[NG20] Initial proposal for the NG20 Fiducial Parameter Space (PS_NG20_Fiducial)

[kayhangultekin]

Summary & Motivation

This PR introduces the first concrete draft of the Fiducial Parameter Space (PS_NG20_Fiducial) intended for the flagship NANOGrav 20-year (NG20) astrophysical interpretation analysis.

The goal of this PR is to provide the Astrophysics Working Group with a specific, fully implemented, and testable model to review, critique, and iterate on, rather than debating parameter choices in the abstract.

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Physical Framework Summary

1. Binary Evolution / Hardening (FixedOuterTime_InnerPL_SAM):

   • Implements Blecha Inside-Out (BIO) Hardening for circular binaries using Model 1 (inner_model_type=1).
   • Outer phase: Galaxy-scale delay hard_outer_time ($t_\mathrm{out} \sim \mathcal{U}(0.1, 10.0),\mathrm{Gyr}$).
   • Inner phase: Starts at mass-dependent transition radius $r_\mathrm{char}(M)$ (hard_rchar_9 $\sim \mathrm{Log}\text{-}\mathcal{U}(0.1, 10.0),\mathrm{pc}$).
   • Bounded boundary velocity hard_log10_dadt_rchar ($\sim \mathcal{U}(1.0, 7.0),\log_{10}(\mathrm{cm/s})$), ensuring all Latin Hypercube samples are strictly physical and subluminal ($|da/dt| \ll c$) without rejecting points.
   • Fixed structural scalings: $\alpha_\mathrm{char} = -2/3$, $r_\mathrm{gw,crit,9} = 10^{2.5}r_g$, $\alpha_\mathrm{gw,crit} = -0.25$, $\beta_\mathrm{gw,crit} = +0.25$.

2. Galaxy Stellar Mass Function (GSMF_Double_Schechter):

   • Full 11-dimensional covariance matrix derived from Leja et al. (2020) anchor points (PD_MVNormal).

3. Galaxy Merger Rate (GMR_Illustris):

   • Simulation-derived merger rates from Rodriguez-Gomez et al. (2015) with formal fit uncertainties.

4. $M_\mathrm{BH}$–$M_\mathrm{bulge}$ Scaling & Redshift Evolution (MMBulge_Redshift_KH2013):

   • $z=0$ Kormendy & Ho (2013) amplitude $\log_{10}(M_\mathrm{amp,0}) \sim \mathcal{N}(8.69, 0.05)$, slope $\alpha_\mathrm{MMB} \sim \mathcal{N}(1.17, 0.08)$, and scatter $\sigma_\mathrm{dex} \sim \mathcal{N}(0.28, 0.05)$.
   • Redshift power-law amplitude evolution $M_\mathrm{amp}(z) = M_\mathrm{amp,0}(1+z)^{\gamma_z}$ with uniform prior $\gamma_z \sim \mathcal{U}(-2.0, +2.0)$ (Matt et al. 2026a).
   • Bulge fraction parameterized via BF_Sigmoid.

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Parameter Table & Prior Distributions

