# Planck Trispectrum Constraints on Primordial Non-Gaussianity at Cubic Order

###### Abstract

Non-Gaussianity of the primordial density perturbations provides an important measure to constrain models of inflation. At cubic order the non-Gaussianity is captured by two parameters and that determine the amplitude of the density perturbation trispectrum. Here we report measurements of the kurtosis power spectra of the cosmic microwave background (CMB) temperature as mapped by Planck by making use of correlations between square temperature-square temperature and cubic temperature-temperature anisotropies. In combination with noise simulations, we find the best joint estimates to be and . If , we find .

Introduction.—Existing cosmological data from cosmic microwave background (CMB) and large-scale structure (LSS) are fully consistent with a simple cosmological model involving six basic parameters describing the energy density components of the universe, age, and the amplitude and spectral index of initial perturbations. The perturbations depart from a scale-free power spectrum and are Gaussian. These facts support inflation as the leading paradigm related to the origin of density perturbations Guth (1981a); Linde (1982); Albrecht and Steinhardt (1982). Under inflation a nearly exponential expansion stretched space in the first moments of the early universe and promoted microscopic quantum fluctuations to perturbations on cosmological scales today Guth (1981b); Bardeen (1980). Moving beyond simple inflationary models with a single scalar field, models of inflation now involve multiple fields and exotic objects such as branes that have non-trivial interactions. Such inflationary models produce a departure from Gaussianity in a model-dependent manner Byrnes et al. (2010); Engel et al. (2009); Chen et al. (2009); Boubekeur and Lyth (2006). The amplitude of non-Gaussianity therefore is an important cosmological parameter that can distinguish between the plethora of inflationary models Komatsu et al. (2009).

The first order non-Gaussian parameter, , has been measured with increasing success using the bispectrum - the Fourier analog of the three-point correlation function of the CMB temperature. Such studies have found to be consistent with zero Yadav and Wandelt (2008); Smith et al. (2009); Komatsu et al. (2011); Smidt et al. (2009), with the strongest constraint coming from Planck given by Planck Collaboration et al. (2014a). The inflationary model expectation is that and a constraint at such a low amplitude level may be feasible in the future with large scale structure data and with 21-cm intensity fluctuations. Alternatively, with the trispectrum or four point correlation function of CMB anisotropies Hu (2001), we can measure the second and third order non-Gaussian parameters and . While these higher order parameters generally lead to a trispectrum that has a lower signal-to-noise ratio than the bispectrum, there may be models in which the situation is reversed with the trispectrum dominating over the bispectrum contribution. An example of such a model is an inhomogeneous end to thermal inflation discussed in Ref. Suyama et al. (2013).

A previous analysis using WMAP data out to using the kurtosis power spectra involving two-to-two and three-to-one temperature correlations Smidt et al. (2010a); Munshi et al. (2011a), found and at the 95% confidence level (C.L.). Other measures of the WMAP trispectrum have been presented in Smidt et al. (2010b); Sekiguchi and Sugiyama (2013); Fergusson et al. (2010); Regan et al. (2015). While the Planck data have been used to constrain at the 95% C.L. such a constraint ignored the signal associated with Planck Collaboration et al. (2014a). Using all of the Planck data, the expectation is that can be constrained with a 68% CL uncertainty of Sekiguchi and Sugiyama (2013) with , while can be constrained down to 560 if Kogo and Komatsu (2006). Here we present an analysis of the Planck temperature anisotropy maps by making use of kurtosis power spectra to constrain and jointly.

Theory.— We begin the discussion with the temperature trispectrum defined as Okamoto and Hu (2002)

(1) | |||||

where we have introduced the Wigner 3- symbol. The angular trispectrum, , can be further expressed in terms of sums of the products of Wigner 3- or 6- symbols times the so-called reduced trispectrum, Hu (2001).

To derive the angular trispectrum given by we assume that the curvature perturbations of the universe generated by inflation follow as:

(2) |

where the curvature perturbation and the initial gravitational potential are related by and .

We refer the reader to Ref. Kogo and Komatsu (2006) for intermediate steps in our derivation. Using the above form the full trispectrum can be reduced to two forms involving the two amplitudes (associated with term in above) and coming from .

Defining , Regan et al. (2010), where

(3) |

we find that the reduced trispectrum is

(4) |

The two terms are

(5) | |||||

and

(6) | |||||

Here and .
The primordial curvature power spectrum is with no “running”Planck Collaboration
et al. (2014b).
Here is the pivot scale set at .
We use the public code ^{2}^{2}2http://www.mpa-garching.mpg.de/~komatsu/CRL/nongaussianity/localform/ to compute ,
and the temperature transfer function .

In the part, we define the function as

(7) |

Following the efficient algorithm in Liguori et al. (2007), we define , and compress and into one dimension such that

(8) |

Here . We validate that this fast algorithm gives the same results as Eq. 7.

The first part of the trispectrum associated with approximates to at . This is due to the fact that the integrand peaks at and Pearson et al. (2012). Here is the comoving distance at last scattering surface and . For the comparison with the data, however, we perform an exact calculation defined in Eqs. 5, 6. The adaptive -grid is used for the integration.

The estimators of the connected trispectrum are constructed in Refs. Munshi et al. (2011a, b) and they are given by

(9) |

and

(10) |

In Eqs. 9, 10, the reduced trispectrum is evaluated at and . The estimators and are parametrized by these two parameters. The denotes the full trispectrum from data or simulation.

