arXiv:0805.3580v2 [astro-ph] 6 Aug 2008 rdcinwudb upiei edo nexplana- an of need of this detection in from a surprise deviation Second, a any tion. and be slowly would chang- very prediction be scale only should with slope inflation ing primordial Moreover, the pre- spectrum. that some blue predicts and slope, some and actual shape red the potential dicting on inflationary differ a models the gives inflationary of spectrum first, measure power effort: primordial direct multi-pronged the a as of done measurement At is inflation. this of moment, models the actu- different might between that distinguish be observables ally to constraining seem is into effort which put of of lot being both a Consequently, 10], observations. 9, ini- by 8, confirmed a of 7, predict spectrum [6, scale-invariant fluctuations generically tial nearly models and Inflationary in universe data-sets successful flat [5]). distinct tremendously very e.g. of been number (see has large which very inflation a 4], of describing cosmol- 3, that current is 2, in paradigm [1, standard topics The debated research. hotly ogy most the of one seprtcmdl 1,1,1,1] ie rminflation- from differ 14], such 13, inflation, 12, to [11, Alternatives models [11]. ekpyrotic occurs scale as inflation energy predict low major a that a at models out rule inflationary and of de- inflation effectively class of will scale energy Universe, the early termine the gravita- from as waves interpreted tional if background, microwave cosmic h rgno tutr omto nteuies is universe the in formation structure of origin The 1 osrit nlclpioda o-asint rmlarg from non-Gaussianity primordial local on Constraints eklyCne o omlgclPyis hsc Departm Physics Physics, Cosmological for Center Berkeley Anˇze Slosar, flresaedt optalmto h au of value the on predictions limit these a use put par we to In Next data halos. scale suppressed. large of is of history effect t and merging the violated as find be we such We may properties, mass. assumption other on this on depends that only argue depen bias to only halo formalism red it assuming first function, that we mass show Here of and scale. formalism with split growing peak-background is amplitude whose bias, cuaindpnsol nhl as egtalmtof limit a get we mass, halo on only depends occupation inlfo h DSpooercqaa ape ftelatte the If to sample. weaken our quasar datasets, limits photometric of the SDSS range wide the a from use signal we While confidence. (99.7%) urn iisfo h MP5ya nlss ihn evidenc no with analysis, year 5 WMAP the from limits current ASnmes 88.k 98.80.Cq an 98.80.Jk, observationally numbers: both PACS pursued further scale th be indicate large should results against and our CMB tested models, to thoroughly formation galaxy be and to quasar needs method the eetwr a hw httelclnnGusaiyparame non-Gaussianity local the that shown has work Recent 6 .INTRODUCTION I. arneBree ainlLbrtr,nvriyo Cali of Laboratory,University National Berkeley Lawrence 4 hsc eatet nvriyo aiona ekly C Berkeley, California, of University Department, Physics 3 nttt o hoeia hsc,Uiest fZrc,Z Zurich, of University Physics, Theoretical for Institute 1 hitpe Hirata, Christopher B rneo nvriy rneo,NwJre 84,USA 08544, Jersey New Princeton, University, Princeton md oaiaini the in polarization -mode nvriyo aiona eklyClfri 42,USA 94720, California Berkeley California, of University 2 atc / 3-3 aaea aiona915 USA 91125, California Pasadena, 130-33, M/C Caltech − 5 1( 31 eateto srpyia cecs etnHall, Peyton Sciences, Astrophysical of Department − 96) f < NL 2 Dtd uut6 2008) 6, August (Dated: Uroˇs Seljak, < 7 +6.Teelmt r oprbet h strongest the to comparable are limits These (+96). +70 f NL ny tnadprmtrzto fteprimordial potential the scale-independent the of so-called of parameterization the value is standard local non-Gaussianity the A on mean- type, depends only. local it the that to of ing is accessible non-Gaussianity models not that the of and predict departures Most small constraints. the observational very current Stan- that the are predicts Gaussianity conditions. inflation from initial field single in dard non-Gaussianity is simplest paper, the 18]. out 17, 16, rule of [15, would inflation detected mode of if main models subdom- a and the at level present as inant be out could these ruled formation, structure now while be detected. per- isocurvature can turbations; are generate they could waves models Thus, multifield gravitational Third, primordial negligible. wave always if gravitational is falsified expected CMB the in that signal in predictions ary eto oteptnil[9 20]: [19, cor- potential quadratic the a to includes rection one which in parameterization, inrno edand field random sian orcin yia au of value typical A correction. where naini fteodro lwrl aaee n thus and parameter roll slow 10 of order order of the of is inflation h rmrilfil utain(sue ob Gaussian be to (assumed fluctuation field between primordial transformation the nonlinear from contribution the ,4 3, hncmiigaldt suigta halo that assuming data all combining When . orhdrcin n h n efcso nthis in on focus we one the and direction, fourth A n n arneBree ainlLaboratory, National Berkeley Lawrence and ent hre Ho, Shirley − φ tti sacmeiiemto relative method competitive a is this at 9( 29 stepioda oeta sue ob Gaus- a be to assumed potential primordial the is r oelda eetmresthen mergers recent as modelled are r iua,i h ii frcn mergers recent of limit the in ticular, rv hsrsl ihntecnetof context the within result this erive so h supino universality of assumption the on ds ona eklyC 42,USA 94720, CA Berkeley fornia, obndrsl sdmntdb the by dominated is result combined esaedpnetba ildepend will bias dependent scale he tutr iuain ihrealistic with simulations structure theoretically. d ncnucinwt compendium a with conjunction in − fapstv inlin signal positive a of e − hnueetne Press-Schechter extended use then ter 2 65) 2] u hsi ieyt esapdby swamped be to likely is this but [21], 5 f lfri 42,USA 94720, alifornia NL rc,Switzerland urich, n ihlPadmanabhan Nikhil and f < nue scale-dependent a induces = Φ NL f NL < φ cl structure scale e 7 +3 t95% at (+93) +70 ecie h mltd fthe of amplitude the describes + f f NL NL φ f 2 NL o tnadso roll slow standard for , While . 