Component	Parameter Name	Description	Default Value	Prior Distribution / Range	Source / Motivation
BIO-Hardening	hard_outer_time	Outer galactic inspiral timescale delay	$1.0\text{ Gyr}$	$\mathcal{U}(0.1, 10.0)\text{ Gyr}$	Galactic dynamical friction delay
(Model 1: Safe)	hard_rchar_9	Transition radius $r_\text{char}$ for $10^9,M_\odot$ BHs	$1.0\text{ pc}$	$\text{Log-}\mathcal{U}(0.1, 10.0)\text{ pc}$	Onset of loss cone / gas hardening
	hard_log10_dadt_rchar	$\log_{10}|da/dt|$ at $r_\text{char}$ (equal-mass $10^9M_\odot$)	$5.0\text{ [log10(cm/s)]}$	$\mathcal{U}(1.0, 7.0)\text{ [log10(cm/s)]}$	Bounded physical hardening speed ($10^{-2}$–$10^2\text{ km/s}$)
(Fixed defaults)	hard_alpha_char	Mass scaling slope of $r_\text{char}$	$-0.67$ ($-2/3$)	Fixed	Galaxy core scaling
	hard_r_gw_crit_9	GW transition radius for $10^9,M_\odot$ BHs	$10^{2.5}\approx 316.2,r_g$	Fixed	GW vs environmental crossover
	hard_alpha_gw_crit	Mass scaling of $r_\text{gw,crit}$	$-0.25$	Fixed	Theoretical mass scaling
	hard_beta_gw_crit	Mass-ratio scaling of $r_\text{gw,crit}$	$+0.25$	Fixed	Mass ratio dependence
GSMF (Leja+2020)	gsmf_log10_phi_one_z0	1st Schechter normalization at $z=0$	$-2.386$	Covariant $\mathcal{MVN}$	Leja et al. (2020) joint posterior
(11 covariant params)	gsmf_log10_phi_one_z1	1st Schechter norm linear redshift term	$-0.259$	Covariant $\mathcal{MVN}$	Leja et al. (2020) joint posterior
	gsmf_log10_phi_one_z2	1st Schechter norm quadratic redshift term	$-0.110$	Covariant $\mathcal{MVN}$	Leja et al. (2020) joint posterior
	gsmf_log10_phi_two_z0	2nd Schechter normalization at $z=0$	$-2.821$	Covariant $\mathcal{MVN}$	Leja et al. (2020) joint posterior
	gsmf_log10_phi_two_z1	2nd Schechter norm linear redshift term	$-0.368$	Covariant $\mathcal{MVN}$	Leja et al. (2020) joint posterior
	gsmf_log10_phi_two_z2	2nd Schechter norm quadratic redshift term	$+0.046$	Covariant $\mathcal{MVN}$	Leja et al. (2020) joint posterior
	gsmf_log10_mstar_z0	Characteristic mass $M_*$ at $z=0$	$+10.765$	Covariant $\mathcal{MVN}$	Leja et al. (2020) joint posterior
	gsmf_log10_mstar_z1	Characteristic mass $M_*$ linear $z$ term	$+0.130$	Covariant $\mathcal{MVN}$	Leja et al. (2020) joint posterior
	gsmf_log10_mstar_z2	Characteristic mass $M_*$ quadratic $z$ term	$-0.034$	Covariant $\mathcal{MVN}$	Leja et al. (2020) joint posterior
	gsmf_alpha_one	1st Schechter low-mass slope	$-0.278$	Covariant $\mathcal{MVN}$	Leja et al. (2020) joint posterior
	gsmf_alpha_two	2nd Schechter low-mass slope	$-1.479$	Covariant $\mathcal{MVN}$	Leja et al. (2020) joint posterior
GMR (Illustris)	gmr_norm0_log10	Merger rate normalization at $z=0$	$-2.2287$	$\mathcal{N}(-2.2287, 0.0045)$	Rodriguez-Gomez et al. (2015)
	gmr_normz	Redshift scaling $\eta$	$+2.4644$	$\mathcal{N}(+2.4644, 0.0128)$	Rodriguez-Gomez et al. (2015)
	gmr_malpha0	Mass scaling slope $\alpha_0$	$+0.2241$	$\mathcal{N}(+0.2241, 0.0038)$	Rodriguez-Gomez et al. (2015)
	gmr_malphaz	Mass-redshift coupling $\alpha_1$	$-1.1759$	$\mathcal{N}(-1.1759, 0.0316)$	Rodriguez-Gomez et al. (2015)
	gmr_mdelta0	High-mass transition slope $\delta_0$	$+0.7668$	$\mathcal{N}(+0.7668, 0.0202)$	Rodriguez-Gomez et al. (2015)
	gmr_mdeltaz	High-mass redshift coupling $\delta_1$	$-0.4695$	$\mathcal{N}(-0.4695, 0.0440)$	Rodriguez-Gomez et al. (2015)
	gmr_qgamma0	Mass ratio power law $\beta_0$	$-1.2595$	$\mathcal{N}(-1.2595, 0.0026)$	Rodriguez-Gomez et al. (2015)
	gmr_qgammaz	Mass ratio redshift coupling $\beta_1$	$+0.0611$	$\mathcal{N}(+0.0611, 0.0021)$	Rodriguez-Gomez et al. (2015)
	gmr_qgammam	Mass ratio mass coupling $\gamma$	$-0.0477$	$\mathcal{N}(-0.0477, 0.0013)$	Rodriguez-Gomez et al. (2015)
$M_\text{BH}$–$M_\text{bulge}$ & Bulge	mmb_mamp_log10	$z=0$ Amplitude $\log_{10}(M_\text{amp,0}/M_\odot)$	$8.69$	$\mathcal{N}(8.69, 0.05)$	Kormendy & Ho (2013) Table 2
	mmb_plaw	$z=0$ Power-law slope	$1.17$	$\mathcal{N}(1.17, 0.08)$	Kormendy & Ho (2013) Table 2
	mmb_scatter_dex	Intrinsic scatter in dex	$0.28$	$\mathcal{N}(0.28, 0.05)$	Kormendy & Ho (2013)
	mmb_zplaw_amp	Normalization redshift evolution $\gamma_z$	$0.0$	$\mathcal{U}(-2.0, +2.0)$	Matt et al. (2026a)
	bf_frac_lo	Bulge fraction lower plateau	$0.4$	$\mathcal{U}(0.1, 0.4)$	Morphological sigmoid
	bf_frac_hi	Bulge fraction upper plateau	$0.8$	$\mathcal{U}(0.6, 1.0)$	Morphological sigmoid
	bf_width_dex	Transition width in dex	$1.0$	$\mathcal{U}(0.5, 1.5)$	Morphological sigmoid

