SPHINX Galaxy Formation Simulation

• SPHINX is a state-of-the-art simulation suite that models galaxy formation and reionization using cosmological radiation hydrodynamics.
• It employs advanced methodologies like adaptive mesh refinement, non-equilibrium thermochemistry, and detailed radiative transfer to capture small-scale ISM physics.
• The results calibrate nebular emissions and quantify LyC escape fractions, providing benchmarks that align with observations from JWST, HST, and other ground-based surveys.

The SPHINX galaxy formation simulation suite comprises state-of-the-art cosmological radiation hydrodynamics models designed to trace the evolution of galaxies, the physical drivers of reionization, and key emission observables in the high-redshift universe. SPHINX simulations deliver fully coupled treatments of gas dynamics, non-equilibrium chemistry, radiative transfer, stellar population synthesis, and feedback mechanisms at parsec-scale resolution, enabling direct confrontation with modern JWST, HST, and ground-based datasets. Results from SPHINX underpin theoretical calibrations for nebular lines (e.g., Hα, [O III]), resolve the mechanisms regulating the Lyman-continuum escape fraction ($f_\mathrm{esc}$), and forecast the impact of primordial physics—including magnetic fields and binary stellar evolution—on cosmic reionization.

1. Simulation Framework and Physics Modules

SPHINX simulations deploy periodic cosmological volumes (typically $(20~\mathrm{cMpc})^3$) evolved from high redshifts ($z\gtrsim100$) to $z\sim4.6$ using adaptive mesh refinement (AMR). Hydrodynamics utilizes RAMSES-RT, an explicit second-order Godunov solver with HLLC (or HLLD for MHD) Riemann solvers, coupled to on-the-fly multi-group radiative transfer (M1 closure). Radiation transport subcycles up to 500 times per hydro step per AMR level. The code supports non-equilibrium thermochemistry for H, He ionization states, and metal-line cooling using pre-computed CLOUDY tables (for $T\geq10^4$~K) plus fine-structure cooling below $10^4$~K.

Star formation follows a (magneto-)thermo-turbulent prescription. Cells exceeding density thresholds ($n_H\gtrsim10-100~\mathrm{cm}^{-3}$, typical for ISM in high-z galaxies) and unresolvable local Jeans length generate star particles stochastically, with masses $400-10^3~M_\odot$ and an efficiency per free-fall time $\epsilon_\ast\sim1-2\%$ (or variable, if magnetized; see SPHINX-MHD). Feedback incorporates mechanical supernova injection—calibrated to stellar-to-halo mass and UV luminosity relations—distributing energy ($10^{51}$~erg per SN) and metal yields via momentum-driven snow-plow algorithms. The SN rate may be boosted (e.g., by a factor $4$ versus canonical Kroupa IMF) for high-redshift calibrations.

The full radiation hydrodynamics module treats ionizing photon emission using BPASS v2.2 (binary population and spectral synthesis), tracks non-equilibrium photo-ionization heating and direct radiation pressure, and models photon escape. Metal enrichment operates as passive scalar advection with SNII/SNIa delay kernels, yielding physically motivated ISM compositions.

2. Box Size, Resolution, and Initial Conditions

SPHINX adopts $(20~\mathrm{cMpc})^3$ volumes (SPHINX-20) for statistical samples of resolved galaxies, with smaller $(5~\mathrm{cMpc})^3$ boxes (SPHINX-MHD) for studies of primordial magnetic fields and early structure formation. Each simulation begins from Planck-like $\Lambda$CDM cosmology (see Rosdahl et al. 2018 for fiducial parameters) and initial conditions generated using MUSIC. DM particle masses are $m_{\rm DM}\sim2.5\times10^5~M_\odot$ (SPHINX-20), AMR achieves minimum cell widths of $76~\mathrm{cpc}$ (comoving), corresponding to $\sim11~\mathrm{pc}$ at $z=6$.