6 f (1) NL 2 if it started from the pure Bunch-Davies vacuum) and the large scale structure (LSS) of the universe. The main the observable (such as CMB temperature fluctuation), problem is that non-linearities add their own phase cor- which generically gives fNL of order unity (see e.g. [22]). relations between Fourier modes that can very quickly Models where fNL is significantly higher include multi- swamp the primordial signal. Historically, the focus was field inflation [23, 24, 25] as well as models where non- on the mass function of very massive virialised structures Gaussianity arises during reheating [26, 27] or preheating [42, 43, 44, 45, 46]. The motivation was the notion that [28, 29, 30, 31]. Non-slow roll inflation models may also very massive virialised objects correspond to very rare lead to a significant non-Gaussianity, but are constrained peaks in the initial density field and therefore their num- because they may not lead to enough inflation in the first ber density should be an exponentially sensitive probe −5 place. Note that since φ 10 even fNL 100, compa- of those peaks at the high mass end, allowing a unique rable to the present limits,∼ generates non-Gaussian∼ sig- probe of the primordial peak structure. While results natures only at a 10−3 level, so the non-Gaussian signal generally agree with this picture, the observational task one is searching for is very small. Overall, any detection is made very difficult by the low number statistics of such of fNL above unity would be a major surprise in need of objects, uncertainties in the mass-observable relation and an explanation within the inflationary paradigm. its scatter, and selection effects. Very recently, non-Gaussianity in ekpyrotic models has A different method has been recently proposed by also been studied with the results suggesting that non- Dalal et al. [47]. By extending the classical calculation Gaussianity in these models is generically large [32] and for calculating the clustering of rare peaks in a Gaus- often correlated with the spectral slope ns [33, 34], in sian field [48] to the fNL-type non-Gaussianity, they have the sense that the redder the spectrum the higher non- shown that clustering of rare peaks exhibits a very dis- Gaussianity one may expect. Thus, non-Gaussianity is tinct scale-dependent bias on the largest scales. The an- emerging as one of the strongest discriminators among alytical result has been tested using N-body simulations, the models attempting to explain the origins of structure which confirm this basic picture. in the universe. The purpose of this paper is two-fold, first to provide Traditionally, the cleanest method for detecting the a better theoretical understanding of the effect and the non-Gaussianity has been to measure the bispectrum or range of its applicability, and second, to apply it to the 3-point function of the Cosmic Microwave Background real data. We begin in Section II by providing a new, (CMB). The initial 3-year Wilkinson Anisotropy Probe more general, derivation of the nonlinear bias induced by (WMAP) result gave a limit on fNL of 54
B. Application to universal mass functions C. Halo merger bias
We now apply Equation (14) to halo abundance models The above statements apply to biasing of objects whose with a universal mass function. Universal mass functions HOD depends only on the mass of the halo. However this are those that depend only significance ν(M), i.e. may not be true for the quasars; in particular there are many lines of evidence that suggests that quasar activity d ln ν is triggered by recent mergers [53, 54, 55]. Therefore we n(M)= n(M, ν)= M −2νf(ν) , (15) d ln M should consider the standard bias b and large-scale bias ∆b for recent mergers, which is in general not the same 2 2 as the bias of all haloes of the final mass [56, 57]. where we define ν = δc /σ (M) and f(ν) is the fraction of mass that collapses into haloes of significance between This section considers the simplified case in which ν and ν + dν. Here δc = 1.686 denotes the spherical quasars are triggered by a merger between a halo of mass collapse linear over-density and σ(M) is the variance of M1 and one of mass M2, after which the quasar lives for a −1 time t H in the new halo of mass M0 = M1 + M2. the density field smoothed with a top-hat filter on the Q ≪ scale enclosing mass M. Universality of the halo mass This requires us to understand the dependence of the function has been tested in numerous simulations, with number of recent mergers on amplitude, which we will results generally confirming the assumption even if the again express as σ8. Unlike the case of the mass func- specific functional forms for f(ν) may differ from one tion there are no accurate fitting formulae to the merger another. rate that have been tested against N-body simulations The significance of a halo of mass M depends on for a variety of cosmologies. Therefore we will take two approaches here. The first will be to consider the recent the background density field δl, so one can compute merger probabilities from the extended Press-Schechter ∂n/∂δl(x) and insert it into Equation (4) [51], (ePS) formalism. With ePS, we will find that for haloes of a given