Effects and Detectability of Quasi-Single Field Inflation
in the Large-Scale Structure and Cosmic Microwave Background
Quasi-single field inflation predicts a peculiar momentum dependence in the squeezed limit of the primordial bispectrum which smoothly interpolates between the local and equilateral models. This dependence is directly related to the mass of the isocurvatons in the theory which is determined by the supersymmetry. Therefore, in the event of detection of a non-zero primordial bispectrum, additional constraints on the parameter controlling the momentum-dependence in the squeezed limit becomes an important question. We explore the effects of these non-Gaussian initial conditions on large-scale structure and the cosmic microwave background, with particular attention to the galaxy power spectrum at large scales and scale-dependence corrections to galaxy bias. We determine the simultaneous constraints on the two parameters describing the QSF bispectrum that we can expect from upcoming large-scale structure and cosmic microwave background observations. We find that for relatively large values of the non-Gaussian amplitude parameters, but still well within current uncertainties, galaxy power spectrum measurements will be able to distinguish the QSF scenario from the predictions of the local model. A CMB likelihood analysis, as well as Fisher matrix analysis, shows that there is also a range of parameter values for which Planck data may be able distinguish between QSF models and the related local and equilateral shapes. Given the different observational weightings of the CMB and LSS results, degeneracies can be significantly reduced in a joint analysis.
Quasi-single field (QSF) inflation models Chen and Wang (2010, 2010); Baumann and Green (2011) are a natural consequence of inflation model-building in string theory and supergravity. In addition to the inflaton field, these models have extra fields with masses of order the Hubble parameter. Such masses are stabilized by the supersymmetry. A distinctive observational signature of these massive fields is a one-parameter family of large non-Gaussianities whose squeezed limits interpolate between the local and the equilateral shape. Therefore, by measuring the precise momentum-dependence of the squeezed configurations in the non-Gaussianities, in principle, we are directly measuring the parameters of the theory naturally determined by the fundamental principle of supersymmetry.
The possibility of detecting a non-Gaussian component in the initial conditions of Early Universe has been the subject of considerable attention in recent years both from an observational perspective and theoretically through inflation model-building (Komatsu et al., 2009; Liguori et al., 2010; Chen, 2010). Current constraints from measurements of the bispectrum of temperature fluctuations in the cosmic microwave background (CMB) from the WMAP satellite (Komatsu et al., 2011) are still consistent with Gaussianity. However, the Planck mission (The Planck Collaboration, 2006) will soon significantly improve the errors on non-Gaussian parameters, leading to strong new constraints or what could be a major breakthrough in cosmology.
The effects of non-Gaussian initial conditions on the large-scale matter and galaxy distributions have been the subject of several studies for more than a decade (see (Desjacques and Seljak, 2010a) and (Liguori et al., 2010) for recent reviews). The most direct of such effect consists in the additional contribution to the matter bispectrum due to the linearly evolved initial component. Measurements of the galaxy bispectrum in upcoming, large-volume galaxy surveys are expected to improve even over ideal CMB limits, for any non-Gaussian model (Scoccimarro et al., 2004; Sefusatti and Komatsu, 2007).
In addition, the relatively recent discovery of a significant scale-dependent corrections to galaxy bias due to non-Gaussian initial conditions (Dalal et al., 2008) has led to constraints on the local non-Gaussian parameter from current Large-Scale Structure (LSS) data-sets, already comparable to those from the CMB (Slosar et al., 2008). Such bias corrections are present for models where the curvature bispectrum takes large values in the squeezed limit and precisely for this reason, the case of Quasi-Single Field inflation is of particular interest. The results of ref. (Dalal et al., 2008) motivated a significant number of further works aimed at a rigorous theoretical description of the effect (Slosar et al., 2008; Matarrese and Verde, 2008; McDonald, 2008; Taruya et al., 2008; Afshordi and Tolley, 2008; Desjacques et al., 2009; Valageas, 2009; Pillepich et al., 2010; Giannantonio and Porciani, 2010; Nishimichi et al., 2010; Schmidt and Kamionkowski, 2010; Shandera et al., 2011; Gong and Yokoyama, 2011; Cyr-Racine and Schmidt, 2011; Desjacques et al., 2011a, b; Scoccimarro et al., 2012). At the same time several groups ran new sets of simulations with non-Gaussian initial conditions of the local (Desjacques et al., 2009; Grossi et al., 2009; Pillepich et al., 2010; Nishimichi et al., 2010) and other types of models, such as equilateral, orthogonal and folded (Wagner et al., 2010; Wagner and Verde, 2012; Scoccimarro et al., 2012; Ganc and Komatsu, 2012; Agullo and Shandera, 2012), as well as generalized local models (e.g. cubic or scale-dependent models) (Desjacques and Seljak, 2010b; Tseliakhovich et al., 2010; LoVerde and Smith, 2011; Shandera et al., 2011). The picture emerging from such extensive investigations is that the relatively simple expression sufficient to describe the effect of local non-Gaussianity, requires additional corrections to accurately describe models presenting a squeezed limit of the curvature bispectrum different from the local one (Desjacques et al., 2011a; Scoccimarro et al., 2012). More generally for such models, the overall correction to bias induced by a generic non-Gaussian model will depend on the halo mass in a non-trivial way and will include a scale-independent component (Desjacques et al., 2009). The extension of these predictions to nonlinear bias and to the description of the galaxy bispectrum have been studied in (Sefusatti, 2009; Jeong and Komatsu, 2009; Baldauf et al., 2011; Giannantonio and Porciani, 2010) and recently tested in simulations by (Nishimichi et al., 2010; Sefusatti et al., 2011) in the context of the local model.
