THE METALLICITY DEPENDENCE OF THE CO → H₂ CONVERSION FACTOR IN z ⩾ 1 STAR-FORMING GALAXIES^*

In this paper, we discuss recent galaxy-integrated measurements of the ¹²CO 3–2 and 2–1 lines in z ∼ 1–3 ''normal'' massive SFGs. Most were drawn from the CO 3–2 observations of z ∼ 1–2 SFGs of the Tacconi et al. (2010) and Tacconi & Combes IRAM Large Programs (henceforth ''LP''). The LP data and the smaller CO 2–1 z ∼ 1.5 sample of Daddi et al. (2010a) were taken with the IRAM Plateau de Bure millimeter Interferometer (Guilloteau et al. 1992; Tacconi et al. 2010 and L. J. Tacconi et al., in preparation). The LP SFGs are drawn from two samples with median redshifts of 〈z〉 = 1.2 and 〈z〉 = 2.2. They are matched to cover the same ranges of stellar mass (M_* = 3–30 × 10¹⁰ M_☉) and star formation (20–300 M_☉ yr⁻¹). The LP sample used in this paper has 21 detections between z = 1 and 1.5, and 11 detections and 5 upper limits between z = 2 and 2.4. We also include four detections from Daddi et al. (2010a) at 〈z〉 = 1.5, with comparable selection criteria as in the LP sample. To these sets we add the detections of three somewhat lower-mass (M_* = 5–30 × 10⁹ M_☉), strongly lensed SFGs (cB58: z = 2.7, Baker et al. 2004; ''Cosmic Eye": z = 3.1, Coppin et al. 2007; ''Eyelash'': z = 2.3, Swinbank et al. 2010; Danielson et al. 2011). For a description of the observations and the data analysis, refer to the papers above.

The galaxies we are analyzing exhibit a correlation between stellar mass (M_*) and star formation rate (SFR), or stellar mass and specific star formation rate (sSFR = SFR/M_*), the so-called star formation "main sequence" (Schiminovich et al. 2007; Noeske et al. 2007). Galaxies near the ''main-sequence'' make up 90% of the cosmic star formation density at z ∼ 1.5–2.5 (Rodighiero et al. 2011). The relation has a substantial scatter of rms ± 0.3 dex at z ∼ 0.5–2 (Figure 1; Elbaz et al. 2007; Noeske et al. 2007; Daddi et al. 2007; Rodighiero et al. 2010; Mancini et al. 2011). ''Main-sequence'' SFGs have disk-like morphologies with low Sérsic indices (n_S ∼ 1) and, compared to off-main-sequence systems, have relatively large effective radii (Wuyts et al. 2011). Most of the galaxies in our sample are extended rotating disks in Hα integral field spectroscopy data sets, Hubble Space Telescope (HST) rest-frame UV/optical images, or CO high-resolution interferometry maps (Förster Schreiber et al. 2009; Law et al. 2009; Tacconi et al. 2010; Daddi et al. 2010a; Mancini et al. 2011; F. Combes et al., in preparation). A few are compact, dispersion-dominated systems (Law et al. 2009; Förster Schreiber et al. 2009; Tacconi et al. 2010). Their SFRs thus are 5–30 times smaller than bright submillimeter-selected galaxies in the same redshift range that are starbursts and often appear to be major mergers (Greve et al. 2005; Engel et al. 2010). Based on their CO or Hα kinematics, and/or their HST high-resolution morphology, only one galaxy in our current sample (BX528) is a merger (Tacconi et al. 2010; L. J. Tacconi et al., in preparation; Shapiro et al. 2008; Förster Schreiber et al. 2009).

