 or x2 themselves and pairs izes the k-dependence of two-point propagators agrees connecting them. Therefore, one is always able to re- with the data with acceptable accuracy, group Eq.(20) in the following way

(n) (n, tree) Γ (k1, ··· , kn; η) = Γ (k1, ··· , kn; η) m m hx0(1) ··· x0(2n)i = hx0(l)x0(m)i (1) k in between x1 internal lm Γm (| 1···n|) X  X Y  × (1, tree) . (17) Γm (|k |) ′ ′ 1···n hx0(l )x0(m )i hx0(i)x0(j)i . ′ ′ For gaussian initial condition, the nonlinear power spec- x2 internalX lYm ij inY between  trum can then be expressed with the help of Eq.(16), (21) 5

Combining above equations (19, 20 and 21) together, one For the matter distribution, one usually assumes that the ′ is able to reformulate hx1(λ)x2(λ )i as a sum of two-point initial field is sufficiently uniform δ0 ≈ 0, so that above cross correlations of x0 belonging to x1 and x2 respec- equation (Eq. 24) can be further simplified. In Fourier tively with some pre-summed nonlinear quantity involv- space, ing internal pairs internal lmhx0(l)x0(m)i, which can be formally defined as the average of functional derivative 3 −ik·q −ik·Ψ(q) P Q δm(k) = d q e e − 1 1 δnx Z (n) i ∞ n h i Γi = . (22) (−i) 3 n! δx0(1) ··· δx0(n) = d p [k · Ψ(p )] ··· [k · Ψ(p )]   n! 1···n 1 n n=1 Z Therefore, as long as Γ(n) is well estimated, one is always X i × δD(k − p1···n). (25) able to carry out the Γ−expansion in such situation. In the following, we will demonstrate that with the help of Similarly, the above equation can also be represented by the density propagator of (Eq. 16), diagrams as shown in Fig. (3). Here, wavy lines denote many similar quantities of this kind can be calculated. the displacement field Ψ, and the thin solid line together And we will extensively utilize such expansion in the fol- with grey open circle stands for the linear density per- lowing of the paper. turbation, i.e. Eq.(23).

=

+ + + ······

(1) FIG. 3: Diagrammatic representation of nonlinear matter FIG. 4: Two-point propagator of matter density Γm (k) in density perturbation in Lagrangian perturbation theory, sim- the context of Lagrangian perturbation theory up to one-loop ilar to the one introduced in [4]. The wavy line denotes the order. The grey ellipse shown in each contribution symbol- displacement field Ψ, and the thin solid lines together with izes the quantity of Eq (26). Note here we have omitted the grey open circles stand for the linear density perturbations summation sign over the displacement field Ψ presented in δ0. [10].

Starting from Eq.(25), [4] performed a resummation of the power spectrum of the density contrast, known as La- C. Lagrangian Perturbation Theory grangian Resummed Perturbation Theory. Although not been explicitly discussed in [4], it is also possible to define As discussed in the beginning of this section, the dy- the same propagator within this framework. First of all, namics of the system in Lagrangian picture is fully char- as pointed out in [10] for biased tracers1, the following acterized by the displacement field Ψ(q, η). Combining quantity can be resummed the equation of motion (Eq. 8) and the Poisson equation, ∞ the displacement field can be solved perturbatively. In (−i)n d3p [k ··· k ] hΨ (p ) ··· Ψ (p )i Fourier space, the n−th order perturbation Ψ(n) can be n! 1···n i1 in i1 1 in n n=0 generally expressed as X Z ∞ (−i)n n iDn = d3p k · Ψ(p) = he−ik·Ψ(0)i, (26) Ψ(n)(k)= d3p L(n)(p )δ (p ) ··· δ (p ), n! n! 1···n 1···n 0 1 0 n n=0 Z   Z X (23) where Ψ(0) = d3p Ψ(p) is the displacement field at the L(n) origin. It is represented as a grey ellipse in Fig. (4), and where is the perturbation kernel. R Due to the matter conservation, the Eulerian density contrast δm(x) can be expressed as

1 b b 1+ δ (x)= d3q [1 + δ (q)] δ (x − q − Ψ(q)). (24) Setting 1 = 1 and n = 0 otherwise in Eq. (71) of [10], one m 0 D recovers the following equation. Z 6

can be further simplified as [4, 10] ~p1,1 e−ik·Ψ(0) = Π(0)(k) = exp e−ik·Ψ(0) ~p1,2 c F (3) ~ D E ∞ nhD 2n E i k1 ~p1,3 (−1) h|Ψ(0)| ic 2n = exp k F (2) (2n + 1)(2n)! ~ bn "n=1 # k X2 2 k D 2 ~pn,1 ≈ exp − σΨ (27) 2 ~k (m)   n F with the variance of displacement field along one direc- tion D2σ2 = h|Ψ(0)|2i/3 = d3qD2P (q)/(3q2). Here, Ψ 0 ~p we have used the cumulant theorem n,m R P j x P j x e[ n≥0 n n] = exp e[ n≥0 n n] − 1 , (28) c FIG. 5: Diagrammatic representation of δE (k, η) in Eulerian D E hD E i where h· · · ic denotes the connected part of the ensemble bias model. average. As shown in [10], it leads us to a partial resum- mation of the propagator as represented in Fig. (4) up to the one-loop order. Again, this is just a simplified version where the nonlinear mass density field δm(x) is solved of Fig. (17) in [10] for the Lagrangian bias model. The by perturbation theory mentioned in the last section. As grey ellipse represents all possible graphs attached with discussed by [10], this functional is generally non-local. various number of wavy lines, i.e. Eq. (26). Compared Therefore in Fourier space, bias coefficient bn depends on to the diagram introduced in [10], we have omitted the wavenumbers summation sign over external wavy lines. Mathemati- ∞ 1 cally, such diagrammatical series is equivalent to δ (k, η) = d3k δ (k − k )b (k , ··· , k ) E n! 1···n D 1···n n 1 n n=1 δ X Z Γ(1)(k) = e−ik·Ψ(0) d3p −ik · Ψ(p) × [δ (k , η) ··· δ (k , η)] . (32) m δφ(k′) m 1 m n  Z   δ In the local bias model, b then becomes constant = Π(0)(k) exp [−ik · Ψ(0)] n ′ times a smoothing window function W (k, R) in Fourier δφ(k ) c    space. In this model, the linear bias relation at large (29) 2 scale P (k) ≈ b Pm(k) is not only attributed to the first Note that we have used an alternative version of the cu- term in Eq. (32), but also from the higher orders bn. As mulant theorem in Eq. (28). first pointed out by [17], the effective large scale bias is