The detectability of the bias correction, both in galaxy and cluster surveys, has also been the subject of several works in recent years (Slosar et al., 2008; Carbone et al., 2008; Seljak, 2009; Slosar, 2009; Verde and Matarrese, 2009; Carbone et al., 2010; Cunha et al., 2010; Sartoris et al., 2010; Hamaus et al., 2011; Giannantonio et al., 2011; Pillepich et al., 2012). Typical forecasted errors for upcoming galaxy redshift surveys on , marginalized over cosmological parameters, are of the order of few (Carbone et al., 2010; Giannantonio et al., 2011), while cluster surveys including information on the cluster spatial correlations should provide similar results (Cunha et al., 2010; Sartoris et al., 2010; Pillepich et al., 2012). Beyond local non-Gaussianity, ref. (Giannantonio et al., 2011) provided forecasts for equilateral and orthogonal models as well, combining information from galaxy and weak lensing observables. When only the 3D galaxy power spectrum is considered, therefore relying mostly on corrections to galaxy bias, the expected errors are of the order of and for orthogonal and equilateral non-Gaussianity, respectively, including CMB priors from the Planck mission. The case of an initial bispectrum described in terms of two parameters, and the possibility of determining them both, a case complimentary to the one considered here, has been considered by (Sefusatti et al., 2009) in the context of running non-Gaussianity. In this phenomenological model, theoretically motivated in (Chen, 2005; Byrnes et al., 2010), the non-Gaussian parameter presents a scale-dependence parametrized in terms of a running parameter .
Along with these predictions, as already mentioned, other papers derived constraints on non-Gaussian parameters from current observations. In particular, ref. (Slosar et al., 2008) find at 95% C.L. combining different data-sets in the Sloan Digital Sky Survey (SDSS), with a dominating contribution from the photometric quasar sample (Ho et al., 2008). Ref. (Xia et al., 2010) finds instead the 2- limits and from high-redshift radio sources from the NRAO VLA Sky Survey (NVSS) (Condon et al., 1998) and the SDSS quasar sample (Richards et al., 2009). More recently ref. (Xia et al., 2011) considers the analysis of high-redshift probes for the equilateral and folded models, in addition to the local one, finding, respectively , and at 95% C.L. It should be remarked that the model assumed to describe the bias correction in (Xia et al., 2011) is quite approximate as it neglects the dependence on the halo mass and further corrections considered for instance in (Desjacques et al., 2011a, b; Scoccimarro et al., 2012), so the limits derived in the equilateral and folded cases are to be considered, as pointed-out by the authors, as limits on “effective” non-Gaussian parameters and .
In the first part of this work, we study the effects of QSF models of inflation on Large-Scale Structure with special attention given to the scale-dependent correction to the linear halo bias. The interesting aspect of this correction is its direct dependence on squeezed configurations of the curvature bispectrum. Since, as we will see, the momentum-dependence of these configuration is directly related to one important parameter of the theory, this model constitutes a veritable case study for non-Gaussian effects on halo bias. We will consider the relative importance of both scale-dependent and scale-independent corrections as a function of the halo mass. We will perform a Fisher matrix analysis to assess the detectability of the effect from measurements of the galaxy power spectrum in large-volume redshift surveys and provide a first estimate of the expected uncertainty on the two parameters controlling the initial bispectrum. In addition, we will consider as well the constraints on these parameters expected from measurements of the CMB bispectrum, with particular reference to the upcoming Planck satellite.
