---
title: 'PRIYA Emulator: Lyman-α Simulation Inference'
url: https://www.emergentmind.com/topics/priya-emulator
type: topic
---

# PRIYA Emulator: Lyman-α Simulation Inference

PRIYA Emulator denotes the emulator built on the PRIYA suite of hydrodynamical Lyman-\(\alpha\) forest simulations and, in wider usage, the associated forward-modeling framework for cosmological inference from the one-dimensional flux power spectrum. In the PRIYA literature, the term can refer narrowly to the multi-fidelity surrogate for \(P_F(k,z)\) and the mean intergalactic-medium temperature, or more broadly to the coupled stack of simulations, emulator, likelihood, and MCMC pipeline used for eBOSS, high-resolution quasar spectra, and related analyses [2306.05471], [2309.03943].

## 1. Definition and conceptual scope

PRIYA was developed to provide a simulation-calibrated forward model of the Lyman-\(\alpha\) forest accurate enough for modern survey inference while avoiding rerunning expensive hydrodynamical simulations at every likelihood evaluation. Its principal emulated observable is the one-dimensional Lyman-\(\alpha\) flux power spectrum, denoted \(P_F(k,z)\) or \(P_{1\mathrm{D}}(k,z)\), and a companion emulator is constructed for the mean IGM temperature. The framework is intended for analyses of SDSS/BOSS/eBOSS data and future DESI measurements, and later work extends its use to smaller scales probed by XQ100 and KODIAQ-SQUAD [2306.05471], [2509.18271].

A central feature of PRIYA is that it is not merely a generic interpolation layer over summary statistics. The simulations include galaxy formation, star formation, stellar feedback, AGN feedback, patchy hydrogen reionization, and patchy helium reionization, and therefore model the forest with explicit thermal and reionization astrophysics rather than purely phenomenological rescalings. This gives the emulator a dual role: it is simultaneously a cosmological surrogate and an astrophysical nuisance model for IGM thermal history, mean flux evolution, and high-column-density absorber treatment [2306.05471].

In the eBOSS cosmology analysis, PRIYA is explicitly described as a percent-level, multi-fidelity Gaussian-process emulator of the 1D Lyman-\(\alpha\) flux power spectrum and the IGM temperature at mean density over an 11-dimensional parameter space. Later analyses also use a compressed likelihood derived from PRIYA posterior constraints on the local linear-power amplitude and slope, \((\Delta_L^2,n_{\rm eff})\), at a pivot \(z_p=3\) and \(k_p=0.009\ \mathrm{s/km}\) for beyond-\(\Lambda\)CDM applications such as neutrino self-interactions [2309.03943], [2503.15592].

## 2. Simulation basis and parameter space

The original PRIYA simulation suite consists of 48 low-fidelity simulations with \(1536^3\) particles in a \(120\ \mathrm{Mpc}/h\) box and 3 high-fidelity simulations with \(3072^3\) particles in the same volume. The suite was later extended for the eBOSS analysis to 60 low-fidelity simulations plus 3 high-fidelity simulations, preserving the \((120\ \mathrm{Mpc}/h)^3\) box and the same multi-fidelity design [2306.05471], [2309.03943].

PRIYA’s original simulation design spans a 9-dimensional parameter space. In practical likelihood use, this becomes an 11-dimensional emulator because two mean-flux parameters are introduced in post-processing rather than by rerunning simulations.

| Group | Parameters | Status |
|---|---|---|
| Cosmology | \(n_P\), \(A_P\), \(h\), \(\Omega_M h^2\) | Simulated |
| Thermal and feedback | \(z_{Hei}\), \(z_{Hef}\), \(\alpha_q\), \(z_{HI}\), \(\epsilon_{AGN}\) | Simulated |
| Mean flux | \(\tau_0\), \(d\tau_0\) | Post-processing |

