Perturbation-Induced Likelihood Curvature in Cosmology

• The paper introduces a perturbative likelihood curvature formalism that incorporates one-loop and bispectrum corrections to enhance parameter inference.
• The methodology computes higher-order corrections consistent with the perturbation theory order, accounting for mode coupling and non-Gaussian effects.
• The approach improves cosmological analyses by breaking parameter degeneracies and enabling systematic exploitation of higher-order statistical information.

Perturbation-induced likelihood curvature refers to the modification of the curvature (Hessian) of the log-likelihood function for cosmological summary statistics—such as the power spectrum and bispectrum—when the likelihood is constructed directly from first principles in perturbation theory (PT), rather than by assuming a Gaussian or other ad hoc form. This approach ensures that the statistical inference from large-scale structure surveys faithfully incorporates the mode coupling and non-Gaussian features encoded in cosmological initial conditions and their nonlinear evolution. By systematically computing higher-order corrections in the PT expansion parameter, the formalism yields likelihoods and their curvatures (and therefore Fisher matrices) that are fully consistent with the chosen order of perturbative accuracy (Voivodic, 29 May 2025).

1. Perturbative Likelihood Construction

The perturbative likelihood formalism expands the probability distribution of cosmological observables around the Gaussian initial conditions, incorporating corrections order-by-order in the PT parameter $\lambda$. After binning over Fourier modes (e.g., in rings of radius $k$ for the power spectrum, and triangle-bins $\chi$ for the bispectrum), the joint discrete likelihood for observed summary statistics $P_o(k_i)$ and $B_o(\chi_j)$ is given through second order ($\mathcal{O}(\lambda^2)$) as

$$\mathcal{P}^{\rm d}[\{P_o(k_i)\},\{B_o(\chi_j)\}| \theta] \approx \mathcal{P}^{(0),\rm d}[\{P_o\}|\theta] \bigg\{ 1 - \sum_{i=1}^{N^P} N(k_i) \frac{P_{t,(2,2)}(k_i|\theta) P_o(k_i)}{P_t(k_i|\theta)^2} + 2 \sum_{j=1}^{N^B} N_t(\chi_j) \frac{B_t(\chi_j|\theta) B_o(\chi_j)}{P_t(q_{1,j}|\theta)P_t(q_{2,j}|\theta)P_t(q_{12,j}|\theta)} \bigg\}$$

where $P_t$ and $B_t$ are the tree-level theoretical expectations, $P_{t,(2,2)}$ and $F_2$ encode higher-order/PT corrections, and $N(k_i)$, $N_t(\chi_j)$ count the number of independent modes in each bin.

The tree-level power spectrum is $P_t(k|\theta) = F_1(k)^2 P_L(k)$, while $P_{t,(2,2)}$ is a mode-coupling one-loop correction. This construction yields expressions for the log-likelihood, $\ln \mathcal{L}$, through one-loop order, strictly consistent with the perturbation theory employed (Voivodic, 29 May 2025).

2. Log-Likelihood Curvature and the Hessian

The curvature of the log-likelihood surface with respect to the parameter vector $\theta$ is captured by the Hessian: $$H_{\alpha\beta}(\theta) = -\frac{\partial^2}{\partial \theta_\alpha \partial \theta_\beta} \ln \mathcal{L} \bigg|_{\theta=\bar{\theta}}$$ where the derivatives are taken at a fiducial parameter point $\bar{\theta}$.

This curvature can be split into tree-level ($H^{(0)}$) and higher-order (one-loop, denoted $H^{(1\text{-loop})}$) pieces: $$H_{\alpha\beta} = H_{\alpha\beta}^{(0)} + H_{\alpha\beta}^{(1\text{-loop})}$$ The leading term $H^{(0)}$ represents the Fisher-like information from the Gaussian power spectrum; higher-order terms encode mode-coupling and non-Gaussianity corrections arising from $P_{t,(2,2)}$ and from bispectrum contributions.

