In Fig. 8, we show the observed $L_{\rm[C_{II}]}$-SFR relation of the rest-UV-
selected galaxy samples at $z{}\mathbin{\lower 3.0pt\hbox{$\hbox
to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}{}5$
(see Table 5 for the details) together with the result of the FIRE galaxies at
$z{}={}4$, $z{}={}6$ and $z{}={}8$ in the two panels. For the observed
galaxies having no detection of dust, we show the relation between their
$L_{\rm[C_{II}]}$ (for the $\rm[C_{II}]$-undetected galaxies, the $3\sigma$
upper limit of their $L_{\rm[C_{II}]}$) and the lower and upper bound of their
SFR, respectively, in the left and right panels of the figure. For reference,
we also show in Fig. 8 the observed $L_{\rm[C_{II}]}$-SFR relation of the
local star-forming galaxies by L11, L14 and H15.

It can be seen that the FIRE galaxies at $z{}={}4-8$ lie systematically below
the observed local $L_{\rm[C_{II}]}$-SFR relations (and thus also the FIRE
galaxies at $z{}={}0$) over the broad SFR range of
$\approx{}0.1-10^{3}\,M_{\odot}\,\rm yr^{-1}$, showing a $\rm[C_{II}]$
deficit. This appears to be in agreement with the observational data.

At $\rm SFR{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}{}100\,M_{\odot}\,\rm
yr^{-1}$, most of the observed galaxies at $z{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}5$ have both $\rm[C_{II}]$ and dust detections and thus their
(dust-obscured) SFR is more reliably constrained. The mean
$L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio of these galaxies is lower than the L11
relation (solid green line) by 0.22 dex, which is close to the $1\sigma$
scatter of the L11 relation (see Table 2). The FIRE galaxies at $z\geq 4$ are
about $2\sigma$ below the L11 relation in the same SFR range, which seem to
show a slightly more prominent ‘deficit’ than the observed samples.

At $\rm SFR{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 60\relax$}\hss}\mathchar 29208\relax$}}{}100\,M_{\odot}\,\rm
yr^{-1}$, most of the $z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}{}5$ galaxies do not
have confirmed dust detection with the current ALMA observations, and a large
fraction of them do not have confirmed $\rm[C_{II}]$ detections neither
(marked by downward arrows). The uncertainty in the SFR estimate of these
dust-undetected galaxies can be as large as a factor of $\sim 5$
($\approx{}20-100\,M_{\odot}\,\rm yr^{-1}$, see Fig. 8). Such a large
uncertainty is due to the high $T_{\rm eqv}$ of galaxies at
$z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}5$ ($T_{\rm eqv}{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}45$ K for $\delta_{\rm dzr}{}={}0.4$, see equation (7)), so
that even a low noise level (typically $\sigma\sim 10\mu$Jy, see Table 5) of
the ALMA observations is converted to a relatively high upper bound of $L_{\rm
IR}$ (and hence $\rm SFR_{IR}$). From Fig. 8, it can be seen that the
predicted $L_{\rm[C_{II}]}$-SFR relation of the FIRE galaxies does not
conflict with the observational constraints over ${\rm
SFR}{}\approx{}10-100\,M_{\odot}\,\rm yr^{-1}$. In particular, for the
$\rm[C_{II}]$-undetected galaxies, the $3\sigma$ upper limit of their
$L_{\rm[C_{II}]}$ (marked by downward arrows) appears to be above the data
points of the FIRE galaxies at similar SFR when their dust emission is
insignificant, namely, $\rm SFR{}\approx{}SFR_{UV}$ (see the left panel of
Fig. 8).

At ${\rm SFR}{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 60\relax$}\hss}\mathchar 29208\relax$}}{}10\,M_{\odot}\,\rm
yr^{-1}$, we lack enough observational data for a reliable constraint on the
$L_{\rm[C_{II}]}$-SFR relation at $z{}\mathbin{\lower 3.0pt\hbox{$\hbox
to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}{}5$
because galaxies having such low SFR are intrinsically faint. The galaxy
having the lowest SFR (${\rm SFR}{}\approx{}1\,M_{\odot}\,\rm yr^{-1}$) that
has had $\rm[C_{II}]$ measurement to date at $z{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}5$ is MS0451-H (Knudsen et al., 2016), a strongly lensed
galaxy at $z{}={}6.7$ with an estimated magnification factor of $\mu=100\pm
20$. MS0451-H has no confirmed $\rm[C_{II}]$ detection yet. The upper bound of
its $L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio is more than 1.5 dex below the L11
relation (even with the most conservative, UV-based SFR, see the left panel of
Fig. 8), showing a strong $\rm[C_{II}]$ deficit. This appears to be in
agreement with the FIRE sample. It can be seen from the figure that the
$\rm[C_{II}]$ deficit of the FIRE galaxies extends to ${\rm
SFR}{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
60\relax$}\hss}\mathchar 29208\relax$}}{}10\,M_{\odot}\,\rm yr^{-1}$ at
$z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}5$, which is even slightly more
prominent than at higher SFR. Encouragingly, some of the FIRE galaxies at
$z{}\geq{}4$ show similarly low $L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio as
MS0451-H.

The $L_{\rm[C_{II}]}$-SFR relation of the observed galaxies at
$z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}5$ reported in this work seems to
have lower normalization than a number of the recent observational studies,
including e.g. Schaerer et al. (2020) (ALPINE paper), Ferrara et al. (2022)
(REBELS paper), Matthee et al. (2017, 2019), Carniani et al. (2018a), Harikane
et al. (2020) and Fujimoto et al. (2021). This is due to the fact that these
studies have assumed a lower $T_{\rm eqv}$ than what we use for this study as
derived using equation (7). As has been mentioned in some of these studies,
the largest uncertainty of the derived galaxy $L_{\rm[C_{II}]}$-SFR relation
at $z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}5$ is the assumed $T_{\rm eqv}$. In
Table 7, we explicitly show the difference in the mean $T_{\rm eqv}$ adopted
by the ALPINE/REBELS projects and this work (for $\delta_{\rm dzr}{}={}0.4$),
as well as the resulting difference in the derived mean
$L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio (<$L_{\rm[C_{II}]}{}/{}\rm SFR$>) of the
galaxies. Note that Ferrara et al. (2022) has used very similar $T_{\rm eqv}$
compared to what is used in our work as fiducial (with $\delta_{\rm
dzr}{}={}0.4$), whereas Schaerer et al. (2020) has used significantly lower
$T_{\rm eqv}$ ($<T_{\rm eqv}>{}={}42$ K) for the ALPINE galaxies than us
($<T_{\rm eqv}>{}={}52.1$ K). Our estimate of the $L_{\rm[C_{II}]}$-SFR
relation of the ALPINE galaxies is therefore about 0.3 dex below the
originally reported result.

#####  $L_{\rm[C_{II}]}{}/{}L_{\rm IR}$ of IR-luminous galaxies

Figure 9: $L_{\rm[C_{II}]}{}/{}L_{\rm IR}$ vs. $L_{\rm IR}$ relation of
galaxies at high redshifts. Filled coloured symbols indicate the data of the
FIRE galaxies (magenta circles for $z{}={}4$, green diamonds for $z{}={}6$ and
purple downward triangles for $z{}={}8$). Red vertical and blue diagonal
crosses represent the observational data of the REBELS (<$z$>${}\approx{}7$)
and ALPINE (<$z$>${}\approx{}4.5$) galaxy samples, respectively. Black symbols
represent the observational data of the other galaxy samples at
$z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}5$. Specifically, black diagonal
crosses, black circles (filled and unfilled) and black stars correspond to the
UV-selected galaxies, SMGs and quasar hosts, respectively. For the galaxies
whose dust continuum is measured at only single ALMA band, $L_{\rm IR}$ is
derived using $T_{\rm eqv}$ that follows equation (7) assuming $\delta_{\rm
dzr}{}={}0.4$ except for the REBELS galaxies, for which we show two different
sets of data that are produced by using $\delta_{\rm dzr}{}={}0.4$ (semi-
transparent red crosses) and $\delta_{\rm dzr}{}={}0.1$ (non-transparent red
crosses). The lower $\delta_{\rm dzr}$ yields higher $T_{\rm eqv}$ (and hence
$L_{\rm IR}$) estimates for the galaxies. The black arrow indicates the
direction along which the data points of these galaxies move on the diagram
with increasing $T_{\rm eqv}$. For the SMGs, filled circles indicate the
galaxies that are either confirmed as un-lensed or have observationally
determined lensing magnification factor $\mu$, whereas unfilled circles
indicate the lensed SPT galaxies having no determined $\mu$ yet (see Section
4.2). Grey symbols in the background represent the observational data of the
local $z{}={}0$ galaxy samples, as is shown in Fig. 6 (right panel). Black
horizontal line indicates the median $L_{\rm[C_{II}]}{}/{}L_{\rm IR}$ ratio
(<$L_{\rm[C_{II}]}{}/{}L_{\rm IR}$>${}={}0.002$) of the local galaxies at
$L_{\rm IR}{}<{}10^{11}\,L_{\odot}$. Galaxies at $z{}>{}5$ show a trend of
declining $L_{\rm[C_{II}]}{}/{}L_{\rm IR}$ ratio with $L_{\rm IR}$ at $L_{\rm
IR}{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}10^{11}\,L_{\odot}$ similar to the
local samples. The FIRE simulations successfully reproduced the observed
$\rm[C_{II}]$ deficit at high $L_{\rm IR}$ at $z{}>{}5$.

In addition to the LBGs/LAEs having moderate SFRs, there have been studies
probing the more extreme systems at $z{}\mathbin{\lower 3.0pt\hbox{$\hbox
to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}{}5$,
in particular, the quasar hosts. These systems are gas/dust-rich and very IR-
luminous ($L_{\rm IR}{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}10^{12}\,L_{\odot}$). They typically are also bright
$\rm[C_{II}]$ emitters, having $L_{\rm[C_{II}]}$ that spans across the range
of $\approx{}10^{8}-10^{10}\,L_{\odot}$. We summarize the properties of the
quasar hosts at $z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}{}5$ having had
$\rm[C_{II}]$ line detections to date in Table 8 ($>{}65$ galaxies in total).
Observations targeting the quasar hosts have a high successful detection rate
for $\rm[C_{II}]$ line (e.g. Decarli et al., 2017; Venemans et al., 2020).

Like most of the LBGs/LAEs at this epoch, the selected quasar hosts typically
have one or two data points in their dust continuum (measured with ALMA band 6
or 7) and their $L_{\rm IR}$ is converted from a single broad-band sub-mm flux
density in the literature using the standard MBB function with an assumed
$T_{\rm eqv}$. $L_{\rm IR}$ has generally been considered as a crude estimate
of their SFR by the observational studies assuming that these quasar hosts are
gas and dust-rich and the stellar radiation of these galaxies is significantly
dust-obscured. It is, however, unknown to what degree the radiation from the
accreting supermassive black hole affects the shape of the IR SED and the
total IR luminosity of these early galaxies. Observations of galaxies at lower
redshifts ($z{}\approx{}0-3$) demonstrate that the IR SED shape of galaxies
becomes ‘warmer’ (indicating higher $T_{\rm eqv}$) with increasing AGN power
(Kirkpatrick et al., 2015). A similar conclusion was reached in the early
study by Younger et al. 2009 with hydrodynamic simulations of galaxy mergers
that include AGN modelling. Note, however, that some recent studies (e.g.
Symeonidis, 2016; McKinney et al., 2021) also suggest that AGN radiation may
even dominate the cold-dust emission of the host galaxies at high redshifts.

In Fig. 9, we show the $L_{\rm[C_{II}]}{}/{}L_{\rm IR}$ vs. $L_{\rm IR}$
relation of the quasar hosts, along with other galaxy populations at
$z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}5$, including the few SMGs (listed in
Table 3), the ALPINE and REBELS galaxies and other rest-UV-selected galaxies
at $z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}5$ (we only show the galaxies having
confirmed dust detection, which have more reliable constraints on $L_{\rm IR}$
than the dust-undetected galaxies). We convert the reported single-band sub-mm
flux density of all the quasar hosts to $L_{\rm IR}$ using the standard MBB
function and $T_{\rm eqv}$ that follows equation (7) with the best-fit
parameters derived by Liang et al. (2019). We note that for the quasar hosts,
this is likely to be an underestimate because the best-fit parameters of Liang
et al. (2019) are derived using FIRE simulations which do not include AGN
feedback. Having a higher $T_{\rm eqv}$, the data points of the quasar hosts
(black stars) will shift in the diagonal direction toward the bottom-right
corner of the diagram (marked by the black arrow in Fig. 9).

Looking at the observational data, we can see a clear trend of declining
$L_{\rm[C_{II}]}{}/{}L_{\rm IR}$ ($\sim L_{\rm[C_{II}]}{}/{}\rm SFR$) ratio of
the galaxies with $L_{\rm IR}$ ($\rm[C_{II}]$ deficit) at $L_{\rm
IR}{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}10^{11.5}\,L_{\odot}$ at
$z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}5$, similar to the trend seen at
lower redshifts. The $L_{\rm[C_{II}]}{}/{}L_{\rm IR}$-$L_{\rm IR}$ relation of
these early galaxies appears to consistent with the local samples (grey
symbols) in the overlapping $L_{\rm IR}$ regime and show similarly large
scatter.

We also show in Fig. 9 the $L_{\rm[C_{II}]}{}/{}L_{\rm IR}$-$L_{\rm IR}$
relation of the FIRE galaxies at $z{}={}4-8$. The result of the FIRE galaxies
is in good agreement with the observational data in overlapping $L_{\rm IR}$
range, except for the REBELS sample (<$z$>${}\approx{}7$, indicated by red
vertical crosses in Fig. 9). Using $\delta_{\rm dzr}{}={}0.4$, the REBELS
galaxies (semi-transparent red crosses) show systematically higher
$L_{\rm[C_{II}]}{}/{}L_{\rm IR}$ than the rest of the observed galaxy samples
(blue and black diagonal crosses) as well as the FIRE galaxies at similar
$L_{\rm IR}$ ($\approx{}10^{12}\,L_{\odot}$) by $\sim 0.5$ dex. Using
$\delta_{\rm dzr}{}={}0.1$ instead, the expected mean $T_{\rm eqv}$ of the
REBELS sample increases by $\approx{}20\%$ (from 57 K to 71 K), and the
derived mean $L_{\rm IR}$ ($L_{\rm[C_{II}]}{}/{}L_{\rm IR}$ ratio) of the
galaxies increases (decreases) by a factor of $\sim{}3$. The data of the
REBELS sample for $\delta_{\rm dzr}{}={}0.1$ (non-transparent red crosses)
appears to be consistent with the other observed samples as well as the FIRE
galaxies.

The FIRE galaxies at $z{}\geq{}4$ show a trend of declining
$L_{\rm[C_{II}]}{}/{}L_{\rm IR}$ ratio with $L_{\rm IR}$, which agrees with
the observational data. It is also clear to see that the
$L_{\rm[C_{II}]}{}/{}L_{\rm IR}$ ratio of the FIRE galaxies decreases with
redshift at fixed $L_{\rm IR}$ at $z{}\geq{}4$. The trend of decreasing
$L_{\rm[C_{II}]}{}/{}L_{\rm IR}$ ratio with both redshift and $L_{\rm IR}$
persists up to $z{}={}8$ in the FIRE simulations.

Finally, we note that it is unclear whether AGN activity is directly related
to the $\rm[C_{II}]$ deficit at high $L_{\rm IR}$ based on the current data,
despite the large number of quasar hosts at $z{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}5$ showing strong $\rm[C_{II}]$ deficit. This is because most
of the selected SMGs in the literature ($2{}\mathbin{\lower 3.0pt\hbox{$\hbox
to0.0pt{\raise 4.0pt\hbox{$\char 60\relax$}\hss}\mathchar
29208\relax$}}{}z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 60\relax$}\hss}\mathchar 29208\relax$}}{}7$), having similar
$L_{\rm IR}$ to the quasar hosts, have no identified AGN feature (see Table 3)
but show similarly strong $\rm[C_{II}]$ deficit as the quasar hosts. In
addition, the FIRE simulations, which do not include AGN physics, have also
successfully reproduced similarly low $L_{\rm[C_{II}]}{}/{}L_{\rm IR}$ ratio
at high $L_{\rm IR}$.

Table 8: Characteristics of the high-$z$ quasar host galaxies.

