Halo statistics in non-Gaussian cosmologies: the collapsed fraction, conditional mass function, and halo bias from the path-integral excursion set method.
Characterizing the level of primordial non-Gaussianity (PNG) in the initial conditions for structure formation is one of the most promising ways to test inflation and differentiate among different scenarios. The scale-dependent imprint of PNG on the large-scale clustering of galaxies and quasars has already been used to place significant constraints on the level of PNG in our observed Universe. Such measurements depend upon an accurate and robust theory of how PNG affects the bias of galactic halos relative to the underlying matter density field. We improve upon previous work by employing a more general analytical method - the path-integral extension of the excursion set formalism - which is able to account for the non-Markovianity caused by PNG in the random-walk model used to identify halos in the initial density field. This non-Markovianity encodes information about environmental effects on halo formation which have so far not been taken into account in analytical bias calculations. We compute both scale-dependent and -independent corrections to the halo bias, along the way presenting an expression for the conditional collapsed fraction for the first time, and a new expression for the conditional halo mass function. To leading order in our perturbative calculation, we recover the halo bias results of Desjacques et. al. (2011), including the new scale-dependent correction reported there. However, we show that the non-Markovian dynamics from PNG can lead to marked differences in halo bias when next-to-leading order terms are included. We quantify these differences here. We find that the next-to-leading order corrections suppress the amplitudes of both the scale-dependent and -independent bias by for massive halos with , and for halos with . The corrections appear to be more significant as the halo mass is lowered, though we caution that the apparently large effects we observe in the low-mass regime likely signal a breakdown of the perturbative approach taken here.
keywords:cosmology: theory, large-scale structure of the Universe, inflation - galaxies: statistics
The inflationary paradigm provides a robust framework for explaining key aspects of our observable Universe such as its geometric flatness, features of the cosmic microwave background, and the initial conditions for structure formation. Despite these remarkable successes, we still know very little about the physics behind it, and currently cannot distinguish among a wide variety of viable inflationary models. One of the most promising ways to differentiate among these models is to probe the statistics of the initial density fluctuations (see Desjacques & Seljak, 2010a, and references therein). While the simplest scenarios - the canonical single-field slow-roll models - predict an almost perfectly Gaussian distribution of initial fluctuations, more general inflationary models predict significant deviations from Gaussianity that observations might yet be sensitive enough to detect. There is therefore great interest in developing ways to measure primordial non-Gaussianity (PNG) since its detection would have profound implications for inflationary theory.
There are presently two methods that have so far been applied with some success to place significant constraints on the level of PNG in our observed Universe. The first is the statistical imprint of PNG in the temperature anisotropies of the cosmic microwave background (CMB), which directly probe the initial fluctuations while they are still in the linear regime (e.g. Komatsu, 2010). The second is the imprint on the large-scale structure that develops as the initial fluctuations grow to the highly non-linear point of forming galaxies, which serve as tracers of the underlying matter. In general, non-linear growth can complicate the interpretation of the observed structure, both because density fluctuations develop non-Gaussianity even when the initial conditions are purely Gaussian, and because the theoretical predictions in this regime ultimately depend upon N-body simulations. On large enough scales, however, the density fluctuations filtered on these scales are still linear, and the problem reduces to the theory of how well galaxies trace the large-scale mass distribution - the so-called “bias.”
The prospect of probing PNG with large-scale structure improved dramatically when it was discovered that the mode coupling effects of PNG induce a scale-dependent signature in the power spectrum of biased tracers (such as galactic halos) on large scales (Dalal et al., 2008; Matarrese & Verde, 2008; Afshordi & Tolley, 2008). The first and best studied example of this signature involves the local-quadratic model of PNG, in which the primordial Bardeen potential fluctuation in the matter-dominated epoch, , is obtained from a quadratic transformation of the local Gaussian fluctuation field, , according to
where is the so-called non-linearity parameter (Salopek & Bond, 1990; Komatsu & Spergel, 2001). In this case, while the Gaussian field is entirely characterized by its power spectrum, , information about higher-order correlations, for example through the bispectrum, is required to characterize the statistics of the non-Gaussian potential. To first order in , equation (1) gives a bispectrum with the form111The bispectrum in equation (2) is more general than equation (1), as it can be generated in number of different models that do not involve the latter (see footnote 34 in Komatsu et al., 2011, for example). It is nonetheless customary to refer to this form for the bispectrum as the local template. ,
Observational constraints have been placed on this form of the bispectrum by finding the range of allowed amplitudes, expressed in terms of . For example, the CMB anisotropy measurements by the Wilkinson Microwave Anisotropy Probe seven-year data analysis (WMAP7) find a limit of (Komatsu et al., 2011). The bispectrum in equation (2) is important in the phenomenology of PNG because a detection of non-zero would rule out standard single-field inflation (Creminelli & Zaldarriaga, 2004; Seery & Lidsey, 2005; Chen et al., 2007; Cheung et al., 2008).