The primordial spectrum is parameterized as
$$
P(k) = A_P \left(\frac{k}{0.78\, \mathrm{Mpc}^{-1}}\right)^{n_P - 1}.
$$
For the eBOSS likelihood, the simulated ranges are \(0.8 \le n_P \le 1.05\), \(1.2\times10^{-9} \le A_P \le 2.6\times10^{-9}\), \(0.65 \le h \le 0.75\), \(0.14 \le \Omega_M h^2 \le 0.146\), \(3.5 \le z_{Hei} \le 4.1\), \(2.6 \le z_{Hef} \le 3.2\), \(1.3 \le \alpha_q \le 3.0\), \(6.5 \le z_{HI} \le 8\), and \(0.03 \le \epsilon_{AGN} \le 0.07\). The mean-flux nuisance parameters are varied as \(0.75 \le \tau_0 \le 1.25\) and \(-0.4 \le d\tau_0 \le 0.25\) [2309.03943].

The mean transmitted flux is modeled through
$$
\mathcal{F}=e^{-\tau},
$$
with the Kim et al. reference evolution
$$
\tau^{\mathrm{Kim}}_{\mathrm{H\,I}}(z)=0.0023(1+z)^{3.65},
$$
and the PRIYA rescaling
$$
\tau^{\mathrm{eff}}_{\mathrm{H\,I}}(z)=\tau_0 \left( \frac{\tau^{\mathrm{Kim}}_{\mathrm{H\,I}}(z)} {\tau^{\mathrm{Kim}}_{\mathrm{H\,I}}(3)} \right)^{d\tau_0} \tau^{\mathrm{Kim}}_{\mathrm{H\,I}}(z).
$$
This post-processing strategy is important because it allows dense mean-flux sampling without rerunning the hydrodynamical suite [2309.03943].

The simulation outputs are converted into mock spectra with `fake_spectra` or Fake Spectra Flux Extractor. For each snapshot, the pipeline generates \(3\times 480^2 = 691{,}200\) sightlines with 10 km s\(^{-1}\) pixels. The original suite stores snapshots from \(z=5.4\) down to \(z=2.2\) in steps of \(\Delta z=0.2\); the eBOSS inference pipeline uses \(z=2.2\) to \(4.6\) in the same step size [2306.05471], [2309.03943].

## 3. Emulation methodology

PRIYA uses a multi-fidelity Gaussian-process construction rather than a single-fidelity interpolator. The low-fidelity simulations map the broad parameter dependence, while the high-fidelity simulations calibrate the resolution correction. In the eBOSS formulation, the high-fidelity flux power is modeled as
$$
P_F^{\mathrm{HF}}(k,\boldsymbol{\theta}\mid z) = \rho_z\,P_F^{\mathrm{LF}}(k,\boldsymbol{\theta}\mid z) + \delta(k,\boldsymbol{\theta}\mid z),
$$
where \(\rho_z\) is a redshift-dependent coefficient and \(\delta\) is a Gaussian process describing the discrepancy between low and high fidelity. The same structure is applied to the mean-temperature emulator [2309.03943].

In the original suite paper, the Gaussian-process kernel is written as a sum of RBF and linear terms,
$$
k_\mathrm{RBF}(\boldsymbol{\theta}, \boldsymbol{\theta}'; \sigma_0, \boldsymbol{l}) + k_\mathrm{LIN}(\boldsymbol{\theta}, \boldsymbol{\theta}'; \boldsymbol{\sigma}) = \sigma_0^2 \exp{\left( \sum_{i=1}^{d} -\frac{(\theta_i - \theta_i')^2}{2 l_i^2} \right)} +  \sum_{i=1}^{d} \sigma_i^2 \theta_i \theta_i',
$$
with the outputs first rescaled by the median low-fidelity spectrum so that a zero-mean GP prior is appropriate. The implementation uses GPy and EmuKit [2306.05471].