1 1 1 1 1 1 determined by infinite number of higher-point local vari- ej x x = exp ej x − 1 ej x x . (30) 2 c 2 c ances of the field. Therefore, similar to the situation in which is obtained by taking the derivative  of Eq. (28) the standard perturbation theory of matter density field, respect to j2 and then setting it to zero. We further cal- at later time when mass density variance grows larger, culate these contributions explicitly in the Appendix A. a naive perturbative expansion fails. In this case, such The result is not the same as the form originally proposed a breakdown affects both the large scale amplitude and by [2]. However, the matching prescription between high- the high-k regime of the power spectrum. k and one-loop calculation in [2] seems more or less ad Given the definition of δE(k) in Eq.(32) and the dia- hoc. Therefore, an alternative way of constructing the grammatic representation of the density perturbation in propagators was proposed by [20], to which our calcu- Fig. (1), it is also possible to draw diagrams representing lation do agree. See [20] for the discussion of different δE(k). As depicted in Fig.(5), each diagram contains two versions of calculating multipoint propagators. levels of nonlinear interactions. The thick solid n-branch tree represents the nonlinear convolution, and their inter- action vertex, shown as a solid square, carries the bias co- III. THE NON-LINEAR BIAS MODEL efficient bn/n!, where n is the number of branches. Each branch represents one nonlinearly evolved density con- A. Eulerian Bias Model trast δm(kn, η) at time η, which is followed by another m thin branches, to express its gravitational nonlinearity (m) In Eulerian bias model, the galaxy number density per- in terms of initial fields and perturbative kernels F . turbation at time η is considered as some general func- At the end of each branch, small open circles represent the initial conditions δ . When expanding δ (k, η) to tional F [δm] of the underlying matter density distribu- 0 E different orders, one should also include the information tion characterized by its Taylor coefficient bn about the number of topologically equivalent diagrams, b δ (x, η)= F [δ (x, η)] = n [δ (x, η)]n , (31) which essentially comes from the multinomial coefficient E m n! m n n of (δm) . The final field δE(k, η) is then a summation of X 7 all possible diagrams.

B. Lagrangian Bias Model

To incorporate the halo bias with nonlinear pertur- bation theory, [12] proposed an alternative bias model involving the initial Lagrangian density fluctuation δ0(q) instead of final Eulerian density δm(x, η). In this La- grangian bias model, the non-linear objects, typically FIG. 6: Comparison between Lagrangian bias model (left) halos, were formed at some initial time with the num- discussed in [4] and our galaxy formation model (right). The ber density is a functional F [δ0(q)] of underlying linear thin solid denotes the linear density perturbation δ0, and thick density distribution δ0. Then the Eulerian observable line represents nonlinear overdensity δm(η). Note that the δL(x, η) at later time is described by the continuity rela- right diagram has not revealed very clearly the process of tion continuous galaxy formation as shown in Eq. (40).

3 1+ δL(x) = d q [1 + δL(q)] δD[x − q − ΨL(q)] Z two sections, in the following of this section, we will try to construct a model by combining Eulerian and Lagrangian = d3q F [δ (q)] δ [x − q − Ψ (q)], 0 D L pictures together. Z (33) First, we still assume that the complicated nonlinear process leading to the formation of galaxies or other type where ΨL(q) is the displacement field of such tracer. In of observable tracers can be described by a general non- Fourier space, linear functional F [δm(η)] at the moment of galaxy cre- ation. In general this can be nonlocal, however, we will 3 −ik·q −ik·Ψ(q,η) δL(k, η)= d q e F [δ0(q)] e − 1 .(34) assume F [δm] to be local for simplicity. As what will be seen in the following, the locality assumption here is Z h i In the limits that nonlinear objects trace exactly the un- not as problematic as in the local Eulerian bias model since it only relates to the physical process during the derlying dark matter, i.e. ΨL = Ψm, one can further proceed to calculate the power spectrum P (k) using La- moment of galaxy formation and could be easier to gen- grangian perturbation theory of the matter field. In Fig. eralize. Furthermore, such functional can also explicitly (6), we reproduced the diagram of this model introduced depends on the formation time, i.e. we have F [δm(η), η] by [10] on the left, where the wavy line represents the and bn(k1··· ,n, η). Assuming at time η, there are ∆ρg(q, η) new galax- displacement field ΨL and the n branches of thin solid ies were created during ∆η at a Lagrangian-like position lines with open circle symbolize the nth order of F [δ0]. The great advantage of this approach is that it natu- q, and then evolved to Eulerian position x where have rally takes account of the non-locality induced by gravita- been observed by a survey at time η0. Analogous to the tional evolution. As shown by [10], after perturbatively Lagrangian bias model (Eq. 33), this contributes to the expanding both sides of Eq.(34) and comparing terms Eulerian galaxies a number density ρg(x, η0) order by order, one obtains the relationship between Eu- 3 lerian and Lagrangian bias parameters. To the first two d q ∆ρg(η, q) δD[x − q − Ψg(q,η,η0)], (36) orders [10], Z E L Ψ q b1 (k) = 1+ b1 (k), where g( ,η,η0) is the displacement field of galaxy E L L from η to η0. Note that ∆ρ here only comprises galax- b2 (k1, k2) = b2 (k1, k2) − F2(k1, k2)b1 (k) + [k · L1(k1)] L L ies that are finally selected in the sample. Therefore, the × b (k2) + [k · L1(k2)] b (k1). (35) 1 1 galaxy number density at observed time η0 and comoving x Therefore, even if the Lagrangian bias bL is local ini- position is a sum of all contributions that formed before n x Ψ q tially, non-linear and non-local gravitational evolution η0 and then moved to with a displacement g( ,η,η0), (n) E F (p1, ··· , pn) would render the Eulerian bias b2 non- η0 3 ′ local. ρg(x, η0) = dη d q ρ¯g(η)F [δm(q, η), η] Zηmin Z × δD[x − q − Ψg(q,η,η0)]. (37) C. Continuous Galaxies Formation Model