Name | $z$ |  $S_{\nu}$ ($\rm mJy$)$\parallel$ |  ${\rm log}\,(L_{\rm IR}{}/{}L_{\odot})$$\mathsection$ |  ${\rm log}\,(L_{\rm[C_{II}]}{}/{}L_{\odot})$$\parallel$ |  References$\ast$
---|---|---|---|---|---
SDSS J1015+0020 |  4.407 |  0.60 (ALMA 7) | 12.3 |  8.46 (ALMA 7) |  [1]
BRI 1335-0417 |  4.41 |  9.03 (ALMA 6) | 14.0 |  10.21 (APEX/FLASH) |  [2, 3]
BR 1202-0725 N |  4.691 |  18.8 (ALMA 7) | 13.8 |  10.00 (ALMA 7) |  [4, 5]
BR 1202-0725 S |  4.694 |  18.0 (ALMA 7) | 13.8 |  9.81 (ALMA 7) |  [4, 5]
SDSS J0338+0021 |  5.027 |  2.98 (ALMA 6) | 13.5 |  9.76 (ALMA 6) |  [6]
SDSS J0129-0035 |  5.779 |  2.61 (ALMA 6) | 13.5 |  9.28 (ALMA 6) |  [7, 8, 9]
SDSS J1044-0125 |  5.785 |  3.00 (ALMA 6) | 13.5 |  9.21 (ALMA 6) |  [7, 8, 9]
PSO J004+17 |  5.817 |  0.88 (ALMA 6) | 13.0 |  8.31 (ALMA 6) |  [10]
PSO J352-15 |  5.832 |  0.34 (ALMA 7) | 12.1 |  9.09 (ALMA 7) |  [11]
HSC J1202-0057 |  5.929 |  0.25 (ALMA 6) | 12.4 |  8.79 (ALMA 7) |  [12]
PSO J056+16 |  5.967 |  0.17 (ALMA 6) | 12.3 |  7.11 (ALMA 6) |  [10]
PSO J007+04 |  6.001 |  2.07 (ALMA 6) | 13.4 |  9.20 (ALMA 6) |  [9, 13]
SDSS J2310+1855$\dagger$ |  6.003 | $-$ | 13.2 |  9.94 (ALMA 6) |  [7, 14]
PSO J009-10 |  6.004 |  3.66 (ALMA 6) | 13.6 |  9.95 (ALMA 6) |  [9, 13]
CFHQS J0055+0146 |  6.006 |  0.21 (ALMA 6) | 12.4 |  8.92 (ALMA 6) |  [15]
CFHQS J0216-0455 |  6.01 |  $<0.04$ (ALMA 6) | $<11.6$ |  $<7.85$ (ALMA 6) |  [16]
PSO J265+41 |  6.026 |  3.61 (ALMA 6) | 13.6 |  9.96 (ALMA 6) |  [10]
SDSS J1306+0356 |  6.033 |  0.74 (ALMA 6) | 12.9 |  9.05 (ALMA 6) |  [9, 13]
ULAS J1207+0630 |  6.037 |  0.50 (ALMA 6) | 12.7 |  9.13 (ALMA 6) |  [13]
SDSS J2054-0005 |  6.039 |  3.15 (ALMA 6) | 13.5 |  9.49 (ALMA 6) |  [7, 9]
VDESJ0454-4448 |  6.058 |  0.71 (ALMA 6) | 12.9 |  8.86 (ALMA 6) |  [13]
PSO J158+14 |  6.068 |  3.46 (ALMA 6) | 13.6 |  9.22 (ALMA 6) |  [10]
SDSS J0842+1218 |  6.075 |  0.68 (ALMA 6) | 12.9 |  8.88 (ALMA 6) |  [9, 13, 17]
HSC J2228+0152 |  6.081 |  $<0.05$ (ALMA 6) | $<11.7$ |  8.39 (ALMA 6) |  [18]
CFHQS J2100-1715 |  6.081 |  0.56 (ALMA 6) | 12.8 |  9.12 (ALMA 6) |  [9, 13, 17, 19]
HSC J2216-0016 |  6.096 |  0.14 (ALMA 6) | 12.2 |  9.01 (ALMA 6) |  [12]
PSO J239+07 |  6.110 |  0.23 (ALMA 6) | 12.4 |  8.37 (ALMA 6) |  [10]
HSC J1208-0200 |  6.117 |  0.09 (ALMA 6) | 12.0 |  8.43 (ALMA 6) |  [18]
CFHQS J1509-1749 |  6.123 |  1.72 (ALMA 6) | 13.3 |  9.37 (ALMA 6) |  [13]
PSO J065-19 |  6.125 |  0.46 (ALMA 6) | 12.7 |  8.97 (ALMA 6) |  [13]
CFHQS J0221-0802 |  6.13 |  0.25 (ALMA 6) | 12.4 |  $<8.08$ (ALMA 6) |  [16]
ULAS J1319+0950 |  6.135 |  5.13 (ALMA 6) | 13.8 |  9.61 (ALMA 6) |  [7, 9, 20]
VIK J2318-3029 |  6.146 |  3.11 (ALMA 6) | 13.5 |  9.35 (ALMA 6) |  [9, 13]
VIMOS2911 |  6.149 |  0.77 (ALMA 6) | 12.9 |  9.41 (ALMA 6) |  [16]
PSO J217-16 |  6.150 |  0.37 (ALMA 6) | 12.6 |  9.00 (ALMA 6) |  [13]
CFHQS J2229+1457 |  6.152 |  0.05 (ALMA 6) | 11.8 |  8.78 (ALMA 6) |  [15]
PSO J359-06 |  6.172 |  0.79 (ALMA 6) | 12.9 |  9.42 (ALMA 6) |  [9, 10, 13]
PSO J065-26 |  6.187 |  1.37 (ALMA 6) | 13.2 |  9.23 (ALMA 6) |  [9, 13]
PSO J308-21 |  6.236 |  1.18 (ALMA 6) | 13.1 |  9.53 (ALMA 6) |  [9, 13, 17]
HSC J2239+0207 |  6.250 |  1.11 (ALMA 6) | 13.1 |  8.98 (ALMA 6) |  [18]
SDSS J0100+2802 |  6.327 |  1.37 (ALMA 6) | 13.2 |  9.58 (ALMA 6) |  [21, 22]
ATLAS J025-33 |  6.338 |  2.49 (ALMA 6) | 13.4 |  9.75 (ALMA 6) |  [9, 13]
VIK J2211-3206 |  6.339 |  0.57 (ALMA 6) | 12.8 |  8.98 (ALMA 6) |  [13]
PSO J083+11 |  6.340 |  5.10 (ALMA 6) | 13.8 |  10.02 (ALMA 6) |  [23]
VIK J1152+0055 |  6.364 |  0.22 (ALMA 6) | 12.4 |  8.81 (ALMA 6) |  [12, 13]
PSO J159-02 |  6.381 |  0.65 (ALMA 6) | 12.9 |  9.05 (ALMA 6) |  [13]
HSC J0859+0022 |  6.390 |  0.16 (ALMA 6) | 12.2 |  8.66 (ALMA 6) |  [12]
J2329-0301 |  6.417 |  0.04 (ALMA 6) | 11.6 |  8.59 (ALMA 6) |  [16]
SDSS J1148+5251$\dagger$ |  6.42 | $-$ | 13.3 |  9.64 (NOEMA) |  [22, 24, 25, 26]
CFHQS J0210-0456 |  6.432 |  0.12 (ALMA 6) | 12.1 |  8.48 (ALMA 6) |  [27]
PSO J183+05 |  6.439 |  4.79 (ALMA 6) | 13.7 |  9.85 (ALMA 6) |  [9, 13]
VIK J2318-3113 |  6.443 |  0.36 (ALMA 6) | 12.6 |  9.20 (ALMA 6) |  [9, 13]
PSO J011+09 |  6.469 |  1.20 (ALMA 6) | 13.1 |  8.47 (ALMA 6) |  [10]
PSO J167-13 |  6.514 |  0.89 (ALMA 6) | 13.0 |  9.75 (ALMA 6) |  [9, 13, 16]
J043947+163415 (lensed$\ddagger$) |  6.519 |  3.27 (ALMA 6) | 13.6 |  9.54 (ALMA 6) |  [28, 29]
PSO J036+03 |  6.542 |  2.55 (ALMA 6) | 13.5 |  9.53 (ALMA 6) |  [9, 30]
PSO J231-20 |  6.587 |  4.37 (ALMA 6) | 13.7 |  9.55 (ALMA 6) |  [9, 13, 17]
PSO J323+12 |  6.587 |  0.23 (ALMA 6) | 12.4 |  9.16 (ALMA 6) |  [9, 31]
PSO J006+39 |  6.610 |  0.55 (NOEMA) | 12.8 |  8.95 (NOEMA) |  [32]
VIK J030516-315056 |  6.614 |  5.34 (ALMA 6) | 13.8 |  9.77 (ALMA 6) |  [9, 32, 33]
PSO J338+29 |  6.658 |  0.97 (NOEMA) | 13.0 |  9.30 (NOEMA) |  [31]
VIK J1048-0109 |  6.676 |  2.84 (ALMA 6) | 13.5 |  9.32 (ALMA 6) |  [9, 13]

(Continue on next page)

Table 9: continued

Name | $z$ |  $S_{\nu}$ ($\rm mJy$) |  ${\rm log}\,(L_{\rm IR}{}/{}L_{\odot})$$\mathsection$ |  ${\rm log}\,(L_{\rm[C_{II}]}{}/{}L_{\odot})$ |  References$\ast$
---|---|---|---|---|---
HSC J1205-0000 |  6.723 |  1.17 (ALMA 6) | 13.1 |  8.58 (ALMA 6) |  [34]
VIK J0109-3047 |  6.791 |  0.52 (ALMA 6) | 12.8 |  9.38 (ALMA 6) |  [9, 33]
VIK J2348-3054 |  6.901 |  2.28 (ALMA 6) | 13.4 |  9.25 (ALMA 6) |  [9, 33]
HSC J1243+0100 |  7.075 |  1.52 (ALMA 6) | 13.2 |  9.40 (ALMA 6) |  [35]
ULAS J1120+0641 |  7.085 |  0.64 (ALMA 6) | 12.9 |  9.08 (ALMA 6) |  [9, 36]
ULAS J1342+0928 |  7.541 |  0.34 (ALMA 6) | 12.6 |  9.12 (ALMA 6) |  [9, 37]

  * $\dagger$

$L_{\rm IR}$ of SDSS J2310+1855 and SDSS J1148+5251 are derived by SED fitting
(e.g. Casey, 2012; Casey et al., 2014) to multiple data points at both Wien
and Rayleigh-Jeans sides of the dust IR SED.

  * $\ddagger$

J043947+163415 has been confirmed to be gravitationally-lensed, and its
luminosities have been de-magnified by $\mu{}={}4.6\pm 2.0$, estimated based
on the lensing configuration from HST imaging by Fan et al. (2019).

  * $\parallel$

NOEMA: NOrthern Extended Millimeter Array
(Website: https://www.iram-institute.org/EN/content-
page-235-3-235-0-0-0.html).

  * $\mathsection$

$L_{\rm IR}$ (or its upper $3\sigma$ limit) is converted from $S$ (its
$3\sigma$ upper limit) using the standard MBB function and with $T_{\rm eqv}$
that follows equation (7) (assuming $\beta_{\rm dust}{}={}2.0$ and
$\delta_{\rm dzr}{}={}0.4$), except for SDSS J2310+1855 and SDSS J1148+5251.

  * $\ast$

References: [1]: Bischetti et al. (2018), [2]: Wagg et al. (2010), [3]: Lu et
al. (2018), [4]: Wagg et al. (2012), [5]: Iono et al. (2006), [6]: Leipski et
al. (2014), [7]: Wang et al. (2013), [8]: Wang et al. (2019), [9]: Venemans et
al. (2020), [10]: Eilers et al. (2020): [11]: Rojas-Ruiz et al. (2021), [12]:
Izumi et al. (2018), [13]: Decarli et al. (2018), [14]: Shao et al. (2019),
[15]: Willott et al. (2015a), [16]: Willott et al. (2017), [17]: Decarli et
al. (2017), [18]: Izumi et al. (2019), [19]: Walter et al. (2018), [20]: Shao
et al. (2017), [21]: Wang et al. (2016), [22]: Leipski et al. (2013), [23]:
Andika et al. (2020), [24]: Walter et al. (2009), [25]: Maiolino et al.
(2005), [26]: Meyer et al. (2022), [27]: Willott et al. (2013b), [28]: Yang et
al. (2019b), [29]: Yue et al. (2021), [30]: Bañados et al. (2015), [31]:
Mazzucchelli et al. (2017), [32]: Venemans et al. (2019) [33]: Venemans et al.
(2016), [34]: Izumi et al. (2021a), [35]: Izumi et al. (2021b), [36]: Venemans
et al. (2012), [37]: Venemans et al. (2017).

##  5 The physics of the $L_{\rm[C_{II}]}$-SFR scaling relation of galaxies

In the previous section, we have shown that the $L_{\rm[C_{II}]}$-SFR relation
of the FIRE galaxies predicted using our model is in good agreement with the
observational data of local and high-$z$ galaxies. In particular, our model
reproduces the observed $\rm[C_{II}]$ deficit of galaxies at high $L_{\rm IR}$
and high redshifts. In this section, we explore the origin(s) of the
$\rm[C_{II}]$ deficit of galaxies using the FIRE galaxy sample.

In Section 5.1, we present the analytic solution of $\rm[C_{II}]$ line flux
emerging from a plane-parallel gas slab. The toy model provides useful
insights for understanding the $\rm[C_{II}]$ emission of galaxies. In Section
5.2, we derive an important scaling relation of galaxies between their
$L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio and other physical properties. Based on
this scaling relation, we investigate the cause of the $\rm[C_{II}]$ deficit
of galaxies in Section 5.3. Finally, in Section 5.4, we show the presence of
two distinct physical regimes where the main reason for the $\rm[C_{II}]$
deficit of galaxies is different.

###  5.1 Insights from the plane parallel slab model

The $\rm[C_{II}]$ line flux emerging from a plane-parallel slab that is
irradiated by an external radiation field has recently been studied by Ferrara
et al. (2019, hereafter F19). In this section, we summarize the key points of
the F19 model. We refer interested readers to F19 for the details.

The plane-parallel slab can be characterized by three distinct zones based on
the ionization structures of gas, as has been discussed in Section 3.1. Right
beneath the surface of the slab, ionizing radiation ($E_{\gamma}{}>{}13.6$ eV)
creates a $\rm H_{II}$ region extending to a gas column density $N_{\rm s}$
(Zone I), where both hydrogen and carbon are ionized. Beyond $N_{\rm s}$,
hydrogen becomes neutral but LW ($11.2<E_{\gamma}<13.6$ eV) photons maintain
carbon in the singly ionized state (Zone II). The LW photons become fully
absorbed by dust and $\rm H_{2}$ at a column density $N_{\rm F}$, beyond which
hydrogen turns into $\rm H_{2}$ and carbon becomes neutral (Zone III). We have
shown in Fig. 2 the ionization structures of a plane-parallel slab calculated
by CLOUDY as an example (see also Fig. 1 of F19 for a schematic plot).

$N_{\rm s}$ can be estimated by equating the photo-ionization rate to the
recombination rate of hydrogen inside the $\rm H_{II}$ region (Zone I)
assuming that dust extinction is negligible, which can be expressed as (see
Appendix C for the details)

$N_{\rm s}=n_{\rm H}l_{\rm s}=\frac{Uc}{\alpha_{\rm B}}\approx 10^{23}U\;\rm
cm^{-2},$ (8)

where $l_{\rm s}$ is the distance from the surface of the slab to the end of
Zone I, $U$ parameter represents the ionizing photon-to-gas density ratio,
i.e.

$U=\frac{n_{\gamma}}{n_{\rm H}},$ (9)

$c$ represents the speed of light, and $\alpha_{\rm B}{}={}2.6\times
10^{-13}\,\rm cm^{3}\,s^{-1}$ is the Case-B recombination coefficient at gas
temperature $T{}\approx{}10^{4}\,\rm K$ (Ferland et al., 1992). For a slab
with density $n_{\rm H}{}={}50\,\rm cm^{-3}$ that is exposed to a radiation
field having $G{}={}200\,G_{0}$, we obtain $U{}={}n_{\gamma}/n_{\rm
H}{}\approx{}1.3\times 10^{-3}$ at and near the surface of the slab. Using
equation (8), we obtain $N_{\rm s}{}\approx{}1.3\times 10^{20}\,\rm cm^{-2}$.
We can see from Fig. 2 that this estimated $N_{\rm s}$ is in good agreement
with the result computed by CLOUDY, in particular, for the metal-poor model
(with $Z_{\rm gas}{}={}0.1\,Z_{\odot}$; right panels of Fig. 2), where dust
extinction in the $\rm H_{II}$ (Zone I) region is negligible. $N_{\rm s}$ of
the metal-rich model (with $Z_{\rm gas}{}={}Z_{\odot}$; left panels of Fig. 2)
is smaller by about 1/4 due to higher absorption of ionizing photons by dust.

$N_{\rm F}$ can be estimated using

$N_{\rm F}=n_{\rm H}l_{\rm F}=\bar{\sigma}^{-1}_{\rm d}\ln(1+10^{5}\omega
U),\\\ $ (10)

which is obtained by performing a RT calculation (Sternberg et al., 2014) that
accounts for the absorption of LW photons by dust grains and $\rm H_{2}$ as
light propagates through the slab. In equation (10), $l_{\rm F}$ represents
the distance between the surface of the slab and the end of Zone II,

$\bar{\sigma}_{\rm d}=5.9\times 10^{-22}\left(\frac{\delta_{\rm
dgr}}{\delta_{\rm dgr,\,MW}}\right)\,\rm cm^{2}$ (11)

represents the flux-weighted dust extinction cross section per H-atom, and

$\omega=\frac{1}{1+0.9(\delta_{\rm dgr}/\delta_{\rm dgr,\,MW})^{1/2}},$ (12)

where $\delta_{\rm dgr,\,MW}=10^{-2}$ represents the Galactic dust-to-gas
ratio (see e.g. Gilmore et al., 1989; Sodroski et al., 1997; Zubko et al.,
2004; Rémy-Ruyer et al., 2014; McKinnon et al., 2016; Li et al., 2019). For
the two models where $Z_{\rm gas}{}={}Z_{\odot}$ and $Z_{\rm
gas}{}={}0.1Z_{\odot}$, $N_{\rm F}$ is expected to be $\sim 10^{21}\,\rm
cm^{-2}$ and $\sim 10^{22}\,\rm cm^{-2}$ (according to equation 10),
respectively. This result is again in good agreement with the prediction of
CLOUDY as shown in Fig. 2.

Now we can derive the $\rm[C_{\rm II}]$ line flux ($F_{\rm[C_{II}]}$) emerging
from a plane-parallel slab following the three-zone model. $F_{\rm[C_{II}]}$
can be calculated using

$F_{\rm[C_{II}]}=\,\Lambda^{\rm(1)}_{\rm[C_{II}]}\,l_{\rm
s}+\Lambda^{\rm(2)}_{\rm[C_{II}]}\,(l_{\rm F}-l_{\rm s}),$ (13)

where the first and second terms correspond to the contribution of
$\rm[C_{II}]$ line flux by Zone I and Zone II, respectively.
$\Lambda^{\rm(1)}_{\rm[C_{II}]}$ ($\Lambda^{\rm(2)}_{\rm[C_{II}]}$) in the
above equation represents the $\rm[C_{II}]$ cooling rate ($\rm
erg\,s^{-1}\,cm^{-3}$) of gas in Zone I (II). In the above and the following
equations, the superscript “(1)" (“(2)") indicates the properties of gas in
Zone I (II). We neglect the $\rm[C_{II}]$ emission from the $\rm H_{2}$ region
(Zone III).

Equation (13) can be rewritten as (see Appendix D for the details)

$\displaystyle F_{\rm[C_{II}]}\approx$ $\displaystyle\,h_{\rm
P}\nu_{\rm[C_{II}]}\left(\frac{g_{\rm u}}{g_{\rm l}}\right)R^{\rm e}_{\rm
ul}(T^{(1)})n^{(1)}_{\rm C_{II}}n^{(1)}_{\rm e}l_{\rm s}$
$\displaystyle+\frac{2}{5}h_{\rm P}\nu_{\rm[C_{II}]}\left(\frac{g_{\rm
u}}{g_{\rm l}}\right)R^{\rm H_{I}}_{\rm ul}(T^{(2)})n^{(2)}_{\rm
C_{II}}n^{(2)}_{\rm H_{I}}(l_{\rm F}-l_{\rm s}),$ (14)

where $h_{\rm P}$ is the Planck constant, $\nu_{\rm[C_{II}]}{}={}1900.5\,\rm
GHz$ is the rest-frame frequency of the $\rm[C_{II}]$ line, $g_{\rm u}{}={}4$
($g_{\rm l}{}={}2$) is the statistical weight of the ${}^{2}P_{3/2}$
(${}^{2}P_{1/2}$) state, $R^{\rm e}_{\rm ul}$ ($R^{\rm H_{I}}_{\rm ul}$) is
the downward rate coefficient ($\rm s^{-1}$) for ${\rm C_{II}}+e^{-}$ ($\rm
C_{II}+H^{0}$) collision, and $n^{(1)}_{\rm C_{II}}$ (and $n^{(2)}_{\rm
C_{II}}$), $n^{(1)}_{\rm e}$ and $n^{(2)}_{\rm H_{I}}$ represent the number
density of $\rm C_{II}$ ion, electron and $\rm H$ atom, respectively. Equation
(14) implies that in Zone I (II), the main collision partner of $\rm C_{II}$
ion is electron ($\rm H$ atom). Knowing that $n^{(1)}_{\rm e}\approx n_{\rm
H}$ and $n^{(2)}_{\rm H_{I}}\approx n_{\rm H}$ (see the upper panels of Fig.
2), we can rewrite equation (14) to be

$F_{\rm[C_{II}]}=h_{\rm P}\nu_{\rm[C_{II}]}\left(\frac{g_{\rm u}}{g_{\rm
l}}\right)\left[R^{\rm e}_{\rm ul}n^{(1)}_{\rm C_{II}}N_{\rm
s}+\frac{2}{5}R^{\rm H_{I}}_{\rm ul}n^{(2)}_{\rm C_{II}}(N_{\rm F}-N_{\rm
s})\right],$ (15)

where $N_{\rm F}{}={}n_{\rm H}\,l_{\rm F}$ and $N_{\rm s}{}={}n_{\rm
H}\,l_{\rm s}$. Furthermore, $n^{(1)}_{\rm C_{II}}$ and $n^{(2)}_{\rm C_{II}}$
in the above equation can be rewritten as

$n^{(1)}_{\rm C_{II}}=n_{\rm H}x^{(1)}_{\rm C_{II}}\mathcal{A}_{\rm
C}\;\;\;{\rm and}\;\;\;n^{(2)}_{\rm C_{II}}=n_{\rm H}x^{(2)}_{\rm
C_{II}}\mathcal{A}_{\rm C},$ (16)

where

$\mathcal{A}_{\rm C}=2.5\times 10^{-4}\left(\frac{Z_{\rm
gas}}{Z_{\odot}}\right)$ (17)

represents the abundance of carbon. The numerical factor $2.5\times 10^{-4}$
in equation (17) is the abundance of carbon in the solar photosphere (Asplund
et al., 2009). $x^{(1)}_{\rm C_{II}}$ ($x^{(2)}_{\rm C_{II}}$) in equation
(16) represents the fraction of carbon in $\rm C_{II}$ form in Zone I (II).
$x^{(1)}_{\rm C_{II}}$ is roughly inversely proportional to $U$ (see Appendix
E), whereas $x^{(2)}_{\rm C_{II}}{}\approx{}1$ (see the middle panels of Fig.
2). By inputting equation (16) to equation (15), we get