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For 14 z ⩾ 1 SFGs we have individual determinations of gas-phase metallicities based on the [N ii]/Hα estimator of Pettini & Pagel (2004: Z_O = 8.9 + 0.57 log (F([N ii])/F(Hα))). Of these 14, 6 galaxies were observed with the SINFONI integral field spectrometer as part of the SINS program (Förster Schreiber et al. 2009), and 8 galaxies with long slit spectroscopy (Teplitz et al. 2000; Erb et al. 2006b; D. K. Erb et al. 2009, private communication; Richard et al. 2011; P. Buschkamp et al., in preparation). The rms dispersion of the Pettini & Pagel relation is ±0.18 dex for 7.5 < Z_O < 8.6. For the rest we determined metallicities from the stellar-mass–metallicity relation at the respective redshifts (Erb et al. 2006b; P. Buschkamp et al., in preparation; Shapley et al. 2005; Liu et al. 2008; Z_O = a + 2.18 log(M_*) − 0.0896 log(M_*)², with a = −4.51 for z = 1.5–3, and a = −4.45 for z ∼ 1–1.5). This includes two active galactic nuclei with measured [N ii]/Hα (Erb et al. 2006b), for which we also adopt the metallicities estimated from the mass–metallicity relation. The rms dispersion of the z ∼ 2 mass–metallicity relation is ±0.09 dex, which is a measure of the statistical error. Metallicities derived from the mass–metallicity relation should thus not be much more uncertain than those inferred from [N ii]/Hα. Systematic uncertainties between different metallicity indicators and calibrations are larger and can exceed 0.3 dex (e.g., Kewley & Ellison 2008). The [N ii]/Hα estimator is known (e.g., Pettini & Pagel 2004) to saturate above solar metallicity (Z_☉ ∼ 8.69 (±0.05); Asplund et al. 2009). To minimize this systematic effect, we converted all metallicities to the Denicoló et al. (2002) calibration system (also based on [N ii]/Hα), with the conversion function given in Table 3 of Kewley & Ellison (2008). The transformation onto the Denicoló et al. scale delivers the best agreement (i.e., smallest scatter) between different metallicity calibrators. It optimizes the comparison to the z ∼ 0 metallicity estimates, especially at the high M_* end (Kewley & Ellison 2008), which is particularly important for our study. The systematic uncertainties within the Denicoló et al. (2002) system and over the observed range should be within ±0.2 dex (Kewley & Ellison 2008). Relative uncertainties from the measurement errors in [N ii]/Hα and the mass–metallicity relation are much smaller (see typical red error bar at the bottom of Figure 3). For the 14 SFGs with metallicity estimates from both methods the rms scatter between the two methods is ±0.1 dex.

Our final sample of 44 z ⩾ 1 SFGs covers inferred oxygen abundances from Z_O ∼ 8.4 to 8.9 on the Denicoló et al. (2002) scale. For comparison to z ∼ 0 SFGs of different metallicities we used the recent compilations of Leroy et al. (2011) and Krumholz et al. (2011). Wherever possible, we replaced their quoted metallicities with [N ii]/Hα-based metallicities from the published literature, with the same calibrations as for the high-z data. Most of the [N ii]/Hα-derived metallicities are very similar to the ones given by Leroy et al. and Krumholz et al.

In his seminal 1998 paper, Kennicutt (1998, henceforth K98) established that the star formation surface density Σ_{star form} and the total (atomic + molecular) gas surface density Σ_gas in z ∼ 0 SFGs (including ultra-luminous infrared galaxy mergers, ULIRGs) are related through

$$\begin{equation} \log (\Sigma _{{\rm star\; form}}) = a{\rm } + {\rm }n \times \log (\Sigma _{{\rm gas}}), \end{equation} \tag{ 3 }$$

with a best-fit slope of n ∼ 1.4. More recent studies of galaxy-integrated and spatially resolved KS-relations (on kpc scales) in nonmerger SFGs near the star formation main sequence and near solar metallicity have found that the SFR surface density correlates mainly with the molecular gas surface density, Σ_{mol gas} (Kennicutt et al. 2007; Bigiel et al. 2008; Schruba et al. 2011). The ''molecular'' KS-relation is flatter, with n ∼ 1.0–1.3 for Σ_{mol gas} ∼ 10 to 10^3.5 M_☉ pc⁻² for near-solar metallicity disk galaxies and a constant Galactic conversion factor (Kennicutt 2008; Bigiel et al. 2008; Leroy et al. 2008; Daddi et al. 2010b; Genzel et al. 2010; Schruba et al. 2011).

