Virialization of the Inner CGM in the FIRE Simulations and Implications for Galaxy Disks, Star Formation, and Feedback

Classic cooling flow solutions for the intracluster medium (e.g., Mathews & Bregman 1978; Bertschinger 1989) highlight two characteristic radii of the flow, the "cooling radius" and the "sonic radius." At the cooling radius, the cooling time of shocked gas ${t}_{\mathrm{cool}}^{({\rm{s}})}$ (an exact definition is given below) equals the Hubble time t_H or age of the universe, so only within this radius is substantial cooling expected. At the sonic radius, ${t}_{\mathrm{cool}}^{({\rm{s}})}$ equals the free-fall time t_ff, so within this radius cooling is so rapid that it cannot be balanced by heating due to gravitational compression of the inflow. The sonic radius thus separates between an outer subsonic region where ${t}_{\mathrm{ff}}\lesssim {t}_{\mathrm{cool}}^{({\rm{s}})}\lesssim {t}_{{\rm{H}}}$ and the gas temperature is ∼T_vir and an inner supersonic region where ${t}_{\mathrm{cool}}^{({\rm{s}})}\lesssim {t}_{\mathrm{ff}}$ and the flow is free-falling with a temperature ≪T_vir. The outer subsonic region is expected to be smooth since in subsonic flows thermal instabilities develop on the same timescale as the flow time, while the inner supersonic region is expected to be clumpy since thermal instabilities develop faster than the flow time (e.g., Balbus & Soker 1989; Paper I).

In Papers I and II we adapted the classic cooling flow solutions to gas in galaxy-scale halos, i.e., the CGM. We demonstrated that the cooling radius is typically outside the halo virial radius, while the sonic radius can be at galaxy radii, at halo radii, or beyond the halo. Steady-state CGM solutions thus fall in one of three possible regimes: a fully subsonic CGM, where ${t}_{\mathrm{cool}}^{({\rm{s}})}\gt {t}_{\mathrm{ff}}$ at all halo radii; a transonic CGM, where ${t}_{\mathrm{cool}}^{({\rm{s}})}\sim {t}_{\mathrm{ff}}$ in the halo; and a fully supersonic CGM, where ${t}_{\mathrm{cool}}^{({\rm{s}})}\lt {t}_{\mathrm{ff}}$ at all halo radii (see Figure 3 in Paper II). Since the gas temperature is ∼T_vir only in subsonic regions, where the CGM is "virialized," the condition for a fully virialized CGM is thus

$$\begin{eqnarray}&&{t}_{\mathrm{cool}}^{({\rm{s}})}({R}_{\mathrm{circ}})\gt {f}_{t}{t}_{\mathrm{ff}}({R}_{\mathrm{circ}}),\end{eqnarray} \tag{ 1 }$$

where f_t is a factor of order unity and we use the gas circularization radius R_circ to define the inner radius of the CGM (see Paper II). The value of R_circ can be calculated from

$$\begin{eqnarray}&&j={v}_{{\rm{c}}}({R}_{\mathrm{circ}}){R}_{\mathrm{circ}},\end{eqnarray} \tag{ 2 }$$

where j is the gas specific angular momentum, v_c is the circular velocity

$$\begin{eqnarray}&&{v}_{{\rm{c}}}=\sqrt{\displaystyle \frac{{GM}(\lt r)}{r}},\end{eqnarray} \tag{ 3 }$$

and M(<r) is the total mass within r. The value of R_circ can be approximated by assuming an isothermal potential and that j is independent of radius and similar to the average specific angular momentum of the dark matter. This gives ${R}_{\mathrm{circ}}\approx \sqrt{2}\lambda {R}_{\mathrm{vir}}\sim 0.05{R}_{\mathrm{vir}}$, where the characteristic spin parameter λ ∼ 0.035 is independent of halo mass and redshift (e.g., Rodríguez-Puebla et al. 2016).

To estimate Equation (1), we use the definition of the free-fall time: ¹⁴

$$\begin{eqnarray}&&{t}_{\mathrm{ff}}=\displaystyle \frac{\sqrt{2}r}{{v}_{{\rm{c}}}}\approx 160\displaystyle \frac{r}{0.05{R}_{\mathrm{vir}}}{f}_{{v}_{{\rm{c}}}}^{-1}{\left(1+z\right)}^{-1.4}\,\mathrm{Myr},\end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray}&&{f}_{{v}_{{\rm{c}}}}(r)\equiv \displaystyle \frac{{v}_{{\rm{c}}}(r)}{{v}_{{\rm{c}}}({R}_{\mathrm{vir}})}\end{eqnarray} \tag{ 5 }$$

and the numerical evaluation is based on the virial relation ${t}_{\mathrm{ff}}({R}_{\mathrm{vir}})=2/(\sqrt{{{\rm{\Delta }}}_{{\rm{c}}}}H)$. We define the virial overdensity Δ_c as in Bryan & Norman (1998) and use the following approximation of $\sqrt{{{\rm{\Delta }}}_{{\rm{c}}}}H$ in our assumed cosmology:

$$\begin{eqnarray}&&\sqrt{{{\rm{\Delta }}}_{{\rm{c}}}}H\approx 0.6{\left(1+z\right)}^{1.4}\,{\mathrm{Gyr}}^{-1}.\end{eqnarray} \tag{ 6 }$$

The cooling time of shocked gas ${t}_{\mathrm{cool}}^{({\rm{s}})}$ in condition (1) is defined assuming that gas at ≳R_circ forms a cooling flow in the pressure-supported limit, i.e., that it satisfies v_r ≈ − r/t_cool and ${v}_{r}^{2}\ll {c}_{{\rm{s}}}^{2}$, where v_r and c_s are the radial velocity and sound speed, respectively. These conditions yield

$$\begin{eqnarray}&&{t}_{\mathrm{cool}}^{({\rm{s}})}\equiv {t}_{\mathrm{cool}}({T}^{({\rm{s}})},{n}_{{\rm{H}}}^{({\rm{s}})})=\displaystyle \frac{(3/2)\cdot 2.3{k}_{{\rm{B}}}{T}^{({\rm{s}})}}{{n}_{{\rm{H}}}^{({\rm{s}})}{\rm{\Lambda }}},\end{eqnarray} \tag{ 7 }$$

where T^(s) and ${n}_{{\rm{H}}}^{({\rm{s}})}$ are the temperature and hydrogen number density of a cooling flow in the pressure-supported limit, respectively, Λ is the cooling function defined such that ${n}_{{\rm{H}}}^{2}{\rm{\Lambda }}$ is the cooling per unit volume, and we assume 2.3 particles per hydrogen particle. The value of T^(s) can be calculated from Equation (24) in Paper I:

$$\begin{eqnarray}&&{T}^{({\rm{s}})}\equiv \displaystyle \frac{3\mu {m}_{{\rm{p}}}{v}_{{\rm{c}}}^{2}}{5{{Ak}}_{{\rm{B}}}}=4.5\cdot {10}^{5}{A}^{-1}{v}_{100}^{2}\,{\rm{K}},\end{eqnarray} \tag{ 8 }$$

where μ = 0.62 is the molecular weight, v_c = 100v₁₀₀ km s⁻¹, and the factor A is equal to

$$\begin{eqnarray}&&A=\displaystyle \frac{9}{10}\left(1-2\displaystyle \frac{d\mathrm{log}{v}_{{\rm{c}}}}{d\mathrm{log}r}\right)\approx 1.\end{eqnarray} \tag{ 9 }$$

This temperature is comparable to the virial temperature defined as

$$\begin{eqnarray}&&{T}_{\mathrm{vir}}=\displaystyle \frac{\mu {m}_{{\rm{p}}}}{2{k}_{{\rm{B}}}}\displaystyle \frac{{{GM}}_{\mathrm{halo}}}{{R}_{\mathrm{vir}}}\end{eqnarray} \tag{ 10 }$$

and hence

$$\begin{eqnarray}&&{T}^{({\rm{s}})}=\displaystyle \frac{6}{5A}{f}_{{v}_{{\rm{c}}}}^{2}{T}_{\mathrm{vir}}.\end{eqnarray} \tag{ 11 }$$

The density of a cooling flow ${n}_{{\rm{H}}}^{({\rm{s}})}$ can be calculated from the assumption of pressure support of the weight of the overlying gas:

$$\begin{eqnarray}&&2.3{n}_{{\rm{H}}}^{({\rm{s}})}{k}_{{\rm{B}}}{T}^{({\rm{s}})}={\int }_{r}\frac{\rho {v}_{{\rm{c}}}^{2}}{r^{\prime} }{\rm{d}}r^{\prime} ,\end{eqnarray} \tag{ 12 }$$

where $-{v}_{{\rm{c}}}^{2}/r^{\prime}$ is the gravitational acceleration at radius $r^{\prime}$. Below we use Equation (12) to estimate ${n}_{{\rm{H}}}^{({\rm{s}})}$ in the FIRE simulations. To derive a numerical approximation of ${n}_{{\rm{H}}}^{({\rm{s}})}({R}_{\mathrm{circ}})$, we assume that the gas distribution in the halo follows a power law $\rho =\rho {({R}_{\mathrm{vir}})(r/{R}_{\mathrm{vir}})}^{-a}$ with slope −a and normalization ρ(R_vir) = (1 − a/3)f_gas · (0.158Δ_c ρ_crit), where ρ_crit is the critical density and f_gas is the ratio of the halo gas mass to the cosmic halo baryon budget 0.158M_halo. We thus get

$$\begin{eqnarray}&&\rho (r)\approx 4.9{f}_{\mathrm{gas}}{g}^{-1}(a){\left(\displaystyle \frac{r}{0.05{R}_{\mathrm{vir}}}\right)}^{-a}{{\rm{\Delta }}}_{{\rm{c}}}{\rho }_{\mathrm{crit}},\end{eqnarray} \tag{ 13 }$$

where g is defined such that g(3/2) = 1:

$$\begin{eqnarray}&&g(a)={20}^{3/2-a}\left(\displaystyle \frac{3}{6-2a}\right).\end{eqnarray} \tag{ 14 }$$

Further using n_H ≈ 0.7ρ/m_p, ρ_crit = 3H²/8π G, and Equation (6), we get

$$\begin{eqnarray}&&{n}_{{\rm{H}}}^{({\rm{s}})}(r)\approx 2.0\cdot {10}^{-3}{\left(1+z\right)}^{2.8}{f}_{\mathrm{gas}}{g}^{-1}(a){\left(\displaystyle \frac{r}{0.05{R}_{\mathrm{vir}}}\right)}^{-a}\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 15 }$$

While our choice of T^(s) and ${n}_{{\rm{H}}}^{({\rm{s}})}$ in the definition of ${t}_{\mathrm{cool}}^{({\rm{s}})}$ in Equation (7) assumes that the CGM forms a cooling flow in the limit of full thermal pressure support, in practice only the pressure support condition is crucial for our results. Any pressure-supported CGM will have a temperature ∼T_vir and a pressure set by the weight of the overlying gas. Our conclusions are hence not altered if we use, e.g., T_vir instead of T^(s) in Equations (7) and (12), beyond changing the exact value of the order-unity factor f_t .

