Primordial Non-Gaussianity Predictions

• Primordial non-Gaussianity predictions are measurable deviations from Gaussian initial conditions in the early universe that inform the analysis of CMB bispectrum and trispectrum data.
• The methodology employs skewness and kurtosis power spectrum estimators, optimized with weighting functions and pseudo-Cℓ corrections to mitigate complex survey masks.
• WMAP constraints and Fisher forecasts for Planck and EPIC enable discrimination between single-field and multi-field inflation models through consistency tests like Aₙₗ.

Primordial non-Gaussianity (PNG) predictions constitute one of the most discriminating signatures for early universe physics, especially the inflationary paradigm and its alternatives. PNG refers to measurable deviations from Gaussian random initial conditions in the primordial curvature perturbations, typically quantified through the amplitude and shape of higher-order correlations, primarily the bispectrum (three-point function) and trispectrum (four-point function) of the CMB and large-scale structure. The constraints and forecasted sensitivities on key non-Gaussian parameters, such as $f_{NL}$ (bispectrum amplitude), $\tau_{NL}$ and $g_{NL}$ (trispectrum amplitudes), directly inform the space of viable inflationary and multi-field models, as well as the physics of structure formation at the largest observable scales.

1. Parameterizations and Theoretical Framework

The local model of PNG expands the primordial curvature perturbation as

$$\zeta(\mathbf{x}) = \zeta_g(\mathbf{x}) + \frac{3}{5}f_{NL} \big[\zeta_g^2(\mathbf{x}) - \langle \zeta_g^2 \rangle \big] + \frac{9}{25}g_{NL} \zeta_g^3(\mathbf{x})$$

where $\zeta_g$ is a Gaussian field, $f_{NL}$ describes the quadratic (bispectrum) amplitude, and $g_{NL}$ the cubic (trispectrum) amplitude. The trispectrum parameter $\tau_{NL}$ is defined via the square of the quadratic nonlinearity in the $\delta N$ formalism: $$f_{NL} = \frac{5}{6}\frac{N''}{(N')^2}, \quad \tau_{NL} = \frac{(N'')^2}{(N')^4}, \quad g_{NL} = \frac{25}{54}\frac{N'''}{(N')^3}$$ with $N$ the number of e-folds and primes denoting derivatives with respect to the scalar field(s). The dimensionless consistency ratio

$A_{NL} = \frac{\tau_{NL}}{(6f_{NL}/5)^2}$

serves as a stringent test for single-field inflation, which predicts $A_{NL}=1$ exactly; deviations signal multi-field or nontrivial dynamics.

2. Estimators and Statistical Methodology

To extract primordial non-Gaussian signals from the CMB, the paper introduces and applies higher-order power spectrum estimators: the skewness power spectrum for $f_{NL}$ and kurtosis power spectra (2-to-2 and 3-to-1) for $\tau_{NL}$ and $g_{NL}$. Weighted maps $A(r, \hat n)$ and $B(r, \hat n)$ are constructed using harmonic coefficients with optimal filter functions given by integrals over the radiation transfer function and primordial power spectrum: $$A_{\ell m}(r) = \frac{\alpha_\ell(r)}{\mathcal{C}_\ell} b_\ell a_{\ell m}, \quad B_{\ell m}(r) = \frac{\beta_\ell(r)}{\mathcal{C}_\ell} b_\ell a_{\ell m}$$ where $\mathcal{C}_\ell = C_\ell b_\ell^2 + N_\ell$. The estimators are

• Skewness: two-to-one and ab-b power spectra (for $f_{NL}$)
• Kurtosis: 2-to-2 and 3-to-1 spectra (for $\tau_{NL}$, $g_{NL}$), integrating combinations of $A$, $B$ over $r$ and harmonics.

Mask and partial sky corrections are implemented via pseudo-$C_\ell$ techniques, using a coupling matrix $M_{\ell\ell'}$ derived from the window (mask) harmonic coefficients and Wigner 3j symbols. This approach systematically corrects estimated statistics in the presence of complex survey masks without requiring cumbersome linear term corrections.

3. Empirical Constraints from WMAP

Applying the described estimators to WMAP 5-year V+W band data (multipoles $l_{max} \simeq 600$), the following 95\% confidence level bounds are obtained:

• $-7.4 < g_{NL}/10^5 < 8.2$
• $-0.6 < \tau_{NL}/10^4 < 3.3$

These bounds, particularly on $\tau_{NL}$, represent an improvement by nearly four orders of magnitude compared to limits set by COBE ($\tau_{NL} < 10^8$). The $f_{NL}$ constraint used for testing $A_{NL}$ is from an earlier paper: $-36.4 < f_{NL} < 58.4$. All values are consistent with no detection of primordial non-Gaussianity, but the upper limits are now low enough to exclude broad classes of multi-field and strongly interacting inflationary models.

4. Consistency Tests and Model Discrimination

A central result is the $A_{NL}$ consistency test: $A_{NL} = \tau_{NL} / (6f_{NL}/5)^2 = 1$ for all single-field inflation models. Significant observational deviation from unity would immediately falsify all single-source mechanisms. Thus, the trispectrum and its ratio to the bispectrum serve as a key axis for discriminating between inflationary scenarios. The authors' analysis indicates that trispectrum measurements can become more sensitive than the bispectrum in some regimes, due to enhanced small-scale signal contributions.

5. Forecasts for Future CMB Experiments

Using Fisher information forecasts incorporating realistic noise, beam, and masking effects, the following detection prospects are established:

• Planck: Detection threshold for $\tau_{NL}$ is $\sim 3000$ for local-type configurations, possibly lower with strong $A_{NL}$. For $g_{NL}$, error bars will improve but the constraint remains less tight compared to $\tau_{NL}$.
• EPIC (next-generation): Sensitivity expected down to $\tau_{NL} \sim 600$.
• Any measured $A_{NL}\ne 1$ at these precision levels would rule out single-source slow-roll inflation across the board, making the trispectrum a decisive probe of early universe physics.

Table: Summary of Constraints from WMAP 5-Year Data

Parameter	Scaled Constraint	95% C.L. Bound
$g_{NL}$	$g_{NL}/10^5$	$-7.4$ to $8.2$
$\tau_{NL}$	$\tau_{NL}/10^4$	$-0.6$ to $3.3$
$A_{NL}$	$\tau_{NL}/(6f_{NL}/5)^2$	Test at $\sim 1$

6. Methodological Advancements

By leveraging power spectrum–based estimators for higher-order moments, the paper circumvents deficiencies of earlier moment-based statistics (e.g., need for linear term corrections, sensitivity to cut-sky). The full statistical power of modern CMB datasets is retained even in the presence of complex observational masks.

7. Implications and Future Directions

The stringent WMAP constraints—especially on $\tau_{NL}$—substantially tighten the viable inflationary model space. The methods and results lay the groundwork for Planck and next-generation CMB missions to access trispectrum amplitudes relevant to distinguishing among single-field, multi-field, and exotic inflationary models. Detection of $A_{NL} \ne 1$ remains a "smoking gun" for multi-field or non-canonical physics. The scale-dependence (and $k$-shape configurations) in the higher-order spectra remain a crucial observational frontier, as they can be used to disentangle primordial from secondary and systematic contributions, offering a robust pathway to revealing the dynamical physics of the early universe.

The described consistency framework and higher-order CMB estimators represent the contemporary frontier for PNG constraints, with clear paths for exploiting the increased sensitivity of new cosmological surveys.