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Mon. Not. R. Astron. Soc. 000, 000–000 (0000)

Printed 5 February 2008

A (MN L TEX style ?le v1.4)

Tests for primordial non-Gaussianity

Licia Verde1 , Raul Jimenez2 , Marc Kamionkowski3 & Sabino Matarrese4

1 2 3 4

Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08540–1001 USA (lverde@astro.princeton.edu) Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road, Piscataway, NJ 08854–8019 USA (rauj@physics.rutgers.edu) Mail Code 130–33, California Institute of Technology, Pasadena, CA 91125 USA (kamion@tapir.caltech.edu) Dipartimento di Fisica ”Galileo Galilei”, via Marzolo 8, I-35131 Padova, Italy. (matarrese@pd.infn.it)

arXiv:astro-ph/0011180v2 2 Mar 2001

5 February 2008

ABSTRACT

We investigate the relative sensitivities of several tests for deviations from Gaussianity in the primordial distribution of density perturbations. We consider models for non-Gaussianity that mimic that which comes from in?ation as well as that which comes from topological defects. The tests we consider involve the cosmic microwave background (CMB), large-scale structure (LSS), high-redshift galaxies, and the abundances and properties of clusters. We ?nd that the CMB is superior at ?nding non-Gaussianity in the primordial gravitational potential (as in?ation would produce), while observations of high-redshift galaxies are much better suited to ?nd non-Gaussianity that resembles that expected from topological defects. We derive a simple expression that relates the abundance of high-redshift objects in non-Gaussian models to the primordial skewness. Key words: cosmology: theory - galaxies - clusters of galaxies - large scale structures cosmic microwave background - methods: analytical

1 INTRODUCTION Now that cosmic-microwave-background (CMB) experiments (de Bernardis et al. 2000; Jaffe et al. 2000; Balbi et al 2000; Lange et al. 2000) have veri?ed the in?ationary predictions of a ?at Universe and structure formation from primordial adiabatic perturbations, we are compelled to test further the predictions of the simplest single-scalar-?eld slow-roll in?ation models and to look for possible deviations. Measurements of the distribution of primordial density perturbation afford such tests. If the primordial perturbations are due entirely to quantum ?uctuations in the scalar ?eld responsible for in?ation (the “in?aton”), then their distribution should be very close to Gaussian (e.g., Guth & Pi 1982; Starobinski 1982; Bardeen, Steinhardt & Turner 1983; Falk, Rangarajan & Srednicki 1993; Gangui et al. 1994; Gangui 1994; Wang & Kamionkowski 2000; Gangui & Martin 2000). However, multiple-scalar-?eld models of in?ation allow for the possibility that a small fraction of primordial perturbations are produced by quantum ?uctuations in a second scalar ?eld. If so, the distribution of these perturbations could be non-Gaussian (e.g., Allen, Grinstein & Wise 1987; Kofman & Pogosyan 1988; Salopek, Bond & Bardeen 1989; Linde & Mukhanov 1997; Peebles 1999a; Peebles 1999b; Salopek 1999). Moreover, it is still possible that some component of primordial perturbations are due to topological defects or some other exotic causal mechanism (Bouchet et al. 2000), and if so, their distribution should be non-Gaussian (e.g., Vilenkin 1985; Vachaspati 1986; Hill, Schramm & Fry 1989; Turok 1989; Albrecht & Stebbins 1992). Detection of any non-Gaussianity would thus be invaluable for appreciating the nature of the ultra-high-energy physics

