Implications for Primordial Non-Gaussianity (f_NL) from weak lensing masses of high-z galaxy clusters

The recent weak lensing measurement of the dark matter mass of the high-redshift galaxy cluster XMMUJ2235.3-2557 of (8.5 +- 1.7) x 10^{14} Msun at z=1.4, indicates that, if the cluster is assumed to be the result of the collapse of dark matter in a primordial gaussian field in the standard LCDM model, then its abundance should be<0.002 clusters in the observed area. Here we investigate how to boost the probability of XMMUJ2235.3-2557 in particular resorting to deviations from Gaussian initial conditions. We show that this abundance can be boosted by factors>3-10 if the non-Gaussianity parameter f^local_NL is in the range 150-200. This value is comparable to the limit for f_NL obtained by current constraints from the CMB. We conclude that mass determination of high-redshift, massive clusters can offer a complementary probe of primordial non-gaussianity.

The recent weak lensing measurement of the dark matter mass of the high-redshift galaxy cluster XMMUJ2235.3-2557 of (8.5 ± 1.7) × 10 14 M⊙ at z = 1.4, indicates that, if the cluster is assumed to be the result of the collapse of dark matter in a primordial gaussian field in the standard LCDM model, then its abundance should be < 2 × 10 −3 clusters in the observed area. Here we investigate how to boost the probability of XMMUJ2235.3-2557 in particular resorting to deviations from Gaussian initial conditions. We show that this abundance can be boosted by factors > 3 − 10 if the non-Gaussianity parameter f local NL is in the range 150 − 200. This value is comparable to the limit for fNL obtained by current constraints from the CMB. We conclude that mass determination of high-redshift, massive clusters can offer a complementary probe of primordial non-gaussianity.

PACS numbers: cosmology
Introduction.-It has been recognized for almost a decade that the abundance of the most massive and/or high-redshift collapsed objects could be used to constraint the nature of the primordial fluctuation field [1,2,3,4]. The subject has recently received renewed attention [5,6,7,8,9] possibly sparked by a claimed detection of deviations from Gaussianity on CMB maps [10]. Depending on the sign of the non-Gaussian perturbation, the abundance of rare objects will be enhanced or depleted. In [1] we developed the necessary theoretical tools to interpret any enhancement (depletion) in the abundance of rare-peak objects over the gaussian initial conditions case. Working with ratios of non-Gausian over the Gaussian case makes the theoretical predictions more robust. Later on, Ref. [5] generalised the procedure to more modern mass-functions and type of non-gaussianity including scale dependence. The validity of the analytical formulas developed in [1] has been recently confirmed by detailed N-body numerical simulations with non-gaussian initial conditions [7]. These authors have shown that the analytical findings in [1] provide an excellent fit to the non-Gaussian mass function found in N-body simulations with a simple "calibration" procedure.
Ref. [12] have recently reported a weak-lensing analysis of the z = 1.4 galaxy cluster XMMU J2235.3-2557 based in HST (ACS) images. Assuming a NFW [11] dark matter profile for the cluster, they estimate a projected mass within 1 Mpc of (8.5 ± 1.7) × 10 14 M ⊙ . Adopting a LCDM cosmology with cosmological parameters given by WMAP 5 yr data ( [13]) and assuming Gaussian initial conditions they estimate that in the surveyed 11 sq. deg. there should be 0.005 clusters above that mass. Therefore the observed cluster is unlikely a the 3σ level. In this Letter we explore what level of non-Gaussianity is required to boost this abundance by a factor ∼ 10 and how this relates to the available constraints obtained from the CMB. We show that with f local N L in the range 150 − 200 it is possible significantly enhance the abundance expected for such a massive cluster. This value of f N L is comparable with current limits from the CMB [13] , [10].
High Redshift and/or Massive Objects.-While there are in principle infinite types of possible deviations from Gaussianity, it is common to parameterize these deviations in terms of the dimensionless parameter f N L (e.g., [1,14,15,16]).
where Φ denotes the primordial Bardeen potential [29] and φ denotes a Gaussian random field. With this convention a positive value of f local N L will yield to a positive skewness in the density field and an enhancement in the number of rare, collapsed objects.