mass the probability of being a recent merger 2 d −1 b =1 ν ln[νf(ν)]. (16) is proportional to σ8 . In this picture, the bias of the − δc dν quasars in this case is the same as the halo bias b(M0), but the f -induced bias ∆b is less for recent mergers (This is > 1 for massive haloes since the last derivative NL than for all haloes of mass M . However there is no rig- is negative in this case.) 0 orous error bound on ePS calculations, so it is desirable The derivative ∂ ln n/∂ ln σ8 appearing in Equation to have an independent way to get the dependence of (14) can be obtained under the same universality assump- merger histories on σ8. We therefore consider a second tion. In fact, the calculation is simpler. The definition method of getting the recent merger probability during of the significance implies ν σ−2, so that d ln ν/d ln M ∝ 8 the matter-dominated era by using redshift scaling rela- does not depend on σ8 at fixed M. Therefore n νf(ν) ∝ tions from N-body simulation results. The latter method and −1 confirms the ePS σ8 relation. ∂ ln n ∂ ln ν ∂ ln[νf(ν)] d = = 2ν ln[νf(ν)]. (17) ∂ ln σ8 ∂ ln σ8 ∂ ln ν − dν 1. Extended Press-Schechter calculation Thus by comparison to Equation (16), we find: We will work in the ePS formalism in which the merger 2 history seen by a given dark matter particle is con- Ωm H0 ∆b(M, k)=3fNL(b 1)δ (18) trolled by the linear density field δ(M) measured to- − c k2T (k)D(z) c day, spherically smoothed on a mass scale M in La- grangian space [58]. At time t a particle is inside a This is equivalent to the previously derived expressions halo of mass M if δ(M ′) > ω(t) for any M ′ > M, ≥ [47, 52]. However, it is more general, because it is in- where ω(t) = δcD(t0)/D(t) is the ratio of the threshold dependent of the form of f(ν). It is therefore valid for over-density for collapse δc to the growth function D(t). the Press-Schechter mass function as well as for the more In this picture it is convenient to replace the smooth- accurate Sheth-Tormen function. It is also valid for any ing scale M with the variance of the density field on that object that obeys a halo occupation distribution (HOD) scale, S(M)= δ(M)2 . The smoothed density field δ(S) that depends only on the halo mass, N (M), since in then follows a randomh i walk as a function of S; this ran- h i this case both b and ∆b are linearly averages of their dom walk is usually assumed to be Markovian because values for individual masses: (i) each Fourier mode is independent for Gaussian initial conditions, and (ii) one neglects the difference between b(M)n(M) N (M) dM b = h i , (19) smoothing with a top-hat in Fourier space (in which each n(M) N (M) dM R mode is independent) and the physically motivated top- h i hat in real space. and similarly for ∆bR. In this formalism we would like the probability that a 5 halo of mass M0 at time t was actually of mass M1 at an (14) to recent mergers we need only understand how the earlier time t t and experienced a merger with a halo of number density of recent mergers varies with σ8. Since − Q mass M2 = M0 M1. As argued by Lacey & Cole [58], the number density of recent mergers is the product of the − the probability for a particular dark matter particle in number density of haloes of mass M0 and the probability this halo to have been in an object of mass ω(t tQ) ω(t) From Equations (16) and (17), the first term evaluates Pparticle( 1 tQ ω˙ M0 dS1 P (M1 M0) dM1 = | | dM1. | √ (S S )3/2 M dM 2. Scaling from N-body simulations 2π 1 0 1 1 − (22) This formula was first derived by Lacey & Cole [58], The ePS formalism predicts that the probability of a albeit in a slightly different form [they computed halo of mass M0 being a recent merger, P (M1 M0)dM1, −1 | P (M0 M1)]. It has two well-known deficiencies. One is proportional to σ . While the qualitative result that | 8 is that it is not symmetric under exchange of M1 and massive haloes are more likely to be recent mergers in M2, especially for extreme mass ratios [59, 60]. Another low-σ8 than high-σ8 cosmologies is supported by N-body is that it does not contain merger bias in the Gaussian simulations [62, 63], the quantitative validity of the 1 − case, i.e. the bias of mergers is simply b(M0) [56]. This exponent does not appear to be well-tested. Neverthe- is because of the assumption that the trajectory δ(S) is less, in the matter-dominated era it is possible to deter- a Markovian random walk, which is not quite correct. mine the exponent from the redshift dependence of the For example, the analytic explanations for merger bias merger rate. of high-mass haloes [61] are based on non-Markovian be- The key is that there is no preferred timescale in haviour due to the fact that the physically meaningful the Einstein-de Sitter cosmology; scale factor and linear smoothing in real space does not correspond to a sharp growth factor are both proportional to t2/3 and therefore cutoff at some kmax. The corresponding merger history the rescaling of initial amplitude is mathematically iden- bias is based on the correlation coefficient γ between tical to rescaling of time. Hence two N-body simulations δ(S) and dδ(S)/dS; one would have γ = 0 if one used whose initial conditions differ only by the normalization the Fourier-space rather than real-space top-hat filter. of the primordial perturbations will evolve through ex- This subtlety is however not required to understand why actly the same sequence of halo formation and mergers, the merger bias in fNL cosmologies is significant on large except that the scale factor of each merger is re-scaled −1 scales. according to amerger σ8 . For our application we would also need to integrate Note that for ΛCDM∝ cosmologies this correspondence over the range of masses M1 that define a major merger, between scaling time and scaling normalisation breaks as but since the result does not actually depend on this we the amplitude of fluctuations at the onset of cosmic ac- will not explicitly write it. In order to apply Equation celeration will be different for different initial amplitudes. 