We emphasize that below we measure ${t}_{\mathrm{cool}}^{({\rm{s}})}$ in the FIRE simulations rather than the standard t_cool, since the latter depends on the actual gas temperature T in the simulation. The reason for this is twofold. First, since ${t}_{\mathrm{cool}}^{({\rm{s}})}$ does not depend on T, the relation between ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ and T in FIRE is nontrivial (see Section 3 and Figure 9). Second, the cooling function at temperatures ≪T_vir is immaterial for the question of whether the gas shock-heated to ∼T_vir can remain pressure supported or rather cools and free-falls. For example, if ${t}_{\mathrm{cool}}^{({\rm{s}})}$ is short and the gas does cool, then it could reach the minimum of the cooling curve where t_cool is quite long, and hence t_cool would not be a useful indication of the strong cooling. A comparison of ${t}_{\mathrm{cool}}^{({\rm{s}})}$ with other ways of evaluating the cooling time in the simulations is presented in Appendix A.

Using Equations (8) and (15) in Equation (7), we get

$$\begin{eqnarray}&&{t}_{\mathrm{cool}}^{({\rm{s}})}\approx 14\,{v}_{100}^{3.4}{Z}_{1}^{-1}{\left(1+z\right)}^{-2.8}{f}_{\mathrm{gas}}^{-1}g(a){\left(\displaystyle \frac{r}{0.05{R}_{\mathrm{vir}}}\right)}^{a}\,\mathrm{Myr},\end{eqnarray} \tag{ 16 }$$

where we used A = 1 and approximate the cooling function as ${\rm{\Lambda }}\approx 1.4\cdot {10}^{-22}{Z}_{1}{T}_{6}^{-0.7}\,\mathrm{erg}\,{\mathrm{cm}}^{3}\,{{\rm{s}}}^{-1}$, where Z = Z₁ Z_⊙ is the gas metallicity and T = 10⁶ T₆ K. This approximation is roughly valid for Z₁ ∼ 1 and 0.1 < T₆ < 10. Using Equation (4), the ratio of the two timescales is hence

$$\begin{eqnarray}&&\displaystyle \frac{{t}_{\mathrm{cool}}^{({\rm{s}})}}{{t}_{\mathrm{ff}}}\approx 0.09\,{v}_{100}^{3.4}{f}_{{v}_{{\rm{c}}}}{Z}_{1}^{-1}{\left(1+z\right)}^{-1.4}{f}_{\mathrm{gas}}^{-1}g(a){\left(\displaystyle \frac{r}{0.05{R}_{\mathrm{vir}}}\right)}^{a-1}.\end{eqnarray} \tag{ 17 }$$

To express this ratio in terms of halo mass, we replace v_c(r) with ${f}_{{v}_{{\rm{c}}}}{v}_{{\rm{c}}}({R}_{\mathrm{vir}})$ and use the virial relation ${v}_{{\rm{c}}}^{3}({R}_{\mathrm{vir}})\,=(\sqrt{{{\rm{\Delta }}}_{{\rm{c}}}/2}){{HGM}}_{\mathrm{halo}}$. With Equation (6) this gives

$$\begin{eqnarray}&&\displaystyle \frac{{t}_{\mathrm{cool}}^{({\rm{s}})}}{{t}_{\mathrm{ff}}}\approx 0.17\,{M}_{12}^{1.1}{f}_{{v}_{{\rm{c}}}}^{4.4}{Z}_{1}^{-1}{\left(1+z\right)}^{0.2}{f}_{\mathrm{gas}}^{-1}g(a){\left(\displaystyle \frac{r}{0.05{R}_{\mathrm{vir}}}\right)}^{a-1},\end{eqnarray} \tag{ 18 }$$

where M_halo = 10¹² M₁₂ M_⊙. Note that the ratio ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ increases outward for a > 1 and that it is almost independent of redshift if other parameters are held fixed.

The results of Paper I–II suggest that if ${t}_{\mathrm{cool}}^{({\rm{s}})}\gtrsim {t}_{\mathrm{ff}}$ at R_circ, then we expect the CGM to be virialized down to the galaxy scale. If this condition is violated but ${t}_{\mathrm{cool}}^{({\rm{s}})}\gtrsim {t}_{\mathrm{ff}}$ in the outer halo (say, at 0.5R_vir), then we expect the CGM to be "transonic" with a cool and free-falling inner region and a virialized CGM farther out. If ${t}_{\mathrm{cool}}^{({\rm{s}})}\lesssim {t}_{\mathrm{ff}}$ also in the outer halo, then we expect the volume filling phase to be cool and free-falling throughout the halo.

We test the above idealized theory against cosmological "zoom-in" simulations run as part of the Feedback In Realistic Environments project, ¹⁵ using the second version of these simulations (FIRE-2). The simulation methods are described in detail in Hopkins et al. (2018), while the main aspects are summarized here.

The FIRE-2 simulations use the multimethod gravity and hydrodynamics code GIZMO ¹⁶ (Hopkins 2015) in its meshless finite-mass mode (MFM). MFM is a Lagrangian, mesh-free, finite-mass method that combines advantages of traditional smooth particle hydrodynamics (SPH) and grid-based methods. Gravity is solved using a modified version of the Tree-PM solver similar to GADGET-3 (Springel 2005) but with adaptive softening for gas resolution elements. Radiative heating and cooling rates account for metal line cooling, free–free emission, photoionization and recombination, Compton scattering with the cosmic microwave background, collisional and photoelectric heating by dust grains, and molecular and fine-structure cooling at low temperatures (<10⁴ K). The relevant ionization states are derived from precomputed cloudy (Ferland et al. 1998) tables including the effects of the cosmic UV background from Faucher-Giguère et al. (2009) and local radiation sources. Star formation occurs in self-gravitating, self-shielded molecular gas with n_H > 1000 cm⁻³ (Hopkins et al. 2013). The subgrid implementation of feedback processes from stars includes radiation pressure, heating by photoionization and photoelectric processes, and energy, momentum, mass, and metal deposition from supernovae (SNe; core collapse and Type Ia) and stellar winds. Feedback parameters and their time dependence are based on the stellar evolution models in Leitherer et al. (1999) assuming a Kroupa (2001) initial mass function.

The subset of FIRE-2 simulations analyzed in this work are listed in Table 1. They span a broad range in the mass of the central halo and how it evolves with time, as shown in the top left panel of Figure 1. The simulations include six "m11's" where the main halo at z = 0 has a mass of ∼10¹¹ M_⊙ and hosts a dwarf galaxy, five "m12's" with M_halo ∼ 10¹² M_⊙ and an ∼L^* galaxy at z = 0, and five "m13's" in which M_halo exceeds 10¹² M_⊙ at high redshift (2 < z < 6). ¹⁷ We use this diverse range of halo mass assembly histories to demonstrate that for a given ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ the virialization of the inner CGM is independent of redshift.

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Table 1. FIRE-2 Cosmological "Zoom" Simulations Used in This Work

Name	mb	${z}_{\min }$	${M}_{\mathrm{halo}}({z}_{\min })$	${M}_{* }({z}_{\min })$	md	References
	(M⊙)		(M⊙)	(M⊙)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
m11's
m11b	2100	0	0.4 · 1011	1.2 · 108	no	A
m11i	7100	0	0.7 · 1011	1.0 · 109	yes	B
m11e	7100	0	1.6 · 1011	1.7 · 109	yes	B
m11h	7100	0	1.9 · 1011	4.0 · 109	yes	B
m11v	7100	0	2.6 · 1011	5.8 · 109	no	C
m11d	7100	0	3.0 · 1011	5.1 · 109	yes	B
m12's
m12z	4200	0	0.8 · 1012	2.5 · 1010	yes	D
m12i	7100	0	1.1 · 1012	7.3 · 1010	yes	E
m12b	7100	0	1.3 · 1012	1.0 · 1011	yes	D
m12m	7100	0	1.5 · 1012	1.4 · 1011	no	C
m12f	7100	0	1.6 · 1012	1.0 · 1011	no	F
m13's
m13A1	33000	1	0.4 · 1013	2.8 · 1011	no	G
m13A4	33000	1	0.5 · 1013	2.7 · 1011	no	G
m13A2	33000	1	0.8 · 1013	5.1 · 1011	no	G
m13A8	33000	1	1.3 · 1013	8.0 · 1011	no	G
z5m13a	57000	4.5	0.4 · 1013	2.0 · 1011	no	H

Note. Column (1): galaxy name. Column (2): initial baryonic particle mass. Column (3): final redshift of the simulation. Column (4): central halo mass at the final redshift. Column (5): stellar mass of central galaxy at the final redshift. Column (6): whether a prescription for subgrid metal diffusion is included in the simulation. Column (7): reference papers for simulations. A: Chan et al. (2018); B: El-Badry et al. (2018b); C: Hopkins et al. (2018); D: Garrison-Kimmel et al. (2018); E: Wetzel et al. (2016); F: Garrison-Kimmel et al. (2017); G: Anglés-Alcázar et al. (2017b); H: Ma et al. (2018).

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In the m11 and m12 simulations the initial mass of a baryonic resolution element is m_b = 7100 M_⊙ or better. This implies that subgrid physics are applied at the giant molecular cloud level (or better), while ∼L^⋆ galaxy disks are well resolved with ∼10⁶ particles. The more massive m13 simulations have a mass resolution of m_b = 33,000–57,000 M_⊙. Implications of resolution for our results are discussed in Appendix B.