3. Explicit Expressions for the Curvature

At tree-level: $$H^{(0)}_{\alpha\beta} = \sum_{i=1}^{N^P} N(k_i) \frac{1}{P_t(k_i)^2} \frac{\partial P_t(k_i)}{\partial \theta_\alpha} \frac{\partial P_t(k_i)}{\partial \theta_\beta}$$

For one-loop corrections, $H^{(1\text{-loop})}$ further splits into contributions from the power spectrum (the $(2,2)$ term) and the tree-level bispectrum:

• Power spectrum $(2,2)$ correction:

$$H^{(P_{22})}_{\alpha\beta} =\sum_i N(k_i)\Bigg[ \frac{\partial_\alpha\partial_\beta P_{t,(2,2)}}{P_t} - \frac{P_{t,(2,2)}}{P_t^2}\partial_\alpha\partial_\beta P_t - \frac{\partial_\alpha P_{t,(2,2)}\,\partial_\beta P_t+ \partial_\beta P_{t,(2,2)}\,\partial_\alpha P_t}{P_t^2} + 2\frac{P_{t,(2,2)}}{P_t^3}\partial_\alpha P_t\,\partial_\beta P_t \Bigg]$$

• Bispectrum Fisher structure:

$$H^{(B)}_{\alpha\beta} = 2\sum_j N_t(\chi_j)\Bigg\{ \frac{2\partial_\alpha B_t \partial_\beta B_t + 2 B_t \partial_\alpha\partial_\beta B_t}{\Pi} - \frac{B_t^2}{\Pi} \sum_{n=1}^3 \frac{\partial_\alpha\partial_\beta P_t(q_{n,j})}{P_t(q_{n,j})} + \cdots \Bigg\}$$

with $\Pi = P_t(q_{1,j}) P_t(q_{2,j}) P_t(q_{12,j})$ and the omitted terms representing cross-derivatives.

The total Fisher information matrix is the expectation value of the Hessian,

$$F_{\alpha\beta} = \langle H_{\alpha\beta}\rangle = H_{\alpha\beta}^{(0)} + H_{\alpha\beta}^{(P_{22})} + H_{\alpha\beta}^{(B)}$$

with integral forms for all terms in the continuum limit (Voivodic, 29 May 2025).

4. Physical Interpretation and Impact of Higher-Order Kernels

In the Gaussian (zero-order) approximation, only $H_{\alpha\beta}^{(0)}$ is present, and the log-likelihood curvature is determined solely by sensitivities of the power spectrum to model parameters. This coincides with the standard Fisher information derived from Gaussian likelihood assumptions.

Including higher-order corrections, such as the one-loop term $P_{t,(2,2)}$ and the tree-level bispectrum $B_t$, introduces both diagonal and off-diagonal blocks in the Fisher matrix. The power spectrum correction $H^{(P_{22})}$ arises from mode coupling, tightening constraints on parameters by accessing information in the mildly nonlinear regime. The bispectrum contribution $H^{(B)}$ further introduces mixed curvature terms, exploiting dependencies among different Fourier modes and helping to break degeneracies not addressed by the power spectrum alone.

Iteratively including higher-order PT kernels ($F_n$ for $n > 2$) brings in successive higher-point statistics (e.g., trispectrum, two-loop power spectrum) and their corresponding likelihood curvature blocks, guaranteeing consistency between the order of the mean theory and its statistical covariances. The sign and magnitude of each contribution are set by the functional derivatives of these higher-point spectra, with total curvature always increasing provided $\lambda$ remains within perturbative control (Voivodic, 29 May 2025).

5. Summary Table: Decomposition of Likelihood Curvature Contributions

Term	Physical Content	Mathematical Form
$H^{(0)}_{\alpha\beta}$	Gaussian power spectrum only	$$\sum_i N_i P_t^{-2} \partial_\alpha P_t \partial_\beta P_t$$
$H^{(P_{22})}_{\alpha\beta}$	One-loop power spectrum correction	Multiple derivative terms involving $P_{t,(2,2)}$ and $P_t$
$H^{(B)}_{\alpha\beta}$	Tree-level bispectrum	Multiple derivative terms involving $B_t$ and $P_t$

These entries correspond, respectively, to: the base information from the tree-level power spectrum; new parameter sensitivities and constraints from the one-loop (mode-coupling) corrections; and further mixed (power spectrum–bispectrum) information from the lowest-order non-Gaussianity.

6. Consistency and Advantages of the Perturbative Approach

The formalism ensures full consistency between the likelihood model and the chosen PT order by directly enforcing that only the PT kernels ($F_n$) and linear spectrum $P_L$ up to that order contribute. This removes ad hoc or inconsistent mixing of loop orders between the mean summary statistics and their covariance structures.

A principal advantage is that the likelihood curvature at each order is precisely determined by the underlying theoretical expansion, eliminating the need for empirical adjustment of covariance matrices or likelihood forms. A plausible implication is a systematic and improvable pathway to next-generation analyses of cosmological data, in which higher-order statistics and their information content can be robustly exploited, bounded only by the validity of the perturbative regime (Voivodic, 29 May 2025).