We show in Figure 3 an update of the galaxy-integrated KS-relations shown in Figures 2 and 3 of Genzel et al. (2010), amended with the additional new z ⩾ 1 data discussed in this paper. We used α_{CO 1–0} = α_G for all data. To our knowledge, Figure 3 is the largest compilation of both low- and high-z galaxy-integrated KS-measurements so far in the literature. The z ∼ 0 SFG sample (all observed in CO 1–0) includes many more SFGs, moderate starburst galaxies (e.g., M82, NGC 253, NGC 3256), and luminous infrared galaxies (LIRGs) than K98, but no ultraluminous LIRG (ULIRG) mergers (see Genzel et al. 2010 for details). These additional z ∼ 0 SFGs span more than three orders of magnitude in molecular gas surface density, from the lower limit of the mostly molecular gas-dominated regime at 5–10 M_☉ pc⁻² (Blitz & Rosolowsky 2006) to dense circumnuclear starbursts with >10⁴ M_☉ pc⁻². The z ∼ 0 SFGs also overlap with the entire range of the z ⩾ 1 SFGs (observed in 3–2 for the z ∼ 1–1.5 and z ∼ 2–2.4 IRAM galaxies, and the lenses, and in CO 2–1 for the z ∼ 1.5 Daddi et al. 2010a SFGs). The spatially resolved KS-studies of Kennicutt et al. (2007), Bigiel et al. (2008), Leroy et al. (2008), and Schruba et al. (2011) cover the range of 10–100 M_☉ pc⁻² and are in excellent agreement with the galaxy-integrated data shown here. For computing surface densities, half light radii were taken from molecular observations themselves at z ∼ 0, and from a mixture of CO, Hα-, and rest-frame UV-continuum sizes for z > 1 (Genzel et al. 2010).

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For the best examination of the slope of the KS-relation, we plot in Figure 3 on the vertical axis the ratio of SFR (or its surface density) to molecular gas mass (or its surface density), inferred from the CO line observations with α = α_G. This is the molecular gas depletion rate (t_depletion(CO))⁻¹. An n = 1 KS-relation (a constant depletion rate independent of gas surface density) corresponds to a horizontal line. The best fit to all z ∼ 0 SFGs and the z ⩾ 1 SFGs above solar metallicity (Z_O > 8.7) yields a slope of n = 1.1 (±0.06), in the middle of the range of slopes found in the literature from both spatially resolved and galaxy-integrated z ∼ 0 studies. The slope of the z ⩾ 1 SFGs with Z_O > 8.7 is marginally flatter (n = 0.9 ± 0.12). Given the fairly large (±0.3 dex; K98; Kennicutt et al. 2007, Figure 3) scatter of the relation, as well as the systematic uncertainties in the slope determinations of ±0.2 to ±0.25 (see discussion in Genzel et al. 2010), Occam's razor suggests that all data sets are consistent with a universal, near-unity slope, assuming that α_CO does not vary systematically with Σ_{mol gas} (see the discussion at the end of this section).

There does appear to be a small but statistically significant offset in the average depletion rates at z ∼ 0 and z > 1 (Figure 3, Genzel et al. 2010). The average depletion rate is 0.4–0.5 Gyr⁻¹ at z ∼ 0 and ∼1 Gyr⁻¹ at z = 1–2.5 (Leroy et al. 2008; Bigiel et al. 2011; Tacconi et al. 2010; Daddi et al. 2010a, 2010b; Genzel et al. 2010; Bauermeister et al. 2010; Saintonge et al. 2011). Saintonge et al. (2011) find a positive correlation of the depletion rate with sSFR, with an ∼0.5 dex difference in depletion rate between the highest and smallest sSFR SFGs in Figure 3. This variation is consistent with the scatter in Figure 3 (rms ± 0.3 dex for z ∼ 0 SFGs, and ±0.2 dex for Z_O > 8.7, z > 1 SFGs).

