Constraints on local primordial non-Gaussianity from large scale structure Anˇze Slosar,1 Christopher Hirata,2 UroˇsSeljak,3, 4 Shirley Ho,5 and Nikhil Padmanabhan6 1Berkeley Center for Cosmological Physics, Physics Department and Lawrence Berkeley National Laboratory, University of California, Berkeley California 94720, USA 2Caltech M/C 130-33, Pasadena, California 91125, USA 3Institute for Theoretical Physics, University of Zurich, Zurich, Switzerland 4Physics Department, University of California, Berkeley, California 94720, USA 5Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, New Jersey 08544, USA 6Lawrence Berkeley National Laboratory,University of California, Berkeley CA 94720, USA (Dated: August 6, 2008) Recent work has shown that the local non-Gaussianity parameter fNL induces a scale-dependent bias, whose amplitude is growing with scale. Here we first rederive this result within the context of peak-background split formalism and show that it only depends on the assumption of universality of mass function, assuming halo bias only depends on mass. We then use extended Press-Schechter formalism to argue that this assumption may be violated and the scale dependent bias will depend on other properties, such as merging history of halos. In particular, in the limit of recent mergers we find the effect is suppressed. Next we use these predictions in conjunction with a compendium of large scale data to put a limit on the value of fNL. When combining all data assuming that halo occupation depends only on halo mass, we get a limit of 29 ( 65) < fNL < +70 (+93) at 95% (99.7%) confidence. While we use a wide range of datasets,− our combined− result is dominated by the signal from the SDSS photometric quasar sample. If the latter are modelled as recent mergers then the limits weaken to 31 ( 96) < fNL < +70 (+96). These limits are comparable to the strongest − − current limits from the WMAP 5 year analysis, with no evidence of a positive signal in fNL. While the method needs to be thoroughly tested against large scale structure simulations with realistic quasar and galaxy formation models, our results indicate that this is a competitive method relative to CMB and should be further pursued both observationally and theoretically. PACS numbers: 98.80.Jk, 98.80.Cq I. INTRODUCTION ary predictions in that the expected gravitational wave signal in CMB is always negligible. Thus, they can be falsified if primordial gravitational waves are detected. The origin of structure formation in the universe is Third, multifield models could generate isocurvature per- one of the most hotly debated topics in current cosmol- turbations; while now ruled out as the main mode of ogy research. The standard paradigm is that of inflation structure formation, these could be present at a subdom- [1, 2, 3, 4], which has been tremendously successful in inant level and if detected would rule out the simplest describing a very large number of very distinct data-sets models of inflation [15, 16, 17, 18]. (see e.g. [5]). Inflationary models generically predict a A fourth direction, and the one we focus on in this flat universe and nearly scale-invariant spectrum of ini- paper, is non-Gaussianity in initial conditions. Stan- tial fluctuations [6, 7, 8, 9, 10], both of which seem to be dard single field inflation predicts that the departures confirmed by observations. Consequently, a lot of effort is from Gaussianity are very small and not accessible to being put into constraining observables that might actu- arXiv:0805.3580v2 [astro-ph] 6 Aug 2008 the current observational constraints. Most of the models ally distinguish between different models of inflation. At predict that non-Gaussianity is of the local type, mean- the moment, this is done as a multi-pronged effort: first, ing that it depends on the local value of the potential measurement of the primordial power spectrum gives a only. A standard parameterization of the primordial direct measure of the inflationary potential shape and non-Gaussianity is the so-called scale-independent f inflationary models differ on the actual slope, some pre- NL parameterization, in which one includes a quadratic cor- dicting red and some blue spectrum. Moreover, inflation rection to the potential [19, 20]: predicts that the primordial slope should only be chang- 2 ing with scale very slowly and any deviation from this Φ= φ + fNLφ , (1) prediction would be a surprise in need of an explana- tion. Second, a detection of B-mode polarization in the where φ is the primordial potential assumed to be a Gaus- cosmic microwave background, if interpreted as gravita- sian random field and fNL describes the amplitude of the tional waves from the early Universe, will effectively de- correction. A typical value of fNL for standard slow roll termine the energy scale of inflation and rule out a major inflation is of the order of slow roll parameter and thus class of inflationary models that predict inflation occurs of order 10−2 [21], but this is likely to be swamped by at a low energy scale [11]. Alternatives to inflation, such the contribution from nonlinear transformation between as ekpyrotic models [11, 12, 13, 14], differ from inflation- the primordial field fluctuation (assumed to be Gaussian 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 <fNL < 134 non-Gaussianity, highlighting more clearly its underlying [35] (all limits reported at 95% confidence− limit) from assumptions. We then extend the basic derivation using the bispectrum of the WMAP data at ℓ< 400. This has the extended Press-Schechter formalism and show that been improved subsequently to 36 <fNL < 100 by the for certain classes of halos, such as those that have un- 3 year analysis [36]. However, Yadav− & Wandelt recently dergone a recent merger, the results may be substantially claimed a detection of fNL > 0 at 99.5% significance, with modified. 2σ range 27 < fNL < 147 [? ]. This result is surprising We then use this formalism and apply it to a wide se- as, taken at its face value, it implies a very non-standard lection of publicly available large scale structure data. In inflation or something else all together. Interestingly, a Sections III–IV we discuss the data, methodology, main similar result has been obtained by Jeong & Smoot using results and systematic issues, including application of the one-point distribution function of the CMB [38]. The Section II to the derived observational constraints. In WMAP 5-year results also prefer positive fNL from their Section V we discuss the results and present some direc- bispectrum analysis, although zero fNL is within its 2- tions for the future. sigma significance [5], namely 9 <fNL < 111.