MNRAS
520,
722–739
(2023)
https://doi.org/10.1093/mnras/stad171
Advance
Access
publication
2023
January
16
Stellar
feedback-regulated
black
hole
gro
wth:
dri
ving
factors
from
nuclear
to
halo
scales
Lindsey
Byrne
,
1
‹
Claude-Andr
́
e
Faucher-Gigu
`
ere,
1
Jonathan
Stern
,
1
,
2
Daniel
Angl
́
es-Alc
́
azar,
3
,
4
Sarah
Wellons
,
1
,
5
Alexander
B.
Gurvich
1
and
Philip
F.
Hopkins
6
1
Department
of
Physics
and
Astronomy
and
CIERA,
Northwestern
University,
Evanston,
IL
60201,
USA
2
School
of
Physics
&
Astronomy,
Te l
Aviv
University,
Te l
Aviv
69978,
Israel
3
Department
of
Physics,
University
of
Connecticut,
196
Auditorium
Road,
U-3046,
Storrs,
CT
06269-3046,
USA
4
Center
for
Computational
Astrophysics,
Flatiron
Institute,
162
5th
Ave,
New
Yo r k ,
NY
10010,
USA
5
Department
of
Astronomy,
Va n
Vleck
Observatory,
We s l ey a n
University,
96
Fo s s
Hill
Drive,
Middletown,
CT
06459,
USA
6
TAPIR,
Mailcode
350-17,
California
Institute
of
Technology,
Pasadena,
CA
91125,
USA
Accepted
2023
January
11.
Received
2022
December
23;
in
original
form
2022
October
14
A
B
S
T
R
A
C
T
Several
recent
simulations
of
galaxy
formation
predict
two
main
phases
of
supermassive
black
hole
(BH)
accretion:
an
early,
highly
intermittent
phase
(during
which
BHs
are
undermassive
relative
to
local
scaling
relations),
followed
by
a
phase
of
accelerated
growth.
We
investigate
physical
factors
that
drive
the
transition
in
BH
accretion
in
cosmological
zoom-in
simulations
from
the
FIRE
project,
ranging
from
dwarf
galaxies
to
galaxies
sufficiently
massive
to
host
luminous
quasars.
The
simulations
model
multichannel
stellar
feedback,
but
neglect
AGN
feedback.
We
show
that
multiple
physical
properties,
including
halo
mass,
galaxy
stellar
mass,
and
depth
of
the
central
gravitational
potential
correlate
with
accelerated
BH
fuelling:
constant
thresholds
in
these
properties
are
typically
crossed
within
∼
0.1
Hubble
time
of
accelerated
BH
fuelling.
Black
hole
masses
increase
sharply
when
the
stellar
surface
density
in
the
inner
1
kpc
crosses
a
threshold
1
kpc
≈
10
9
.
5
M
kpc
−
2
,
a
characteristic
value
abo
v
e
which
gravity
prevents
stellar
feedback
from
ejecting
gas,
and
similar
to
the
value
abo
v
e
which
galaxies
are
observed
to
quench.
We
further
show
that
accelerated
BH
growth
correlates
with
the
emergence
of
long-lived
thin
gas
discs,
as
well
as
with
virialization
of
the
inner
circumgalactic
medium.
The
halo
mass
M
halo
∼
10
12
M
and
stellar
mass
M
∗
∼
10
10.5
M
at
which
BH
growth
accelerates
correspond
to
∼
L
galaxies.
The
fact
that
stellar
feedback
becomes
inefficient
at
ejecting
gas
from
the
nucleus
abo
v
e
this
mass
scale
may
play
an
important
role
in
explaining
why
AGN
feedback
appears
to
be
most
important
in
galaxies
abo
v
e
L
.
K
ey
words:
galaxies:
e
volution
– galaxies:
formation
– galaxies:
disc
– quasars:
supermassive
black
holes.
1
INTRODUCTION
Supermassive
black
holes
(BHs)
in
galactic
nuclei
co-evolve
with
their
host
galaxies
in
ways
which
may
significantly
affect
those
galaxies,
but
which
are
not
well
understood.