To explore the dependence of the KS-slope on CO excitation and the rotational quantum number J of the observed line, Narayanan et al. (2011a) have carried out numerical hydro-simulations of star formation in the molecular ISM with an input volumetric star formation relation, $\dot{\rho} _{\rm star\; form} \propto {\rho _{\rm mol\; gas}} / {t_{\rm ff}} \propto \rho _{\rm mol\; gas}^m$. Here t_ff is the local free-fall time in the molecular clouds. Narayanan et al. find that the resulting gas–star-formation–surface-density relation depends on the rotational quantum number J of the CO line used for tracing the molecular gas. For m = 1.5 (t_ff∝ρ^−1/2) and conditions similar to z ⩾ 1 SFGs or z ∼ 0 starbursts the simulations yield n(J = 1) ∼ 1.47 and n(J = 3) ∼ 1.08. The data in Figure 3 are clearly not consistent with this result, since the z ∼ 0 data (observed in CO 1–0) and the z > 1 data (observed in CO 2–1 and 3–2) both have n ∼ 1 and the difference in slope is less than 0.3, even if the fit uncertainties are included. A better match to the simulations occurs for m = 1 (t_ff = const.), for which Narayanan et al. predict n(J = 1) = 0.96 and n(J = 3) = 0.76. This case would apply if the molecular interstellar media in the different SFGs consist of collections of clouds with similar properties (Bigiel et al. 2008), or if the SFGs are in a state of marginal gravitational stability. For the latter case the Toomre Q-parameter is near unity, and the local free-fall timescale above becomes similar to the global dynamical timescale (cf., Genzel et al. 2010). The former explanation may be relevant for the z ∼ 0 disks, while the latter is applicable to the z ⩾ 1 SFGs.

Could there be a dependence of α_CO on Σ_{mol gas}? There are plausible reasons discussed in Section 3.3 and Tacconi et al. (2008) (due to the n(H₂)^1/2/T dependence of α_CO) that such a dependence should be present to some extent (see Appendix A and Figure 10 in Tacconi et al. 2008). Based on the available empirical data the possible change of α_CO in the range sampled in Figure 3 may be about a factor of two (0.3 dex). To explore the impact of such a change we have applied an artificial shift of −0.3 dex to all galaxies with surface densities >10³ M_☉ pc⁻² (M82 is at 10^3.4), and then refitted the KS-relation. This results in an overall KS-slope of 1.22 instead of 1.1, again within the systematic slope uncertainties.

A final point relates to the issue of atomic hydrogen in the KS-relation. We have mentioned that all galaxies in Figure 3 are above ∼5–10 M_☉ pc^—2, where the z ∼ 0 pressure–H₂ relation of Blitz & Rosolowsky (2006) indicates that almost all hydrogen is in molecular form. However, the combined effects of strong stellar feedback, shocks, and turbulence in starbursts and high-z SFGs may lower the fraction of H₂ to total (star-forming) gas at a given gas surface density, relative to that in normal z ∼ 0 disks. In this case, the gas surface densities are lower limits to the total (star-forming) gas in each galaxy. It is plausible that this correction decreases from low to high gas surface densities, suggesting that the intrinsic KS-relation (with these corrections added) is still flatter than indicated by the data in Figure 3.