$\displaystyle F_{\rm[C_{II}]}=\,$ $\displaystyle n_{\rm H}\mathcal{A}_{\rm
C}N_{\rm F}h_{\rm P}\nu_{\rm[C_{II}]}\left(\frac{g_{\rm u}}{g_{\rm
l}}\right)\left[R^{\rm e}_{\rm ul}x^{(1)}_{\rm C_{II}}\left(\frac{N_{\rm
s}}{N_{\rm F}}\right)+\frac{2}{5}R^{\rm H_{I}}_{\rm ul}\left(\frac{N_{\rm
F}-N_{\rm s}}{N_{\rm F}}\right)\right]$ $\displaystyle=\,$ $\displaystyle
n_{\rm H}\mathcal{A}_{\rm C}N_{\rm F}\bar{\epsilon}_{\rm[C_{II}],\,slab},$
(18)

where we define

$\displaystyle\bar{\epsilon}_{\rm[C_{II}],\,slab}$ $\displaystyle=h_{\rm
P}\nu_{\rm[C_{II}]}\left(\frac{g_{\rm u}}{g_{\rm l}}\right)\left[R^{\rm
e}_{\rm ul}x^{(1)}_{\rm C_{II}}\left(\frac{N_{\rm s}}{N_{\rm
F}}\right)+\frac{2}{5}R^{\rm H_{I}}_{\rm ul}\left(\frac{N_{\rm F}-N_{\rm
s}}{N_{\rm F}}\right)\right]$ $\displaystyle\equiv\alpha\,x^{(1)}_{\rm
C_{II}}\left(\frac{N_{\rm s}}{N_{\rm F}}\right)+\gamma\left(\frac{N_{\rm
F}-N_{\rm s}}{N_{\rm F}}\right)$ (19)

as the specific $\rm[C_{II}]$ cooling rate of the slab ($\rm
erg\,s^{-1}\,cm^{3}$). It can be shown that (see Appendix D for the details)

$\displaystyle\alpha$ $\displaystyle\equiv h_{\rm
P}\nu_{\rm[C_{II}]}\left(\frac{g_{\rm u}}{g_{\rm l}}\right)R^{\rm e}_{\rm
ul}(T^{(1)})$ $\displaystyle\approx 10^{-21}\,{\rm
erg\,s^{-1}\,cm^{3}}\;(T^{(1)}\approx 10^{4}\,\rm K)$ (20)

and

$\displaystyle\gamma$ $\displaystyle\equiv\frac{2}{5}h_{\rm
P}\nu_{\rm[C_{II}]}\left(\frac{g_{\rm u}}{g_{\rm l}}\right)R^{\rm H_{I}}_{\rm
ul}(T^{(2)})$ $\displaystyle\approx 10^{-23}\,{\rm
erg\,s^{-1}\,cm^{3}}\;(T^{(2)}\approx 10^{2}\,\rm K)$ (21)

From equation (19), we see that $\bar{\epsilon}_{\rm[C_{II}],\,slab}$ depends
on $x^{(1)}_{\rm C_{II}}$, $N_{\rm s}$ and $N_{\rm F}$, and varies typically
within the range $10^{-23}-10^{-21}\;{\rm erg\,s^{-1}\,cm^{3}}$.

Likewise, we can derive the $\rm[C_{II}]$ luminosity of a spherical uniform
gas cloud ($L_{\rm[C_{II}],\,cl}$). $L_{\rm[C_{II}],\,cl}$ can be expressed as

$L_{\rm[C_{II}],\,cl}=\begin{cases}4\pi\bigints_{\,0}^{R_{\rm
cl}}\Lambda^{(1)}_{\rm[C_{II}]}r^{2}{\rm d}r\;\;\;\;({\rm if}\;l_{\rm s}\geq
R_{\rm cl})\\\ \\\ 4\pi\left[\bigints_{R_{\rm cl}-l_{\rm s}}^{R_{\rm
cl}}\Lambda^{(1)}_{\rm[C_{II}]}r^{2}{\rm d}r+\bigints_{R_{\rm cl}-{\rm
min}(l_{\rm F},R_{\rm cl})}^{R_{\rm cl}-l_{\rm
s}}\Lambda^{(2)}_{\rm[C_{II}]}r^{2}{\rm d}r\right].\\\ \;\;({\rm if}\;l_{\rm
s}<R_{\rm cl})\end{cases}$ (22)

The first condition of equation (22) (i.e. $l_{\rm s}{}\geq{}R_{\rm cl}$)
corresponds to when the cloud is fully ionized, while the second condition
(i.e. $l_{\rm s}{}<{}R_{\rm cl}$) corresponds to when neutral hydrogen region
(Zone II) forms in the cloud. Through simple re-arrangement,
$L_{\rm[C_{II}],\,cl}$ can be expressed as

$L_{\rm[C_{II}],\,cl}=f_{\rm[C_{II}],\,cl}\left(\frac{M_{\rm cl}}{\mu_{\rm
H}m_{\rm H}}\right)n_{\rm H}\mathcal{A}_{\rm
C}\bar{\epsilon}_{\rm[C_{II}],\,cl},$ (23)

where $f_{\rm[C_{II}],\,cl}$ represents the fraction of the gas mass that is
in $\rm H_{II}$ or $\rm H_{I}$ phases (Zone I and Zone II), $M_{\rm cl}$
indicates the mass of the gas cloud, $\mu_{\rm H}$ is the mean molecular
weight of the gas and $m_{\rm H}$ represents the proton mass. By definition,
$f_{\rm[C_{II}],\,cl}{}={}1$ when $l_{\rm F}{}>{}R_{\rm cl}$ and the cloud
becomes $\rm H_{2}$-free. $\bar{\epsilon}_{\rm[C_{II}],\,cl}$ in equation (22)
represents the specific $\rm[C_{II}]$ cooling rate of the spherical uniform
cloud, which accounts for the relative contribution of the $\rm[C_{II}]$
emission from $\rm H_{II}$ and $\rm H_{I}$ regions ($10^{-23}{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 60\relax$}\hss}\mathchar
29208\relax$}}{}\bar{\epsilon}_{\rm[C_{II}],\,cl}{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 60\relax$}\hss}\mathchar
29208\relax$}}{}10^{-21}\,\rm erg\,s^{-1}\,cm^{3}$). Like
$\bar{\epsilon}_{\rm[C_{II}],\,slab}$ for the plane-parallel slab (equation
19), $\bar{\epsilon}_{\rm[C_{II}],\,cl}$ depends on $x^{(1)}_{\rm C_{II}}$,
$N_{\rm s}$ and $N_{\rm F}$ but have different functional relation with these
parameters due to the difference in geometry. We refer the readers to Appendix
F, where we present the derivation for $\bar{\epsilon}_{\rm[C_{II}],\,cl}$.

Note that we do not take into account the effects of the CMB background on the
$\rm[C_{II}]$ cooling rate of gas in the analytic solution for the toy models
presented in this section. While the CMB sets a floor for the excitation (or
spin) temperature of gas and boosts the upper level (${}^{2}P_{3/2}$)
population of the $\rm[C_{II}]$ transition (‘CMB heating’), it acts as a
background against which the $\rm[C_{II}]$ line is measured (‘CMB
attenuation’). The CMB effects (both heating and attenuation) can be important
for the $\rm[C_{II}]$ emission from the low-density and low-temperature gas in
galaxies at high redshifts ($z{}\mathbin{\lower 3.0pt\hbox{$\hbox
to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}{}6$)
(see Appendix D). We find, however, that the total $\rm[C_{II}]$ luminosity of
the FIRE sample is not significantly affected by the CMB (in agreement with
Lagache et al. 2018). This is due to the fact that the bulk of the
$\rm[C_{II}]$ luminosity of the high-$z$ ($z{}\geq{}6$) galaxies in our sample
originates from the gas of densities in excess of the densities where the CMB
effects become important.

###  5.2 A scaling relation for the $L_{\rm[C_{II}]}$/SFR ratio of galaxies

Figure 10: The relation between the $L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio and
$f_{\rm[C_{II}]}\,\bar{Z}_{\rm gas}\,t_{\rm dep}\bar{n}_{\rm
gas}\bar{\epsilon}_{\rm[C_{II}]}$ of the FIRE galaxies at different redshifts
(cyan stars for $z{}={}0$, yellow hexagons for $z{}={}1$, red triangles for
$z{}={}2$, blue squares for $z{}={}3$, magenta circles for $z{}={}4$, green
diamonds for $z{}={}6$ and purple downward triangles for $z{}={}8$). The
orange shaded area indicates the $L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio of the
local star-forming galaxy sample measured by Herrera-Camus et al. (2015). The
solid black line shows the best linear fit to the data of the FIRE galaxies.
The FIRE galaxies show a strong linear correlation (Pearson correlation
coefficient $\rho{}=0.96$) between $L_{\rm[C_{II}]}{}/{}\rm SFR$ and
$f_{\rm[C_{II}]}\,\bar{Z}_{\rm gas}\,t_{\rm dep}\bar{n}_{\rm
gas}\bar{\epsilon}_{\rm[C_{II}]}$.  Figure 11: The gas density PDFs of two
selected FIRE galaxies at $z{}={}0$ (upper panel) and $z{}={}6$ (lower panel).
The $z{}={}6$ galaxy has a relatively denser ISM. In the two panels, magenta
lines indicate the luminosity-weighted PDFs. Solid, dotted and dashed magenta
lines represent the result of the total gas, $\rm H_{II}$ gas (Zone I) and
$\rm H_{I}$ gas (Zone II) in the ISM, respectively. In the two panels, shaded
areas show the mass-weighted gas density PDFs. Grey, red, green and blue areas
represent the result of total gas, $\rm H_{II}$ gas (Zone I), $\rm H_{I}$ gas
(Zone II) and $\rm H_{2}$ gas (Zone III), respectively.

We have summarized the key points of the F19 model for the structures of a
plane-parallel gas slab that is exposed to an external radiation field. We
then derive the $\rm[C_{II}]$ luminosity of a uniform spherical gas cloud
(equation 23). Following the results of the toy models, we now present a
scaling relation for the $\rm[C_{II}]$ luminosity of galaxies, based on which
we will explore the origins of the $\rm[C_{II}]$ deficit of galaxies.

From equation (23), one would expect that the $\rm[C_{II}]$ luminosity
($L_{\rm[C_{II}]}$) of galaxy has a similar expression, i.e.

$L_{\rm[C_{II}]}\sim f_{\rm[C_{II}]}\left(\frac{M_{\rm gas}}{\mu m_{\rm
H}}\right)\bar{n}_{\rm gas}\bar{\mathcal{A}}_{\rm
C}\bar{\epsilon}_{\rm[C_{II}]},$ (24)

where we have replaced $M_{\rm cl}$ in equation (23) by $M_{\rm gas}$, i.e.
the gas mass of galaxy191919We calculate the gas mass of galaxy using the gas
particles within $0.1R_{\rm vir}$ around the DM halo centre having
$T{}<{}10^{5}$ K.. $f_{\rm[C_{II}]}$ ($={}1-f_{\rm H_{2}}$) in the above
equation represents the fraction of the total gas mass in ionized or neutral
atomic hydrogen forms (Zone I and Zone II), and $\bar{n}_{\rm gas}$,
$\bar{\mathcal{A}}_{\rm C}$ and $\bar{\epsilon}_{\rm[C_{II}]}$ represent the
statistical average of gas density, carbon abundance and specific
$\rm[C_{II}]$ cooling rate of the galaxy, respectively. We can then divide the
two sides of equation (24) by galaxy SFR, and obtain

$\frac{L_{\rm[C_{II}]}}{\rm SFR}\sim f_{\rm[C_{II}]}t_{\rm dep}\bar{n}_{\rm
gas}\bar{\mathcal{A}}_{\rm C}\bar{\epsilon}_{\rm[C_{II}]}(\mu m_{\rm H})^{-1}$
(25)

where

$t_{\rm dep}\equiv\frac{M_{\rm gas}}{\rm SFR}$ (26)

is the gas depletion time of the galaxy (e.g., Genzel et al., 2015; Tacchella
et al., 2016; Semenov et al., 2017; Scoville et al., 2017; Tacconi et al.,
2018; Feldmann, 2020). Through further re-arrangement, equation (25) can be
expressed as

$\displaystyle\frac{L_{\rm[C_{II}]}/L_{\odot}}{{\rm SFR}/(M_{\odot}\,\rm
yr^{-1})}{}\sim{}$ $\displaystyle 4\times
10^{6}f_{\rm[C_{II}]}\left(\frac{\bar{Z}_{\rm gas}}{Z_{\odot}}\right)$
$\displaystyle\times\left(\frac{t_{\rm dep}}{\rm
Gyr}\right)\left(\frac{\bar{n}_{\rm gas}}{\rm
cm^{-3}}\right)\left(\frac{\bar{\epsilon}_{\rm[C_{II}]}}{10^{-22}\,\rm
erg\,s^{-1}\,cm^{3}}\right)$ (27) $\displaystyle\propto$ $\displaystyle
f_{\rm[C_{II}]}\,\bar{Z}_{\rm gas}\,t_{\rm dep}\,\bar{n}_{\rm
gas}\,\bar{\epsilon}_{\rm[C_{II}]},$ (28)

where we have replaced the carbon abundance $\bar{\mathcal{A}}_{\rm C}$ in
equation (25) by metallicity $\bar{Z}_{\rm gas}$ using equation (17).

Equation (27) indicates that the $L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio of galaxy
is determined by five physical parameters, $f_{\rm[C_{II}]}$, $\bar{Z}_{\rm
gas}$, $t_{\rm dep}$, $\bar{n}_{\rm gas}$ and $\bar{\epsilon}_{\rm[C_{II}]}$.
Whilst $f_{\rm[C_{II}]}$ and $t_{\rm dep}$ are global properties of galaxy,
which are well defined, the other three parameters are the statistical average
of the corresponding physical properties of all different ‘gas clouds’ in the
ISM. This contrasts with the toy models (uniform plane-parallel slab or
spherical cloud), where each of these properties (gas density, gas
metallicity, and the specific $\rm[C_{II}]$ cooling rate) has a single,
definite value.

In Fig. 10, we show the relation between the $L_{\rm[C_{II}]}{}/{}\rm SFR$
ratio of the FIRE sample at $z{}={}0-8$ and their
$f_{\rm[C_{II}]}\,\bar{Z}_{\rm gas}\,t_{\rm dep}\,\bar{n}_{\rm
gas}\bar{\epsilon}_{\rm[C_{II}]}$, where $\bar{n}_{\rm gas}$, $\bar{Z}_{\rm
gas}$ and $\bar{\epsilon}_{\rm[C_{II}]}$ are the luminosity-weighted gas
density 202020Note that we use the ‘luminosity-weighted median gas density’,
i.e. the gas density at the $50^{\rm th}$ percentile of $\rm[C_{II}]$
luminosity, instead of the ‘luminosity-weighted mean gas density’. This is
because the gas density PDF of galaxy resembles a lognormal function,
exhibiting an elongated tail at the high density end. Under certain
circumstances, the ‘mean gas density’ can be strongly biased by the
$\rm[C_{II}]$-emitting gas at the highest density ($n_{\rm H}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}10^{3}\,\rm cm^{-3}$, see the lower panel of Fig. 11), and hence
is not statistically representative for the part of the gas that contributes
the bulk of the $\rm[C_{II}]$ emission of galaxy. Throughout this paper, we
use the term ‘luminosity-weighted’ for simplicity when we refer to
‘luminosity-weighted median’. Similarly, ‘mass-weighted’ in this paper refers
to ‘mass-weighted median’, i.e. value at the $50^{\rm th}$ percentile of mass.
In Appendix G, we show explicitly the difference between the ‘luminosity-
weighted median gas density’ and the ‘luminosity-weighted mean gas density’ of
the FIRE galaxy sample. The former is higher by a factor of $\sim{}5$ on
average., gas metallicity212121Unlike the gas densities, the luminosity-
weighted mean and the luminosity-weighted median gas metallicity are similar.
Both are close to the mass-weighted gas metallicity (see Appendix H). and
specific $\rm[C_{II}]$ cooling rate of the galaxies, respectively. Our FIRE
sample follows a clear linear scaling relation on the diagram (Pearson
correlation coefficient $\rho{}={}0.96$), which is in agreement with equation
(27).

Figure 12: The relation between the $\rm[C_{II}]$ luminosity-weighted gas
density ($\bar{n}_{\rm gas}$) and the mass-weighted density of the $\rm
H_{II}$ ($\bar{n}_{\rm H_{II},\,MW}$), $\rm H_{I}$ ($\bar{n}_{\rm
H_{I},\,MW}$) and $\rm H_{2}$ gas ($\bar{n}_{\rm H_{2},\,MW}$) of the FIRE
galaxies at $z{}={}0-8$. Filled, empty and semi-transparent symbols correspond
to the $\bar{n}_{\rm H_{I},\,MW}$ vs. $\bar{n}_{\rm gas}$, the $\bar{n}_{\rm
H_{II},\,MW}$ vs. $\bar{n}_{\rm gas}$ and the $\bar{n}_{\rm H_{2},\,MW}$ vs.
$\bar{n}_{\rm gas}$ relations, respectively. The diagonal line indicates the
one-to-one relationship. It can be seen that $\bar{n}_{\rm gas}$ appears to be
close to $\bar{n}_{\rm H_{I},\,MW}$, both being systematically lower (higher)
than $\bar{n}_{\rm H_{2},\,MW}$ ($\bar{n}_{\rm H_{II},\,MW}$).

Figure 13: The fraction of the total $\rm[C_{II}]$ luminosity of the FIRE
galaxy sample that originates from the $\rm H_{II}$ gas region (Zone I) as a
function of their SFR. In the left and right panels, the data points are
colour-coded by the redshift and the effective $\rm[C_{II}]$ cooling rate
($\bar{\epsilon}_{\rm[C_{II}]}$) of the galaxies, respectively.

One important question is which part of the ISM contributes the most
$\rm[C_{II}]$ emission of a galaxy. The ISM of a galaxy spans a wide range of
density over several orders of magnitude, with the dense (diffuse) regions
being dominated by $\rm H_{2}$ ($\rm H_{II}$) gas. In Fig. 11, we show the
$\rm[C_{II}]$ luminosity-weighted (magenta lines) and gas mass-weighted (grey
and coloured shaded areas) probability density functions (PDFs) of $n_{\rm H}$
for two selected FIRE galaxies at $z{}={}0$ (upper panel) and $z{}={}6$ (lower
panel). It can be seen in the figure that the $\rm[C_{II}]$ emission of FIRE
galaxies originates from gas spanning a wide range of density across several
orders of magnitude. Interestingly, we find that the luminosity-weighted gas
density ($\bar{n}_{\rm gas}$) of FIRE galaxies is close to the mass-weighted
density of the $\rm H_{I}$ gas ($\bar{n}_{\rm H_{I},\,MW}$) of the ISM, both
of which are much higher (lower) than the mass-weighted density of the $\rm
H_{II}$ ($\rm H_{2}$) gas. This can be more clearly seen from Fig. 12, where
we show the relation between $\bar{n}_{\rm gas}$ and the mass-weighted gas
density of the $\rm H_{II}$, $\rm H_{I}$ and $\rm H_{2}$ gas for the FIRE
sample at $z{}={}0-8$.

This can be understood as follows. The bulk of the diffuse, ionized $\rm
H_{II}$ gas is inefficient at producing $\rm[C_{II}]$ emission due to the low
gas density ($L_{\rm[C_{II}],\,cl}{}/{}M_{\rm cl}{}\propto{}n_{\rm H}$, see
equation 23). On the other hand, in the densest regions of the ISM, where gas
becomes mostly in molecular hydrogen form (Zone III), not much $\rm[C_{II}]$
emission is produced due to the scarcity of the amount of ionized carbon
(which exists mostly in Zone I and Zone II) in those regions. As a result, the
bulk of the $\rm[C_{II}]$ luminosity of the FIRE galaxies at $z{}={}0-8$
originates from the gas at the intermediate gas density range.

