Appendix K. The three-point correlation function (bispectrum) of density peaks

Transcription

1 Appedix K The three-poit correlatio fuctio (bispectrum) of desity peaks Cosider the smoothed desity field, ρ (x) ρ [ δ (x)], with a geeral smoothig kerel W (x) δ (x) d yw (x y)δ(y). (K.) We defie the peaks as regios i the space where the smoothed desity cotrast exceed a certai threshold value, δ c : p (x) V θ [δ (x) δ c ], (K.) where θ(x) is a step-fuctio, ad V is the volume of the smoothed regio. I this chapter, we shall calculate the three-poit correlatio fuctio of peaks whe the desity field follows geeral probability distributio. We shall use the fuctioal itegratio method adopted by Matarrese et al. (98), but, istead of presetig the geeral formula, we explicitly calculate the two poit correlatio fuctio ad the three poit correlatio fuctio. Let us deote the desity cotrast of the peaks as δ p (x) p (x)/ p. The probability P (x, x ) of fidig two peaks at two differet locatios x ad x, ad the probability P (x, x, x ) of fidig three peaks at three differet locatios x, x ad x are related to the two-poit correlatio fuctio ξ p (x, x ), ad the three-poit correlatio For a spherical top-hat filter W (x), x < V 0, otherwise, the volume of the smoothed regio is V 4π /. However, V may ot be well-defied for differet filters, e.g. Gaussia filter, where smoothig fuctio is exteded to ifiity. Fortuately, it is ot importat to calculate the correlatio fuctio from the method we use here, as mea umber desity cacels out i equatio (K.) ad (K.4).

2 fuctio ζ p (x, x, x ) of desity cotrast of peaks as (Peebles, 980): P (x, x ) P P (x, x, x ) P p(x ) p (x ) p ( δ p (x ))( δ p (x )) ξ p (x, x ) (K.) p(x ) p (x ) p (x ) p ( δ p (x ))( δ p (x ))( δ p (x )) ξ p (x, x )ξ p (x, x )ξ p (x, x )ζ p (x, x, x ) (K.4) Therefore, i order to calculate the two-poit correlatio fuctio ad the three-poit correlatio fuctio, we have to calculate P N up to N. I Appedix K. we calculate the probability P N : startig from the geeral fuctioal itegratio method for calculatig P N, we explicitly calculate P (Appedix K..), P (Appedix K..) ad P (Appedix K..). The, i Appedix K., we calculate the two-poit correlatio fuctio (power spectrum) of peaks by usig equatio (K.). Fially, We calculate the three-poit correlatio fuctio (bispectrum) of peaks i Appedix K. by usig equatio (K.4). K. Probability of fidig N distict peaks The probability fuctioal P[δ(x)] is the probability distributio fuctio of δ(x) at all poit x i space. It is ormalized to be [Dδ] P [δ(x)], with a suitable measure [Dδ]. The probability distributio fuctio of the desity field δ 0 as a specific positio x 0 ca be calculated as P (δ 0 ) [Dδ] P [δ(x)] δ D (δ(x 0 ) δ 0 ), (K.5) which meas that fixig the desity at x 0 to be δ 0 ad margialize over all other poits. As defied i equatio (K.), the peaks are the regios where the smoothed desity field [Eq. (K.)] exceeds δ c. By usig a P[δ(x)] we ca formulate the probability

3 P N (x,, x N ), which is the probability of fidig N peaks at x,, x N, as P N (x,, x N ) N dα dα N [Dδ] P [δ(x)] δ D (δ (x r ) α r ) δ c δ c N dφ r [Dδ]P[δ(x)] dα r e iφr[ d yw (x r y)δ(y) α r]. (K.) π νσ Here, i the secod equality we use δ D (x) /(π) dφe iφx represetatio of the Dirac delta fuctio ad the defiitio of the smoothed desity field. I order to simplify the otatio later, we defie a variable ν δ c /σ which quatify the desity threshold i a uit of the root-mea-squared (r.m.s.) value of the smoothed desity cotrast. Let us defie the partitio (geeratig) fuctioal Z[J] as Z[J] [Dδ] P [δ(x)] e i d yδ(y) N φrw ( x r y ) exp i d yδ(y)j(y), (K.7) with followig source fuctio: N J(y) φ r W (x r y). (K.8) The, from equatio (K.7), we ca defie the -poit coected correlatio fuctio as ξ () (y,, y ) δ(y ) δ(y ) c δ l Z[J] i δj(y ) δj(y ). (K.9) J0 I other words, if we kow all -poit correlatio fuctios, we ca recostruct the partitio fuctio as a Taylor expasio: l Z[J] i! d y d y ξ () (y,, y )J(y ) J(y ). By substitutig the source fuctio i equatio (K.8), the partitio fuctioal becomes (K.0) i l Z[J] d y d y ξ () (y!,, y ) N N φ r W (x r y ) φ r W (x r y ). (K.) r r 4

