Analytic Minkowski Functionals of the Cosmic Microwave Background: Second-order Non-Gaussianity with Bispectrum and Trispectrum Takahiko Matsubara ∗ Department of Physics, Nagoya University, Chikusa, Nagoya, 464-8602, Japan (Dated: August 10, 2018) Analytic formulas of Minkowski functionals in two-dimensional random fields are derived, including e ff ects of second-order non-Gaussianity in the presence of both the bispectrum and trispectrum. The set of formulas provides a promising method to constrain the primordial non-Gaussianity of the universe by temperature fluctu- ations in the cosmic microwave background radiation. In a case of local-type non-Gaussianity, the Minkowski functionals are analytically given by powers of quadratic and cubic parameters, f$_{NL}$ and g$_{NL}$ . Our formulas are not restricted to this particular model, and applicable to a wide class of non-Gaussian models. The analytic formulas are compared to numerical evaluations from non-Gaussian realizations of temperature maps, showing very good agreements. PACS numbers: 98.80.-k, 98.70.Vc, 98.80.Jk, 98.80.Es polyspectra may be primary methods to constrain higher-order non-Gaussianity, it is desirable to have as many alternative methods as possible. Among others, the set of Minkowski functionals (MFs) [25–27] has been proved to be a sim- ple and powerful tool in analyzing non-Gaussianity in CMB [15, 16, 18, 28–33]. Detecting the primordial non-Gaussianity will play a key role in discriminating the models of the early universe. All models of inflation predict the primordial non-Gaussianityto some extent in principle. While a simple inflationary model with a single slow-rolling scalar field produces too small no n- Gaussianity to be observed [1–6], many other models such as the curvaton scenario [7–9], models with non-standard kine tic term [10, 11], cyclic or ekpyrotic universes without inflati on [13], etc. predict large non-Gaussianity which could be de- tected in any future [14]. Since the hierarchy of polyspectra contains all the informa- tion of the statistical property of random field, the expecta tion values of MFs should be expressible by the polyspectra. Such relations were analytically derived for weakly non-Gaussian fields in the lowest-order approximation [34]. It was shown that deviations of the non-Gaussian MFs from the Gaussian MFs are proportional to a linear combination of the bispec- trum. The analytic formulas are incorporated with the CMB bispectrum [35] and are applied to the analysis of CMB, suc- cessfully constraining non-Gaussianity parameter f$_{NL}$ without relying on massive numerical simulations to generate non- Gaussian CMB maps [36–38]. The statistical properties of random Gaussian fields are completely characterized by the two-point correlation func- tion in configuration space, or the power spectrum in Fourier space. Thus the information of non-Gaussianity is con- tained in higher-order polyspectra, such as the bispectrum, trispectrum, and so forth. Recently, much work is devoted to analyzing the temperature fluctuations in the cosmic mi- crowave background (CMB) to constrain the primordial non- Gaussianity [15–20]. So far most of the constraints are plac ed to a phenomenological parameter f$_{NL}$ , which is largely re- sponsible for the primordial bispectrum. However, the bispec- trum is not su ffi cient to fully discriminate the wide variety of models of the early universe. Some models, such as the curva- ton scenario [21–23], ekpyrotic universes [24], etc. can ge n- erate a large trispectrum with a relatively small bispectrum. The trispectrum is often characterized by phenomenological parameters g$_{NL}$ (and τ$_{NL}$ in some models). When g$_{NL}$ is large enough such that g$_{NL}$ ≫ f 2 $_{NL}$, the bispectrum is not signifi- cant and the primordial non-Gaussianity can be detected only through the e ff ect of trispectrum. So far the analytic formulas of non-Gaussian MFs are known only in the lowest-order approximation, which we call first-order non-Gaussianity. The contributions from t he trispectrum and other polyspectra to the geometrical de- scriptors, such as the MFs, appear in higher-order non- Gaussianities [34, 39, 40]. To extract any information from MFs beyond the bispectrum, such as g$_{NL}$ and τ$_{NL}$ , it is par- ticularly useful to have analytic formulas with higher-order approximations. The purpose of this paper is to derive the analytic formulas with second-order approximation of non- Gaussianity, including the e ff ects up to the trispectrum. These primary formulas are applicable to a wide class of non- Gaussian fields in 2-dimensional space. We also focus on how the formulas are applied to analyses of CMB temperature fluctuations. Assuming a local model of non-Gaussianity, we derive the explicit dependence of the MFs on parameters f$_{NL}$ and g$_{NL}$ (and τ$_{NL}$ ). Straightforward measurements of the polyspectra become progressively complicated for higher-order statistics beyond the bispectrum, since the higher-order polyspectra have many arguments with complicated dependence on shapes and scales. Even though direct calculations of the higher-order This paper is organized as follows: In § 2, the analytic for- mulas of MFs with second-order non-Gaussianity are derived. Non-Gaussianity parameters to describe the analytic MFs are explicitly given by the bispectrum and trispectrum in § 3. In § 4, the polyspectra in the local model of non-Gaussianity are summarized for applications to the analytic MFs. Numerical ∗ Electronic address: taka@a.phys.nagoya-u.ac.jp arXiv:1001.2321v2[astro-ph.CO]4 Mar 2010 I. INTRODUCTION