6 Therefore results of this rescaling do not apply to the low- These results provide an independent calculation of est redshifts, where the dark energy becomes important, ∂ ln P (M1 M0)/∂ ln σ8 that is on a more solid footing but since in our analysis the only sample where recent than ePS.| The agreement of the logarithmic derivatives mergers may be relevant is the quasar sample, which has at the 10% level is remarkable, especially given that a redshift distribution peaking at z 1.7, this is a minor ePS does∼ not do so well at predicting the absolute merger deficiency. ∼ rate. We want to test this relation from the merger his- tory statistics [64] in the Millenium Simulation [65]. The recent merger probability, which we have denoted 3. Summary P (M1 M0)dM1, is related to the merger rate B/n de- fined by| Ref. [64] by: We can write a generalised expression for the fNL in- duced scale dependent bias as B(M0, ξ) dz ∂M1 P (M1 M0)dM1 = tQ dξ, (25) | n(M0) dt ∂ξ 2 Ωm H0 ∆b(M, k)=3fNL(b p)δc , (31) where ξ > 1 is the mass ratio of the progenitors. Here − k2T (k)D(z) c B/n is the merger rate per final halo of mass M0 per unit redshift per unit ξ. The derivative with respect to σ8 is where 1 2 values of fNL and Q, the non-Gaussian bias ∆b is present where ∆ (k) is the linear matter power spectrum today. only at large scales and the Q-term is present only at The window functions are given by small scales; there is no range of scales over which both D(r) dN are important. We explicitly confirmed that there is no W (k) = (b + ∆b) j (kr) dr and ℓ D(0) dr ℓ correlation between Q and fNL present in our MCMC Z chains and we let the data to determine the two param- D(r) dN 2ℓ2 +2ℓ 1 W r(k) = Ω0.6(r) − j (kr) eters. ℓ m D(0) dr (2ℓ 1)(2ℓ + 3) ℓ Z h − As discussed above if the halos in which objects re- ℓ(ℓ 1) − j −2(kr) side have undergone recent mergers and thus depend on −(2ℓ 1)(2ℓ + 1) ℓ − properties other than halo mass then the simple scaling (ℓ + 1)(ℓ + 2) with (b 1) may not be valid. This is unlikely to be rel- jℓ+2(kr) dr, (35) − −(2ℓ + 1)(2ℓ + 3) evant for LRGs, which are old red galaxies sitting at the i center of group and cluster sized halos. Moreover, num- where dN/dr is the redshift distribution normalised to ber density of LRGs is so high that it is reasonable to unity and written in terms of comoving distance r, and D assume that almost every group sized halo contains one, is the growth function. The code automatically switches since otherwise it is difficult to satisfy both the number to the Limber approximation when this becomes accu- density and high bias at the same time [98]. For LRGs rate. Note that for the low multipoles, it is essential to we thus do not expect there is a second variable in addi- include the redshift-space distortion even for a photomet- tion to halo mass and halo occupation models find that ric survey because a significant amount of power comes 13 populating all halos with mass above 10 M⊙/h with an from Fourier modes that are not transverse to the line of LRG is consistent with all the available data [99]. Hence sight. we will only use p = 1 in Equation (31). The same also We use an independent bias parameter for each red- holds for the photometric LRGs discussed below. shift slice. In addition to the bias dependence, we also use the Equation (32) to take into account non-linear cor- rections. While strictly speaking the value of Q should be different for each slice, we use a single free parameter Q for all slices. We have explicitly checked that non-linear B. Photometric LRGs from SDSS corrections are negligible for k < 0.1h/Mpc and there- fore this is not a major issue. Since there is a strong overlap between the spectroscopic and photometric sam- We use data from Padmanabhan et al. [94], who pro- ple for z < 0.45, we use only slices 5–7 when combining vide the LRG angular power spectrum measured in 8 this data with spectroscopic sample. redshift slices (denoted 0–7) covering the range 0.2 < zphoto < 0.6 in slices of width ∆zphoto =0.05. The power spectrum is based on 3500 square degrees of data. We C. Photometric quasars from SDSS use only data-points that correspond to k < 0.1h/Mpc at the mean redshift of each individual slice. We use We use the power spectrum of the high redshift the full Bessel integration to calculate the angular power quasar photometric sample that has recently been con- spectrum on largest scales and account for the redshift- structed for ISW and CMB lensing cross-correlation stud- distortion power as described in Padmanabhan et al. [94]: ies [95, 100]. The sample covers 5800 square degrees of sky and originally had two photometric redshift ranges, 0.65 < zphoto < 1.45 (“QSO0”) and 1.45 < zphoto < gg gv vv Cℓ = Cℓ + Cℓ + Cℓ , (33) 2.00 (“QSO1”). It consists of ultraviolet-excess (UVX; u g < 1.0) point sources, classified photometrically as quasars− [101], and with photometric redshifts [102]. The where superscripts g and v denote galaxies over-density classification and photometric redshifts were at the time and velocity terms respectively. The bias and β depen- of sample construction only available over a subset of the dence has been put back into the Bessel integral as it now survey region; they were extended to the remaining re- depends on the value of k. The three terms are given by gion using a nearest-neighbor algorithm in color space the integrals: [95]. As described below, we only used the QSO1 sam- ple as QSO0 appears to suffer from systematic errors on large scales. dk Cgg = 4π ∆2(k) W (k) 2, The largest angular scales in the quasar data are sub- ℓ k | ℓ | Z ject to at least three major sources of systematic error: dk Cgv = 8π ∆2(k) [W ∗(k)W r(k)] , and stellar contamination, errors in the Galactic extinction ℓ k ℜ ℓ ℓ maps, and calibration errors. All of these are potentially Z dk much worse than for the LRGs: some stars (e.g. M dwarf- Cvv = 4π ∆2(k) W r(k) 2, (34) ℓ k | ℓ | white dwarf binaries) can masquerade as quasars, and we Z 9 are relying on the u band where extinction is most severe QSO0-star power spectrum QSO1-star power spectrum and the photometric calibration is least well understood. 