Given this fairly universal KS-relation for massive SFGs at both low- and high-redshifts, we can now ask whether there is a systematic dependence of the depletion rate on metallicity. In Figure 3, we plot the depletion rate of the z > 1 SFGs as a function of Σ_{mol gas} for three different metallicity bins, assuming α_G = 4.36 for all bins. Super-solar, high-metallicity galaxies (Z_O > 8.76) are marked by blue filled circles, SFGs with 8.76 > Z_O > 8.62 are denoted by black triangles, and the lowest metallicity galaxies in our sample (Z_O < 8.62) are marked by filled red squares. There is a significant difference in the inferred depletion rates for these three metallicity bins. At least for the highest and middle bins the number of galaxies is large enough to see that the galaxies in each bin follow a fairly flat KS-relation but are offset from one another. The offset in the lowest metallicity bin is even larger but the number of galaxies is too small to estimate the KS-slope. This appearance of flat KS-relations ''peeling off'' vertically as a function of Z_O is exactly what is expected if the variations were due to differences in α_CO(Z_O) between the metallicity bins.

Another representation of the same data is given in Figure 4, where we plot the depletion rate as a function of gas phase oxygen abundance in the z ∼ 1–2.5 SFGs of our sample. At or above solar metallicity all z ⩾ 1 SFGs approach SFR/(α_GL'_{CO 1–0}) ∼ 1  Gyr⁻¹, in agreement with the discussion above, and with a scatter that is consistent with the measurement uncertainties. Below solar metallicity, the data exhibit a trend of rapidly increasing SFR/(α_GL'_{CO 1–0}) with decreasing oxygen abundance. The trend at z ⩾ 1 is similar to that found at z ∼ 0. Data points with metallicities derived from [N ii]/Hα and from the mass–metallicity relation agree within the uncertainties discussed in Section 2.2.

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Can the trends in Figures 3 and 4 be driven by a physical change in depletion rate or in the ISM properties, and is metallicity the primary underlying variable? At z ∼ 0 the lowest metallicity star-forming systems are dwarf/irregular galaxies, such as the SMC and NGC 6822. Given the evidence for time variable star formation histories in such systems (Tolstoy et al. 2009), their much greater depletion rates compared to normal disk galaxies might well be the result of recent short-duration starbursts. However, combined spatially resolved studies of H i, infrared dust, and CO emission in the SMC and a number of the other z ∼ 0 SFGs plotted in Figure 4—which yield the CO conversion factor and the molecular gas mass without relying on the KS-relation—strongly suggest that it is the absence of CO emission, and not the presence of starburst events that dominate the apparently high depletion rates (Leroy et al. 2011; Bolatto et al. 2011).

With the exception of cB58, the starburst explanation is even more unlikely for the high-z SFGs in our sample. Almost all are massive systems on or near the star formation main sequence (Figure 1). Galaxies near the main-sequence exhibit exponential light profiles with fairly large (R_1/2 ∼ 3–6 kpc) disk radii (Wuyts et al. 2011). They exhibit high star formation duty cycles (30%–100%; Adelberger et al. 2005; Noeske et al. 2007; Daddi et al. 2007). High-z main-sequence SFGs are forming stars at high rates (20–300 M_☉ yr⁻¹) primarily because of their high gas fractions (Daddi et al. 2010a; Tacconi et al. 2010) and not because they are undergoing short-duration ''starburst'' events. The high gas fractions are plausibly driven by the semi-continuous large gas accretion rates predicted by all cold dark matter models (e.g., Kereš et al. 2005; Dekel & Birnboim 2006; Ocvirk et al. 2008; Genel et al. 2008; Bouché et al. 2010). At fixed redshift the star formation surface densities are almost constant as a function of stellar mass (and thus metallicity) along the main sequence (∼0.6 and ∼3 M_☉ yr^—1 kpc⁻² for z ∼ 1 and 2; Wuyts et al. 2011).

With the possible exception of the lensed ''Eyelash'' galaxy (Swinbank et al. 2010) all galaxies in our sample are within the ±0.3 dex dispersion of the M_*–sSFR main-sequence relations at z ∼ 1.2 and 2.2, including the individual uncertainties in stellar masses and SFRs (Figure 1). Merger-induced starbursts (e.g., z ∼ 0 ULIRGs) with internally enhanced depletion rates typically lie an order of magnitude or more above the main-sequence line (Daddi et al. 2010b; Genzel et al. 2010; Combes et al. 2011; Saintonge et al. 2011).