In order to constrain equation (2) from observations of large-scale structure, it is necessary to consider the expected halo bias for this model, i.e. the ratio of the fractional halo number density to the fractional matter density. In Fourier space, this ratio has been found to depart from the Gaussian expectation by a correction term, , containing two parts: one that depends on the wavenumber (scale-dependent) and one that does not (scale-independent). In the limit of small wavenumber, the correction approaches the form , where is the expected Gaussian bias (Dalal et al., 2008; Matarrese & Verde, 2008; Afshordi & Tolley, 2008). Based upon this assumed scale-dependence, Slosar et al. (2008) have already constrained to be in the range ( limit) using the clustering of massive galaxies and quasars in the Sloan Digital Sky Survey; a result that is competitive with the WMAP7 constraints. There is naturally a great interest in this method as future large-scale structure surveys of ever-increasing volume utilizing both the galaxy power spectrum and bispectrum may surpass the CMB measurements in constraining (Scoccimarro et al., 2004; Jeong & Komatsu, 2009; Nishimichi et al., 2010; Baldauf et al., 2011; Giannantonio et al., 2012).
Numerical N-body methods have confirmed that the halo bias scales as in the small- limit in the local-quadratic model, with a redshift dependence inversely proportional to the linear perturbation growth factor, , as predicted by the analytical theory (Dalal et al., 2008; Desjacques et al., 2009; Pillepich et al., 2010; Grossi et al., 2009; Nishimichi et al., 2010; Smith & LoVerde, 2011). However, the amplitude of the bias is still somewhat uncertain, as the N-body simulations so far produce a range of values differing at the level (Desjacques & Seljak, 2010b). On the other hand, the analytical predictions disagree significantly with results from N-body simulations in models beyond the local-quadratic case (Desjacques & Seljak, 2010c; Shandera et al., 2011; Wagner & Verde, 2012). This motivated Desjacques et al. (2011b) to re-examine three analytical derivations of the bias (see also Smith et al., 2012): 1) An approach based on the statistics of thresholded regions in the density field (Matarrese & Verde, 2008). 2) A peak-background split (PBS) approach based on the separation of uncorrelated long- and short-wavelength contributions to the Gaussian perturbations (Dalal et al., 2008; Slosar et al., 2008; Schmidt & Kamionkowski, 2010). 3) A second PBS approach (Desjacques et al., 2010) using the conditional halo mass function, which they derive by an extension of the Press & Schechter (1974) method to non-Gaussian initial conditions (also see Matarrese et al., 2000; Lo Verde et al., 2008). This approach is conceptually different from the previous one because it does not involve a separation of scales in the Gaussian perturbations, but instead considers through the conditional mass function how the local halo-abundance depends on the large-scale non-Gaussian density contrast. In addition to showing that the thresholding method cannot be reconciled with N-body simulations, Desjacques et al. (2011b) use the last two approaches to derive a new scale-dependent contribution to the bias that was previously overlooked in the literature. In a companion paper (Desjacques et al., 2011a), they showed that the new term is critical for improving the analytical predictions that were previously discrepant with N-body simulations.
