InterPrior: EFT Prior Calibration

• InterPrior is a multi-survey, data-driven effective-field-theory calibration framework that infers nuisance parameters directly from PFS-DESI overlap data.
• It employs multi-tracer cross-spectra to separate deterministic cosmological signals from stochastic noise, sharply constraining key bias and stochastic parameters.
• By exporting calibrated priors to the full DESI footprint, the method significantly improves forecasts on growth rates, dark energy, and neutrino mass limits.

InterPrior is the name used in "Multi-tracers, multi-surveys: data-driven EFT prior calibration from the PFS--DESI overlap" for an inter-survey, data-driven effective-field-theory prior calibration framework, also described as a multi-survey prior method. Its purpose is to calibrate nuisance-parameter priors directly from the overlap region between two independent spectroscopic surveys—in this case Prime Focus Spectrograph (PFS) and the Dark Energy Spectroscopic Instrument (DESI)—and then export those calibrated priors to the much larger DESI footprint. In the paper’s formulation, the method is “inter-” because the priors are learned between surveys rather than from simulations (Nguyen, 28 Apr 2026).

1. Definition and motivation

InterPrior is designed for one-loop EFTofLSS full-shape galaxy power-spectrum analyses, where the dominant limitation is often the marginalization over nuisance parameters rather than statistical noise. The paper emphasizes that a realistic full-shape model uses roughly 12 nuisance parameters per tracer per redshift bin. These include bias parameters $b_1$, $b_2^2$, $b_{G_2}^2$, $b_{\Gamma_3}$; counterterms $c_0$, $c_2$, $c_4$, $c$, $c_1$; and stochastic amplitudes consisting of one constant shot-noise departure plus $a_0$, $b_2^2$0 (Nguyen, 28 Apr 2026).

The motivation for calibrating these priors is quantitative. The paper states that conservative priors degrade constraints relative to the idealized case of perfect nuisance knowledge by about a factor of $b_2^2$1 in $b_2^2$2 and a factor of $b_2^2$3 in $b_2^2$4 for the combined six-sample DESI forecast. In that setting, prior calibration is not a secondary refinement but a direct route to improving constraints on growth, dark energy, and especially neutrino mass. InterPrior addresses this by replacing simulation-derived calibration with overlap-region inference from real data.

2. Calibration geometry in the PFS--DESI overlap

The framework is enabled by the $b_2^2$5 PFS--DESI overlap at $b_2^2$6. Within that common volume, the analysis can jointly use up to four tracers: PFS-ELG, DESI-ELG, DESI-LRG, and DESI-QSO. This yields up to 10 auto- and cross-spectra per redshift bin, with

$b_2^2$7

The overlap is therefore treated as a calibration laboratory in which the data themselves constrain nuisance parameters that would otherwise need to be imported from simulations (Nguyen, 28 Apr 2026).

The redshift range is partitioned into five bins: $b_2^2$8 This binning structures the multi-tracer Fisher analysis used to learn nuisance information in the shared survey volume. The method is explicitly formulated as model-independent with respect to halo-occupation-distribution assumptions, and the calibrated priors are then transferred to the full $b_2^2$9 DESI footprint.

3. Multi-tracer mechanism and the role of cross-spectra

The central statistical mechanism is the joint fit of auto- and cross-spectra for tracers that respond differently to the same underlying matter fluctuations. For an auto-spectrum, the paper writes schematically

$b_{G_2}^2$0

where the first term is the deterministic EFT signal and the second is stochastic noise, including Poisson shot noise and EFT corrections. For the cross-spectrum,

$b_{G_2}^2$1

The key structural fact is that for different galaxy populations with no shared objects,

$b_{G_2}^2$2

Cross-spectra between distinct tracers are therefore clean measurements of the deterministic cosmological signal without stochastic contamination (Nguyen, 28 Apr 2026).

This zero-cross-shot property is what allows InterPrior to separate signal from noise more effectively than a single-tracer analysis. Cross-spectra such as PFS-ELG $b_{G_2}^2$3 DESI-LRG and PFS-ELG $b_{G_2}^2$4 DESI-QSO strongly constrain relative bias amplitudes, the linear bias $b_{G_2}^2$5, the constant stochastic amplitude, and some tidal-bias combinations through the one-loop $b_{G_2}^2$6-dependence. By contrast, $b_{G_2}^2$7-dependent stochastic terms such as $b_{G_2}^2$8 and $b_{G_2}^2$9 remain more degenerate with counterterms like $b_{\Gamma_3}$0 and $b_{\Gamma_3}$1. The paper notes one special case: for PFS-ELG $b_{\Gamma_3}$2 DESI-ELG, a small shared-object fraction $b_{\Gamma_3}$3 can make the cross-shot term nonzero, but the final cosmological impact is found to be negligible when this is varied.

