Integrated Perturbation Theory and One-loop Power Spectra of Biased Tracers
aa r X i v : . [ a s t r o - ph . C O ] A ug Integrated Perturbation Theory and One-loop Power Spectra of Biased Tracers
Takahiko Matsubara ∗ Department of Physics, Nagoya University, Chikusa, Nagoya, 464-8602, Japan; andKobayashi-Maskawa Institute for the Origin of Particles and the Universe,Nagoya University, Chikusa, Nagoya, 464-8602, Japan (Dated: July 23, 2018)General and explicit predictions from the integrated perturbation theory (iPT) for power spectra and correla-tion functions of biased tracers are derived and presented in the one-loop approximation. The iPT is a generalframework of the nonlinear perturbation theory of cosmological density fields in presence of nonlocal bias,redshift-space distortions, and primordial non-Gaussianity. Analytic formulas of auto and cross power spectraof nonlocally biased tracers in both real and redshift spaces are derived and the results are comprehensivelysummarized. The main di ff erence from previous formulas derived by the present author is to include e ff ectsof generally nonlocal Lagrangian bias and primordial non-Gaussianity, and the derivation method of the newformula is fundamentally di ff erent from the previous one. Relations to recent work on improved methods ofnonlinear perturbation theory in literature are clarified and discussed. PACS numbers: 98.80.-k, 98.65.-r, 98.80.Cq, 98.80.Es
I. INTRODUCTION
Density fluctuations in the universe contain invaluable in-formation on cosmology. For example, the history and ingre-dients of the universe are encoded in detailed patterns of thedensity fluctuations. The large-scale structure (LSS) of theuniverse is one of the most popular ways to probe the densityfluctuations in the universe. Spatial distributions of galaxiesand other astronomical objects which can be observed reflectthe underlying density fluctuations in the universe.In cosmology, it is crucial to investigate the spatial distri-butions of dark matter, which dominates the mass of the uni-verse. Unfortunately, distributions of dark matter are di ffi cultto directly observe, because the only interaction we know thatthe dark matter surely has is the gravitational interaction. Con-sequently, we need to estimate the density fluctuations of theuniverse by means of indirect probes such as galaxies, whichhave electromagnetic interactions.Relations between distributions of observable objects andthose of dark matter are nontrivial. On very large scales wherethe linear theory can be applied, the relations are reasonablyrepresented by the linear bias; the density contrasts of darkmatter δ m and those of observable objects δ X are proportionalto each other, δ X = b δ m , where b is a constant called the biasparameter. However, nonlinear e ff ects cannot be neglectedwhen we extract cosmological information as much as possi-ble from observational data of LSS, and bias relations in non-linear regime are not as simple as those in linear regime.Observations of LSS play an important role in cosmology.Shapes of power spectra of galaxies and clusters contain infor-mation on the density parameters of cold dark matter Ω CDM ,baryons Ω b and neutrinos Ω ν in the universe. Precision mea-surements of baryon acoustic oscillations (BAO) in galaxypower spectra or correlation functions can constrain the natureof dark energy [1–3], which is a driving force of the acceler- ∗ Electronic address: [email protected] ated expansion of the present universe. The non-Gaussianityin the primordial density field induces a scale-dependent biasin biased tracers of LSS on very large scales [4–8]. Cosmo-logical information contained in detailed features in LSS is sorich that there are many ongoing and future surveys of LSS,such as BOSS [9], FMOS FastSound [10], BigBOSS [11],LSST [12], Subaru PFS [13], DES [14], Euclid [15], etc.Elucidating nonlinear e ff ects on observables in LSS hascrucial importance in the precision cosmology. Whilestrongly nonlinear phenomena are di ffi cult to analyticallyquantify, the perturbation theory is useful in understandingquasi-nonlinear regime. The traditional perturbation theorydescribes evolutions of mass density field on large scaleswhere the density fluctuations are small. However, spatialdistributions of astronomical objects such as galaxies do notexactly follow the mass density field, and they are biased trac-ers. Formation processes of astronomical objects are governedby strongly nonlinear dynamics including baryon physics etc.,which cannot be straightforwardly treated by the traditionalperturbation theory.Even though the tracers are produced through strongly non-linear processes, it is still sensible to apply the perturbationtheory to study LSS on large scales. For example, the biasinge ff ect in linear theory is simply represented by a bias param-eter b as described above. However, biasing e ff ects in higher-order perturbation theory are not that simple. A popular modelof the biasing in the context of nonlinear perturbation theoryis the Eulerian local bias [16–20]. This model employs freelyfitting parameters in every orders of perturbations, and is justa phenomenological model because the Eulerian bias is notdefinitely local in reality.The integrated perturbation theory (iPT) [21] is a frame-work of the perturbation theory to predict observable powerspectra and any higher-order polyspectra (or the correlationfunctions) of nonlocally biased tracers. In addition, the e ff ectsof redshift-space distortions and primordial non-Gaussianityare naturally incorporated. This theory is general enough sothat any model of nonlocal bias can be taken into account. Pre-cise mechanisms of bias are still not theoretically understoodwell, and are under active investigations. The framework ofiPT separates the known physics of gravitational e ff ects onspatial clustering from the unknown physics of complicatedbias. The unknown physics of nonlocal bias is packed into“renormalized bias functions” c ( n ) X in the iPT formalism. Oncethe renormalized bias functions are modeled for observabletracers, weakly nonlinear e ff ects of gravitational evolutionsare taken care of by the iPT. The iPT is a generalization of aprevious formulation called Lagrangian resummation theory(LRT) [22–25] in which only local models of Lagrangian biascan be incorporated.In recent developments, the model of bias from the halo ap-proach has turned out to be quite useful in understanding thecosmological structure formations [26–32]. The halo bias isnaturally incorporated in the framework of iPT. Predictions ofiPT combined with the halo model of bias do not contain anyfitting parameter once the mass function and physical massof halos are specified. This property is quite di ff erent fromother phenomenological approaches to combine the perturba-tion theory and bias models.A concept of nonlocal Lagrangian bias has recently at-tracted considerable attention [33–35]. Extending the haloapproach, a simple nonlocal model of Lagrangian bias is re-cently proposed [36] for applications to the iPT. Applying thisnonlocal model of halo bias to evaluating the scale-dependentbias in the presence of primordial non-Gaussianity, not onlythe results of peak-background split are reproduced, but alsomore general formula is obtained. In this paper, the usage ofthis simple model of nonlocal halo bias in the framework ofiPT is explicitly explained.The bias in the framework of iPT does not have to be a halobias. There are many kinds of tracers for LSS, such as var-ious types of galaxies, quasars, Ly- α absorption lines, 21cmabsorption and emission lines, etc. Once the bias model foreach kind of objects is given, it is straightforward to calculatebiased power spectra and polyspectra of those tracers in theframework of iPT. As described above, it is needless to saythat detailed mechanisms of bias for those tracers have notbeen fully understood yet. As emphasized above, the iPT sep-arates the di ffi cult problems of fully nonlinear biasing fromgravitational evolutions in weakly nonlinear regime.While the basic formulation of iPT is developed inRef. [21], explicit calculations of the nonlinear power spec-tra are not given in that reference. The purpose of this paperis to give explicit expressions of biased power spectra withan arbitrary model of nonlocal bias in the one-loop approxi-mations, in which leading-order corrections to the nonlinearevolutions are included. The expressions are given both inreal space and in redshift space. Three-dimensional integralsin the formal expressions of one-loop power spectra are re-duced to one- and two-dimensional integrals, which are easyand convenient for numerical integrations. Contributions fromprimordial non-Gaussianity are also taken into account in thegeneral expressions. Explicit formulas of the renormalizedbias functions are provided for a simple model of nonlocalhalo bias. In this way, general formulas of power spectra ofbiased objects in the one-loop approximation is provided inthis paper. Since the iPT framework is based on the Lagrangian per-turbation theory (LPT) [37–42], a scheme of resummations ofhigher-order perturbations in terms of the Eulerian perturba-tion theory (EPT) [44] is naturally considered [22]. In this pa-per, we clarify the relations of the present formula of iPT andsome previous methods of resummation technique such as therenormalized perturbation theory [45, 46], the Gamma expan-sions [47–52], the Lagrangian resummation theory [22–25],and the convolution perturbation theory [53]. Some aspectsfor the future developments of iPT are suggested.This paper is organized as follows. In Sec. II, formal ex-pressions of power spectra in the framework of iPT with anarbitrary model of bias are derived. A simple model of renor-malized bias functions for a nonlocal Lagrangian bias in thehalo approach are summarized. In Sec. III, explicit formulasof biased power spectra, which are the main results of thispaper, are derived and presented. Relations to other previouswork in literature are clarified in Sec. IV, and conclusions aregiven in Sec. V. In App. A, diagrammatic rules of iPT used inthis paper are briefly summarized. II. THE ONE-LOOP POWER SPECTRA IN THEINTEGRATED PERTURBATION THEORY
In this first section, the formalism of iPT [21] is brieflyreviewed (without proofs), and formal expressions of powerspectra in the one-loop approximation are derived.