The volume resolves halos down to the atomic-cooling limit ($M_{\rm vir, min}\simeq3\times10^7~M_\odot$), supports $10^7$–$10^8$ total resolution elements, and ensures refinement is triggered by mass thresholds and Jeans criteria. Initial metallicity floors are set to $\sim3\times10^{-4}~Z_\odot$ for cooling, with metal cooling and self-shielding evolved self-consistently.

3. Galaxy Formation, Feedback, and Statistical Outcomes

Galaxy identification utilizes halo finders (ADAPTAHOP), with stellar and gas properties tracked per virial radius. Each star-forming galaxy is characterized by extensive catalog entries: $M_\mathrm{vir}$, $M_\ast$, SFR histories ($1$–$100$~Myr bins), mass- and LyC-weighted metallicities, and ISM diagnostics.

Star formation is bursty, with sSFR ($\mathrm{sSFR}_\tau$) defined over $10$ and $100$~Myr windows. The global SFR density follows observations at $z=5$–$10$ if integrated to $M_{1500}\sim-11$. Stellar-to-halo mass relations at $z=6$ match abundance-matching and Local Group dwarf measurements within a factor $\lesssim3$. The UV luminosity function and mass–metallicity relations are in excellent agreement with HST/JWST datasets, particularly for $M_\mathrm{UV}<-17$ (Katz et al., 2023, Rosdahl et al., 2022).

Supernova feedback is the principal regulator of ISM porosity and LyC escape. The escape fraction $f_\mathrm{esc}$ fluctuates rapidly (3–10 Myr timescales) in individual galaxies, with bursts reaching $f_\mathrm{esc}\sim20\%$ after SN events and quenching to $\lesssim1\%$ during quiescence. The luminosity-weighted galaxy sample shows $f_\mathrm{esc}$ peaking at $M_\ast\sim10^7~M_\odot$, $M_{1500}\sim-17$, and low metallicity ($Z\lesssim5\times10^{-3}~Z_\odot$); higher and lower masses yield suppressed escape fractions. Empirically, $f_\mathrm{esc}$ at $z=6$ is $5$–$10\%$ in average, with $\sim55\%$ of escaping ionizing photons sourced from galaxies fainter than $M_{1500}=-17$—just below HST’s detection threshold, but accessible to JWST (Rosdahl et al., 2022).

4. Nebular Line Emission, Forward Models and JWST Comparisons

SPHINX simulates intrinsic H$\alpha$ luminosity on-the-fly for each gas cell via CLOUDY v17.03 grid interpolation when Strömgren spheres are unresolved, and direct non-equilibrium recombination rates otherwise (case B). Equivalent widths (EW) include both stellar and nebular continuum. Dust radiative transfer is handled in post-processing using RASCAS (SMC dust model), yielding attenuated $L_{\mathrm{H}\alpha,\mathrm{obs}}$ and $\mathrm{EW}_{\mathrm{H}\alpha,\mathrm{obs}}$.

Standard SFR–H$\alpha$ conversions adopt canonical coefficients:

• Kennicutt (1998, Salpeter): $C_{\mathrm{H}\alpha}\simeq -41.10$
• Kroupa: $C_{\mathrm{H}\alpha}\simeq -41.30$
• BPASS solar-Z: $C_{\mathrm{H}\alpha}\simeq -41.35$
• BPASS $Z=0.1Z_\odot$: $C_{\mathrm{H}\alpha}\simeq -41.64$

Recent SPHINX work provides new metallicity- and EW-dependent calibrations that reduce prediction RMSE by $0.04$–$0.06$ dex compared to classical relations at $z>3$, shifting cosmic SFRD by $-12\%$ and the slope of the star-formation main sequence by $$\Delta\partial\,\log \mathrm{SFR}/\partial\,\log M_\ast=0.08\pm0.02$$ (Kramarenko et al., 5 Sep 2025).