Recent Herschel PACS observations have revealed a remarkable uniformity of the infrared spectral energy distributions of massive main-sequence SFGs at all redshifts (Hwang et al. 2010; Elbaz et al. 2011; Nordon et al. 2012). Main-sequence galaxies with SFRs from a few to a few hundred solar masses per year (with the exception of z ∼ 0 ULIRGs) have similar dust temperatures, T_dust ∼ 27–38 K, between z ∼ 0 and 2 (Hwang et al. 2010). High-z SFGs are merely 2–5 K colder than z ∼ 0 SFGs of the same luminosity.

Observations of multiple CO rotational lines in a number of z ⩾ 1 SFGs and submillimeter galaxies show that the CO ladder distributions are similar to those of local starburst galaxies, such as M82 and NGC 253, with inferred local molecular hydrogen volume densities (for the CO 3–2 emission) of n_H2 = 10^2.5 to 10^4.5 cm⁻³ (Weiss et al. 2007; Dannerbauer et al. 2009; Danielson et al. 2011; Riechers et al. 2011; F. Combes et al., in preparation). Average molecular hydrogen densities in the giant star-forming clumps in z ⩾ 1 SFGs may be ∼10² to 10³ cm⁻³ (Genzel et al. 2011). However, the fortuitous cancellation of the density and temperature dependencies may break down very close to massive star formation sites and may drive the conversion factor downward, as in the case of the central starburst region in M82 (α_{CO 1–0} ∼ 2; Wild et al. 1992; Weiss et al. 2001). This means that the average ratio (〈n(H₂)〉)^1/2/T_R1 in Equation (2) probably is comparable in z ⩾ 1 SFGs to that in moderate z ∼ 0 starbursts, or the Milky Way GMC population (T_{R 1} ∼ 7–30 K, 〈n_H2〉 ∼ 10^1.5 to 10² cm⁻³).

Could the large turbulence in high-z SFGs affect the observed metallicity trend in α_{CO 1–0}? As a rule z ⩾ 1 SFGs near the ''main sequence'' exhibit 4–10 times larger velocity dispersions than z ∼ 0 SFGs (Förster Schreiber et al. 2009; Law et al. 2009; Epinat et al. 2009). Recent theoretical work suggests that increased turbulence may profoundly affect the local density and temperature structure, and in turn also the conversion factor (Narayanan et al. 2011b; Shetty et al. 2011a, 2011b). However, the observed velocity dispersions in z ⩾ 1 main-sequence SFGs appear to depend little on galaxy mass, SFR, or surface density (Genzel et al. 2011, and references therein). In addition, for most of the z ⩾ 1 SFGs in Figures 1 and 4, rotation dominates over random motions, in contrast to many z ∼ 0 ULIRGs and z ⩾ 1 submillimeter galaxies (Tacconi et al. 2008; Engel et al. 2010). Star formation in z ⩾ 1 main-sequence SFGs is plausibly driven by gravitational instabilities in giant star-forming clouds, similar to GMCs but scaled-up to the large gas fractions at z ⩾ 1 (Genzel et al. 2011). In any case, the effect of turbulent compression and the presence of nonvirialized gas components with chaotic motions drive the conversion factor downward, not upward, both in the empirical data (Solomon et al. 1997; Scoville et al. 1997; Downes & Solomon 1998; Tacconi et al. 1999, 2008), as well as in the simulations of Narayanan et al. (2011b) and Shetty et al. (2011b).