It should be noted, however, that though the luminosity-weighted gas density
of the FIRE galaxies coincide with $\bar{n}_{\rm H_{I}}$ and is systematically
higher than $\bar{n}_{\rm H_{II}}$, more of the $\rm[C_{II}]$ emission of the
FIRE galaxies actually originates from the $\rm H_{II}$ gas (Zone I) instead
of the $\rm H_{I}$ gas (Zone II). Specifically, it is the $\rm H_{II}$ gas
layer at the surface of the $\rm H_{I}$-rich clouds having
$n\approx\bar{n}_{\rm gas}$ that contributes most of the $\rm[C_{II}]$
emission of the galaxies. In Fig. 13, we show the fraction of the total
$\rm[C_{II}]$ luminosity of the FIRE galaxies produced by the $\rm H_{II}$
gas, i.e. $L_{\rm[C_{II}],\,H_{II}}{}/{}L_{\rm[C_{II}]}$ (note:
$L_{\rm[C_{II}],\,H_{II}}$ is the sum of the $\rm[C_{II}]$ luminosity
originating from Zone I in each ‘gas cloud’), as a function of the galaxy SFR.
It can be seen that the $\rm H_{II}$ gas contributes about $60\%{}-{}80\%$ of
the total luminosity of the galaxies, except for the few massive starburst
galaxies at ${\rm SFR}{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}{}100\,M_{\odot}\,\rm
yr^{-1}$, which show a reduced fractional contribution by the $\rm H_{II}$ gas
down to $\sim 50\%$. The remaining fraction of the $\rm[C_{II}]$ luminosity
originates mainly from the $\rm H_{I}$ gas. The contribution of the
$\rm[C_{II}]$ luminosity by the $\rm H_{2}$ gas (Zone III) is negligible
($<{}2\%$ of the total luminosity for all the FIRE galaxies).

Figure 14:  The relation between $L_{\rm[C_{II}]}{}/{}{\rm SFR}\,t^{-1}_{\rm
dep}$ (upper left), $L_{\rm[C_{II}]}{}/{}{\rm SFR}\,\bar{Z}^{-1}_{\rm gas}$
(upper right), $L_{\rm[C_{II}]}{}/{}{\rm SFR}\,\bar{n}^{-1}_{\rm gas}$ (lower
left), $L_{\rm[C_{II}]}{}/{}{\rm SFR}\,f^{-1}_{\rm[C_{II}]}$ (lower middle)
and $L_{\rm[C_{II}]}{}/{}{\rm SFR}\,\bar{\epsilon}^{-1}_{\rm[C_{II}]}$ (lower
right) against SFR of the FIRE galaxies at different redshifts (cyan stars for
$z{}={}0$, yellow hexagons for $z{}={}1$, red triangles for $z{}={}2$, blue
squares for $z{}={}3$, magenta circles for $z{}={}4$, green diamonds for
$z{}={}6$ and purple downward triangles for $z{}={}8$). In each panel, large
(small) symbols represent the galaxies having $L_{\rm
IR}{}\geq{}10^{11}\,L_{\odot}$ ($L_{\rm IR}{}<{}10^{11}\,L_{\odot}$). The
$\rm[C_{II}]$ deficit at high $L_{\rm IR}$ (at redshift $z{}>{}5$) vanishes in
the parameter space of $L_{\rm[C_{II}]}{}/{}{\rm SFR}\,t^{-1}_{\rm dep}$
($L_{\rm[C_{II}]}{}/{}{\rm SFR}\,\bar{Z}^{-1}_{\rm gas}$), indicating that low
$t_{\rm dep}$ (low gas metallicity) is the main driver of the $\rm[C_{II}]$
deficit at high $L_{\rm IR}$ (at high redshifts).

The relatively high $\rm[C_{II}]$ cooling rate in the $\rm H_{II}$ phase
(note: $\alpha{}\approx{}100\,\gamma$, see equation 20 and 21) explains why a
relatively small amount of $\rm H_{II}$ gas at $n_{\rm
H}{}\approx{}\bar{n}_{\rm gas}$ can produce a larger fraction of the
$\rm[C_{II}]$ luminosity of the galaxies than the $\rm H_{I}$ gas. Also, one
would expect a correlation between the fractional contribution of the
$\rm[C_{II}]$ luminosity by the $\rm H_{II}$ gas (Zone I) and the effective
$\rm[C_{II}]$ line cooling rate (luminosity-weighted),
$\bar{\epsilon}_{\rm[C_{II}]}$, of the galaxies. This is indeed the case, as
is shown in the right panel of Fig. 13.

###  5.3 The physical origins of $\rm[C_{II}]$ deficit of galaxies

Figure 15: $\Delta({\rm log}\,L_{\rm[C_{II}]})$ as a function of $t_{\rm dep}$
(upper left), $\bar{Z}_{\rm gas}$ (upper right), $\bar{n}_{\rm gas}$ (lower
left), $f_{\rm[C_{II}]}$ (lower middle) and $\bar{\epsilon}_{\rm[C_{II}]}$
(lower right) of the FIRE galaxies at different redshifts, where $\Delta({\rm
log}\,L_{\rm[C_{II}]})$ represents the offset between the
$L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio of the galaxies and the observed mean
value of the local star-forming sample of H15 ($4.3\times
10^{7}\,L_{\odot}\,M^{-1}_{\odot}\,\rm yr$). In each panel, large (small)
symbols correspond to the FIRE galaxies having $L_{\rm
IR}{}\geq{}10^{11}\,L_{\odot}$ ($L_{\rm IR}{}<{}10^{11}\,L_{\odot}$).

In the previous section, we have presented a simple analytic expression for
the $L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio of galaxies (equation 27) found with
the FIRE galaxy sample. Based on this result, we will probe in this section
the origins of the observed $\rm[C_{II}]$ deficit of galaxies.

Equation (27) indicates that the $L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio of the
galaxies depends on five parameters: the fraction of gas in the
$\rm[C_{II}]$-emitting regions (Zone I and Zone II), the depletion time (i.e.
gas mass per unit SFR), gas density, gas metallicity and the specific
$\rm[C_{II}]$ cooling rate. Hence, the $\rm[C_{II}]$ deficit of the galaxies
can, in principle, be due to a strong deficit of one or few of the five
parameters with respect to the observed local star-forming samples (e.g., L11,
L14 and H15). It should be noted that the observed $\rm[C_{II}]$ deficit in
the two regimes, high redshifts and high $L_{\rm IR}$, may not be due to the
same reason. We will separately discuss the origin of the $\rm[C_{II}]$
deficit in these two regimes in this section.

To reveal what parameters drive the $\rm[C_{II}]$ deficit of the FIRE sample,
we divide $L_{\rm[C_{II}]}{}/{}\rm SFR$ by each of these parameters and check
whether in the new parameter spaces (i.e. $L_{\rm[C_{II}]}{}{\rm
SFR}^{-1}{}f^{-1}_{\rm[C_{II}]}$, $L_{\rm[C_{II}]}{}{\rm
SFR}^{-1}{}\bar{Z}^{-1}_{\rm gas}$, $L_{\rm[C_{II}]}{}{\rm
SFR}^{-1}{}\bar{n}^{-1}_{\rm gas}$, $L_{\rm[C_{II}]}{}{\rm
SFR}^{-1}{}t^{-1}_{\rm dep}$ and $L_{\rm[C_{II}]}{}{\rm
SFR}^{-1}{}\epsilon^{-1}_{\rm[C_{II}]}$), the $\rm[C_{II}]$ deficit becomes
alleviated (or even vanishes). We show in Fig. 14 the relation between the new
parameters and the SFR for the galaxies in our sample at different redshifts.
In Fig. 15, we also show how $L_{\rm[C_{II}]}{}/{}\rm SFR$ of the FIRE sample
depends on $f_{\rm[C_{II}]}$, $\bar{Z}_{\rm gas}$, $\bar{n}_{\rm gas}$,
$t_{\rm dep}$ and $\bar{\epsilon}_{\rm[C_{II}]}$ in separate panels. Readers
can find the mean of $f_{\rm[C_{II}]}$, $\bar{Z}_{\rm gas}$, $\bar{n}_{\rm
gas}$, $t_{\rm dep}$ and $\bar{\epsilon}_{\rm[C_{II}]}$ and the five new
parameters of the FIRE sample at each redshift in Table 11 and Table 10,
respectively.

Table 10: The change of the mean $L_{\rm[C_{II}]}{}/{}\rm SFR$, $L_{\rm[C_{II}]}{}/{}{\rm SFR}\,t^{-1}_{\rm dep}$, $L_{\rm[C_{II}]}{}/{}{\rm SFR}\,\bar{Z}^{-1}_{\rm gas}$, $L_{\rm[C_{II}]}{}/{}{\rm SFR}\,\bar{n}^{-1}_{\rm gas}$, $L_{\rm[C_{II}]}{}/{}{\rm SFR}\,f^{-1}_{\rm[C_{II}]}$ and $L_{\rm[C_{II}]}{}/{}{\rm SFR}\,\bar{\epsilon}^{-1}_{\rm[C_{II}]}$ of the FIRE sample from redshift $z$ to $z{}={}0$.  $z$ | $\Delta\,{\rm log}\,\left(\frac{L_{\rm[C_{II}]}}{\rm SFR}\right)$ | $\Delta\,{\rm log}\,\left(\frac{L_{\rm[C_{II}]}}{\rm SFR}t^{-1}_{\rm dep}\right)$ | $\Delta\,{\rm log}\,\left(\frac{L_{\rm[C_{II}]}}{\rm SFR}\bar{Z}^{-1}_{\rm gas}\right)$ | $\Delta\,{\rm log}\,\left(\frac{L_{\rm[C_{II}]}}{\rm SFR}\bar{n}^{-1}_{\rm gas}\right)$ | $\Delta\,{\rm log}\,\left(\frac{L_{\rm[C_{II}]}}{\rm SFR}f^{-1}_{\rm[C_{II}]}\right)$ | $\Delta\,{\rm log}\,\left(\frac{L_{\rm[C_{II}]}}{\rm SFR}\bar{\epsilon}^{-1}_{\rm[C_{II}]}\right)$
---|---|---|---|---|---|---
| (dex) | (dex) | (dex) | (dex) | (dex) | (dex)
0 | / | / | / | / | / | /
1 | $-0.38$ | 0.13 | $-0.16$ | $-0.78$ | $-0.38$ | $-0.35$
2 | $-0.67$ | 0.14 | $-0.19$ | $-1.37$ | $-0.67$ | $-0.48$
3 | $-0.78$ | $-0.08$ | $-0.08$ | $-1.64$ | $-0.79$ | $-0.55$
4 | $-0.92$ | $-0.19$ |  $-$0.08 | $-1.95$ | $-0.93$ | $-0.74$
6 | $-1.12$ | $-0.32$ | 0.07 | $-2.33$ | $-1.12$ | $-0.75$
8 | $-1.21$ | $-0.40$ |  $-$0.02 | $-2.62$ | $-1.21$ | $-0.83$
Table 11: The mean of $t_{\rm dep}$, $\bar{Z}_{\rm gas}$, $\bar{n}^{-1}_{\rm gas}$, $f_{\rm[C_{II}]}$ and $\bar{\epsilon}_{\rm[C_{II}]}$ of the FIRE galaxy sample at different redshifts.  $z$ | <$\frac{t_{\rm dep}}{\rm Gyr}$> | <$\frac{\bar{Z}_{\rm gas}}{Z_{\odot}}$> | <$\frac{\bar{n}_{\rm gas}}{\rm cm^{-3}}$> | <$f_{\rm[C_{II}]}$> | <$\frac{\bar{\epsilon}_{\rm[C_{II}]}}{10^{-22}\,{\rm erg\,s^{-1}\,cm^{3}}}$>
---|---|---|---|---|---
|  |  | Total |  |
0 | 6.30 | 1.69 | 0.76 | 0.93 | 3.2
1 | 2.02 | 1.08 | 2.29 | 0.96 | 2.6
2 | 1.02 | 0.56 | 4.17 | 0.98 | 2.0
3 | 1.10 | 0.43 | 5.75 | 0.99 | 2.1
4 | 1.14 | 0.24 | 7.59 | 1.00 | 1.8
6 | 0.86 | 0.12 | 11.75 | 1.00 | 1.5
8 | 0.72 | 0.09 | 16.26 | 1.00 | 1.2
|  | $L_{\rm IR}{}\geq{}10^{11}\,L_{\odot}$ |
0 | 1.88 | 2.40 | 1.32 | 0.88 | 2.5
1 | 1.32 | 1.45 | 2.89 | 0.90 | 2.1
2 | 0.52 | 1.07 | 4.37 | 0.95 | 1.9
3 | 0.83 | 0.54 | 7.59 | 0.99 | 1.7
4 | 0.69 | 0.36 | 10.97 | 1.00 | 1.5
6 | 0.51 | 0.32 | 22.87 | 1.00 | 0.9
8 | 0.09 | 0.59 | 41.34 | 0.94 | 0.5

#####  $\rm[C_{II}]$ deficit at high redshifts

The normalization of the $L_{\rm[C_{II}]}$-SFR relation of the FIRE sample
decreases monotonically with redshift. The mean $L_{\rm[C_{II}]}{}/{}\rm SFR$
ratio of the galaxies decreases by 1.2 dex (a factor of $\sim{}15$) from
$z{}={}0$ to $z{}={}8$ (see col. 2 of Table 10).

It can be seen from Table 10 (and also Fig. 14) that the redshift evolution of
the $L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio of the galaxies is mainly driven by
$\bar{Z}_{\rm gas}$ and $t_{\rm dep}$ since the $\rm[C_{II}]$ deficit is
significantly alleviated (or even vanishes at some redshifts) in the parameter
space of $(L_{\rm[C_{II}]}{}/{}{\rm SFR}){}t^{-1}_{\rm dep}$ and
$(L_{\rm[C_{II}]}{}/{}{\rm SFR}){}\bar{Z}^{-1}_{\rm gas}$. This result
indicates that the $\rm[C_{II}]$ deficit of galaxies at high redshifts is due
to either low gas metallicity or a deficiency of gas (that is able to produce
$\rm[C_{II}]$ emission) per unit SFR.

Looking into the details, we see from Table 10 that while $t_{\rm dep}$ is the
key parameter driving the evolution of the $L_{\rm[C_{II}]}$-SFR relation at
$z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
60\relax$}\hss}\mathchar 29208\relax$}}{}2$, $\bar{Z}_{\rm gas}$ plays a more
important role at $z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}{}4$. This is because
$t_{\rm dep}$ of the FIRE sample decreases more at $z{}={}0-2$ (from 6.30 to
1.02 Gyr, by a factor of $\sim{}6$) than at $z{}={}2-8$ (from 1.02 to 0.72
Gyr, by only $\sim{}30\%$) (see Table 11). At $z{}={}2-8$, $\bar{Z}_{\rm gas}$
of the FIRE sample decreases sharply with redshift (from $0.56\,Z_{\odot}$ to
$0.09\,Z_{\odot}$, by a factor of $\sim{}6$) and it thus has a stronger impact
on the evolution of $L_{\rm[C_{II}]}{}/{}\rm SFR$ of the FIRE galaxies than
$t_{\rm dep}$.

Unlike $t_{\rm dep}$ and $\bar{Z}_{\rm gas}$, $\bar{\epsilon}_{\rm[C_{II}]}$
has only a mild impact on the redshift evolution of $L_{\rm[C_{II}]}{}/{}\rm
SFR$. From $z{}={}0$ to $z{}={}8$, the mean $\bar{\epsilon}_{\rm[C_{II}]}$ of
the FIRE sample has a slight decrease with redshift by less than a factor of 3
(Table 11). The $\rm[C_{II}]$ deficit persists at high redshifts in the space
of $(L_{\rm[C_{II}]}{}/{}{\rm SFR})\,\bar{\epsilon}^{-1}_{\rm[C_{II}]}$ (Table
10).

The other two parameters, $\bar{n}_{\rm gas}$ and $f_{\rm[C_{II}]}$, have
completely no contribution to the $\rm[C_{II}]$ deficit at high redshifts
since both increase with redshift instead of decreasing. The increase of
$\bar{n}_{\rm gas}$ indicates that earlier galaxies have a more compact ISM.
Naively, because $L_{\rm[C_{II}],\,cl}{}/{}M_{\rm cl}{}\propto{}n_{\rm H}$
(equation 23), an increase of gas density would result in a rise of
$L_{\rm[C_{II}]}{}/{}\rm SFR$ with redshift. This effect, however, is
overwhelmed by the combined effect of $t_{\rm dep}$ and $\bar{Z}_{\rm gas}$ on
$L_{\rm[C_{II}]}{}/{}\rm SFR$.

The increase of $f_{\rm[C_{II}]}$ with redshift indicates that our sample
includes more $\rm H_{2}$ gas-poor galaxies at higher redshift where a larger
fraction of carbon in the ISM gas becomes ionized. Nonetheless, the impact of
$f_{\rm[C_{II}]}$ on the evolution of $L_{\rm[C_{II}]}{}/{}\rm SFR$ is
negligible since $f_{\rm[C_{II}]}$ of the galaxies in our sample differs by no
more than a factor of 2 (ranging between $\approx{}60\%$ and unity, see the
lower middle panel of Fig. 15).

To summarize, the decrease of $L_{\rm[C_{II}]}/{\rm SFR}$ of the FIRE sample
with redshift is mainly driven by the decrease of their $t_{\rm dep}$ and gas
metallicity. While $t_{\rm dep}$ plays a more important role at $z{}\leq{}2$,
gas metallicity becomes the key parameter driving the $\rm[C_{II}]$ deficit of
the galaxies at higher redshifts. The redshift evolution of $\bar{n}_{\rm
gas}$, $f_{\rm[C_{II}]}$ and $\bar{\epsilon}_{\rm[C_{II}]}$ have no or limited
contribution.

#####  $\rm[C_{II}]$ deficit at high $L_{\rm IR}$

The FIRE sample exhibits a trend of declining $L_{\rm[C_{II}]}{}/{}L_{\rm IR}$
ratio with $L_{\rm IR}$ at each redshift. To find the main driver of the
$\rm[C_{II}]$ deficit at high $L_{\rm IR}$, we check how each of the five
physical parameters ($f_{\rm[C_{II}]}$, $t_{\rm dep}$, $\bar{n}_{\rm gas}$,
$\bar{Z}_{\rm gas}$ and $\bar{\epsilon}_{\rm[C_{II}]}$) depends on $L_{\rm
IR}$.

In Table 11, we explicitly show the mean of $f_{\rm[C_{II}]}$, $t_{\rm dep}$,
$\bar{n}_{\rm gas}$, $\bar{Z}_{\rm gas}$ and $\bar{\epsilon}_{\rm[C_{II}]}$ of
the galaxies having $L_{\rm IR}{}\geq{}10^{11}\,L_{\odot}$ (where galaxies are
observed to show $\rm[C_{II}]$ deficit) in our sample at different redshifts,
in addition to the mean of the five parameters of the entire sample (including
fainter galaxies). We can see from the table (also Fig. 15) that the IR-
luminous galaxies ($L_{\rm IR}{}\geq{}10^{11}\,L_{\odot}$) have lower $t_{\rm
dep}$, $f_{\rm[C_{II}]}$ and $\bar{\epsilon}_{\rm[C_{II}]}$ but higher
$\bar{Z}_{\rm gas}$ and $\bar{n}_{\rm gas}$ than the rest of the sample — they
are relatively metal and $\rm H_{2}$ gas-rich, and have more compact ISM and
shorter depletion time than the normal SFGs having lower SFR. The fact that
they have a reduced $\bar{\epsilon}_{\rm[C_{II}]}$ is associated with a
stronger ISRF in these galaxies (we will discuss this more in Section 6.2).

