Testing non-Gaussianity in cosmic microwave background maps by morphological statistics

Mon. Not. R. Astron. Soc. 331, 865–874 (2002) Testing non-Gaussianity in cosmic microwave background maps by morphological statistics Sergei F. ShandarinP The University of Kansas, Department of Physics and Astronomy, 1082 Malott Hall, 1251 Wesco Hall Drive, Lawrence, KS 66045, USA Accepted 2001 December 3. Received 2001 December 2; in original form 2001 July 30 A B S T R AC T Key words: methods: data analysis – cosmic microwave background – cosmology: theory. 1 INTRODUCTION The quest for the physical mechanism of the generation of the initial inhomogeneities along with the measurements of the major cosmological parameters [H0, VL, VCDM, Vb, P(k), etc.] is one of the most important problems in modern cosmology. The standard inflationary model predicts the primordial fluctuations were Gaussian random fields (Guth & Pi 1982; Hawking 1982; Starobinski 1982; Bardeen, Steinhardt & Turner 1983). In agreement with the theory the current observations provide little evidence for deviations from Gaussianity. The majority of the tests of Gaussianity in the COBE maps (Colley, Gott & Park 1996; Kogut et al. 1996; Ferreira, Magueijo & Górski 1998; Pando, VallsGabaud & Fang 1998; Novikov, Feldman & Shandarin 1999; Bromley & Tegmark 1999; Banday, Zaroubi & Górski 2000; Mukherjee, Hobson & Lasenby 2000; Barreiro et al. 2000; Aghanim, Forni & Bouchet 2001; Phillips & Kogut 2001) have resulted in the general agreement that all non-Gaussian signals P E-mail: q 2002 RAS were of non-cosmological origin.1 This was perhaps not a surprise because of a very large physical scale corresponding to the COBE resolution (< 78). Recent studies of the DT/T maps on degree and sub-degree scales also showed no significant deviations from Gaussianity (Park et al. 2001; Wu et al. 2001; Shandarin et al. 2002). Nevertheless, the question of possible non-Gaussianity in cosmic microwave background (CMB) maps is very important for the following reasons. First, a detection of a non-Gaussian component in the primordial fluctuations may profoundly affect modern cosmology ruling out some models of the early universe and boosting the others (see e.g. Turner 1997; Vilenkin & Shellard 1994). Secondly, Gaussianity is a key underlying assumption of all experimental power spectrum analyses to date, entering into the computation of error bars (Tegmark 1997; Bond & Jaffe 1998), and therefore needs to be observationally tested. In addition, the hypothesis of the Gaussianity of the initial perturbations enters in 1 However, Magueijo (2000) still has a 97 per cent confidence level that the signal is not due to systematics. The assumption of the Gaussianity of primordial perturbations plays an important role in modern cosmology. The most direct test of this hypothesis consists of testing the Gaussianity of cosmic microwave background (CMB) maps. Counting the pixels with the temperatures in given ranges and thus estimating the one-point probability function of the field is the simplest of all the tests. Other usually more complex tests of Gaussianity generally use a great deal of the information already contained in the probability function. However, the most interesting outcome of such a test would be the signal of non-Gaussianity independent of the probability function. It is shown that the independent information has purely morphological character i.e. it depends on the geometry and topology of the level contours only. As an example we discuss in detail the quadratic model v ¼ u þ aðu 2 2 1Þ (u is a Gaussian field with u ¼ 0 and ku 2 l ¼ 1, a is a parameter) that may arise in slow-roll or two-field inflation models. We show that in the limit of small amplitude a the full information about the non-Gaussianity is contained in the probability function. If other tests are performed on this model they simply recycle the same information. A simple procedure allowing us to assess the sensitivity of any statistics to the morphological information is suggested. We provide an analytic estimate of the statistical limit for detecting the quadratic non-Gaussianity ac as a function of the map size in the ideal situation when the scale of the field is resolved. This estimate is in a good agreement with the results of the Monte Carlo simulations of 2562 and 10242 maps. The effect of resolution on the detection quadratic non-Gaussianity is also briefly discussed. 866 S. F. Shandarin about every n-point function. Reversing this statement one can say that no n-point function carries information that is completely independent from the PF. Thus, one may also ask what information is stored in n-point functions that is independent of the PF and whether it is possible to extract it or at least to assess it. Obviously, the same question must be addressed not only to the n-point functions but also to all other statistics. These issues are discussed below. The PF or equivalently the cumulative probability function (CPF)2 is not only the simplest conceptually but also the most efficient numerically. Computing this statistic requires only O(Npix) operations. The only problem is that the Gaussian PF does not guarantee the Gaussianity of the field. Therefore, some additional statistical information is badly needed in the case when the PF of the field is Gaussian, because if the PF is non-Gaussian the non-Gaussianity is already detected. The next step obviously would be the identification of the physical process responsible for the non-Gaussianity but first it must be detected. Thus, if the PF is Gaussian, the additional information must be independent of that contained in the PF. We will show that such information has purely morphological character. This means that it is completely determined by the geometric and topological statistic of the excursion sets. Thus, a set of morphological parameters based on Minkowski functionals becomes a natural choice of the statistics, one that is sensitive to non-Gaussianity and completely independent of the PF provided that proper parametrization is used. A particular kind of non-Gaussianity known as the quadratic model has recently attracted much attention (Coles & Barrow 1987; Luo & Schramm 1993; Matarrese, Verde & Jimenez 2000; Verde et al. 2000; Verde 2001; Verde et al. 2001). One reason is that it could be generated by plausible physical mechanisms in the early universe (Falk, Rangarajan & Srednicki 1993; Gangui et al. 1994; Luo 1994). The other is the relative ease of its analysis. In this paper we show that the simplest test for Gaussianity, the probability function, provides also the complete statistical information in the most interesting case of small amplitudes. It means that other tests if applied to this model at best only recycle a part (probably small) of this non-Gaussian information. In the general case of arbitrary amplitude, the set of global Minkowski functionals completely characterize the statistical properties of this field. The rest of the paper is organized as follows. We de (...truncated)