This paper is organized as follows. In Section II we introduce the Quasi-Single Field model of inflation and present the template assumed to describe the predicted curvature bispectrum. In Section III we study the effect of the QSF model on the matter bispectrum and skewness, deriving some basic results useful for the following analysis. In Section IV we consider non-Gaussian corrections to the linear halo bias and show the results for a Fisher matrix analysis of the galaxy power spectrum. In Section V we discuss instead the expected constrains from CMB observations and discuss their combination with LSS forecasts. Finally, we present our conclusions in Section VI.
Ii Quasi-Single Field Inflation
Inflation model building in supergravity and string theory naturally leads to models of quasi-single field inflation. In this class of multifield models, there is one field direction, the inflaton direction, that satisfies the slow-roll conditions through either symmetry or fine-tuning, and many other directions, the isocurvaton directions, that have masses of order the Hubble parameter, . Supersymmetry plays an essential role in determining the masses of these isocurvatons. Without supersymmetry, at the tree level, the coupling of the scalar fields to the space-time curvature would also lead to masses of order , but these masses will run away due to loop corrections, analogous to the situation of the Higgs mass in particle physics. In cosmology, supersymmetry provides the only dynamical mechanism for the masses to maintain this value. The radiative corrections to the mass from the scalar and fermion loops automatically cancel down to the supersymmetry breaking scale . Therefore, despite the Hubble parameter being determined by a sector independent of the isocurvatons, and no matter how large or small is, the mass of light scalars will always trace the value of through the universal gravitational coupling. For the inflaton, this mass is the origin of the -problem in supergravity inflation models, and needs to be tuned away. For the isocurvatons, they become signatures of supersymmetry in the primordial universe. Finding observational evidence of such scalars constitutes an outstanding theoretical and experimental challenge.
The generic couplings between these isocurvatons and the inflaton have several possible consequences on the primordial density perturbation. The simplest one is the correction to the two-point correlation function, the power spectrum, of the density perturbation. Since the power spectrum is a function of one momentum, the observable effects only appear if such a correction is non-scale-invariant. Furthermore, we expect these effects easily can be degenerate with other types of corrections. The most distinctive signatures of these isocurvatons come from the three or higher-point correlation functions. Unlike the inflaton, the self-interactions of these isocurvaton fields are free of any slow-roll conditions and can be very large. These become the sources of large non-Gaussianities. More importantly, these non-Gaussianities turn out to have very special properties. For example for the scalar three-point function in the simplest QSF model, the momentum dependence in the squeezed limit is given by Chen and Wang (2010, 2010),
This is a property of the shape of the non-Gaussianity and is present even if the non-Gaussianity is perfectly scale-invariant. For QSF, the stability of the inflaton requires , so that . As , the isocurvatons gradually become too massive to have significant effects on density perturbations in the absence of sharp features. So we are mainly interested in , i.e. . Such a momentum dependence lies between that of the equilateral shape () which arises in single field models with large non-Gaussianity or its direct multifield generalization, and that of the local shape () which arises in multifield models with light isocurvatons (see Chen (2010); Bartolo et al. (2004); Creminelli et al. (2011) for reviews.)
While the detailed dependence of on the isocurvaton masses may be model-dependent in more general situations, the signature intermediate momentum dependence in the squeezed limit is a robust evidence for the existence of such isocurvatons. This can be seen qualitatively as follows Chen and Wang (2010, 2010). The fluctuations of the massive scalars decay after the horizon-exit. For heavy scalars they decay immediately after the horizon-exit, and for lighter scalars they decay more slowly. The scalar interactions, responsible for the large non-Gaussianities, are therefore generated between the horizon scale and the superhorizon scales. The former is responsible for the equilateral-like shapes, and the latter for local-like shapes. As a consistency check, if we look at the special limit of massless scalars, the superhorizon fluctuations do not decay, and we recover the characteristic local shape in the squeezed limit. This momentum dependence can be also seen more quantitatively as follows Baumann and Green (2011), at least for close to . Ignoring the physics within and near the horizon scale, the squeezed limit of the three-point function can be regarded as the modulation of the two-point function of two short-wavelength modes from a long-wavelength mode. After horizon exit, we know that the amplitude of a massive scalar decays as as a function of the scale factor . So the amplitude of the long-wavelength mode has decayed by a factor of by the time the short-wavelength modes start to exit the horizon. The amplitude of the modulation is proportional to the amplitude of the long-wavelength mode. Taking the massless limit as the reference point, at which we know the squeezed limit behavior from the simple locality argument Lyth and Rodriguez (2005); Starobinskiǐ (1985); Sasaki and Stewart (1996), in the massive case we get the momentum dependence .