Hence, the $\rm[C_{II}]$ deficit at high $L_{\rm IR}$ is driven by the
combined effect of $t_{\rm dep}$, $f_{\rm[C_{II}]}$ and
$\bar{\epsilon}_{\rm[C_{II}]}$. We see from Fig. 14 that the $\rm[C_{II}]$
deficit of the FIRE sample at high $L_{\rm IR}$ vanishes only in the space of
$(L_{\rm[C_{II}]}{}/{}{\rm SFR})\,t^{-1}_{\rm dep}$ (upper left panel) but not
in that of $(L_{\rm[C_{II}]}{}/{}{\rm SFR})\,f^{-1}_{\rm[C_{II}]}$ (lower
middle panel) or $(L_{\rm[C_{II}]}{}/{}{\rm
SFR})\,\bar{\epsilon}^{-1}_{\rm[C_{II}]}$ (lower right panel). This indicates
that $t_{\rm dep}$ plays a major role in driving the $\rm[C_{II}]$ deficit at
high $L_{\rm IR}$. That is, the $\rm[C_{II}]$ deficit of the IR-luminous
galaxies in our sample is mainly due to the reduced amount of gas that is
available for producing $\rm[C_{II}]$ emission per unit SFR of the galaxies.

###  5.4 The two regimes of $\rm[C_{II}]$ emission of galaxies

In the previous section, we have shown with the FIRE sample that the main
driver of the $\rm[C_{II}]$ deficit at high redshifts and high $L_{\rm IR}$ is
different. The observed $\rm[C_{II}]$ deficit of the galaxies at
$z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}4$ (at $L_{\rm IR}{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}10^{11}\,L_{\odot}$) may be due to their low gas metallicity
(gas depletion time). In this section, we explore the fundamental reason for
galaxies having different origin of $\rm[C_{II}]$ deficit in the two regimes.

We at first discuss the $L_{\rm[C_{II}]}{}/{}\rm SFR$ vs. $t_{\rm dep}$
relation of the FIRE galaxies (Section 5.4.1). We subsequently explore the
reason for galaxies showing two distinct regimes on the $\Delta\,({\rm
log}\,L_{\rm[C_{II}]})$ vs. $t_{\rm dep}$ diagram (Section 5.4.2). Finally, we
discuss how this is related to the distinct origin of $\rm[C_{II}]$ deficit at
high redshifts and high $L_{\rm IR}$ (Section 5.4.3).

####  5.4.1 The $L_{\rm[C_{II}]}{}/{}\rm SFR$ vs. $t_{\rm dep}$ relation

The FIRE galaxies exhibit two distinct regimes on the $\Delta\,({\rm
log}\,L_{\rm[C_{II}]})$ vs. $t_{\rm dep}$ diagram. While a considerable number
of the galaxies show a tight linear correlation between their ${\rm
log}\,(t_{\rm dep}{}/{}\rm Gyr)$ and $\Delta\,({\rm log}\,L_{\rm[C_{II}]})$,
exhibiting a linear sequence (we hereafter refer to it as the ‘deficit-
depletion time sequence’, or DDS), others show larger scatter on the diagram
and fall systematically below the DDS.

The galaxies on the DDS appear to be more $\rm H_{2}$ gas-rich. In Fig. 16, we
show the same $L_{\rm[C_{II}]}{}/{}\rm SFR$ vs. $t_{\rm dep}$ relation of the
FIRE sample as in Fig. 15 (upper left panel), but colour-code the data points
by the $\rm H_{2}$ gas mass fraction, $f_{\rm H_{2}}$, of the galaxies instead
of their redshift. It can be seen from Fig. 16 that the galaxies along the DDS
tend to be more $\rm H_{2}$ gas-rich, having $f_{\rm H_{2}}{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}10\%$ (equivalent to $f_{\rm[C_{II}]}{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 60\relax$}\hss}\mathchar
29208\relax$}}{}90\%$). Besides, we see from the two figures that the majority
of the low-redshift ($z{}={}0-2$, shown by cyan stars, yellow hexagons and red
triangles in Fig. 15) and IR-luminous ($L_{\rm IR}{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}10^{11}\,L_{\odot}$, indicated by large symbols in Fig. 15 and
Fig. 16) galaxies locate on or close to the DDS.

We derive the best-fit linear scaling relation between ${\rm log}\,(t_{\rm
dep}{}/{}\rm Gyr)$ and $\Delta\,({\rm log}\,L_{\rm[C_{II}]})$ for the $\rm
H_{2}$ gas-rich galaxies in our sample having $f_{\rm H_{2}}{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}10\%$, i.e.

$\Delta({\rm log}\,L_{\rm[C_{II}]})=(-0.38\pm 0.01)+(0.71\pm 0.03)\;{\rm
log}\left(\frac{t_{\rm dep}}{\rm Gyr}\right),$ (29)

which can be rewritten as

$\frac{L_{\rm[C_{II}]}/L_{\odot}}{{\rm SFR}/(M_{\odot}\,\rm
yr^{-1})}=1.78\times 10^{7}\left(\frac{t_{\rm dep}}{\rm Gyr}\right)^{0.71}.$
(30)

The coefficient of determination is $R^{2}{}={}0.936$.

####  5.4.2 The two regimes on the $\Delta\,({\rm log}\,L_{\rm[C_{II}]})$ vs.
$t_{\rm dep}$ diagram

Figure 16: The relation between $t_{\rm dep}$ and $\Delta\,({\rm
log}\,L_{\rm[C_{II}]})$ of the FIRE sample at $z{}={}0-8$ (same as the upper
left panel of Fig. 15 except for the colour). The data points are coloured-
coded by $f_{\rm H_{2}}$ of the galaxies. The large (small) symbols represent
the galaxies having $L_{\rm IR}{}\geq{}10^{11}\,L_{\odot}$ ($L_{\rm
IR}{}<{}10^{11}\,L_{\odot}$). The $\rm H_{2}$ gas-rich galaxies ($f_{\rm
H_{2}}{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}10\%$) exhibit a linear correlation
between ${\rm log}\,(t_{\rm dep}{}/{}\rm Gyr)$ and $\Delta\,({\rm
log}\,L_{\rm[C_{II}]})$ (indicated by the black dashed line), which can be
converted to a power-law relation $L_{\rm[C_{II}]}{}/{}{\rm
SFR}{}\propto{}t^{0.71}_{\rm dep}$ (equation 30). Figure 17: The relation
between $\bar{Z}_{\rm gas}$ and $f_{\rm[C_{II}]}$ of the FIRE sample at
different redshifts. The large (small) symbols represent the galaxies having
$L_{\rm IR}{}\geq{}10^{11}\,L_{\odot}$ ($L_{\rm IR}{}<{}10^{11}\,L_{\odot}$).
The black dotted lines indicate the relation of $f_{\rm[C_{II}]}\bar{Z}_{\rm
gas}{}={}0.02$, 0.1, 0.5, 1, 2 and 4 (from left to right). At $\bar{Z}_{\rm
gas}{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
60\relax$}\hss}\mathchar 29208\relax$}}{}Z_{\odot}$, where galaxies are $\rm
H_{2}$ gas-poor, $f_{\rm[C_{II}]}{}\approx{}1$ and
$f_{\rm[C_{II}]}\bar{Z}_{\rm gas}{}\approx{}\bar{Z}_{\rm gas}$ (c.f. equation
32). At larger $\bar{Z}_{\rm gas}$, $f_{\rm[C_{II}]}$ scales roughly inversely
with $\bar{Z}_{\rm gas}$ and hence $f_{\rm[C_{II}]}\bar{Z}_{\rm
gas}{}\approx{}$constant (c.f. equation 34).

The reasons for the $\rm H_{2}$ gas-rich galaxies ($f_{\rm
H_{2}}{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}10\%$) showing a linear sequence on
the $\Delta\,({\rm log}\,L_{\rm[C_{II}]})$ vs. $t_{\rm dep}$ diagram are
threefolds: i) Their $f_{\rm[C_{II}]}\bar{Z}_{\rm gas}$ ‘saturates’, meaning
that it becomes almost like a constant and hence $L_{\rm[C_{II}]}{}/{}{\rm
SFR}$ of the galaxies simply scales to $t_{\rm dep}\bar{n}_{\rm gas}$, ii)
their $t_{\rm dep}$ and $\bar{n}_{\rm gas}$ anti-correlate with each other,
and iii) $\bar{\epsilon}_{\rm[C_{II}]}$ has relatively small variation among
different galaxies.

Figure 18: The relation between $\bar{n}_{\rm gas}$ and $t_{\rm dep}$ of the
FIRE sample at different redshifts. The data points in the left (right) panel
are colour-coded by the redshift ($f_{\rm H_{2}}$) of the galaxies. Large
(small) symbols represent the galaxies having $L_{\rm IR}\geq
10^{11}\,L_{\odot}$ ($L_{\rm IR}<10^{11}\,L_{\odot}$). The FIRE galaxies show
a clear anti-correlation between $t_{\rm dep}$ and $\bar{n}_{\rm gas}$, in
particular, the $\rm H_{2}$ gas-rich galaxies in the sample.

Let us at first understand the $f_{\rm[C_{II}]}$ vs. $\bar{Z}_{\rm gas}$
relation. In Fig. 17, we show the $f_{\rm[C_{II}]}$ vs. $\bar{Z}_{\rm gas}$
relation for the FIRE sample. It can be seen that at $\bar{Z}_{\rm
gas}{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
60\relax$}\hss}\mathchar 29208\relax$}}{}Z_{\odot}$, $f_{\rm[C_{II}]}$ barely
declines from unity ($f_{\rm[C_{II}]}{}\approx{}1$) with increasing
$\bar{Z}_{\rm gas}$, whereas at higher $\bar{Z}_{\rm gas}$, $f_{\rm[C_{II}]}$
declines sharply and $f_{\rm[C_{II}]}\bar{Z}_{\rm gas}$ becomes approximately
a constant (‘saturates’) with increasing $\bar{Z}_{\rm gas}$ (or decreasing
$f_{\rm[C_{II}]}$).

The shape of the $f_{\rm[C_{II}]}$ vs. $\bar{Z}_{\rm gas}$ relation of the
FIRE galaxies can be understood as follows. Consider a spherical gas cloud
having a radius $R_{\rm cl}$ and a surface-to-centre column density $N_{\rm
cl}$ ($=n_{\rm H}R_{\rm cl}$). When the cloud is metal and dust-poor (having
very low $Z_{\rm gas}$ and $\delta_{\rm dgr}$), the LW photons from the
radiation field can penetrate the entire cloud (i.e. $l_{\rm F}{}>{}R_{\rm
cl}$) and dissociate all the molecular hydrogen ($\rm H_{2}$) and neutral
carbon ($\rm C_{I}$ and $\rm CO$) in the cloud. In such a low-metallicity (or
$\delta_{\rm dgr}$) regime, we have

$f_{\rm[C_{II}],\,cl}\approx 1$ (31)

and

$f_{\rm[C_{II}],\,cl}\,Z_{\rm gas}\propto Z_{\rm gas}.$ (32)

Since $N_{\rm F}{}\propto{}l_{\rm F}{}\propto{}\delta^{-1}_{\rm dgr}\propto
Z^{-1}_{\rm gas}$ (equation 10 and 11), indicating stronger dust absorption of
UV photons with increasing gas metallicity, $l_{\rm F}$ decreases with $Z_{\rm
gas}$ and will become equal or less than $R_{\rm cl}$ when $Z_{\rm gas}$
becomes sufficiently large. Through simple mathematics, it can be derived that
for a spherical geometry, $f_{\rm[C_{II}]}Z_{\rm gas}$ increases sub-linearly
with $Z_{\rm gas}$ until when $l_{\rm F}\ll R_{\rm cl}$, we have

$f_{\rm[C_{II}],\,cl}\propto\frac{N_{\rm F}}{N_{\rm cl}}\propto(Z_{\rm
gas}N_{\rm cl})^{-1}$ (33)

or

$f_{\rm[C_{II}],\,cl}\,Z_{\rm gas}=\rm constant.$ (34)

It is not surprising to find similar scaling relations with the FIRE galaxies,
$f_{\rm[C_{II}]}\,\bar{Z}_{\rm gas}{}\approx{}\bar{Z}_{\rm gas}$ at low
$\bar{Z}_{\rm gas}$ and $f_{\rm[C_{II}]}\,\bar{Z}_{\rm gas}{}\approx{}$const.
at high $\bar{Z}_{\rm gas}$ (as shown in Fig. 17), given that the ISM of the
galaxies can be viewed as being made up of numerous such idealized gas
‘clouds’. The ‘saturation’ of $f_{\rm[C_{II}]}\,\bar{Z}_{\rm gas}$ at high
$\bar{Z}_{\rm gas}$ indicates that the $\rm[C_{II}]$ cooling rate of the
galaxies does not increase much with gas metallicity due to the shrinking of
the size of the $\rm[C_{II}]$-emitting region (Zone I + Zone II).

Another important reason for the $\rm H_{2}$ gas-rich galaxies showing a clear
sequence on the $\Delta\,({\rm log}\,L_{\rm[C_{II}]})$ vs. $t_{\rm dep}$
diagram is that their $t_{\rm dep}$ and $\bar{n}_{\rm gas}$ have clear anti-
correlation. In Fig. 18, we show the $t_{\rm dep}$ vs. $\bar{n}_{\rm gas}$
relation of the FIRE sample. This anti-correlation is due to the fact that the
local free-fall timescale of star-forming clouds decreases with gas density
($t_{\rm ff}{}\propto{}\rho^{-1/2}$), and hence gas is converted into stars
more rapidly in the galaxies having denser ISM. It also accounts for the sub-
linearity (power law index $n{}={}0.71$) of the $L_{\rm[C_{II}]}/\rm SFR$ vs.
$t_{\rm dep}$ scaling relation of the $\rm H_{2}$ gas-rich galaxies on the DDS
(equation 30).

For the $\rm H_{2}$ gas-poor galaxies, the fact that they lie below the DDS on
the $\Delta\,({\rm log}\,L_{\rm[C_{II}]})$ vs. $t_{\rm dep}$ diagram (Fig. 16)
is because of their low gas metallicity (and hence low
$f_{\rm[C_{II}]}\bar{Z}_{\rm gas}$). From equation (27), we see that at fixed
$L_{\rm[C_{II}]}{}/{}{\rm SFR}$ (equivalently, at fixed $\Delta\,{\rm
log}\,L_{\rm[C_{II}]}$), their $t_{\rm dep}$ has to be higher than that of the
galaxies on the DDS so as to compensate for their having lower
$f_{\rm[C_{II}]}\bar{Z}_{\rm gas}$. Besides, the fact that the $\rm H_{2}$
gas-poor galaxies show a larger scatter of $t_{\rm dep}$ at given $\Delta({\rm
log}\,L_{\rm[C_{II}]})$ (Fig. 16) than the $\rm H_{2}$ gas-rich galaxies is
due to the non-trivial scatter of $f_{\rm[C_{II}]}\bar{Z}_{\rm gas}$ among
these galaxies, as opposed to $f_{\rm[C_{II}]}\bar{Z}_{\rm gas}$ being like a
constant for the $\rm H_{2}$ gas-rich galaxies (Fig. 17).

####  5.4.3 The physical origins of $\rm[C_{II}]$ deficit of galaxies (a
revisit)

The important consequence of $f_{\rm[C_{II}]}\,\bar{Z}_{\rm gas}$ being
‘saturated’ for the $\rm H_{2}$ gas-rich galaxies is that the overall
$L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio of the galaxies shows a tight and steep
dependence on $t_{\rm dep}$ (equation 30). As a result, $t_{\rm dep}$ becomes
the dominating parameter that determines the $L_{\rm[C_{II}]}{}/{}\rm SFR$
ratio of these galaxies. Their $L_{\rm[C_{II}]}{}/{}\rm SFR$, in contrast,
does not shows a clear correlation with any of the other four parameters
($f_{\rm[C_{II}]}$, $\bar{Z}_{\rm gas}$, $\bar{n}_{\rm gas}$ or
$\bar{\epsilon}_{\rm[C_{II}]}$).

Now we should be able to understand the fundamental reason for $t_{\rm dep}$
being the main driver of the $\rm[C_{II}]$ deficit at high $L_{\rm IR}$. The
IR-luminous galaxies are $\rm H_{2}$ gas-rich (due both to their being dust-
rich and having high gas column density). Hence, they are in the regime where
the $L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio of galaxies is determined primarily by
$t_{\rm dep}$ (i.e. they lie on the DDS in the $\Delta\,({\rm
log}\,L_{\rm[C_{II}]})$ vs. $t_{\rm dep}$ diagram) and their $\rm[C_{II}]$
deficit is due to their low $t_{\rm dep}$.

Besides, we can now understand the redshift evolution of the
$L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio of the FIRE sample at $z{}={}0-2$. At
these low redshifts, our sample includes more galaxies that are $\rm H_{2}$
gas-rich as a result of their being more metal and dust-rich than the galaxies
at higher redshifts. The $L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio of these low-$z$
galaxies therefore depends more sensitively on $t_{\rm dep}$.

At higher redshifts, in contrast, our sample includes a large fraction of
metal and dust-poor galaxies that are also $\rm H_{2}$ gas-poor. They are off
the DDS in the $\Delta\,({\rm log}\,L_{\rm[C_{II}]})$ vs. $t_{\rm dep}$
diagram. For these galaxies, $f_{\rm[C_{II}]}\,\bar{Z}_{\rm
gas}{}\approx{}\bar{Z}_{\rm gas}$ (Fig. 17) and hence $L_{\rm[C_{II}]}{}/{}\rm
SFR$ of the galaxies depends more sensitively on $\bar{Z}_{\rm gas}$. As a
result, gas metallicity becomes the main driver of the $\rm[C_{II}]$ deficit
of the high-$z$ galaxies in our sample.

##  6 Discussions

###  6.1 The effect of mass resolution

The fact that the ISM is treated as an aggregate of spherical gas ‘clouds’ in
our model (and in those of the previous studies like e.g. Vallini et al. 2015,
2018; Olsen et al. 2017; Narayanan & Krumholz 2017; Lagache et al. 2018; Li et
al. 2018; Pallottini et al. 2019; Leung et al. 2020; Yang et al. 2021) is
certainly an idealization — the ISM in real galaxies is a continuous medium,
and has complex spatial configurations at and below the scale of these
idealized ‘clouds’. Nonetheless, such treatment offers a (crude) sampling of
the column density of gas in the ISM, enabling us to capture the essential
physics causing the $\rm[C_{II}]$ deficit of galaxies.

We note that the predicted $\rm[C_{II}]$ luminosity of galaxies can depend on
the mass resolution of the simulations (mass of the ‘gas clouds’) in this
model. To assess this dependency, we adopt two additional models (‘HighRes’
and ‘LowRes’) in post-processing, where we increase and decrease the mass of
each individual ‘cloud’ by a factor of $2^{3}{}={}8$ (equivalent to increasing
and reducing the surface-to-centre column density by a factor of 2). For the
HighRes model, we simply split each ‘cloud’ in the fiducial model into 8 with
equal mass, and assume that the 8 HighRes ‘clouds’ have the same density and
metallicity and are exposed to the same radiation field as the parent ‘cloud’.
For the LowRes model, we calculate the luminosity of each ‘cloud’ assuming as
if the ‘cloud’ is 8 times more massive than it is in the fiducial model. The
total $\rm[C_{II}]$ luminosity of galaxy is calculated by summing the
luminosity of each massive ‘cloud’ and then dividing the sum by 8 (equivalent
to having 8 times lower number of ‘clouds’, each being 8 times more massive
than that in the fiducial model).

Figure 19: The effect of mass resolution on the predicted $\rm[C_{II}]$
luminosity of galaxies. The upper (middle) panel shows the
$L_{\rm[C_{II}]}$-SFR relation of the FIRE sample at $z{}={}0-8$ predicted by
the HighRes (LowRes) model (see Section 6.1 for details). For reference, we
also show in the two panels the observed $L_{\rm[C_{II}]}$-SFR relation of
local galaxies by L11, L14 and H15. In the lower panel, we explicitly show the
difference between the $\rm[C_{II}]$ luminosity predicted by the HighRes
(unfilled symbols) and LowRes (filled symbols) models and that predicted by
the fiducial model as a function of galaxy SFR. The shaded grey area indicates
a factor of two difference.