A subset of the simulations also include a mechanism for subgrid metal diffusion between neighboring resolution elements as described in Hopkins (2017) and Escala et al. (2018). However, as shown in Appendix B, the inclusion of this prescription does not appear to affect our conclusions.

In this section we describe how we estimate ${t}_{\mathrm{cool}}^{({\rm{s}})}$ and t_ff in the FIRE simulations. The ratio of these two timescales is expected to determine whether the volume filling phase of the CGM has virialized into a hot and subsonic medium (Section 2.1).

For each snapshot in the simulation, we use the Amiga Halo Finder (Knollmann & Knebe 2009) to identify the center, virial mass M_vir, and virial radius R_vir of the main halo, utilizing the virial overdensity definition of Bryan & Norman (1998). We calculate v_c(r) from M(<r) using Equation (3), where M(<r) is the sum of all types of resolution elements (gas, stars, dark matter) with centers within r. We then use v_c and Equation (4) to derive t_ff.

Calculating ${t}_{\mathrm{cool}}^{({\rm{s}})}(r)$ in the simulations requires calculating T^(s), ${n}_{{\rm{H}}}^{({\rm{s}})}$, and Λ (see Equation (7)). For T^(s) we use Equation (8) and the estimated v_c(r). For ${n}_{{\rm{H}}}^{({\rm{s}})}$ we use Equation (12), i.e., we evaluate the weight of the overlying gas to estimate the expected thermal pressure in a virialized CGM, and then we divide by T^(s) to get ${n}_{{\rm{H}}}^{({\rm{s}})}$. To compute ρ in the integrand in Equation (12), we divide the gas in each snapshot into radial shells with a width of 0.05 dex and divide the total mass of all gas resolution elements within the shell by the shell volume. The integration is then carried out from the desired r out to R_vir. Since the weight of the overlying gas (which enters in the integrand) is typically largest at the smallest radii in the integration range, the exact choice of outer integration limit does not significantly affect the result. The values of T^(s) and ${n}_{{\rm{H}}}^{({\rm{s}})}$ we derive are comparable to the actual mean temperatures and densities in the simulation after the inner CGM virializes, though they can differ substantially prior to virialization (see Section 3.1 and Appendix A). We derive Λ from the Wiersma et al. (2009) tables, using T^(s), ${n}_{{\rm{H}}}^{({\rm{s}})}$, z, and the mass-weighted metallicity of gas in a shell with width 0.05 dex around r. Our calculation of Λ thus assumes that the metals at each radius are uniformly distributed at that radius, as assumed in the spherical steady-state solutions in Section 2.1.

Estimating R_circ (Equation (2)) in FIRE is not straightforward. In Section 4 below we show that only after the inner CGM virializes there is a well-defined radius where the gas circularizes, with typical values in the range R_circ ∼0.02R_vir–0.07R_vir. Therefore, to estimate condition (1) uniformly at all epochs, we instead evaluate ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ at 0.1R_vir, i.e., condition (1) becomes

$$\begin{eqnarray}&&{t}_{\mathrm{cool}}^{({\rm{s}})}(0.1{R}_{\mathrm{vir}})={f}_{t}^{\prime} {t}_{\mathrm{ff}}(0.1{R}_{\mathrm{vir}}),\end{eqnarray} \tag{ 19 }$$

where ${f}_{t}^{\prime}$ is expected to be somewhat larger than f_t (since 0.1R_vir > R_circ and ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ increases outward; see Equation (17)). Using a constant fraction of R_vir rather than R_circ has the advantages of being conceptually simpler and independent of jitter in the specific angular momentum profile. Also, using a radius that is a factor of ∼2 larger than R_circ avoids the unwanted increase of our CGM density estimate by gas in the disk. ¹⁸ The substitution of R_circ with 0.1R_vir, however, implies that we expect scatter in ${f}_{t}^{\prime}$ due to halo-to-halo variance in R_circ.

We experimented also with other choices for the radius at which to estimate ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$, including the maximum radius where j exceeds some order-unity factor of v_c r, and the radius where v_c r equals the average specific angular momentum of all gas at 0.1R_vir–1R_vir. Up to order-unity differences in the derived ${f}_{t}^{\prime}$, these alternative choices yielded similar results to estimating ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ at 0.1R_vir.

Figure 1 plots the results of the above estimates versus time in the 16 FIRE simulations. The top panels show M_halo and 0.1R_vir, while the bottom four panels show Z, v_c, ${n}_{{\rm{H}}}^{({\rm{s}})}$, and ${t}_{\mathrm{cool}}^{({\rm{s}})}$ measured at 0.1R_vir and smoothed with a gigayear-wide boxcar to avoid clutter. The value of T^(s) is shown in the right axis of the v_c panel. In the bottom right panel we also plot the median value of t_ff, based on the median of all simulations run at the relevant redshift. In individual simulations t_ff is within a factor of two of the median, so for clarity individual t_ff curves are not shown. The curves in Figure 1 are colored by the ratio ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$, as noted in the color bar in the top right panel.

Several trends are apparent in Figure 1. The bottom right panel shows that t_ff increases rather slowly with time, as expected from Equation (4). In contrast, ${t}_{\mathrm{cool}}^{({\rm{s}})}$ increases significantly faster, so a typical simulation spans more than three orders of magnitude in ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$. The increase in ${t}_{\mathrm{cool}}^{({\rm{s}})}$ is a result of the decrease in ${n}_{{\rm{H}}}^{({\rm{s}})}$ and increase in v_c, while the general increase in Z with time slows the increase in ${t}_{\mathrm{cool}}^{({\rm{s}})}$ (see Equation (16)). The value of ${n}_{{\rm{H}}}^{({\rm{s}})}$ at 0.1R_vir is naively expected to scale with the mean halo density Δ_c ρ_crit (see Equation (13)), though in practice they decrease somewhat faster, from typical ratios ρ^(s)/Δ_c ρ_crit of 1.2–4 at z ≳ 2 to 0.4–1 at z = 0 (see bottom left panel). Note also that when ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff, the trend in metallicity reverses and the metallicity starts to decrease with time, suggesting a change in physical conditions in the CGM. We show below that this is likely due to the suppression of outflows once the inner CGM virializes, causing low-metallicity inflowing gas to reduce the overall CGM metallicity.

The top left panel of Figure 1 shows that for a fixed halo mass ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ at 0.1R_vir tends to increase with time. This is evident in ∼10¹² M_⊙ halos, where ${t}_{\mathrm{cool}}^{({\rm{s}})}\sim {t}_{\mathrm{ff}}$ at high redshift in contrast with ${t}_{\mathrm{cool}}^{({\rm{s}})}\gg {t}_{\mathrm{ff}}$ at z ∼ 0, and also in ∼10¹¹ M_⊙ halos, where ${t}_{\mathrm{cool}}^{({\rm{s}})}\ll {t}_{\mathrm{ff}}$ at high redshift in contrast with ${t}_{\mathrm{cool}}^{({\rm{s}})}\sim {t}_{\mathrm{ff}}$ at z ∼ 0. The increase in ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ with time at fixed halo mass is mainly due to the decrease in ρ^(s)/Δ_c ρ_crit seen in the bottom left panel and due to the increase in ${f}_{{v}_{{\rm{c}}}}={v}_{{\rm{c}}}(0.1{R}_{\mathrm{vir}})/{v}_{{\rm{c}}}({R}_{\mathrm{vir}})$ as a result of the larger concentration of low-redshift halos. We further address this trend in the discussion.

To identify the virialization of the volume filling CGM phase, we measure the volume-weighted temperature in radial shells with centers r and thickness ${\rm{\Delta }}\mathrm{log}r=0.05\,\mathrm{dex}$, calculated via

$$\begin{eqnarray}&&\mathrm{log} \langle T(r) \rangle \equiv \frac{\sum {V}_{i}\mathrm{log}{T}_{i}}{\sum {V}_{i}},\end{eqnarray} \tag{ 20 }$$

where T_i and V_i = m_i /ρ_i are the temperature and volume of resolution element i, respectively, with m_i and ρ_i its mass and density. The summations in Equation (20) are over all resolution elements whose centers are within the shell. This weighting by volume deemphasizes the effects of satellites and filaments and focuses on the properties of the volume filling phase. We average the logarithm of the temperature in order to give similar weights to a hot phase with T ≈ T_vir and a cool phase with T ≈ 10⁴ K, in contrast with a linear average which would overweight the hot phase relative to the cool phase.

The top panel of Figure 2 shows the volume-weighted temperature profiles in the last 8 Gyr of the m12i simulation (0 < z < 1). Each curve plots the median temperature profile of all snapshots in a Δt = 0.5 Gyr window (individual snapshots are separated by ≈25 Myr). Taking the median reduces the variability induced by transient CGM heating events due to outflows and allows us to focus on the time-steady effects of virialization (see van de Voort et al. 2016). The distribution of temperatures in individual snapshots is discussed below. We color each line in Figure 2 according to the median ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ at 0.1R_vir during the time window, using the same color scheme as in Figure 1. This ratio increases by a factor of 50 over the plotted time period, from 0.3 at z = 1 to 16 at z = 0. For comparison, the halo mass increases only mildly over this time from 0.9 × 10¹² M_⊙ to 1.1 × 10¹² M_⊙. The middle panel shows the temperature profiles normalized by T_vir. The figure shows that in the z = 0 snapshot where ${t}_{\mathrm{cool}}^{({\rm{s}})}\gg {t}_{\mathrm{ff}}$ (reddest curve) the volume-weighted temperature profile decreases from T ≈ 1.5T_vir = 10⁶ K at 0.1R_vir to ≈0.3T_vir = 2 × 10⁵ K at R_vir. At ≳R_vir the temperature profile tends to steepen, while at galaxy radii (<0.1R_vir) the volume-weighted temperature is somewhat lower than at 0.1R_vir owing to the cool gas disk (see below). At higher redshift when ${t}_{\mathrm{cool}}^{({\rm{s}})}\lesssim {t}_{\mathrm{ff}}$ (blue-gray curves) the temperature is significantly lower at 0.1R_vir, roughly equal to 0.1T_vir, indicating that a time-steady volume filling phase with T ≈ T_vir has not formed in the inner CGM, i.e., the inner CGM has not virialized. In contrast, 〈T〉/T_vir at ≳0.5R_vir in all shown snapshots is similar to that at z = 0. If one moves forward in time from blue-gray curves to red curves, the temperature profiles within R_vir tend to join the z = 0 profile first at large halo radii and later at small halo radii, indicating that virialization proceeds from the outside in.