Indeed,
active
galactic
nucleus
(AGN)
feedback
is
a
core
element
of
current
galaxy
for-
mation
theories,
especially
at
the
massive
end
(e.g.
Somerville
&
Dav
́
e
2015
;
Naab
&
Ostriker
2017
).
Observations
have
linked
many
properties
of
supermassive
black
holes
to
properties
of
their
host
galaxies,
including
kiloparsec-scale
outflows
of
gas
from
galaxies
hosting
luminous
quasars
(e.g.
Feruglio
et
al.
2010
;
Rupke
&
Veilleux
2013
;
Cicone
et
al.
2014
;
Fiore
et
al.
2017
;
Fluetsch
et
al.
2019
),
and
scaling
relations
between
black
hole
mass
M
BH
and
the
stellar
properties
of
the
host
galaxy,
such
as
galaxy
mass,
bulge
mass,
or
velocity
dispersion
(e.g.
Magorrian
et
al.
1998
;
Ferrarese
&
Merritt
2000
;
Tremaine
et
al.
2002
;
Kormendy
&
Ho
2013a
;
Sahu,
Graham
&
Davis
2019
).
Moreo
v
er
,
A
GN
feedback
is
the
primary
suspected
driver
of
star
formation
quenching
in
massive
galaxies,
E-mail:
byrnelin@u.northwestern.edu
which
is
necessary
to
explain
the
blue/red
colour
bi-modality
(e.g.
Springel,
Di
Matteo
&
Hernquist
2005
;
Croton
et
al.
2006
;
Hopkins
et
al.
2008
).
AGN
feedback
can
operate
through
se
veral
dif
ferent
mechanisms,
including
kinetic
winds,
radiation,
and
powerful
radio
jets
(e.g.
Fabian
2012
).
In
this
paper,
we
use
cosmological
zoom-in
simulations
of
galaxy
formation
from
the
FIRE
project
1
(Hopkins
et
al.
2014
,
2018
)
to
investigate
some
of
the
physical
processes
that
limit
and/or
drive
the
growth
of
massive
black
holes.
The
FIRE
simulations
provide
a
specific,
high-resolution
model
to
study
the
growth
of
massive
black
holes
in
the
cosmological
context,
but
our
study
is
more
broadly
moti
v
ated
by
general
trends
that
have
been
found
in
several
recent
simulations
based
on
different
codes
and
subgrid
models.
Specifically,
a
number
of
different
simulations
have
found
that
SMBH
growth
is
strongly
inhibited
at
high
redshift
due
to
repeated
gas
ejection
by
stellar
feedback
(Dubois
et
al.
2015
;
Bower
et
al.
2017
;
Habouzit,
Volonteri
&
Dubois
2017
;
Habouzit
et
al.
2021
;
Lapiner,
Dekel
&
Dubois
2021
).
This
results
in
nuclear
black
1
See
the
FIRE
project
website
at:
ht
tp://fire.nort
hwestern.edu
.
© 2023
The
Author(s)
Published
by
Oxford
University
Press
on
behalf
of
Royal
Astronomical
Society
Downloaded from https://academic.oup.com/mnras/article/520/1/722/6988194 by California Institute of Technology user on 15 May 2023
Supermassive
black
hole
growth
in
FIRE
723
MNRAS
520,
722–739
(2023)
holes
that,
at
low
galaxy
or
bulge
mass,
can
be
order-of-magnitude
undermassi
ve
relati
ve
to
scaling
relations
measured
at
low
redshift
and
primarily
for
more
massive
galaxies.
It
is
only
after
some
time,
or
after
the
host
galaxy
has
gro
wn
suf
ficiently,
that
black
holes
‘catch
up’
to
masses
expected
from
observed
relationships
in
the
local
universe.
Angl
́
es-Alc
́
azar
et
al.
(
2017b
)
and
C
̧ atmabacak
et
al.