A. Fundamental equations of the integrated perturbationtheory
In evaluating the power spectra in iPT, a concept of multi-point propagator [47, 48, 54] is useful. The ( n + Γ ( n ) X of any biased objects, which are labeled by X ingeneral, is defined by [21] * δ n δ X ( k ) δδ L ( k ) · · · δδ L ( k n ) + = (2 π ) − n δ ( k − k ··· n ) Γ ( n ) X ( k , . . . , k n ) , (1)where δ X ( k ) is the Fourier transform of the number den-sity contrast of biased objects in Eulerian space, δ L ( k ) is theFourier transform of linear density contrast, δ is the Dirac’sdelta function in three-dimensions, and we adopt a notation k ··· n = k + · · · + k n , (2)throughout this paper. The left-hand side of Eq. (1) is an en-semble average of n th-order functional derivative. The num-ber density field is considered as a functional of the initialdensity field. In the basic framework of iPT, the biased ob-jects can be any astronomical objects which are observed astracers of the underlying density field in the universe.The method how to evaluate multi-point propagators of bi-ased objects in the framework of iPT is detailed in Ref. [21].In the most general form of iPT formalism, both Eulerianand Lagrangian pictures of dynamical evolutions can be dealtwith, and both pictures give equivalent predictions for observ-ables. The models of halo bias fall into the category of La-grangian bias, i.e., the number density field of halos is relatedto the mass density field in Lagrangian space. In such a case,the Lagrangian picture is a natural way to describe evolutionsof halo number density field. In the models of Lagrangianbias, the renormalized bias functions [21] are the key elementsin iPT, which are defined by c ( n ) X ( k , . . . , k n ) = (2 π ) n Z d k (2 π ) * δ n δ L X ( k ) δδ L ( k ) · · · δδ L ( k n ) + , (3)where δ L X ( k ) is the Fourier transform of halo number densitycontrast in Lagrangian space. We allow the bias to be nonlocalin Lagrangian space. In fact, the halo bias is not purely localeven in Lagrangian space [36]. For a mass density field, theLagrangian number density contrast δ L X is identically zero, andthe bias functions are identically zero, c ( n ) X = n = , , . . . .Assuming statistical homogeneity in Lagrangian space, therenormalized bias functions in Eq. (3) is equivalently definedby [36] * δ n δ L X ( k ) δδ L ( k ) · · · δδ L ( k n ) + = (2 π ) − n δ ( k − k ··· n ) c ( n ) X ( k , . . . , k n ) . (4)The similarity of this equation with Eq. (1) is apparent in thisform. The information on dynamics of bias in Lagrangianspace is encoded in the set of renormalized bias functions.Assuming statistical isotropy in Lagrangian space, the renor-malized bias functions c ( n ) X ( k , . . . , k n ) depend only on magni-tudes k , . . . , k n and relative angles ˆ k i · ˆ k j ( i > j ) of wavevec-tors.Applying the vertex resummation of iPT, the multi-pointpropagators of biased objects X are given by a form, Γ ( n ) X ( k , . . . , k n ) = Π ( k ··· n ) ˆ Γ ( n ) X ( k , . . . , k n ) , (5)where Π ( k ) = D e − i k · Ψ E = exp ∞ X n = ( − i ) n n ! h ( k · Ψ ) n i c , (6)is the vertex resummation factor in terms of the displacementfield Ψ , and h· · · i c indicates the connected part of ensembleaverage. The displacement fields Ψ ( q ) are the fundamentalvariables in LPT, where q is the Lagrangian coordinates andthe Eulerian coordinates are given by x = q + Ψ ( q ). Thecumulant expansion theorem is used in the second equalityof Eq. (6). Cumulants of the displacement fields with oddnumber vanish from the parity symmetry, thus the summationin the exponent of Eq. (6) is actually taken over n = , , , . . . .The normalized multi-point propagators of the biased objects,ˆ Γ ( n ) X , are naturally predicted in the framework of iPT.In the one-loop approximation of iPT, the vertex resumma-tion factor is given by Π ( k ) = exp ( − Z d p (2 π ) h k · L (1) ( p ) i P L ( p ) ) , (7) = + ++ ++ + FIG. 1: The diagrammatic representation of the two-point propagatorwith partially resummed vertex up to one-loop contributions. and the normalized two-point propagator is given byˆ Γ (1) X ( k ) = c (1) X ( k ) + k · L (1) ( k ) + Z d p (2 π ) P L ( p ) (cid:26) c (2) X ( k , p ) h k · L (1) ( − p ) i + c (1) X ( p ) h k · L (1) ( − p ) i h k · L (1) ( k ) i + k · L (3) ( k , p , − p ) + c (1) X ( p ) h k · L (2) ( k , − p ) i + h k · L (1) ( p ) i h k · L (2) ( k , − p ) i(cid:27) , (8)where L ( n ) is the n th-order displacement kernel in LPT.Each term in Eq. (8) respectively corresponds to each di-agram of Fig. 1 in the same order. Diagrammatic rules iniPT [21] with the Lagrangian picture, which are explained inApp. A, are applied in the correspondence. The normalizedtwo-point propagator of mass density field, ˆ Γ (1)m , is obtainedby putting c ( n ) X = Ψ ( k ) is given by˜ Ψ ( k ) = ∞ X n = in ! Z k ··· n = k L ( n ) ( k , . . . , k n ) δ L ( k ) · · · δ L ( k n ) , (9)where we adopt a notation, Z k ··· n = k · · · = Z d k (2 π ) · · · d k n (2 π ) (2 π ) δ ( k − k ··· n ) · · · . (10)Such notation as Eq. (10) is commonly used throughout thispaper.In real space, the kernels of LPT in the standard theory ofgravity (in the Newtonian limit) are given by [40] L (1) ( k ) = k k , (11) L (2) ( k , k ) = k k − k · k k k ! , (12) L (3) ( k , k , k ) = h L (3a) ( k , k , k ) + perm . i ; (13) L (3a) ( k , k , k ) = k k − k · k k k ! − k · k k k ! − − k · k k k ! + k · k )( k · k )( k · k ) k k k + k k × T ( k , k , k ) , (14)where a vector function T represents a transverse part whoseexplicit expression will not be used in this paper. Completeexpressions of the displacement kernels of LPT up to 4th or-der, including transverse parts are given in, e.g., Ref. [41, 42].Eqs. (7) and (8) remain valid even when the non-standard the-ory of gravity is assumed as long as the appropriate form ofkernels L n in such a theory is used.One of the benefits in the Lagrangian picture is thatredshift-space distortions are relatively easy to be incorpo-rated in the theory. A displacement kernel in redshift space L s( n ) is simply related to the kernel in real space at the sameorder by a linear mapping [22], L ( n ) → L s( n ) = L ( n ) + n f (cid:16) ˆ z · L ( n ) (cid:17) ˆ z , (15)where f = d ln D / d ln a = ˙ D / HD is the linear growthrate, D ( t ) is the linear growth factor, a ( t ) is the scale fac-tor, and H ( t ) = ˙ a / a is the time-dependent Hubble parame-ter. The distant-observer approximation is assumed in redshiftspace, and the unit vector ˆ z denotes the line-of-sight direc-tion. Strictly speaking, the mapping of Eq. (15) is exact onlyin the Einstein-de Sitter universe. However, this mapping isa good approximation in general cosmology. The expressionsof Eqs. (7) and (8) apply as well in redshift space when thedisplacement kernels in redshift space L s( n ) are used insteadof the real-space counterparts L ( n ) .The three-point propagator at the tree-level approximationin iPT is given byˆ Γ (2) X ( k , k ) = c (2) X ( k , k ) + c (1) X ( k ) h k · L (1) ( k ) i + c (1) X ( k ) h k · L (1) ( k ) i + h k · L (1) ( k ) i h k · L (1) ( k ) i + k · L (2) ( k , k ) , (16)where each term respectively corresponds to each diagram ofFig. 2 in the same order. When the mapping of Eq. (15) isapplied to every displacement kernels in Eq. (16), the expres-sion of three-point propagator in redshift space is obtained.The three-point propagator of mass, Γ (2)m , is given by just sub-stituting c ( n ) X = = + ++ + FIG. 2: The diagrammatic representation of the three-point propaga-tor with partially resummed vertex at the tree-level contribution. + ++
FIG. 3: The diagrammatic representation of the power spectrum upto one-loop approximation.
In terms of the multi-point propagators, the power spectrumof biased objects, up to the one-loop approximation, is givenby P X ( k ) = Π ( k ) (h ˆ Γ (1) X ( k ) i P L ( k ) + Z k = k h ˆ Γ (2) X ( k , k ) i P L ( k ) P L ( k ) + ˆ Γ (1) X ( k ) Z k = k ˆ Γ (2) X ( k , k ) B L ( k , k , k ) ) , (17)where P L ( k ) and B L ( k , k , k ) are the linear power spectrumand the linear bispectrum, respectively. The diagrammaticrepresentations of Eq. (17) are shown in Fig. 3. Crossed cir-cles correspond to the linear power spectrum or the linear bis-pectrum, depending on number of lines attached to them. Thefirst two terms in Eq. (17) corresponds to the first two dia-grams in Fig. 3. The last two diagrams in Fig. 3 are contribu-tions from the primordial non-Gaussianity. The two diagramsgive the same contribution because of the parity symmetry,and the sum of the two diagrams corresponds to the last termin Eq. (17).The matter power spectrum P m ( k ) is simply given by re-placing Γ ( n ) X by Γ ( n )m in Eq. (17), or equivalently, setting c ( n ) X = n ≥
1. The cross power spectrum between two typesof objects, X and Y , is similarly obtained as P XY ( k ) = Π ( k ) ( ˆ Γ (1) X ( k ) ˆ Γ (1) Y ( k ) P L ( k ) + Z k = k ˆ Γ (2) X ( k , k ) ˆ Γ (2) Y ( k , k ) P L ( k ) P L ( k ) +
12 ˆ Γ (1) X ( k ) Z k = k ˆ Γ (2) Y ( k , k ) B L ( k , k , k ) +
12 ˆ Γ (1) Y ( k ) Z k = k ˆ Γ (2) X ( k , k ) B L ( k , k , k ) ) . (18)The diagrams for the above equations are similar to the ones inFig. 3, where the left and right multi-point propagators corre-spond to those of X and Y , respectively. When X = Y , Eq. (18)apparently reduces to Eq. (17).The predictions of biased power spectra in the one-loop ap-proximation of iPT are given by Eq. (17) for the auto powerspectrum, and by Eq. (18) for the cross power spectrum. Oncea model of the renormalized bias functions c ( n ) X is given, it isstraightforward to numerically evaluate those equations. Theabove results are general and do not depend on bias models.Any bias model can be incorporated in the expression of iPTthrough the renormalized bias functions. In the next subsec-tion, we explain a simple model of the renormalized bias func-tion based on the halo approach. B. Renormalized bias functions in a simple model of haloapproach
The renormalized bias functions c ( n ) X are not specified in thegeneral framework of iPT. Precise modeling of bias is a non-trivial problem, depending on what kind of biased tracers areconsidered. In this subsection, we consider a simple modelof halo bias as an example. The expressions of renormalizedbias functions in a simple model of the halo approach are re-cently derived in Ref. [36]. We summarize the consequencesof this model below. It should be emphasized that the generalframework of iPT does not depend on this specific model ofbias.Without resorting to approximations such as the peak-background split, the halo bias is shown to be nonlocal even inLagrangian space. As a result, the renormalized bias functionshave nontrivial scale dependencies. For the halos of mass M ,the renormalized bias functions are given by [36] c ( n ) M ( k , . . . , k n ) = b L n ( M ) W ( k R ) · · · W ( k n R ) + A n − ( M ) δ c n dd ln σ M [ W ( k R ) · · · W ( k n R )] , (19)where δ c = π/ / / ≃ .
686 is the critical overdensityfor spherical collapse and W ( kR ) is a window function. In ausual halo approach, the window function is chosen to be atop-hat type in configuration space, which corresponds to W ( x ) = x − x cos xx , (20)in Fourier space. In this case, the Lagrangian radius R is nat-urally related to the mass M of halo by M = π ¯ ρ R , (21)where ¯ ρ is the mean density of mass at the present time, or R = " M . × h − M ⊙ Ω m0 / h − Mpc , (22)where M ⊙ = . × kg is the solar mass, Ω m0 isthe density parameter of mass at the present time, and h = H / (100 km s − Mpc − ) is the normalized Hubble constant. Empirically, one can also use other types of window func-tion. Direct evaluations of the renormalized bias functionssuggest that Gaussian window function W ( x ) = e − k R / givesbetter fit [43]. In the latter case, the relation between thesmoothing radius R and mass M is not trivial and should alsobe empirically modified from the relation of Eq. (22). How-ever, the shapes of one-loop power spectrum on large scalesare not sensitive to the choice of window function.The variance of density fluctuations on the mass scale M isdefined by σ M = Z d k (2 π ) W ( kR ) P L ( k ) . (23)The radius R is considered as a function of σ M throughEq. (19). The functions A n ( M ) are defined by A n ( M ) ≡ n X j = n ! j ! δ c j b L j ( M ) , (24)where b L n is the scale-independent Lagrangian bias parameterof n th-order. For example, first three functions are given by A = , A = + δ c b L1 , A = + δ c b L1 + δ c2 b L2 . (25)When the halo mass function n ( M ) takes a universal form n ( M ) dM = ¯ ρ M f MF ( ν ) d νν , (26)where ν = δ c /σ M , the Lagrangian bias parameters are givenby b L n ( M ) = − σ M ! n f ( n )MF ( ν ) f MF ( ν ) , (27)where f ( n )MF = d n f MF / d ν n .Once the model of the mass function f MF ( ν ) is given, thescale-independent bias parameters b L n ( M ) and the functions A n ( M ) are uniquely given by Eqs. (27) and (24). In Table I,those functions are summarized for popular models of massfunction, i.e., the Press-Schechter (PS) mass function [26], theSheth-Tormen (ST) mass function [30], Warren et al. (W + )mass function [55].In the simplest PS mass function, it is interesting to notethat general expressions of the parameters for all orders canbe derived [36]: b L n = ν n − H n + ( ν ) /δ c n , A n = ν n H n ( ν ), where H n ( ν ) are the Hermite polynomials. The ST mass functiongives a better fit to numerical simulations of halos in cold-dark-matter type cosmologies with Gaussian initial condi-tions. The values of parameters in Table I are p = . q = . A ( p ) = [1 + π − / − p Γ (1 / − p )] − is thenormalization factor. When we put p = q =
1, the STmass function reduces to the PS mass function. The W + mass function is represented by a parameter σ = δ c /ν , whichis also a function of M , and parameters are A = . a = . b = . c = . A ( z ) = PS ST W + , MICE ( σ = δ c /ν ) f MF ( ν ) r π ν e − ν / A ( p ) r π " + q ν ) p √ q ν e − q ν / A (cid:0) σ − a + b (cid:1) e − c /σ b L1 ( M ) ν − δ c δ c " q ν − + p + ( q ν ) p δ c c σ − a + b σ a ! b L2 ( M ) ν − ν δ c2 δ c2 " q ν − q ν + p (2 q ν + p − + ( q ν ) p δ c2 c σ − c σ − a (cid:16) c /σ − a + (cid:17) + b σ a A ( M ) ν q ν + p + ( q ν ) p c σ + − a + b σ a A ( M ) ν ( ν − q ν ( q ν − + p (2 q ν + p + + ( q ν ) p c σ + c σ + − a (cid:16) c /σ − a + (cid:17) + b σ a Parameters - A ( p ) = [1 + π − / − p Γ (1 / − p )] − p = . q = .
