Galaxy × X-ray joint fit: hot gas + AGN across BGS samples

This page documents the temperature-resolved joint fit of the galaxy × X-ray angular cross-correlation $w(\theta)$ for the seven LS10-BGS stellar-mass threshold samples (S1–S7), combining a full-APEC hot-gas model with a duty-cycle AGN model. It covers the forward model, the emulator that makes the fit tractable, the broad-band (0.5–2 keV) results, and the energy-band (Phase B) extension that constrains the gas temperature.

The measurement is the Comparat et al. 2025 GALxEVT product (arXiv:2503.19796, zenodo:15111974): $w(\theta) = (S^{G}_X - S^{R}_X)/S^{R}_X$ (Davis–Peebles), where $S^{R}_X$ is the random×events background surface brightness, so the physical surface brightness is $S_X = (1+w)\,S^{R}_X$.

Code: hod_mod.scripts.fitting.fit_xray_joint (broad band), hod_mod.scripts.fitting.fit_xray_joint_bands (energy bands), hod_mod.scripts.fitting.make_xray_diagnostics (figures).

Forward model 

For each halo mass $M$ the two X-ray emitters are projected onto the sky and combined with the galaxy occupation through the halo model.

Hot gas — full APEC emissivity 

The X-ray volume emissivity is

\[\varepsilon(r\,|\,M) = n_e^2(r)\;\Lambda_b\!\big(T(r), Z(r)\big),\]

with the DPM profiles (Oppenheimer et al. 2025, arXiv:2505.14782):

electron density $n_e(r|M,z)$ — generalised NFW, amplitude $n_{e,0.3}$ at $0.3\,R_{200}$, mass slope $\beta_{\rm gas}$ (this is the $L_X$–$M$ scaling), outer slope set by p2, truncated at r_max × $R_{200}$ (GasDensityDPM);
pressure $P(r|M,z)$ → temperature $T = P/n_e$ (ideal gas) (PressureProfileDPM);
metallicity $Z(r)$ (MetallicityProfileDPM).

$\Lambda_b(T,Z)$ is the APEC band-integrated cooling function from ApecCoolingTable (built with soxs / AtomDB, integrating the plasma spectrum over the energy band $[e_{\min}, e_{\max}]$). Because $\Lambda_b$ depends on $T$, the predicted $w(\theta)$ is temperature dependent — the physical basis of the energy-band fit below.

AGN — duty cycle 

The AGN component is the duty-cycle model (DutyCycleAGNModel): the hard-band X-ray luminosity function (Aird et al. 2015) is K-corrected to the observed soft band and convolved with the sample redshift kernel; the free normalisation is the duty cycle log10DC (AGN-host fraction, prior $[10^{-3}, 0.5]$).

From 3-D to $w(\theta)$

Profile FT $\tilde X(k|M) = 4\pi\!\int_0^{r_{\max}} \varepsilon(r)\,j_0(kr)\,r^2\,\mathrm{d}r$ by Gauss–Legendre quadrature (hod_mod.gas.conversions._profile_uk_gl()).
Halo-model cross-power $P_{gX}(k,z)$ — 1-halo (central + satellite, NFW-weighted) + 2-halo ($b_{\rm eff}\,P_{\rm lin}$), with the galaxy occupation from the Zu & Mandelbaum 2015 iHOD (arXiv:1505.02781) held fixed at its $w_p+n_{\rm gal}$ MAP, the Tinker 2008 HMF (arXiv:0803.2706) + Tinker 2010 bias (arXiv:1001.3162) and Diemer & Joyce 2019 concentrations (arXiv:1809.07326) (_pk_tables_gX()).
Limber → $C_\ell^{gX}$, then Hankel $w(\theta) = \int \frac{\ell\,\mathrm{d}\ell}{2\pi} C_\ell J_0(\ell\theta)$, multiplied by the eROSITA King PSF window (core $\theta_c = 8.64''$, CalDB on-axis).

eROSITA energy-conversion factor 

Because the fit is to $w(\theta)$ — a ratio that is independent of the measurement’s fixed 1 keV/photon ECF assumption — each predicted component folds in the true eROSITA response (TM0 survey ARF × RMF from the eSASS4DR1 ARF/RMF page, eROSITA DR1) via ErositaResponse.

The emulator (making the fit tractable)

The single expensive step is the emissivity FT: at the production grid ($N_k = N_M = 512$, $n_{gl}=200$) it is a 52-million-element $(N_k, N_M, n_{gl})$ cube, originally ≈ 5.7 s per redshift, ≈ 13 s per angular_cl_gX call — far too slow for a 7-sample MAP/MCMC. Three changes make it fast without changing the result:

Vectorised emissivity mass loop (GasDensityDPM._ne_grid) — bit-exact.
``np.sinc`` + ``np.einsum`` rewrite of the FT (hod_mod.gas.conversions._profile_uk_gl()): no dense product cube, cached Legendre rule — bit-exact ($\lesssim 10^{-10}$), ~1.3 s/call, half the peak memory. n_gl cannot be lowered (Gauss–Legendre cannot resolve the oscillatory $j_0(kr)$ at high $k$).
X_uk emulator — x_uk_override on _pk_tables_gX() / angular_cl_gX: the raw $\tilde X(k|M)/\Lambda_{\rm ref}$ is cached per (p2, r_max, β_gas) cell (rebuilt exactly per $\beta$ — no post-FT tilt approximation, which only captures $n_e^2$ and misses the $T=P/n_e$ shift of $\Lambda(T)$), and the exact $(n_e/n_{e,\rm fid})^2$ density scaling is applied analytically. The override reproduces a direct call bit for bit; a fit evaluation drops from ~13 s to ~2 s (interpolation only).