Here we have written the number density ∆ρg(q, η) = As briefly mentioned in the introduction, we will try to ∆ρg(η)F [δm(η), η], where δm(q, η) is the nonlinear generalize the Eulerian bias model in two different ways. evolved matter density field at η. In principle, the dis- As the resummation technique will be discussed in next placement field of galaxies Ψg should follow the one of 8 dark matter Ψm, i.e. (1) ΓE = + Ψg(q; η, η0)= bΨ [Ψm(qin, η0) − Ψm(qin, η)] , (38) where qin is some initial Lagrangian position of the mat- ter field when density perturbation is still linear. In + + ······ the following of this paper, we will assume bΨ = 1. In the simplest case where all galaxies formed at ηf , i.e. ′ ρ¯g(η)=ρ ¯gδD(η − ηf ), a similar form of Eq.(33) is re- covered. Taking the average of both sides of Eq.(37), FIG. 7: Diagrammatical demonstration of the two-point non- one obtains the average number densityρ ¯g(η0) as a in- (1) tegration of the average density over the whole galaxy linear propagator ΓE (k) for Eulerian bias mode. The grey formation history ellipse includes all possible ensemble averages among initial density perturbation δ0. η0 η0 ′ ′ ρ¯g(η0)= dη ρ¯g(η)hF [δm, η]i = dη ρ¯g(η)(39) Zηmin Zηmin IV. RESUMMED PERTURBATION THEORY OF EULERIAN BIAS MODEL Therefore, the perturbation of galaxy number density can then be expressed as, In this section, we will show that the Eulerian bias η0 model can be partially resummed with the help of ex- δ (k, η ) = dη d3q f(η) e−ik·qF [δ (q, η), η] (n) g 0 m ponential decay of multipoint matter propagators Γm . Zηmin Z Starting from the definition of this model (Eq. 32), one −ik·Ψg(q,η,η0) × e − δD(k). (40) (n) can define the (n + 1)-point nonlinear propagator ΓE of ′ the biased-tracer overdensity δE, where f(η)=ρ ¯g(η)/ρ¯g(η0) denotes the average galax- ies formation rate, with the normalization relation (n) Γ (k1, ··· , kn; η)δD(k − k1···n) dη f(η) = 1. To first order, we have E, b1···bn n 1 δ ΞE(k) R η0 = . (44) n! δφb1 (k1) ··· δφbn (kn) δg(η0) = dη f(η)[b1D(η)+ bΨ [D(η0) − D(η)]] δ0   Zηmin η0 Before substituting δE (Eq. 32) as well as the pertur- = bΨD(η0)+ dη f(η)D(η)(b1 − bΨ) δ0, bative solution of δm (Eq. 16, 18) into above definition, ηmin (n)  Z  one first notices that, ΓE can be expanded in terms (41) of the number of nonlinear density fields δm involved, just as depicted in the schematic diagram of Fig.(7) Therefore, the linear bias of galaxies evolves as for the two-point propagator. Mathematically, it means (n) (n, m) η0 ΓE = m ΓE , where g D(η) b1(η0)= bΨ + dη f(η) [b1 − bΨ]. (42) D(η ) (n, mP) ηmin 0 Γ (k , ··· , k ; η)δ (k − k ) Z E, b1···bn 1 n D 1···n

In the case that galaxies exactly trace cold dark matter, 1 3 = d q1···m δD(k − q1···m)bm(q1, ··· , qm) i.e. bΨ = 1, we recover the result calculated by [11] if n!m! Z only the growing mode is considered. As first discussed δn by [18], for the model galaxies burst at a single time ηf , [Ξm(q1) ··· Ξm(qm)] .(45) δφb1 (k1) ··· δφbn (kn) f(η)= δD(η − ηf ),   For the two-point propagator Γ(1)(k), let us write down g D(ηf ) E b1(η0)=1+ [b1 − 1]. (43) all contributions formally. First of all, it is clear that D(η0) the one-δm term, i.e. the first diagram in Fig.(7), equals (1) So the gravitational growth will always debias the clus- the two-point propagator of the matter field itself Γm (k) g multiplied with the linear bias parameter b1(k). For the tering of galaxies towards b1 = 1. Because of this Lagrangian-like approach adopted here, two-δm contribution, i.e. the second diagram, one reads this framework automatically incorporates the gravita- from Eq. (45) tional non-locality induced after the formation of the b (q , q ) galaxies. Furthermore, as will be seen in Section V., Eq. Γ(1, 2)(k; η) = 2 d3q δ (k − q ) 2 1 2 g, a 12 D 12 2 (40) also allows the application of the resummation tech- Z nique so that the perturbative convergence of such model δΞm(q1, η) × Ξ (q , η) (46) is guaranteed. δφ m 2  a  9

As discussed in Section II, the ensemble average at the proceed, we expand Eq. (46) explicitly second line can be expanded by a Γ−like series, which (n) in this case, is the propagator for matter field Γm . To

(1, 2) 3 3 3 ′ ΓE, a (k; η) = d q12 δD(k − q12) b2(q1, q2) (n1 + 1) d p1···n1 d p1···n2 δD(q1 − k − p1···n1 ) n1,n2 Z X Z ′ (n1+1) (n2) ′ ′ × δ (q − p ) F 1 (k, p , ··· , p 1 ; η)F (p , ··· , p ; η) D 2 1···n2 m, ac ···cn1 1 n m, d1···dn2 1 n2 ′ ′ × φc1 (p1) ··· φcn1 (pn1 ) φd1 (p1) ··· φdn2 (pn2 ) . (47)

As we have shown, Wick’s theorem ensure the de- order n, composition of joint ensemble average at the last line into the combinations of two-point correlations. Each Γ(1, n)(k; η)= g(1, n) d3pij E, a {tij } 1···tij of these terms can be labeled by three indices r1, r2,t, {tij } Z 1≤i

where the index tij denotes the number of connections between the i-th and the j-th density field δm (e.g. Γ(1, 2)(k; η)= (n + 1)! d3p P (p ) ··· P (p ) {tij } = {t12,t13,t14,t23,t24,t34} for n = 4), so the total E, a 1···n 0 1 0 n number of the indices is then n(n − 1)/2. The coefficient n X Z (1, n) (n+1) g{t } is then given by × b2(k + p1···n, −p1···n) Γm, a (k, p1, ··· , pn; η) ij (n) × Γ (−p1, ··· , −pn; η), (48) t m g(1, n) = n (t + 1) i {tij } 1 ti1 ··· tin 1≤Yi≤n   where we have introduced the notation × (tij )! . (51) 1≤i

(n) (n) Here, ti = m tim is the total number of connections Γ (p1, ··· , pn; η)=Γm, c1···c (p1, ··· , pn; η) uc1 ··· uc , m n n linked between the i-th density field δm and others, with P (49)tij = tji. Eq. (51) expresses the products of multino- mial coefficients of choosing {ti1, ··· ,tin} from ti for each density field, times all possible permutations within tij (n) for pair matching. Each connection carries a momen- and used the definition of Γ in terms of the perturba- ij m tum p , (1