To illustrate how close these intermediate shapes can get to the local shape but with qualitatively different values of the fundamental parameter, we look at the example of . This mass is still of order , qualitatively different from the massless isocurvaton () in multifield slow-roll inflation models. But the resulting shape has the momentum dependence , very close to the local shape characteristic of the massless isocurvatons. Therefore, how well we can measure the squeezed limit behavior, i.e. the parameter , is an important question, if any large non-Gaussianities are discovered.
ii.2 Initial curvature bispectrum
We will assume throughout the following template for the bispectrum of the Bardeen potential (with ) (Chen and Wang, 2010)
where is the Neumann function of order . For simplicity, we will further assume scale invariance for the curvature power spectrum111Our Fourier transform convention implies the following definitions for power spectra and bispectra and with ., with the constant defined as . From now on we will generically refer with to the non-Gaussian amplitude parameter for QSF models as described by the template above. If need be, non-Gaussian parameters for other models will be denoted explicitly with a superscript as, for instance, or for the local and equilateral shapes, respectively.
As shown by (Chen and Wang, 2010), the template of Eq. (4) well reproduces the main features of the bispectrum predicted by QSF models, and, in particular, provides the correct scale-dependence for squeezed configurations as a function of the parameter . In fact, in the squeezed limit, i.e. , , the leading order expression for the template becomes, for ,
For the special case , the limit is given by
being the Euler constant222For the next to leading order term goes like , therefore for should be taken into account.. In particular assuming and , we have for ,
while for we have
In the upper panels of Fig. 1 we compare the squeezed limit of flattened triangular configurations of the curvature bispectrum, that is with constant as , as described by the template above with the numerical evaluation performed in (Chen and Wang, 2010) for , , and . The lower panels of Fig. 1 show the same comparison but for squeezed, isosceles triangles, i.e. . It is evident that the template provides an accurate description of the numerical results for while it presents an increasing (but asymptotically constant) discrepancy for larger values of , of fews tens of percent in the squeezed limit. We remark that this is a minor problem as long as the templates provides the corrected momentum dependence in the squeezed limit. As discussed, this momentum dependence is the most general consequence of the models and is related to the most interesting underlying fundamental physics. In the squeezed limit, this discrepancy simply rescales the definition of . Nonetheless, more accurate comparisons between observations and a specific theoretical model will clearly require a direct evaluation of the bispectrum from the model. So far for the simplest models, direct analytical expressions for the entire shapes are not written in a closed form, and explicit evaluation is only available numerically.
A “local limit” is attained for , leading to , as for the usual local model. It should be noted, however, that even for and for squeezed configurations, the amplitude of the QSF template does not coincide with the amplitude of the local model for the same configurations. The upper panels of Fig. 2 show a comparison between the QSF template for and the local model bispectrum for both flattened (left) and isosceles (right) squeezed configurations. The shape function is normalised in such a way to obtain for any value of , as it happens for the local model, hence the agreement for in the plots. For all other configurations this is generically not true. Even for the particular case of we will denote by the amplitude parameter for the QSF bispectrum, while we will indicate with the analogous parameter for the local model. We find that for squeezed configurations, the two model present the same amplitude if , with the factor of representing the discrepancy shown in the plots as . This relation is useful in comparing the errors on the parameters expected from observations of the galaxy power spectrum since, as we will see in section IV, the scale-dependent non-Gaussian corrections depend almost exclusively on squeezed configurations. For instance, an expected error of the order of would correspond to an error on of . More importantly, current LSS constraints at 95% C.L. (Slosar et al., 2008) translate to for , so that values of are still allowed by galaxy power observations within the 2- limit. Comparisons with CMB results are less simple, as we will see, since the CMB bispectrum is naturally sensitive to all triangular configurations.
The lower panels of Fig. 2 compare instead the QSF template for with the “orthogonal” template proposed by (Senatore et al., 2010) in their Eq. (3.2). It should be noted that such template represents a good approximation to the exact orthogonal shape for configurations far from the squeezed limit, and it is therefore a viable choice only for CMB analysis. Its peculiar squeezed limit, in fact, leads to a scale-dependent correction to galaxy bias, not predicted by the true orthogonal shape, but of some phenomenological interest, as it determines an effect on bias intermediate between the local and equilateral models (Wagner and Verde, 2012; Scoccimarro et al., 2012). On the other hand, QSF inflation naturally predicts this kind of behavior. As shown by the figure, for the asymptotic behavior in the squeezed limit for the two models is the same, with a difference in amplitude of about a factor of two. Notice that for the orthogonal case we assume the same absolute value for as the QSF model, but opposite sign as the orthogonal bispectrum is negative for such triangles. Far from the squeezed limit, as it is evident in particular from the plot showing isosceles triangles, the two templates are significantly different. For this reason, we will not consider further comparisons between the approximate orthogonal template, but we will instead confront QSF predictions with those of both the local and equilateral shapes.