0.004 0.004 These errors were discussed in the context of ISW stud- 0.003 0.003 0.002 0.002 π π ies [95], but if one wishes to use the quasar auto-power 0.001 0.001 /2 /2 qs ℓ 0 qs ℓ 0 spectrum on the largest angular scales the situation is -0.001 -0.001 +1)C +1)C ℓ ℓ more severe. ( -0.002 ( -0.002 ℓ -0.003 ℓ -0.003 We investigated this subject by computing the cross- -0.004 -0.004 power spectrum of each QSO sample with the SDSS 2 3 5 10 20 30 50 2 3 5 10 20 30 50 18.0 ( -0.002 ( -0.002 quasar and stellar locus in nontrivial ways. The results ℓ ℓ -0.003 -0.003 of this correlation are shown in Figure 1, with error bars -0.004 -0.004 estimated from the usual harmonic space method, 2 3 5 10 20 30 50 2 3 5 10 20 30 50 ℓ ℓ qq −1 ss (C + nq )C qs ℓ ℓ FIG. 1: The correlation of each quasar sample with stars and σ(Cℓ )= 2 2 , (36) [(ℓmaxq + 1) ℓ ]fsky with the red stars. − min qq where Cℓ is the quasar autopower spectrum, nq is the ss 2. We then assume that the functional form for b(z) number of quasars per steradian, and Cℓ is the star au- −1 is either topower spectrum; the ns term is negligible. (Aside from boundary effects, this is the same error that one 1+ z 5 b(z) 1+ (38) would obtain by correlating random realizations of the ∝ 2.5 quasar field with the actual star field.) From the fig- ure, QSO1 appears clean, but QSO0 appears contami- as measured in [103], or that its form is given by nated: the first bin (2 ℓ < 12) has a correlation of qs ≤−4 b(z) 1/D(z), (39) Cℓ = (2.9 1.0) 10 with the red stars. This is ∝ a 2.9σ−result± and strongly× suggests some type of sys- as would be valid if the clustering amplitude is not tematic− in the QSO0 signal on the largest angular scales. changing with redshift. In both cases the constant We are not sure of the source of this systematic, but the of proportionality is determined from the normali- amount of power in the quasar map that is correlated sation condition with the red stars is dn bdn 1 dz = dz =1. (40) qs 2 dz dz b(z) ℓ(ℓ + 1) qq ℓ(ℓ + 1) Cℓ −4 Z Z Cℓ (corr) = ss 2 10 . (37) 2π 2π Cℓ ∼ × 3. Once both dn/dz and b(z) are known we calculate The variation in QSO0 density that is correlated with the the theoretical angular power spectrum using the red stars is thus at the 1.4% level. This is consistent same code as for photometric LRGs, taking into with the excess auto-power∼ in QSO0 in the largest scale account all available ℓ points. This is the theoreti- qq −4 cal spectrum that is used to calculate the χ2 that bin, which is at the level of [ℓ(ℓ + 1)/2π]Cℓ 3 10 (see figure 10 in [95]) and is comparable to∼ what× one goes into the MCMC procedure. In principle, all might expect from the 1–2% calibration errors in SDSS free parameters that determine the shape of bdn/dz [84], although other explanations such as the extinction should be varied through the MCMC, rather than map are also possible. Because of this evidence for sys- being fixed at the best-fit point. However, in the tematics, we have not used the QSO0 autopower spec- limit of Gaussian likelihood, the two procedures are trum in our analysis; in what follows we only use QSO1. equivalent, while the latter offers significant speed QSO0 may be added in a future analysis if our under- advantages. standing of the systematics improves. 4. We calculate χ2 using two different methods. Our Because the redshift distribution is poorly known and standard method is to use all points and the full needs to be determined internally from the data sample covariance matrix assuming a Gaussian likelihood: itself the data is analyzed in a two step procedure: χ2 = (d t) C−1(d t), (41) 1. Power spectrum points with ℓ = 30 . . . 200 are used − · − to constrain the product (bdn/dz)(z), where b is the where d and t are data and theory vectors of Cℓ’s, linear bias at redshift z and dn/dz is the normalised respectively. This is likely to be a good approxima- radial window function. Although the fNL is taken tion except on the largest scales, where small num- into account at this step, its effect is sub-dominant. ber of modes leads to corrections that may affect 10 D. Cross-correlation between galaxies and dark QSO autopower window functions matter via Integrated Sach-Wolfe effect 0.18 L =6 0.16 ctr Lctr=18 0.14 One can also look for the fNL using the cross- 0.12 correlation between a tracer like galaxies or QSOs and 0.1 dark matter. Since we do not have dark matter maps L 0.08 from large scales we can use cosmic microwave back- W 0.06 ground (CMB) maps as a proxy. If the gravitational 0.04 potential is time dependent, as is the case in a universe 0.02 dominated by dark energy or curvature then this leads 0 to a signature in CMB, the so-called integrated Sach- -0.02 Wolfe (ISW) effect [105]. This signature can easily be 0 10 20 30 40 50 related to the dark matter distribution, but part of the L signal is not coming from ISW but from the primary CMB anisotropies at the last scattering surface. These FIG. 2: The window