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that gave rise to primordial perturbations. Ruling such exotic possibilities in or out will also be necessary to test the assumptions that underly the new era of precision cosmology. There are several observables that can be used to look for primordial non-Gaussianity. CMB maps probe cosmological ?uctuations when they were closest to their primordial form, and many authors have developed various mathematical tools to test the Gaussian hypothesis. The statistics of present-day large scale structure (LSS) in the Universe can also be used (e.g., Coles et al. 1993; Luo & Schramm 1993; Lokas et al. 1995; Chodorowski & Bouchet 1996; Stirling & Peacock 1996; Durrer et al. 2000; Verde & Heavens 2000). The properties and abundances of the most massive and/or highest-redshift objects in the Universe also contain precious information about the nature of the initial conditions (e.g., Chiu, Ostriker & Strauss 1998; Robinson, Gawiser & Silk 1999; Robinson, Gawiser & Silk 2000; Willick 2000; Matarrese, Verde & Jimenez 2000 (MVJ); Verde et al. 2000b). In Verde et al. (2000a; VWHK00), the relative sensitivities of the CMB and LSS to several broad classes of primordial non-Gaussianity were compared, and it was found that forthcoming CMB maps can provide more sensitive probes of primordial non-Gaussianity than galaxy surveys. Here we extend the results of that paper to include comparisons to the abundances of high-redshift galaxies as well as the abundance and properties of clusters. One of our original aims was to determine whether any of these probes would be able to detect the miniscule deviations from Gaussianity that arise from quantum ?uctuations in the in?aton; unfortunately, we have been unable to ?nd any. Nevertheless, some detectable deviations from Gaussianity are conceivable with multiple-?eld models of in?ation and/or some secondary

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contribution to primordial perturbations from topological defects. We will follow VWHK00 and parameterize the primordial nonGaussianity with a parameter that can be dialed from zero (corresponding to the Gaussian case) for two different classes of nonGaussianity. We will then compare the smallest value for the parameter that can be detected with each of the different approaches.

2 THE METHOD 2.1 Models for primordial non-Gaussianity There are in?nite types of possible deviations from Gaussianity, and it is unthinkable to address them all. However, we can consider plausible physical mechanisms that produce small deviations from the Gaussian behavior and thus analyze the following two models for the primordial non-Gaussianity (e.g., Coles & Barrow 1987, VWHK00, MVJ). In the ?rst model, we suppose that the fractional density perturbation δ(x) is a non-Gaussian random ?eld that can be written in terms of a Gaussian random ?eld φ(x) through (Model A) δ = φ + ?A (φ2 ? φ2 ). (1)

In the second model, we suppose that the primordial gravitational potential Φ(x) is a non-Gaussian random ?eld that can be written in terms of a Gaussian random ?eld φ(x) through (Model B) Φ = φ + ?B (φ2 ? φ2 ). (2)

Figure 1. Mmax as a function of redshift. At a given redshift one should only consider those masses (≤ Mmax ) for which at least one object is expected in the whole sky for Gaussian initial conditions. The shaded region encloses predictions for Mmax (z) from different mass functions in the literature; we adopted the currently favored cosmological model with parameters: ?0 = 0.3, Λ0 = 0.7, h = 0.65, σ8 = 0.99 and transfer function of Sugiyama (1995) with ?b = 0.015/h2 (ΛCDM).

Non-Gaussianity in the density ?eld is then obtained from that in the potential through the Poisson equation. Here, Φ and δ refer to the primordial gravitational potential and density perturbation, before the action of the transfer function that takes place near matterradiation equality. Although not fully general, these models may be considered as the lowest-order terms in Taylor expansions of more general ?elds, and are thus quite general for small deviations from Gaussianity. The scale-dependence of the non-Gaussianity in the two models differs. Model A produces deviations from Gaussianity that are roughly scale-independent on large scales, while Model B produces deviations from non-Gaussianity that become larger at larger distance scales. Although we choose these models essentially in an ad hoc way, the non-Gaussianity of Model B is precisely that arising in standard slow-roll in?ation and in non-standard (e.g., multi?eld) in?ation (Luo 1994; Falk, Rangarajan & Srednicki 1993; Gangui et al. 1994; Fan & Bardeen 1992; see also below). Model A more closely resembles the non-Gaussianity that would be expected from topological defects (e.g., VWHK00). In either case, the lowestorder deviations from non-Gaussianity (and those expected generically to be the most easily observed) are the three-point correlation function (including the skewness, its zero-lag value) or equivalently the bispectrum, its Fourier-space counterpart. It is straightforward to calculate these quantities for both Models A and B.