4 The strategy of calculatig P N is followig. For give -poit correlatio fuctios of desity field, we ca calculate the partitio fuctioal from equatio (K.). By usig the partitio fuctioal, P N becomes P N (x,, x N ) νσ dα νσ dα N dφ (π) dφ N N (π) e i αrφr Z[J], (K.) which ivolves oly ordiary itegratio. Matarrese et al. (98) provide the geeral solutio for P N, from there the authors reach the geeral formula for the N-poit correlatio fuctio of peaks. We however fid that the formula i Matarrese et al. (98) is too abstract to be directly adopted without justificatio from the explicit calculatio. Therefore, we shall explicitly show the solutio for N, ad i the followig sectios. For the otatioal simplicity we deote the smoothed -th order correlatio fuctio as ξ () : ξ () (x,, x ) d y r w (x r y r ) ξ () (y,, y ). (K.) K.. Calculatio of P Let s cosider N case. The geeratig fuctioal becomes l Z[J] i! d y d y ξ () (y,, y ) φ W (x y r ), (K.4) ad the itegratio over y i s are simply a smoothed correlatio fuctio: d y ξ () x,, x times d y ξ () (y,, y ) W (x y r ). (K.5) As we have oly oe argumet i the smoothed correlatio fuctio, we simply deote it as ξ () without argumets. The, the geeratig fuctioal is simplified as l Z[J] i! φ ξ (), (K.) 5

5 ad we ca calculate P from equatio (K.). P (x ) i dα dφ exp iα φ (π) νσ! φ ξ (). (K.7) Let us first cosider the φ i itegratio. I dφ exp We first calculate the quadratic term I 0 iα φ dφ exp iα φ φ i! φ ξ (). (K.8) π exp α, (K.9) the, we calculate the other terms i I, whichicludeφ ( >), by takig -th derivative of α o I 0 as followig. ( ) I exp ξ ()! π ( ) exp! σ α ξ () dφ e φ σ / iαφ α exp α (K.0) Now, P becomes ( ) P (x ) dα exp π νσ! dα ( ) ξ () exp π ν! ξ () α α where i the secod equality, we chage the variable α i α i/σ. exp α σ exp α, (K.) We use the Hermite polyomial, H (x) to simplify the equatio further. Especially, followig two properties of Hermite polyomial is useful for our purpose. H (x) ( ) e x lim H (x) x x d dx e x (K.) (K.) From equatio (K.), we fid ( ) d dα e α / / e α α / H,

6 ad whe the argumet is large, (α ) we approximate the derivative as a d ( ) dα e α / e α / (α). Therefore, by usig the asymptotic formula of Gaussia itegratio f(x)e x / f(a) a e a / O a, (K.4) we further simplify P as P (x ) for the high peak limit (ν ). exp πν ν! ξ () e ν /. (K.5) K.. Calculatio of P I this sectio, we focus oly o N case. The geeratig fuctioal becomes i l Z[J] d y d y ξ () (y!,, y ) φ W (x y r )φ W (x y r ) i d y d y ξ () (y!,, y ) m m0 m W (x y r ) W (x y r ), r r m φ m φ m (K.) where, i the secod lie, we use the biomial expasio ad the symmetry of ξ (), amely, the correlatio fuctio does ot deped o the order of argumet. We ca replace the itegratio over y i s as a smoothed correlatio fuctio. m d y d y ξ () (y,, y ) W (x y r ) ξ () x,, x, x,, x m times m times r r m W (x y r ) (K.7) For otatioal simplicity we deote such a smoothed correlatio fuctio as ξ (),m. The, the geeratig fuctioal becomes l Z[J] i m0 φ m φ m m!( m)! ξ(),m. (K.8) 7