4. Constrained nuisance parameters and the centrality of $b_{\Gamma_3}$4

The overlap calibration significantly tightens three nuisance parameters in particular: $b_{\Gamma_3}$5, the constant stochastic term, and $b_{\Gamma_3}$6. The standout result is the prior on $b_{\Gamma_3}$7. In the broad analysis, $b_{\Gamma_3}$8 has a flat prior, while the overlap data constrain it to roughly

$b_{\Gamma_3}$9

The abstract summarizes the corresponding multi-tracer Fisher result as a calibration of the $c_0$0 prior from a flat prior to $c_0$1 (Nguyen, 28 Apr 2026).

This matters because the paper identifies a strong degeneracy between $c_0$2 and growth, and an especially strong degeneracy between $c_0$3 and the neutrino-mass-dependent suppression of clustering. In the single-tracer Fisher matrix, $c_0$4 is described as nearly perfectly degenerate with $c_0$5, with a correlation of about $c_0$6. The paper therefore emphasizes that the principal cosmological gain comes from breaking the $c_0$7--$c_0$8 degeneracy intrinsic to single-tracer analyses.

The calibration also reduces the constant stochastic amplitude from $c_0$9 to $c_2$0, and tightens $c_2$1 from $c_2$2 to about $c_2$3. At the same time, parameters such as $c_2$4, $c_2$5, and most counterterms remain close to their broad priors, because the overlap volume at $c_2$6 has little leverage on those higher-order terms. The paper’s parameter-importance decomposition identifies calibration of the $c_2$7 prior as the dominant driver of the forecast improvement.

5. Export procedure and forecast impact on DESI cosmology

The calibration procedure is formulated in Fisher-matrix terms. The overlap Fisher matrix $c_2$8 is regularized with broad external information,

$c_2$9

then inverted,

$c_4$0

and converted into marginalized calibration uncertainties,

$c_4$1

These $c_4$2 values are then imposed as Gaussian priors in the full-area DESI forecast. The paper describes the logic as a volume lever arm,

$c_4$3

where the small overlap region constrains nuisance parameters and the full footprint uses that information to sharpen cosmological inference (Nguyen, 28 Apr 2026).

The headline forecast is reported at $c_4$4. Exporting the calibrated priors to the full $c_4$5 DESI footprint of the DESI LRG1--3, ELG1--2 and QSO samples improves $c_4$6 by 8\% and $c_4$7 by 54\%, with per-sample gains of 7--25\% and 46--71\%, respectively. For a single DESI-ELG full-footprint analysis, the paper reports about a 19\% improvement in $c_4$8 and a 55\% improvement in $c_4$9. It also reports $c$0-dependent behavior: at $c$1, growth and neutrino-mass improvements are 11\% and 59\%; at $c$2, 8\% and 54\%; and at $c$3, 6\% and 49\%. The weakening at larger $c$4 is attributed to the increasing weight of counterterms and $c$5-dependent stochastic terms that the overlap cannot calibrate well.

The paper also addresses the double-counting issue created by using the overlap both for calibration and as part of the full-area analysis, and finds the effect negligible at the sub-percent level in the final forecast.

6. Relation to simulation-based priors and methodological scope

InterPrior is presented as the observational counterpart to simulation-based priors (SBPs). Both approaches seek to tighten EFT nuisance priors, improve full-shape cosmological constraints, and reduce the impact of nuisance marginalization on $c$6. The difference is methodological. SBPs are derived from HOD mocks built on $c$7-body simulations, encode the full joint distribution of EFT parameters, and can strongly constrain counterterms and stochasticity, but depend on the fidelity of the galaxy--halo model. InterPrior, by contrast, is based on real data only, uses overlap-region multi-tracer cross-spectra, is model-independent with respect to HOD assumptions, and generalizes to arbitrary combinations of overlapping spectroscopic surveys (Nguyen, 28 Apr 2026).

The paper treats the two approaches as complementary rather than mutually exclusive. InterPrior is strongest for $c$8, bias ratios, and constant stochasticity; SBPs are strongest for counterterms and $c$9-dependent stochastic terms. On that basis, InterPrior functions as a model-independent consistency check on SBPs. Agreement between SBPs and overlap-calibrated priors on parameters such as $c_1$0, $c_1$1, and the stochastic amplitudes would support the HOD-based calibration. Disagreement would suggest possible issues in galaxy--halo modeling or real astrophysical or systematic differences.

A common misconception would be to interpret InterPrior as a full replacement for all prior-calibration strategies. The paper does not make that claim. Instead, it argues that the method is most effective where cross-survey multi-tracer information cleanly isolates deterministic structure from stochastic contamination, and less effective for nuisance sectors whose signatures require more small-scale reach. A plausible implication is that overlap-based calibration can become a standard external prior layer in future full-shape analyses whenever sufficiently rich spectroscopic overlaps exist.