(
2022
)
showed
that
a
similar
effect
is
seen
in
simulations
of
galaxies
evolved
with
the
FIRE
model
which,
in
detail,
implements
galaxy
formation
physics
using
quite
different
numerical
models
than
the
other
simulations
which
found
this
same
effect.
The
fact
that
delayed
supermassive
BH
growth
appears
generic
to
very
different
numerical
models
suggests
that
it
is
due
to
fundamental
processes
that
are
common
to
most
galaxy
formation
simulations.
Thus,
this
appears
to
be
a
relatively
robust
prediction
which
may
be
realized
in
the
real
universe.
There
is
in
fact
some
reported
observ
ational
e
vidence
for
a
‘break’
in
the
relationship
between
black
hole
mass
and
either
the
total
galaxy
stellar
mass
or
the
mass
of
the
stellar
spheroid
at
M
∗
∼
5
×
10
10
M
(e.g.
Graham
&
Scott
2013
;
Reines
&
Volonteri
2015
;
Sa
v
orgnan
et
al.
2016
;
Sahu
et
al.
2019
).
Ho
we
ver,
the
presence
of
this
break
in
the
observations
may
depend
on
which
galaxies
are
included
in
the
analysis
(e.g.
early
versuslate
type
galaxies)
and
exactly
which
galaxy
property
(e.g.
stellar
mass
versus
stellar
velocity
dispersion
σ
)
BH
masses
are
compared
to.
As
a
result,
some
studies
have
reported
no
evidence
of
a
break
in
scaling
relations
(e.g.
Schutte,
Reines
&
Greene
2019
;
Baldassare
et
al.
2020
),
though
these
studies
were
based
on
relatively
small
data
sets
of
dwarf
galaxies,
and
the
interpretation
is
subject
to
significant
uncertainty
due
to
non-trivial
selection
effects.
Depending
on
the
stochasticity
of
BH
fuelling
in
the
early
phase,
a
break
in
the
scaling
relation
could
appear
as
increased
scatter
in
scaling
relations
(e.g.
L
̈
asker
et
al.
2016
;
Nguyen
et
al.
2019
).
A
recent
study
by
Tillman
et
al.
(
2021
)
suggests
that
a
break
in
the
M
BH
–
M
∗
relation
at
a
mass
consistent
with
the
FIRE
simulations
(see
Fig.
1
)
can
provide
a
good
fit
to
the
observed
quasar
luminosity
function,
and
may
in
fact
be
fa
v
ored
o
v
er
models
in
which
the
scaling
relation
is
purely
linear.
Ultimately,
additional
observations
of
low-
mass
galaxies
will
be
required
to
resolve
the
issue
of
whether
there
are
breaks
in
observed
scaling
relations.
In
this
work,
we
focus
on
understanding
the
physics
that
drives
the
delayed
SMBH
growth
predicted
by
simulations.
The
potential
implications
of
delayed
SMBH
growth
are
wide-
ranging
and
include
predictions
for:
(i)
the
redshift
and
mass
evo-
lution
of
black
hole-galaxy
scaling
relations;
(ii)
the
demographics
of
nuclear
BHs
in
dwarf
galaxies;
(iii)
AGN
demographics;
and
(iv)
the
mergers
of
massive
black
hole
that
future
gravitational
wave
experiments
may
detect
(Baileset
al.
2021
).
Furthermore,
understanding
the
physical
factors
that
eventually
enable
nuclear
black
holes
to
grow
at
an
accelerated
pace
may
shed
light
on
the
galaxy
mass
scale
at
which
AGN
feedback
becomes
most
important.
F
or
e
xample,
Bower
et
al.
(
2017
)
analysed
simulations
from
the
EAGLE
project
(Schaye
et
al.
2015
),
and
found
that
accelerated
BH
growth
occurs
at
roughly
constant
dark
matter
halo
mass
M
h
∼
10
12
M
,
corresponding
to
the
mass
scale
abo
v
e
which
haloes
become
filled
with
hot,
virialized
gas
(the
usual
transition
mass
between
‘cold’
and
‘hot’
accretion;
e.g.