Mock JWST products from SPHINX include $>14,000$ synthetic spectra/images per galaxy, covering NIRCam filters and nebular lines ([O III], Hβ, Lyα), subjected to angle-dependent ISM/dust radiative transfer. Observed distributions in UV slopes $\beta$, color–magnitude, and equivalent widths reproduce high-EW tails and the bluest photometric systems from JWST Cycle 1 (Katz et al., 2023). Lyα radiative transfer utilizes MCRT with $10^6$ photon packets per halo, processed for surface brightness and spectral profiles.

5. Magnetohydrodynamics and the Impact of Primordial Physics

The SPHINX-MHD suite incorporates ideal MHD with constrained transport at $\sim7$~pc physical resolution in $(5~\mathrm{cMpc})^3$ volumes. Primordial magnetic fields (PMFs) are introduced with controlled strengths ($B_0=5\times10^{-11}~\mathrm{G}$) and spectral indices ($n_B$). PMFs substantially boost halo abundance at $M_\mathrm{vir}\lesssim3\times10^7~M_\odot$ for $n_B>-2.7$, but their impact on reionization history, UV LF, and SFRD is mild unless $\tau_e$ constraints are strongly violated.

Strong PMF runs yield galaxy sizes $\sim44\%$ smaller and escape fractions up to $25\%$ higher, with notable but not transformative effects on reionization timing. The global 21 cm absorption trough is highly sensitive to PMF heating—flattening or removing the first absorption feature for $n_B\gtrsim-2.6$. Observationally, the main ionizing sources at $z=6$ are reachable with surveys to $M_\mathrm{UV}=-13$ (Katz et al., 2021).

6. Binary Stellar Populations and Reionization Efficiency

SPHINX robustly demonstrates that binary stellar evolution is critical for early reionization. BPASS models yield instantaneous ionizing photon rates and cumulative production $\sim2$ times higher vs single-star SEDs (BC03), with a shallower post-starburst decline. Simulations employing BPASS reach $99.9\%$ ionized volume by $z\sim7$ ($Q_{\mathrm{HII}}\simeq0.999$) while single-star runs do not achieve reionization by $z=6$.

The binary-enhanced output aligns periods of high LyC luminosity with SN-cleared escape channels, raising $\langle f_\mathrm{esc}\rangle$ by a factor $3.5$ (to $7$–$10\%$ for $z\lesssim9$) (Rosdahl et al., 2018), driving a strong reionization signature. Photo-ionization rates, galaxy scaling relations, and predicted number densities of reionization-era galaxies for JWST agree well with recent empirical constraints.

7. Public Data Products, Limitations, and Future Directions

The SPHINX$^{20}$ data release (Katz et al., 2023) provides comprehensive catalogs, SEDs, nebular lines, and mock JWST observables for $\sim1400$ galaxies per snapshot at $z=4.6$–$10$, each with detailed physical and observational properties over ten viewing angles. Data are distributed in CSV/JSON/FITS formats with supporting Jupyter notebooks and Python scripts (Astropy, sedpy, Photutils, RASCAS readers).

Major limitations include finite-volume cosmic variance (missing brightest galaxies), SFR selection ($>0.3~M_\odot~\mathrm{yr}^{-1}$) biasing sample against quiescent systems, single-scalar metallicity tracking, lack of on-the-fly AGN or H$_2$ network physics, and phenomenological dust curves. Stromgren spheres and ISM turbulence are not fully resolved at 10 pc scales, and IR/sub-mm lines require external models.

A plausible implication is that, as JWST and ALMA surveys advance, forward-modeling with the SPHINX framework will be pivotal for interpreting line diagnostics, evolutionary trends in escape fractions, and galaxy scaling laws up to the highest redshifts. Future enhancements may integrate multi-element chemical evolution, AGN feedback, and higher-resolution turbulent ISM physics for further fidelity in simulating reionization-era sources and observables.