The model→data gas amplitude C_total is re-anchored for the full-APEC path on S1 (its unconstrained best-fit $A_{\rm gas}$ ⇒ density_norm = 1), then scaled by $1/S^R_X$ per sample; the AGN uses the analogous c_obs_total.

Phase A — broad-band (0.5–2 keV) fit 

Five shared physical parameters are fit per sample (or jointly): log10_ne_03 (density normalisation), beta_gas ($L_X$–$M$ slope), p2 (outer slope), r_max (extent), log10DC (AGN duty cycle). The MAP uses multi-start Nelder–Mead; posteriors use emcee (64 walkers). Run with:

HOD_MOD_RESULTS=$HOD_MOD_RESULTS JAX_PLATFORMS=cpu python -m \
    hod_mod.scripts.fitting.fit_xray_joint --samples S1 --mcmc

_images/xray_joint__params_vs_mass.png — Posterior physical parameters vs stellar-mass threshold (S1–S4). The **AGN duty cycle increases monotonically and tightly with stellar mass** ($\log_{10}\mathrm{DC}: -1.89 \to -1.51$), a clean physical trend. Gas parameters are noisier and the MAP $\chi^2/\mathrm{dof}$ is uneven (S2, S4 good; S1, S3 poor — see below).

_images/xray_joint__decomp_overview.png — Data vs the gas / AGN / total decomposition at the MAP. The AGN dominates the small-scale ($\lesssim 30''$) signal and the gas the large scales.

_images/xray_joint__S3_decomp.png — S3 residuals (pulls). A **coherent** $+/-$ **oscillation** across $\theta$ — a systematic model-shape mismatch at the gas→AGN “knee” ($\sim 20\text{--}50''$), not statistical scatter. With only the broad-band integrated signal the two components are degenerate, so for S1/S3 the MAP leaves this wiggle. This degeneracy is what the energy bands break.

Phase B — energy bands (gas temperature)

The signal is measured in 15 narrow bands (0.5–0.6 … 1.9–2.0 keV, 0.1 keV steps). The gas is thermal — its band ratios encode $\Lambda_b(T)$ — while the AGN is a near power law, so the bands separate gas from AGN spectrally and constrain $kT$.

Data — reconstructed from the per-field measurements with hod_mod.scripts.fitting.reconstruct_band_wtheta (Landy–Szalay merge over the eROSITA-DE fields; validated: $\Sigma$(15 bands) matches the zenodo broad band to 0.4–3.3 %).
Model — all 15 bands share $n_e, T, Z$ and the FT geometry, so emissivity_full_uk_bands() computes them in ONE batched FT (the $j_0$ cube is reused; bit-exact vs looping single bands). The fit adds a temperature normalisation kT_norm (grid axis) to the Phase-A shape parameters (hod_mod.scripts.fitting.fit_xray_joint_bands).

_images/xray_joint__band_energy.png — Measured band $w_b(\theta)$ vs X-ray energy at three angular scales — the energy dependence (peaking near 0.9–1.0 keV) that carries the temperature information.

_images/xray_joint__band_ratios.png — Soft/hard band ratio vs $\theta$ — a direct, model-independent gas-temperature tracer (higher ratio = cooler gas).

_images/xray_joint__band_validation.png — Reconstruction check: $\Sigma$(15 bands) vs the zenodo broad band.

Data & results layout (environment variables)

All paths resolve through hod_mod.paths (no hardcoded paths):

Helper	Env var	Holds
`data_root()`	`$HOD_MOD_DATA_DIR`	`zenodo/`, `erosita/`, `xray_bands/`
`results_root()`	`$HOD_MOD_RESULTS`	emulator caches, chains, figures
`repo_root()`	`$HOD_MOD_REPO`	`configs/`, package `data/`

Reconstruct the band data ($HOD_MOD_DATA_DIR/xray_bands/<basename>/), then:

JAX_PLATFORMS=cpu python -m hod_mod.scripts.fitting.reconstruct_band_wtheta
JAX_PLATFORMS=cpu python -m hod_mod.scripts.fitting.fit_xray_joint_bands \
    --samples S1 S2 S3 S4 --map-only
JAX_PLATFORMS=cpu python -m hod_mod.scripts.fitting.make_xray_diagnostics

References 

Comparat et al. 2025 — GALxEVT BGS × eROSITA (arXiv:2503.19796; data zenodo:15111974)
Oppenheimer et al. 2025 — DPM gas profiles (arXiv:2505.14782)
Zu & Mandelbaum 2015 — iHOD (arXiv:1505.02781)
Tinker et al. 2008 HMF (arXiv:0803.2706); Tinker et al. 2010 bias (arXiv:1001.3162)
Diemer & Joyce 2019 concentrations (arXiv:1809.07326)
eROSITA DR1 (erosita.mpe.mpg.de/dr1) and the eSASS4DR1 ARF/RMF
APEC / AtomDB via soxs