Primordial Non-Gaussianities in Cosmology

• Primordial non-Gaussianities are deviations from Gaussian distributions in cosmic density fluctuations, directly probing the physics of inflation and interactions at ultrahigh energies.
• They are quantified via higher-order correlations like the bispectrum and trispectrum, with specific shapes (local, equilateral, etc.) guiding observational constraints.
• Advanced simulation techniques and theoretical methods, including EFT and separate universe approaches, refine our understanding and constraints on inflationary models.

Primordial non-Gaussianities (PNGs) are deviations from Gaussian statistics in the initial distribution of cosmological density fluctuations, originating during the inflationary era. These non-Gaussian features encode information about the physics of inflation, the mechanism for generating primordial curvature perturbations, and the possible presence of additional particle content or non-trivial interactions at ultrahigh energies. PNGs are typically parameterized by higher-order correlation functions, most notably the bispectrum (three-point function) and trispectrum (four-point function), with specific "shapes" (local, equilateral, orthogonal, resonant, etc.) characterizing their physical origin and observable imprint.

1. Theoretical Framework and Parameterization

Primordial non-Gaussianity is most commonly quantified via the primordial gravitational potential Φ and its local expansion,

$$\Phi(\mathbf{x}) = \Phi_\text{l}(\mathbf{x}) + f_\mathrm{NL} \left[\Phi_\text{l}^2(\mathbf{x}) - \langle \Phi_\text{l}^2 \rangle \right] + g_\mathrm{NL} \left[\Phi_\text{l}^3(\mathbf{x}) - 3\Phi_\text{l} \langle \Phi_\text{l}^2 \rangle \right]+\dots$$

where Φₗ is the linear (Gaussian) part, and the nonlinearity parameters fₙₗ, gₙₗ, etc., describe quadratic, cubic, and higher-order contributions.

The bispectrum,

$$B_\Phi(k_1,k_2,k_3) = 2f_\mathrm{NL}\left[P_\phi(k_1)P_\phi(k_2) + \text{cyc.}\right]$$

is the lowest-order diagnostic of PNG; different models of inflation predict characteristic bispectrum "shapes," such as local (squeezed), equilateral, orthogonal, and resonant.

The observationally accessible curvature perturbation ζ relates to Φ via

$$\zeta = -\frac{5}{3}\Phi \quad \text{(during matter domination)},$$

and the power spectrum and higher-order correlators of ζ then fully characterize the statistical content of the primordial perturbations. For multi-field and non-standard models (e.g., non-canonical kinetic terms, excited initial states), the form and scale-dependence of PNG can be considerably richer (Renaux-Petel, 2015).

Current constraints from Planck and large-scale surveys set stringent upper limits on fₙₗ for the canonical shapes: $$\begin{array}{l} f_\mathrm{NL}^\mathrm{local} = 0.8 \pm 5.0, \ f_\mathrm{NL}^\mathrm{equilateral} = -4 \pm 43, \ f_\mathrm{NL}^\mathrm{orthogonal} = -26 \pm 21 \end{array}$$ (68% CL) (Renaux-Petel, 2015).

2. Physical Origins and Model Dependence

The physical mechanisms able to generate observable PNG include:

• Single-field slow-roll inflation: Predicts negligible PNG (fₙₗ ∼ slow-roll parameters, i.e., ≪ 1), and the bispectrum satisfies the Maldacena consistency relation in the squeezed limit.
• Single-field models with non-canonical kinetic terms (e.g., DBI, k-inflation): Allow for large equilateral-type PNG if the sound speed cₛ ≪ 1, with

$f_\mathrm{NL}^\mathrm{eq} \sim \frac{1}{c_s^2}.$

• Multi-field inflation (e.g., curvaton, modulated reheating): Easily yield large local/squeezed PNG, often through classical transfer of isocurvature perturbations after horizon exit, with

$$f_\mathrm{NL}^\mathrm{local}\sim \text{nonlinear transfer function coefficients from the δN expansion}.$$

• Quasi-single-field and "cosmological collider" scenarios: Feature extra fields with masses m ∼ H during inflation, inducing bispectra that scale as (k_L/k_S)^{3/2-ν} (with ν related to m/H), interpolating between the local and equilateral shapes. Oscillatory and resonant features, as well as particle production, introduce complex and scale-dependent PNG (Anbajagane et al., 2 Sep 2025).
• Non-local inflation, higher-derivative, and non-Bunch-Davies scenarios: Can produce orthogonal or highly scale-dependent/oscillatory bispectra and significant deviations from the single-field consistency relations (Koshelev et al., 2022).
• Primordial Black Hole (PBH) formation: Sharply peaks the curvature power spectrum at small scales, inherently invokes strong PNG effects, and is extremely sensitive to the sign and amplitude of local non-linearity parameters (Young et al., 2013, Taoso et al., 2021).