7 We calculate P from equatio (K.) P (x, x ) νσ dα νσ dα dφ (π) Let s first cosider the φ i itegratio. I dφ dφ exp i α r φ r dφ (π) e i(αφαφ) Z[J]. i m0 φ m φ m m!( m)! ξ(),m (K.9) (K.0) As we have doe i Sectio K.., we first calculate the quadratic itegratio, I 0 dφ dφ exp i α r φ r φ φ π exp αr. (K.) The, φ i itegratio simplifies to be a sum of successive derivatives o I 0 as below. I π exp ( ) ξ () (x ξ () ) ( ),m α α m!( m)! α m m0 α m exp αr (K.) σ Now, we have to calculate the α i itegratio. For the otatioal simplicity, we defie αi α i/σ, ad w m () as followig. w m () ξ () (x )/ (m ) w m () 0 (m 0 or ) w () m ξ (),m /σ ( >) Now the two-poit probability P (x, x ) becomes P (x, x ) π dα ν ν exp dα exp ( ) αr m0 w () m m!( m)! α m α m (K.), (K.4) Agai, we use the Hermite polyomial, ad take the high peak limit of ν P (x, x ) πν exp w m () ν e ν m!( m)! m0 8

8 K.. Calculatio of P Let s calculate for P (x, x, x ): P (x, x, x ) (π) i i νσ dα i dφ i e i αrφr Z[J]. (K.5) We first calculate the partitio fuctioal i l Z[J] d y i ξ () (y!,, y ) φ j W (x j y r ). (K.) j By usig a multiomial expasio theorem, φ j W (x j y r ) j m m 0 m 0 r [m ]! m!m!( m m )! φm φm φ m m W (x y r ) r [m ] W (x y r ) r [m ] W (x y r ) (K.7) where, m m m, ad [m i ] stads for the set of idexes [m i ]{a k a k, k,,m i }, ad empty whe m i 0. Note that, [m i ] are mutually exclusive, that is [m i ] [m j ]{}, ad [m ] [m ] [m ]{,,,}. For example, whe, it becomes φ j W (x j y r ) j φ W (x y )W (x y )φ W (x y )W (x y ) φ W (x y )W (x y )φ φ W (x y )W (x y ) φ φ W (x y )W (x y )φ φ W (x y )W (x y ) φ φ W (x y )W (x y )φ φ W (x y )W (x y ) φ φ W (x y )W (x y ), (K.8) 9

9 ad we ca write compoet of l Z[J] as d y d y ξ () (y, y ) eq.[k.8] (φ φ φ ) Therefore, we rewrite Z[J] as φ φ ξ () (x, x )φ φ ξ () (x, x )φ φ ξ () (x, x ). (K.9) l Z[J] i! m m 0 m 0! m!m!m! φm φm φm ξ(),m m m, (K.40) where ξ (),m m m ξ () x,, x, x,, x, x,, x, m times m times m times For example, for case we write i! m m 0 m 0! m!m!m! φm φm φm ξ(),m m m φ ξ (),00 φ ξ (),00 φ ξ (),00 φ φ ξ (),0 φ φ ξ (),0 φ φ ξ (),0 We follow the same procedure as we have calculated P ad P i the previous sectio. First, we calculate φ i itegratio by usig a quadratic itegratio of I 0 dφ dφ dφ exp i α r φ r φ r (π)/ exp αr. (K.4) The, we ca replace the φ r -itegratio by applyig successive differetiatios of ( iα r ). That is,.. I exp ij ξ () (x i, x j ) α i m m 0 m 0 α j ( ) ξ (),m m m m!m!m! α m αm αm I 0. (K.4) 0

10 As we do ot apply the derivative operator to the terms appears i I 0,wedefiew () m m m as followig. w m (),m,m ξ () (x ij)/ (m,m,m ) w m (),m,m 0 (otherwise) w () m,m,m ξ (),m m m / (>) (K.4) By usig the ew otatio, ad chagig the variable to α i α i/σ, P becomes P (x, x, x ) dα (π) / r ν exp ( ) w m m!m!m!,m () m m m α m α m α m Usig the Hermite polyomial, ad, fially, we imposig the high peak coditio of ν, we fid P (x, x, x ) exp (πν ) / w m,m,m exp () m m m ν m!m!m! e ν. α r. (K.44) K. The two poit correlatio fuctio of peaks We calculate the two poit correlatio fuctio of peaks by substitutig P from equatio (K.5) ad P from equatio (K.5) ito equatio (K.): ξ p (x, x ) P (x, x ) P exp m0 w m () ν m!( m)! ν! ξ (), (K.45) where w () m ad ξ () are defied i equatio (K.) ad equatio (K.5), respectively. We further simplify the otatio by usig w () 0 w () ξ () /σ for, ad w() 0 w () 0.