The predicted $\rm[C_{II}]$ luminosity of the HighRes (LowRes) model appears
to be higher (lower) than that of our fiducial model by 0.13 dex on average
(see Fig. 19). This is because with decreasing (increasing) gas column
density, a higher (lower) fraction of the gas becomes in the
$\rm[C_{II}]$-emitting phase (Zone I + Zone II) and besides, an increased
(reduced) fraction of the $\rm[C_{II}]$-emitting gas becomes in the $\rm
H_{II}$ phase (note: $\rm H_{II}$ gas has on average higher $\rm[C_{II}]$
emissivity than $\rm H_{I}$ gas). Overall, by changing the mass resolution by
about a decade results in a difference in the predicted $\rm[C_{II}]$
luminosity of the galaxies by $\mathbin{\lower 3.0pt\hbox{$\hbox
to0.0pt{\raise 4.0pt\hbox{$\char 60\relax$}\hss}\mathchar
29208\relax$}}{}0.15$ dex. The difference does not show a strong dependence on
redshift or SFR of the galaxies (see the lower panel of Fig. 19).

Comparing to the observations, it is clear that the $L_{\rm[C_{II}]}$-SFR
relation of the FIRE galaxies at $z{}={}0$ predicted by the HighRes model
becomes systematically offset from the data of L14 and H15 (see the upper
panel of Fig. 19), indicating a too low mean gas column density of the
galaxies in the HighRes model. The $L_{\rm[C_{II}]}$-SFR relation of the
LowRes model is below that of the fiducial model, but still appears to be
within the scatter of the L14 and H15 data. The LowRes model predicts a
slightly stronger (by $\sim{}$0.13 dex) $\rm[C_{II}]$ deficit of galaxies at
high redshifts than the fiducial model.

Nonetheless, our conclusions regarding the causes of $\rm[C_{II}]$ deficit of
galaxies at high $L_{\rm IR}$ (low $t_{\rm dep}$) and at high redshifts (low
gas metallicity) does not change with the chosen mass resolution of the
$\rm[C_{II}]$ model, despite that the predicted $L_{\rm[C_{II}]}$ of galaxies
shows moderate dependence on it.

###  6.2 Comparison with the previous studies

Here we discuss the relation between the findings of the previous studies to
this from this work. Specifically, we will discuss the conclusions regarding
the origin of the $\rm[C_{II}]$ deficit at high $L_{\rm IR}$ in Section 6.1,
whereas in Section 6.2, we will compare the predictions of the
$L_{\rm[C_{II}]}$\- SFR relation of galaxies at redshift $z{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}5$ from the recent studies with ours.

####  6.2.1 The $\rm[C_{II}]$ deficit at high $L_{\rm IR}$

#####  $\rm[C_{II}]$ deficit due to a strong ISRF

A number of studies suggest that the observed $\rm[C_{II}]$ deficit at high
$L_{\rm IR}$ is due to a strong ISRF in IR-luminous galaxies. This can lead to
large positive grain charge and thus inefficient heating of gas via photo-
electric processes in the neutral galactic medium (Tielens & Hollenbach, 1985;
Kaufman et al., 1999). Consequently, the rate of gas cooling via $\rm[C_{II}]$
line drops. Apart from that, a strong ISRF (and hence high $U$) may also lead
to “dust-bounded" $\rm H_{II}$ regions near the newly formed young stars
(Bottorff et al., 1998; Abel et al., 2009), i.e. $N_{\rm s}{}\approx{}N_{\rm
F}$ (note: $N_{\rm s}$ increases about linearly with $U$ until $N_{\rm
s}{}\approx{}N_{\rm F}$). In this case, gas cooling via $\rm[C_{II}]$ can
become inefficient due to a lack of $\rm C_{II}$ ions in the $\rm H_{II}$
regions — a significant fraction of carbon can be ionized further to $\rm
C_{III}$ ions when $U$ is large (In Zone I, $x^{(1)}_{\rm
C_{II}}{}\approx{}1-x^{(1)}_{\rm C_{III}}{}\propto U^{-1}$, see Appendix E).
Overall, both mechanisms can lead to a reduced $\bar{\epsilon}_{\rm[C_{II}]}$
in galaxies.

Looking at the FIRE sample, we notice that the few massive starburst galaxies
(${\rm SFR}{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}{}100\,M_{\odot}\,\rm
yr^{-1}$) in our sample do show noticeably lower
$\bar{\epsilon}_{\rm[C_{II}]}$ than the normal star-forming galaxies having
lower SFR (see the right panel of Fig. 13 and the bottom right panel of Fig.
15), which can be associated with a strong ISRF (and high $U$) in these
galaxies. Note that CLOUDY (version 17.01) takes into account grain charging
physics (Baldwin et al., 1991; van Hoof et al., 2004; Abel et al., 2005) and
our approach of performing dust RT calculation with SKIRT provides an improved
estimate of the ISRF (and hence $U$) distribution in galaxies than the
previous studies. Our result suggests that the $\rm[C_{II}]$ deficit at high
$L_{\rm IR}$ may in part be driven by a strong ISRF in the massive starburst
galaxies, and yet its impact does not seem to be as prominent as $t_{\rm
dep}$.

#####  $\rm[C_{II}]$ deficit due to high gas density

It has also been suggested that the $\rm[C_{II}]$ deficit in IR-luminous
galaxies can be driven by the high density of the star-forming gas in these
galaxies (e.g. Narayanan & Krumholz, 2017). With increasing density, ISM gas
becomes more shielded from ionizing radiation of massive young stars and more
carbon in the ISM gas becomes neutral (in $\rm CO$ or $\rm C_{I}$). The
$\rm[C_{II}]$ deficit is thus due to a lack of $\rm C_{II}$ ions in the ISM
gas in this scenario (i.e. due to a low $f_{\rm[C_{II}]}$).

This, however, does not seem to be exactly like what we find with the FIRE
simulations. The ISM of the FIRE galaxies spans a very wide range of density
(see Fig. 11), and even for the most massive starburst galaxies in our sample,
a large fraction of their $\rm[C_{II}]$ luminosity originates from the gas
having intermediate density ($\bar{n}_{\rm gas}{}\approx{}\bar{n}_{\rm
H_{I},\,MW}$, see Fig. 11). Overall, the luminosity-weighted gas density
($\bar{n}_{\rm gas}$) of the IR-luminous galaxies ($L_{\rm
IR}{}\geq{}10^{11}\,L_{\odot}$) is not significantly higher than that of the
IR-fainter galaxies in our sample at any given redshift (see Table 11), and
the difference is not as strong as that in $t_{\rm dep}$. Hence, the
$\rm[C_{II}]$ deficit of the IR-luminous galaxies in FIRE simulations doe not
appear to be mainly due to their having too dense ISM gas.

####  6.2.2 The $L_{\rm[C_{II}]}$-SFR relation at redshift $z{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}5$

Figure 20: The $L_{\rm[C_{II}]}$-SFR relation at $z{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}5$ predicted by different simulation groups. Red, yellow, blue
and cyan lines indicate the mean result of Yang et al. (2021) ($4.5<{}z{}<6$),
Leung et al. (2020) ($z{}={}6$), Lagache et al. (2018) (dashed blue line for
$z{}\approx{}6$ and dotted blue line for $z{}\approx{}8$), and Kannan et al.
(2022) ($6{}<z{}<10$). These studies use statistically significant samples.
The corresponding coloured shaded areas represent the $1\sigma$ dispersion of
the data around the mean relation of each sample. In addition, we also show
the data of individual galaxies of the Olsen et al. (2017) ($z{}={}6$) and
Pallottini et al. (2019) ($z{}={}8$) samples by grey diamonds and grey
downward triangles, respectively. For reference, we show the observed
$L_{\rm[C_{II}]}$-SFR relation of the local star-forming samples of H15 (solid
orange line) and L11 (solid green line) as well as the the data of the FIRE
sample at $z{}={}4$ (magenta circles), $z{}={}6$ (green diamonds) and
$z{}={}8$ (purple downward triangles). A $\rm[C_{II}]$ deficit at
$z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}5$ is generally predicted by various
simulation groups.

As mentioned in the Introduction, several planned ground-based $\rm[C_{II}]$
line intensity mapping (LIM) experiments will target the emitting sources at
redshift $z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}{}5$ (Kovetz et al.,
2017), including CCAT-prime, CONCERTO and TIME. Predicting the
$L_{\rm[C_{II}]}$-SFR relation of galaxies at this early epoch has thus become
extremely important for interpreting the upcoming data of these experiments
(see e.g. Visbal et al., 2011; Gong et al., 2012; Serra et al., 2016; Fonseca
et al., 2017; Padmanabhan, 2019; Yue & Ferrara, 2019; Chung et al., 2020;
Padmanabhan et al., 2022; Sun et al., 2022; Karoumpis_2022; Murmu et al.,
2023).

In Fig. 20, we present the results from a number of recent studies. These
include the ones using SAMs (Lagache et al., 2018; Yang et al., 2021) as well
as those using hydrodynamic simulations (Olsen et al., 2017; Pallottini et
al., 2019; Leung et al., 2020; Kannan et al., 2022). It can be seen that
different studies have generally predicted a clear $\rm[C_{II}]$ deficit at
$z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}{}5$ with respect to the local samples
of L11 and H15, similar to this work using the FIRE simulations. Some have
also predicted a mild trend of growing deficit with increasing redshift (e.g.
Lagache et al., 2018; Kannan et al., 2022). The predicted $1\sigma$ scatter of
the $L_{\rm[C_{II}]}$-SFR relation at a given redshift of these studies is
typically as large as $0.3-0.5$ dex (except Kannan et al. 2022, which shows
noticeably smaller scatter than the others).

There is, however, a clear difference in the normalization and slope of the
$L_{\rm[C_{II}]}$-SFR relation predicted by the different groups. In
particular, Yang et al. (2021) (Kannan et al. 2022) produce the highest
(lowest) normalization among all different groups at ${\rm
SFR}{}\approx{}1-100\,M_{\odot}\,\rm yr^{-1}$. Note that both also produce a
considerably steeper power-law slope ($\approx{}1.5$) than the others.
Interestingly, our prediction with the FIRE galaxies appears to be in good
agreement with the result of Lagache et al. (2018) (in both normalization and
slope).

The difference in the $L_{\rm[C_{II}]}$-SFR relation indicates that the
predicted ISM properties (e.g. $\bar{Z}_{\rm gas}$, $t_{\rm dep}$) of the
galaxies at $z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}{}5$ are not well
converged between the current simulations. We highlight that the data of the
upcoming LIM experiments may provide useful constraints on the ISM properties
of the galaxies in this early epoch, given that direct measurement of these
properties is very challenging using the current techniques.

##  7 Summary and conclusions

The $\rm 158\,\mu m$ fine structure line of singly ionized carbon
($\rm[C_{II}]$) has been considered as a SFR indicator since observations of
nearby star-forming galaxies found a linear correlation between their
$L_{\rm[C_{II}]}$ and SFR. There is, however, evidence showing that IR-bright
($L_{\rm IR}{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}10^{11}\,L_{\odot}$), starburst galaxies as well as early
galaxies at $z{}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
4.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}{}5$ have reduced
$L_{\rm[C_{II}]}{}/{}\rm SFR$ with respect to the local star-forming samples
(so-called ‘$\rm[C_{II}]$ deficit’ problem). Different models have been
posited to explain the origin of the $\rm[C_{II}]$ deficit of galaxies at high
$L_{\rm IR}$ or at high redshifts and yet no consensus has been reached at
both regimes.

In this work, we present a comprehensive analysis on the $L_{\rm[C_{II}]}$-SFR
relation of galaxies using a galaxy sample at $z{}={}0-8$
($M_{*}{}={}10^{7}-5\times 10^{11}\,M_{\odot}$) extracted from the
cosmological hydrodynamic simulations, which are part of the FIRE project
(Hopkins et al., 2014, 2018, 2022), coupled with CLOUDY (Ferland et al., 1998;
Ferland et al., 2017) models. The sample consists mainly of galaxies ($N_{\rm
gal}{}\sim{}500$) from FIREbox (Feldmann et al., 2022), a high-resolution
cosmological-volume hydrodynamic simulation run with FIRE-2 physics, and is
supplemented with a few dozen of high-$z$ massive galaxies from the
cosmological ‘zoom-in’ simulations of the MassiveFIRE suite (Feldmann et al.,
2016; Feldmann et al., 2017; Anglés-Alcázar et al., 2017). The sample covers
an unprecedentedly broad dynamic range among all studies on $\rm[C_{II}]$,
including normal star-forming galaxies, (U)LIRG and SMG candidates as well as
UV-bright galaxies at EoR, which can be used to study the full range of the
observational data on $\rm[C_{II}]$ currently available.

The predicted $L_{\rm[C_{II}]}$-SFR relation of the FIRE sample agrees well
with the observational data. In particular, we successfully reproduce the
observed linear correlation of the local star-forming samples over the SFR
range $\approx{}0.1-10\,M_{\odot}\,\rm yr^{-1}$ (Fig. 4 and Fig. 6). Apart
from that, we also reproduce the sharp decline of $L_{\rm[C_{II}]}{}/{}\rm
SFR$ with $L_{\rm IR}$ ($\sim{}\rm SFR$) at $L_{\rm IR}{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}10^{11}\,L_{\odot}$ at low and high redshifts, which is
consistent with the data of the (U)LIRGs and SMGs in this $L_{\rm IR}$ regime
(Fig. 7 and Fig. 9).

Our sample shows a general decline of $L_{\rm[C_{II}]}{}/{}\rm SFR$ with
redshift, in particular, at low SFR (Fig. 8). The mean
$L_{\rm[C_{II}]}{}/{}\rm SFR$ ratio of the early EoR galaxies at $z{}>{}5$ in
our sample is about one order of magnitude below the local galaxies, showing a
clear $\rm[C_{II}]$ deficit, similar to what has been previously found with
other simulations (Section 6.2.2). Observations of galaxies at EoR have drawn
divergent conclusions on their $L_{\rm[C_{II}]}$-SFR relation, which is
largely due to the uncertainty in the dust SED shape (or ‘dust temperature’)
of the galaxies at these high redshifts. We analyze the sub-mm data of all the
observed EoR galaxies and derive their dust-obscured $\rm SFR$ using the ‘dust
temperature’ estimated from the SED templates of the FIRE samples self-
consistently. We conclude that the $L_{\rm[C_{II}]}$-SFR relation of the FIRE
galaxies at $z{}>{}5$ is in no conflict with the current observational
constraints, including those placed by the recent ALPINE and REBELS surveys.

The $L_{\rm[C_{II}]}{}/{}{\rm SFR}$ ratio of the FIRE sample roughly follows a
simple linear scaling relationship (equation 27)

$\frac{L_{\rm[C_{II}]}}{\rm SFR}\propto f_{\rm[C_{II}]}\bar{Z}_{\rm gas}t_{\rm
dep}\bar{n}_{\rm gas},$

where $f_{\rm[C_{II}]}$ is the mass fraction of ionized or neutral atomic
hydrogen gas in the ISM, $t_{\rm dep}$ is the gas depletion time ($={}M_{\rm
gas}{}/{}\rm SFR$), and $\bar{Z}_{\rm gas}$ and $\bar{n}_{\rm gas}$ indicate
the gas metallicity and gas density that are weighted by $\rm[C_{II}]$
luminosity. Following this scaling relationship, we find that the key driver
of the $\rm[C_{II}]$ deficit is different at high $L_{\rm IR}$ and high
redshifts (Section 5.3). At high $L_{\rm IR}$, the $\rm[C_{II}]$ deficit is
mainly due to the low $t_{\rm dep}$ of galaxies, indicating that IR-luminous,
starburst galaxies have less amount of gas that is able to produce
$\rm[C_{II}]$ emission per unit SFR than the normal star-forming galaxies with
moderate SFR. The $\rm[C_{II}]$ deficit at $z{}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 4.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}{}5$, in contrast, is mainly driven by the low gas metallicity
of galaxies at this epoch.

The underlying reason for $\rm[C_{II}]$ deficit being driven by different
physical parameters at high $L_{\rm IR}$ and high redshifts is as follows. In
the low-metallicity regime (corresponding to high-$z$ galaxies),
$L_{\rm[C_{II}]}$ of galaxies depends sensitively on metallicity because line
emissivity scales linearly with metallicity. In the high-metallicity regime
(corresponding to low-$z$, massive and starburst galaxies), however, such
dependence can become weak. This is because dust-to-gas ratio ($\delta_{\rm
dgr}$) in the ISM increases with metallicity, which leads to the shrinking of
the size of $\rm[C_{II}]$-emitting region (Section 5.4). The shrinking of its
size almost cancels out the effect of increasing emissivity with metallicity
(in this case, $f_{\rm[C_{II}]}\bar{Z}_{\rm gas}{}\approx{}$constant). As a
result, $L_{\rm[C_{II}]}{}/{}\rm SFR$ of galaxies does not depend much on
metallicity — but instead, on $t_{\rm dep}{}={}M_{\rm gas}{}/{}\rm SFR$, see
equation (30) — for massive, metal (dust) and $\rm H_{2}$ gas-rich starburst
galaxies at low redshifts.

Our study shows that $\rm[C_{II}]$ deficit may be a common phenomenon of
galaxies. This would be particularly important for interpreting the
observational data from several major upcoming $\rm[C_{II}]$ line intensity
mapping experiments, such as EXCLAIM (Ade et al., 2020), TIME (Sun et al.,
2021), CCAT-prime (CCAT-Prime collaboration et al., 2021) and CONCERTO
(CONCERTO Collaboration et al., 2020; Gkogkou et al., 2022). Our result
suggests that by using a constant linear $L_{\rm[C_{II}]}$-SFR relation
derived using nearby star-forming galaxies (e.g. De Looze et al., 2011; De
Looze et al., 2014; Herrera-Camus et al., 2015) may lead to systematic
overestimate of the cosmic star formation rate density of the high-$z$
Universe.

## Acknowledgements

LL acknowledges financial support from the Swiss National Science Foundation
(hereafter SNSF) (grant no P2ZHP2_199729) and the University of Toronto
Faculty of Arts and Science. RF acknowledges financial support from the SNSF
(grant no PP00P2_194814, 200021_188552). DN acknowledges funding from the NSF
via AST-1909153. DAA acknowledges support by NSF grants AST-2009687 and
AST-2108944, CXO grant TM2-23006X, and Simons Foundation award CCA-1018464.
CAFG was supported by NSF through grants AST-1715216, AST-2108230, and CAREER
award AST-1652522; by NASA through grants 17-ATP17-0067 and 21-ATP21-0036; by
STScI through grants HST-AR-16124.001-A and HST-GO-16730.016-A; by CXO through
grant TM2-23005X; and by the Research Corporation for Science Advancement
through a Cottrell Scholar Award. LB acknowledge financial support from the
SNSF (grant no PP00P2_194814). The Flatiron Institute is supported by the
Simons Foundation.

We acknowledge PRACE for awarding us access to MareNostrum at the Barcelona
Supercomputing Center (BSC), Spain. This research was partly carried out via
the Frontera computing project at the Texas Advanced Computing Center.
Frontera is made possible by National Science Foundation award OAC-1818253.
This work was supported in part by a grant from the Swiss National
Supercomputing Centre (CSCS) under project IDs s697 and s698. We acknowledge
access to Piz Daint at the Swiss National Supercomputing Centre, Switzerland
under the University of Zurich’s share with the project ID uzh18. This work
made use of infrastructure services provided by S3IT (www.s3it.uzh.ch), the
Service and Support for Science IT team at the University of Zurich.

## data availability statement

The data underlying this article will be shared on reasonable request to the
corresponding author.