Universe today grew via gravitational infall from primordial perturbations in the early Universe, and this process alters the mass distribution in a calculable way. Cosmological perturbation theory allows the bispectrum for the mass distribution in the Universe today to be related to that for the primordial distribution. VWHK00 calculated the smallest values of ?A and ?B that would be accessible with the CMB and with LSS. For the CMB calculation, it was assumed that a temperature map could be measured to the cosmic-variance limit only for multipole moments ? < 100; ? it was assumed (quite conservatively) that no information would be obtained from larger multipole moments. The LSS calculation were made under the very optimistic assumption that the distribution of mass could be determined precisely from the galaxy distribution (i.e., that there was no biasing) in a survey of the size of SDSS and/or 2dF. VWHK00 found that the smallest values of ? that can be detected with the CMB under these assumptions is ?A ? 10?2 and ?B ? 20 (Komatsu & Spergel 2000 including noise and foreground but neglecting dust contamination found that ?B > 5 from ? the Planck experiment ), while the smallest values measurable with LSS are ?A ? 10?2 and ?B ? 103 . More realistically, the galaxy distribution will be biased relative to the mass distribution, and this will degrade the sensitivities to nonzero ?A and ?B obtainable with LSS. VWHK00 thus concluded that the CMB will provide a keener probe of primordial non-Gaussianity for the class of models considered.

2.3 High-redshift and/or massive objects 2.2 Cosmic Microwave Background and Large Scale Structure Temperature ?uctuations in the CMB come from density perturbations at the surface of last scatter, so the distribution of temperature ?uctuations re?ects that in the primordial density ?eld. It is thus straightforward to relate the density-?eld bispectra of Models A and B to the bispectrum of the CMB. Density perturbations in the According to the Press-Schechter theory, the abundance of highredshift and/or massive objects is determined by the form of the high-density tail of the primordial density distribution function. A probability distribution function (PDF) that produces a larger number of > 3σ peaks than a Gaussian distribution will lead to a larger abundance of rare high-redshift and/or massive objects. Since small deviations from Gaussianity have deep impact on those statistics that probe the tail of the distribution (e.g. ?, MVJ), rare

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Tests for primordial non-Gaussianity

high-redshift and/or massive objects should be powerful probes of primordial non Gaussianity. The number densities of high-redshift galaxies and/or of clusters (at either low or high redshifts) provides a very sensitive probe of the PDF. Since the Gaussian tail is decaying exponentially at higher densities, even a small deviation from Gaussianity can lead to huge enhancements in the number densities. The non-Gaussianity parameters ?A,B are effectively “tail enhancement” parameters (c.f., MVJ)? In order to determine the minimum value of ?A,B that can be detected using high-redshift objects, one needs to compute by how much the observed number density of objects changes with respect to the Gaussian case when the primordial ?eld is described by equations (1) and (2). We calculate this enhancement using the results for the mass function for mildly non-Gaussian initial conditions obtained analytically in MVJ. Conservatively, we make the assumption that objects form at the same redshift at which they are observed (zc = z); since for some objects the dark halo will have collapsed before we observe them, the assumption gives a lower limit to the amount of non-Gaussianity. The directly observed quantity, however, is not the mass function, but is N (≥ M, z), the total number of objects –in the survey area– of mass ≥ M that collapse at redshift z. In fact it is extremely dif?cult to obtain an accurate estimate of the mass of high-redshift objects, what is a more robust quantity is the minimum mass that these objects must have in order to be detected at that redshift. This quantity is related to the mass function, n(M, z), by

∞

3

N (≥ M, z) =

M

n(M, z)dM .