11 ξ p (x, x )exp exp exp m0 m m w m () ν m!( m)! w () m ν m!( m)! m ν! ξ (),m σ ν w () 0! ν w ()! (K.4) This result was first derived i Gristei & Wise (98) with a followig asatz p (x) C exp T d yδ(y)w (x y). (K.47) It was oly after the full calculatio of Matarrese et al. (98) that T ν/σ is idetified. Note that for Gaussia case, where all the higher order correlatio fuctio vaishes (ξ () 0 for >) equatio (K.4) coicides with the result i Politzer & Wise (984): ξp G ξ () (x, x )exp (x ) ν ν ξ () (x ), (K.48) ad, i the secod equality, we ca also reproduce the Lagragia bias, b L ν/σ,by takig the large scale limit (ξ ). ecetly, i the light of the scale depedet bias iduced by primordial o- Gaussiaity (see, Sectio I.), Matarrese & Verde (008) study the large scale limit of equatio (K.4). For the large separatio, we ca expad the expoetial i equatio (K.4), ad the two-poit correlatio fuctio icludig the leadig order correctio due to the o-gaussiaity becomes ξ p ( x x ) ν ξ () (x, x ) ν ξ () (x, x, x ). (K.49) Fourier trasform of the o-gaussia correctio term is related to the matter bispectrum as ξ () (x, x, x ) d q d q (π) (π) d q (π) δ (q ) δ (q ) δ (q )e iq x e iq x iq e x d q (π) d q (π) B (q, q, (q q ))e i(q q ) (x x). (K.50)

12 Therefore, by takig Fourier trasformatio of equatio (K.49), we fid the power spectrum of peaks as P p (k) d rξ p (r)e ik r ν where r x x. ν σ d rξ () (r)e ik r d q d r ν P (k) ν (π) d q (π) B (q, q, (q q ))e i(q q k) r d q (π) B (q, k, k q) (K.5) K. The three poit correlatio fuctio of peaks We calculate the three poit fuctio from the relatio betwee P ad correlatio fuctios i equatio (K.4). By usig the calculatio of P i equatio (K.5) ad P i equatio (K.44), we calculate P /P (i the high peak limit ν, ad o large scales,

13 ξ () (x,, x ) ) up to four-poit fuctio order: P (x, x, x ) exp P m m 0 m 0 ν m m 0 m 0 σ ν σ ν w m () m m m!m!( m m )! ν w m () m m m!m!( m m )! ξ () (x )ξ () (x )ξ () (x ) () ξ,00 ξ(),0 ξ(),0 ξ(),00 ξ(),0 ν4 (4) ξ, ν ξ(4),0 σ ν ν4 σ 4 ξ(),00 ξ(4),0 ξ(4),0 4 ν σ ξ(4),0 4 ξ(4), ξ () ξ () (x )ξ () (x )ξ () ξ(4),0 ξ(4),0 4 (x ) ξ(),0 ξ(4),040 4 ξ(4),0 ν! ν! ξ () ξ () σ ξ(), ξ(4),0 ξ(4),0 ξ () (x, x, x )ξ () (x, x, x )ξ () (x, x, x )ξ () ξ(),0 ξ(4), ξ(4),400 4 ξ(),0 ξ(4), ν4 σ 4 (x, x, x ) ξ(4) (x, x, x, x ) ξ(4) (x, x, x, x ) ξ(4) (x, x, x, x ) 4 ξ(4) (x, x, x, x ) ξ(4) (x, x, x, x ) 4 ξ(4) (x, x, x, x ) ξ(4) (x, x, x, x ) 4 ξ(4) (x, x, x, x ) ξ(4) (x, x, x, x ). (K.5) I order to calculate the three-poit correlatio fuctio of peaks, we eed to subtract the two-poit correlatio fuctios from equatio (K.5). As we expad P /P up to four-poit ξ (4) 4 4