Although not fully general, this model (called localtype) may be considered as the lowest-order terms in Taylor expansions of more general fields. Local nongaussianity arises in standard slow roll inflation (although in this case f local N L is unnmeasurably small), and in multi-field models (e.g., [17,18,19,20]). For other types of non-gaussianity (as we will see below) an "effective" f N L can be defined and related to this model.
The abundance of rare events (high-redshift and/or massive objects) is determined by the form of the highdensity 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 events. Since small deviations from Gaussianity have deep impact on those statistics that probe the tail of the distribution (e.g. [1,22]), rare events should be powerful probes of primordial non Gaussianity.
As shown in [4,5,7] when using an analytical approach to compute the mass function a robust quantity to use is the fractional non-gaussian correction to the Gaussian mass function R N G (M, z). This quantity was calibrated on non-gaussian N-body simulations in [7]. For our purpose here we want to compute a closely related quantity: the non-gaussianity enhancement i.e. ratio of the nongaussian to gaussian abundance of halos above a mass threshold [4]. As the mass function is exponentially steep for rare events here we can safely make the identification of the non-gaussianity enhancement with R N G .
To understand the effect of non-gaussianity on halo abundance let us recall that to first order the nongaussianity enhancement is given by [5,7]: where S 3,M denotes the skewness of the density field linearly extrapolated at z = 0 and smoothed on a scale R corresponding to the comoving Lagrangian radius of the halo of mass M , σ M denotes the rms if the -linearly extrapolated at z = 0-density field also smoothed on the same scale R; δ ′ c (z f ) = √ qδ c (z f ) and δ c (z f ) denotes critical collapse density at the formation redshift of the cluster z f . Note that δ c (z) = ∆ c D(z = 0)/D(z) with D(z) denoting the linear growth factor and ∆ c is a quantity slightly dependent on redshift and on cosmology, which only for an Einstein-de-Sitter Universe is constant ∆ c = 1.68. The constant q ≃ 0.75 (which we will call "barrier factor") can be physically understood as the effect of non-spherical collapse [24,25] lowering the critical collapse threshold of a diffusing barrier [26] see also [27], and has been calibrated on N-body simulations in Ref. [7]. The full expression for R N G is (cf Eqs. 6 and 7 in Ref. [7]): Let us re-iterate that in principle the enhancement factor should be computed by integrating the mass function n(M, z, f N L ) between the minimum and the maximum mass and for redshifts above the observed one [4]: but since the mass function, in the regime we are interested in, is exponentially steep, we can identify R N G = R N G . Note that for the quoted values of f local NL it is possible to obtain enhancements of order 10 in the cluster number abundance. This enhancement brings the expected abundance of such massive clusters in better agreement with the observations. Note that for masses above the estimated central value (8.5 × 10 14 M⊙) one expects to find zero such objects in the whole sky (one expects 7 objects in the whole sky at the lowest value of the mass estimate) which emphasizes the need of an enhancement as the one provided by primordial nongaussianity studied here.
Small deviations from Gaussian initial conditions will lead to a non-zero skewness and in particular for local non Gaussianity S 3,M = f local N L S 1 3,M where S 1 3,M denotes the skewness produced by f local N L = 1. Since non-Gaussianity comes in the expression for R N G only through the skewness, the same expression can be used for other types of non-Gaussianity such as the equilateral type (see e.g. Ref. [5,6] for example of applications). For example, at the scales of interest R = 13M pc/h, S 1,local 3,R = 3.4S 1,equil 3,R thus when working on these scales to obtain the same non-Gaussian enhancement as a local model, an equilateral model needs a higher effective f N L : we can make the identification f equil N L = 3.4f local N L . Here we will use the full [1] expression, corrected for the "barrier factor", for the non-gaussian mass function to compute the non-gaussianity enhancement. Note that the estimated mass and redshift of XMMUJ2235.3-2557, places it just outside the range where the mass function expressions of [1,5] have been directly reliably tested with non-Gaussian N-body simulations. Simulations seems to indicate that the [1] expression is a better fit than [5] at high masses/redshift and large f N L , this is also supported by theoretical considerations [7].