A notable modeling choice is that PRIYA trains one GP per redshift that outputs the entire \(k\)-vector of flux-power values, rather than training a separate emulator for each \(k\)-bin. In later high-resolution work, the emulator remains multi-fidelity and is described as providing few-percent-level predictions over \(k=0.003\)–\(0.06\ \mathrm{s/km}\), with about \(\sim 1\%\) median interpolation accuracy at \(k=0.01\)–\(0.06\ \mathrm{s/km}\), degrading to about \(\sim 2\%\) at \(z=4\)–4.2. Scales above \(0.065\ \mathrm{s/km}\) are excluded because convergence worsens beyond that point [2509.18271].

The mean-flux augmentation is also integral to the emulator design. In the small-scale analysis, each simulation is post-processed with 10 mean-flux values, giving 600 low-fidelity and 30 high-fidelity flux-power spectra. A cosmology-independent correction is then applied at \(k=0.02\)–\(0.06\ \mathrm{s/km}\) using the ratio between two very high-resolution small-box simulations, improving convergence at the smallest included scales [2509.18271].

## 4. Likelihood construction and inference

PRIYA’s full inference pipeline maps parameters to simulated observables and then to a survey likelihood. The standard eBOSS pipeline is
\[
\boldsymbol{\theta}\rightarrow \text{simulation outputs} \rightarrow \text{synthetic spectra} \rightarrow P_F(k,z),\ T_0(z)\rightarrow \text{emulator interpolation} \rightarrow \text{likelihood}.
\]
The flux contrast is defined from transmitted flux, and the 1D flux power spectrum is evaluated from its Fourier modes. Inference is then performed with Cobaya, while CLASS is used for derived linear-theory quantities such as \(\sigma_8\) [2309.03943].

The eBOSS likelihood uses a redshift-binned Gaussian form,
$$
\log \mathcal{L} = -\frac{1}{2} \sum_z \left[ (\boldsymbol{P}_F^{\mathrm{diff}})^T\boldsymbol{K}^{-1}\boldsymbol{P}_F^{\mathrm{diff}} + \log\det \boldsymbol{K} \right],
$$
with \(\boldsymbol{P}_F^{\mathrm{diff}}=\boldsymbol{P}_F^{\mathrm{sim}}-\boldsymbol{P}_F^{\mathrm{obs}}\). The covariance includes the observational covariance and an empirical sample-variance term from the finite simulation box. The likelihood also incorporates nuisance corrections for residual DLA/LLS contamination and Si III contamination [2309.03943].

A second operational mode is the compressed PRIYA likelihood. Instead of evaluating the full flux-power emulator at each chain point, this approach fits a 2D Gaussian distribution to posterior constraints on the local amplitude and slope of the linear matter power spectrum at
$$
z_p = 3, \qquad k_p = 0.009\ \mathrm{s/km}\sim 1\ \mathrm{Mpc}^{-1}.
$$
The compressed variables are \(\Delta_L^2\) and \(n_{\rm eff}\), defined at that pivot, and the resulting Gaussian likelihood is then combined with Planck or other datasets. This reduced form was used to study neutrino self-interactions without rerunning a dedicated non-\(\Lambda\)CDM hydrodynamical suite [2503.15592].

## 5. Scientific applications and published results

PRIYA’s first major cosmological application was the eBOSS DR14 Lyman-\(\alpha\) forest analysis. Using the preferred redshift range \(z=2.6\)–4.6, because the lowest-redshift bins were found to be in internal tension with the rest of the dataset, the flux-power-only chain yielded
$$
n_P = 1.009^{+0.027}_{-0.018}, \qquad
\sigma_8 = 0.733^{+0.026}_{-0.029},
$$
together with
$$
\Delta_L^2 = 0.302^{+0.024}_{-0.027}, \qquad
n_\mathrm{eff} = -2.264^{+0.026}_{-0.018}.
$$
Adding IGM temperature data shifted the inference to
$$
n_P = 0.983 \pm 0.020, \qquad
\sigma_8 = 0.703^{+0.023}_{-0.027},
$$
and favored an early and extended He II reionization episode [2309.03943].