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As another estimate of CGM virialization, we measure the fraction of the volume in each shell with supersonic radial velocities, either inflowing or outflowing (again, taking the median of snapshots in a Δt = 0.5 Gyr window). As discussed in Section 2.1 and in Fielding et al. (2017), in a virialized CGM we expect most of the volume to have subsonic radial velocities, i.e., the kinetic energy should be subdominant to the thermal energy. In contrast, prior to virialization we expect a significant fraction of the volume to be supersonic, i.e., the kinetic energy should dominate. This supersonic fraction profile is shown in the bottom panel of Figure 2. At low redshifts where ${t}_{\mathrm{cool}}^{({\rm{s}})}\gg {t}_{\mathrm{ff}}$ the volume is dominated (>70%) by subsonic gas from galaxy scales out to ≈0.8R_vir. At ≈R_vir a relatively sharp increase in the supersonic fraction is evidence for an accretion shock. Thus, at z = 0 the halo gas in m12i is almost entirely subsonic, i.e., to zeroth order supported against gravity by thermal pressure. In contrast, at higher redshifts at which ${t}_{\mathrm{cool}}^{({\rm{s}})}\lesssim {t}_{\mathrm{ff}}$ at 0.1R_vir the gas is predominantly supersonic at small radii and predominantly subsonic at larger radii, i.e., the halo gas is "transonic." This panel therefore also supports an outside-in virialization scenario, since the outer CGM is subsonic before the inner CGM becomes subsonic.

Another interesting result of Figure 2 is the temperature at r > R_vir, i.e., outside the halo. At these large radii there appears to be a mild reverse trend relative to the trend at 0.1R_vir, with the temperature decreasing with time as ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ at 0.1R_vir increases. This trend is apparent even if we plot the radius in physical units rather than as a fraction of R_vir. The bottom panel shows that this decrease in temperature at r ≳ R_vir is associated with a prominent accretion shock forming at ∼R_vir after the inner CGM virializes (the causal relationships are not addressed here), similar to a classic virial shock (e.g., Birnboim & Dekel 2003). Note, however, that prior to the formation of this shock the kinematics are subsonic at all radii outside ≳0.2R_vir in the FIRE simulations, in contrast with the idealized simulations of Birnboim & Dekel in which the kinematics are supersonic at all radii prior to shock formation. In the discussion we address the possible origin of this reverse trend outside the halo.

Figure 3 compares the volume-weighted temperature profiles of the z = 0 snapshots in the m12 and m11 simulations. As in Figure 2 the panels show from top to bottom temperature, normalized temperature, and supersonic fraction, while the curves are colored by ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ at 0.1R_vir. The blue-grayish curves include the m11 subgroup and m12z, with halo masses spanning the range 4 × 10¹⁰−8 × 10¹¹ M_⊙, while the red curves include the remaining four m12's with halo masses spanning the range (1.1–1.6) × 10¹² M_⊙. The trends of temperature and supersonic fraction versus ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ are similar to those seen in Figure 2. The value of 〈T〉/T_vir at 0.1R_vir is 1T_vir–2T_vir in the red group (${t}_{\mathrm{cool}}^{({\rm{s}})}\gg {t}_{\mathrm{ff}}$, higher M_halo), compared to 0.1T_vir–0.5T_vir in the blue-gray group (${t}_{\mathrm{cool}}^{({\rm{s}})}\lesssim {t}_{\mathrm{ff}}$, lower M_halo). In contrast, there is almost no difference between the two groups in 〈T〉/T_vir at 0.5R_vir. A parallel trend is also seen in the supersonic fraction profile shown in the bottom panel. At 0.1R_vir the supersonic fraction is 0.05–0.2 in the red group, substantially lower than the fraction of 0.4–0.8 in the blue-gray group. In contrast, at ∼0.5R_vir there is almost no supersonic gas (<20% of the volume) in either group. Also evident in this plot is the reverse trend of decreasing 〈T〉/T_vir at >R_vir with increasing ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ at 0.1R_vir. Figure 3 thus supports the conclusion from Figure 2 that the CGM is fully virialized when ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}\gg 1$ at 0.1R_vir, while halos with ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}\lesssim 1$ at 0.1R_vir are transonic. Furthermore, Figure 3 demonstrates that in all simulations in our sample the CGM at large radii is virialized at z ∼ 0, regardless of their halo mass. Virialization of the CGM in FIRE thus begins at halo masses well below the classic threshold of ∼10¹² M_⊙.

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Further insight into the differences between snapshots with ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}\lesssim 1$ and ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}\gg 1$ can be gained by exploring the gas temperature distribution within radial shells. To this end, Figure 4 plots a temperature map and 2D temperature histograms of the z = 0 snapshot of m12i, in which ${t}_{\mathrm{cool}}^{({\rm{s}})}=16{t}_{\mathrm{ff}}$ at 0.1R_vir. The images in the two left panels are oriented such that the total angular momentum vector of gas within 0.05R_vir is oriented upward, i.e., edge-on to the galaxy disk. The first panel spans ±R_vir, while the second panel zooms in on the central ±0.2R_vir. The images are derived by averaging $\mathrm{log}T$ perpendicular to the image plane over a depth equal to 10% of the image size. The figure shows that in this snapshot hot gas (>10⁵ K) dominates at practically all locations within the halo except in the galaxy disk. The accretion shock is evident as a temperature drop along a nonspherical contour roughly at a distance ∼R_vir from the center.

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The two right panels in Figure 4 show temperature histograms of gas in shells at different radii, weighted by volume (third panel from the left) and weighted by mass (rightmost panel). Color denotes the volume or mass fraction of the shell in each temperature bin. As suggested by the images on the left, the hot phase dominates the volume out to ≈R_vir, at which there is a break in the temperature profile due to the accretion shock. In contrast, the cool phase (T ≲ 10⁴ K) is entirely negligible in the halo in terms of volume and becomes significant (but still subdominant) only at disk radii. In the mass-weighted histogram on the right the hot phase dominates at halo radii (>0.1R_vir) while the cold phase dominates at disk radii (<0.1R_vir). The transition between halo and galaxy radii is sharp (as seen also in the zoomed image in the second panel). We show below that this sharp transition corresponds to the radius R_circ where gas in the CGM circularizes (Equation (2)).

For comparison, Figure 5 plots temperature maps and 2D temperature histograms for the z = 0 snapshot of m11d, in which ${t}_{\mathrm{cool}}^{({\rm{s}})}=0.2{t}_{\mathrm{ff}}$ at 0.1R_vir. As in Figure 4 the orientation of the two left panels is such that the total angular momentum vector of gas within 0.05R_vir is oriented upward. The figure shows that beyond 0.3R_vir the hot phase dominates also in this snapshot, similar to the m12i snapshot shown in Figure 4 albeit with a lower absolute temperature due to the lower T_vir and a larger dispersion in temperature at a given radius. The gas temperatures of m11d in the inner CGM and at galaxy radii are, however, completely different from those in m12i. In m11d the cool phase dominates the volume out to 0.3R_vir, while in m12i the volume of the cool phase is subdominant at all radii and negligible beyond 0.1R_vir. By mass, the hot phase is completely negligible in m11d out to ≈0.2R_vir, in contrast with m12i, where the hot phase mass is significant at <0.1R_vir and dominant at >0.1R_vir. The zoomed image also shows that m11d lacks the clear sharp disk seen in m12i. We conclude that m12i and m11d, which at z = 0 are, respectively, above and below the threshold for virialization of the inner CGM, differ mainly in the temperature distribution at inner halo and galaxy radii. This conclusion reinforces the conclusion from Figures 2–3 based on the 1D temperature profiles.

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Figures 6–7 repeat the above analysis for simulations in which ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff at high redshift. Figure 6 shows the temperature, normalized temperature, and supersonic fraction of m13A8 for different snapshots in the redshift range 1.1 < z < 3.4, at which ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ increases from 0.1 to 10. We plot a wider range of r/R_vir in this plot than in Figures 2–3 since the corresponding galaxy sizes are a smaller fraction of R_vir (see below). Figure 6 demonstrates that the trends versus ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ in m13A8 are similar to the trends in m12i (Figure 2). The value of 〈T〉/T_vir tends to reach its final value first at large CGM radii and later at small CGM radii and at galaxy radii. Similarly, when ${t}_{\mathrm{cool}}^{({\rm{s}})}\lesssim {t}_{\mathrm{ff}}$ the supersonic fraction is high at the inner CGM and at galaxy radii while it is low at the outer CGM, i.e., the halo gas is transonic. When ${t}_{\mathrm{cool}}^{({\rm{s}})}\gg {t}_{\mathrm{ff}}$, the supersonic fraction is low at all radii within R_vir. These trends again indicate that the outer CGM becomes steadily hot and subsonic prior to the inner CGM, i.e., the CGM virializes from the outside inward.

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Figure 7 shows temperature maps and 2D temperature histograms of the z = 2.5 snapshots of m13A1 (top) and m13A8 (bottom). This redshift is chosen since it is after the inner CGM virializes in m13A1 but before it virializes in m13A8. In m13A1 at radii larger than 0.05R_vir the halo gas is clearly separated into a hot phase with T ≳ 10⁶ K and cool streams with T ≲ 10^4.5 K, where the hot phase dominates by volume but the two phases are comparable in mass. Within 0.05R_vir the hot phase continues to dominate by volume, but the cool disk evident in the second panel dominates by mass. For comparison in m13A8, gas in the outer halo (>0.3R_vir) is similar to gas in the outer halo of m13A1, with a hot phase that dominates by volume and is comparable in mass to the cool streams. At inner halo and galaxy radii, however, cool gas fills most of the volume and there is no clear disk, similar to m11d and in stark contrast with gas in the inner halos of m13A1 and m12i. Figures 6–7 thus suggest that the behavior of the volume filling phase with respect to ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ at high redshift is similar to its behavior at low redshift (Figures 2–5). This suggests that the process of inner CGM virialization does not strongly depend on redshift or on the existence of cool streams.

One may wonder how the existence of cool streams does not significantly affect our calculation of ${t}_{\mathrm{cool}}^{({\rm{s}})}$, for which we assume that the CGM mass is distributed spherically (Equation (12)). This follows since after virialization the mass in the cool T < 10⁵ K streams is typically ≲30% of the total CGM mass at 0.1R_vir–1R_vir. The implied change in the expected density of the virialized phase due to the existence of cool streams is thus small.