707 W + : A = . a = . b = . c = . A ( z ) = . + z ) − . a ( z ) = . + z ) − . b ( z ) = . + z ) − . c ( z ) = . + z ) − . TABLE I: Functions b L n ( M ), A n ( M ) derived from several models of mass function. . + z ) − . , a ( z ) = . + z ) − . , b ( z ) = . + z ) − . , c ( z ) = . + z ) − . . When the redshift-dependent param-eters are adopted, the W + mass function is sometimes referredto as the “MICE mass function”. In the latter case, the multi-plicity function f MF ( ν ) explicitly depends on the redshift, andthe mass function is no longer ’universal’.The nonlocal nature of the halo bias in Lagrangian space isencoded in the second term in the RHS of Eq. (19), since thesimple dependence on the window function of the first termappears even in the local bias models through the smoothedmass density field. In the large-scale limit, k , k , . . . , k n → c ( n ) M ≃ b L n ( M ). This property is consistent withthe peak-background split. However, the loop corrections inthe iPT involve integrations over the wavevectors of the renor-malized bias functions, and there is no reason to neglect thesecond term which represents nonlocal nature of Lagrangianbias of halos.The Eq. (19) is shown to be equivalent to the following ex-pression [36], c ( n ) M ( k , . . . , k n ) = A n ( M ) δ c n W ( k R ) · · · W ( k n R ) + A n − ( M ) σ Mn δ c n dd ln σ M " W ( k R ) · · · W ( k n R ) σ Mn . (28)For the PS mass function, there is an interesting relation, A n = ν δ c n − b L n − , and in this case, the renormalized bias func-tion c L n is expressible by lower-order parameters b L n − and b L n − ,which is a reason why the scale-dependent bias in the presenceof primordial non-Gaussianity is approximately proportionalto the first-order bias parameter, b L1 rather than the second-order one, b L2 [36]. However, this does not mean that c ( n ) M isindependent on b L n , because b L n can be expressible by a linearcombination of b L n − and b L n − in the PS mass function.In the expressions of renormalized bias functions, Eqs. (19) and (28), all the halos are assumed to have the same mass, M .These expressions apply when the mass range of halos in agiven sample is su ffi ciently narrow. When the mass range isfinitely extended, the expressions should be replaced by [36] c ( n ) φ ( k , . . . , k n ) = R dM φ ( M ) n ( M ) c ( n ) M ( k , . . . , k n ) R dM φ ( M ) n ( M ) , (29)where n ( M ) is the halo mass function of Eq. (26), φ ( M ) is aselection function of mass. For a simple example, when themass of halos are selected by a finite range [ M , M ], we have c ( n )[ M , M ] ( k , . . . , k n ) = R M M dM n ( M ) c ( n ) M ( k , . . . , k n ) R M M dM n ( M ) . (30) III. EXPLICIT FORMULAS
The auto power spectrum P X ( k ) of Eq. (17) is a special caseof the cross power spectrum P XY ( k ) of Eq. (18) as the formeris given by setting X = Y in the latter. It is general enough togive the formulas for the cross power spectrum below. In thefollowing, we decompose Eq. (18) into the following form: P XY ( k ) = Π ( k ) [ R XY ( k ) + Q XY ( k ) + S XY ( k )] , (31)where Π ( k ) is given by Eq. (7) and R XY ( k ) = ˆ Γ (1) X ( k ) ˆ Γ (1) Y ( k ) P L ( k ) , (32) Q XY ( k ) = Z k = k ˆ Γ (2) X ( k , k ) ˆ Γ (2) Y ( k , k ) P L ( k ) P L ( k ) , (33) S XY ( k ) =
12 ˆ Γ (1) X ( k ) Z k = k ˆ Γ (2) Y ( k , k ) B L ( k , k , k ) + ( X ↔ Y ) . (34)Three-dimensional integrals appeared in the above compo-nents of Eqs. (32)–(34) can be reduced to lower-dimensionalintegrals both in real space and in redshift space. Such dimen-sional reductions of the integrals are useful for practical calcu-lations. The purpose of this section is to give explicit formulasfor the above components Π , R XY , Q XY , S XY in terms of two-dimensional integrals at most. The results of this section areapplicable to any bias models, and do not depend on specificforms of renormalized bias functions, e.g., those explained inSec. II B. A. The power spectra in real space
In real space, the power spectrum is independent on the di-rection of wavevector k , and thus the components above Π ( k ), R XY ( k ), Q XY ( k ), S XY ( k ) are also independent on the direc-tion. In this case, dimensional reductions of the integrals inEqs. (32)–(34) are not di ffi cult, because of the rotational sym-metry. The vertex resummation factor Π ( k ) of Eq. (7) is givenby Π ( k ) = exp " − k π Z d p P L ( p ) . (35)On small scales, this factor exponentially suppresses thepower too much, and such a behavior is not physical. Thisproperty is a good indicator of which scales the perturbationtheory should not be applied. However, the resummation ofthe vertex factor is not compulsory in the iPT. When the ver-tex factor is not resummed, one can expand the factor as Π ( k ) = − k π Z d p P L ( p ) , (36)instead of Eq. (35) in the case of one-loop perturbation the-ory. In a quasi-linear regime, the resummed vertex factor ofEq. (35) gives better fit to N -body simulations in real space[22, 25].The expression of two-point propagator in Eq. (8) isstraightforwardly obtained, substituting the Lagrangian ker-nels of Eqs. (11)–(14). Taking the z -axis of p as the di-rection of k , integrations by the azimuthal angle are trivial.Transforming the rest of integration variables as r = p / k and x = ˆ p · ˆ k , we have two equivalent expressions,ˆ Γ (1) X ( k ) = + c (1) X ( k ) + k π Z ∞ dr Z − dx ˆ R X ( k , r , x ) P L ( kr )(37) = + c (1) X ( k ) + k π Z ∞ dr ˜ R X ( k , r ) P L ( kr ) , (38)whereˆ R X ( k , r , x ) = r (1 − x ) + r − rx +
37 (1 − rx )(1 − x )1 + r − rx h rx + r c (1) X ( kr ) i − rx c (2) X ( k , kr ; x ) , (39) and˜ R X ( k , r ) = + r + r − r r + (1 − r ) (2 + r )168 r ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − r + r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + " + r − r + − r ) r ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − r + r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c (1) X ( kr ) − r Z − dx x c (2) X ( k , kr ; x ) . (40)In the above expressions, rotationally invariant arguments for c (2) X are used, i.e., c (2) X ( k , k ) = c (2) X ( k , k ; x ) , (41)where x = ˆ k · ˆ k is the direction cosine between k and k .The second expression of Eq. (38) is obtained by analyticallyintegrating the variable x in the first expression of Eq. (37).Both expressions are suitable for numerical evaluations. Withthe expression of Eq. (37) or (38), we have R XY ( k ) = ˆ Γ (1) X ( k ) ˆ Γ (1) Y ( k ) P L ( k ) . (42)Evaluating the convolution integrals in Eqs. (33) and (34)with the three-point propagator of Eq. (16) is also straight-forward in real space. Substituting the Lagrangian kernels ofEqs. (11) and (12) into Eq. (16), and transforming the integra-tion variables as r = k / k , x = ˆ k · ˆ k , we have Q XY ( k ) = k π Z ∞ dr Z − dx r ˆ Γ (2) X ( k , r , x ) ˆ Γ (2) Y ( k , r , x ) × P L ( kr ) P L ( k y ) (43)and S XY ( k ) = k π ˆ Γ (1) X ( k ) Z ∞ dr Z − dx r ˆ Γ (2) Y ( k , r , x ) × B L ( k , kr , k y ) + ( X ↔ Y ) , (44)where y = √ + r − rx , (45)andˆ Γ (2) X ( k , r , x ) = −
47 1 − x y + xr h + c (1) X ( k y ) i + − rx y h + c (1) X ( kr ) i + c (2) X ( kr , k y ; x ) , (46)The factor ˆ Γ (2) Y ( k , r , x ) is similarly given by substituting X → Y in Eq. (46). The function ˆ Γ (2) X ( k , r , x ) is just the normalizedthree-point propagator ˆ Γ (2) X ( k , k − k ) as a function of trans-formed variables.All the necessary components to calculate the power spec-trum of Eq. (31) in real space, P XY ( k ) = Π ( k ) [ R XY ( k ) + Q XY ( k ) + S XY ( k )] , (47) F ( k , p ) Z d p (2 π ) F ( k , p ) P L ( p ) Diagram L (3) ( k , p , − p ) 1021 k k R ( k ) L (1) i ( − p ) L (2) j ( k , p ) 314 k i k j − k δ ij k R ( k ) + k i k j k R ( k ) L (2) ( k , p ) c (1) X ( p ) 37 k k R X ( k ) L (1) ( − p ) c (2) X ( k , p ) − k k R X ( k )TABLE II: Integral formulas for one-loop corrections, which are re-lated to the two-point propagator. We denote R ( k ) = R X ( k ) and R ( k ) = R X ( k ), as these functions are independent on the bias. are given above, i.e., Eqs. (35) [or (36)], (42), (43) and (44).Numerical integrations of Eqs. (38) [or (37)], (43) and (44) arenot di ffi cult, once the model of renormalized bias functions c ( n ) X and primordial spectra P L ( k ), B L ( k , k , k ) are given. Thelast factor S XY ( k ) is absent in the case of Gaussian initial con-ditions. B. Kernel integrals
Evaluations of power spectra in redshift space are moretedious than those in real space. The reason is that thepower spectra depend on the lines-of-sight direction in red-shift space. One cannot arbitrary choose the direction of z -axisin the three-dimensional integrations of Eqs. (8) and (32)–(34), because the rotational symmetry is not met. Even insuch cases, an axial symmetry around the lines of sight re-mains, and the three-dimensional integrations can be reducedto two- or one-dimensional integrations as shown below. Allthe necessary techniques for such reductions are the same withthose presented in Refs. [22, 23], making use of rotational co-variance. We summarize useful formulas for the reduction inthis subsection. We assume the standard theory of gravity inthe formula below, although the same technique may be ap-plicable to other theories such as the modified gravity, etc.The first set of formulas is related to the two-point propa-gator Γ (1) X of Eq. (8). The results are summarized in Table II.The integrals of a form, Z d p (2 π ) F ( k , p ) P L ( p ) , (48)where F ( k , p ) consists of LPT kernels L ( n ) and renormalizedbias functions c ( n ) X , are reduced to one-dimensional integrals, R Xn ( k ). The explicit formulas are given in Table II. In this Ta-ble, we denote R ( k ) = R X ( k ) and R ( k ) = R X ( k ), as thesefunctions are independent on the bias. The functions R Xn ( k ) are defined by three equivalent sets of equations, R Xn ( k ) = Z d p (2 π ) R Xn ( k , p ) P L ( p ) = k π Z ∞ dr Z − dx ˆ R Xn ( r , x ) P L ( kr ) = k π Z ∞ dr ˜ R Xn ( r ) P L ( kr ) , (49)where integrands R Xn ( k , p ), ˆ R Xn ( r , x ), and ˜ R Xn ( r ) are given inTable III. The last expression of Eq. (49) is the formula whichis practically useful for numerical evaluations. The other ex-pressions are shown to indicate origins of the integrals.If the second-order bias function c (2) X ( k , k ) only dependson magnitudes of wavevectors k and k , and not on the rela-tive angle µ = ˆ k · ˆ k , the fourth function generically van-ishes: R X ( k ) =
