Co-Evolution of Halos, Galaxies & SMBHs in IllustrisTNG – Phase Framework & Observational Signposts

Abstract and Central Aims

• IllustrisTNG (TNG100-1) is used to dissect co-evolution of dark-matter halos (DMHs), central galaxies, and super-massive black holes (SMBHs).

• Individual MBH–M★ tracks decompose into four time-ordered phases, each separated by a transition point that can be quantitatively linked to physical criteria.

• Overarching goal: derive necessary & sufficient conditions for phase changes, expose governing processes, and map them onto observable scaling relations that can test sub-grid physics.

ΛCDM Structure Formation Primer

• Growth is hierarchical: small structures merge to make larger objects, gas cools, forms stars, occasionally produces central BHs.

• Feedback is bidirectional: small-scale (stellar/AGN) energy input reshapes halo-scale gas; conversely halo accretion regulates what baryons are available.

• Key observational constraints: $M<em>★{-}M</em>h$, $M<em>{BH}{-}M</em>★$, $M<em>{BH}{-}M</em>h$, galaxy/AGN luminosity & mass functions, clustering, CGM thermodynamics.

Simulation & Sample

• IllustrisTNG100-1

  • Box $\approx 110.7\,\mathrm{cMpc}$; $2\times1820^3$ elements; $m<em>{b}=1.4\times10^{6}\,M</em>\odot$.

  • Cosmology: $h=0.6774,\;\Omega<em>m=0.3089,\;\Omega</em>b=0.0486,\;\Omega_\Lambda=0.6911$ (Planck-2015).

• Group finding: FoF ($b=0.2$); Subfind for sub-halos; central = most massive subhalo.

• Merger trees: baryonic SubLink.

• Derived quantities

  • $M<em>h$ inside $R</em>{200\bar\rho}$; $V<em>{h}=\sqrt{GM</em>h/R_h}$.

  • $M<em>★$ within $2R</em>{★,1/2}$.

  • $M_{BH}$ = sum of BH particles in subhalo.

  • Specific rates $\dot M / M$ → sHAR, sSFR, sBHAR.

  • Quenching criterion: \mathrm{sSFR}<10^{-11}\,\mathrm{yr^{-1}} for ≥3 consecutive snapshots.

• Final analysis sample (1425 galaxies)

  1. Central at $z=0$(no backsplash);

  2. $M<em>h≥5\times10^{11}M</em>\odot,\;M<em>★≥10^{9}M</em>\odot$ and host an SMBH;

  3. Quenched at $z=0$;

  4. Tree traced to $z≈7$.

Phenomenological Fit & Phase Boundaries

• Fit each MBH–M★ history with piece-wise function:
  
  y(x)=\begin{cases}
  \frac{(a+b)^3}{2}\left[\frac{1}{(x-x0-a)^2}-\frac{1}{a^2}\right]+y0,&x<x0\[4pt] k(x-x0)+y0,&x\ge x0\end{cases}
  
  where $y\equiv\log M<em>{BH},\;x\equiv\log M</em>★$.

• Three transition abscissae

  • $x<em>{12}=x</em>0-b$ (slope becomes 1).

  • $x<em>{23}=x</em>0$ (end of rapid BH growth).

  • $t<em>{34}=t</em>q$ (quenching; linear piece ends).

• Phases: 1-SF dominated, 2-Rapid BH accretion, 3-Self-regulated, 4-Merger dominated.

Phase-1: Star-Formation Dominated

• SMBH seed: $M<em>{seed}=1.18\times10^{6}M</em>\odot$ planted once Mh>7.38\times10^{10}M\odot.

• Bondi accretion $\dot M<em>{BH}=A</em>s M<em>{BH}^2$ with $A</em>s=\frac{4\pi G^2\rho<em>{gas}}{c</em>s^3}$ constant because star-forming gas follows effective EOS.

• Analytical growth: $M<em>{BH}(t)=\frac{M</em>{seed}}{1-A<em>sM</em>{seed}(t-t<em>{seed})}$ ⇒ blow-up at $\tau</em>1^{max}=(A<em>sM</em>{seed})^{-1}\approx1\text{–}2\,\mathrm{Gyr}$.

• Transition when $\mathrm{sBHAR}=\mathrm{sSFR}$ (feedback energy ratio $E<em>{AGN}/E</em>{SF}≈3000\,\dot M<em>{BH}/\dot M</em>★$ reaches unity around $M<em>{BH}/M</em>★\sim10^{-3}$).

Phase-2: SMBH-Accretion Dominated

• Exponential BH growth (Bondi) until AGN thermal feedback unbinds galactic gas.

• Condition: $E<em>{AGN}^{therm}\gtrsim\eta</em>{eff}^{-1}E<em>{bind}$ where $\eta</em>{eff}\approx0.1$.

  • $E<em>{AGN}=\eta</em>{therm}M<em>{BH}c^2,\;\eta</em>{therm}=0.02$.

  • $E<em>{bind}\approx\tfrac12\sum m</em>{gas}\phi<em>{gas}$ within $2R</em>{★,1/2}$.

• Duration $\tau<em>2\lesssim1\,\mathrm{Gyr}$; inside-out gas depletion: cold gas drops within $0.2R</em>{★,1/2}$, halo gas unchanged.