It is important to confirm the above findings with more general methods. Here, we consider an independent analytical approach to the halo bias - the excursion set formalism (Bond et al., 1991; Lacey & Cole, 1993). Until recently, the excursion set method had been analytically tractable only in the case with Gaussian initial conditions and, even then, only when a sharp k-space filter was used. In this case, the dynamics in the excursion-set random-walk model for identifying halos in the initial density field are Markovian. Maggiore & Riotto (2010a, b, c) recently showed how to extend the excursion set model to include non-Markovian dynamics by formulating it with a path integral. This breakthrough opens the door to non-Gaussian initial conditions and/or more general filter functions (for a different approach to non-Markovian dynamics, see Paranjape & Sheth, 2012; Paranjape et al., 2012; Musso & Paranjape, 2012; Musso & Sheth, 2012; Musso et al., 2012). The path-integral approach has been successfully applied in a number of contexts involving PNG (Maggiore & Riotto, 2010c; D’Amico et al., 2011a; de Simone et al., 2011b; D’Amico et al., 2011b; D’Aloisio & Natarajan, 2011; de Simone et al., 2011a). However, it has not yet been used to derive the scale-dependent correction to the halo bias.
In what follows, we use the path-integral excursion set method to derive expressions for the conditional collapsed fraction and conditional mass function for non-Gaussian models with general bispectra. We then use these expressions to compute both scale-dependent and -independent corrections to the halo bias. In comparison with the results of Desjacques et al. (2011b), our results will come closest to the conclusions of their third approach described above. In the process of deriving our result, we will be able to investigate more specifically under what circumstances their result is valid. In particular, the excursion-set approach shows through the non-Markovian dynamics that environmental effects on halo formation can lead to marked differences in halo bias. We will also quantify these differences here by a perturbative calculation, applied for illustrative purposes to the familiar local-quadratic model, as well as a second example - the so-called orthogonal template.
The remainder of this paper is organized as follows. In 2, we review statistics of the linear density field and define the bispectrum templates used in this work for plotting purposes. In 3, we outline the path-integral excursion set method of Maggiore & Riotto (2010a, b, c). In 4, we use the formalism to calculate the conditional collapsed fraction to leading-order, and also compute the next-to-leading order environmental corrections. We then use the collapsed fraction to obtain expressions for the conditional mass function in 5. From the conditional mass function, we obtain scale-dependent and -independent linear bias parameters in 6. Finally, we summarize our results and offer concluding remarks in 7.
2 Statistics of density fluctuations in models with primordial non-Gaussianity
2.1 The density contrast and its two- and three-point functions
The excursion set model is formulated in the Lagrangian picture; it is a method for computing halo statistics from the statistics of the linearly extrapolated initial density fluctuations. We quantify fluctuations in the linearly extrapolated density field, , with the density contrast smoothed on scale about a point ,
where is a spherically symmetric filter function with characteristic scale and the un-smoothed density contrast is . The Fourier transform of the smoothed density contrast is related to the primordial Bardeen potential in the matter-dominated epoch through the cosmological Poisson equation222Strictly speaking, this relation holds only in the synchronous comoving gauge (Jeong et al., 2012).,
Here, we define
where is the matter transfer function normalized to unity on large scales, is the linear growth factor of the potential normalized to unity during the epoch of matter domination ( for our fiducial cosmology), is Hubble’s constant, and is the present-day matter density in units of the critical density.
Gaussian initial density fluctuations are uniquely characterized by their variance,
More generally, higher-order correlation functions are required to characterize non-Gaussian initial fluctuations. In what follows, we consider non-Gaussianity that is characterized solely through the three-point function,
where is the Dirac delta function. We neglect the effects of all higher-order spectra.
The excursion set method is unique among other analytical techniques for treating non-Gaussianities because it depends on for the full range of smoothing scales down to the Lagrangian radius of the halo. As we will see, the mixed three-point correlator will be particularly important in our calculations; it is the source of the scale-dependence of the bias. We may rewrite this correlator in a way that is convenient for manifesting the scale-dependence by collapsing the delta function, rearranging, and relabeling the integration variables to obtain
where . The quantity in the brackets is a form factor denoted by in the notation of Desjacques et al. (2011b). We adopt this notation and express the mixed correlator as
The advantages of rewriting in this form will become apparent in the following sections.
2.2 Phenomenological templates of the primordial bispectrum
Primordial bispectra generated by inflationary models vary considerably and can be quite complicated. For the purpose of computing the effects of a primordial bispectrum on large-scale structure, it is convenient to employ commonly used phenomenological templates. We will use the local template extensively when plotting our results, where the primordial bispectrum is given by equation (2). At times, we will find it desirable to bring out the effects of the form factor, , in our final results. The local template is not ideal for this task because, in this case, asymptotes to in the low- limit. As a result, we will also use the orthogonal template (Senatore et al., 2010), where the bispectrum has the form
and is a constant parameter that determines the amplitude of the bispectrum. In this case, the form factor scales as in the low- regime.
