\[ f_{\rm NL}^{\rm loc}=\tfrac{5}{12}\,(1-n_s)\simeq 1.5\times10^{-2} \]

single-clock + adiabatic + attractor

\( \sigma\!\left(f_{\rm NL}^{\rm loc}\right)\lesssim 1 \)

• A measured \(|f_{\rm NL}^{\rm loc}|\sim\mathcal{O}(1)\) would mean an extra field, nonlinear superhorizon evolution, or a non-attractor phase.
• \(n_s\) sets where the relation becomes testable.

Rules out standard attractor single-clock dynamics, not literally every one-field model [Green, TASI 2212.08685 §5.2].

\[ f_{\rm NL}^{\rm equil}\sim\mathcal{O}(1)\!\left[\,c_s^{-2}\!-\!1,\ \tfrac{\dot\phi^{2}}{\Lambda^4},\ \ldots\right] \]

\(c_s\) = sound speed of the fluctuations  ·  \(\dot\phi\) = inflaton velocity  ·  \(\Lambda\) = EFT cutoff of inflation

\( \bigl|f_{\rm NL}^{\rm equil}\bigr|\sim 1\!-\!10,\ \ \sigma\!\left(f_{\rm NL}^{\rm equil}\right)\lesssim 1 \)

• Canonical slow-roll gives \(f_{\rm NL}^{\rm eq}\ll1\) (weakly coupled). A measured \(f_{\rm NL}^{\rm eq}>1\) rules that out.
• Unity asks a different question: is the inflaton weakly coupled?

\(1\!-\!10\) is a natural regime, not a universal prediction; pure Einstein-gravity terms stay slow-roll suppressed. A heavy field \(m\sim H\) gives an orthogonal / quasi-single-field shape instead [Green, TASI 2212.08685 §5.3].

\(f_{\rm NL}^{\rm local} = -0.9 \pm 5.1\)

Planck 2018 T+E bispectra [Planck Collaboration 2019]

• One 2D last-scattering surface, so \(N_{\rm modes}\sim \ell_{\rm max}^2\)
• Temperature is already cosmic-variance limited

\(f_{\rm NL}^{\rm local} = -3.6^{+9.0}_{-9.1}\)

DESI DR1 P analysis [Chaussidon+ 2024]

independent \(P{+}B\) re-analysis: \(-0.1\pm7.4\) [Chudaykin, Ivanov and Philcox 2025]

• A 3D volume, so \(N_{\rm modes}\sim V k_{\rm max}^3 \gg \ell_{\rm max}^2\)
• \(\sigma(f_{\rm NL})\sim1\) is within the reach of SPHEREx, Spec-S5 [Chen, Green and Lee 2026]
• Also sensitive to the cosmological collider [Green, Han and Wallisch 2026]
• …and galaxies are biased tracers, so \(f_{\rm NL}\) shows up as a scale-dependent bias

Peak-background split: a long density mode \(\delta_L\) lifts the local background, so more peaks cross \(\delta_{\rm cr}\) and more galaxies form.

\[ \delta_g = b_1\,\delta_m \]

\(b_1\) = how galaxy abundance responds to the long density mode. Local PNG adds a second response: the long potential mode also modulates the small-scale power.

\[ \Delta b(k)\;\propto\; b_\phi\,f_{\rm NL}\,k^{\,\Delta-2} \]

local \(k^{-2}\)  ·  orthogonal \(k^{-1}\)  ·  equilateral \(k^{0}\)

\(\Delta\) is the squeezed-limit exponent of the primordial bispectrum.

\[ I_{ij}\propto\sum_p w_p\,(x_{i,p}-x_{i,g})(x_{j,p}-x_{j,g}) \]

size \(=\) trace \(t\equiv\mathrm{tr}\,I_{ij}\);   field \(T(\mathbf{x})=\sum_g t\,\delta_D(\mathbf{x}-\mathbf{x}_g)\)

\[ \delta_T=\frac{T}{\bar T}-1,\qquad \boxed{\;\delta_s \equiv \delta_T-\delta_g\;}=b_s\,\delta_m \]

Same 2-point formalism as counts, on galaxies you already have.

