A Modal Approach to Constrain Inflation through Numerical Bispectra

TL;DR

Confronting inflation with high-precision bispectra is computationally intensive, motivating a template-free pipeline that combines Primodal's efficient numerical bispectrum generation with the Planck/Modal analysis. By expressing numerically computed bispectra as separable mode expansions and projecting them into the Planck CMB basis, the authors introduce a consistency-level indicator $c_{NL}$ to assess model-data agreement without relying on analytic templates. The method is validated on IR DBI inflation, yielding $c_s$ constraints of $\ge 0.073$ and $\beta$ constraints of $\le 0.39$ (95% CL) and demonstrating robust constraints from both the power spectrum and bispectrum. This approach preserves amplitude information, enables direct model testing beyond standard templates, and is poised to exploit forthcoming high-resolution data from experiments like the Simons Observatory.

Abstract

Constraining inflationary models with high precision bispectra across broad parameter ranges is a challenging task, requiring intensive computations at all stages, first, predicting the primordial inflation bispectrum from quantum field theory, secondly, projecting this forward with transfer functions to the late universe and, finally, comparing with the bispectrum extracted from the observational data and matching mock catalogues. Here, the longstanding separable \texttt{Modal} pipeline for constraining primordial bispectrum templates using WMAP and Planck CMB data has been supplemented by the more recently developed \texttt{Primodal} code to accurately calculate bispectra numerically from inflation models, showing great potential for enhanced computational efficiency; \texttt{Primodal} exploits the in-in separability of the tree-level in-in formalism, together with a separable mode-expansion technique to bypass the need for point-by-point bispectrum calculations. Building upon this progress, we propose a bispectrum pipeline that systematically explores the parameter space of inflationary Lagrangians, numerically computing the tree-level bispectrum (and power spectrum) for each scenario and comparing with the \texttt{Modal} bispectrum decompositions obtained from the Planck 2018 data. Our pipeline identifies and excludes disfavored scenarios through this analysis, providing direct constraints on the parameter space, the sound speed and other quantities from the surviving observationally viable scenarios. This is preparatory work for a planned analysis using much higher-resolution CMB data from the Simons Observatory. To validate our pipeline, we perform a proof-of-concept analysis of the IR DBI inflation model, obtaining constraints of $c_s \geq 0.073$ for the sound speed and $β\leq 0.39$ for the parameter space, demonstrating the pipeline's accuracy and effectiveness.

Figures (8)

• Figure 1: (a) Three-dimensional plot of the numerical DBI bispectrum computed with Primodal; (b) The differences between the numerical DBI bispectrum and templates with scaling (solid lines) and without scaling (dashed line) at folded, equilateral and squeezed limits.
• Figure 2: Graphic representation of the calculation and transformation steps involved in the Modal pipeline. The template-free analysis starts with the numerical computation of the In-In formalism with Primodal and ends with measurements of the consistency-level indicator $c_{\mathrm{NL}}$ defined in eqn (\ref{['E53']}). A simple template-based analysis starts with the mode-decomposition of analytical templates (the second block) and ends with the apparent "nonlinearity parameter" $f_{\mathrm{NL}}$.
• Figure 3: Scenarios in the sub-parameter space of $\lambda$, $V_0$ and $\beta$, at (a) $\phi_i = 0.457$, $\Delta N_* = 55.4$; (b) $\phi_i = 0.457$, $\Delta N_* = 56.2$; (c) $\phi_i = 0.457$, $\Delta N_* = 55.4$. Scatters (red and green) correspond to scenarios with $A_s$ and $n_s$ located inside the $1 \sigma$ region of Planck 2018 results for the scalar power spectrum. Red dots label scenarios ruled out by bispectrum analysis (with $c_{\mathrm{NL}} = 1$ being outside their $2 \sigma$ region). Green dots label scenarios which cannot be ruled out by $2\sigma$.
• Figure 4: All secnarios in the sub-parameter space of $\lambda$, $V_0$ and $\beta$ favoured by power spectrum, marginalizing over $\phi_i$ from 0.455 to 0.465 and $\Delta N_*$ from 55 to 56.5. Green scatters label scenarios that survive after the bispectrum analysis, and red scatters label those inconsistent with the Planck bispectrum by $2\sigma$ and are therefore ruled out. The Blue plane corresponds to $\beta = 0.36$, which is interpreted as the constraint on $\beta$.
• Figure 5: The quantity $c_{\mathrm{NL}}$ is plotted as a function of $c_s$. The "+" symbols mark the values of $c_{\mathrm{NL}}$ for the benchmark scenarios, in which only $\beta$ varies from 0.16 to 0.6, while all other parameters are fixed according to Eqs. (\ref{['6E1.1']})--(\ref{['6E1.5']}). (Note that some of these benchmark scenarios are excluded by the power-spectrum constraints.) The corresponding $1\sigma$ and $2\sigma$ confidence regions are shown as the dark--blue and light--blue shaded areas, respectively. The green scatter points indicate the values of $c_{\mathrm{NL}}$, together with $c_{\mathrm{NL}}\pm\sigma$ and $c_{\mathrm{NL}}\pm2\sigma$, obtained from the full five--dimensional parameter scan after imposing the power-spectrum constraints. The red horizontal line corresponds to $c_{\mathrm{NL}} = 1$. It intersects the boundary of the $2\sigma$ region of the power-spectrum--allowed scenarios at $c_s = 0.073$, which is therefore interpreted as the lower bound on the sound speed.