For all the subsequent calculations in this paper we will assume a flat CDM cosmological model with the following parameters: , , , and , leading to . The matter transfer function is computed with the CAMB code333http://camb.info. Notice that, for simplicity we are assuming scale invariance, consistent with the original expression for the bispectrum template proposed by (Chen and Wang, 2010).
Iii The matter bispectrum and skewness
The most direct effect of primordial non-Gaussianity on large-scale structure is given by its linear contribution to the matter bispectrum. At large-scales, in fact, it is possible to approximately describe the matter bispectrum by its tree-level expression in Eulerian Perturbation Theory (see (Bernardeau et al., 2002) for a general review on perturbation theory and (Liguori et al., 2010) for the case of non-Gaussian initial conditions). This is given by the sum of the primordial component and the contribution induced by gravitational instability. We have
where the primordial component, linearly evolved to redshift , is given by
with the function
expressing the Poisson equation as and representing the matter density contrast in Fourier space. The gravity-induced component is (see e.g. (Bernardeau et al., 2002))
where is the linear matter power spectrum and the second-order kernel of the perturbative expansion of is given by
On the left panel of Fig. 3 we compare the primordial component to the matter bispectrum due to QSF models to the component due to the local and equilateral non-Gaussian models as well as to the one due to gravitational instability for squeezed triangular configurations, i.e. as function of for fixed . In particular we consider for the QSF models the cases given by , , and corresponding to the increasingly long-dashed curves from bottom to top. We notice how as increases the curves approach the local model, reaching the same dependence on the scale for . The right panel shows the reduced matter bispectrum, defined as where the different curves correspond to the same, different NG models, but where they include in all cases the gravity contribution. For the choice of triangles considered in the limit , with two sides fixed at close values and , this corresponds to the squeezed limit, where different values of lead to distinctly different behaviors.
A quantity directly related to the linear matter bispectrum, and useful for the following calculations is given by the reduced skewness of the matter density field smoothed on the scale defined as
where the third order moment is given by the integral
with being the Fourier transform of a top-hat function. We notice that the reduced skewness does not depend on redshift as well as on the normalization of primordial fluctuations and it is generically mildly depends on the other cosmological parameters.444In the subsequent calculation we make use of a fit for the reduced skewness of QSF models, function of the parameter and of given by , with , , and , accurate at the 1% level for . In addition, it is almost constant even with respect to same variable . This evident from Fig. 4 where we plot for local, equilateral and QSF models with , , and assuming in all cases .
The effect of primordial non-Gaussianity on matter correlators is not limited to the large-scale bispectrum, but also involves corrections to the small-scales nonlinear evolution of both power spectrum (Taruya et al., 2008; Wagner et al., 2010; Smith et al., 2011) and bispectrum (Sefusatti et al., 2010; Figueroa et al., 2012). In the case of the bispectrum, such small scale corrections can be significant, but can be directly accessible only via weak lensing measurements. A proper assessment of the possibility offered by future weak lensing surveys to constrain primordial non-Gaussianity with the measurement of the shear higher-order correlation functions is not yet available. Instead, studies of non-Gaussian effects on galaxy correlators have witnessed recently a great deal of activity, mainly due to the scale-dependent corrections to galaxy bias which we will consider in the next section.
Iv Linear halo bias
Only relatively recently, N-body simulations with local non-Gaussian initial conditions have shown that the bias of dark matter halos receives a significant scale-dependent correction at large scales (Dalal et al., 2008; Desjacques et al., 2009; Grossi et al., 2009; Desjacques and Seljak, 2010b; Pillepich et al., 2010; Wagner and Verde, 2012; Scoccimarro et al., 2012). Several papers assumed different approaches in the theoretical description of this effect, mostly based on the peak-background split framework or on the theory of peak correlations (see for instance (Desjacques and Seljak, 2010c) and references therein).
In this work, we will consider the approach of (Scoccimarro et al., 2012), based on the peak-background split argument (Bardeen et al., 1986; Cole and Kaiser, 1989), tested in N-body simulations assuming different non-Gaussian models for the initial conditions. We assume that the Eulerian relation, in Fourier space, between the halo density contrast for a given halo mass and the matter density contrast is given, at the linear level, by
with the linear halo bias function
where we distinguish a scale-independent contribution from the scale-dependent correction .