PRIYA was then used for small-scale cosmology with high-resolution spectra from XQ100 and KODIAQ-SQUAD. In that analysis, XQ100 yielded constraints on \((A_P,n_P)\) consistent with eBOSS DR14 and Planck while also constraining thermal history without external IGM temperature data. By contrast, KODIAQ-SQUAD preferred a significantly higher \(A_P\), which the paper attributes to selection bias toward high-column-density absorbers, especially LLSs and sub-DLAs. A key conclusion of that study is that thermal and cosmological parameters are “largely sensitive to different scales,” with \(k \gtrsim 0.045\ \mathrm{s/km}\) carrying especially strong thermal-history and LLS-contamination information [2509.18271].

In neutrino-cosmology applications, the compressed PRIYA likelihood was combined with Planck to revisit neutrino self-interactions. The resulting posterior favored a negligible level of self-interaction rather than the strong interaction suggested by older compressed forest likelihoods, with
$$
\log_{10}(G_\mathrm{eff}\,\mathrm{MeV}^2)=-5.26_{-1.49}^{+0.87}
$$
at 68% confidence for Planck + PRIYA Lyman-\(\alpha\). The paper attributes the shift relative to earlier results to improved simulations, more accurate interpolation techniques such as Gaussian processes or neural networks, and the inclusion of patchy helium reionization, which broadens the allowed slope of the matter power spectrum [2503.15592].

## 6. Accuracy, limitations, and development trajectory

PRIYA’s principal technical strength is its combination of scale, fidelity, and astrophysical scope. The original suite was presented as the first to resolve the Lyman-\(\alpha\) forest in a \((120\ \mathrm{Mpc}/h)^3\) volume, and it was designed to avoid the splicing strategy used in earlier analyses by simultaneously providing large volume and sufficient resolution. The resulting emulator achieves \(<1\%\) interpolation error in the original suite paper, with low-fidelity leave-one-out errors around \(0.2\%\) and high-fidelity multi-fidelity errors around \(1\%\), while the underlying simulations themselves are converged at about the percent level over the redshift range \(z=5.4\)–2.2 relevant for the forest [2306.05471].

The framework’s main limitations are equally well documented. First, the emulator is an interpolator within a defined training domain and is not intended for extrapolation. Second, finite-volume and sample-variance effects remain important on the largest scales, particularly around the epochs where helium reionization is active. Third, some astrophysical parameters, especially those controlling patchy He II reionization, are degenerate with the slope of the linear matter power spectrum and therefore broaden cosmological constraints. Fourth, the original suite does not include massive neutrinos as an emulator dimension, because their impact was treated as effectively degenerate with \(A_P\) over the forest scales considered [2306.05471], [2309.03943].

Operationally, published analyses also identify dataset-specific caveats. The eBOSS DR14 application found an internal tension between the \(z=2.2\)–2.4 bins and the higher-redshift data, leading later work to exclude the lowest-redshift interval when using the PRIYA-based likelihood. The high-resolution analysis excludes \(k>0.065\ \mathrm{s/km}\) because resolution convergence errors become too large. The compressed PRIYA likelihood for nonstandard cosmologies further assumes that the relevant beyond-\(\Lambda\)CDM model can be mimicked by a suitably rescaled \(\Lambda\)CDM linear spectrum over the eBOSS forest scales [2309.03943], [2509.18271], [2503.15592].

Despite these limitations, PRIYA has become a methodological reference point for simulation-based Lyman-\(\alpha\) inference. Its published trajectory runs from the original simulation suite and multi-fidelity emulator, through a full eBOSS DR14 likelihood, to high-resolution XQ100/KODIAQ analyses and compressed-likelihood applications beyond minimal \(\Lambda\)CDM. The framework is explicitly positioned for future DESI analyses, and its public software ecosystem includes the `lya_emulator` code base and accompanying data products [2306.05471], [2309.03943].

Source: https://www.emergentmind.com/topics/priya-emulator