14 fuctio order, we also expad equatio (K.4) i the same order as ξ p (x, x ) m0 ν ξ () (x ) ν ν ξ () (x ) ν ν4 σ 4 ν w m () m!( m)! ξ (), ν! (), ξ ξ () ξ () (x, x, x ) ν4 σ 4 ξ (4), ξ(4) (x, x, x, x ) 4 ξ(4) (x, x, x, x ) (4), (4), ξ 4 ξ (K.5) Subtractig the two-poit correlatio fuctio, we fid that the three-poit correlatio fuctio is ζ p (x, x, x ) ν ξ () (x, x, x ) ν4 4 ν 4 4 ξ (4) (x, x, x, x )(cyclic) ξ () (x )ξ () (x )(cyclic). (K.54) Bispectrum is the Fourier trasform of the three-poit correlatio fuctio. uiverse is statistically isotropic, the three-poit correlatio fuctio oly depeds o the shape ad size of triagles costructed by three poits (x, x, x ), which ca be fully specified by usig two vectors, r x x ad s x x. That is, As ζ(r, s) ζ(r, s, r s ). (K.55) I order to see this, let s Fourier trasform the three poit correlatio fuctio. ζ(x, x, x ) d q d q d q (π) (π) (π) δ(q )δ(q )δ(q )e iq x e iq x iq e x d q d q d q (π) (π) (π) (π) B(q,q,q )δ D (q )e iq x e iq x iq e x d q d q (π) (π) B(q,q, q q )e iq (x x) iq e (x x) d q (π) d q (π) B(q, q )eiq r e iq s ζ(r, s) (K.5) Therefore, we calculate the bispectrum by iverse-fourier trasform of the three-poit correlatio fuctio: B(q, q ) d r d sζ(r, s)e iq r e iq s (K.57) 5

15 Fially, let s thik about the Fourier trasformatio of coected four poit fuctio, ξ (4) (x, x, x, x ) d q d q d q d q 4 (π) (π) (π) (π) δ (q ) δ (q ) δ (q ) δ (q 4 ) c e iq x e iq x e iq x iq e 4 x d q d q d q d q 4 (π) (π) (π) (π) (π) T (q, q, q, q 4 )δ D (q 4 ) e iq x e iq x e iq x iq e 4 x d q d q d q (π) (π) (π) T (q, q, q, q )e iq (x x) e iq (x x) e iq (x x) d q (π) d q (π) d q (π) T (q, q, q, q )e i(q q ) r iq e s (K.58) Similarly, ξ (4) (x, x, x, x ) d q d q d q (π) (π) (π) T (q, q, q, q )e iq r e i(q q ) s ξ (4) (x, x, x, x ) d q d q d q (π) (π) (π) T (q, q, q, q )e iq (r s) e i(q q ) s (K.59) (K.0) By usig the Fourier relatios above, we ca easily calculate the Fourier trasfor-

16 matio of equatio (K.54). B p (k, k ) d r d sζ p (r, s)e ik r e ik s ν B (k, k ) ν4 4 ν4 4 d r d s P (k )P (k )P (k )P (k )P (k )P (k ) d q d q d q (π) (π) (π) T (q, q, q, q )e iq r e iq s e ik r e ik s T (q, q, q, q )e iq r e iq s e ik r e ik s T (q, q, q, q )e iq (r s) e iq s e ik r e ik s ν B (k, k ) ν4 4 ν4 σ 4 where k k k. d q (π) P (k )P (k )(cyclic) T (q, k q, k, k )(cyclic), (K.) Note that, eve i the absece of the primordial o-gaussiaity, matter bispectrum B (k, k, k ) is o-zero due to the gravitatioal istability, ad that is give by B (k, k, k )F (s) (k, k )P (k )P (k )(cyclic). (K.) Therefore, the galaxy bispectrum is give by F (s) B p (k, k, k ) ν σ ν4 σ 4 (k, k )P (k )P (k )(cyclic) P (k )P (k )(cyclic). (K.) i the large scale where we expect that halo/galaxy bias is liear. 7