3. Observational Signatures and Statistical Probes

PNGs introduce distinctive signatures in a range of cosmological observables:

• Cosmic Microwave Background (CMB):
  • The temperature/polarization bispectrum measures PNG at z ∼ 1100. The optimal estimator decomposes maps into harmonic space, and matches compact templates for local, equilateral, and orthogonal shapes, as well as oscillatory and feature bispectra (Renaux-Petel, 2015).
  • Planck and other CMB experiments set the tightest constraints on large scales, but are limited by cosmic variance and foregrounds.
• Large-Scale Structure (LSS):
  • Scale-dependent bias: For local PNG, the effect is a correction,

  $$\Delta b(k) = \frac{2\delta_c(b_1-1)f_\mathrm{NL}}{\mathcal{M}(k)}$$

  with $\mathcal{M}(k) \propto k^2 T(k) D(z)$, resulting in ∼ fₙₗ/k² enhancement at large scales for biased tracers (halos, galaxies) (Sullivan et al., 27 Mar 2025). - Galaxy clustering: The largest scales (small k) are most sensitive, but survey volume, systematics, and uncertainties in tracer bias are limiting factors. Constraints using ∼800,000 photometric quasars have yielded competitive bounds on both fₙₗ and gₙₗ (Leistedt et al., 2014). - Cosmic voids: The bias of large voids can be negative, and including void scale-dependent bias can substantially improve PNG constraints, tightening limits on fₙₗ by up to a factor of two in multi-tracer analyses (Chan et al., 2018). - Halo abundances and cluster scaling relations: Modifications to mass functions, concentrations, and scaling parameters (e.g., Y-M, T-M, S-M) have been observed in high-resolution N-body/SPH simulations with varying fₙₗ; PNG affects normalizations and redshift evolutions, particularly in models where fₙₗ is scale-dependent (Trindade et al., 2016, Stahl et al., 17 Jan 2024). - Weak Lensing Convergence Field: Higher-order lensing statistics (moments, CDFs, and the convergence 3-point function) are sensitive to non-linear and low-redshift PNG signatures. Forecasts show that LSST-like surveys achieve competitive constraints for both local and non-local shapes, especially on small (non-linear) scales (Anbajagane et al., 2023, Anbajagane et al., 2 Sep 2025, Anbajagane et al., 2 Sep 2025).

• 21 cm Tomography and Reionization Structure:

  • The 21 cm brightness temperature power spectrum during the epoch of reionization directly traces the imprint of PNG in the clustering of ionized patches, inheriting the PNG-induced 1/k² scale-dependent bias (D'Aloisio et al., 2013, Mao et al., 2013).
  • Future radio surveys (e.g., SKA, Omniscope) are expected to reach sensitivity Δfₙₗ ≲ 3–0.2, complementing CMB bounds on large scales.
• Non-relativistic Probes and PBHs:
  • The non-Gaussian tail of the density PDF governs PBH formation; even small PNG can enhance or suppress PBH production by many orders of magnitude, and the sign of the quadratic or cubic non-linearity parameters is crucial (Young et al., 2013, Taoso et al., 2021, Gong et al., 12 Apr 2024).
  • The non-Gaussianity from Poissonian fluctuations in PBH number can further imprint higher cumulants (skewness, kurtosis) and highly scale-dependent bias in the matter and halo fields (Gong et al., 12 Apr 2024).

4. Advanced Theoretical Developments and Numerical Methods

Modern approaches for modeling PNGs in structure formation and observable maps include:

• Excursion-Set Models / Peak-Background Split: Used for halo/void bias calculations, these techniques relate number density responses to global background modulations, connecting the time evolution of tracer abundance directly to PNG bias coefficients (Sullivan et al., 27 Mar 2025).
• Separate Universe Simulations: Simulations where long-wavelength modes are absorbed as a local change in the background (effectively as a change in σ₈ or growth history), which allows validation of theoretical predictions for scale-dependent bias and its dependence on tracer selection effects (Sullivan et al., 27 Mar 2025).
• Diagrammatic and Modal Decomposition: For arbitrary and non-separable bispectrum shapes (such as those in cosmological collider or resonant templates), decomposition into separable kernels and an FFT-based implementation allows accurate and efficient generation of non-Gaussian initial conditions in N-body codes (Anbajagane et al., 2 Sep 2025, Anbajagane et al., 2 Sep 2025).
• Effective Field Theory (EFT) of Large-Scale Structure: Accounting for small-scale uncertainties and counterterms in the bispectrum modeling, the EFT approach provides forecasted improvements in PNG sensitivity, especially for local-type PNG. However, theoretical uncertainties remain a limiting factor for equilateral modes even in the EFT framework (Welling et al., 2016).
• Foreground Systematics and Relativistic Corrections: For 21 cm and galaxy/weak lensing surveys, proper modeling of foreground cleaning, relativistic projection, and selection function effects is crucial to avoid spurious PNG signatures (systematics can easily generate O(1) biases in fₙₗ estimates) (Dio et al., 2016, Cunnington et al., 2020).
• Gravitational-Wave Probes: The spectrum and anisotropy of scalar-induced gravitational waves (SIGWs) are sensitive to primordial PNG of arbitrary order. Bispectrum and trispectrum of the SIGW energy density can act as null diagnostics of Gaussianity, with diagrammatic approaches permitting computation of the full hierarchy of PNG corrections (Li et al., 22 May 2025).