3 The path-integral generalization of the excursion set formalism to include non-Gaussian initial conditions
Here we summarize the non-Markovian extension of the excursion set formalism by Maggiore & Riotto (2010a, c). For more details on the original formulation of the excursion set model, we refer the reader to the pioneering paper by Bond et al. (1991) (also see the review of Zentner (2007) and references therein).
It is convenient to linearly extrapolate the initial density fluctuations to the present day. With this choice, the density field stays fixed in time, and therefore so do the variance and three-point function, while the linear over-density threshold for collapse, , acquires an additional redshift dependence, , where is normalized to unity at the present day. In the excursion set procedure, a filter function with characteristic scale is centered on a fiducial point in space. The density contrast is smoothed about that point with some large initial scale, , to obtain the smoothed density contrast, , with corresponding variance . In what follows, the initial filtering scale always corresponds to the limit, , so that and . The scale of the filter function is decreased and the corresponding and are again calculated. As this procedure is repeated, the stochastic variable executes a random walk. When first exceeds the collapse threshold , the fiducial point is assumed to reside within a halo with mass set by the filter scale. When numerically evaluating our results, we will use the coordinate-space top-hat filter, where is assumed to be the initial comoving radius of the collapsed over-density, so that the mass of the halo is to a good approximation given by . The Fourier transform of the coordinate-space top-hat filter is
However, for simplicity, and since we are mainly interested in the effects due to non-Gaussianity, we will neglect non-Markovian correction terms due to the use of this filter function. In terms of the derived expressions, this is equivalent to using the sharp -space filter in the formalism.
Maggiore & Riotto (2010a, c) showed how to formulate the above model in terms of a path integral. Consider the “trajectory” traced out in -space. The starting point is to discretize the “time” interval so that , where . The probability density in the space of trajectories may be written as
The integral representation of the Dirac delta function,
is then used to write
where we have used the notation,
From here on, we will use the notation . The probability density for a trajectory to obtain a value at the independent coordinate , without having crossed , is obtained by integrating over the space of possible trajectories,
Note the dependence of on the connected correlators for scales spanning the “time” interval. Through this dependence, the path-integral excursion set formalism contains additional information about the relationship between halo formation and the environment compared to the classic Press-Schechter approach and its simple extensions to the case with PNG.
The upper limits on the integrals in (16) limit trajectories to stay below the collapse barrier. In the excursion set model, the fraction of mass contained within halos with masses greater than is equal to the fraction of trajectories that cross before . This fraction is calculated by integrating (16) over all possible to obtain the fraction of trajectories that reach without having crossed , and then taking the complement,
After taking the continuum limit, , the mass function follows from
So far the discussion has been completely general. Let us now consider the evaluation of the above expressions in specific cases of interest. First, we take the case of Gaussian initial conditions, where the connected correlators with vanish, and we are left with
The situation simplifies even more if we use the sharp k-space filter function, for which . In this case let us define
Maggiore & Riotto (2010a) have shown that the corresponding probability density,
yields the standard excursion set result upon taking the continuum limit (),
In equations (20), (21) and (22), the superscripts “g” and “m” represent Gaussian and Markovian respectively; the latter meaning that the corresponding random walks are Markovian processes - a property exhibited only in the case of Gaussian initial conditions and the sharp k-space filter. Maggiore & Riotto (2010a) have also shown how to handle non-Markovian effects from the coordinate-space top-hat filter in an approximate way by perturbing over the Markovian expression, though we do not consider such effects here (see however Ma et al., 2011).
Let us now turn our attention to the case with PNG - the focus of this work. As stated in the last section, we neglect the effects of the higher-order connected N-point functions, where (i.e. we consider only the primordial bispectrum from PNG in our calculations) so that
This expansion333D’Amico et al. (2011a) point out in their mass function calculation that the second term on the right-hand side of equation (24) yields a term that goes as , which can be of order of unity for very large masses (see their Figure 2), indicating that the expansion breaks down in that regime. (Maggiore & Riotto, 2010c) forms the backbone of our perturbative calculation of the collapsed fraction described in the next section.