\(b_1^{s}\approx 0\)
but \(b_\phi^{s}\neq 0\),
opposite sign.

The \(b_\phi/b_1\) ratio and the sign flip maximize the signal in multi-tracer analyses.

Sample-variance cancellation beats the few-modes floor.

Opposite-sign \(b_\phi^{s}\) gives a near-maximal split, and it also loosens the \(b_\phi\) degeneracy.

Empirical halo-to-galaxy map, fixed by small-scale clustering.

The same data that fix these curves carry information about \(b_\phi\).

A Bayesian fit of the HOD to the projected correlation function \(w_p(r_p)\).

The posterior spread on the HOD becomes the spread on \(b_\phi\).

LRG1 \(\mathcal{N}(3.19,1.85^2)\) · LRG2 \(\mathcal{N}(3.99,2.19^2)\) · LRG3 \(\mathcal{N}(4.18,1.95^2)\), for \(0.4

Consistent with universality on average — honest about the spread.

With prior: truth within \(1\sigma\), always.

Without: posteriors pile up at 0. Gains up to 55% at high \(z\).

SUMMARY  ·  \(d=\widehat P_k=\tfrac{1}{N_k}\!\sum_{\mathbf k\in\,\rm bin}|\delta(\mathbf k)|^2\)

\[ \ln\mathcal{P}(\widehat P\mid\theta)=-\tfrac12\!\!\sum_{i\,\in\,\text{bins}}\!\frac{\big[\widehat P_{k_i}-P_{k_i}(\theta)\big]^2}{\sigma^2_{k_i}(\theta)} \]

average \(|\delta(\mathbf k)|^2\) in each bin, then sum over bins

\(\sigma^2_{k_i}(\theta)\) = sample (cosmic) variance + shot noise · phases gone

FIELD LEVEL  ·  \(d=\delta(\mathbf k)\)

\[ \ln\mathcal{P}(\delta\mid\delta_{\rm in},\theta)=-\tfrac12\!\sum_{\mathbf k}\frac{\big|\delta^{\rm obs}(\mathbf k)-\delta^{\rm det}(\mathbf k;\theta,\delta_{\rm in})\big|^2}{\sigma_\epsilon^2(\mathbf k)} \]

compare each mode, then sum over modes

\(\sigma_\epsilon^2(\mathbf k)\) = free EFT noise, not Poisson [eq 5] · sample \(\delta_{\rm in}\) · phases kept

Identical modes, identical \(k_{\rm max}\) — the gain is pure information that \(P{+}B\) throws away.

Inject any primordial bispectrum into the initial field — infer its amplitude directly.

No per-shape template. Equilateral has no \(k^{-2}\) handle, so field level may be its only clean route in LSS.

Open — for all of us:

• Is the \(P{+}B\) forecast setup optimal?
• How much do modeling assumptions affect this comparison?

\[ \delta_g(\theta,\hat s)\;=\;\sum_O b_O\,O(\theta,\hat s)\;+\;\epsilon \]

\(O\in\{\delta,\ \delta^2,\ K_{ij}^2,\ \nabla^2\delta,\ \ldots\}\), built from the initial field \(\hat s\)

bias expansion: free coefficients \(b_O\), marginalized · equivalence principle: fixes which \(O\) are allowed · stochasticity: \(\epsilon\), variance \(\sigma_\epsilon^2(k)\)

• Decorated HOD, SDSS-LRG environment assembly bias (\(B_{\rm cen}{=}{-}0.04,\,B_{\rm sat}{=}{-}0.17\))
• \(b_1\) shifts 2–3\(\sigma\) — absorbs the amplitude change
• \(f_{\rm NL}\) posterior essentially unchanged
• Targeted check (LRG3) — not a general immunity claim