In this section we will present theoretical results regarding the linear halo bias, keeping in mind that the effects described naturally translate into effects on the linear bias of the galaxy distribution. Galaxy bias can in fact be described as an integral of the halo bias weighted by the halo mass function and by a prescription on how to populate halos with galaxies, the Halo Occupation Distribution (HOD). We will return to this issue in Sec. IV.3 where we will discuss the detectability of non-Gaussian corrections to the galaxy power spectrum at large-scales.
iv.1 Mass function and scale-independent corrections to bias
The scale-independent contribution can be derived from the halo mass function with the usual relation (Bardeen et al., 1986)
under the assumption of the universality of the mass function and Markovianity in the excursion set derivation, with representing the large-scale component of the matter fluctuations. In the case of non-Gaussian initial conditions, the mass function receives a correction that leads in turn to a scale-independent non-Gaussian correction. Assuming the non-Gaussian mass function to be described as (Sefusatti et al., 2007) we can write
We will assume the Sheth & Tormen (Sheth and Tormen, 1999) expression for the Gaussian mass function while for the non-Gaussian relative correction we assume the simple description of (Lo Verde et al., 2008) based on the Edgeworth expansion of the non-Gaussian matter probability distribution function in the Press-Schechter framework (Press and Schechter, 1974). At linear order in this is given by
where the variable is defined as with being the linear threshold for spherical collapse and the r.m.s. of the matter perturbations smoothed on the radius , with denoting the mean matter density. In addition, represents the reduced skewness of the initial matter density field, Eq. (14). To improve the agreement between the measurements of the mass function correction in numerical simulations with the prediction of Eq. (21), a scaling parameter defined by has been considered (Grossi et al., 2009; Maggiore and Riotto, 2010; Paranjape et al., 2011). We assume here as derived in (Sefusatti et al., 2011) from the simulations of (Desjacques et al., 2009) (but see also (Tinker et al., 2008), for a similar correction in the context of Gaussian initial conditions).
In the left panel of Fig. 4 we plot the non-Gaussian correction to the halo mass function at . Both the skewness and consequently (in this description) the mass function, at fixed , do not depend strongly on , at least for . A mild dependence can be noticed as approaches the limiting value of 1.5. On the right panel of Fig. 4 we show the relative scale-independent bias corrections given by as a function of for the same models and . Such corrections are typically negative and below one percent for moderate values of .
iv.2 Scale-dependent corrections to bias
For the scale-dependent correction to the linear halo bias due to non-Gaussian initial conditions we assume the following expression from (Scoccimarro et al., 2012)
and with a top-hat filter function of radius corresponding to the halo mass as . This description of the linear bias correction is valid under the assumptions of Markovianity and universality of the mass function which we adopt here for simplicity.
For local non-Gaussianity it is easy to show that in the large-scale limit
so that the second term in Eq. (22) vanishes and the first term gives
that is the usual expression first proposed in (Dalal et al., 2008; Slosar et al., 2008) with for small . The first term of Eq. (22) however generalizes to any non-Gaussian model as in (Schmidt and Kamionkowski, 2010), while the second term accounts for additional corrections studied in (Desjacques et al., 2011a, b). In addition, the full result of (Scoccimarro et al., 2012) accounts as well for non-Markovian effects and departures from universality of the mass function. Such effects will have to be taken into account in a proper comparison with numerical simulations results but we can neglect them here as they will not affect our results.
In the case of the QSF model as described by the template for the curvature bispectrum in Eq. (4), from the squeezed limit, Eq. (5), we can derive the large-scale approximation of the integral , which is given, for , by
where, following the notation of (Scoccimarro et al., 2012),
In the case , we have
Since at large scales , the expected scale-dependent correction to halo bias presents in this case the behavior
Over the allowed range of values of , , such corrections will therefore interpolate between and , the latter corresponding to the effect of local NG.
In Fig. 6 we show the quantity , Eq. (23), as a function of evaluated for different halo masses and divided by its asymptotic value, Eq.s (26) and (28) (see also Fig. 2 and 3 in (Scoccimarro et al., 2012)). We notice that different masses correspond to specific dependence on scale for intermediate values of . In the case of the QSF model for this aspect is more evident than in the otherwise similar local model. It is evident that the asymptotic approximation is able to describe the effect only at the very largest scales, while the full evaluation of the integral is required already for wavenumbers above .