functions for the ℓ = 6 and ℓ = 18 bins of the quasar power spectrum. act as a noise and lead to a large sampling variance on large scales and as a result the statistical power of this technique is weakened. At the moment ISW is only de- tected at 4σ level [106]. Our procedure closely follows the outer limits (e.g. 3σ). To test this we model that of [95]∼ and is in many respects very similar to that 2 the first QSO bin with an inverse-χ distribution, of the Section III C. We use all 9 samples present in Ho but neglecting covariance of this bin with higher ℓ et al., although the discriminating power is mostly com- bins: ing from the NVSS-CMB cross-correlation because NVSS sample is available over 27 361 deg2 area and the tracers, 2 dℓ +1/n tℓ +1/n radio galaxies, are biased with b 2. First, bdn/dz and χ = N ln + 1 + b(z) are determined for each sample.∼ Here we always use tℓ +1/n dℓ +1/n − b(z) 1/D(z) for all samples except the quasar sample. Gaussian χ2 for other points, (42) Then∝ we calculate the ISW Limber integral: where N 21.5 is the effective number of modes 3Ω H2T contributing∼ to the first bin and n is the number of CgT = m 0 CMB dz [b(z) + ∆b(k(z),z)] ℓ c3(ℓ +1/2)2 quasars per steradian. (This was called the “equal Z dn d D(z) variance” case in Ref. [104], and is appropriate if (1 + z) D(z)P (k(z)) , (43) there is equal power in all modes; this is the case × dz dz D(0) here to a first approximation since we find that the Poisson noise 1/n dominates.) where k(z) = (ℓ +1/2)/χ. Again, we used the full win- dow function for the NVSS-CMB correlation to avoid an unnecessary bias and we assume p = 1 for all the samples. Since the power rises dramatically at low ℓ in the fNL models, we have included the full window function in our calculation of the binned power spectrum, Cbin = IV. RESULTS ℓ WℓCℓ. The window functions for the lowest two QSO bins are shown in Figure 2, and the algorithm for their P computation is presented in Appendix A. We begin by plotting data-points and theoretical pre- As shown in the Section II C 1, if halos have undergone dictions for six of the datasets and values of fNL in Figure recent mergers then we would expect p 1.6 instead of 3. This plot deserves some discussion. The easiest and p 1 in Equation (31). Recent mergers∼ could be a plau- most intuitive to understand is the case of spectroscopic ∼ sible model for QSOs, whose activity could be triggered LRGs. We note that the inclusion of the fNL param- by a merger [53, 54, 55]. For QSOs used in this work we eter modifies the behaviour of the power spectrum on do not find that their number density places a significant the largest measured scales. Naively, one would expect constraint: at z 1.8 the number density of halos with LRGs to be very competitive at constraining fNL. In ∼ b 2.5 3 is one to two orders of magnitude higher than practice, however, fNL is degenerate with matter den- the∼ measured− number density, hence we can easily pick sity, which also affects the shape of the power spectrum and choose the halos with a recent merger and still satisfy and hence the constraints are somewhat weaker. Since the combined number density and bias constraint. This the effect of fNL raises very strongly, just a couple of is in agreement with conclusions of [103], which looked points on largest scales might break this degeneracy, im- at a similar QSO sample from 2dF. Because of this un- proving constraints by a significant factor. The effect certainty we therefore run another separate analysis for in the photometric LRG samples is similar, although we QSOs with p =1.6 (QSO merger case). are now looking the angular space, where the dependence 11 FIG. 3: This figure shows 6 datasets that are most relevant for our constraints on the value of fNL. In the left column show the NVSSxCMB Integrate Sach Wolfe Cross correlation, the QSO1 power spectrum, the spectroscopic LRG power spectrum, while the right column shows the last three slices of the photometric LRG sample. The lines show the best fit fNL = 0 model (black, solid) and two non-Gaussian models: fNL = 100 (blue, dotted), fNL = 100 (red, dashed). The ISW panel additionally − shows the fNL = 800 model as green, dot-dashed line. While changing fNL, other cosmological parameters were kept fixed. See text for further discussion. 12 FIG. 4: This figure shows the median value (red points) and 1,2 and 3-sigma limits on fNL obtained from different probes (vertical lines). The data set used are, from top to bottom: Photometric LRGs, Photometric LRGs with only slices 0–4 used, Spectroscopic LRGs, Integrated Sach Wolfe effect, photometric QSO, photometric QSOs using b(z) 1/D(z) biasing scheme (see Section III C), photometric QSOs using alternative χ2 calculation scheme (see Section III C), using∝ a scale dependent bias formula appropriate for recently merged halos (Section II C), Combined sample, Combined sample using a scale dependent bias formula appropriate for recently merged halos (for QSO), the last two resoluts to which a statistically independent WMAP 5 bispectrum fNL constraint was added. See text for discussion. has been smeared out. The QSO plot again shows simi- of fNL produce anomalously large power in the angular lar behaviour, with two caveats. First, the changes in the power spectrum if bdn/dz has significant contributution predicted power spectrum on small scales are a result of at high-z tail, which probes large scales. Therefore, the the fact that bdn/dz is perturbed with changing fNL, al- bdn/dz fitting procedure skews the distribution towards though this is a minor effect. Second, the increase in the lower redshifts, leading to a lower bias overall. At very power at smallest ℓ for negative fNL is due to the