To depict the virialization of the inner CGM in a single simulation, the top row of Figure 8 plots temperature maps of the inner 0.2R_vir in different snapshots of m12i, as in the second panel of Figure 4. From left to right the redshifts are z = 1, 0.4, 0.1, and 0, while ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ is 0.25, 1, 4, and 16. As suggested by the comparison above of different halos at the same redshift, in snapshots with ${t}_{\mathrm{cool}}^{({\rm{s}})}\lesssim {t}_{\mathrm{ff}}$ a large fraction of the volume is filled with cool gas, and there is no prominent disk. In contrast, in snapshots with ${t}_{\mathrm{cool}}^{({\rm{s}})}\gg {t}_{\mathrm{ff}}$ the inner halo volume is filled with hot gas, and a prominent cool disk is apparent.

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The bottom row of Figure 8 plots spatial pressure fluctuations in the same snapshots of m12i shown in the top row. Color indicates the pressure in each pixel relative to the average pressure 〈P(r)〉 in a shell of the same distance as the pixel. The calculation of 〈P(r)〉 is similar to that of the average temperature in Equation (20):

$$\begin{eqnarray}&&\mathrm{log} \langle P(r) \rangle \equiv \frac{\sum {V}_{i}\mathrm{log}{P}_{i}}{\sum {V}_{i}},\end{eqnarray} \tag{ 21 }$$

where P_i is the thermal pressure of resolution element i and the summation is over all resolution elements whose center is within a shell with thickness ${\rm{\Delta }}\mathrm{log}r=0.05\,\mathrm{dex}$. Normalizing the pressure by the mean removes the radial pressure gradient and allows focusing on the differences between different solid angles. The figure shows that in the left panels before the inner CGM virializes different directions differ substantially in thermal pressure, with overpressurized angles having a factor of up to ∼100 higher pressure than underpressurized angles. In contrast, in the right panels after virialization fluctuations are more mild, and the pressure distribution is closer to being spherically symmetric.

Figures 9 and 10 demonstrate the dependence of inner CGM virialization on ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ in all 16 FIRE simulations. Figure 9 plots 〈T〉/T_vir at 0.1R_vir against ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ at the same radius, while Figure 10 plots the supersonic fraction against ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ at 0.1R_vir. Each gray marker corresponds to a single snapshot, while the plots include all snapshots with ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}\gt 0.01$ from all 16 simulations. Since ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ increases with time in individual simulations (see bottom right panel of Figure 1), the tracks of individual simulations proceed from left to right in these plots. The colored lines show the median value versus ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ for each of the three simulation subgroups, derived using a Gaussian kernel density estimator.

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Figure 9 shows that the typical 〈T〉 is ≪T_vir when ${t}_{\mathrm{cool}}^{({\rm{s}})}\lesssim {t}_{\mathrm{ff}}$, with occasional snapshots with 〈T〉 > T_vir. In contrast, when ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff, the median temperature increases to ≳T_vir and the scatter at a given ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ substantially decreases. This transition is paralleled by a sharp transition in the supersonic fraction seen in Figure 10, where the median supersonic fraction is ≈0.6 when ${t}_{\mathrm{cool}}^{({\rm{s}})}\lesssim {t}_{\mathrm{ff}}$, i.e., gas in the inner halo is predominantly supersonic and drops abruptly when ${t}_{\mathrm{cool}}^{({\rm{s}})}\gtrsim {t}_{\mathrm{ff}}$, i.e., the inner CGM becomes predominantly subsonic. These plots therefore demonstrate that in the majority of snapshots with ${t}_{\mathrm{cool}}^{({\rm{s}})}\ll {t}_{\mathrm{ff}}$, most of the volume in the inner CGM is occupied by subvirial gas that is flowing supersonically. In contrast, when ${t}_{\mathrm{cool}}^{({\rm{s}})}\gg {t}_{\mathrm{ff}}$, the bulk of the inner CGM is constantly occupied by virial-temperature gas that is flowing subsonically. This transition is the virialization of the inner CGM.

Figures 9–10 suggest that the inner CGM virializes when ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ is in the range of 1–4, indicating that the factor ${f}_{t}^{\prime}$ defined in Equation (19) is larger than unity. We choose ${f}_{t}^{\prime} =2$ as an intermediate value, and in Table 2 we list for each simulation several properties of the galaxy and halo at the redshift where ${t}_{\mathrm{cool}}^{({\rm{s}})}=2{t}_{\mathrm{ff}}$.

Table 2. Properties of the Galaxy and Halo at the Redshift Where ${t}_{\mathrm{cool}}^{({\rm{s}})}=2{t}_{\mathrm{ff}}$ at 0.1R_vir and the Inner CGM Virializes

Name	z	Mhalo	vc	${f}_{{v}_{{\rm{c}}}}$	Z	ρ(s)
		(1012 M⊙)	(km s−1)		( Z⊙)	(Δc ρcrit)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
m12's
m12z	⋯	⋯	⋯	⋯	⋯	⋯
m12i	0.32	0.9	190	1.4	1.2	0.8
m12b	0.70	0.8	210	1.4	1.0	0.7
m12m	0.44	1.2	240	1.6	2.3	1.1
m12f	0.26	1.5	220	1.4	1.2	1.5
m13's
m13A1	3.6	1.4	320	1.2	1.0	1.2
m13A4	2.2	2.4	300	1.0	0.8	1.4
m13A2	2.7	2.4	390	1.3	1.8	1.5
m13A8	1.7	1.6	310	1.3	0.9	3.4
z5m13a	4.9	2.9	530	1.3	1.4	1.5

Note. Column (1): galaxy name. Column (2): redshift. Column (3): halo mass. Column (4): circular velocity at 0.1R_vir. Column (5): ratio of circular velocity at 0.1R_vir to virial velocity. Column (6): metallicity at 0.1R_vir. Column (7): gas density at 0.1R_vir divided by the mean halo density (Δ_c is the Bryan & Norman 1998 virial overdensity, and ρ_crit is the total critical density of the universe).

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The median ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ of the m11 simulations (purple lines in Figures 9–10) does not extend much beyond unity, since this is the maximum ratio reached by these relatively low mass halos (see Figure 1). However, the medians of the different subgroups show similar relations of 〈T〉/T_vir and supersonic fraction versus ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$. This reinforces the conclusion above that the virialization of the inner CGM is mainly a result of the change in ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$, rather than of other parameters of the system that differ between the simulation subgroups, such as the relation between halo mass and redshift.

Figure 11 compares the ratio ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ estimated at 0.1R_vir with the same ratio estimated ¹⁹ at 0.5R_vir. The figure shows that ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ is almost always larger at 0.5R_vir than at 0.1R_vir, typically by a factor of ∼30 when ${t}_{\mathrm{cool}}^{({\rm{s}})}\ll {t}_{\mathrm{ff}}$ at 0.1R_vir and by a factor of ≈3 when ${t}_{\mathrm{cool}}^{({\rm{s}})}\sim 10{t}_{\mathrm{ff}}$ at 0.1R_vir. The median relation between the vertical and horizontal axes is rather similar in the m11, m12, and m13 subgroups. This result is consistent with the expectation of steady-state inflow solutions that ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ increases outward (Papers I and II). Furthermore, since ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ generally increases with time, this result implies that ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff first in the outer halo and then in the inner halo.

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In Figure 12 we compare the supersonic fractions at 0.1R_vir and 0.5R_vir. Gray markers denote individual snapshots in all 16 simulations, while the lines and arrows plot the tracks of three simulations, one from each simulation subgroup. The tracks are calculated using median values in 1 Gyr time windows for m11d and m12i and using 500 Myr windows for m13A1. In m12i the supersonic fraction decreases first in the outer CGM and then in the inner CGM. In m11d the supersonic fraction decreases in the outer CGM but remains high (≈65%) down to z = 0 in the inner CGM. In m13A1 the supersonic fraction is always below 50% in the outer CGM and decreases with time in the inner CGM. All three tracks, and the small number of snapshots in the upper left quadrant, suggest that the volume filling phase becomes predominantly subsonic first in the outer CGM and then in the inner CGM, again indicating that the CGM virializes from the outside inward.

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To explore the formation of the gaseous disk, we calculate the specific angular momentum profile of gas at galaxy and CGM radii. As above, we divide the gas in each snapshot into radial shells and calculate

$$\begin{eqnarray}&&\langle \vec{j}\rangle =\displaystyle \frac{\sum {m}_{i}\left({{\boldsymbol{r}}}_{i}\times {{\boldsymbol{v}}}_{i}\right)}{\sum {m}_{i}},\end{eqnarray} \tag{ 22 }$$

where m_i , r _i , and v _i are the mass, position, and velocity of resolution element i, respectively. We then project the total angular momentum onto the axis of rotation z,

$$\begin{eqnarray} &&\langle {j}_{z} \rangle = \langle {\boldsymbol{j}} \rangle \cdot \hat{z},\end{eqnarray} \tag{ 23 }$$

where $\hat{z}$ is defined as the direction of the total angular momentum vector of all gas resolution elements within 0.05R_vir in the snapshot. The rotational velocity is hence

$$\begin{eqnarray} &&\langle {V}_{\mathrm{rot}}(r) \rangle =\frac{\sum {m}_{i}\left({{\boldsymbol{v}}}_{i}\cdot \hat{\phi }\right)}{\sum {m}_{i}},\end{eqnarray} \tag{ 24 }$$

where $\hat{\phi }$ is the corresponding azimuthal coordinate. The summations in Equations (22) and (24) are over all resolution elements within a shell centered at r and with thickness ${\rm{\Delta }}\mathrm{log}r=0.05\,\mathrm{dex}$.

The top panel of Figure 13 shows the specific angular momentum profiles 〈j_z 〉 in the m12i simulation. As above, we plot the median profiles of snapshots in Δt = 0.5 Gyr windows and use line color to denote the median ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ at 0.1R_vir. For each time window we also mark the specific angular momentum corresponding to circular orbits v_c r using thin dashed lines with the same color. The middle panel shows 〈V_rot〉/v_c, while the lines span the redshift range at which $-1.5\lt \mathrm{log}{t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}\lt 1.5$. Similar plots for the other simulations are available online.