0. If the first-order bias function c (1) X is scale-independent, it is explicitly shown from the last expressionsthat R X ( k ) = [ R ( k ) + R ( k )] c (1) X . Specifically, the functions R X ( k ) and R X ( k ) are redundant in the Lagrangian local biasmodels, in which renormalized bias functions c ( n ) X are scale-independent. This is the reason only two functions R ( k ) and R ( k ) are needed in Ref. [23]. In general situations with La-grangian nonlocal bias models, all four functions are needed.In a simple model of halo bias in this paper, the second-order bias function c (2) X does not depend on the angle µ and R ( k ) = Γ (2) X in calculating theone-loop power spectrum. The integrals of the form, Z k = k F ( k , k ) P L ( k ) P L ( k ) , (50)where F consists of LPT kernels L n and renormalized biasfunctions c ( n ) X and c ( n ) Y , are reduced to two-dimensional inte-grals, Q XYn ( k ). The explicit formulas are given Table IV. Forthe third and fifth formulas in this Table, the indices of theLPT kernels are symmetrized, since only symmetric com-binations are used in this paper. In this Table, we denote Q n ( k ) = Q XYn ( k ) for n = , , ,
4, as these functions are inde-pendent on the bias, and Q Xn ( k ) = Q XYn ( k ) for n = , , , , X .The functions Q XYn ( k ) are defined by two equivalent sets ofequations, Q XYn ( k ) = Z k = k Q XYn ( k , k ) P L ( k ) P L ( k ) = k π Z ∞ dr Z − dx ˜ Q XYn ( r , x ) P L ( kr ) × P L (cid:16) k √ + r − rx (cid:17) , (51)where integrands Q XYn ( k , k ), ˜ Q XYn ( r , x ) are given in Table V.The last expression of Eq. (51) is the formula which is prac-tically useful for numerical evaluations. The first expressionsare shown to indicate origins of the integrands. n R Xn ( k , p ) ˆ R Xn ( r , x ) ˜ R Xn ( r )1 k | k − p | − k · p kp ! r (1 − x ) + r − rx − (1 + r )(3 − r + r )24 r − (1 − r ) r ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − r + r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k · p ) (cid:2) k · ( k − p ) (cid:3) p | k − p | − k · p kp ! rx (1 − rx )(1 − x )1 + r − rx (1 − r )(3 − r + r )24 r + (1 − r ) (1 + r )16 r ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − r + r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k · ( k − p ) | k − p | − k · p kp ! c (1) X ( p ) r (1 − rx )(1 − x )1 + r − rx c (1) X ( kr ) " + r − r + (1 − r ) r ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − r + r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c (1) X ( kr )4 k · p p c (2) X ( k , p ) rx c (2) X ( k , kr ; x ) r Z − dx x c (2) X ( k , kr ; x )TABLE III: Integrands for functions R Xn ( k ) of Eq. (49). The third set of formulas is related to the initial bispectrum,which is an indicator of primordial non-Gaussianity. The in-tegrals of the form, Z k = k F ( k , k ) B L ( k , k , k ) , (52)where F consists of LPT kernels L n and renormalizedbias functions c ( n ) X , are reduced to two-dimensional integrals, S Xn ( k ). The explicit formulas are given in Table VI. In thisTable, we denote S ( k ) = S X ( k ) and S ( k ) = S X ( k ), as thesefunctions are independent on the bias. The functions S Xn ( k )are defined by two equivalent sets of equations, S Xn ( k ) = Z k = k S Xn ( k , k ) B L ( k , k , k ) = k π Z ∞ dr Z − dx ˜ S Xn ( r , x ) × B L (cid:16) k , kr , k √ + r − rx (cid:17) , (53)where integrands S Xn ( k , k ), ˜ S Xn ( r , x ) are given in Table VII. C. The power spectra in redshift space
As all the necessary integral formulas are derived in the pre-vious subsection, we are ready to write down the explicit for-mula of the power spectrum in redshift space. The decompo-sition of Eq. (31) is applicable in redshift space, and it is suf-ficient to give the explicit expressions for the functions Π ( k ), R XY ( k ), Q XY ( k ), S XY ( k ) in redshift space. These functionsdepends on not only the magnitude k but also the directionrelative to the lines of sight.We employ the distant-observer approximation for theredshift-space distortions, and the lines of sight are fixed inthe direction of the third axis, ˆ z . Lagrangian kernels are re-placed according to Eq. (15) in the formulas of propagatorsEqs. (8) and (16). In those formulas, the Lagrangian kernelsappear only in the form of k · L ( n ) . With the linear mapping ofEq. (15), we have k · L ( n ) → k · L s( n ) = ( k + n f µ k ˆ z ) · L ( n ) , (54) where µ = ˆ k · ˆ z , (55)and ˆ k = k / k . Thus, in the distant-observer approximation ofthis paper, the direction dependence comes into the formulasonly through the direction cosine of Eq. (55). We denote thefunctions of Eqs. (32)–(34) as R XY ( k , µ ), Q XY ( k , µ ), S XY ( k , µ )in the following.Substituting Eq. (54) into Eqs. (7), (8) and (16), one cansee that evaluations of Eqs. (32)–(34) are straightforward bymeans of the integral formulas in the previous subsection. Theresults are explicitly presented in the following.The vertex resummation function of Eq. (7) can be evalu-ated by applying the same technique of the previous section.The relevant integral is Z d p (2 π ) h k · L s(1) ( p ) i P L ( p ) = ( k i + f µ k ˆ z i )( k j + f µ k ˆ z j ) Z d p (2 π ) p i p j p P L ( p ) , (56)and the last integral is proportional to the Kronecker’s delta.The proportional factor is evaluated by taking contraction ofthe indices. Consequently, we have Π ( k , µ ) = exp ( − h + f ( f + µ i k π Z d pP L ( p ) ) . (57)The two-point propagator of Eq. (8) with the substitutionof Eq. (54) is evaluated by means of Table II, where R n ( k )functions are defined by Eq. (49) and Table III. The result isgiven byˆ Γ (1) X ( k , µ ) = + c (1) X + R + R + R X − R X + " + R + R + R X − R X f µ − R f µ + " R + R f µ . (58)The quantities c (1) X , R n , R Xn on LHS are functions of k , althoughthe arguments are omitted. The component R XY of Eq. (32)0 F ( k , k ) Z k = k F ( k , k ) P L ( k ) P L ( k ) Diagram L (2) i ( k , k ) L (2) j ( k , k ) 949 k i k j k Q ( k ) L (1) i ( k ) L (1) j ( k ) L (2) k ( k , k ) 314 ( k i k j − k δ ij ) k k k Q ( k ) + k i k j k k k Q ( k ) L (1)( i ( k ) L (1) j ( k ) L (1) k ( k ) L (1) l ) ( k ) 38 k i k j k k k l − k δ ( ij k k k l ) + k δ ( ij δ kl ) k Q ( k ) − k i k j k k k l − k δ ( ij k k k l ) k Q ( k ) + k i k j k k k l k Q ( k ) L (1) i ( k ) L (2) j ( k , k ) c (1) X ( k ) 37 k i k j k Q X ( k ) L (1)( i ( k ) L (1) j ( k ) L (1) k ) ( k ) c (1) X ( k ) − k i k j k l − k δ ( ij k k ) k Q X ( k ) + k i k j k k k Q X ( k ) L (2) ( k , k ) c (2) X ( k , k ) 37 k k Q X ( k ) L (1) i ( k ) L (1) j ( k ) c (2) X ( k , k ) 12 k i k j − k δ ij k Q X ( k ) + k i k j k Q X ( k ) L (1) i ( k ) L (1) j ( k ) c (1) X ( k ) c (1) Y ( k ) 12 k i k j − k δ ij k Q XY ( k ) + k i k j k Q XY ( k ) L (1) i ( k ) L (1) j ( k ) c (1) X ( k ) c (1) Y ( k ) − k i k j − k δ ij k Q XY ( k ) + k i k j k Q XY ( k ) L (1) ( k ) c (1) X ( k ) c (2) Y ( k , k ) k k Q XY ( k ) c (2) X ( k , k ) c (2) Y ( k , k ) Q XY ( k )TABLE IV: Integral formulas for one-loop corrections, which are related to convolving three-point propagators. In the third and fifth formulas,the spatial indices are completely symmetrized. We denote Q n ( k ) = Q XYn ( k ) for n = , , ,
4, as these functions are independent on the bias,and Q Xn ( k ) = Q XYn ( k ) for n = , , , ,
9, as these functions are only dependent on the bias of objects X . is straightforwardly obtained by the above result of the two-point propagator: R XY ( k , µ ) = ˆ Γ (1) X ( k , µ ) ˆ Γ (1) Y ( k , µ ) P L ( k ) . (59)The tree-level contribution of the above equation is given by( b X + f µ )( b Y + f µ ) P L ( k ) where b X = + c (1) X , and Kaiser’s lin-ear formula of redshift-space distortions for the power spec-trum [57] is exactly reproduced. In calculating the mass powerspectrum, X = Y = m, we only need terms with R ( k ) and R ( k ) and other terms R X ( k ) and R X ( k ) vanish since c ( n ) X = Q XY ( k , µ ) of Eq. (33) is similarly evaluated,while the number of terms are larger. The result is given by Q XY ( k , µ ) = X n , m µ n f m h q XYnm ( k ) + q YXnm ( k ) i , (60) where q XY = Q + Q + Q + Q X + Q X + Q X + Q X + Q XY + Q XY + Q XY + Q XY , (61) q XY = Q + Q + Q + Q X + Q X + Q X + Q X + Q XY + Q XY + Q XY , (62) q XY = − Q + Q + Q X − Q X − Q XY + Q XY , (63) q XY = Q + Q − Q + Q + Q X − Q X + Q X + Q X + Q X + Q XY + Q XY − Q XY + Q XY , (64) q XY = − Q + Q + Q X , (65)1 n Q XYn ( k , k )[ k = k + k ] ˜ Q XYn ( r , x ) " y = (1 + r − rx ) / ,µ = ( x − r ) /y − k · k k k ! r (1 − x ) y k · k )( k · k ) k k − k · k k k ! rx (1 − rx )(1 − x ) y k − k · k )( k · k ) k k − k · k k k ! (1 − rx + r x )(1 − x ) y k · k ) ( k · k ) k k x (1 − rx ) y k · k k − k · k k k ! c (1) X ( k ) rx (1 − x ) y c (1) X ( k y )6 k − k · k k − k · k k k ! c (1) X ( k ) (1 − rx )(1 − x ) y c (1) X ( k y )7 k · k k ! k · k k c (1) X ( k ) x (1 − rx ) y c (1) X ( k y )8 − k · k k k ! c (2) X ( k , k ) r (1 − x ) y c (2) X ( kr , k y ; µ )9 ( k · k )( k · k ) k k c (2) X ( k , k ) rx (1 − rx ) y c (2) X ( kr , k y ; µ )10 − k · k k k ! c (1) X ( k ) c (1) Y ( k ) r (1 − x ) y c (1) X ( kr ) c (1) Y ( k y )11 ( k · k )( k · k ) k k c (1) X ( k ) c (1) Y ( k ) rx (1 − rx ) y c (1) X ( kr ) c (1) Y ( k y )12 k k − k · k kk ! c (1) X ( k ) c (1) Y ( k ) (cid:16) − x (cid:17) c (1) X ( k y ) c (1) Y ( k y )13 k · k k ! c (1) X ( k ) c (1) Y ( k ) x c (1) X ( k y ) c (1) Y ( k y )14 k · k k c (1) X ( k ) c (2) Y ( k , k ) rx c (1) X ( k y ) c (2) Y ( kr , k y ; µ )15 c (2) X ( k , k ) c (2) Y ( k , k ) r c (2) X ( kr , k y ; µ ) × c (2) Y ( kr , k y ; µ )TABLE V: Integrands for functions Q XYn ( k ) of Eq. (51). F ( k , k ) Z k = k F ( k , k ) B L ( k , k , k ) Diagram L (2) ( k , k ) 37 k k S ( k ) L i ( k ) L j ( k ) 12 k i k j − k δ ij k S ( k ) + k i k j k S ( k ) L (1) ( k ) c (1) X ( k ) k k S X ( k ) c (2) X ( k , k ) S X ( k )TABLE VI: Integral formulas for one-loop corrections, which arerelated to convolving three-point propagators with the linear bispec-trum. We denote S ( k ) = S X ( k ) and S ( k ) = S X ( k ), as these func-tions are independent on the bias. n S Xn ( k , k )[ k = k + k ] ˜ S Xn ( r , x ) " y = (1 + r − rx ) / ,µ = ( x − r ) /y − k · k k k ! r (1 − x ) y k · k )( k · k ) k k rx (1 − rx ) y k · k k c (1) X ( k ) rx c (1) X ( k y )4 c (2) X ( k , k ) r c (2) X ( kr , k y ; µ )TABLE VII: Integrands for functions S Xn ( k ) of Eq. (53). q XY = Q , (66) q XY = Q + Q − Q + Q − Q X + Q X , (67) q XY = − Q + Q , (68) q XY = Q − Q + Q , (69)and other q XYnm ( k )’s which are not listed above all vanish. Thequantities Q n , Q Xn , Q XYn are functions of k , although the ar-guments are omitted. The Q n functions of n = , . . . , , . . . , ,
15 are symmetric with respect to X ↔ Y , whilethose of n = , . . . , ,
14 are not. In calculating cross powerspectra, X , Y , the symmetrization with respect to XY inEq. (60) is necessary. In calculating auto power spectra, X = Y , two terms in the square bracket in Eq. (60) are thesame, and can be replaced by 2 q XXnm ( k ). In calculating themass power spectrum, X = Y = m, we only need terms with Q ( k ) , . . . , Q ( k ) and other terms Q X ( k ) , . . . , Q XY ( k ) all vanishsince c ( n ) X = S XY ( k , µ ) of Eq. (34) is similarly evaluated.The result is given by S XY ( k , µ ) =
