Limits on non-Gaussianities from WMAP data

TL;DR

The paper addresses constraints on equilateral-type primordial non-Gaussianity in the CMB, proposing a practical factorizable shape template to represent the equilateral bispectrum and an optimal estimator that accounts for partial-sky coverage and anisotropic noise. It derives a near-optimal estimator that includes a linear term to saturate the Cramer-Rao bound, enabling efficient computation via A,B,C,D maps and a normalization that incorporates sky masking. Applying the method to WMAP 1-year data yields no detection but sets strong 95% CL limits: |f_NL^equil| < 238 and |f_NL^local| < 121, with the linear correction notably tightening the local constraint. The work demonstrates a scalable approach to extracting equilateral non-Gaussian signals from current data and forecasts substantial improvements for Planck, especially when polarization is included.

Abstract

We develop a method to constrain the level of non-Gaussianity of density perturbations when the 3-point function is of the "equilateral" type. Departures from Gaussianity of this form are produced by single field models such as ghost or DBI inflation and in general by the presence of higher order derivative operators in the effective Lagrangian of the inflaton. We show that the induced shape of the 3-point function can be very well approximated by a factorizable form, making the analysis practical. We also show that, unless one has a full sky map with uniform noise, in order to saturate the Cramer-Rao bound for the error on the amplitude of the 3-point function, the estimator must contain a piece that is linear in the data. We apply our technique to the WMAP data obtaining a constraint on the amplitude f_NL^equil of "equilateral" non-Gaussianity: -366 < f_NL^equil < 238 at 95% C.L. We also apply our technique to constrain the so-called "local" shape, which is predicted for example by the curvaton and variable decay width models. We show that the inclusion of the linear piece in the estimator improves the constraint over those obtained by the WMAP team, to -27 < f_NL^local < 121 at 95% C.L.

Figures (6)

• Figure 1: Plot of the function $F(1,\, k_2/k_1, \, k_3/k_1) (k_2/k_1)^2 (k_3/k_1)^2$ for the equilateral shape used in the analysis (top) and for the local shape (bottom). The functions are both normalized to unity for equilateral configurations $\frac{k_2}{k_1}= \frac{k_3}{k_1}=1$. Since $F(k_1,k_2,k_3)$ is symmetric in its three arguments, it is sufficient to specify it for $k_1\ge k_2\ge k_3$, so $\frac{k_3}{k_1} \le \frac{k_2}{k_1} \le 1$ above. Moreover, the triangle inequality says that no side can be longer than the sum of the other two, so we only plot $F$ in the triangular region $1-\frac{k_2}{k_1}\leq \frac{k_3}{k_1} \leq\frac{k_2}{k_1} \le 1$ above, setting it to zero elsewhere.
• Figure 2: Top. Plot of the function $F(1,\, k_2/k_1, \, k_3/k_1) (k_2/k_1)^2 (k_3/k_1)^2$ predicted by the higher-derivative Creminelli:2003iq and the DBI models Alishahiha:2004eh. Bottom. Difference between the above plot and the analogous one (top of fig. \ref{['fig:shapes']}) for the factorizable equilateral shape used in the analysis.
• Figure 3: The functions $\alpha_l(r)$ (in units of ${\rm Mpc}^{-3}$), $\beta_l(r)$ (dimensionless), $\gamma_l(r)$ (in units of ${\rm Mpc}^{-2}$), and $\delta_l(r)$ (in units of ${\rm Mpc}^{-1}$) are shown for various radii $r$ as functions of the multipole number $l$. The cosmological parameters are the same ones used in the analysis: $\Omega_b h^2 = 0.024$, $\Omega_m h^2 = 0.14$, $h =0.72$, $\tau = 0.17$. With these parameters, the present conformal time $\tau_0$ is $13.24$ Gpc and the recombination time $\tau_R$ is $0.27$ Gpc.
• Figure 4: Standard deviation for estimators of $f_{\rm NL}^{\rm local}$ as a function of the maximum $l$ used in the analysis. Lower curve: lower bound deduced from the full sky variance. Lower data points: standard deviations for the trilinear + linear estimator. Upper data points: the same for the estimator without linear term, for which the divergence at high $l$'s caused by noise anisotropy had already been noticed in Komatsu:2003fd.
• Figure 5: Top: $S_{AB}(\hat{n},r)$ map for $r$ around recombination calculated with noise only. Bottom: number of observations as a function of the position in the sky for the Q1 band (the plot is quite similar for the other bands). A lighter color indicates points which are observed many times.