Birnboim
&
Dekel
2003
;
Kere
ˇ
s
et
al.
2005
,
2009a
;
Faucher-Gigu
`
ere,
Kere
ˇ
s
&
Ma
2011
;
van
de
Voort
et
al.
2011
).
Bower
et
al.
(
2017
)
explained
this
phenomenon
in
terms
of
the
suppression
of
star
formation-driven
outflows
as
they
lose
buoyancy
when
halo
gas
becomes
hot.
Bower
et
al.
(
2017
)
identified
enhanced
BH
accretion
at
this
halo
mass
scale
as
the
cause
of
strong
AGN
feedback
(see
also
the
recent
paper
by
Lapiner
et
al.
2021
).
Previous
studies
have
analysed
SMBH
growth
in
FIRE
simulations
(Angl
́
es-Alc
́
azar
et
al.
2017a
,
2021
;
C
̧ atmabacak
et
al.
2022
).
This
study
is
complementary
in
its
focus
on
exploring
different
physical
factors
that
may
play
a
role
in
explaining
when
and
why
SMBH
fuelling
becomes
more
efficient.
In
this
paper,
we
also
analyse
a
broader
range
of
galaxy
masses,
including
dwarf
galaxies.
In
another
complementary
study,
Wellons
et
al.
(
2022
)
present
a
broad
surv
e
y
of
FIRE
simulations
including
multichannel
AGN
feedback.
The
plan
of
this
paper
is
as
follows.
We
describe
our
simulations
and
analysis
methodology
in
Section
2
.
Our
main
results
are
presented
in
Section
3
,
and
we
discuss
them
in
Section
4
.
Our
conclusions
are
summarized
in
Section
5
.
2
METHODOLOGY
2.1
FIRE
simulations
We
analyse
a
set
of
FIRE-2
cosmological
zoom-in
simulations.
All
simulations
were
run
with
the
meshless
finite
mass
hydrodynamics
code
GIZMO
(Hopkins
2015
).
Details
of
the
FIRE-2
methods
and
physics
are
explained
in
Hopkins
et
al.
(
2018
).
Throughout,
we
assume
a
standard
flat
CDM
cosmology
consistent
with
recent
measurements
(Planck
Collaboration
et
al.
2020
).
The
simulations
include
multiple
forms
of
stellar
feedback,
including
feedback
from
supernovae
of
Type
I
and
II,
stellar
winds,
photoionization,
and
radiation
pressure
on
dust
grains.
We
examined
four
massive
galaxies
with
halo
mass
M
halo
≈
10
12
.
5
M
at
z
=
2,
which
were
initially
studied
by
Angl
́
es-Alc
́
azar
et
al.
(
2017b
).
The
initial
conditions
for
these
galaxies
were
first
pre-
sented
in
Feldmann
et
al.
(
2017
),
and
they
were
then
re-simulated
by
Angl
́
es-Alc
́
azar
et
al.
(
2017b
)
with
a
gravitational
torque
model
for
black
hole
accretion,
described
below.
The
simulations
have
mass
res-
olution
m
b
=
3
.
3
×
10
4
M
for
baryons
and
m
DM
=
1
.
7
×
10
5
M
for
dark
matter
particles,
and
were
run
to
z
=
1.
We
also
examined
five
‘m12’
Milky
Way-mass
galaxies
(
M
halo
=
10
12
M
at
z
=
0)
and
six
‘m11’
galaxies
with
M
halo
=
10
11
M
at
z
=
0
from
the
FIRE-2
simulation
suite
(Wetzel
et
al.
2016
;
Hopkins
et
al.
2018
),
all
of
which
were
run
to
z
=
0.
The
m11
and
m12
simulations
have
a
baryonic
mass
resolution
of
m
b
=
7100
M
,
with
the
exceptions
of
m11b,
which
has
a
mass
resolution
of
m
b
=
2100
M
,
and
m12z,
which
has
a
mass
resolution
of
m
b
=
4200
M
.