(1, tree) (1) The formula can be further generalized to arbitrary ΓE, a (k; η)= b1(k) Γm, a(k; η; R) (52) 10

~q a straight path through the diagram carrying the same ~k ~k ~k ~k + 4 ~k − ~q =

~q −~q

~k ~k +3 + ······ + + + ······

(1) FIG. 8: Nonlinear propagator ΓE (k) of Eulerian bias model up to one-loop order. The integer in front of each diagram indicates the number of all topologically equivalent diagrams. FIG. 9: Each diagram contains an infinite number of loop contributions. where we have introduced the notation momentum k at both the start and end. Along this (n) p (n) p p p Γm ( 1···n; η; R)=Γm ( 1 ··· n; η)W (| 1···n|R), (53) path, there is one convolution bias vertex bn, as well as one (n + 1)-point propagator Γ(n). For n = 1 (e.g. where W (kR) is some smoothing window function. m the third diagram in Fig.(8)), this contribution recovers At one-loop level, from Eq.(50), the contribution would (1) be nonzero only when n ≤ 3, since otherwise there would the same k-dependence as Γm (k), rescaled by a constant from loop integration. When n > 1, Eq.(17) suggests a exist at least one Γm with zeroth order, which would (0) similar damping of k given p ··· p . Meanwhile every vanish because Γ = hδ i = 0. For n = 2, t = 1, the 1 n−1 m m 12 propagator associates with a smoothing window function coefficient equals 4. For n = 3, t =1,t = t = 0, the 23 12 13 W (kR). Therefore one should expect that, in the large-k coefficient equals 3. So we have (1) limit, ΓE will decay as a combined effect of a Gaussian (1) damping of Γm and the smoothing window function. (1, 1−loop) 3 b3(k, q, −q) ΓE, a (k; η)= d q P0(q) Besides the tree level result of two-point propagator 2 (1, tree) Z  ΓE (k), which equals the one of dark matter field (1) q 2 (1) k k q q (1) × Γm ( ; η; R) Γm a( ; η; R)+2 b2( − , ) times the linear bias coefficient b1(k)Γm (k), the loop integration from higher-order calculation, e.g. the third  (2)  (1) × Γm, a(k, −q; η; R)Γm (q; η; R) . (54) diagram in Fig.(8), will also change the small-k normal-  (1) ization of ΓE (k) and eventually the effective large-scale ∗ To make the Eq.(50) easier to understand, one can bias of the power spectrum b1. As we will see at the end draw every contribution diagrammatically. Starting from (1) ∗ of this section, ΓE (k) dominates the contribution of b1, (n) ∗ the diagram representing δE in Fig.(5), we change all hence the accuracy of the b1 depends on the precision kernels into n-point propagators. After selecting one par- (1) of estimating the ΓE (k). Due to the exponential decay ticular branch out, we glue the rest of the initial states (n) of matter propagators Γm , our resummed formula will (open circles) together. Every resultant topologically in- ∗ equivalent diagram represents one or several terms in help us in improving the convergence of b1. Eq.(50). Since all ordinary kernels have already been substituted by propagators, the ensemble average (glu- ~p −~q ~p + ~q ing) is only performed among different density fields δm. ~p (1) ~k ~k In Fig.(8), we show all diagrams of ΓE up to one-loop or- der. They correspond one-to-one to Eq.(52) and Eq.(54). ~ ~ k + ~q ~k − ~p − ~q k − ~p We also present all two-loop diagrams and equations of ~k ~p (1) − ΓE in Appendix B. It should be emphasized that, by substituting ordinary kernels into propagators, each dia- gram in fact contains an infinite number of loop contribu- FIG. 10: Examples of two different contributions of three- tions at every substituting position, as shown in Fig.(9). point propagator, full set of diagrams is shown in the Ap- Numerically, the Gaussian damping of the multipoint pendix B. (n) matter propagator Γm will eventually contribute to the improvement of the convergence in calculating nonlin- Now, we can further derive the three-point propagator ear matter power spectrum. Therefore, it is important (n) similarly, except that two distinct contributions have to to check whether ΓE share the similar feature. From be taken into account because of the second order func- (1) (2) diagrams representing ΓE (k), we see that there exists tional derivative in the definition of ΓE . 11

n b (q , ··· , q ) δ2Ξ (q , η) Γ(2, n)(k , k ; η) = d3q δ (k − q ) n 1 n m 1 Ξ (q , η) ··· Ξ (q , η) E, ab 1 2 2 1···n D 1···n n! δφ δφ m 2 m n Z  a b  δΞ (q , η) δΞ (q , η) + (n − 1) m 1 m 2 Ξ (q , η) ··· Ξ (q , η) (55) δφ δφ m 3 m n  a b 

As shown in Eq.(55), the first term takes the joint aver- category includes the first, third and sixth diagram. The age of the products of a second derivative with n−1 den- rest of the diagrams then correspond to the second term sity fields. Diagrammatically speaking, it corresponds to of Eq.(55), where two branches originate from two differ- selecting two branches from a single δm, i.e. the first ent density fields. Both terms can be written in the same diagram in Fig. (10). In Fig. (18) of Appendix C, this form of Eq.(50),

ij (2, n) 3 ij bn(k, p ) (2, n) (t1+2) 11 1n Γ (k1, k2; η) = d p g Γ (k1, k2; p , ··· , p ) E, ab 1···tij n! {tij } m, ab 1···t11 1···t1n {Xtij } Z 1≤i

The difference between the two terms can be δm term with t12 = 0 for the second term of Eq.(56), clearly seen from their orders of the matter prop- agators. With the same labeling system, the (2, tree) (2) ΓE, ab (p, k − p; η; R)= b1(k) Γm, ab(p, k − p; η; R) (t1+2) (t2) (tn) first gives Γm Γm ··· Γm , while the second gives b2(k − p, p) (1) (1) (t1+1) (t2+1) (tn) + Γ (p; η; R)Γ (k − p; η; R). Γm Γm ··· Γm . Meanwhile, the two g coefficients 2 m, a m, b equal (58)

All the one-loop contributions can be found in the Ap- pendix C. (2, n) n ti g = (t + 2)(t + 1) Finally, the non-linear power spectrum of the field δE {tij } 1 1 2 ti1 ··· tin can be expressed as 1≤Yi≤n  