12 ˆ Γ (1) X ( k , µ ) " S + S + S Y + S Y + S + S + S Y ! f µ − S f µ + S + S ! f µ + ( X ↔ Y ) . (70)The quantities S n , S Xn and S Yn are functions of k , although thearguments are omitted. The normalized two-point propagatorˆ Γ (1) X ( k , µ ) in Eq. (70) can be replaced by the tree-level term,1 + c (1) X + f µ , because the rest of the factor is already of one-loop order.All the necessary components to calculate the power spec-trum of Eq. (31) in redshift space, P XY ( k , µ ) = Π ( k , µ ) (cid:2) R XY ( k , µ ) + Q XY ( k , µ ) + S XY ( k , µ ) (cid:3) , (71)2are provided above, i.e., Eqs. (57), (59), (60) and (70). Numer-ical integrations of Eqs. (49), (51) and (53) are not di ffi cult,once the model of renormalized bias functions c ( n ) X and pri-mordial spectra P L ( k ), B L ( k , k , k ) are given. The last term S XY ( k ) is absent in the case of Gaussian initial conditions. D. Evaluating correlation functions
We have derived full expressions of power spectra of biasedtracers in the one-loop approximation. The correlation func-tions are obtained by Fourier transforming the power spec-trum. In real space, the relation between the correlation func-tion ξ XY ( r ) and the power spectrum P XY ( k ) is standard: ξ XY ( r ) = Z ∞ k dk π j ( kr ) P XY ( k ) , (72)where j l ( z ) is the spherical Bessel function. For a numeri-cal evaluation, it is convenient to first tabulate the values ofpower spectrum P XY ( k ) of Eq. (47) in performing the one-dimensional integration of Eq. (72).In redshift space, multipole expansions of the correlationfunction are useful [58–60]. For reader’s convenience, wesummarize here the set of equations which is useful to nu-merically evaluate the correlation functions in redshift spacefrom the iPT formulas of power spectra derived above. Themultipole expansion of the power spectrum in redshift space, P XY ( k , µ ), with respect to the direction cosine relative to linesof sight has a form, P XY ( k , µ ) = ∞ X l = p lXY ( k ) P l ( µ ) , (73)where P l ( µ ) is the Legendre polynomial. Inverting the aboveequation by the orthogonal relation of Legendre polynomials,the coe ffi cient p lXY ( k ) is given by p lXY ( k ) = l + Z − d µ P l ( µ ) P XY ( k , µ ) . (74)Because of the distant-observer approximation, the index l only takes even integers.The dependence on the direction µ of our power spectrum, P XY ( k , µ ) of Eq. (71), appears in forms of µ n e − αµ where n = , , , . . . are non-negative integers. It is possible to ana-lytically reduce the integral of Eq. (74) by using an identity Z − d µ µ n e − αµ = α − n − / γ n + , α ! , (75)where γ ( z , p ) is the lower incomplete gamma function definedby γ ( z , p ) = Z p e − t t z − dt . (76)Although the number of terms is large, it is straightforward toobtain the analytic expression of p lXY ( k ) of Eq. (74) in terms of Q n ( k ), R n ( k ), S n ( k ), c (1) X ( k ), and the lower incomplete gammafunction. Computer algebra like M athematica should be use-ful for that purpose. Alternatively, it is feasible to numer-ically integrate the one-dimensional integral of Eq. (74) foreach k , once the functions Q n ( k ), R n ( k ), S n ( k ), c (1) X ( k ) are pre-computed and tabulated. The latter method is much simplerthan the former.The multipole expansion of the correlation function in red-shift space, ξ XY ( r , µ ), with respect to the direction cosine rela-tive to lines of sight is given by ξ XY ( r , µ ) = ∞ X l = ξ lXY ( r ) P l ( µ ) , (77) ξ lXY ( r ) = l + Z − d µ P l ( µ ) ξ XY ( r , µ ) . (78)Since the power spectrum P XY ( k , µ ) and the correlation func-tion ξ XY ( k , µ ) are related by a three-dimensional Fourier trans-form, corresponding multipoles are related by [58] ξ lXY ( r ) = i − l Z ∞ k dk π j l ( kr ) p lXY ( k ) . (79)Since l is an even integer, the above equation is a real number.Once the multipoles of power spectrum p lXY ( k ) are evaluatedby either method described above and tabulated as a functionof k , we have a multipoles of the correlation function ξ lXY ( r ) bya simple numerical integration of Eq. (79). Because the ver-tex resummation factor exponentially damps for high- k , thenumerical integration of Eq. (79) is stable enough. E. A sample comparison with numerical simulations
The purpose of this paper is to analytically derive explicitformulas of one-loop power spectra in iPT, and detailed anal-ysis of numerical consequences of derived formulas is beyondthe scope of paper. In this subsection, we only present a sam-ple comparison with halos in N -body simulations. In Fig. 4,correlation functions in real space are presented.The numerical halo catalogs in this figure are the same asthe ones used in Sato & Matsubara (2011; 2013) [25, 61].The N -body simulations are performed by a publicly avail-able tree-particle mesh code, Gadget2 [62] with cosmologicalparameters Ω M = . Ω Λ = . Ω b = . h = . n s = . σ = .
80. Other simulation parameters are givenby the box size L box = h − Mpc, the number of particles N p = , initial redshift z ini =
36, the softening length r s = h − kpc, and the number of realizations N run = z = .
0, and the mass range of the selected halos is4 . × h − M ⊙ ≤ M ≤ . × h − M ⊙ .In the upper panel, the auto- and cross-correlation func-tions of mass and halos, ξ hh , ξ mh , ξ mm , are plotted. Since3 r ξ XY ( r ) [ h - M p c ] M h = (4.11-12.32) x 10 [ h -1 M ⊙ ] z = 1 ξ mm ξ mh ξ hh b i a s r [ h -1 Mpc]
FIG. 4: The correlation functions in real space. The prediction ofone-loop iPT is compared with numerical simulations. The results ofmass auto-correlation, ξ mm , halo auto-correlation, ξ hh , and mass-halocross-correlation, ξ mh are compared in the above panel. Dashed linesrepresent the predictions of linear theory, solid lines represent thoseof one-loop iPT, and symbols with error bars represent the results ofnumerical simulations. In the bottom panel, scale-dependent bias pa-rameters, which are defined by p ξ hh /ξ mm for auto-correlations and ξ mh /ξ mm for cross-correlations, are plotted. Predictions of iPT aregiven by solid line for auto-correlations and by dotted line for cross-correlations. These two lines are almost overlapped and indistin-guishable. The horizontal dashed line corresponds to the predictionof linear theory with a constant bias factor. the amplitude of linear halo bias, b L1 , predicted by the peak-background split in the simple halo model, does not accuratelyreproduce the value of halo bias in numerical simulations, weconsider the value of smoothing radius R (or mass M ) in thesimple model of the renormalized bias function as a free pa-rameter. We approximately treat this freely fitted radius as arepresentative value, and ignore the finiteness of mass range,e.g., Eq. (29). The same value of radius is used both in auto-and cross-correlations, ξ hh and ξ mh . We use a Gaussian win-dow function W ( kR ) = e − k R / , while the shape of the win-dow function does not change the predictions on large scales.There is no fitting parameter for the mass auto-correlationfunction ξ mm . As obviously seen in the Figure, the predictionsof one-loop iPT agree well with N -body simulations on scales & h − Mpc where the perturbation theory is applicable.In the lower panel, scale-dependent bias parameters areplotted. Two definitions of linear bias factor, p ξ hh /ξ mm and ξ mh /ξ mm , are presented. The iPT predicts almost similar curves for both definitions, and slight scale-dependence of lin-ear bias on BAO scales is suggested. Such scale-dependenceis already predicted also in models of Lagrangian local bias[23]. Unfortunately, the N -body simulations used in this com-parison are not su ffi ciently large to quantitatively confirm theprediction for the scale-dependent bias. However, a recent N -body analysis of the MICE Grand Challenge run [67] showsqualitatively the same scale-dependence. This observation ex-emplifies unique potentials of the method of iPT. IV. RELATION TO PREVIOUS WORKA. Lagrangian resummation theory
It is worth mentioning here the relation between the aboveformulas and previous results of Ref. [23], in which the La-grangian resummation theory (LRT) with local Lagrangianbias is developed. The iPT is a superset of LRT. The resultsof Ref. [23] can be derived from the formulas in this paperby restricting to the local Lagrangian bias and by neglectingcontributions from the primordial non-Gaussianity, althoughthe way to derive the same results is apparently di ff erent. Thedefinitions of Q n and R n functions are somehow di ff erent inRef. [23] from those in this paper. The notational correspon-dences are summarized in Table VIII.In Ref. [23], the linear density field δ L and the biased den-sity field in Lagrangian space δ L X are related by a local relation δ L X ( q ) = F ( δ L ( q )) in Lagrangian configuration space. Fouriertransforming this relation, the renormalized bias functions ofEq. (3) in models of local Lagrangian bias reduce to scale-independent parameters, c ( n ) X = D F ( n ) E , (80)where F ( n ) = ∂ n F /∂δ L n is the n th derivative of the function F ( δ L ). Thus the renormalized bias functions are independenton wavevectors in the case of local bias, and we have h F ′ i = c (1) X and h F ′′ i = c (2) X , etc.It is explicitly shown that the results of Ref. [23] are exactlyreproduced by setting X = Y and S XY =
0, expanding theproduct ( ˆ Γ (1) X ) in R XX and adopting the replacement of vari-ables according to the Table VIII. In making such a compari-son, the product Γ (1) X Γ (1) Y should be expanded up to the second-order terms in P L ( k ) (i.e., one-loop terms). Thus, Eq. (31) isconsidered as a nontrivial generalization of the previous for-mula of Ref. [23]. Another previous formula of Ref. [22] is aspecial case of Ref. [23] without biasing. As a consequence,setting c ( n ) X = S XY = B. Scale-dependent bias and primordial non-Gaussianity