The
gravitational
softenings
for
gas
particles
were
again
adaptive;
the
mean
softening
length
in
star-forming
gas
in
the
m12i
simulation
was
gas
=
4.6
pc
(Hopkins
et
al.
2018
).
Finally,
we
analysed
a
massive,
high-redshift
galaxy
(
M
halo
≈
10
12
.
5
M
at
z
=
5)
first
studied
by
Ma
et
al.
(
2020
).
We
use
the
Amiga
Halo
Finder
Knollmann
&
Knebe
(
2009
)
to
identify
the
halo
centre,
virial
mass
M
halo
,
and
virial
radius
R
vir
of
the
main
halo
for
each
simulation,
adopting
the
virial
o
v
erdensity
definition
of
Bryan
&
Norman
(
1998
).
We
define
the
stellar
mass
M
∗
as
the
total
stellar
mass
within
0.1
R
vir
.
As
the
simulations
other
than
those
from
Angl
́
es-Alc
́
azar
et
al.
(
2017b
)
did
not
include
on-the-fly
BH
accretion
calculations,
we
modelled
BH
growth
in
post-processing
in
all
of
the
simulations,
including
the
massive
galaxies
for
the
sake
of
consistency.
Angl
́
es-
Alc
́
azar
et
al.
(
2017b
)
compared
the
on-the-fly
and
post-processing
models
and
found
o
v
erall
good
agreement,
thus
validating
the
post-processing
approach.
C
̧ atmabacak
et
al.
(
2022
)
analysed
how
different
post-processing
assumptions
(such
as
the
placement
of
BHs
and
the
treatment
of
mergers)
affect
BH
growth
in
a
study
focused
on
massive
galaxies.
Downloaded from https://academic.oup.com/mnras/article/520/1/722/6988194 by California Institute of Technology user on 15 May 2023
724
L.
Byrne
et
al.
MNRAS
520,
722–739
(2023)
2.2
Model
for
black
hole
growth
The
black
hole
accretion
rate
is
calculated
in
post-processing
assum-
ing
a
model
in
which
inflows
are
driven
by
gravitational
torques,
using
the
same
methodology
as
in
in
Angl
́
es-Alc
́
azar
et
al.
(
2017a
).
Black
hole
seeds
with
mass
M
seed
=
1
.
4
×
10
4
M
are
introduced
in
galaxies
with
stellar
masses
abo
v
e
1000
×
M
seed
.
The
black
holes
are
assumed
to
be
located
at
the
centre
of
the
halo,
determined
using
the
Amiga
Halo
Finder
as
the
point
in
the
halo,
where
the
combined
stellar
and
dark
matter
density
is
highest
(Knollmann
&
Knebe
2009
).
The
black
holes
are
treated
as
collisionless
particles
and
allowed
to
grow
through
accretion
and
mergers.
BH
accretion
rates
are
calculated
at
each
snapshot.
With
the
exception
of
the
first
few
snapshots
before
z
=
15,
all
snapshots
are
spaced
between
10–
27
Myr
apart
from
one
another,
and
we
have
600
snapshots
for
each
simulation
evolved
to
z
=
0.
This
spacing
is
small
enough
that
it
does
not
introduce
major
uncertainties
in
our
analysis,
as
it
is
shorter
than
the
time-scales
o
v
er
which
the
main
physical
changes
we
will
be
discussing
in
this
paper
occur.
The
accretion
rate
is
calculated
as
̇
M
BH
=
(1
−
η
)
̇
M
Torque
,
where
η
=
0.1
is
the
constant
radiative
efficiency.
̇
M
Torque
is
calculated
based
on
properties
of
the
galaxy
within
a
distance
R
0
enclosing
256
gas
particles
(up
to
a
maximum
value
of
R
0
=
100
pc
h
−
1
)
as
̇
M
Torque
=
T
f
5
/
2
d
M
1
/
6
BH
,
8
M
tot
,
9
R
−
3
/
2
0
(1
+
f
0
/f
gas
)
−
1
,
(1)
(Hopkins
&
Quataert
2011
),
where
T
=
2.5
is
a
normalization
factor,
f
d
is
the
disc
mass
fraction,
M
tot
is
the
total
baryonic
mass
(gas
+
stars),
and
f
0
≈
0
.