fact that large values, e.g. fNL = 800, this effect is so severe that for sufficiently negative fNL (or sufficiently large scales), the b 1/D(z) scaling forces b < 1 at the low-redshift δb < 2b and hence power spectrum rises again above end. This∝ implies ∆b < 0, and the large-scale ISW sig- what− is expected in the Gaussian case. The more unex- nal actually goes negative (see top-left panel of Fig. 3). pected is the NVSS-CMB cross-correlation. Naively, one Therefore, the ISW is surprisingly bad at discriminating would expect that the first point of that plot will produce fNL and we were unable to fit the first NVSS ISW data a very strong fNL “detection”. However, the CMB cross- point with a positive fNL. This behavior is however only correlation signal is only linearly dependent on fNL, while of academic interest because the other data sets strongly cross-correlations of NVSS with other tracers of struc- rule out these extreme values of fNL. ture are quadratically dependent on fNL. Large values We ran a series of MCMC chains with base cosmolog- 13 Data set fNL strains fNL in both directions. The second and third +54+101+143 point have some excess power relative to the best fit Photometric LRG 63− − − 85 331 388 fNL = 0 model and so give rise to a slightly positive value +115+215+300of fNL in the final fits. This is however not a statistically Photometric LRG (0-4) 34−194−375−444 − significant deviation from fNL = 0. +74+139+202 Spectroscopic LRG 70−83−191−371 We also test the robustness of fNL constraints from +647+755+933 the quasar sample power spectrum by performing the ISW 105−337−1157−1282following tests. First we replace the form of evolution of +26+47+65 bias from that in Eq. (38) to that in Eq. (39). Second we QSO 8−37−77−102 replaced the details of the likelihood shape for the first +28+49+69 QSO(b=1/D) 8−38−81−111 points, we replace Eq. (41) with Eq. (42). In both cases, 2 +27+52+72 the effect on the constraints was minimal, as shown in QSO alternative χ 10−40−74−101 the Figure 4 and the Table I. +30+58+102 The large-scale quasar power spectrum could also be QSO merger 12−44−94−138 affected by spurious power (e.g. calibration fluctuations +23+42+65 Combined 28−24−57−93 or errors in the Galactic extinction map) or 1-halo shot +16+39+65 noise, either of which would dominate at large scales over Comb. merger 31−27−62−127 the conventional P (k) k autocorrelation. At the level ∝ +18+33+52 of the current data these are not affecting our results: Combined + WMAP5 bispectrum fNL 36−17−36−57 uncorrelated power adds to the power spectrum Cℓ, and +13+29+53 Combined merger + WMAP5 bispectrum fNL 36−17−36−57 if present it would only would only tighten the upper limits on the power spectrum given by our ℓ = 6 and 18 points. The 1-halo shot noise would contribute an added TABLE I: Marginalised constraints on fNL with mean, 1σ constant to C since z 1.7 haloes are unresolved across (68% c.l.), 2σ (95% c.l.) and 3σ (99.7%) errors. Ordering ℓ ∼ exactly matches that of the Figure 4. See text for discussion. our entire range of ℓ; the observed power at ℓ 250 constrains the 1-halo noise to be much less than the∼ error bars at ℓ =6, 18. Nevertheless, if we had detected excess power in the quasars at the largest scales, a much more ical data and one of the four data sets considered above, detailed analysis would have been necessary to show that as well as a run in which all data were combined (with it was in fact due to fNL and not to systematics. the exception of slices 0–4 of photometric LRGs as de- If quasars are triggered by mergers, then they do scribed above). The results are summarized in Table I not reside in randomly picked halos and the use of the and visualized in the Figure 4. Eq. (18) may not be appropriate. As discussed in Sec- We note several interesting observations. When only tion IIC, one can replace the (b 1) factor in Eq. (18) slices 0–4 of the photometric LRG sample are used, the with a modified factor (b 1.6).− We quote the results results are weaker than those of the spectroscopic LRGs. in this case in the Table I− as “QSO merger.” The error The two trace comparable volume, but photometric sam- bars have increased by about 40%, which is somewhat ple does not use radial mode information and as a re- less than naively expected from a population with an av- sult its errors are expectedly larger. On the other hand, erage bias of around 2.5. This is due to various feedback the overall photometric sample performs somewhat bet- related to bdn/dz fitting. Moreover, this is likely to be ter than the spectroscopic LRG sample, due to its larger an extreme case since it assumes that all quasars live in volume: it traces LRGs up to z 0.6 as opposed to recently merged halos with short lifetimes, so we would z 0.45 for spectroscopic sample. ∼ ∼ expect that the true answer is somewhere in between the We find that the ISW is constraining the fNL param- two results. On the other hand, it is also unclear how eter rather weakly. This is somewhat disappointing, but accurate extended Press-Schechter formalism is for this not surprising, since the cross-correlation between LSS application, so there is some uncertainty associated with and CMB is weak and has only been detected at a few this procedure. Recent progress in understanding quasar sigma overall. In addition, as mentioned above, fNL en- formation and its relation to the underlying halo popu- ters only linearly (rather than quadratically) in the ISW lation [54] gives us hope that this can be solved in the expressions and is strongly degenerate with determina- future. tion of bdn/dz. Given that ISW S/N can only be im- We also quote results for the combined analysis. In this proved by another