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Deliverables Included

1. Parameter Space Class: _PS_NG20_Base and PS_NG20_Fiducial defined in holodeck/librarian/param_spaces.py and registered in param_spaces_dict.
2. Hardening Bugfix: Fixed TypeError in FixedOuterTime_InnerPL_SAM.__str__ when formatting Model 1 (nu_inner is None).
3. Demonstration Notebook: notebooks/devs/ng20_fiducial_param_space.ipynb demonstrating:
   • Parameter definitions and default values.
   • Side-by-side prior distribution comparisons against the NANOGrav 15yr classic phenomenological extended parameter space (PS_Classic_Phenom_Astro_Extended).
   • Leja+2020 GSMF covariance joint distributions.
   • Full production grid (sam_shape=[100, 100, 100]) GWB spectrum calculation ($h_c(f)$).
   • Commands for library production (gen_lib_sams.py and SLURM).

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Discussion Points for the Collaboration

• BIO-Hardening parameterization: We use Model 1 (sampling $|da/dt|{r\mathrm{char}}$) to ensure Latin Hypercube Sampling is free of holes / unphysical superluminal points. Is this preferred over Model 0 (sampling $\nu_\mathrm{inner}$ with a hard speed cap)?
• Prior bounds: Are the current prior bounds for hard_outer_time (0.1–10 Gyr), hard_rchar_9 (0.1–10 pc), and mmb_zplaw_amp (-2 to +2) acceptable as our baseline?
• Fixed vs. Free Scalings: Confirming agreement on fixing secondary scalings ($\alpha_\mathrm{char} = -2/3$, $r_\mathrm{gw,crit,9} = 10^{2.5}r_g$, etc.) to baseline theoretical values.

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Testing & Validation

• test_param_spaces.py & test_lib_tools__param_space.py: Passed 100%.
• test_host_relations__mmbulge.py: Passed 100%.
• Verified top-to-bottom execution of ng20_fiducial_param_space.ipynb under native Apple Silicon (arm64).

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AI Usage Statement

This pull request was developed in a collaborative human–AI pair programming session using Google Antigravity:

• Human Author: Guided the astrophysical choices (Blecha Inside-Out hardening, circular binaries, Leja double-Schechter with joint covariance, Illustris merger rate, Kormendy & Ho baseline with Matt et al. redshift evolution), directed the sampling strategy to prevent superluminal hardening while preserving rectangular Latin Hypercube Sampling, specified library grid resolutions, and reviewed all code and documentation.
• AI Assistant: Researched the codebase, implemented _PS_NG20_Base and PS_NG20_Fiducial in param_spaces.py, resolved the FixedOuterTime_InnerPL_SAM.__str__ formatting bug, constructed and verified the demonstration notebook (ng20_fiducial_param_space.ipynb), compiled Cython extensions across environments, and drafted this PR description.