31
f
2
d
(
M
d
(
R
0
)
10
9
M
)
−
1
/
3
.
(2)
Although
(Hopkins
&
Quataert
2011
)
originally
tested
their
gravitational
torque
estimator
on
simulations,
which
used
a
smooth
subgrid
model
for
the
ISM
based
on
Springel
&
Hernquist
(
2003
),
Hopkins
et
al.
(
2016
)
showed
that
this
prescription
is
also
much
more
accurate
at
predicting
BH
accretion
than
Bondi-like
prescrip-
tions
in
subparsec
resolution
simulations
of
multiphase
galactic
nuclei,
in
which
the
central
gas
reservoir
is
dominated
by
cold
gas
supported
by
angular
momentum
(see
also
Angl
́
es-Alc
́
azar
et
al.
2021
).
Angl
́
es-Alc
́
azar
et
al.
(
2017b
)
explored
several
BH
growth
pre-
scriptions,
including
models
based
on
the
gas
mass
and
free-fall
time
(
t
ff
)
near
the
central
BHs,
and
found
that
the
‘kinked’
relationship
between
BH
mass
and
host
mass
(see
Fig.
1
for
results
on
this
from
our
analysis)
was
generic
to
prescriptions
in
which
the
accretion
rates
were
based
on
the
gas
content
in
the
inner
galactic
regions
around
the
black
holes.
Our
results
are
therefore
not
specific
to
the
gravitational
torque
model.
2
The
normalization
of
the
resulting
scaling
relation,
ho
we
ver,
depends
on
the
normalization
of
the
accretion
rate
prescription
(the
value
of
T
for
the
gravitational
torque
model).
Angl
́
es-Alc
́
azar
et
al.
(
2017b
)
also
found
that
the
depth
of
the
break
in
the
scaling
relation
depends
on
the
radius
within
which
the
accretion
estimator
is
e
v
aluated
in
galaxies,
such
that
the
break
becomes
weaker
for
larger
apertures.
This
indicates
that
the
physical
conditions
in
the
central
regions
of
galaxies,
as
opposed
to
purely
galaxy-integrated
properties,
play
an
important
role
in
shaping
2
We
will
see
that
in
the
simulations,
the
onset
of
accelerated
BH
fuelling
correlates
with
the
emergence
of
steady,
thin
gas
discs
(Section
4.3
).
We
stress
that
this
result
is
not
simply
a
consequence
of
the
f
d
factors
in
equations
(
1
)
and
(
2
),
because
our
main
results
are
robust
to
changes
in
the
accretion
model
in
which
the
disc
fraction
does
not
enter
(see
Section
3.3
).
scaling
relations
(see
also
C
̧ atmabacak
et
al.
2022
).
We
assume
that
the
black
holes
are
at
the
centres
of
the
simulated
galaxies.
This
may
result
in
black
holes
which
can
grow
more
efficiently
at
early
times
than
in
reality.
F
or
e
xample,
Ma
et
al.
(
2021
)
showed
that
black
hole
seeds
can
take
a
long
time
to
physically
sink
to
the
galaxy
centres,
especially
in
early
galaxies
with
clumpy
potentials,
delaying
gro
wth.
Ho
we
ver,
this
ef
fect
goes
in
the
direction
of
further
limiting
BH
fuelling
in
early
galaxies,
so
it
most
likely
accentuates
the
relati
vely
inef
ficient
fuelling
found
by
neglecting
this
sinking
problem.
The
simulations
in
this
study
do
not
include
any
form
of
black
hole
feedback.
This
allows
us
to
isolate
effects
on
BH
fuelling
that
are
due
to
other
physics,
namely
stellar
feedback.