(3)

In calculating the enhancement of high-redshift objects due to primordial non-Gaussianity, we restrict ourselves to consider, at any given redshift, only those masses, M ≤ Mmax (z) for which at least one object is expected in the whole sky for Gaussian initial conditions (N (≥ Mmax , z) = 1 in 4π radians)? . This is illustrated in Fig. 1 for a ΛCDM model (hereafter we adopt the currently fa? In fact, when looking on a particular scale, it is always possible to parameterize the deviation of the PDF from Gaussianity, with some “effective” ?A or ?B , if the PDF is not too non-Gaussian. It is easy to understand this statement if one thinks in terms of skewness. Physical mechanisms that produce non-Gaussianity generically produce non-zero skewness in the PDF for the simple reason that underdense region cannot be more empty than voids while overdense regions can become arbitrarily overdense. Skewness can be scale dependent, but for a given value of the skewness there is oneto-one correspondence to ?A,B parameters (see the Appendix). ? This choice for the threshold N (≥ Mmax , z) = 1 is motivated by the following considerations. Of course it is not robust to detect a nonGaussianity that suppresses the number of objects with respect to the Gaussian prediction, since one can always argue that one did not look hard enough, or that the objects are there but are somewhat “invisible”. So we set to detect a non-Gaussianity that enhances the number of objects relative to the Gaussian case. If within Gaussian initial conditions we expect N (> M, z) ? 0 in the whole sky, and observations ?nd N (> M, z) > 1 in the survey area, we can say that we have detected non-Gaussianity. However, the non-Gaussianity (or tail enhancement) parameter is directly related to the ratio of observed Nng (> M, z) to the Gaussian predicted N (> M, z) (see Eq. 4). Obviously this ratio is well de?ned for any Nng > 0 and N > 0, but the observed Nng can only be an integer ≥ 1. The tail enhancement parameter will then make Nng ≥ N (and we consider only cases where Nng > 10N ). It is reasonable therefore to consider ? only those masses and redshift for which the theoretical prediction for the Gaussian N is ≥ 1. c 0000 RAS, MNRAS 000, 000–000

vored cosmological model with parameters: ?0 = 0.3, Λ0 = 0.7, h = 0.65, σ8 = 0.99 and transfer function of Sugiyama (1995) with ?b = 0.015/h2 ) where the shaded region encloses predictions for Mmax (z) from different mass functions (e.g., Press & Schechter 1974; Lee & Shandarin 1998; Sheth & Tormen 1999; Jenkins et al. 2000). Given the rapidly dying tail of the Gaussian PDF, small uncertainties in the mass determination of high-redshift objects could lead to overestimate the value of ?A,B . An overabundance of galaxies of estimated mass Me , which in principle can be attributed to a non-zero value of ?A,B , can also be explained under the hypothesis of Gaussian initial conditions if the actual galaxy mass Mtrue is Mtrue < Me . We thus include conservative values for the uncertainty ?M in the mass determination of high-redshift objects and we then calculate the minimum change ?N in the number density of objects over the Gaussian case that cannot be attributed to the uncertainty in the mass determination. For a given uncertainty in the mass, this can be computed by using the standard Press-Schechter (PS) theory (Press & Schechter 1974). Observationally it is dif?cult to measure the mass of high-redshift clusters with accuracy better than 30%, with either weak lensing or the X-ray temperature, and of high-redshift galaxies better than a factor 2 (?M = M ; at least of their stellar mass). Although the calculations in this section are obtained using the standard PS theory, our conclusions will be essentially unchanged if we had used modi?ed PS theories (e.g., Lee & Shandarin 1998; Sheth & Tormen 1999; Sheth, Mo & Tormen 1999; Jenkins et al. 2000; see below). With the mass uncertainties discussed above, we obtain that the minimum ?N that cannot be attributed to ?M is a factor 10 for clusters and a factor 100 for galaxies (see, e.g., Fig. 6 of MVJ00). We therefore estimate the minimum ?A,B that can be measured from the abundance of high-redshift objects as the one that corresponds respectively to a factor 100 and 10 change in the observed number density of objects (N (≥ M, z)) over the Gaussian case. This condition can be written as Nng (≥ M, z)/N (≥ M, z) ≡ R(M, z) ≥ R? , (4)

where N is obtained using the Gaussian mass function while Nng is obtained using the non-Gaussian mass function as in MVJ, and R? is set to be 100 for galaxies and 10 for clusters. For small primordial non-Gaussianity (i.e., for small values for ?A,B ), it is possible to derive an expression for R(M, z) using the analytical approximation for the mass function nng found in MVJ. Doing so we ?nd

∞

R(M, z) ? Here,

M

(σM M )?1 exp[?