Contributions from the primordial bispectrum, if any, areincluded in S XY . In the cases of X = Y and X , Y = m,the relations between the primordial bispectrum and scale-dependent bias are already analyzed in Ref. [36] with gen-4 This paper Ref. [23] Ref. [36] R ( k ) R ( k ) / P L ( k ) - R ( k ) R ( k ) / P L ( k ) - R X ( k ) h F ′ i [ R ( k ) + R ( k )] / P L ( k ) - R X ( k ) 0 - Q ( k ) Q ( k ) - Q ( k ) Q ( k ) - Q ( k ) Q ( k ) − Q ( k ) - Q ( k ) Q ( k ) - Q X ( k ) h F ′ i Q ( k ) - Q X ( k ) h F ′ i Q ( k ) - Q X ( k ) h F ′′ i Q ( k ) - Q X ( k ) h F ′′ i Q ( k ) - Q XX ( k ) h F ′ i Q ( k ) - Q XX ( k ) h F ′ i Q ( k ) - Q XX ( k ) h F ′ i Q ( k ) - Q XX ( k ) h F ′ i Q ( k ) - Q XX ( k ) h F ′ ih F ′′ i Q ( k ) - Q XX ( k ) h F ′′ i Q ( k ) - S ( k ) - R ( k ) S ( k ) - 2 R ( k ) − R ( k ) S X ( k ) - Q ( k ) / S X ( k ) - Q ( k )TABLE VIII: When the local Lagrangian bias is employed, andprimordial non-Gaussianity is not considered, the expression of theauto power spectrum ( X = Y ) in this paper reproduces the result ofRef. [23]. When contributions from the primordial non-Gaussianityare extracted, the results of Ref. [36] are reproduced. Correspon-dences of the functions defined in this paper and those defined inRefs. [23, 36] are provided in this Table. The renormalized bias func-tions are constants in local bias models, and denoted by h F ′ i = c (1) X and h F ′′ i = c (2) X in Ref. [23]. erally nonlocal Lagrangian bias. In the presence of primordialbispectrum, the scale-dependent bias emerges on very largescales [4, 5]. The iPT generalizes the previous formulas ofthe scale-dependent bias with less number of approximations.The previous formulas of scale-dependent bias [4, 68–70],which are derived in the approximation of peak-backgroundsplit for the halo bias, are exactly reproduced as limiting casesof the formula derived by iPT [36]. It should be noted that theformula of scale-dependent bias in the framework of iPT is notrestricted to the particular model of halo bias. Therefore theiPT provides the most general formula of the scale-dependentbias among previous work. The correspondence between thefunctions defined in Ref. [36] and those in this paper is sum-marized in Table VIII.In this paper, the cross power spectrum of two di ff erentlybiased objects, X and Y are considered in general. Onecan derive the scale-dependent bias of cross power spec-trum P XY ( k ) as illustrated below. In the following argument,the redshift-space distortions are neglected for simplicity, al-though it is straightforward to include them. We define thescale-dependent bias ∆ b XY of cross power spectrum by P XY ( k ) = [ b XY ( k ) + ∆ b XY ( k )] P m ( k ) , (81) where P m ( k ) is the matter power spectrum, and b XY ( k ) is thelinear bias factor of the cross power spectrum without con-tributions from primordial non-Gaussianity. In the lowest-order approximation, b XY ( k ) = [ b X ( k ) b Y ( k )] / , where b X ( k )and b Y ( k ) are linear bias factors of objects X and Y , respec-tively. When higher orders of ∆ b XY are neglected, we have ∆ b XY = b XY ( k ) ∆ P XY ( k ) P G XY ( k ) − ∆ P m ( k ) P Gm ( k ) , (82)where P G XY ( k ) and P Gm ( k ) are the Gaussian parts of crosspower spectrum and the auto power spectrum of mass, respec-tively, and ∆ P XY ( k ) and ∆ P m ( k ) are corresponding contribu-tions from primordial non-Gaussianity so that the full spec-tra are given by P XY ( k ) = P G XY ( k ) + ∆ P XY ( k ) and P m ( k ) = P Gm ( k ) + ∆ P m ( k ).On su ffi ciently large scales, nonlinear gravitational evolu-tions are not important, and dominant contributions to themulti-point propagators are asymptotically given by [36]ˆ Γ (1) X ( k ) ≈ b X ( k ) , (83)ˆ Γ (2) X ( k , k ) ≈ c (2) X ( k , k ) , (84)where b X ( k ) = + c (1) X ( k ) is the linear bias factor of object X .In this limit, Eq. (34) reduces to S XY ( k ) ≈ b X ( k ) Z k = k c (2) Y ( k , k ) B L ( k , k , k ) . (85)In the lowest-order approximation with a large-scale limit, thepredictions of iPT are given by P Gm ( k ) ≈ P L ( k ) , P G XY ( k ) ≈ b X ( k ) b Y ( k ) P L ( k ) , (86) ∆ P m ( k ) ≈ , ∆ P XY ( k ) ≈ S XY ( k ) , (87)and we have b XY ( k ) = [ b X ( k ) b Y ( k )] / as previously noted.Substituting these equations into Eq. (82), we have ∆ b XY ( k ) ≈ S XY ( k )2 √ b X ( k ) b Y ( k ) P L ( k ) . (88)This equation gives the general formula of the scale-dependent bias for cross power spectra in general.In a case of the auto-power spectrum with X = Y , theabove equation reduces to a known result [36], ∆ b X ≈ S XX ( k ) / [2 b X ( k ) P L ( k )]. Previous formulas of the scale-dependent bias in the approximation of peak-background splitare reproduced in limiting cases of this result, adopting therenormalized bias functions c ( n ) X in the nonlocal model ofhalo bias described in Sec. II B. The integral of Eq. (85) isscale-dependent according to the squeezed limit of the pri-mordial bispectrum, B L ( k , k , k ) with k ≪ k , k . Thus,the scale-dependencies of the bias in cross power spectra aresimilar to those in auto power spectra. Amplitudes of thescale-dependent bias are di ff erent. When the primordial non-Gaussianity are actually detected, scale-dependent biases ofcross power spectra of multiple kinds of objects would be use-ful to cross-check the detection.5 CC k k n k FIG. 5: Diagrammatic representation of the resummation scheme ofCLPT [53]. The original CLPT does not include the e ff ects of nonlo-cal bias, and can be easily extended to include them by applying theformalism of iPT and the resummation of this type of diagrams. = +C + +++ · · · + FIG. 6: Ingredients of the displacement correlator. All the diagramsup to one-loop approximation are shown. These diagrams are re-summed in CLPT.
C. Convolution Lagrangian perturbation theory
Recently, a further resummation method, called the con-volution Lagrangian perturbation theory (CLPT) [53], is pro-posed on the basis of LRT. The implementation of the CLPTactually improves the nonlinear behavior on small scaleswhere the original LRT breaks down. The proposed CLPTis based on the LRT in which only local Lagrangian bias canbe incorporated.Under the light of iPT, the resummation scheme of CLPTcorresponds to resumming the diagrams depicted by Fig. 5.The shaded ellipse with the symbol ‘C’ represents a summa-tion of all the possible connected diagrams. The actual ingre-dients are shown in Fig. 6 up to the one-loop approximation.The corresponding function of this Figure is given by˜ Λ i j ( k ) = − L (1) i ( k ) L (1) j ( k ) P L ( k ) − Z d p (2 π ) L (1)( i ( k ) L (3) j ) ( k , p , − p ) P L ( p ) P L ( k ) − Z k = k L (2) i ( k , k ) L (2) j ( k , k ) P L ( k ) P L ( k ) − L (1)( i ( k ) Z k = k L (2) j ) ( k , k ) B L ( k , k , k ) . (89)The indices i , j are symmetrized on RHS of the above equa-tion. This function is the same as C i j ( k ) in Ref. [23], and − C i j ( k ) in Ref. [22]. We refer to the graph of Fig. 6 andEq. (89) as “displacement correlator” below. To the full or- ders, the displacement correlator ˜ Λ i j ( k ) is given by D ˜ Ψ i ( k ) ˜ Ψ j ( k ′ ) E c = − (2 π ) δ ( k + k ′ ) ˜ Λ i j ( k ) . (90)The expression of Eq. (89) is also obtained from this equa-tion, adopting the one-loop approximation in the perturbativeexpansion of Eq. (9).Using the displacement correlator, the diagrams of Fig. 5can be represented by a convolution integral of the form ∞ X n = ( − n n ! k i · · · k i n k j · · · k j n Z k ··· n = k ′ ˜ Λ i j ( k ) · · · ˜ Λ i n j n ( k n ) = Z d q e − i k ··· n · q exp h − k i k j Λ i j ( q ) i , (91)where k is the wave vector of the nonlinear power spectrum P XY ( k ) to evaluate, k ··· n = k + · · · k n is the total wave vectorthat flows through the resummed part of Fig. 5, and Λ i j ( q ) = Z d k (2 π ) e i k · q ˜ Λ i j ( k ) (92)is the displacement correlator in configuration space. The con-volution integral of Eq. (91) contributes multiplicatively to theevaluation of the power spectrum P XY ( k ).The displacement correlator in configuration space,Eq. (92), is given by the full-order displacement field Ψ ( q )as Λ i j ( q ) = − D Ψ i ( q ) Ψ j ( q ) E c , (93)where q = q − q . This function is denoted as C i j ( q ) / C CLPT i j ( q ) = Λ i j ( q ) (94)In the CLPT, the vertex resummation factor is included in afunction A i j ( q ) = B i j + C i j ( q ) of their notation, where B i j = σ η δ i j and σ η = h| Ψ | i /
3. Thus we have a correspondence, A CLPT i j ( q ) = σ η δ i j + Λ i j ( q ) . (95)The first term in the LHS corresponds to the vertex resum-mation in iPT, and is kept exponentiated in both original LRTand CLPT. The second term is kept exponentiated in CLPTand expanded in the original LRT formalism.When the Lagrangian local bias is assumed ( c (1) X = h F ′ i , c (2) X = h F ′′ i ,...), and the convolution resummation of Fig. 5is taken into account in the iPT, the formalism of CLPT isexactly reproduced. When the Lagrangian nonlocal bias is al-lowed in the iPT with the convolution resummation, we obtaina natural extension of the CLPT without restricting to modelsof local Lagrangian bias.Extending this diagrammatic understanding of CLPT in theframework of iPT, it is possible to consider further convo-lution resummations that are not included in the formula-tion of CLPT. In the CLPT, only connected diagrams withtwo wavy lines (i.e., Fig. 6) are resummed. We define the6 = +C ++ + · · · FIG. 7: Connected diagrams with three wavy lines up to tree-levelapproximation. These diagrams are not resummed in CLPT. three-point correlator of displacement, ˜ Λ i jk ( k , k , k ) where k + k + k = by the connected diagrams with three wavylines as shown in Fig. 7. This function is given by D ˜ Ψ i ( k ) ˜ Ψ j ( k ) ˜ Ψ j ( k ) E c = − i (2 π ) δ ( k + k + k ) ˜ Λ i jk ( k , k , k ) . (96)to the full order. This three-point correlator ˜ Λ i jk is the same as − C i jk in Ref. [23] and − iC i jk in Ref. [22]. In a similar way asFig. 5 and Eq. (91), including resummations of the three-pointcorrelator modifies the convolution integral of Eq. (91) as Z d q e − i k ′ · q exp h − k i k j Λ i j ( q ) + k i k j k k Λ i jk ( q ) i , (97)where k ′ is the total wave vector that flows through the re-summed part, and Λ i jk ( q ) = Z d k (2 π ) e i k · q Z d p (2 π ) ˜ Λ i jk ( k , − p , p − k ) . (98)One can similarly consider four- and higher-point convolutionresummations, which naturally arise in two- or higher-loopapproximations. However, it is not obvious whether or notprogressively including such kinds of convolution resumma-tions actually improves the description of the strongly non-linear regime. Comparisons with numerical simulations arenecessary to check. Detailed analysis of this type of exten-sions in the iPT is beyond the scope of this paper, and can beconsidered as an interesting subject for future work. D. Renormalized perturbation theory
Recent progress in improving the standard perturbation the-ory (SPT) was triggered by a proposition of the renormalizedperturbation theory (RPT) [45, 46]. Although this theory isformulated in Eulerian space, there are many common fea-tures with iPT in which resummations in terms of Lagrangianpicture play an important role. Below, we briefly discuss thesecommon features. However, one should note that purposesof developing RPT and iPT are not the same. The RPT for-malism mainly focuses on describing nonlinear evolutions of density and velocity fields of matter, extrapolating the pertur-bation theory in Eulerian space. The iPT formalism mainlyfocuses on consistently including biasing and redshift-spacedistortions into the perturbation theory from the first principleas possible. The RPT (and its variants) is properly applicableonly to unbiased matter clustering in real space (even thoughthere are phenomenological approaches with freely fitting pa-rameters, such as the model of Ref. [52], for example). Thus,the resummation methods in RPT can be compared only witha degraded version of iPT without biasing and redshift-spacedistortions.