2.3
Comparing
the
time
of
accelerated
BH
growth
with
thresholds
in
other
physical
properties
To
investigate
the
physical
factors
that
may
drive
BH
growth,
we
estimate
the
cosmic
time
at
which
the
BH
accretion
rate
undergoes
a
transition
to
rapid
growth
in
our
simulations
(
t
growth,
BH
).
Of
the
six-
teen
galaxies
in
our
sample,
seven
did
not
experience
a
BH
accretion
transition
(including
all
of
the
dwarf
galaxies
as
well
as
m12z),
eight
did
undergo
the
transition
(including
all
the
massive
galaxies
–A1,
A2,
A4,
A8,
HL09
–as
well
as
m12i,
m12b,
and
m12f),
and
one
(m12m)
was
ambiguous.
For
the
eight
galaxies
that
experience
two
phases
of
black
hole
growth,
we
fit
a
simple
step
function
(
y
=
A
×
((
t
−
t
growth,
BH
)/(
|
t
−
t
growth,
BH
|
)
+
C
)
to
the
BH
accretion
rate,
smoothed
with
a
moving
time-average
over
300
Myr.
As
the
BH
accretion
rate
is
highly
time-variable,
the
smoothing
time-scale
of
300
Myr
was
chosen
to
be
long
enough
to
smooth
o
v
er
large
fluctuations
such
as
those
caused
by
mergers
while
still
being
short
enough
to
accurately
capture
the
step-like
behaviour
of
the
transition;
our
o
v
erall
results
are
not
sensitive
to
the
exact
value
chosen.
An
example
step-function
fit
for
one
of
our
Milky
Way-mass
galaxies,
m12b,
is
shown
in
Fig.
2
.
We
use
the
horizontal
shift
t
growth,
BH
determined
by
the
step
function
model
as
the
BH
growth
transition
point
in
subsequent
analysis.
We
will
analyse
how
t
growth,
BH
compares
with
when
constant
thresholds
in
other
physical
properties
of
the
galaxy
and
its
halo
are
crossed.
For
each
property
and
each
galaxy,
we
calculate
t
=
t
threshold
−
t
growth,
BH
,
the
time
interval
between
the
‘predicted’
onset
of
accelerated
by
growth
based
on
different
galaxy-
or
halo-based
thresholds
and
the
actual
time
of
accelerated
BH
growth
identified
using
the
simulation
data.
The
distributions
of
these
time
intervals,
which
we
examine
in
Section
3.2.1
,
can
in
principle
distinguish
between
‘better’
or
‘worse’
predictors.
For
most
properties
we
will
consider,
we
will
find
the
threshold
value
that
minimizes
the
median
t
.
An
exception
is
the
time
t
bursty
at
which
the
galactic
star
forma-
tion
rate
(SFR)
transitions
from
‘bursty’
(order-of-magnitude
time-
variable)
to
time-steady.
This
SFR
transition,
when
it
occurs,
is
rather
sharp
in
the
FIRE
simulations
and
can
be
identified
from
the
SFR
time-series
alone.
Therefore,
we
do
not
attempt
to
‘optimize’
t
bursty
,
but
rather
for
each
simulation
we
determine
it
following
the
method
used
in
Gurvich
et
al.
(
2022a
).
Specifically,
we
e
v
aluate
the
scatter
in
300
Myr
windows
of
SFR,
σ
300
Myr
(log
(SFR)).
The
time
t
bursty
is
then
defined
as
the
earliest
time
after
which
σ
300
Myr
(log
(SFR))
is
al
w
ays
below
0.3
dex.
The
time
of
transition
from
bursty
to
steady
SFR
is
of
interest
because
it
corresponds
closely
to
changes
in
the
properties
of
the
ISM.
Before
t
bursty
,
the
ISM
is
highly
dynamic
and
frequently
ejected
by
bursts
of
stellar
feedback.
After
t
bursty
,
the
simulated
galaxies
tend
to
sustain
a
long-lived
gas
reservoir
in
the
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