Results. -Fig. 1 shows the enhancement factor R N G as a function of the mass of the galaxy cluster for different values of f local N L and the redshift of collapse. The shaded area shows the error band for the mass determination of XMMUJ2235.3-2557 from Ref. [12] and the different lines have been computed using the [1] mass function, with the "barrier factor" correction. Ref. [7] show that it fits very well the N-body numerical simulations for the case of rare peaks, which is the one we are concerned with. The solid lines correspond to f N L = 260, the lower one is for a cluster collapse redshift of z f = 1.4 (i.e. assuming that the cluster forms at the observed redshift) and the upper one for z f = 2. The two dashed lines also depict the mentioned collapse redshifts but for f N L = 150. We see that the galaxy cluster abundance can be enhanced by a factor up to 10. In the mass range of interest, the same enhancement factor can be obtained for an equilateral-type non-gaussianity for f equil N L = 884 and 510 respectively.
We should bear in mind that XMMUJ2235.3-2557 is an extremely rare object, sampling the tail of the mass function which may not be well known and may be strongly affected by cosmology. Using the [23] mass function we estimate that in the WMAP5 LCDM model [30] one should find 7 galaxy clusters in the whole sky with mass greater or equal than the lower mass estimate of XMMUJ2235.3-2557 M = 5 × 10 14 M ⊙ and z > 1.4 corresponding to a probability of 0.002 for the 11 deg 2 of the survey. This should be compared with the reported number of 0.005 obtained by [12] for a different cosmology and different mass function. Thus the effects of cosmology and uncertainty in the mass function can account for a factor ∼ 2 uncertainty in the predicted halo abundance.
Note that in all our calculations we have used a conservative lower limit for the mass of the cluster. If instead we use the central or upper value for the mass, using the WMAP5 cosmology and the [23] mass function we expect to find zero such clusters in the whole sky, which will make our conclusions even stronger.
The survey area used in Ref. [12] is 11 deg 2 , but the XMM serendipitus survey in 2006 covered 168 deg 2 and today covers ∼ 400 deg 2 . Below we report the Ref. [12] numbers and in parenthesis the numbers we obtain. The probability of finding XMMUJ2235.3-2557 is thus 0.005 (0.002) if using 11 deg 2 ; to avoid as much as possible biases due to a posteriori statistics one could use 168 deg 2 obtaining a probability of 0.07 (0.03), or, as a limiting case, even 0.17 (0.07) if using 400 deg 2 . Note that it is likely that there are more clusters as massive in the survey area [28] and therefore these numbers are conservative. If we use from Fig. 1 the factor 3 to10 enhancement, we find that it would help boost the probability to ∼ 1 in the surveyed areas.
The latest WMAP compilation [13] reports −9 < f local N L < 111 and −151 < f equil N L < 253 at 95% confidence, [10] reports 27 < f local N L < 147. The CMB however probes much larger scales (R > 120M pc/h) than those probed by clusters such as XMMUJ2235.3-2557 R ∼ 13Mpc/h: a scale-dependent f N L with k ∼ −0.3 can yield an effective f N L on dependence XMMUJ2235.3-2557 scales that is larger than the CMB one by a factor of 3.
The f local N L values needed to accomodate the observed cluster at z = 1.4 is in the range 150 to 260. This is comparable to the limits quoted by Ref. [13] and [10].
Conclusions.-Accurate masses of high-redshift clusters are now becoming available through weak lensing analysis of deep images. As already discussed in previous papers [1,5], their abundance can be used to put constraints on primordial non-gaussianity. f local N L in the range 150 − 260 can boost the expected number of massive (> 5 × 10 14 M ⊙ ) high redshift (z > 1.4) clusters by factors of 3 to 10. Such large numbers would help make clusters like XMMUJ2235.3-2557 much more probable. The scales probed by clusters are smaller than the CMB scales, and in principle non-Gaussianity may be scaledependent, making this a complementary approach. The adopted error range in the mass determination of XMMUJ2235.3-2557 is 100%; even with such a large mass uncertainty and considering the pessimistic estimate of 7 such objects expected in the entire sky with a Poisson error of ±2.6, if the entire sky could be covered, f local N L ∼ 150 could be detected at > 4σ level. RJ and LV acknowledge support of MICINN grant AYA2008-03531. LV acknowledges support of FP7-PEOPLE-2002IRG4-4-IRG#202182. RJ is supported by a FP7-PEOPLE-IRG grant.