× (tij )! , 3 PE(k; η) = n! d q1···n δD(k − q1···n)P0(q1) ··· P0(qn) 1≤i

dependence is, roughly speaking, determined by the rel- ative value of linear bias b1 and high-order bn. So for a z . fixed linear bias, a bigger high-order b leads to a larger PE (k n 105 PL (k scale-dependent deviation, while a smaller b1 results in b PL (k the similar trend if bn are fixed. Not surprisingly, the same bias model will give a larger deviation at lower red- shift. k ( 4 P 10 V. RESUMMED PERTURBATION THEORY OF CONTINUOUS GALAXY FORMATION MODEL ( E ( E Based on the construction of resummed perturbation theory of Eulerian bias model in the last section, we will 103 further extend such model to incorporate a continuous 10-2 10-1 k(Mpc/h galaxies formation history. As shown in Eq.(40), the per- turbation of galaxy number density observed at η0 can be expressed as, FIG. 11: The power spectrum of Eulerian bias model at z = η0 k k k 0.5 with b1 = 1.5, b2 = 0.5 and b3 = 0.2, assuming a spherical δg( , η0) = Ξg( , η0)= dη f(η)∆g( ,η,η0). top-hat smoothing window function with R = 2Mpc/h. The Zηmin (1) (60) solid line gives the full result PE(k) including both ΓE and Γ(2) contributions, while each of them are represented in long E where the integrand ∆g(k; η, η0) is the density contrast (1) dashed as well as short dashed line respectively. Here, ΓE of galaxies formed at η and then been observed at η0 (2) is calculated up to two-loop order and ΓE to the one-loop order. The dotted line shows the linear matter power spec- trum PL(k), and the dot-dashed line illustrates the linear bias ∆ (k) = d3q e−ik·q F [δ (q, η), η]e−ik·Ψg (q,η,η0) − 1 2 g m contribution b PL(k). The offset between the dot-dashed and 1 Z solid curve indicates that higher order bias parameters con- h i −^ik·Ψg tribute to the large-scale amplitude of the power spectrum. = F ∗ e (k,η,η0) − δD(k) ∞ h (−i)mi = e d3k d3k′ n! m! 1···n 1···m clearly that higher order bias parameters contribute to n+m≥1 X Z the large-scale amplitude of the power spectrum. This ′ × δD(k − k1···n − k1···m)bn(k1, ··· , kn; η) can also be seen in the left panel of Fig. (14), where we ′ × [δm(k1, η) ··· δm(kn, η)] [k · Ψg(k ,η,η0)] plot the normalized two-point propagator Γ(1)(k,z)/D(z) 1 E × · · · [k · Ψ (k′ ,η,η )] (61) for the same bias model but at various redshifts, starting g m 0 from z = 6 to z = 0.5. At lower redshift, the variance ^ where F and e−ik·Ψg are Fourier transforms of F and of the local density field grows larger, so does the effec- −ik·Ψ ∗ e g respectively, and ∗ denotes the convolution. tive linear bias b1 as shown in the figure. For the same reason, the exponential damping length decrease towards Apart frome the weighted average of ∆g with the galaxy higher redshift. As expected, all of lines at large scale lie formation rate f(η), this model differs from the work of [12] in two ways. First, the non-linear bias func- above the true linear bias b1, i.e. the dotted horizontal line. We want to remind here that the accuracy of such tional F [δm] depends on the nonlinear evolved matter effective large scale bias calculated by our formalism has field δm(η) instead of initial state δ0. Secondly, the dis- already been verified with the simulation in [16] for a placement field Ψg(η, η0) characterizes the movement of logarithmic transformation. It is guaranteed by incorpo- the galaxy from location q at time η to Eulerian posi- rating the matter propagator, since the loop integrations, tion x of time η0. In the following, we will construct the ∗ propagator of δg(k, η0) instead of following the work of that contribute to b1, are reduced and therefore the series expansion is regulated. [12] exactly. Another distinguishing feature provided by our for- Starting from Eq.(60), one can define the multi-point propagator of galaxies similarly. For the two-point prop- mula is the scale-dependent bias. In Fig.(12), we give (1) several examples of such scale-dependent deviation at agator Γg (k), we have baryonic acoustic oscillation sclaes compared to the lin- δΞ (k, η ) ear matter power spectrum at two different redshifts for g 0 = Γ(1)(k, η )δ (k − k′) δφ(k′) g 0 D various bias parameters. They are calculated by dividing   η0 δ∆ (k,η,η ) out the no-wiggle power spectrum [19] with appropriate = dη f(η) g 0 (62) normalizations. As can be seen in the figure, such scale δφ(k′) Zηmin   13

1.6 1.6 z =1 z =3

1.5 1.5

1.4 1.4 )) )) k k ( ( L L

P 1.3 P 1.3 2 2 eff eff b b ( ( / / ) )

k 1.2 k 1.2 ( ( E E P P 1.1 1.1

1.0 1.0

0.9 0.9

0.05 0.10 0.15 0.20 0.25 0.05 0.10 0.15 0.20 0.25 k(Mpc/h) k(Mpc/h)

FIG. 12: Examples of scale-dependent deviation of nonlinear Eulerian bias model at scales of baryonic acoustic oscillation for (1) (2) various bias parameters at redshift z =1(left) and z =3(right). Both ΓE and ΓE are calculated up to the one-loop order. From the top to bottom, we adopt bias parameters: (1) b1 = 0.8, b2 = 1.5, b3 = 0.4; (2) b1 = 0.8, b2 = 1, b3 = 0.2; (3) b1 = 1, b2 = 0.5, b3 = 0.2.

tive expansion gives

∞ b (k, k , ··· k ) Γ(1) = dk n 1 n δ (k ) ··· δ (k ) g, 1 1···n n! m 1 m n n=1 Z  X ∞ δ (−i)m × δ (k) × dk′ [k · Ψ (k′ )] ··· δφ m m! 1···m g 1 m=0  X Z ′ × · · · [k · Ψg(km)] (65)  Without going into detail, one finds the resemblance be- (1) FIG. 13: Diagrams of Γg (k) up to the two-loop order. The tween above equation and Eq. (46, 47), since both F and (1) ^−ik·Ψg first row represents the contribution of Γg, 2(k) and the second e can be written in the form of Eq. (19). There- (1) e row for Γg, 1(k) terms. fore, following the exact same procedure after Eq. (46), (1) the contribution Γg, 1 can be similarly resummed using the Γ−expansion, and one simply reads from Eq. (48) as

Concentrating on the quantity inside the integration, and n we define (1) 3 Γg ,1(k; η, η0)= (n + 1)! d p1···n P0(p1···n) n Z   δ∆g(k; η, η0) X = Γ(1)(k; η, η )δ (k − k′) (63) × Γ(n+1)(k, p , ··· , p ; η)Π(n)(p , ··· , p ; η, η ) (66) δφ(k′) g 0 D E 1 n k 1 n 0   Here the first contribution, arisen from hδF/δφ˜ (k)i, is Substituting the expression of Eq.(61) into this defini- (n) the multipoint propagator of Eulerian biased tracer Γ tion, one immediately obtains two different contributions E which we have discussed in the last section, and the sec- arising from the functional derivative ond contribution is defined as