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View figure set (15 panels)

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The top two panels in Figure 13 demonstrate a clear trend with increasing ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$. In snapshots with ${t}_{\mathrm{cool}}^{({\rm{s}})}\lesssim {t}_{\mathrm{ff}}$ (before the inner CGM virializes; blue-gray lines) there is no range of radii where V_rot follows v_c. In contrast, in snapshots with ${t}_{\mathrm{cool}}^{({\rm{s}})}\gg {t}_{\mathrm{ff}}$ there is an extended region where V_rot ≈ v_c and the gas is on circular orbits. The range of radii where V_rot ≈ v_c corresponds to the range of radii where the gas is predominantly cool by mass (see right panel of Figure 4). These panels thus indicate that a rotating disk forms once ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff and the inner CGM virializes. This association of a rotating thin disk with virialization is also suggested by the images shown in Figures 4, 5, 7, and 8 and is apparent also in the other simulations (see online figures).

The top panels of Figure 13 also show that in snapshots after a disk forms the gas transitions from circular orbits at r < 0.06R_vir to a roughly flat angular momentum profile at 0.06R_vir < r < R_vir. This radius corresponds to R_circ defined in Equation (2), the radius where halo gas can be supported by angular momentum. In snapshots after the inner CGM virialized we find a trend of decreasing R_circ/R_vir and disk-to-halo size ratio with increasing redshift, with R_circ ≈ 0.05R_vir for the m12 subgroup and R_circ ≈ 0.02R_vir for the m13 subgroup. We defer exploring the origin of this trend to future work.

The bottom panel of Figure 13 plots the profile of 〈V_rot〉/σ_g in the same time windows as in the top panels. The ratio 〈V_rot〉/σ_g has been highlighted by recent observations as a measure of "disk settling" (e.g., Kassin et al. 2012; Simons et al. 2017). We calculate σ_g following El-Badry et al. (2018b):

$$\begin{eqnarray}&&{\sigma }_{{\rm{g}}}=\sqrt{\langle {V}_{\mathrm{rot}}^{2}\rangle -\langle {V}_{\mathrm{rot}}{\rangle }^{2}},\end{eqnarray} \tag{ 25 }$$

i.e., σ_g equals the dispersion in the rotational velocity. El-Badry et al. demonstrated that σ_g defined in this way is similar to the velocity dispersion measured in a mock slit aligned along the major axis of the simulated galaxy (see their Figure 2). Figure 13 shows that 〈V_rot〉/σ_g increases rapidly at galaxy radii (r ≲ 0.1R_vir) as ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff, from a value of ≈1 when ${t}_{\mathrm{cool}}^{({\rm{s}})}\ll {t}_{\mathrm{ff}}$ to a value of up to ≈8 when ${t}_{\mathrm{cool}}^{({\rm{s}})}\gg {t}_{\mathrm{ff}}$. This result also indicates that inner CGM virialization coincides with the formation of a rotation-dominated galactic disk.

The calculations of 〈V_rot〉 and σ_g shown in Figure 13 are weighted by gas mass, regardless of whether it is ionized or neutral. This allows us to understand the properties of the angular momentum profile independent of phase changes in the gas. To also explore the gas kinematics integrated over the cool SF disk, we recalculate $\hat{z}$, 〈V_rot〉, and σ_g using Equations (22)–(25), but weighting the resolution elements by their H i mass or by their SFR rather than by the total gas mass. Figure 14 plots H i-weighted quantities versus ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$, while SFR-weighted quantities (not shown) exhibit similar trends. The panels in Figure 14 show H i mass-weighted 〈v_c〉 (top left), 〈V_rot〉 (top right), σ_g (bottom left), and 〈V_rot〉/σ_g (bottom right). As above, gray markers represent all individual snapshots in the 16 FIRE simulations, while colored lines show the medians for the three simulation subgroups derived using a Gaussian kernel density estimator. For comparison, we also plot the median v_c of each subgroup in the 〈V_rot〉 and σ_g panels.

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The top right panel of Figure 14 shows how the median V_rot transitions from ≈0.5v_c when ${t}_{\mathrm{cool}}^{({\rm{s}})}\ll {t}_{\mathrm{ff}}$ to ≈v_c when ${t}_{\mathrm{cool}}^{({\rm{s}})}\gg {t}_{\mathrm{ff}}$, similar to the transition seen in the mass-weighted 〈V_rot〉 at small radii (middle panel of Figure 13). Also, the bottom left panel shows that σ_g decreases relative to v_c when ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff, especially in the m12 subgroup. These two trends combine to a sharp transition in 〈V_rot〉/σ_g when ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff. At small ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ the ratio 〈V_rot〉/σ_g is close to unity with a scatter of ≈0.5 in all subgroups, indicating dispersion-dominated kinematics. In contrast, at ${t}_{\mathrm{cool}}^{({\rm{s}})}\sim {t}_{\mathrm{ff}}$ the ratio 〈V_rot〉/σ_g starts to increase, reaching ≈6 in the m12 subgroup and ≈3 in the m13 subgroup when ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}\approx 10$, indicating rotation-dominated kinematics. The m11s do not reach values of ${t}_{\mathrm{cool}}^{({\rm{s}})}$ significantly larger than t_ff (see Figure 1). This plot thus again shows that the formation of a rotation-dominated disk indicated by 〈V_rot〉/σ_g larger than unity is strongly linked to the virialization of the inner CGM indicated by ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ larger than unity at 0.1R_vir.

In this subsection we discuss how the characteristics of the SFR change with the virialization of the inner CGM. To focus on star formation within the central galaxy, we include only stars formed at r < 0.1R_vir. We refer to the average SFR in 10 Myr windows as the "instant" SFR and to the average SFR in 300 Myr windows as the "mean." These two quantities are plotted in the top row of Figure 15 for the m12i, m12b, and m13A1 simulations. All three simulations show a transition from large fluctuations in the instant SFR at early times to small fluctuations at late times. This transition has been previously identified for the m12 subgroup in Muratov et al. (2015; see also Sparre et al. 2017; Faucher-Giguère 2018) and for the m13 subgroup in Anglés-Alcázar et al. (2017b). The transition, though, occurs at substantially different redshifts in the different simulations, at z < 1 in m12i and m12b compared to at z ≈ 3.5 in m13A1. For comparison, red lines in the bottom row of Figure 15 plot ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ at 0.1R_vir. The transitions from "bursty" to "steady" SFR roughly coincide with where ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds ≈2t_ff.

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Figure 16 plots the dispersion in $\mathrm{log}$ SFR in 300 Myr windows against ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ in all 16 simulations in the sample. Each gray marker corresponds to a single snapshot, with ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ measured at this snapshot and the 300 Myr window centered on the snapshot time. The figure shows that the dispersion in $\mathrm{log}$ SFR tends to be large when ${t}_{\mathrm{cool}}^{({\rm{s}})}\lesssim {t}_{\mathrm{ff}}$, typically 0.2–0.6 dex (a factor of 1.5–4), while it tends to be small when ${t}_{\mathrm{cool}}^{({\rm{s}})}\gg {t}_{\mathrm{ff}}$, typically ≲0.1 dex (≲15%). This result also suggests a physical connection between the burstiness of the SFR and the virialization of the inner CGM.

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The middle row of Figure 15 plots the net mass flow rate $\dot{M}$ versus redshift in the m12i, m12b, and m13A1 simulations. The value of $\dot{M}$ is calculated via

$$\begin{eqnarray}&&\dot{M}(r)=\displaystyle \frac{\sum {m}_{i}{v}_{r,i}}{{\rm{\Delta }}r},\end{eqnarray} \tag{ 26 }$$

where v_r,i is radial velocity of resolution element i and the summation is over all resolution elements whose center is within a shell of thickness ${\rm{\Delta }}\mathrm{log}r=0.05\,\mathrm{dex}$ and center at r = 0.1R_vir. All three simulations show two distinct phases, as previously identified for the m12 subgroup in Muratov et al. (2015) and for the m13 subgroup in Anglés-Alcázar et al. (2017b). At early times the flow shows episodes with strong outflow bursts that exceed the inflow rate (positive net $\dot{M}$). At late times the flow is rather steady and almost always has a net inflow. Note that the qualitative behavior across the transition is independent of the redshift at which the transition occurs. The bottom row demonstrates that the disappearance of outflow bursts roughly coincides with when ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds 2t_ff and the inner CGM virializes. This suggests a physical connection between the properties of stellar-driven galactic outflows and the virialization of the inner CGM.

We find that the inner CGM virializes after the outer CGM is virialized, i.e., with time the radius where ${t}_{\mathrm{cool}}^{({\rm{s}})}\approx {t}_{\mathrm{ff}}$ moves inward with respect to R_vir. ²⁰ This outside-in virialization scenario is opposite to the direction of virialization in the 1D simulations of Birnboim & Dekel (2003) and Dekel & Birnboim (2006), which account for radiative cooling and angular momentum but neglect feedback from the galaxy. In their simulations IGM inflows are initially cool, free-falling, and supersonic down to the disk radius R_circ. When the halo mass exceeds a threshold of ∼10^11.5 M_⊙, a shock forms at R_circ and moves outward, i.e., the postshock subsonic phase forms first in the inner halo and then expands into the outer halo. In contrast, in FIRE the CGM becomes predominantly subsonic near the disk radius after it is already subsonic at larger radii (Figures 2, 6, and 12).

An outside-in CGM virialization scenario is consistent with expectations from spherical steady-state CGM solutions without ongoing heating by feedback (Papers I and II). In such solutions ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ increases with radius and with halo mass. At radii where ${t}_{\mathrm{cool}}^{({\rm{s}})}\gt {t}_{\mathrm{ff}}$ the gas is hot and subsonic since radiative cooling is balanced by compressional heating of the inflow, while at radii where ${t}_{\mathrm{cool}}^{({\rm{s}})}\lt {t}_{\mathrm{ff}}$ radiative cooling is rapid so the gas is cool and supersonic. These solutions thus indicate that the hot phase can be long-lived (i.e., reach steady-state) only at radii where ${t}_{\mathrm{cool}}^{({\rm{s}})}\gt {t}_{\mathrm{ff}}$. If we apply these conclusions to a time-dependent scenario where the halo mass is growing and shocks that seed the hot phase are prevalent, then we expect a long-lived hot phase to form first at large radii and later at small radii. These expectations based on steady-state solutions appear to hold in FIRE: a long-lived hot CGM phase exists only when ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff (Figures 9–10), ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ increases outward (Figure 11), and the fraction of gas moving supersonically drops first in the outer CGM and then in the inner CGM (Figure 12).