## References

  * Abel et al. (2005) Abel N. P., Ferland G. J., Shaw G., van Hoof P. A. M., 2005, ApJS, 161, 65
  * Abel et al. (2008) Abel N. P., van Hoof P. A. M., Shaw G., Ferland G. J., Elwert T., 2008, ApJ, 686, 1125
  * Abel et al. (2009) Abel N. P., Dudley C., Fischer J., Satyapal S., van Hoof P. A. M., 2009, ApJ, 701, 1147
  * Ade et al. (2020) Ade P. A. R., et al., 2020, Journal of Low Temperature Physics, 199, 1027
  * Andika et al. (2020) Andika I. T., et al., 2020, ApJ, 903, 34
  * Anglés-Alcázar et al. (2017) Anglés-Alcázar D., Faucher-Giguère C.-A., Quataert E., Hopkins P. F., Feldmann R., Torrey P., Wetzel A., Kereš D., 2017, MNRAS, 472, L109
  * Aravena et al. (2016) Aravena M., et al., 2016, ApJ, 833, 68
  * Asplund et al. (2009) Asplund M., Grevesse N., Sauval A. J., Scott P., 2009, ARA&A, 47, 481
  * Baes & Camps (2015) Baes M., Camps P., 2015, Astronomy and Computing, 12, 33
  * Baes et al. (2011) Baes M., Verstappen J., Looze I. D., Fritz J., Saftly W., Pérez E. V., Stalevski M., Valcke S., 2011, ApJS, 196, 22
  * Bakx et al. (2020) Bakx T. J. L. C., et al., 2020, MNRAS, 493, 4294
  * Baldwin et al. (1991) Baldwin J. A., Ferland G. J., Martin P. G., Corbin M. R., Cota S. A., Peterson B. M., Slettebak A., 1991, ApJ, 374, 580
  * Bañados et al. (2015) Bañados E., Decarli R., Walter F., Venemans B. P., Farina E. P., Fan X., 2015, ApJ, 805, L8
  * Bañados et al. (2016) Bañados E., et al., 2016, ApJS, 227, 11
  * Barisic et al. (2017) Barisic I., et al., 2017, ApJ, 845, 41
  * Bassini et al. (2022) Bassini L., Feldmann R., Gensior J., Hayward C. C., Faucher-Giguère C. A., Cenci E., Liang L., Bernardini M., 2022, arXiv e-prints, p. arXiv:2211.08423
  * Behrens et al. (2018) Behrens C., Pallottini A., Ferrara A., Gallerani S., Vallini L., 2018, MNRAS, 477, 552
  * Bernal & Kovetz (2022) Bernal J. L., Kovetz E. D., 2022, A&AR, 30
  * Béthermin et al. (2020) Béthermin M., et al., 2020, A&A, 643, A2
  * Bisbas et al. (2022) Bisbas T. G., et al., 2022, ApJ, 934, 115
  * Bischetti et al. (2018) Bischetti M., et al., 2018, A&A, 617, A82
  * Blain (1999) Blain A. W., 1999, MNRAS, 304, 669
  * Boselli et al. (2002) Boselli A., Gavazzi G., Lequeux J., Pierini D., 2002, A&A, 385, 454
  * Bottorff et al. (1998) Bottorff M., LaMothe J., Momjian E., Verner E., Vinković D., Ferland G., 1998, PASP, 110, 1040
  * Bouwens et al. (2007) Bouwens R. J., Illingworth G. D., Franx M., Ford H., 2007, ApJ, 670, 928
  * Bouwens et al. (2011) Bouwens R. J., et al., 2011, ApJ, 737, 90
  * Bouwens et al. (2015) Bouwens R. J., et al., 2015, ApJ, 803, 34
  * Bouwens et al. (2016) Bouwens R. J., et al., 2016, ApJ, 833, 72
  * Bouwens et al. (2019) Bouwens R. J., Stefanon M., Oesch P. A., Illingworth G. D., Nanayakkara T., Roberts-Borsani G., Labbé I., Smit R., 2019, ApJ, 880, 25
  * Bouwens et al. (2022) Bouwens R. J., et al., 2022, ApJ, 931, 160
  * Bowler et al. (2018) Bowler R. A. A., Bourne N., Dunlop J. S., McLure R. J., McLeod D. J., 2018, MNRAS, 481, 1631
  * Bowler et al. (2020) Bowler R. A. A., Jarvis M. J., Dunlop J. S., McLure R. J., McLeod D. J., Adams N. J., Milvang-Jensen B., McCracken H. J., 2020, MNRAS, 493, 2059
  * Bradač et al. (2017) Bradač M., et al., 2017, ApJ, 836, L2
  * Brauher et al. (2008) Brauher J. R., Dale D. A., Helou G., 2008, ApJS, 178, 280
  * Breuck et al. (2014) Breuck C. D., et al., 2014, A&A, 565, A59
  * Brisbin et al. (2015) Brisbin D., Ferkinhoff C., Nikola T., Parshley S., Stacey G. J., Spoon H., Hailey-Dunsheath S., Verma A., 2015, ApJ, 799, 13
  * Burgarella et al. (2013) Burgarella D., et al., 2013, A&A, 554, A70
  * CCAT-Prime collaboration et al. (2021) CCAT-Prime collaboration et al., 2021, arXiv e-prints, p. arXiv:2107.10364
  * CONCERTO Collaboration et al. (2020) CONCERTO Collaboration et al., 2020, A&A, 642, A60
  * Calzetti et al. (2007) Calzetti D., et al., 2007, ApJ, 666, 870
  * Camps & Baes (2015) Camps P., Baes M., 2015, Astronomy and Computing, 9, 20
  * Camps et al. (2016) Camps P., Trayford J. W., Baes M., Theuns T., Schaller M., Schaye J., 2016, MNRAS, 462, 1057
  * Camps et al. (2018) Camps P., et al., 2018, ApJS, 234, 20
  * Cañameras et al. (2018) Cañameras R., et al., 2018, A&A, 620, A61
  * Capak et al. (2015) Capak P. L., et al., 2015, Nature, 522, 455
  * Carilli & Walter (2013) Carilli C., Walter F., 2013, ARA&A, 51, 105
  * Carlstrom et al. (2011) Carlstrom J. E., et al., 2011, PASP, 123, 568
  * Carniani et al. (2017) Carniani S., et al., 2017, A&A, 605, A42
  * Carniani et al. (2018a) Carniani S., et al., 2018a, MNRAS, 478, 1170
  * Carniani et al. (2018b) Carniani S., Maiolino R., Smit R., Amorín R., 2018b, ApJ, 854, L7
  * Carniani et al. (2020) Carniani S., et al., 2020, MNRAS, 499, 5136
  * Casey (2012) Casey C. M., 2012, MNRAS, 425, 3094
  * Casey et al. (2014) Casey C. M., Narayanan D., Cooray A., 2014, Physics Reports, 541, 45
  * Casey et al. (2018a) Casey C. M., et al., 2018a, ApJ, 862, 77
  * Casey et al. (2018b) Casey C. M., Hodge J., Zavala J. A., Spilker J., da Cunha E., Staguhn J., Finkelstein S. L., Drew P., 2018b, ApJ, 862, 78
  * Çatmabacak et al. (2022) Çatmabacak O., Feldmann R., Anglés-Alcázar D., Faucher-Giguère C.-A., Hopkins P. F., Kereš D., 2022, MNRAS, 511, 506
  * Chung et al. (2020) Chung D. T., Viero M. P., Church S. E., Wechsler R. H., 2020, ApJ, 892, 51
  * Cochrane et al. (2019) Cochrane R. K., et al., 2019, MNRAS, 488, 1779
  * Cochrane et al. (2022) Cochrane R. K., Hayward C. C., Anglés-Alcázar D., 2022, ApJ, 939, L27
  * Contursi et al. (2017) Contursi A., et al., 2017, A&A, 606, A86
  * Cooke et al. (2018) Cooke E. A., et al., 2018, ApJ, 861, 100
  * Cooray et al. (2014) Cooray A., et al., 2014, ApJ, 790, 40
  * Cormier et al. (2015) Cormier D., et al., 2015, A&A, 578, A53
  * Cox et al. (2011) Cox P., et al., 2011, ApJ, 740, 63
  * Crites et al. (2014) Crites A. T., et al., 2014, Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VII
  * Curti et al. (2022) Curti M., et al., 2022, arXiv e-prints, p. arXiv:2207.12375
  * Davé et al. (2016) Davé R., Thompson R., Hopkins P. F., 2016, MNRAS, 462, 3265
  * Dayal & Ferrara (2018) Dayal P., Ferrara A., 2018, Physics Reports, 780-782, 1
  * De Looze et al. (2011) De Looze I., Baes M., Bendo G. J., Cortese L., Fritz J., 2011, MNRAS, 416, 2712
  * De Looze et al. (2014) De Looze I., et al., 2014, A&A, 568, A62
  * Decarli et al. (2017) Decarli R., et al., 2017, Nature, 545, 457
  * Decarli et al. (2018) Decarli R., et al., 2018, ApJ, 854, 97
  * Dey et al. (2008) Dey A., et al., 2008, ApJ, 677, 943
  * Díaz-Santos et al. (2013) Díaz-Santos T., et al., 2013, ApJ, 774, 68
  * Díaz-Santos et al. (2015) Díaz-Santos T., et al., 2015, ApJ, 816, L6
  * Díaz-Santos et al. (2017) Díaz-Santos T., et al., 2017, ApJ, 846, 32
  * Dole et al. (2006) Dole H., et al., 2006, A&A, 451, 417
  * Donnan et al. (2022) Donnan C. T., et al., 2022, arXiv e-prints, p. arXiv:2207.12356
  * Draine & Salpeter (1979) Draine B. T., Salpeter E. E., 1979, ApJ, 231, 438
  * Draine et al. (2007) Draine B. T., et al., 2007, ApJ, 663, 866
  * Dullemond et al. (2012) Dullemond C. P., Juhasz A., Pohl A., Sereshti F., Shetty R., Peters T., Commercon B., Flock M., 2012, RADMC-3D: A multi-purpose radiative transfer tool, Astrophysics Source Code Library, record ascl:1202.015 (ascl:1202.015)
  * Dunne et al. (2000) Dunne L., Eales S., Edmunds M., Ivison R., Alexander P., Clements D. L., 2000, MNRAS, 315, 115
  * Dwek (1998) Dwek E., 1998, ApJ, 501, 643
  * Eilers et al. (2020) Eilers A.-C., et al., 2020, ApJ, 900, 37
  * Eisenhardt et al. (2012) Eisenhardt P. R. M., et al., 2012, ApJ, 755, 173
  * Elbaz et al. (2011) Elbaz D., et al., 2011, A&A, 533, A119
  * Ellis et al. (2013) Ellis R. S., et al., 2013, ApJ, 763, L7
  * Faisst et al. (2017) Faisst A. L., et al., 2017, ApJ, 847, 21
  * Faisst et al. (2020a) Faisst A. L., et al., 2020a, ApJS, 247, 61
  * Faisst et al. (2020b) Faisst A. L., Fudamoto Y., Oesch P. A., Scoville N., Riechers D. A., Pavesi R., Capak P., 2020b, MNRAS, 498, 4192
  * Fan et al. (2019) Fan X., et al., 2019, ApJ, 870, L11
  * Farrah et al. (2013) Farrah D., et al., 2013, ApJ, 776, 38
  * Faucher-Giguère et al. (2009) Faucher-Giguère C.-A., Lidz A., Zaldarriaga M., Hernquist L., 2009, ApJ, 703, 1416
  * Feldmann (2017) Feldmann R., 2017, MNRAS: Letters, 470, L59
  * Feldmann (2020) Feldmann R., 2020, Communications Physics, 3
  * Feldmann et al. (2016) Feldmann R., Hopkins P. F., Quataert E., Faucher-Giguère C.-A., Kereš D., 2016, MNRAS:Letters, 458, L14
  * Feldmann et al. (2017) Feldmann R., Quataert E., Hopkins P. F., Faucher-Giguère C.-A., Kereš D., 2017, MNRAS, 470, 1050
  * Feldmann et al. (2022) Feldmann R., et al., 2022, arXiv e-prints, p. arXiv:2205.15325
  * Ferland et al. (1992) Ferland G. J., Peterson B. M., Horne K., Welsh W. F., Nahar S. N., 1992, ApJ, 387, 95
  * Ferland et al. (1998) Ferland G. J., Korista K. T., Verner D. A., Ferguson J. W., Kingdon J. B., Verner E. M., 1998, PASP, 110, 761
  * Ferland et al. (2013) Ferland G. J., et al., 2013, Rev. Mex. Astron. Astrofis., 49, 137
  * Ferland et al. (2017) Ferland G. J., et al., 2017, Rev. Mex. Astron. Astrofis., 53, 385
  * Ferrara et al. (2019) Ferrara A., Vallini L., Pallottini A., Gallerani S., Carniani S., Kohandel M., Decataldo D., Behrens C., 2019, MNRAS, 489, 1
  * Ferrara et al. (2022) Ferrara A., et al., 2022, arXiv e-prints, p. arXiv:2202.07666
  * Finkelstein et al. (2013) Finkelstein S. L., et al., 2013, Nature, 502, 524
  * Finkelstein et al. (2015) Finkelstein S. L., et al., 2015, ApJ, 810, 71
  * Fixsen et al. (1998) Fixsen D. J., Dwek E., Mather J. C., Bennett C. L., Shafer R. A., 1998, ApJ, 508, 123
  * Fonseca et al. (2017) Fonseca J., Silva M. B., Santos M. G., Cooray A., 2017, MNRAS, 464, 1948
  * Fudamoto et al. (2021) Fudamoto Y., et al., 2021, Nature, 597, 489
  * Fudamoto et al. (2022) Fudamoto Y., et al., 2022, ApJ, 934, 144
  * Fujimoto et al. (2016) Fujimoto S., Ouchi M., Ono Y., Shibuya T., Ishigaki M., Nagai H., Momose R., 2016, ApJS, 222, 1
  * Fujimoto et al. (2019) Fujimoto S., et al., 2019, ApJ, 887, 107
  * Fujimoto et al. (2020) Fujimoto S., et al., 2020, ApJ, 900, 1
  * Fujimoto et al. (2021) Fujimoto S., et al., 2021, ApJ, 911, 99
  * Fujimoto et al. (2022) Fujimoto S., et al., 2022, arXiv e-prints, p. arXiv:2212.06863
  * Geach et al. (2016) Geach J. E., et al., 2016, ApJ, 832, 37
  * Genzel & Cesarsky (2000) Genzel R., Cesarsky C. J., 2000, ARA&A, 38, 761
  * Genzel et al. (2015) Genzel R., et al., 2015, ApJ, 800, 20
  * Gill et al. (2004) Gill S. P. D., Knebe A., Gibson B. K., 2004, MNRAS, 351, 399
  * Gilmore et al. (1989) Gilmore G., Wyse R. F. G., Kuijken K., 1989, ARA&A, 27, 555
  * Ginolfi et al. (2020) Ginolfi M., et al., 2020, A&A, 633, A90
  * Gkogkou et al. (2022) Gkogkou A., et al., 2022, arXiv e-prints, p. arXiv:2212.02235
  * Goldsmith et al. (2012) Goldsmith P. F., Langer W. D., Pineda J. L., Velusamy T., 2012, ApJS, 203, 13
  * Gong et al. (2012) Gong Y., Cooray A., Silva M., Santos M. G., Bock J., Bradford C. M., Zemcov M., 2012, ApJ, 745, 49
  * Graciá-Carpio et al. (2011) Graciá-Carpio J., et al., 2011, ApJ, 728, L7
  * Griffin et al. (2010) Griffin M. J., et al., 2010, A&A, 518, L3
  * Gruppioni et al. (2013) Gruppioni C., et al., 2013, MNRAS, 432, 23
  * Guilloteau et al. (1992) Guilloteau S., et al., 1992, A&A, 262, 624
  * Gullberg et al. (2015) Gullberg B., et al., 2015, MNRAS, 449, 2883
  * Gullberg et al. (2018) Gullberg B., et al., 2018, ApJ, 859, 12
  * Habing (1968) Habing H. J., 1968, Bull. Astron. Inst. Netherlands, 19, 421
  * Hahn & Abel (2011) Hahn O., Abel T., 2011, MNRAS, 415, 2101
  * Hailey-Dunsheath (2009) Hailey-Dunsheath S., 2009, PhD thesis, Cornell University, United States
  * Hao et al. (2011) Hao C.-N., Kennicutt R. C., Johnson B. D., Calzetti D., Dale D. A., Moustakas J., 2011, ApJ, 741, 124
  * Harikane et al. (2020) Harikane Y., et al., 2020, ApJ, 896, 93
  * Harikane et al. (2022) Harikane Y., et al., 2022, arXiv e-prints, p. arXiv:2208.01612
  * Harrington et al. (2021) Harrington K. C., et al., 2021, ApJ, 908, 95
  * Hashimoto et al. (2018) Hashimoto T., et al., 2018, Nature, 557, 392
  * Hashimoto et al. (2019a) Hashimoto T., et al., 2019a, PASJ, 71
  * Hashimoto et al. (2019b) Hashimoto T., et al., 2019b, MNRAS, 488, 5029
  * Hayward et al. (2011) Hayward C. C., Kereš D., Jonsson P., Narayanan D., Cox T. J., Hernquist L., 2011, ApJ, 743, 159
  * Hayward et al. (2014) Hayward C. C., et al., 2014, MNRAS, 445, 1598
  * Heintz et al. (2022a) Heintz K. E., et al., 2022a, arXiv e-prints, p. arXiv:2212.02890
  * Heintz et al. (2022b) Heintz K. E., et al., 2022b, arXiv e-prints, p. arXiv:2212.06877
  * Helou et al. (2001) Helou G., Malhotra S., Hollenbach D. J., Dale D. A., Contursi A., 2001, ApJ, 548, L73
  * Herrera-Camus et al. (2015) Herrera-Camus R., et al., 2015, ApJ, 800, 1
  * Herrera-Camus et al. (2018) Herrera-Camus R., et al., 2018, ApJ, 861, 95
  * Heyminck et al. (2006) Heyminck S., Kasemann C., Güsten R., de Lange G., Graf U. U., 2006, A&A, 454, L21
  * Hezaveh et al. (2012) Hezaveh Y. D., Marrone D. P., Holder G. P., 2012, ApJ, 761, 20
  * Hezaveh et al. (2013) Hezaveh Y. D., et al., 2013, ApJ, 767, 132
  * Hildebrand (1983) Hildebrand R. H., 1983, QJRAS, 24, 267
  * Ho et al. (2004) Ho P. T. P., Moran J. M., Lo K. Y., 2004, ApJ, 616, L1
  * Hodge & da Cunha (2020) Hodge J. A., da Cunha E., 2020, Royal Society Open Science, 7, 200556
  * Hollenbach & Tielens (1999) Hollenbach D. J., Tielens A. G. G. M., 1999, Reviews of Modern Physics, 71, 173
  * Hollenbach et al. (1991) Hollenbach D. J., Takahashi T., Tielens A. G. G. M., 1991, ApJ, 377, 192
  * Hopkins et al. (2014) Hopkins P. F., Kereš D., Oñorbe J., Faucher-Giguère C.-A., Quataert E., Murray N., Bullock J. S., 2014, MNRAS, 445, 581
  * Hopkins et al. (2018) Hopkins P. F., et al., 2018, MNRAS, 480, 800
  * Hopkins et al. (2022) Hopkins P. F., et al., 2022, arXiv e-prints, p. arXiv:2203.00040
  * Hopwood et al. (2011) Hopwood R., et al., 2011, ApJ, 728, L4
  * Hu et al. (2002) Hu E. M., Cowie L. L., McMahon R. G., Capak P., Iwamuro F., Kneib J.-P., Maihara T., Motohara K., 2002, ApJ, 568, L75
  * Hughes et al. (2017) Hughes T. M., et al., 2017, A&A, 602, A49
  * Indriolo & McCall (2012) Indriolo N., McCall B. J., 2012, ApJ, 745, 91
  * Indriolo et al. (2007) Indriolo N., Geballe T. R., Oka T., McCall B. J., 2007, ApJ, 671, 1736
  * Inoue et al. (2016) Inoue A. K., et al., 2016, Science, 352, 1559
  * Iono et al. (2006) Iono D., et al., 2006, ApJ, 645, L97
  * Ivison et al. (2010) Ivison R. J., et al., 2010, A&A, 518, L35
  * Izumi et al. (2018) Izumi T., et al., 2018, PASJ, 70
  * Izumi et al. (2019) Izumi T., et al., 2019, PASJ, 71