factor of 2 at most even with perfect ∼ case, the error on fNL shrinks somewhat more than one data we do not expect that it will ever provide competi- would expect assuming that error on fNL measurement tive constraints on fNL. from each individual dataset is independent. This is be- We find that the quasar power spectra give the cause other parameters, notably matter density and am- strongest constraints on fNL. This is due to their large plitude of fluctuations get better constrained when data volume and high bias. The lowest ℓ point in this data are combined. Furthermore, we note that when quasars set at ℓ = 6 is a non-detection and therefore highly con- are assumed to be recent mergers the errors expand by 14 an expected amount at the 3σ level, but hardly at a 2σ level. We have carefully investigated this anomaly and it seems to be due a peculiar shape of the likelihood surface and subtle interplay of degeneracies between fNL, matter density and amplitude of fluctuations. V. DISCUSSION The topic of this paper is signature of primordial non- Gaussianity of local type (the so called fNL model) in the large scale structure of the universe. Specifically, it was recently shown that this type of non-Gaussianity gives rise to the scale dependence of the highly biased trac- ers [47]. We extend this analysis by presenting a new derivation of the effect that elucidates the underlying as- sumptions and shows that in its simplest form it is based on the universality of the halo mass function only. Our FIG. 5: This figure shows 1 and 2σ contours on the ns-fNL derivation also allows for possible extensions of the sim- plane for our best combined data set with additional WMAP plest model, in which tracers of large scale structure de- 5 year bispectrum constraint (assumed to be independent of pend on properties other than the halo mass. One that ns). Red lines are predictions from the ekpyrotic models and we explore in some detail is the effect of recent merg- correspond to values of fixed γ and varying ǫ. Different line correspond to different values of γ, which varies between γ = ers. By using extended Press-Schechter model we show 1 (flat, constant, negative fNL) to γ = 0.2 in steps of 0.1. that in this case the effect has the same functional form, −The dashed green line corresponds to the theoretically− favored but the predicted scaling with fNL for a given bias has a value of γ = 1/√3 according to [107]. smaller amplitude than in the simplest model. − Second, we apply these results to constrain the value of fNL from the clustering of highly biased tracers of In this combined result, fNL = 0 is at just around 2σ, large scale structure at largest scales. In our analysis we which taken at a face value suggests than the evidence find that the best tracers are highly biased photometric for a significant non-Gaussianity found in 3 year WMAP quasars from SDSS at redshifts between 1.5 and 2, fol- data by [? ] may have been a statistical fluctuation lowed by photometric and spectroscopic LRGs at redshift rather than evidence of a real signal. around 0.5. Our final limits at 95% (99.7%) confidence The results are already sufficiently strong to con- are strain the ekpyrotic models of generating the primor- dial structure: these generically predict much higher 29 ( 65) −1 1 −1 −1 −1 p = (F ) q signal = (F ) Tr C C C S , APPENDIX A: WINDOW FUNCTIONS h ii ij h j i 2 ij i (A7) S This appendix describes the computation of the win- where is the signal covariance matrix. Its entries are dow functions. We begin with a brief revisit of the prin- ℓ ciples underlying the window function (see the references ∗ Sαβ = Cℓ Yℓm(α)Y (β), (A8) for more details) and then describe our computational ℓm ℓ m=−ℓ method. X X where α and β are pixels: 1 α, β Npix. It could also be written as ≤ ≤ 1. Principles ℓ S = C Y Y† , (A9) The power spectra in this paper were computed using ℓ ℓm ℓm =− the methodology of Padmanabhan et al. [111], imple- Xℓ mXℓ mented on the sphere as described in Refs. [94, 112]. The where Yℓm is a vector of length Npix containing the val- power spectrum is estimated from a vector x of length ues of the spherical harmonic Yℓm at each pixel. With Npix containing the galaxy overdensities in each of the some algebra the expectation value collapses down to: Npix pixels. Following the notation of Ref. [94], we esti- mate the power spectrum in bins, p = W C , (A10) h ii iℓ ℓ ˜i ℓ Cℓ = piCℓ, (A1) X i where the window function W is: X iℓ where C˜i is 1 if multipole l is in the ith bin, and 0 other- ℓ ℓ 1 −1 † −1 −1 wise, and p are the parameters to be estimated. We can Wiℓ = (F )ij Y C CiC Yℓm. (A11) i 2 ℓm C =− then define the template matrices, i, which are the par- mXℓ tial derivatives of the covariance matrix of x with respect to pi. The quadratic estimators are then defined: 2. Computation 1 T −1 −1 qi = x C CiC x, (A2) 2 Like the other matrix operations with million-pixel maps, direct computation of W using Equation (A11) is where C is an estimate of the covariance matrix used to iℓ not feasible; it is O(N 3 ). We have therefore resorted to weight the data (our choice is described in Ho et al. [95]). pix Monte Carlo methods, analogous to those used for trace We also build a Fisher matrix, estimation, to simultaneously solve for all of the ℓ and 1 m terms in Equation (A11). Define a random vector z F = Tr C−1C C−1C . (A3) ij 2 i j of length Npix and with entries consisting of independent 17 random numbers 1 (i.e. probability 1/2 of being 1 and To do a Monte Carlo evaluation of the average, we can 1/2 of being 1).± Then zzT = 1, so we can write: take a random vector z, compute the quantities C−1z and − h i −1 C Ciz. 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