In our analysis, , and . The trispectrum computing time is proportional to at a single . In order to make these calculations more efficient, we use Monte Carlo integration for , i.e., replacing by . The vector l(=) is uniformly sampled from and . For , we restrict the diagonal elements within and validate that a bigger upper bound negligibly modifies the trispectrum. The Wigner 3- symbols’ intrinsic selection rule also helps reduce the computation time. With all these efficient algorithm, we can achieve a hour-level computation time, which is about three orders of magnitude faster than the brute-force calculation. We show the theoretical predictions of these estimators for the case in Fig. 1 for a fixed set of and values for which non-Gaussian simulated maps are available.

From simulated and real data, spherical harmonic coefficients and are computed by inverse spherical harmonic transformation (SHT). Then the two weighted maps are generated from definitions , and where the angular power spectrum is inclusive of noise. is calculated by anafast of Healpix which removes monopole and dipole. To correct the masking effect, we scale the masked modes and by to match the underlying temperature power spectrum. These masked modes are also beam- and pixel window-deconvolved. In the following text, we neglect “n” for brevity.

From and maps, we construct . Then we make and . We can calculate four types of power spectra:

(11) |

(12) | |||||

(13) |

and

(14) |

When all the power spectra are integrated along the line of sight, they become:

(15) |

(16) |

(17) |

and

(18) |

The trispectrum estimators

(19) |

and

(20) |

are then constructed from the correlations associated with and maps that are either from data or simulations.

These estimators are applied to 143 GHz and 217 GHz temperature datasets, as well as the cross-correlation GHz. For the cross correlation, the estimators are

(21) | |||||

and

(22) | |||||

Simulation Validation:
To validate our estimates of the connected trispectra, we make non-Gaussian CMB signal simulations. The non-Gaussian maps for WMAP are publicly available ^{3}^{3}3http://planck.mpa-garching.mpg.de/cmb/fnl-simulations/ so we simulate maps with and , and all
the WMAP experimental settings, consistent with 5-year observations, are adopted. For the signal part, and we choose , i.e.,
given the expected relation between and , independent of the exact value of . Note that the non-Gaussian
simulations we use assume and in a joint model fit to data we test this expectation.
The WMAP 5-yr noises are then added in the signal simulations. The WMAP simulation is . Here and are provided by WMAP. The estimator of the connected trispectrum is . In Fig. 1
we show that the average connected parts from 100 full-sky realizations are consistent with the theoretical calculations.

Data Analysis and Results: We use Planck 143 GHz and 217 GHz temperature maps for the present analysis. We use the foreground mask to remove the point sources and galactic emissions for both frequencies. The 217 GHz map cleaned after the foreground mask still contains visible emission around the galactic plane, so we use an extended mask to further cut the 217 GHz data around it. The resulting sky fractions for both maps become and . At 143 GHz, the map is convolved with a Gaussian beam and has noise. At 217 GHz, it is and . Following Ref. Planck Collaboration et al. (2014c), point sources (PS) and cosmic infrared background (CIB) are also included in simulated data. The power spectra for these two sources are and , respectively. The foreground power at these frequencies are with the parameters . In addition, a white noise is added into the simulations. The data structure is expressed as where n is a direction on the sky, is the beam transfer function, is the pixel transfer function at , and is the noise simulation. We use 100 signal and noise realizations from the FFP6 simulation set of the Planck collaboration Planck Collaboration et al. (2014d). We use the best-fit cosmological parameters from “Planck+WP+highL” Planck Collaboration et al. (2014b). Specifically, , , , , at pivot scale , and Planck Collaboration et al. (2014b).

We calculate both trispectra and from Gaussian simulations and data for Planck. The Gaussian term in the trispectra is averaged from 100 Planck simulations for frequency combinations GHz, GHz and GHz, and is removed from the raw signal, which is defined as the combination of the connected part and the disconnected part. All the trispectra are shown in Fig. 2. It is seen that the disconnected components dominate the raw signal and our simulations can precisely recover these significant biases. Also, all the trispecta show consistent shapes. From 100 simulations, the full covariance matrix M is obtained for each frequency combination and the vector . Here is index of trispectrum band. We choose five bands for each spectrum: =[2,152], [152,302], [302,452], [452,602], [602,800]. Here we use and . We want to both avoid systematic issues with the high trispectra and get enough signal-to-noise, so we choose this conservative cut here.

We choose a binning function to maximize the sensitivity

(23) |

here is the fiducial model with , and which is the connected trispectrum from the simulation or data.

The likelihood function of the data is given as

(24) |

where the two free parameters are , index of the band, and the index of the frequency combination.

We draw samples for two parameters from Monte Carlo Markov chains with flat priors and . The 217 GHz map is still significantly contaminated by CIB although we use a very conservative cut which removes of the sky, so we do not include GHz into our parameter estimation. The constraints for and are listed in Table 1. In the last row of Table 1, we show the 1-parameter constraint on with . For all the combinations, we find that and are consistent with zero. We check the consistency between different frequency combinations in Fig. 4. From Fig. 4, it is seen that different bin sizes do not change the results. We also check the impact of effective range on the parameters. From Fig.4, we find that adding more range can result in a higher value of and the interpretation is that the high range is systematically contaminated by unresolved point sources and non-Gaussian contribution of CIB beyond the foreground mask. All the results shown in Fig. 4 are summarized in Table 2.

Summary: We present the first joint constraints on using Planck kurtosis power spectra that trace square temperature-square temperature and cubic temperature-temperature power spectra. The Gaussian biases in these statistics are corrected for with simulations and we make use of non-Gaussian simulations to test our pipeline. We find the best joint estimate of the two parameters to be and . If , .

###### Acknowledgements.

AC and CF acknowledge support from NSF AST-1313319 and James B. Ax Family Foundation through a grant to Ax Center for Experimental Cosmology. DR acknowledges support from the Science and Technology Facilities Council [ST/L000652/1] and from the European Research Council [ERC Grant Agreement No. 308082].## References

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