5. Key Results from Simulations and Forecasts

Recent simulation campaigns and Fisher-matrix forecasts demonstrate:

• Halo and Void Bias: Combined use of halos and voids in multitracer analyses can tighten fₙₗ constraints by a factor ≈2 compared to halos alone, provided tracer sampling density is ≳4×10⁻³ h³ Mpc⁻³ (Chan et al., 2018).
• Cluster Scaling and Halo Structure: Positive fₙₗ generally boosts concentration and normalization parameters in cluster scaling relations (T-M, Y-M), while negative fₙₗ has the reverse effect. Deviations from universality in halo sparsity appear most pronounced at z = 1 (Trindade et al., 2016, Stahl et al., 17 Jan 2024).
• Lensing Forecasts: Weak lensing moments (2nd, 3rd, higher) and CDFs from LSST-like surveys can constrain fₙₗ^local to σ(fₙₗ^local) ≈ 40, surpassing galaxy clustering for some shapes. For equilateral/orthogonal PNGs, lensing provides σ(fₙₗ) ≈ 110–120, similar to DESI expectations (Anbajagane et al., 2023).
• Resonant/Feature and Collider Templates: Simulations show that both oscillatory and heavy-particle PNG signatures can be accurately propagated into the fully non-linear regime, and their imprints remain distinguishable in late-time statistics. The amplitude of the resonant features in the power spectrum (A_pk) and that of the bispectrum (fₙₗ) remain effectively independent in their impact on structure, enabling joint constraints (Anbajagane et al., 2 Sep 2025, Anbajagane et al., 2 Sep 2025).
• Systematics and Degeneracies: Marginalizing over foregrounds (as in 21cm intensity mapping) can degrade σ(fₙₗ) constraints by a factor of ∼4–5 unless strong external priors are imposed (Cunnington et al., 2020). Mode projection removes bias from stellar and photometric systematics, making quasar and LSST-like photometric surveys robust competitors to CMB constraints (Leistedt et al., 2014).

6. Open Challenges and Future Directions

Several frontiers remain active in PNG research:

• Ultimate Target Sensitivities: For equilateral/orthogonal PNGs, the theoretically motivated challenge is Δfₙₗ ≈ 1, which would test fundamental aspects of EFT inflationary models and related UV-completions (Renaux-Petel, 2015, Koshelev et al., 2022).
• Systematics Control and Theoretical Error: Progress in constraining PNG at the desired level will hinge on improved modeling of foregrounds, projection/lensing effects, baryonic feedback, and bias evolution—each tightly coupled to advances in both simulation and analytic theory (Dio et al., 2016, Welling et al., 2016, Anbajagane et al., 2023, Cunnington et al., 2020).
• Novel Observables: The anisotropies and higher-order moments (bispectrum, trispectrum) of SIGWs, full non-parametric summary statistics of lensing fields, and cross-correlations between lensing and galaxy surveys offer new paths to isolate specific PNG signatures inaccessible to the CMB (Li et al., 22 May 2025, Anbajagane et al., 2023, Anbajagane et al., 2 Sep 2025).
• Community Resources: Public release of simulation suites with arbitrary PNG initial conditions (e.g., Ulagam, Aarambam) and companion analysis tools now allows the community to test, forecast, and interpret a wide array of PNG signatures in both simulated and real survey data (Anbajagane et al., 2 Sep 2025, Anbajagane et al., 2 Sep 2025, Anbajagane et al., 2023).

7. Implications for Early Universe Physics

The presence, absence, or scale-dependence of PNGs can decisively rule out entire classes of inflationary models. Detection of large local/squeezed PNG, significant running (n_NG), or resonant/oscillatory features would favor multi-field, non-canonical, non-EFT, or non-local gravity scenarios (Koshelev et al., 2022, Anbajagane et al., 2 Sep 2025). Conversely, continued null results with ever-tightening bounds will reinforce the simplest single-field slow-roll paradigm and restrict model ambiguity in the early universe to a progressively shrinking region of parameter space (Renaux-Petel, 2015).

The paper of PNGs is thus a direct probe of the microphysics of inflation and the quantum state of the primordial universe. Modern LSS surveys and simulation-based approaches are poised to further advance this frontier, providing increasingly stringent or, potentially, transformative insights into fundamental cosmology.