(1) (1) (1) n −^ik·Ψg Γg (k; η, η0)=Γg, 1(k; η, η0)+Γg, 2(k; η, η0) (n) 1 δ e (p; η, η0) Πk (p1, ··· , pn; η, η0)= (67) δ ^ δ ^ n! * δφ(p1) ··· δφ(pn) + = F ∗ e−ik·Ψg + F ∗ e−ik·Ψg δφ(k) δφ(k)       where p = p1···n. When k = p, it recovers to e e (64) the definition of multipoint matter propagators in La- grangian perturbation theory with galaxy displacement (1) Concentrating on Γg, 1(k; η, η0) first, a further perturba- field Ψg(η, η0). 14

zform zform z . 0.8 form z z 2.0 obs obs zform zobs zobs . zobs zobs . zform zobs zobs . 0.6 1.5 zobs zobs . form

z zobs zobs . ( /D k (

( k 0.4 form 1.0 k,z ( ( E zform 0.2 0.5 zform

zform 0.0 0.0

10-2 10-1 100 10-2 10-1 100 101 k(Mpc/h k(Mpc/h

(1) FIG. 14: (left): Two-point propagator of Eulerian bias model ΓE (k) at various redshifts zform = 0.5, 1, 2, 3, 4 and 6 from (1) top to bottom. We assume the bias parameters b1 = 1.5, b2 = 0.5, b3 = 0.2 and 0 otherwise. (right): Πk (k; zform ,zobs ) (Eq. 67), from zform = 3 and 6 evolved to zobs = 0, 0.5, 1, 1.5, 2 and 2.5 (top to bottom).

(1) The physical consequence of Eq. (66) is the separation galaxy density perturbation Γg (k) equals between the nonlinear gravitational evolution of galaxies (n) after their creation characterized by Π and the more (1) (1) (1) 3 k Γ (k; η, η0) ≈ Γ (k; η) + Π (k; η, η0)+2 d p P0(p) complicated nonlinear galaxy formation physics parame- g E k Z terized by the unknown functional F [δm]. The final non- Γ(2)(k, p; η)Π(1)(p; η, η )+Γ(1)(p; η)Π(2)(k, p; η, η ) linear growth of galaxy density perturbation Γ(1) relates E k 0 E k 0 g   to these two physical processes in a statistical way. Dia- (70) grammatically, Starting from the second diagram in Fig. (n) (n) (n) (6), one can construct ΓE and Πk simply by gluing To proceed, we have to estimate the Π with galaxy initial states connected to wavy line and thick solid line k displacement field Ψg(η, η0). To the first order, the (1) separately. We illustrate Γg, 1(k) in the second row of Zel’dovich approximation simply gives Fig. (13) up to the two-loop order. Ψ(1)(k; η, η ) = [D(η ) − D(η)]δ (k), (71) Similarly, the second term in Eq. (64) g 0 0 0 and therefore, ∞ (1) bn(k1, ··· kn; η) 2 2 Γg,2 = dk1···n δm(k1) ··· δm(kn) (1) k (D0 − D) 2 n! Π (k; η, η0) ≈ [D0 − D]exp − σ , (72) n=1 k Ψ X Z 2 ∞ m   (−i) ′ ′ ′ × dk1···m [k · Ψg(k1)] ··· [k · Ψg(km)] where D = D(η ), and D = D(η). So it is fully charac- m! 0 0 m=0 Z terized by the gravitational growth between two different X ′ δΨg(k ) moments. Though higher order corrections analogous to × k · m+1 (68) δφ(k′) the calculation in Appendix A are possible, we haven’t   explicitly calculated them here. The caveat is that the galaxy displacement field at q relates to Ψm at some can be resummed as initial Lagrangian position qin since we assume

n Ψg(q; η, η0)= Ψm(qin; η0) − Ψm(qin; η). (73) (1) 3 Γg ,2(k; η, η0)= (n + 1)! d p1···n P0(p1···n) n At the lowest order q = q , but the coordinate trans- X Z   in (n) (n+1) form needs to be carefully incorporated for higher order × Γ (k, p1, ··· , pn; η)Π (k, p1, ··· , pn; η, η0), (69) E k calculation. Furthermore, a more reliable approach is to directly solve the fundamental dynamical equation (Eq. which corresponds to the first row of Fig.(13). Therefore, 8) with the potential determined by the nonlinear matter up to the one-loop order, the two-point propagator of distribution at η. We will leave this in the future work 15

fixed formation time zform fixed observation time zobs

tree + 1-loop tree + 1-loop 2.0 tree tree 1.5

1.5 obs obs z z

( 1.0 ( /D /D

k k 1.0 ( ( ( g ( g

0.5 0.5

Pg (k Pg (k

0.0 0.0

10-2 10-1 100 10-2 10-1 100 k(Mpc/h k(Mpc/h

(1) FIG. 15: The normalized two-point propagator Γg (k; zform , zobs )/D(zobs ) of galaxies which formed at fixed formation redshift (left) zform = 3, and observed at zobs = 2, 1.5, 1 and 0 (top to bottom); and at fixed observation redshift (right) zobs = 0, and formed at zform = 0.5, 1, 2 and 3 (top to bottom). Assuming nonlinear bias model b1 = 1.5, b2 = 0.5, b3 = 0.2. Inner panel: the corresponding power spectra in the opposite/same (left/right) order.