We emphasize that hot subsonic gas often exists in the inner CGM also prior to virialization, in contrast with the steady-state solutions. However, the volume filling fraction of this hot gas fluctuates rapidly, with typical values less than 50% (Figures 5 and 9–10). The rapid fluctuations prior to virialization are likely due to cycles of rapid cooling followed by heating by outflow bursts (see van de Voort et al. 2016). Hot gas powered by outflows will also propagate outward, in contrast with the inward direction in which the CGM virializes. Thus, the FIRE simulations appear to be consistent with the steady-state solutions in the sense that when ${t}_{\mathrm{cool}}^{({\rm{s}})}\gt {t}_{\mathrm{ff}}$ a time-steady hot phase forms with volume filling fraction approaching unity (i.e., the CGM "virializes"). The FIRE simulations, though, exhibit also a transient hot phase prior to virialization that is absent from the steady-state solutions.

One may wonder why the outside-in scenario suggested by cooling flow solutions is not realized in the simulations in Birnboim & Dekel. We emphasize that while steady-state solutions imply that a subsonic hot phase would be long-lived at radii where ${t}_{\mathrm{cool}}^{({\rm{s}})}\gt {t}_{\mathrm{ff}}$, an initial shock is still required to seed a subsonic hot phase at these radii. At large radii such initial shocks could be a result of the interaction of inflows with outflows as seen in the idealized simulations of Fielding et al. (2017), or as a result of merger events (e.g., Shi et al. 2020). As these mechanisms are absent from the 1D simulations of Birnboim & Dekel, a subsonic hot phase may not form even if the conditions for it to be long-lived are satisfied.

We note that while our results indicate that the CGM virializes from the outside inward, we do see evidence for some subtler changes in the properties of the outer CGM following the virialization of the inner CGM. This includes the decrease in temperature beyond R_vir when ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff at 0.1R_vir and the associated appearance of an accretion shock (Figures 2–3). Also, the volume filling fraction of supersonic gas in the outer halo appears to decrease from ≈0.3 to ≈0.1 when the inner CGM virializes (Figures 2 and 12). Potentially, these effects are due to the stifling of outflows when the inner CGM virializes (Figure 15), which affects the physical conditions also in the outer halo. We leave exploring this effect to a future study.

Outside-in virialization has several implications for the X-ray and SZ (Sunyaev & Zeldovich 1970) signals from the halo. As mentioned in the introduction, van de Voort et al. (2016) identified in FIRE a transition from highly variable X-ray emission at halo masses below M_thresh ≈ 10¹² M_⊙ to time-steady X-ray emission above M_thresh, a transition that they associated with CGM virialization. Our results add to their conclusions by demonstrating that it is the inner CGM that virializes at this mass scale in FIRE. Indeed, the X-ray emission is dominated by the densest gas and is thus most sensitive to the physical conditions in the inner CGM. In contrast, the SZ signal is roughly weighted by mass and hence dominated by gas in the outer halo, so a drop in the SZ signal is expected only at lower halo masses ≲10¹¹ M_⊙ where the outer halo has not yet formed (the median halo mass at which ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff at 0.5R_vir in our simulations is 0.8 × 10¹¹ M_⊙). We note that additional effects such as gas depletion in low-mass halos (van de Voort et al. 2016; Oppenheimer et al. 2020) may also affect the SZ and X-ray signals and thus complicate the interpretation. We postpone a more quantitative analysis of the implications of our results for these observational signatures to future work.

Table 2 lists for each simulation the halo mass and other physical properties in the snapshot where ${t}_{\mathrm{cool}}^{({\rm{s}})}$ equals 2t_ff and the inner CGM virializes. Defining the halo mass when ${t}_{\mathrm{cool}}^{({\rm{s}})}=2{t}_{\mathrm{ff}}$ as M_thresh, we find M_thresh ≈ (0.8–1.5) × 10¹² M_⊙ in the m12 simulations, which virialize at redshift z < 1, and a somewhat higher M_thresh ≈ (1.4–2.9) × 10¹² M_⊙ in the m13 simulations, which virialize at higher redshift. The somewhat lower M_thresh for halos that virialize at low redshift are mainly driven by the lower gas density relative to the cosmic mean and the higher ${f}_{{v}_{{\rm{c}}}}$ factor (see Equation (5)), where the latter is a result of the higher concentration of low-redshift halos. This can be seen by comparing the halo and gas properties of the m12 and m13 simulations at virialization (Table 2) with the analytic calculation of ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ (Equation (18); note ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}\propto {f}_{{v}_{{\rm{c}}}}^{4.4}$).

Although we deduce a weak dependence of M_thresh on redshift, we find that in halo masses of 10^10.5–10^11.5 M_⊙ at z ∼ 0 the ratio ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ can be comparable to unity, in contrast with ${t}_{\mathrm{cool}}^{({\rm{s}})}\ll {t}_{\mathrm{ff}}$ at the same halo masses at high redshift. This trend is shown in Figure 17, in which we plot ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ as a function of halo mass for different redshifts. This difference is again due to the lower gas mass and higher concentration of low-redshift halos compared to their high-redshift counterparts. The result that ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ can be close to unity in ≪10¹² M_⊙ halos at z ∼ 0 implies that the properties of their CGM and central galaxies strongly depend on the mass and metallicity of the CGM, which in turn depend on the integrated enrichment and depletion of the CGM by outflows over cosmic time. Halo-to-halo variance in these quantities, or if these quantities are somewhat overpredicted by FIRE, could drive ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ above unity in ∼10¹¹ M_⊙ halos at z ∼ 0, allowing them to develop virialized inner CGM and rotation-dominated disks. Our results thus allow for the possibility that M_thresh decreases to ∼10¹¹ M_⊙ or less at late cosmic times.

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The possibility that M_thresh(z ∼ 0) ∼ 10¹¹ M_⊙ could explain the tension between FIRE and observations of low-redshift SF galaxies discussed in El-Badry et al. (2018a, 2018b). They showed that in FIRE only above a stellar mass of ∼10¹⁰ M_⊙ do galaxies predominantly form rotation-dominated disks, in contrast with observations, which find rotation-dominated disks above ∼10⁹ M_⊙. If, as suggested above, the ratio ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ is somewhat above unity at M_halo ∼ 10¹¹ M_⊙ rather than comparable to unity as in FIRE, then we would expect also V_rot ≫ σ_g (Figure 14) at the corresponding stellar mass of ∼10⁹ M_⊙, which would be more consistent with observations. Similarly, ${t}_{\mathrm{cool}}^{({\rm{s}})}\gt {t}_{\mathrm{ff}}$ at M_halo ∼ 10¹¹ M_⊙ would also suggest a steady rather than bursty SFR at stellar masses ≳10⁹ M_⊙ (Figure 16), which could explain the apparent overprediction of SFR burstiness in FIRE at this mass scale (Sparre et al. 2017; Emami et al. 2019).

It is worth noting also that when ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff and the inner CGM virializes the metallicity at 0.1R_vir stops increasing and in some cases starts to drop with time (middle left panel of Figure 1). This is likely due to the suppression of outflows when the inner CGM virializes (Figure 15), so the CGM becomes dominated by fresh accretion and previously ejected winds, both of which tend to have a lower metallicity than new winds (see Hafen et al. 2019 for the metallicities of different components). This causes the CGM and interstellar medium (ISM) metallicities to diverge, a phenomenon previously identified by Muratov et al. (2017), and our results suggest that this divergence is associated with the virialization of the inner CGM. Also, the drop in CGM metallicity causes ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ to further increase once it crosses the threshold for virialization.

The virialization of the inner CGM is expected to both confine galaxy outflows and change the nature of inflows onto the galaxy. In this subsection we discuss these transformations and how they might cause the associated transitions in galaxy properties shown in Figures 13–16.

The virialized inner CGM extends down to a few density scale heights above the galaxy midplane (see Figure 4 and top row of Figure 7) and provides a homogeneous confining medium for the gaseous galactic disk. The homogeneity of a virialized CGM is evident from the relatively small pressure fluctuations shown in the bottom right panel of Figure 8, in contrast with the large fluctuations prior to virialization when the flow is supersonic and dynamic pressure dominates (bottom left panels of Figure 8). A homogeneous medium is more effective in confining outflows since the outflows cannot expand through paths of least resistance. For example, in the z = 1 snapshot of m12i at which ${t}_{\mathrm{cool}}^{({\rm{s}})}=0.25{t}_{\mathrm{ff}}$ (see Figure 8), half the volume at ∼0.1R_vir is filled with gas with a pressure equal to 15% or less of the mean (volume-averaged) pressure, while 1/10 of the volume is filled with gas with a pressure of 2% or less than the mean pressure. Similar volume filling fractions are found for regions with a low density compared to the mean value. The required outflow ram pressure to expand through such underpressurized and underdense regions is lower, by similar factors, than the required ram pressure to expand in a homogeneous medium with the same mean pressure.

How does confinement affect the physical conditions in the galaxy? Hydrodynamic simulations and analytic considerations suggest that without confinement, a "superbubble" produced by SNe of a single stellar cluster can break out of the disk and drive a strong outflow (e.g., Kim et al. 2017; Fielding et al. 2018), which significantly perturbs the disk vertical structure (Martizzi 2020). The disappearance of outflow bursts when the inner CGM virializes (Figure 15) may suggest that the homogeneous virialized CGM confines such outflows. This can be understood by comparing the CGM thermal pressure with the expected pressure in SN-powered superbubbles. Assume a CGM characteristic of the Milky Way with pressure P_h/k ∼ 1000 cm⁻³ K (estimated from the pressure of high-velocity clouds; Dedes & Kalberla 2010) at the disk scale of R_d ≈ R_circ ≈ 10 kpc. The work required to lift the virialized CGM beyond the scale of the disk is ∼$(4\pi /3){R}_{{\rm{d}}}^{3}{P}_{{\rm{h}}}\sim 2\cdot {10}^{55}\,\mathrm{erg}$. For comparison, the formation of a stellar cluster with mass M_cl = 10⁵ M₅ M_⊙ causes ∼1000M₅ core-collapse SNe to explode and inject an energy of E ∼ 10⁵⁴ M₅ erg into their surroundings. A virialized CGM similar to that of the Milky Way is thus capable of confining a superbubble produced by a stellar cluster as massive as ∼2 × 10⁶ M_⊙, or more if cooling losses in the bubble are substantial. Since ${P}_{{\rm{h}}}{R}_{{\rm{d}}}^{3}$ is roughly a constant fraction of the halo gas gravitational binding energy (assuming R_d ∝ R_vir), which scales as ${M}_{\mathrm{halo}}^{2}/{R}_{\mathrm{vir}}$, confinement is even more effective at higher redshift, where R_vir is smaller. A similar conclusion that the CGM of Milky Way–mass galaxies can confine stellar-driven galactic outflows was reached by Hopkins et al. (2021a), who implicitly assumed that the CGM was virialized (see Section 3.3.1 there).