∞ M

2 δ? (zc ) ]F (M, zc , ?A,B )dM 2 2σM 2

c (z (σM M )?1 exp[? δ2σ 2c ) ] m

dσM dM

dM

.(5)

F (M, zc , ?A,B )

=

6 +

dS3,M 1 ? S3,M δc (zc )/3 dM 1 ? S3,M δc (zc )/3 dσM , σM dM (6)

δc (zc )

and δ? (zc ) = δc (zc ) 1 ? S3,M δc (zc )/3, (7) (8)

δc (zc ) = ?c /D(zc ),

where D(zc ) is the linear-theory growth factor, and ?c is the linear extrapolation of the over-density for spherical collapse.

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In the formulas above, S3,M denotes the primordial skewness, we observe today should have a broader formation redshift distribution and thus a broader ST relation. In Verde et al. (2000b), the non-Gaussianity considered is a log-normal distribution; it is not strictly equivalent to Models A or B considered here. However, for small deviations from Gaussianity, the two models can be identi?ed if, for a given scale, they produce the same skewness in the density ?uctuation ?eld. We thus ?nd that in the ΛCDM model the minimum ?A and ?B detectable with the ST distribution method are 3×10?3 and 500 respectively. These estimates assume that the cosmology and σ8 are well known, but use only the local cluster data set of Mohr et al. (2000). Of course, with improved observational data, the ST method could probably yield stronger constraints.

2 S3,M = ?A,B ?3,M /σM , (1)

(1)

(9)

2 where the expressions for ?3,M and σM can be found in MVJ section 3.2 equations (37) and (38). However, for S3 > 1/δc (zc ), ? the mass function nng (M, z) has to be evaluated numerically and equation (5) is not valid. For the cosmological model considered here and the redshifts of interest, the quantity ?c takes a nearly constant value (≈ 1.686) in the PS theory. A better ?t to the mass function of halos in highresolution N-body simulations is however obtained by lowering ?c for rare objects and giving it an extra mass and redshift dependence (Sheth & Tormen 1999; Bode et al. 2000), as motivated by ellipsoidal collapse (e.g., Lee & Shandarin 1998; Sheth, Mo & Tormen 1999). It is possible to understand the effect of a lower ?c by the following argument. For rare ?uctuations such as high-redshift objects one is probing the mass function above the knee. Since the mass function drops very rapidly as M increases we can approximate N (> M, zc ) ? n(M, zc )M . It is then possible to obtain an analytic expression for r(M, zc ) ≡ nng (M, zc )/n(M, zc ) ? R(M, zc ) if the primordial non-Gaussianity is small:

3 RESULTS We ?nd that the non-Gaussianity of Model A has a bigger effect on high-redshift galaxies than on high-redshift clusters. This can be understood for the following reason. For Model A the skewness S3,M is approximately scale independent (dS3,M /dM = 0). Thus, as found in MVJ, the mass function for non-Gaussian initial conditions is obtained from the PS mass function for Gaussian initial conditions replacing δc (zc ) ?→ δ? (zc ). The effect of a non-zero skewness is therefore to lower the effective threshold for collapse thus allowing more objects to be created. For a given S3 , δ? (zc ) is a monotonically decreasing function of zc . Since galaxies can be observed at zc much bigger than that of clusters, the effect is bigger. On the other hand, clusters are better probes than galaxies for Model B. In fact, for Model B the induced skewness in the density ?eld is scale dependent and the effect of non-Gaussianity is roughly the same for galaxies with 8 < z < 10 and clusters with 1 < z < 3. However since mass determinations are more accurate for clusters than for galaxies, we have R?,clusters < R?,galaxies : clusters are therefore better probes. In Fig. 2 we show the ratio R = Nng (≥ M, z)/N (≥ M, z) (cf. eq. (5)) for galaxies at redshift z = 8, 9 and 10 for ?A = 5 × 10?4 (model A, left panel) and clusters at redshift z = 1, 2 and 3 for ?B = ?200 (model B, right panel), as a function of M . Lines are plotted only for masses where, for Gaussian initial conditions, one would expect to observe at least one object in the whole sky with the most conservative estimate (see Fig. 1). Note that those high-redshift objects represent 3- to 5-σ peaks. If we now require R(M, zc ) > R? , we deduce that the minimum detectable deviation from Gaussian initial conditions will be ?A ? 5×10?4 (from highredshift galaxies) and |?B | ? 200 (from high-redshift clusters). We also estimate that an uncertainty of 10% on σ8 would propagate into an uncertainty of 25% in ?B (from clusters) and of 70% in ?A (from galaxies). The minimum ?B detectable from high-redshift cluster abundances is much larger than the value that can be measured from the CMB (?B ? 5 to 20 for Planck data), while for ?A , high-redshift galaxies are much better probes than the CMB, which can only detect ?A ? 10?2 . We therefore conclude that if future NGST or 30- to 100-m ground-based telescope observations of high-redshift galaxies yield a signi?cant number of galaxies at z ? 10 and are able to determine their masses within a factor 2, these observations will perform better than CMB maps in constraining primordial non-Gaussianity of the form of Model A with positive ?A . Conversely, forthcoming CMB maps will constrain deviations from Gaussianity in the initial conditions much better than observations of high-redshift obc 0000 RAS, MNRAS 000, 000–000

r(M, zc ) ? exp

?3 S 3 c 2 6σM

δc 6 1?

S3 δc 3

dS3 + dσM

1?

S3 δc .(10) 3

For a given mass M , r(M, zc ) slowly decreases when lowering ?c , slightly damping the effect of non-Gaussianity. For example when lowering ?c from the value we assume here 1.686, to the value ≈ 1.5—appropriate to ?t the numerical mass function of Sheth & Tormen (1999) for the range of masses and redshifts considered here—r(M, zc ) decreases by less than a factor 2. However this effect is compensated by the fact that, by lowering ?c , objects are created more easily also with Gaussian initial conditions, and it is therefore possible to consider objects of higher M and/or z, where the effect of non-Gaussianity is bigger. In summary, the conclusions obtained by assuming ?c = 1.686 will not be substantially modi?ed. It is important to note that for Model A, the primordial skewness has the same sign as ?A , while for Model B the primordial skewness has the opposite sign of that of ?B . In detecting non-zero ?A,B from CMB maps, the sign of the skewness does not in?uence the accuracy of the detection of non-Gaussianity, but, when using the abundance of high-redshift objects the sign of the skewness matters. Only a positively skewed primordial distribution will generate more high-redshift objects than predicted in the Gaussian case. Although a negatively skewed probability distribution will generate fewer objects than the Gaussian case, a decrement might be dif?cult to attribute exclusively to a negatively skewed distribution. Therefore in the following we will consider only negative ?B and positive ?A .

2.3.1 Cluster size-temperature distribution Verde et al. (2000b) showed that the size-temperature (ST) distribution of clusters is fairly sensitive to the degree of primordial nonGaussianity. If clusters are created from rare Gaussian peaks, the spread in formation redshift should be small and so should the scatter in the ST distribution. Conversely, if the probability distribution function has long non-Gaussian tails, then clusters of a given mass

Tests for primordial non-Gaussianity

5

Figure 2. Ratio R(M, z) = Nng (≥ M, z)/N (≥ M, z) for galaxies at redshift z = 8, 9 and 10 for ?A = 5 × 10?4 (left panel) and clusters at redshift z = 1, 2 and 3 (right panel), for ?B = 200 as a function of M . Lines are plotted only for masses where, for Gaussian initial conditions, one would expect to observe at least one object in the whole sky with the most conservative estimate (see Fig. 1). Note that these high-redshift objects represent 3- to 5-σ peaks. The values for the number density enhancement R that can safely be attributed to primordial non-Gaussianity are R = 100 for galaxies (left panel) and R = 10 for clusters (right panel). See text for details.