In Fig. 7 we finally show, with thick curves, the relative scale-dependent correction with given by Eq. (22) as a function of for two representative halo masses and . Thin curves correspond instead to sole contribution due to the first term on the r.h.s. of Eq. (22) while light-colored curves denotes negative values. It is evident, in the first place that the case is indeed very close to the local NG case, up to the mentioned numerical factor, both in term of scale-dependence and as a function of mass. As we consider lower values of the dependence on mass, and, in particular, the contribution on the second term on the r.h.s. of Eq. (22) becomes relevant, particularly for the low mass examples. In fact, already for , while in the high mass case such additional corrections amounts to a overall decrease in the amplitude of the scale-dependent correction, for the low mass case, they induce a change in the sign of the correction itself, albeit keeping a similar absolute amplitude. We should remark that this sort of behavior has never been tested in simulations for this specific model, and a dedicated study is clearly required. However, as already mentioned, comparisons with different non-local models (e.g. (Desjacques et al., 2011a; Scoccimarro et al., 2012)) indicate that Eq. (22) does describe correctly the mass dependence of the halo bias corrections to the extent allowed by the error on current numerical results. If this description will find further confirmation, possibly for the specific case of the QSF model, the results shown of Fig. 7 would be particularly interesting in terms of observational constraints. One can imagine, in fact, the possibility of combining the measurements of the power spectrum for different galaxy populations where we expect a correction different in sign, allowing in principle for a significant reduction of the degeneracy between the non-Gaussian parameters and the linear Gaussian bias (see e.g. Seljak (2009); Hamaus et al. (2011) for recent works in this direction). The other panels of Fig. 7 referring to the QSF model, show that a similar situation is recovered for even smaller values of , as for instance corresponding to a scale-dependent correction .
iv.3 Fisher matrix
We perform a Fisher matrix analysis in order to determine the expected, simultaneous constraints on both the and parameters from future galaxy surveys. Similar forecasts, in terms of the single parameter, have been performed, for the local model alone in (Dalal et al., 2008; Carbone et al., 2008, 2010; Cunha et al., 2010; Sartoris et al., 2010) for upcoming galaxy and clusters surveys. The cases of equilateral and orthogonal non-Gaussianity, in addition to the local model, have been studied by (Giannantonio et al., 2011) and (Pillepich et al., 2012), focusing respectively on the EUCLID survey (Laureijs et al., 2011) (combining weak lensing with photometric and spectroscopic data) and the eRosita, X-ray cluster survey (Predehl et al., 2010). In these works, the effects of non-local models include scale-dependent corrections to halo bias, described by expressions analogous to Eq. (22). Running non-Gaussianities described by an additional running parameter , such as those for both the local and equilateral shapes (Chen, 2005; Byrnes et al., 2010), also depend on two parameters. Forecasts for these models have been considered in (Sefusatti et al., 2009; Lo Verde et al., 2008; Giannantonio et al., 2011). Some examples of these running non-Gaussianities may also alter the momentum dependence in the squeezed limit of the bispectrum. But the difference between this case and the QSF inflation is clear. The former is caused by the running of non-Gaussianity and the latter by the shape of non-Gaussianity. For QSF inflation, even if is not much less than one and so the deviation from the local shape is significant, the bispectrum can be still scale-invariant. But for running Gaussianities, such a momentum dependence corresponds to a case with very strong overall scale-dependence.
Our goal is to provide an estimate of the possibility to constrain the parameter assuming a positive detection of non-Gaussianity by Planck. For this purpose we consider two large-volume surveys comparable in size and redshfit range to EUCLID and LSST. Such surveys are already expected to provide constraints to (Carbone et al., 2010; Giannantonio et al., 2011) from measurements of the galaxy power spectrum comparable to those expected by Planck (Yadav et al., 2007, 2008; Fergusson and Shellard, 2007). Being this a first, indicative assessment, we will assume a simplified description, characterized simply by the field of view, the redshift range (i.e. the volume) and the expected linear (Gaussian) bias as a function of redshift. Since we expect most of the signal to come from the largest scale probed, we do not expect a significant impact of shot-noise (see, for instance, (Carbone et al., 2010; Giannantonio et al., 2011)). On the other hand, shot-noise should not affect, by design, the BAO analysis, the primary target of these missions. We will nevertheless include a shot-noise contribution to the power spectrum variance as detailed below. The photometric or spectroscopic nature of redshfit observations is assumed to play a negligible role as, again, the determination of large-scale power does not require high precision in the radial galaxy positions.
For our analysis we assume a (“EUCLID-like”) survey, denoted as V1, with a 20,000 deg field of view and a redshift range of , and a galaxy population with fiducial (Gaussian) linear bias parameter given by (Orsi et al., 2010). This would allow a comparison with the results of (Giannantonio et al., 2011) for the local and equilateral models. As opposed to (Giannantonio et al., 2011), which considers 12 equally populated redshift distributions, we will simply compute the Fisher matrix information in redshift bins of size . As an exercise we extend this galaxy population, described by the same bias evolution , to a larger (“LSST-like”) volume, denoted as V2, given by a field of view of 30,000 deg and redshift range . This corresponds to the second example of (Carbone et al., 2010), although with a lower (therefore more conservative) fiducial value for the linear bias.