4 The conditional collapsed fraction
Consider a large, spherical region of the Universe with Lagrangian radius , containing mass , and with a smoothed linear density contrast, . The conditional collapsed fraction, , is the fraction of mass in halos with masses between some lower threshold, (corresponding to smoothing scale ), and . It is derived within the excursion set formalism from the conditional probability for a trajectory passing through to reach , without ever having crossed the barrier . This probability density is represented in discrete form by (Ma et al., 2011; D’Aloisio & Natarajan, 2011; de Simone et al., 2011a)
On the right hand side of (25), “” and “” are integers given by and respectively and . The conditional collapsed fraction corresponds to the fraction of excursion-set trajectories that first cross between the scales and , which can be obtained by integrating equation (25) over and taking its complement,
We start the evaluation of equation (25) by expanding in the numerator and keeping terms up to first order in the three-point correlator,
where the first terms on the right-hand side correspond to the Gaussian contribution and the non-Gaussian correction respectively. Similarly, in the denominator of (25), we have
In the numerator, we perform a separation between the density contrast smoothed on the halo () and environmental () scales. This is achieved by breaking up the sum in the second term of (27),
The remaining details of the collapsed fraction calculation are reserved for the appendix due to their technical nature. An important detail, however, is that the triple sums in equation (29) become triple integrals over , , and in the continuum limit, which are not analytically tractable. However, we can move forward analytically with an extension of the approach of Maggiore & Riotto (2010c) (also see D’Aloisio & Natarajan (2011) and de Simone et al. (2011a) ). We employ a Taylor expansion of about the point ,
For illustrative purposes, in Figure 1 we show the three-point connected correlators, and , for , corresponding to in our fiducial cosmology. In the remainder of this work we will consider the terms in (30) originating from the and contributions. In keeping with the terminology of Maggiore & Riotto (2010c), we will refer to the terms as “leading-order,” while the terms will be called “next-to-leading order.” We again emphasize that all results in this paper are to first order in the three-point correlator.
4.1 The leading-order terms
The remaining steps of the calculation are detailed in Appendix A.1. The final expression for the conditional collapsed fraction up to leading order is
and the connected correlators evaluated at the two scales enter through
are the standard excursion set expressions for the conditional collapsed fraction and first-crossing rate in the case of Gaussian initial conditions and Markovian random walks. Note that (32) takes the form of the Gaussian and Markovian result plus correction terms proportional to the three-point connected correlators.
Figure 2 shows the conditional collapsed fraction at to leading order for the local template. The top panel on the left shows as a function of for a typical large-scale mass fluctuation with . Here, what we mean by a “typical” fluctuation is that the environmental peak height, is unity. For , this translates to in our fiducial cosmology. PNG has a larger impact on the conditional collapsed fraction when only larger halos are included in the latter (case with larger ). PNG has a smaller effect when the conditional collapsed fraction is dominated by less massive halos (case with smaller ), whose abundances are not as strongly affected by PNG. The plot on the right of Figure 2 shows for a fixed and as a function of the environmental peak height, . It shows that PNG has a larger impact on the conditional collapsed fraction within higher significance (i.e. rarer) peaks.
4.2 The next-to-leading order terms
The next-to-leading order terms in the conditional collapsed fraction are obtained from the terms in equation (30). We adopt a short-hand notation similar to Maggiore & Riotto (2010c) for the derivatives of the three-point correlators,
The details of the calculation are given in Appendix A.2. The next-to-leading order terms in the conditional collapsed fraction are
Figure 3 shows the effect of adding the next-to-leading order terms in the local template. As in Figure 2, the environmental mass and peak height are set to and respectively. The top panel shows the ratios of both (thin, dashed and dotted) and (thick, solid and dot-dashed) to the Gaussian and Markovian collapsed fraction, . The bottom panel shows the fractional change from adding , indicating in this example that the next-to-leading order corrections act to suppress the leading-order result by for and to enhance by the same factor for .
5 The Conditional mass function
The conditional mass function is an important ingredient for deriving the halo bias in the next section. We therefore devote this section towards obtaining leading-order and next-to-leading order expressions for it. The conditional mass function may be written in terms of the first-crossing rate,