jects for Model B (with positive and negative value for ?B ) and for Model A with negative ?A .

observable CMB LSS High z obj. ST relation

min. |?A | 10?3 ? 10?2 (+)5 × 10?4 (gal.) (+)3 × 10?3 10?2

min. |?B | 20 103 ? 104 (–) 200 (clusters) (–) 500

3.1 Slow-roll parameters and primordial skewness The type of non-Gaussianity of Model B is particularly interesting because initial conditions set from standard in?ation show deviations from Gaussianity of this kind. In fact, it is possible to relate the two slow roll parameters, ?? = m2 l P 16π V′ V

2

,

η? =

m2 l P 8π

V ′′ 1 ? V 2

V′ V

2

,

(11)

to the non-Gaussianity parameter ?B . In equation (11) mP l is the Planck mass, V denotes the in?aton potential and V ′ and V ′′ the ?rst and second derivatives with respect to the scalar ?eld. The skewness S3 for ΦB , S3,Φ = Φ3 / Φ2 2 , can be evaluated folB B lowing a similar calculation of Buchalter & Kamionkowski (1999), obtaining S3,Φ = 2?B × 3[1 + γ(n)], (12)

Table 1. Minimum |?A | and |?B | detectable form different observables and their sign when positive skewness is required for detection. For Model A the primordial skewness has the same sign as ?A , while for Model B the primordial skewness has the opposite sign as ?B . In detecting non-zero ?A,B from CMB maps, the sign of the skewness does not in?uence the accuracy of the detection of non-Gaussianity, but, when using the abundance of high-redshift objects it is robust to detect non-Gaussianity that produces an excess rather than a defect in the number density. Only a positively skewed primordial distribution will generate more high-redshift objects than predicted in the Gaussian case.

the shape of the in?aton potential through eq. (11). However, from the present analysis, an error of ?B of about an order of magnitude larger seems to be realistically achievable.

where γ(n) ? 1 and weakly depends on n if n < 0, but diverges for n > 0. For a scale-invariant matter density power spectrum, n = ?3, γ(n) = 0, and so S3,Φ = 6?B . We can then compare this expression with the value for the skewness parameter for the gravitational potential arising from in?ation to infer the magnitude of ?B . Gangui et al. (1994) calculate the CMB skewness for the Sach-Wolfe effect S2 in several in?ationary models; S2 is related to S3,Φ by S2 = S3,Φ A?1 where sw Asw = 1/3. From this it follows that S2 = 3S3,Φ = 18?B . The condition for slow roll from Gangui et al. 1994 is S2 ≤ 20; thus, ?B ≤ 1, and the relation to the slow-roll parameters is (cf., Wang & Kamionkowski 2000) ?B = (5/2)?? ? (5/3)η? . (13)

4 DISCUSSION AND CONCLUSIONS We considered two models for small primordial non-Gaussianity, one in which the primordial density perturbation contains a term that is the square of a Gaussian ?eld (Model A), and one in which the primordial gravitational potential perturbation contains a term proportional to the square of a Gaussian (Model B). The nonGaussianity of Model B is precisely that arising in standard slowroll in?ation and in non-standard in?ation, while Model A more closely resembles the non-Gaussianity that would be expected from topological defects. We investigated the relative sensitivities of several observables for testing for deviations from Gaussianity: CMB, LSS and high-redshift and/or massive objects (e.g., galaxies and clusters). The analytic tools developed above allow us to address the question of whether the abundance of currently known highredshift objects can be accommodated within the framework of in?ationary models for a given cosmology. Recently Willick (2000)

Since this combination of the slow-roll parameters is different from the combination that gives the spectral slope n of the primordial power spectrum (n = 2?? ? 6η? + 1), in principle, if ?B could be measured with an error ? 1, it would be possible to determine