1. Propagators in high-k limit
An important ingredient of RPT is an interpolation schemebetween low- k and high- k limits of the multi-point propagatorof mass Γ ( n )m ( k , . . . , k n ) with k = k ··· n . Based on the Eulerianpicture of perturbation theory, the high- k limit of the propaga-tor is analytically evaluated as [46, 47] Γ ( n )m ( k , . . . , k n ) ≈ exp − k σ d2 ! F n ( k , . . . , k n ) , (99)in the fastest growing mode of density field, where σ d2 = π Z dk ′ P L ( k ′ ) , (100) F n is the n th-order kernel function of SPT, and k = | k ··· n | .Although decaying modes and the velocity sector are also in-cluded in the original RPT formalism [45, 46], we neglectthem for our purpose of comparison between RPT and iPT.The multi-point propagator in the iPT has the form ofEqs. (5) and (6) with full orders of perturbations. In the unbi-ased case, X = m, we have Γ ( n )m ( k , . . . , k n ) = Π ( k ) ˆ Γ ( n )m ( k , . . . , k n ) , (101)where Π ( k ) = h e − i k · Ψ i is the vertex resummation factor. TheEq. (101) should also have the same high- k limit as Eq. (99),since we are dealing with the same quantities. Although ex-plicitly proving this property in the framework of iPT is be-yond the scope of this paper, a natural expectation arises thatthe high- k limit of the resummation factor Π ( k ) is given by theexponential prefactor of Eq. (99), as discussed below.In the high- k limit, the factor e − i k · Ψ which is averaged overin the resummation factor strongly oscillates as a function ofdisplacement field Ψ . Consequently, large values of the dis-placement field do not contribute to the statistical average, anddominant contributions come from a regime | Ψ | . k − . Inthe high- k limit, this condition corresponds to a weak fieldlimit of the displacement field, which is well described by theZel’dovich approximation, ˜ Ψ ( k ) ≈ ( i k / k ) δ L ( k ). Assuminga Gaussian initial condition, higher-order cumulants of dis-placement field in the Zel’dovich approximation are absent inEq. (6). Since h Ψ i Ψ j i c = δ i j h| Ψ | i / h ( k · Ψ ) i c = k h| Ψ | i / = k σ d2 inthe Zel’dovich approximation. Thus we naturally expect Π ( k ) = D e − i k · Ψ E ≈ exp − k σ d2 ! , (102)7in the high- k limit, which agrees with the exponential prefac-tor of Eq. (99).Assuming that the above expectation is correct, Eqs. (99)and (101) suggests the high- k limit of normalized propagatoris given by ˆ Γ ( n )m ( k , · · · , k n ) ≈ F n ( k , · · · , k n ) , (103)i.e., the high- k limit of the normalized propagator is given bytree diagrams, and contributions from whole loop correctionsare subdominant. This is a nontrivial statement, since the nor-malized propagator contains non-zero loop corrections in eachorder. For example, taking the limit k → ∞ in Eq. (37) of theone-loop approximation, we haveˆ Γ (1)m ( k ) ≈ + Z ∞ p d p π P L ( p ) , (104)which is apparently di ff erent from F =
1. Actually the inte-gral in the RHS is logarithmically divergent for a spectrum ofcold-dark-matter type, which has an asymptote P L ( k ) ∝ k − for k → ∞ . Thus Eq. (103) does not apparently hold when theloop corrections are truncated at any order. Thus, Eq. (103)has a highly non-perturbative nature. This situation is natu-ral, because the high- k limit of Eq. (102) is also highly non-perturbative. When the equation is truncated at any order,a high- k limit gives divergent terms, while the whole factorapproaches to zero. The same is true for the high- k limitin the RPT formalism, Eq. (99). Provided that Eq. (99) istrue, Eqs. (102) and (103) are the same statement because ofEq. (101), which is a definition of the normalized propagator.The above argument is readily generalized in the case ofnon-Gaussian initial conditions. In the high- k limit of the RPTformalism, the exponential factor in Eq. (99) is replaced by[48]exp − k σ d2 ! → D e i α ( k ) E = exp ∞ X n = i n n ! h [ α ( k )] n i c , (105)where α ( k ) ≡ − i Z d p (2 π ) k · p p δ L ( p ) . (106)Comparing these equations of RPT with Eqs. (6), (9) of iPT,there are correspondences, α ( k ) = − i k · Z d p (2 π ) L (1) ( p ) δ L ( p ) = − k · Ψ (1) , (107) D e i α ( k ) E = D e − i k · Ψ (1) E , (108)where Ψ (1) is the linear displacement field in configurationspace at the origin. Since Eq. (101) holds in non-Gaussianinitial conditions as well, the high- k limit of iPT, Eq. (102), isreplaced by Π ( k ) ≈ D e − i k · Ψ (1) E = exp ∞ X n = ( − i ) n n ! D(cid:16) k · Ψ (1) (cid:17) n E c , (109)which agrees with the the replacement of RPT, Eq. (105).Since only the exponential factor is replaced in Eqs. (99) and(102), the high- k limit of Eq. (103) does not change even inthe case of non-Gaussian initial conditions.
2. Nonlinear interpolation I: R eg PT In the RPT formalism, the nonlinear propagator is approxi-mated by analytically interpolating the behaviors in the high- k limit and the low- k limit [46, 47, 50]. There are at least twoprescriptions for the interpolation. An interpolation scheme ofRefs. [49, 51, 52], which is called R eg PT, uses an prescriptionfor the multi-point propagator truncated at the N -loop order as Γ ( n )RegPT = (cid:16) F n + δΓ ( n )1-loop + δΓ ( n )2-loop + · · · + δΓ ( n ) N -loop + C . T . (cid:17) × exp − k σ d2 ! (110)where δΓ ( n ) M -loop is the M -loop correction term of the propaga-tor, and C . T . is a counterterm to match the N -loop expressionis exact in both limits, i.e.,C . T . = k σ d2 + k σ d4 + · · · + N ! k σ d2 ! N F n + k σ d2 + · · · + N − k σ d2 ! N − δΓ ( n )1-loop + · · · + k σ d2 δΓ ( n )( N − . (111)The tree-level multi-point propagators are the same as thekernel functions in SPT, i.e., Γ ( n )tree = F n . It is apparent thatEq. (110) has the correct low- k limit. In the high- k limit, wehave δΓ ( n ) M -loop ≈ ( − k σ / M F n / M ! according to Eq. (99). Inthis limit, it can be shown by induction that all the loop correc-tions in the first parentheses of Eq. (110) including the counterterm remarkably cancel each other, leaving only the tree-levelcontribution F n . Thus Eq. (110) also has the correct high- k limit, Γ ( n )RegPT → F n exp( − k σ d2 /
2) for k → ∞ . The R eg PTprescription of Eq. (110) can be re-expressed in a more com-pact form including the counterterm as Γ ( n )RegPT = exp k σ d2 ! ∞ X N = δΓ ( n ) N -loop (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) truncated × exp − k σ d2 ! , (112)where δΓ ( n )0-loop ≡ F n , and [ · · · ] | truncated indicates a truncation upto a given order after completely expanding the exponentialfactor.The R eg PT prescription of Eq. (110) can be compared withEq. (101) in the iPT formalism. On one hand, applying a Tay-lor expansion of the resummation factor Π and truncating atthe n -loop order give the same result as the n -loop SPT. Onthe other hand, the lowest-order approximation of the resum-mation factor is given by Eq. (35) in real space, i.e., Π ( k ) ≈ exp − k σ d2 ! , ( k → , (113)which is accidentally the same as the exponential factor in thehigh- k limit of Eq. (99). From these observations, it is now8clear that the R eg PT prescription of Eq. (110) is equivalentto evaluate the unbiased propagator Γ ( n )m by Eq. (101) in theframework of iPT, keeping only the lowest-order term in thevertex resummation factor Π ( k ) and expanding all the otherhigher-order terms from the exponent. In other words, theR eg PT prescription is equivalent to the restricted iPT formal-ism where the vertex resummations are truncated at the one-loop level (without biasing and redshift-space distortions).
3. Nonlinear interpolation II:
MPT breeze
There is another scheme of interpolating the nonlinearpropagators called MPT breeze [50], which is originally em-ployed in the two-point propagator in the RPT formalism [46].This method is simpler than the R eg PT, in a sense that calcu-lations of interpolated propagators require only one-loop inte-grals. In the MPT breeze prescription, the interpolated propa-gators are given by Γ ( n )MPTbreeze ( k , · · · , k n ) = F n ( k , · · · , k n ) exp h δΓ (1)1-loop ( k ) i , (114)where the one-loop correction term of two-point propagatorin the growing mode is explicitly given by δΓ (1)1-loop ( k ) = Z d q (2 π ) P L ( q )504 k q " k q − k q + q k − kq +
34 ( k − q ) (2 k + q ) ln | k − q | | k + q | . (115)The notations P ( k ), f ( k ) Γ ( n ) δ in Ref. [50] are related toour notations by P L ( k ) = (2 π ) D + ( z ) P ( k ), δΓ (1)1-loop ( k ) = D + ( z ) f ( k ) and Γ ( n )MPTbreeze = Γ ( n ) δ / D n + ( z ), where D + ( z ) is the lin-ear growth factor. Since δΓ (1)1-loop ( k ) → − k σ d2 / k limit and δΓ (1)1-loop ( k ) → k limit, Eq. (114) hascorrect limits.It is worth noting that the prescription of Eq. (114) corre-sponds to replacing all the loop-correction terms of propaga-tors by δΓ ( n ) N -loop ( k , . . . , k n ) → N ! h δΓ (1)1-loop ( k ) i N F n ( k , . . . , k n ) . (116)Both prescriptions of MPT breeze and R eg PT give similar re-sults, and they agree with numerical simulations fairly well inthe mildly nonlinear regime [50, 51]. Thus the approximationof Eq. (116) turns out to be empirically good, although thephysical origin of the goodness in this prescription is some-how unclear.According to Eq. (37) or Eq. (38), the iPT-normalized two-point propagator of mass is related to the function δΓ (1)1-loop ( k )by δΓ (1)1-loop ( k ) = δ ˆ Γ (1)1-loop ( k ) − k σ d2 , (117) where δ ˆ Γ (1)1-loop is the one-loop correction term which corre-sponds to the integral in Eq. (38) without bias, c ( n ) X =
0, or δ ˆ Γ (1)1-loop ( k ) = R ( k ) + R ( k ) , (118)as seen in Eq. (58). Substituting Eq. (117) into Eq. (114), wehave Γ ( n )MPTbreeze = F n exp h δ ˆ Γ (1)1-loop ( k ) i exp − k σ d2 ! . (119)Comparing this form with Eq. (112), the relation between theprescriptions of R eg PT and MPT breeze is explicit. Both pre-scriptions di ff er in the prefactor preceding to the exponentialdamping factor; a truncation scheme is employed in R eg PT,and a simple model of the higher-loop corrections is employedin MPT breeze . V. CONCLUSIONS
The iPT is a unique theory of cosmological perturbationsto predict the observable spectra of biased tracers both in realspace and in redshift space. This theory does not have phe-nomenological free parameter once the bias model is fixed. Inother words, all the uncertainties regarding biasing are packedinto the renormalized bias functions c ( n ) X , and weakly nonlin-ear gravitational evolutions of spatial clustering of biased trac-ers are described by iPT without any ambiguity. In this way,the iPT separates the bias uncertainties from weakly nonlinearevolutions of spatial clustering. The renormalized bias func-tions are evaluated for a given model of bias.Most of physical models of bias, such as the halo biasand peaks bias, fall into the category of the Lagrangianbias. Redshift-space distortions are simpler to describe in La-grangian picture than in Eulerian picture. The iPT is primarilybased on the Lagrangian picture of perturbations, and there-fore e ff ects of Lagrangian bias and redshift-space distortionsare naturally incorporated in the framework of iPT.In this paper, general expressions of the one-loop powerspectra calculated from the iPT are presented for the first time.The cross power spectra of di ff erently biased objects, P XY ( k ),both in real space and in redshift space are explicitly givenin terms of two-dimensional integrals at most up to one-looporder. The final result in real space is given by Eq. (47) withEqs. (35), (42), (43), (44), and that in redshift space is givenby Eq. (71) with Eqs. (57), (59), (60) and (70). When thevertex resummation is not preferred, one can alternatively useEq. (36) instead of Eq. (35). An example of the renormalizedbias functions is given by Eq. (19) for a simple model of halobias.The iPT is a nontrivial generalization of the method ofRef. [23], which is applicable only to the case that the La-grangian bias is local and that the initial condition is Gaussian.Although the derivations are quite di ff erent from each other,it is explicitly shown that the general iPT expression of thepower spectrum exactly reduces to the expression of Ref. [23]9in models of local Lagrangian bias and Gaussian initial con-dition.The e ff ects of primordial non-Gaussianity are included aswell. The consequent results are consistent with those de-rived by popular method of peak-background split. In fact, theiPT provides more accurate evaluations of the scale-dependentbias due to the primordial non-Gaussianity [36]. In the presentpaper, both e ff ects of gravitational nonlinearity and primordialnon-Gaussianity are simultaneously included in an expressionof biased power spectrum. Thus, the most general expressionsof power spectrum with leading-order (one-loop) nonlinearityand non-Gaussianity are newly obtained in this paper.In this paper, comparisons of the analytic expressions withnumerical N -body simulations are quite limited. In an accom-panying paper [61], the results in the present paper are used incalculating the nonlinear auto- and cross-correlation functionsof halos and mass, and are compared with numerical simu-lations, focusing on stochastic properties of bias. We haveconfirmed that e ff ects of nonlocal bias is small in the weaklynonlinear regime for the Gaussian initial conditions. That isnot surprising because nonlocality in the halo bias is e ff ectiveon scales of the halo mass indicated by Eq. (22); for exam-ple, R ≃ .
7, 1 .
4, 3 .
1, 6 . h − Mpc for M = , 10 , 10 ,10 h − M ⊙ , respectively, while one-loop perturbation theoryis applicable on scales ≫ h − Mpc for z .