We consider a two-dimensional Fisher matrix for the parameters obtained as a sum over all redshift bins of the three-dimensional matrix for the parameters , marginalized over the linear bias parameter . Since the scale-dependent correction depends linearly on the product , we can expect a significant degeneracy with and a marginalization over the value of should be taken into account. Such marginalization is performed in (Giannantonio et al., 2011) but not in (Carbone et al., 2010), where, on the other hand, only the very largest scales are considered for the analysis of the local model alone. The matrices are defined respectively as
where the sum runs over the available wavenumbers from the fundamental frequency of the redshfit bin , being the bin volume to , in steps of .
The expression for the galaxy power spectrum is given by
where we include only linear corrections in . It should be noted that primordial non-Gaussianity modifies both the amplitude and the form of the galaxy power spectrum and a large range in helps reducing degeneracies between non-Gaussian parameters and bias. Following (Giannantonio et al., 2011), we choose with the scale obtained from the redshift-dependent equality constant. At this choice provides, for our cosmology, , as in (Giannantonio et al., 2011) and it implies the range of values for . We will later discuss how our results depend on this choice, comparing them with those obtained from a more conservative assumption corresponding to at . For the matter power spectrum we assume the linear expression and consequently ignore nonlinear corrections at small scales. While on one side we do expect such corrections to be there for close to , the final results do not strongly depend on this assumption since we neglect them as well in the power spectrum variance . In fact, in our approximation, their inclusion would only reduce the impact of shot-noise on . The range in wavenumbers is, on the other hand, well within reach of accurate predictions in nonlinear perturbation theory (see e.g. (Crocce and Scoccimarro, 2008)). Notice that, for simplicity, we also ignore possible corrections due to nonlinear bias. This is in part justified by the small value of the quadratic bias parameter corresponding to the linear one (typically ) that can be derived by the expressions of Scoccimarro et al. (2001).
The power spectrum variance is given by (see e.g. (Sefusatti et al., 2009))
which accounts for non-Gaussian corrections to the bias and where is the Gaussian galaxy power spectrum including shot-noise,
Since we are describing the galaxy population simply in terms of the value of the linear bias we derive the mean number density in the redshift bin making the quite drastic but simplifying assumption that each dark matter halo of mass above a certain minimal mass contains a single galaxy. The value of is determined imposing the relation
where we assume the Sheth-Tormen (Sheth and Tormen, 1999) expressions for the Gaussian halo mass function and linear halo bias . The galaxy density is then obtained as
We notice that since the halo bias weights more massive halos with respect to smaller mass halos, this procedure tends to overestimate the value of and therefore underestimates the value of the density when compared to the same calculation performed assuming the proper Halo Occupation Distribution (HOD) for the galaxy sample. In addition, the definition of the minimal mass allows us, perhaps improperly, to define a mean or “characteristic mass” for the halo population of the bin at given by
We assume the mean mass to evaluate the integral , Eq. (23), and its derivatives for each bin. Notice that in the marginalization over the bias, i.e. in the derivative , we not only consider the explicit dependence on the parameter but derive as well all mass-dependent quantities like or the integrals as as ). Obviously, the correct procedure would have involved integrating all bias corrections over the proper range of halo masses and the proper HOD. However, such a drastic solution is still more conservative than neglecting altogether the marginalization over the mass-dependence of such corrections as done in previous similar analysis in the literature. Ref. (Giannantonio et al., 2011) correctly points out that while the mass-dependence of such corrections is relevant in the case of the equilateral model, predictions could not be properly tested in simulations yet for this specific model (see e.g. (Scoccimarro et al., 2012)) and they restrict their expressions for to its asymptotic value at small . However, we assume the full expression in Eq. (22) to be valid, noticing that such issues are less important for our model when and that after marginalization over bias (and mass), our results are, as we will see, consistent with those of (Giannantonio et al., 2011) for both the local and equilateral models.
Finally, we do not consider a full marginalization over cosmological parameters. Forecasted constraints on the local parameter, from measurements of the galaxy power spectrum marginalized over the cosmology (with priors from the Planck CMB power spectrum) have been studied in (Carbone et al., 2010) finding an increase in the error of about 30% for a EUCLID-like experiment. Ref. (Giannantonio et al., 2011) compares instead errors on different NG models marginalized on cosmology with and without Planck priors. They find, as one can expect, a particularly strong degeneracy of the equilateral parameter with cosmological ones, due to the lack of a scale-dependence in the halo bias correction. We might therefore expect an larger impact of the uncertainty on cosmological parameters when low value of are considered.
|V1 survey, ,|
|V2 survey, ,|