and simply utilize the Zel’dovich approximation (Eq. 72) quantity for fixed observation redshift zobs = 0, when within this paper. galaxies are formed at different time with the same non- In the right panel of Fig. (14), we illustrate the Π(1)(k) linear process (same F [δm] and bn). In this case, both k (n) (n) from the moment of galaxy formation zform = 3 and 6 ΓE and Πk are changing and the evolution of large evolve to various observational time zobs . At the large scale bias would in principal depends on the specific pa- scale, it converges to linear result D(zobs ) − D(zform ), rameters bn adopted, since the ΓE and Πk evolve with (1) which relates to the linear debiasing we discussed in Eq. time oppositely. In the examples we shows, ΓE (k) dom- (42), and dampens to zero towards smaller scales with inates the evolution, i.e. galaxies formed at lower redshift 2 the speed proportional to [D(zobs ) − D(zform )] . As can would be higher biased due to the stronger nonlinear ef- be seen in the figure, when galaxies have more time to fects. evolve , i.e. zobs − zform is large, the observed galaxy Finally, we consider the continuous galaxy formation distribution is more influenced by gravitational evolu- model, assuming a simple log-normal model for average tion. This can be easily seen from Eq. (70), since at tree galaxy formation rate [11] (1) (1) (1) level Γg (k)=Γ (k; zform )+Π (k; zform ,zobs ). When E k 2 (1) [log(1 + z) − log(1 + z0)] D(zobs ) − D(zform ) is small, Π (k; zform ,zobs ) vanish f(z)= N exp − , (74) k g 2σ2 and therefore the gravitational evolution is negligible.   This can also be clearly seen in the left panel of Fig. where N is the normalization factor ensuring dzf(z)= (15), where we plot the normalized galaxy propagator g 1. Therefore, we have two parameters characterizing the Γ(1)(k; z ,z )/D(z ) with fixed formation redshift R g form obs obs model: z0 the peak redshift of galaxy formation, and zform = 3 and observed at various zobs . The dashed σ2 the width of the redshift for galaxy formation. In line shows the tree level of Eq. (70) and solid line corre- the lower panel of Fig.(16), we give several examples of (n) sponds to 1-loop results. In this situation, ΓE is fixed, galaxy formation history. and as zobs decreases from 2 to 0, the large scale bias In the left-upper panel of Fig.(16), we present again also decreases as expected. However, the amplitude of the normalized propagator for various galaxy formation the power spectrum itself as shown in the inner panel of models which we shows in the lower panel of the same fig- (1) left Fig. (15), is still increasing at lower redshift due to ure. As discussed previously, the Γg (k; zobs ) here is sim- the linear growth D(z ). From the diagrams of three- (1) obs ply a time average of Γ (k; z ,z ) weighted with point propagator of Eulerian bias model, at large scale g form obs (2) f(zform ). Therefore, for a narrower formation history k → 0, ΓE (k, p) is nonzero, hence the loop integration (e.g. the dashed, dotted and dash-dotted lines), where also contributes to the large scale bias, although it’s still σ2 =0.01, the results are similar to corresponding galaxy dominated by the tree-level value. As for smaller scales, bursting models with different z . On the other hand, (1) (1) 0 Γg dampens qualitatively similar to the Πk (k). when galaxy tends to form in a similar rate during a In the right panel of Fig. (15), we illustrate the same longer time (solid and long-dashed lines), e.g. σ2 = 0.1, 16

k(Mpc/h 10-2 10-1 100

z ., z =1.5,2 =01 2.0 0 2 5 =1 =001 10 0 2 0 =3 =001 1.5 obs z ( /D

k 1.0 ( =0) ( g obs

,z 4 ) 10 k

0.5 (

g P

2 0.0 0 =2 =001 2

1.0 0 =25 =01 0.8 () 0.6 3

10

0.4

( 0.2

0.0 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -2 -1 10 10 k(Mpc/h)

(1) FIG. 16: The normalized two-point propagator (left) Γg (k; zform , zobs )/D(zobs ) and corresponding power spectrum (right) for various models with continuous galaxy formation. (left-lower): The cumulative average galaxy formation history for models shown in the upper panel. we are more likely to observe a galaxy power spectrum galaxies, which would introduce a effective non-locality with medium amplitude and nonlinear damping. We also in the functional F [δm]. In principle, this could be solved illustrate the corresponding power spectrum on the right by introducing another merger contributions in the inte- panel of Fig.(16). gration, but with the cost that could further complicate the model.

VI. CONCLUSION AND DISCUSSION Acknowledgments

An accurate modeling of the statistics of galaxy den- XW thanks for the productive discussion with Donghui sity perturbation is crucial for the success of next gen- Jeong, Mark Neyrinck, Patrick McDonald and Kwan eration galaxy surveys. In this paper, we considered the Chuen Chan. XW and AS are grateful for support from nonlinear bias model in the context of resummed pertur- the Gordon and Betty Moore Foundations. bation theory. In Section III, we discussed the Eulerian bias model and then generalized it to incorporate the continuous galaxy formation in Section IV. By utilizing Appendix A: Matter Density Propagator in the multipoint propagators of the matter field, our for- Lagrangian Perturbation Theory malism for Eulerian bias model is more accurate than the standard approach in both the linear as well as the quasi- Here we consider the generalized multipoint propaga- linear regimes. This has already been verified in the work tor of [16], where a good agreement was achieved comparing to the simulation even for the slowly converging loga- n ^−ik·Ψ (n) 1 δ e (p) rithmic mapping of the density distribution. However, Πk (p1, ··· , pn)= (A1) a detailed comparison with high-precision simulation is n! *δφ(p1) ··· δφ(pn)+ still needed, in at least two different levels: the accuracy of perturbative calculation itself compared to the cost of When p1 + ··· + pn = k, one retrieves the (n + 1)−point (n) (0) its numerical calculation, and the ability of describing propagator Γm . For n = 0, we define Π (k) as the scale-dependent galaxy clustering bias at the epoch k2 of their formation. Π(0)(k)= e−ik·Ψ(0) ≈ exp − h|Ψ(0)|2i (A2) 6 Furthermore, even after separating out the subsequent   gravitational non-locality, the bias parameters could still D E be nonlocal. For example, the ∆ρ in Eq. (36) only counts As shown in Eq.(27). the effective newborn galaxies that finally entered into (1) δ ^ the sample, however, such description is more or less Π (p) = Π(0)(k) e−ik·Ψ(p) k δφ(p) ambiguous due to galaxy merger. Therefore, the identi-  c fied progenitors in our formalism then include all merged = Π(0)(k) T (1)(k, p). (A3) 17

We then explicitly expands the terms inside ensemble and average, which will be denoted as T (1)(k) in the following ˜ 2 2 2 4 R1 = − 2 (1 + r )(3 − 14r +3r ) ∞ r (−i)nDm n T (1)(k, p) = d3q k · Ψ(q) d3p 3 2 1+ r n!(m − 1)! + 3 (r − 1) ln n=0,m=1 r 1 − r X Z  Z 3 ′ ′ (m) ′ 2 2 2 4 d p1···(m−1)δD(p − k − p1···m−1) k · L (p1, R˜ = (1 − r )(3 − 2r +3 r ) 2 r2 p′ k p′ p′  3 1+ r ··· , m−1, ) δ0( 1) ··· δ0( m−1) (A4) + (r2 − 1)3(1 + r2) ln (A9) c r3 1 − r    