3. Therefore,the predictions of iPT in Gaussian initial conditions with one-loop approximation are almost the same as those of LRT withLagrangian local bias [22], which have been compared in de-tail [25] with numerical simulations of halos both in real spaceand in redshift space. The nonlocality of halo bias should beimportant on small scales, and further investigations on therenormalized bias functions are interesting extension of thepresent work.In the framework of iPT, the vertex resummation is nat-urally defined, resulting in the resummation factor Π ( k ) ofEq. (35) in real space or Eq. (57) in redshift space. The vertexresummation of iPT is closely related to other resummationmethods like RPT which are formulated in Eulerian space.When the vertex resummation is truncated up to one-loop or-der, the iPT without bias and redshift-space distortions givesthe equivalent formalism to the R eg PT, a version of RPT withregularized multi-point propagators.Beyond the vertex resummations, the scheme of CLPTis readily applied to the framework of iPT as discussed inSec. IV C. Further resummation scheme of convolution can bealso considered. It might be an interesting application of iPTto include those type of further resummations in the presenceof nonlocal bias and redshift-space distortions.Although the resummation technique has proven to be use-ful in the one-loop approximation, it is not trivial whether thesame is true in arbitrary orders. The vertex resummation isnot compulsory in iPT, rather it is optional. The general formof vertex resummation factor in iPT is given by Eq. (6). Whenthis exponential function is expanded into polynomials, weobtain a perturbative expression of power spectrum withoutresummation, which is an analogue to SPT. However, for eval-uations of the correlation function, the exponential damping ofthe resummation factor stabilizes the numerical integrations k k ⇔ Π ( k ) c ( n ) X ( k , . . . , k n ) k i · · · k i m k k k n ⇔ L ( n ) i ( k , k , . . . , k n ) k n i m i i X FIG. 8: Diagrammatic rules of iPT: dynamics and biasing. ⇔ P L ( k ) k − kk k k n ⇔ P ( n )L ( k , k , . . . , k n ) FIG. 9: Diagrammatic rules of iPT: primordial spectra. of Fourier transform, and therefore the vertex resummation ispreferred.The nonlocal model of halo bias [36] explained in Sec. II Bis still primitive. There are plenty of rooms to improve themodel of nonlocal bias in future work. The iPT provides anatural framework to separate tractable problems of weaklynonlinear evolutions of biased tracers from di ffi cult problemsof fully nonlinear phenomena of biasing. Acknowledgments
I thank Masanori Sato for providing numerical data ofpower spectra and correlation functions from the N -body sim-ulations used in this paper. I acknowledge support from theMinistry of Education, Culture, Sports, Science, and Tech-nology, Grant-in-Aid for Scientific Research (C), 21540267,2012. Appendix A: Diagrammatic rules
A set of diagrammatic rules in iPT which is used in this pa-per is summarized in this Appendix. Full set of rules and theirderivations are found in Ref. [21]. The relevant diagrammaticrules are shown in Fig. 8 and 9. Physical meanings of thegraphs are as follows: a double solid line corresponds to thenumber density field δ X ( k ), a square box represents partial re-summations of dynamics and biasing, a wavy line represents0 XX not allowedallowed FIG. 10: Examples of external vertex which is allowed (left) and notallowed (right) in iPT. the displacement field, a black dot represents nonlinear evo-lutions of the displacement field, a crossed circle representsthe primordial spectra. The procedures for obtaining a crosspolyspectra P ( N ) X ··· X N ( k , . . . , k N ) of di ff erent types of objects X , . . . , X N are listed below. Auto polyspectra are obtained byjust setting X = · · · = X N . The power spectrum is a specialcase of polyspectra with N = N square boxes with labels X i ( i = , . . . N ), eachof which has a double solid line. Label each doublesolid line with an outgoing wavevector that correspondsto an argument of the polyspectra P ( N ) X ··· X N .2. Consider possible ways to connect all the square boxesby using wavy lines, solid lines, black dots and crossedcircles, satisfying following constraints:(a) An end of a wavy line should be connected to asquare box, and the other end should be connectedto a black dot.(b) An end of a solid line should be connected to acrossed circle, and the other end should be con-nected to either a square box or a black dot. (c) Only one wavy line can be attached to a black dotwhile arbitrary number of solid line(s) can be at-tached to a black dot.(d) A piece of graph which is connected to a singlesquare box with only wavy lines or with only solidlines is not allowed.3. Label each (solid and wavy) line with a wavevector andits direction. The wavevectors should be conserved ateach vertex of square box, black dot, and crossed circle.Label each wavy line with spatial index together with awavevector.4. Apply the diagrammatic rules of Figs. 8 and 9 to everydistinct graphs.5. Integrate over wavevectors as R d k ′ i / (2 π ) , where k ′ i are not determined by constraints of wavevector con-servation at vertices.6. When there are m equivalent pieces in a graph, put astatistical factor 1 / m ! for each set of equivalent pieces.7. Sum up all the contributions from every distinct graphsup to necessary orders of perturbations.The rule 2.-(d) is due to partial resummations of the squarebox. For example, the left diagram of Fig. 10 is allowed.There is a piece of graph that is connected to a single squarebox with both wavy and solid lines. However, the right di-agram of Fig. 10 is not allowed, because of double reasons.One is that the upper piece of graph is connected to a sin-gle square box with only wavy lines. The other is that thelower piece of graph with only solid lines connected to a sin-gle square box. Each reason itself prohibits this diagram fromcounted. [1] D. J. Eisenstein, W. Hu, and M. Tegmark, Astrophys. J. Letters, , L57 (1998).[2] T. Matsubara, Astrophys. J., , 573 (2004).[3] D. J. Eisenstein et al., Astrophys. J., , 560 (2005).[4] N. Dalal, O. Dor´e, D. Huterer and A. Shirokov, Phys. Rev. D , 123514 (2008).[5] S. Matarrese and L. Verde, Astrophys. J. Letters, , 677, L77(2008).[6] A. Slosar, C. Hirata, U. Seljak, S. Ho, N. Padmanabhan, J. Cos-mol. Astropart. Phys., , 31 (2008).[7] A. Taruya, K. Koyama and T. Matsubara, Phys. Rev. D ,123534 (2008).[8] V. Desjacques, U. Seljak and I. T. Iliev, Mon. Not. R. As-tron. Soc., , 85 (2009).[9] K. S. Dawson et al. , Astron. J., , 10 (2013).[10] [11] D. Schlegel et al. , arXiv:1106.1706 (2011).[12] LSST Science Collaborations: P. A. Abell, et al. ,arXiv:0912.0201 (2009). [13] R. Ellis et al. , arXiv:1206.0737 (2012).[14] [15] R. Laureijs et al. , arXiv:1110.3193 (2011).[16] A. F. Heavens, S. Matarrese, and L. Verde, Mon. Not. R. As-tron. Soc., , 797 (1998).[17] R. Scoccimarro, H. M. P. Couchman, and J. A. Frieman, Astro-phys. J., , 531 (1999)[18] A. Taruya, Astrophys. J., , 37 (2000).[19] P. McDonald, Phys. Rev. D , 103512 (2006); , 129901(E)(2006).[20] D. Jeong and E. Komatsu, Astrophys. J., , 569 (2009).[21] T. Matsubara, Phys. Rev. D , 083518 (2011).[22] T. Matsubara, Phys. Rev. D , 063530 (2008).[23] T. Matsubara, Phys. Rev. D , 083519 (2008); , 109901(E)(2008)[24] T. Okamura, T., A. Taruya and T. Matsubara, arXiv:1105.1491[25] M. Sato and T. Matsubara, Phys. Rev. D , 043501 (2011).[26] W. H. Press and P. Schechter, Astrophys. J., , 425 (1974).[27] J. R. Bond, S. Cole, G. Efstathiou, and N. Kaiser, Astrophys. J., , 440 (1991).[28] H. J. Mo and S. D. M. White, Mon. Not. R. Astron. Soc., ,347 (1996).[29] H. J. Mo, Y. P. Jing, and S. D. M. White, Mon. Not. R. As-tron. Soc., , 189 (1997).[30] R. K. Sheth and G. Tormen, Mon. Not. R. Astron. Soc., ,119 (1999).[31] R. Scoccimarro, R. K. Sheth, L. Hui, and B. Jain, Astrophys. J., , 20 (2001).[32] A. Cooray and R. Sheth, Phys. Rep., , 1 (2002).[33] K. C. Chan, R. Scoccimarro and R. K. Sheth, Phys. Rev. D ,083509 (2012).[34] K. C. Chan and R. Scoccimarro, Phys. Rev. D , 103519(2012).[35] R. K. Sheth, K. C. Chan and R. Scoccimarro, arXiv:1207.7117(2012).[36] T. Matsubara, Phys. Rev. D , 063518 (2012).[37] T. Buchert, Astron. Astrophys., , 9 (1989).[38] F. Moutarde, J.-M. Alimi, F. R. Bouchet, R. Pellat, and A. Ra-mani, Astrophys. J., , 377 (1991).[39] T. Buchert, Mon. Not. R. Astron. Soc., , 729 (1992). (1993).[40] P. Catelan, Mon. Not. R. Astron. Soc., , 115 (1995).R. Juszkiewicz, Astron. Astrophys., , 643 (1995). (1996).[41] C. Rampf and T. Buchert, J. Cosmol. Astropart. Phys., , 21(2012).[42] T. Tatekawa, Progress of Theoretical and Experimental Physics,2013, id.013E03 (2013).[43] T. Nishimichi, T. Matsubara, A. Taruya, in prep. M. B. Wise, Astrophys. J., , 6 (1986). (1994).[44] F. Bernardeau, S. Colombi, E. Gazta˜naga, and R. Scoccimarro,Phys. Rep., , 1 (2002)[45] M. Crocce and R. Scoccimarro, Phys. Rev. D , 063519(2006).[46] M. Crocce and R. Scoccimarro, Phys. Rev. D , 063520(2006).[47] F. Bernardeau, M. Crocce and R. Scoccimarro, Phys. Rev. D ,103521 (2008).[48] F. Bernardeau, M. Crocce and E. Sefusatti, Phys. Rev. D ,083507 (2010).[49] F. Bernardeau, M. Crocce and R. Scoccimarro, Phys. Rev. D , 123519 (2012).[50] M. Crocce, R. Scoccimarro and F. Bernardeau,Mon. Not. R. Astron. Soc., , 2537 (2012).[51] A. Taruya, F. Bernardeau, T. Nishimichi, S. Codis, Phys. Rev.D , 103528 (2012).[52] A. Taruya, T. Nishimichi and F. Bernardeau, arXiv:1301.3624(2013).[53] J. Carlson, B. Reid and M. White, Mon. Not. R. Astron. Soc., , 1674 (2013)[54] T. Matsubara, Astrophys. J. Suppl. Ser., , 1 (1995)[55] M. S. Warren, K. Abazajian, D. E. Holz, L. Teodoro, Astro-phys. J., , 881 (2006).[56] M. Crocce, P. Fosalba, F. J. Castander and E. Gazt˜anaga,Mon. Not. R. Astron. Soc., , 1353 (2010).[57] N. Kaiser Mon. Not. R. Astron. Soc., , 1 (1987).[58] A. J. S. Hamilton, The Evolving Universe, 231, 185 (1998).(astro-ph / , L5 (1992).[60] S. Cole, K. B. Fisher and D. H. Weinberg, Mon. Not. R. As-tron. Soc., , 785 (1994).[61] M. Sato and T. Matsubara, Phys. Rev. D , 123523 (2013).[62] V. Springel, Mon. Not. R. Astron. Soc., , , 1105 (2005).[63] M. Crocce, S. Pueblas and R. Scoccimarro, Mon. Not. R. As-tron. Soc., , , 369 (2006).[64] P. Valageas and T. Nishimichi, Astron. Astrophys., , , A87(2011).[65] A. Lewis, A. Challinor and A. Lasenby, Astrophys. J., , ,473 (2000).[66] M. Davis, G. Efstathiou, C. S. Frenk and S. D. M. White, As-trophys. J., , , 371 (1985).[67] M. Crocce, F. J. Castander, E. Gaztanaga, P. Fosalba and J. Car-retero, arXiv:1312.2013 (2013).[68] F. Schmidt and M. Kamionkowski, Phys. Rev. D , 103002(2010).[69] V. Desjacques, D. Jeong and F. Schmidt, Phys. Rev. D ,061301 (2011).[70] V. Desjacques, D. Jeong and F. Schmidt, Phys. Rev. D84