If superbubbles produced by individual stellar clusters are confined, then multiple superbubbles produced by different stellar clusters would build up pressure in the ISM until it equals that of the inner CGM. Any additional injected feedback energy not lost to radiation would then be channeled to an outflow, e.g., via buoyant bubbles. ²¹ We thus suspect that inner CGM virialization fundamentally changes how feedback energy is distributed. Prior to virialization the energy injected by individual superbubbles vents relatively easily through the porous inner CGM, causing large spatial and temporal pressure fluctuations in the galaxy disk and inner halo. In contrast, after virialization the feedback energy is confined and will regulate the pressure in the ISM. That is, we suspect that the classic ansatz that SNe keep the characteristic ISM pressures uniform (McKee & Ostriker 1977) applies only after the inner CGM virializes. This transition could explain the sharp increase in V_rot/σ_g when the inner CGM virializes (Figure 14) since the large pressure gradients prior to virialization would induce nonrotational motions and thus enhance σ_g, while after virialization these pressure gradients would drop and σ_g would decrease. Also, the transition to pressure balance can enable galaxies to realize a Kennicutt-Schmidt-type star formation relation as in equilibrium ISM models (Thompson et al. 2005; Ostriker & Shetty 2011; Faucher-Giguère et al. 2013; see Gurvich et al. 2020 for a test of these models in FIRE), which could explain the transition to a steady SFR (Figure 16).

A similar argument for confinement of outflows applies to the outer CGM, during the "transonic" phase of halo gas in which the outer halo is smooth and subsonic while the inner halo is clumpy and supersonic. During this phase, outflows would potentially halt at the "sonic radius" which separates the two layers. It would thus be interesting to compare the sonic radius in simulations (roughly the radius where ${t}_{\mathrm{cool}}^{({\rm{s}})}(r)={t}_{\mathrm{ff}}(r)$) to the typical maximum radius reached by outflows, known as the "recycling radius" (Oppenheimer et al. 2010; Ford et al. 2014; Anglés-Alcázar et al. 2017a; Muratov et al. 2017; Hafen et al. 2019).

We note that a prerequisite for outflows to be confined by the inner CGM is that they manage to break out of the disk, which depends on stellar cluster mass (M_cl ≳ 10⁵ M_⊙ for the fiducial parameters in Fielding et al. 2018). Thus, an alternative possibility for the suppression of outflows following inner CGM virialization is that M_cl decreases below the threshold for breakout. This could be induced by the drop in scale height h when the disk settles, if M_cl scales with the Toomre mass, which in turn scales as h².

Inner CGM virialization initiates the "hot accretion mode," in which the inflow is subsonic (i.e., pressure supported) from the accretion shock down to the galaxy. Prior to inner CGM virialization, the accretion flow resembles the previously discussed "cold accretion mode" (e.g., Kereš et al. 2005, 2009; Dekel et al. 2009a), in the sense that the gas accretes supersonically onto the galaxy. However, our results suggest that prior to inner CGM virialization the thermal history of accreted gas can differ from what has often been envisioned for cold mode accretion. In intermediate-mass (∼10¹¹–10¹² M_⊙) halos in FIRE, the outer CGM is virialized, corresponding to a hot inflow at large radii followed by a cool flow at small CGM radii. In contrast, it is often assumed that in cold mode accretion the gas remains cool throughout the CGM. This difference suggests that the maximum temperature reached by gas prior to accretion, which was the focus of many previous studies of the cold mode versus the hot mode in cosmological simulations (e.g., Nelson et al. 2013), is not in general an accurate indicator of gas properties upon accretion onto the galaxy.

The accretion of cold gas onto galaxies can persist after inner CGM virialization if cool streams penetrate the hot phase, as expected at high redshift when cosmic web filaments are narrow relative to ≳10¹² M_⊙ halos (e.g., Kereš et al. 2005; Dekel & Birnboim 2006), and as seen in high-redshift snapshots in FIRE (Figure 7). The relation between inner CGM virialization and the distribution of physical states of the gas as it accretes onto galaxies is thus redshift dependent. Cold accreting gas is expected to stir turbulence in the gaseous disk when the supersonic accretion flow shocks against the ISM (Dekel et al. 2009b), in contrast with hot mode accretion where the subsonic flow is expected to smoothly connect to the disk velocity field (see Figures 2 and 7 in Paper II; see also Cowie et al. 1980). This difference may explain the decrease in σ_g when ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff at low redshift and the cold mode disappears, which is not seen when ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff at high redshift but accretion via cold streams persists (compare the "m12" curve with the "m13" curve in the bottom panels of Figure 14). However, the transitions in galaxy properties highlighted in Sections 4.1–4.3 do not appear to depend on redshift for a fixed ${t}_{\mathrm{cool}}^{({\rm{s}})}/{t}_{\mathrm{ff}}$ and thus appear to be independent of the existence of cold filaments. This suggests that these transitions are more directly related to inner CGM virialization and the associated confinement of outflows discussed in Section 5.3, rather than to the relative fractions of cold and hot gas accreting onto the central galaxy.

Feedback from the central BH is believed to have a central role in the evolution of galaxies and forms the leading paradigm for quenching star formation in red-and-dead galaxies (e.g., Somerville & Davé 2015; Naab & Ostriker 2017). The FIRE simulations used in this work passively followed BH accretion assuming that it proceeds via gravitational torques in the circumnuclear material (see Anglés-Alcázar et al. 2017b) but neglected AGN feedback, which in principle could limit the applicability of our conclusions regarding CGM virialization. We argue here that, for a wide range of supermassive BH growth and feedback models, AGN feedback can likely be assumed to have only modest or negligible effects on our main results regarding virialization. This follows from the result of Anglés-Alcázar et al. (2017b) that BH growth experiences a transition coincident with the other galactic transitions discussed above. At early times when the SFR is bursty, BH accretion is weak and irregular, and the BH mass remains ≲10⁵ M_⊙. At later times when the SFR becomes steady, BH accretion becomes significantly stronger and more steady, and the BH mass increases to ∼0.1% of the galaxy mass. L. Byrne et al. (2021, in preparation) demonstrate that this transition to significant BH accretion also coincides with when ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff at 0.1R_vir. If, as suggested by the results just described, substantial BH growth commences only after the inner CGM virializes, then the energy released by BH feedback is likely small before and up to the virialization of the inner CGM. We thus do not expect BH feedback to significantly alter the conditions in the CGM at this early phase, and hence do not expect it to alter our conclusions regarding inner CGM virialization.

The properties of halo gas may also be modified by interaction with cosmic rays (CRs), which are not accounted for in the simulations used here, but which some studies have suggested could be an important source of feedback (e.g., Booth et al. 2013; Salem et al. 2016; Pfrommer et al. 2017). Explicit injection of CRs by SNe, CR transport, and CR–gas interactions has been implemented in a separate suite of FIRE simulations (Chan et al. 2019; Hopkins et al. 2020; Ji et al. 2020). Hopkins et al. (2020) demonstrated that in the m12 simulations nonthermal pressure gradients in the CR fluid can potentially support the halo gas against gravity at low redshift, thus effectively increasing the free-fall time and preventing the formation of a pressure-supported virialized CGM. This conclusion is, however, sensitive to the assumed CR transport model, which is uncertain especially in the CGM (Hopkins et al. 2021b). Thus, it is not yet clear whether and how CRs affect our conclusions concerning CGM virialization.

Our results differ from the prescriptions typically employed by semianalytic models (SAMs) of galaxy formation in three main ways. First, SAMs usually determine whether the inner CGM virializes by comparing the cooling time to the dynamical time at the virial radius (or, equivalently, they compare the cooling radius with R_vir; e.g., Somerville et al. 2008; Lu et al. 2011; Croton et al. 2016). Our results in contrast suggest that CGM virialization is complete and hence the nature of accretion onto the galaxy and confinement by the CGM change only when ${t}_{\mathrm{cool}}^{({\rm{s}})}$ exceeds t_ff at the galaxy scale ≈0.1R_vir (Figures 9–10), as expected from the steady-state solutions in Paper II. Second, SAMs typically assume that central galaxies form thin disks with a specific angular momentum comparable to that of their parent halo (e.g., Somerville et al. 2008). As far as we are aware, the thermal properties of halo gas do not play a role in shaping disks in SAMs, and so these models may be missing a critical ingredient that enables the formation of thin rotationally supported disks. It would thus be interesting to incorporate into SAMs the connection between the virialization of the inner CGM and the formation of disks found above (Figures 13–14). The predictions of such SAMs could then be compared to the observed demographics of blue irregulars with V_rot ∼ σ_g and blue thin disks with V_rot ≫ σ_g (e.g., Kassin et al. 2012; Karachentsev et al. 2013; Simons et al. 2017). Third, many SAMs assume that there is a direct connection between CGM virialization and SF quenching, which is implemented by "turning on" radio mode feedback from the BH when the condition for CGM virialization is satisfied, thus offsetting cooling and shutting the gas supply for further SF (Croton et al. 2006; Bower et al. 2006, 2012; Cattaneo et al. 2006, 2008; Somerville et al. 2008; Lu et al. 2011). Our results indicate that inner CGM virialization is instead more directly associated with disk settling, suggesting that the connection between virialization and quenching is not so direct. Potentially, if disk settling enables accelerated BH growth as discussed in the previous section, CGM virialization would remain associated with galaxy quenching by BH feedback, but there may be a nonnegligible delay between inner CGM virialization and the time needed for sufficient BH accretion energy to be released.