Fig(4) shows all the contributions up to the one-loop or- Setting k = p in Eq. (A7), we get the T (1)(k ) up to (1) der. Denotes Tn,m(k) as individual terms been summed one-loop order in above equation, we have (1) Ttree = D(η) (1) (1) 3 (1) 3 d p 7 5 3 3 5 T0,1 = D(η)kiLi , T1−loop(k) = D(η) 3 5 PL(p) 6k p +5k p + 50k p 3 504k p (1) D (η)ki 3 (3) Z  T = d q L (q, −q, p)P0(q) 3 k − p 0,3 2 i − 21kp7 + (k2 − p2)3(2k2 +7p2) ln . Z 2 k + p (1) 3 3 (1) (2)  T = D (η)kikj d q L (q)L (−q, p)P0(q), 1,2 i j (A10) Z (A5) which coincide with the calculation of [20]. For three- point Π(2)(p , p ) (n) k 1 2 where LPT kernel L (p1, ··· , pn) 2 (2) 1 δ ^ Π (p , p ) = Π(0)(k) e−ik·Ψ(p) (1) p k 1 2 L = 2 δφ(p1)δφ(p2) p2  c D2(η) 3 p ≈ Π(0)(k) k L(2)(p , p )+ L(2) = 1 − µ2 2 i i 1 2 7 p2 1,2  5 p   1 p (1) (1) (3a) 2 2 2 + kikj L (p1)L (p2) (A11) L = 1 − µ1,2 [1 − µ12,3] − [1 − 3µ1,2 i i 7 p2 3 p2  +2µ  µ µ ]+ p × T 1,2 2,3 3,1 Assuming k = p1 + p2, One recover the tree-level result 1 (2) L(3) = [L(3a) + perm] (A6) Γm (k), 3 (2) (0) Γm (k1, k2) = Π (k)F2(k1, k2), (A12) (n) Substituting the definition of L (p1, ··· , pn) into Eq.(A4), one can explicit carry out T (1) up to the one- where F2 is the second order Eulerian perturbation ker- loop order, nel.

k p T (1) = D(η) i i Appendix B: Two-loop Order of Γ(1) 0,1 p2 E

(1) 5 3 kipi T0,3 = D (η) R1(p) In this paper, we have calculated the two-point prop- 21 p2 (1) agator Γ up to two-loop order. Seven non-vanishing 2 E (1) 3 k · p contributions are depicted in Fig.(17). From these dia- T = D3(η) [R (p)+2R (p)] 1,2 14 p2 1 2 grams, one can write down all terms explicitly   2 k 7 − R1(p) (A7) p2 Γ(1, 2lp)(k; η)= d3p P (p )P (p ) K(1, 2lp)  E, a 12 0 1 0 2 i "i=1 # Z X where we have defined a slightly different version of the (B1) integral functions Rn(k) introduced by [4] Here we de- notes where

(1, 2lp) (2) 3 ∞ K1 = 6 b2(k + p1 + p2, −p1 − p2) Γm (−p1, −p2; R) 1 k ˜ Rn(k)= 2 dr P0(kr)Rn. (A8) (3) 48 4π × Γm, a(k, p1, p2; R), Z0 18

~q1

~q1 ~q2 ~q2 ~q1 ~k ~k ~k ~k ~k ~k 12 18 24

~q2

~q1 ~q1 ~q2 ~q2 ~q1 ~k ~k ~k ~k ~k ~k 6 24 24

~q1 ~q2

~k ~k 15

~q2

FIG. 17: Two-loop order of the two-point nonlinear propagator.

(1, 2lp) (3) K2 = 3 b3(k + p1 + p2, −p1, −p2)Γm, a(k, p1, p2; R) where (1) (1) × Γm (−p1; R)Γm (−p2; R), (2, 1lp) (3) K1 = 3 b2(k + q, −q)Γm, ab(k − p, p, q; R) (1) q (1, 2lp) (2) × Γm (− ; R) K3 = 4 b3(k − p2, p1 + p2, −p1) Γm (p1, p2; R) (1) (2) × Γm (−p1; R)Γm, a(k, −p2; R)

(2, 1lp) (2) (1, 2lp) (2) K2 = 2 b2(k − p − q, p + q) Γm, a(p, q; R) K4 = b3(k, p1 + p2, −p1 − p2) Γm (p1, p2; R) (2) (2) (1) × Γ (k − p, −q; R) × Γm (−p1, −p2; R)Γm, a(k; R) m, b

2 (1, 2lp) (1) K5 = b4(k − p1, p2, −p2, −p1) Γm (p2; R) K(2, 1lp) = 2 b (k − p, p + q, −q) Γ(1)(q; R) (1) (2) h i 3 3 m × Γ (p1; R)Γ (k, −p1; R) m m, a (1) (2) × Γm, a(k − p; R)Γm, b(p, q; R)

(1, 2lp) (2) K6 = b4(k, p1 + p2, −p1, −p2) Γm (p1, p2; R) (1) (1) (1) × Γ (−p1; R)Γ (−p2; R)Γ (k; R) m m m, a 1 K(2, 1lp) = b (k, q, −q) Γ(2) (p, k − p; R) 4 2 3 m, ab 2 (1, 2lp) 1 2 K = b (k, p , −p , p , −p ) Γ(1)(p ; R) × Γ(1)(q; R) 7 8 5 1 1 2 2 m 1 m 2 h i (1) (1) h i × Γm (p2; R) Γm, a(k; R) (B2) h i (2, 1lp) 1 (1) (2) K5 = b4(k − p, p, q, −q) Γm, a(k − q; R) Appendix C: One-loop Order of ΓE 4 2 (1) (1) × Γm (q; R) Γ (p; R). (C2) At one-loop level, five diagrams are nonzero. m, b h i 5 In Eq.(C2), K1 and K2 are two-Ξm contributions from (2, 1lp) 3 (2, 1lp) ΓE, ab (p, k − p; η) = d p P0(q) Ki . the first and second terms of Eq.(55) respectively, K3 "i=1 # Z X and K4 are three-Ξm contributions, and K5 is the four- (C1) Ξm contribution. 19

~p −~q ~p ~p ~ ~ ~k k k ~ 6 k − ~p ~ ~ k + ~q k − ~p ~k − ~p

~p −~q ~q~p ~q −~q ~p + ~q ~p ~k ~k ~k 4 12 3 ~ ~k − ~p − ~q k − ~p ~k − ~p ~k − ~p

~q −~q ~p ~k 6

~k − ~p

(2) FIG. 18: Three-point nonlinear propagator ΓE up to one-loop order.

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