  * Izumi et al. (2021a) Izumi T., et al., 2021a, ApJ, 908, 235
  * Izumi et al. (2021b) Izumi T., et al., 2021b, ApJ, 914, 36
  * Jones et al. (2010) Jones T. A., Swinbank A. M., Ellis R. S., Richard J., Stark D. P., 2010, MNRAS
  * Kanekar et al. (2013) Kanekar N., Wagg J., Chary R. R., Carilli C. L., 2013, ApJ, 771, L20
  * Kannan et al. (2022) Kannan R., Smith A., Garaldi E., Shen X., Vogelsberger M., Pakmor R., Springel V., Hernquist L., 2022, MNRAS, 514, 3857
  * Katz et al. (2017) Katz H., Kimm T., Sijacki D., Haehnelt M. G., 2017, MNRAS, 468, 4831
  * Katz et al. (2019) Katz H., et al., 2019, MNRAS, 487, 5902
  * Kaufman et al. (1999) Kaufman M. J., Wolfire M. G., Hollenbach D. J., Luhman M. L., 1999, ApJ, 527, 795
  * Kawamata et al. (2016) Kawamata R., Oguri M., Ishigaki M., Shimasaku K., Ouchi M., 2016, ApJ, 819, 114
  * Kennicutt (1998) Kennicutt R. C., 1998, ApJ, 498, 541
  * Kennicutt et al. (2011) Kennicutt R. C., et al., 2011, PASP, 123, 1347
  * Kessler et al. (1996) Kessler M. F., et al., 1996, A&A, 500, 493
  * Khusanova et al. (2021) Khusanova Y., et al., 2021, A&A, 649, A152
  * Kirkpatrick et al. (2015) Kirkpatrick A., Pope A., Sajina A., Roebuck E., Yan L., Armus L., Díaz-Santos T., Stierwalt S., 2015, ApJ, 814, 9
  * Knollmann & Knebe (2009) Knollmann S. R., Knebe A., 2009, ApJS, 182, 608
  * Knudsen et al. (2016) Knudsen K. K., Richard J., Kneib J.-P., Jauzac M., Clément B., Drouart G., Egami E., Lindroos L., 2016, MNRAS, 462, L6
  * Knudsen et al. (2017) Knudsen K. K., Watson D., Frayer D., Christensen L., Gallazzi A., Michałowski M. J., Richard J., Zavala J., 2017, MNRAS, 466, 138
  * Kohandel et al. (2019) Kohandel M., Pallottini A., Ferrara A., Zanella A., Behrens C., Carniani S., Gallerani S., Vallini L., 2019, MNRAS, 487, 3007
  * Kovetz et al. (2017) Kovetz E. D., et al., 2017, arXiv e-prints, p. arXiv:1709.09066
  * Kovetz et al. (2019) Kovetz E., et al., 2019, BAAS, 51, 101
  * Kroupa (2002) Kroupa P., 2002, Science, 295, 82
  * Krumholz (2014) Krumholz M. R., 2014, MNRAS, 437, 1662
  * Lagache et al. (2018) Lagache G., Cousin M., Chatzikos M., 2018, A&A, 609, A130
  * Laporte et al. (2017) Laporte N., et al., 2017, ApJ, 837, L21
  * Laporte et al. (2019) Laporte N., et al., 2019, MNRAS, 487, L81
  * Le Fèvre et al. (2020) Le Fèvre O., et al., 2020, A&A, 643, A1
  * Leech et al. (1999) Leech K. J., et al., 1999, MNRAS, 310, 317
  * Leethochawalit et al. (2022a) Leethochawalit N., Roberts-Borsani G., Morishita T., Trenti M., Treu T., 2022a, arXiv e-prints, p. arXiv:2205.15388
  * Leethochawalit et al. (2022b) Leethochawalit N., et al., 2022b, arXiv e-prints, p. arXiv:2207.11135
  * Lehar et al. (2000) Lehar J., et al., 2000, ApJ, 536, 584
  * Leipski et al. (2013) Leipski C., et al., 2013, ApJ, 772, 103
  * Leipski et al. (2014) Leipski C., et al., 2014, ApJ, 785, 154
  * Leitherer & Heckman (1995) Leitherer C., Heckman T. M., 1995, ApJS, 96, 9
  * Leitherer et al. (1999) Leitherer C., et al., 1999, ApJS, 123, 3
  * Leroy et al. (2008) Leroy A. K., Walter F., Brinks E., Bigiel F., de Blok W. J. G., Madore B., Thornley M. D., 2008, ApJ, 136, 2782
  * Leung et al. (2020) Leung T. K. D., Olsen K. P., Somerville R. S., Davé R., Greve T. R., Hayward C. C., Narayanan D., Popping G., 2020, ApJ, 905, 102
  * Li et al. (2018) Li Q., Narayanan D., Davè R., Krumholz M. R., 2018, ApJ, 869, 73
  * Li et al. (2019) Li Q., Narayanan D., Davé R., 2019, MNRAS, 490, 1425
  * Liang et al. (2018) Liang L., Feldmann R., Faucher-Giguère C.-A., Kereš D., Hopkins P. F., Hayward C. C., Quataert E., Scoville N. Z., 2018, MNRAS:Letters, 478, L83
  * Liang et al. (2019) Liang L., et al., 2019, MNRAS, 489, 1397
  * Liang et al. (2021) Liang L., Feldmann R., Hayward C. C., Narayanan D., Çatmabacak O., Kereš D., Faucher-Giguère C.-A., Hopkins P. F., 2021, MNRAS, 502, 3210
  * Lu et al. (2018) Lu N., et al., 2018, ApJ, 864, 38
  * Luhman et al. (1998) Luhman M. L., et al., 1998, ApJ, 504, L11
  * Luhman et al. (2003) Luhman M. L., Satyapal S., Fischer J., Wolfire M. G., Sturm E., Dudley C. C., Lutz D., Genzel R., 2003, ApJ, 594, 758
  * Lupi & Bovino (2020) Lupi A., Bovino S., 2020, MNRAS, 492, 2818
  * Lupi et al. (2020) Lupi A., Pallottini A., Ferrara A., Bovino S., Carniani S., Vallini L., 2020, MNRAS, 496, 5160
  * Ma et al. (2016) Ma X., Hopkins P. F., Faucher-Giguère C.-A., Zolman N., Muratov A. L., Kereš D., Quataert E., 2016, MNRAS, 456, 2140
  * Ma et al. (2019) Ma X., et al., 2019, MNRAS, 487, 1844
  * Madau & Dickinson (2014) Madau P., Dickinson M., 2014, ARA&A, 52, 415
  * Madden et al. (2013) Madden S. C., et al., 2013, PASP, 125, 600
  * Magdis et al. (2014) Magdis G. E., et al., 2014, ApJ, 796, 63
  * Magnelli et al. (2009) Magnelli B., Elbaz D., Chary R. R., Dickinson M., Borgne D. L., Frayer D. T., Willmer C. N. A., 2009, A&A, 496, 57
  * Maiolino et al. (2005) Maiolino R., et al., 2005, A&A, 440, L51
  * Maiolino et al. (2009) Maiolino R., Caselli P., Nagao T., Walmsley M., Breuck C. D., Meneghetti M., 2009, A&A, 500, L1
  * Maiolino et al. (2015) Maiolino R., et al., 2015, MNRAS, 452, 54
  * Malhotra et al. (1997) Malhotra S., et al., 1997, ApJ, 491, L27
  * Malhotra et al. (2001) Malhotra S., et al., 2001, ApJ, 561, 766
  * Marrone et al. (2018) Marrone D. P., et al., 2018, Nature, 553, 51
  * Matthee et al. (2017) Matthee J., et al., 2017, ApJ, 851, 145
  * Matthee et al. (2019) Matthee J., et al., 2019, ApJ, 881, 124
  * Mazzucchelli et al. (2017) Mazzucchelli C., et al., 2017, ApJ, 849, 91
  * McKinney et al. (2020) McKinney J., Pope A., Armus L., Chary R.-R., Díaz-Santos T., Dickinson M. E., Kirkpatrick A., 2020, ApJ, 892, 119
  * McKinney et al. (2021) McKinney J., Hayward C. C., Rosenthal L. J., Martínez-Galarza J. R., Pope A., Sajina A., Smith H. A., 2021, ApJ, 921, 55
  * McKinnon et al. (2016) McKinnon R., Torrey P., Vogelsberger M., 2016, MNRAS, 457, 3775
  * McLeod et al. (2015) McLeod D. J., McLure R. J., Dunlop J. S., Robertson B. E., Ellis R. S., Targett T. A., 2015, MNRAS, 450, 3032
  * McLure et al. (2013) McLure R. J., et al., 2013, MNRAS, 432, 2696
  * Meyer et al. (2022) Meyer R. A., et al., 2022, ApJ, 927, 152
  * Miettinen et al. (2017) Miettinen O., et al., 2017, A&A, 606, A17
  * Molyneux et al. (2022) Molyneux S. J., et al., 2022, MNRAS, 512, 535
  * Muñoz & Oh (2016) Muñoz J. A., Oh S. P., 2016, MNRAS, 463, 2085
  * Murmu et al. (2023) Murmu C. S., et al., 2023, MNRAS, 518, 3074
  * Nahar & Pradhan (1997) Nahar S. N., Pradhan A. K., 1997, ApJS, 111, 339
  * Naidu et al. (2022) Naidu R. P., et al., 2022, arXiv e-prints, p. arXiv:2207.09434
  * Narayanan & Krumholz (2017) Narayanan D., Krumholz M. R., 2017, MNRAS, 467, 50
  * Narayanan et al. (2015) Narayanan D., et al., 2015, Nature, 525, 496
  * Neri et al. (2014) Neri R., Downes D., Cox P., Walter F., 2014, A&A, 562, A35
  * Neufeld & Wolfire (2017) Neufeld D. A., Wolfire M. G., 2017, ApJ, 845, 163
  * Novak et al. (2020) Novak M., et al., 2020, ApJ, 904, 131
  * Oesch et al. (2012) Oesch P. A., et al., 2012, ApJ, 745, 110
  * Oesch et al. (2016) Oesch P. A., et al., 2016, ApJ, 819, 129
  * Oesch et al. (2018) Oesch P. A., Bouwens R. J., Illingworth G. D., Labbé I., Stefanon M., 2018, ApJ, 855, 105
  * Olsen et al. (2015) Olsen K. P., Greve T. R., Narayanan D., Thompson R., Toft S., Brinch C., 2015, ApJ, 814, 76
  * Olsen et al. (2017) Olsen K., Greve T. R., Narayanan D., Thompson R., Davé R., Rios L. N., Stawinski S., 2017, ApJ, 846, 105
  * Olsen et al. (2018) Olsen K., Greve T. R., Narayanan D., Thompson R., Davé R., Rios L. N., Stawinski S., 2018, ApJ, 857, 148
  * Ota et al. (2014) Ota K., et al., 2014, ApJ, 792, 34
  * Oteo et al. (2016) Oteo I., et al., 2016, ApJ, 827, 34
  * Ouchi et al. (2013) Ouchi M., et al., 2013, ApJ, 778, 102
  * Padmanabhan (2019) Padmanabhan H., 2019, MNRAS, 488, 3014
  * Padmanabhan et al. (2022) Padmanabhan H., Breysse P., Lidz A., Switzer E. R., 2022, MNRAS, 515, 5813
  * Pallottini et al. (2017) Pallottini A., Ferrara A., Bovino S., Vallini L., Gallerani S., Maiolino R., Salvadori S., 2017, MNRAS, 471, 4128
  * Pallottini et al. (2019) Pallottini A., et al., 2019, MNRAS, 487, 1689
  * Pentericci et al. (2016) Pentericci L., et al., 2016, ApJ, 829, L11
  * Pilbratt et al. (2010) Pilbratt G. L., et al., 2010, A&A, 518, L1
  * Pillepich et al. (2018a) Pillepich A., et al., 2018a, MNRAS, 473, 4077
  * Pillepich et al. (2018b) Pillepich A., et al., 2018b, MNRAS, 475, 648
  * Planck Collaboration et al. (2016) Planck Collaboration et al., 2016, A&A, 594, A1
  * Popping et al. (2014) Popping G., Pérez-Beaupuits J. P., Spaans M., Trager S. C., Somerville R. S., 2014, MNRAS, 444, 1301
  * Popping et al. (2016) Popping G., van Kampen E., Decarli R., Spaans M., Somerville R. S., Trager S. C., 2016, MNRAS, 461, 93
  * Popping et al. (2019) Popping G., Narayanan D., Somerville R. S., Faisst A. L., Krumholz M. R., 2019, MNRAS, 482, 4906
  * Priddey & McMahon (2001) Priddey R. S., McMahon R. G., 2001, MNRAS, 324, L17
  * Rawle et al. (2014) Rawle T. D., et al., 2014, ApJ, 783, 59
  * Rémy-Ruyer et al. (2014) Rémy-Ruyer A., et al., 2014, A&A, 563, A31
  * Rhoads et al. (2022) Rhoads J. E., et al., 2022, arXiv e-prints, p. arXiv:2207.13020
  * Richings et al. (2014a) Richings A. J., Schaye J., Oppenheimer B. D., 2014a, MNRAS, 440, 3349
  * Richings et al. (2014b) Richings A. J., Schaye J., Oppenheimer B. D., 2014b, MNRAS, 442, 2780
  * Richings et al. (2022) Richings A. J., Faucher-Giguère C.-A., Gurvich A. B., Schaye J., Hayward C. C., 2022, MNRAS, 517, 1557
  * Riechers et al. (2013) Riechers D. A., et al., 2013, Nature, 496, 329
  * Rigopoulou et al. (2018) Rigopoulou D., Pereira-Santaella M., Magdis G. E., Cooray A., Farrah D., Marques-Chaves R., Perez-Fournon I., Riechers D., 2018, MNRAS, 473, 20
  * Rojas-Ruiz et al. (2021) Rojas-Ruiz S., et al., 2021, ApJ, 920, 150
  * Rybak et al. (2019) Rybak M., et al., 2019, ApJ, 876, 112
  * Safarzadeh et al. (2016) Safarzadeh M., Hayward C. C., Ferguson H. C., Somerville R. S., 2016, ApJ, 818, 62
  * Salim & Narayanan (2020) Salim S., Narayanan D., 2020, ARA&A, 58, 529
  * Sanders et al. (2003) Sanders D. B., Mazzarella J. M., Kim D.-C., Surace J. A., Soifer B. T., 2003, AJ, 126, 1607
  * Sargsyan et al. (2012) Sargsyan L., et al., 2012, ApJ, 755, 171
  * Schaerer et al. (2015a) Schaerer D., Boone F., Zamojski M., Staguhn J., Dessauges-Zavadsky M., Finkelstein S., Combes F., 2015a, A&A, 574, A19
  * Schaerer et al. (2015b) Schaerer D., et al., 2015b, A&A, 576, L2
  * Schaerer et al. (2020) Schaerer D., et al., 2020, A&A, 643, A3
  * Schaerer et al. (2022) Schaerer D., Marques-Chaves R., Barrufet L., Oesch P., Izotov Y. I., Naidu R., Guseva N. G., Brammer G., 2022, A&A, 665, L4
  * Schouws et al. (2022a) Schouws S., et al., 2022a, arXiv e-prints, p. arXiv:2202.04080
  * Schouws et al. (2022b) Schouws S., et al., 2022b, ApJ, 928, 31
  * Scoville (2013) Scoville N. Z., 2013, in Falcón-Barroso J., Knapen J. H., eds, Secular Evolution of Galaxies. Cambridge University Press, Cambridge, UK, p. 491
  * Scoville et al. (2017) Scoville N., et al., 2017, ApJ, 837, 150
  * Semenov et al. (2017) Semenov V. A., Kravtsov A. V., Gnedin N. Y., 2017, ApJ, 845, 133
  * Serjeant (2012) Serjeant S., 2012, MNRAS, 424, 2429
  * Serra et al. (2016) Serra P., Doré O., Lagache G., 2016, ApJ, 833, 153
  * Shao et al. (2017) Shao Y., et al., 2017, ApJ, 845, 138
  * Shao et al. (2019) Shao Y., et al., 2019, ApJ, 876, 99
  * Shen et al. (2022) Shen X., Vogelsberger M., Nelson D., Tacchella S., Hernquist L., Springel V., Marinacci F., Torrey P., 2022, MNRAS, 510, 5560
  * Silva et al. (2015) Silva M., Santos M. G., Cooray A., Gong Y., 2015, ApJ, 806, 209
  * Sklias et al. (2014) Sklias P., et al., 2014, A&A, 561, A149
  * Smit et al. (2018) Smit R., et al., 2018, Nature, 553, 178
  * Sobral et al. (2015) Sobral D., Matthee J., Darvish B., Schaerer D., Mobasher B., Röttgering H. J. A., Santos S., Hemmati S., 2015, ApJ, 808, 139
  * Sodroski et al. (1997) Sodroski T. J., Odegard N., Arendt R. G., Dwek E., Weiland J. L., Hauser M. G., Kelsall T., 1997, ApJ, 480, 173
  * Somerville & Davé (2015) Somerville R. S., Davé R., 2015, ARA&A, 53, 51
  * Somerville & Primack (1999) Somerville R. S., Primack J. R., 1999, MNRAS, 310, 1087
  * Somerville et al. (2015) Somerville R. S., Popping G., Trager S. C., 2015, MNRAS, 453, 4338
  * Sommovigo et al. (2020) Sommovigo L., Ferrara A., Pallottini A., Carniani S., Gallerani S., Decataldo D., 2020, MNRAS, 497, 956
  * Sommovigo et al. (2021) Sommovigo L., Ferrara A., Carniani S., Zanella A., Pallottini A., Gallerani S., Vallini L., 2021, MNRAS, 503, 4878
  * Sommovigo et al. (2022) Sommovigo L., et al., 2022, MNRAS
  * Sparre et al. (2017) Sparre M., Hayward C. C., Feldmann R., Faucher-Giguère C.-A., Muratov A. L., Kereš D., Hopkins P. F., 2017, MNRAS, 466, 88
  * Spilker et al. (2016) Spilker J. S., et al., 2016, ApJ, 826, 112
  * Spitzer (1998) Spitzer L., 1998, Physical Processes in the Interstellar Medium. Wiley, doi:10.1002/9783527617722, https://doi.org/10.1002%2F9783527617722
  * Stacey et al. (1991) Stacey G. J., Geis N., Genzel R., Lugten J. B., Poglitsch A., Sternberg A., Townes C. H., 1991, ApJ, 373, 423
  * Stacey et al. (2007) Stacey G. J., et al., 2007, in Baker A. J., Glenn J., Harris A. I., Mangum J. G., Yun M. S., eds, Astronomical Society of the Pacific Conference Series Vol. 375, From Z-Machines to ALMA: (Sub)Millimeter Spectroscopy of Galaxies. p. 52
  * Stacey et al. (2010) Stacey G. J., Hailey-Dunsheath S., Ferkinhoff C., Nikola T., Parshley S. C., Benford D. J., Staguhn J. G., Fiolet N., 2010, ApJ, 724, 957
  * Sternberg et al. (2014) Sternberg A., Le Petit F., Roueff E., Le Bourlot J., 2014, ApJ, 790, 10
  * Suginohara et al. (1999) Suginohara M., Suginohara T., Spergel D. N., 1999, ApJ, 512, 547
  * Sun et al. (2021) Sun G., et al., 2021, ApJ, 915, 33
  * Sun et al. (2022) Sun G., Mas-Ribas L., Chang T.-C., Furlanetto S. R., Mebane R. H., Gonzalez M. O., Parsons J., Trapp A. C., 2022, arXiv e-prints, p. arXiv:2206.14186
  * Swinbank et al. (2010) Swinbank A. M., et al., 2010, Nature, 464, 733
  * Swinbank et al. (2012) Swinbank A. M., et al., 2012, MNRAS, 427, 1066
  * Swinbank et al. (2014) Swinbank A. M., et al., 2014, MNRAS, 438, 1267
  * Symeonidis (2016) Symeonidis M., 2016, MNRAS, 465, 1401
  * Tacchella et al. (2016) Tacchella S., Dekel A., Carollo C. M., Ceverino D., DeGraf C., Lapiner S., Mandelker N., Joel R. P., 2016, MNRAS, 457, 2790
  * Tacconi et al. (2018) Tacconi L. J., et al., 2018, ApJ, 853, 179
  * Tadaki et al. (2018) Tadaki K., et al., 2018, Nature, 560, 613
  * Tadaki et al. (2020) Tadaki K., et al., 2020, ApJ, 889, 141