Phase-3: Self-Regulation Dominated

• Thermal-mode feedback keeps galaxy in quasi-equilibrium:
  $\eta<em>{eff}\eta</em>{therm}M<em>{BH}\sim f</em>{gas}f<em>{★}^{-5/3}M</em>★^{5/3}$ → predicts $M<em>{BH}\propto M</em>★^{5/3}$ near $M<em>h\sim10^{12}M</em>\odot$.

• Observed in simulation: slope $k≈1.35$ anti-correlates with initial BH-to-stellar mass ratio (self-correcting).

• Eddington ratio threshold for kinetic mode: $\chi<em>{th}=\min\bigl[0.02\bigl(\frac{M</em>{BH}}{10^8M_\odot}\bigr)^2,0.1\bigr]$.

  • Low-mass halos: transition coincides with halo entering slow-accretion phase (peak $V_{max}$ time).

  • Massive halos: fixed $\chi_{th}=0.1$ ⇒ transition at $z\approx2$ irrespective of halo growth.

Phase-3 → Phase-4 & Quenching Criterion

• Kinetic feedback must balance cooling of all non-SF gas in subhalo:
  $\dot E<em>{kin}=\eta</em>{kin}\dot M<em>{BH}c^2\approx\dot E</em>{cool,sub}$.

• Quenching lag $t<em>q-t</em>{kin}\lesssim1\,\mathrm{Gyr}$ for Mh<10^{12.3}M\odot; extended lags at higher masses due to merger-induced gas supply.

• After quenching the balance $\dot E<em>{kin}\simeq\dot E</em>{cool}$ persists.

Phase-4: Merger Dominated Growth

• In-situ SFR and BHAR nearly zero; mass assembly via:

  • Ex-situ stars: fraction $f<em>{★,ex}^{[t</em>q,0]}$ rises with halo mass; can exceed $80\%$ for Mh>10^{12.6}M\odot.

  • BH-BH mergers dominate BH mass growth in quenched centrals.

• Mergers drive $(M<em>{BH},M</em>★)$ toward 1:1 vector, steepening high-mass end of scaling relations.

Evolution of Scaling Relations

• At $t<em>{23}$: $M</em>{BH}\propto M_★^{\gamma}$ with $\gamma\approx5/3$ (thermal feedback threshold).

• At $t_q$: shallower (\gamma<1) because kinetic feedback triggered earlier for massive BHs.

• At $z=0$: near-linear ($\gamma\approx1$) at high masses; low-mass end linear in TNG due to late seeding.

• MBH–Mh relation bends; peak correlation near $M<em>h\approx10^{12}M</em>\odot$ where $f_{gas}$ maximal.

Observable Diagnostics & Tests

• Edges/Slices: scaling relations at successive $z$ provide time slices of flow; compare to predicted phase edges.

• Derivatives: combining BHAR + SFR observations yields motion vectors in MBH–M★ plane → phase identification.

• High-z probes: JWST ‘Little Red Dots’ likely Phase-2 systems; Pop-III star clusters could reveal seeding stage.

• Low-z dissection: IFU kinematics (in-situ vs ex-situ stars) + lensing BH mass can isolate post-quench merger growth.

Key Numerical/Physical Parameters (TNG)

• SMBH seed mass $1.18\times10^6M<em>\odot$ @ Mh>7.38\times10^{10}M_\odot.

• Bondi accretion cap = Eddington.

• Thermal-mode efficiency $\eta<em>{therm}=0.02$; kinetic-mode coupling $\eta</em>{kin}=\min(\rho/0.05\rho_{SF},0.2)$.

• Quenching sSFR threshold $10^{-11}\,\mathrm{yr^{-1}}$.

Limitations & Sensitivities

• Phase-1 duration and low-mass scaling depend on seed choice & gas EOS.

• Phase-2 length responds to star-forming threshold density and Bondi kernel.

• $\chi<em>{th}(M</em>{BH})$ curve shapes $t<em>{kin}$ distribution and $M</em>{BH}{-}M_★$ slope at quenching.

• Efficiencies $\eta<em>{therm},\eta</em>{kin},\eta_{eff}$ dictate normalisations of all scaling laws.

Comparison Across Simulations

• EAGLE: stronger SN winds → prolonged Phase-1, non-linear low-mass MBH–M★ at all z.

• SIMBA: delayed seeding ⇒ non-linear relation persists to z=0.

• Horizon-AGN: weak SN feedback → Phase-1 absent, early linearity.

Study-Guide Take-Aways

• Memorise four phases, their governing physics & duration hierarchy \tau1<\tau2\ll\tau3,\tau4.

• Master transition criteria:

  1. $\mathrm{sBHAR}=\mathrm{sSFR}$ (start rapid BH growth).

  2. $E<em>{AGN}^{therm}\gtrsim\eta</em>{eff}^{-1}E_{bind}$ (end rapid growth).

  3. \chi<\chi_{th} (switch to kinetic mode).

  4. $\dot E<em>{kin}\approx\dot E</em>{cool}$ (quenching).

• Be able to derive $M<em>{BH}\propto M</em>★^{5/3}$ from energy balance.

• Recognise observational proxies (e.g., entropy, Pop-III clusters, merger signatures) for each phase.

• Understand how sub-grid choices (seeding, feedback efficiencies) propagate into macroscopic observables.

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