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Verde, Jimenez, Kamionkowski & Matarrese

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has studied in detail the mass determination of the cluster MS105403 concluding that its mass lies in the range 1.4±0.3×1015 M⊙ for ?m = 0.3 (similar to the independent mass estimates by, e.g., Tran et al. (2000) and Newmann & Arnaud (2000)). As already pointed out by Willick (2000), for ?m ≥ 0.3 the expected number of objects like MS1054-03 in the survey area is ≤ 0.01; i.e., it must be a 3-σ ?uctuation or larger. Using the formalism we have described here, a primordial non-Gaussianity parameterized by ?B ≥ 400 would be required to account for MS1054-03 as a 1σ ?uctuation in the ΛCDM model described above. This value is much too large to be consistent with slow-roll in?ation. Our calculation shows that if such a non-Gaussianity exists, it would be easily detectable from forthcoming CMB maps.

ACKNOWLEDGMENTS LV and RJ thank the Caltech theoretical astrophysics group for hospitality. MK was supported in part by NSF AST-0096023, NASA NAG5-8506, and DoE DE-FG03-92-ER40701.

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APPENDIX In this Appendix we quote the expressions for the primordial bispectrum and skewness for the two non-Gaussian models considered in this paper. The large-scale structure (LSS) bispectrum for model A is (e.g. VWHK00) B(k1 , k2 , k3 ) = 2?A P (k1 )P( k2 ) + cyc. (14)

where P denotes the power spectrum. The cosmic microwave background (CMB) bispectrum for model A is (e.g. VWHK00) B?1 ?2 ?3 ? (2?1 + 1)(2?2 + 1)(2?3 + 1) 4π × 2?A g

?1 ?2 ?3 0 0 0

2 ?2 ?2 C?1 C?2 1 2 2 + cyc. 3 ?3

(15)

where C? denotes the cosmic microwave background power spectrum, g denotes the radiation transfer function and (. . .) denotes the Wigner 3J symbol. The LSS bispectrum for model B is (e.g. VWHK00) B(k1 , k2 , k3 ) ? P (k1 )P (k2 )2?B M k3 M k1 M k2 + cyc. (16)

3 where? Mk ? (2k2 T (k)(1+z))/(3H0 ) and T denotes the matter transfer function. The CMB bispectrum for model B is (e.g. Luo

? This expression is strictly valid only for an Einstein de Sitter Universe, for a more general model M is de?ned by δk (z) = Mk (z)Φ(k) where Φ denotes the gravitational potential ?eld. c 0000 RAS, MNRAS 000, 000–000

Tests for primordial non-Gaussianity

1994; Wang & Kamionkowski 2000,VWHK00,Komatsu & Spergel 2000): B?1 ?2 ?3 = (2?1 + 1)(2?2 + 1)(2?3 + 1) ?1 ?2 ?3 0 0 0 4π 2?B (C?1 C?2 + cyc.) × g

7

(17)

The corresponding primordial skewness S3 = δ 3 / δ 2 2 where δ denotes δρ/ρ for the large-scale structure case and ?T /T for the cosmic microwave background is easily obtained from the consideration that δ 3 is given by: δ3

LSS

=

d 3 k1 d 3 k2 3 d k3 B(k1 , k2 , k3 )δ D (k1 +k2 +k3 )(18) (2π)3 (2π)3

(in the absence of spatial ?ltering) and 1 4π (2?1 + 1)(2?2 + 1)(2?3 + 1) 4π ×

?1 ?2 ?3 0 0 0

δ3

CM B

=

?1 ?2 ?3

B?1 ?2 ?3

(19)

for LSS and CMB respectively. For example in the large scale structure case, model A, for a power law power spectrum and in the absence of spatial ?ltering§ we have that S3 = 6?A .

§ The expression for δ3 LSS in the general case can easily be derived following the calculations of Buchalter & Kamionkowski (1999) by setting b1 = 0 and b2 /2 = ?A . c 0000 RAS, MNRAS 000, 000–000

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