Another issue is whether the CO rotational ladder excitation might vary between galaxies in a systematic way such that a constant brightness temperature scaling factor R_1J from J to 1, as used in this study, is not appropriate. The first studies of the CO rotational ladder distributions indeed show variations in CO rotational excitations, but for J ⩽ 3 these are by far too small to account for the magnitude of the variations in Figure 4 (Weiss et al. 2007; Dannerbauer et al. 2009; Danielson et al. 2011; Riechers et al. 2011; Ivison et al. 2011; F. Combes et al., in preparation).

Mannucci et al. (2010) have shown that the gas phase oxygen abundance in z ∼ 0 SFGs depends on not only stellar mass (i.e., the mass–metallicity relation discussed in section 2.2) but also SFR. At fixed stellar mass a galaxy with a higher SFR has a lower oxygen abundance,

$$\begin{equation} 12 + \log ({\rm O/H}) = b{\rm } + {\rm }\beta \times ({\log (M_*) - \gamma \log ({\rm SFR})}),\end{equation} \tag{ 4 }$$

where b = 4.2, β = 0.47, and γ = 0.32 for metallicities on the Maiolino et al. (2008) scale (Mannucci et al. 2010). As we discuss in this section, this ''fundamental metallicity relation'' (FMR) in combination with a nonlinear (n > 1) KS-relation qualitatively yields a depletion rate—metallicity relation similar to that shown in Figure 4.

For z ∼ 0 galaxies in the Sloan Digital Sky Survey the FMR has an rms scatter of ±0.05 to ±0.06 dex, which is smaller than the ±0.08 dex scatter in the mass–metallicity relation (Mannucci et al. 2010), suggesting it may be the more fundamental relation. Mannucci et al. (2010) interpret the FMR as being driven by the interplay of star formation increasing metallicity and accretion delivering fresh, low-metallicity gas. Mannucci et al. (2010) also show that the 10 z ∼ 1–2.5 SFGs with reliable metallicity estimates available to their study fall on the same relation, indicating that Equation (4) may also apply at z ∼ 1–2.5. If the (molecular) KS-relation is nonlinear (n > 1 in Equation (3)), the combination of Equations (3) and (4) yields

$$\begin{equation} \begin{array}{*{20}c} {t_{{\rm depletion}}^{ - 1} ({\rm mol\; gas}){\rm } \propto {\rm }M_*^\delta \times 10^{ - \chi Z_O } \times R_{1/2} ^{ - (2n - 2)/n}\quad {\rm with}} \hfill \\ {\delta = (1 - {1 \mathord{/{\vphantom {1 n}} \kern-\nulldelimiterspace} n})/\gamma\quad {\rm and } \quad\chi = {\rm }(1 - {1 \mathord{/{\vphantom {1 n}} \kern-\nulldelimiterspace} n})/(\beta \gamma).} \hfill \\ \end{array} \end{equation} \tag{ 5 }$$

For n = 1.1 and the FMR parameters found by Mannucci et al. (2010: β = 0.47, γ = 0.32) δ = 0.28 and χ = 0.6. For a fixed stellar mass Equation (5) then predicts that the depletion rate increases by a factor of two between solar metallicity and 0.5 times solar metallicity. This is in qualitative agreement with the observations shown in Figure 4. The FMR predicts that at fixed stellar mass a lower metallicity galaxy has a higher SFR. If the KS-relation is nonlinear, a higher SFR automatically implies a greater depletion rate. But, can Equation (5) also quantitatively account for the observations shown in Figure 4?

The average amplitude of the FMR is already captured in our analysis by using a redshift-dependent mass–metallicity relation, as described in Section 2.2. Additional changes in metallicity estimates from the scatter of the stellar-mass–SFR relation per se (±0.3 dex in SFR at fixed stellar mass) are smaller (<0.1 dex) than other uncertainties. To estimate the best FMR parameters at z ∼ 2 we used a recent, yet unpublished compilation of z ∼ 1.4–2.5 SFGs observed in [N ii] and Hα with integral field and long-slit spectroscopy as part of the SINS, OSIRIS, and LUCIFER surveys (Förster Schreiber et al. 2009; Law et al. 2009; P. Buschkamp et al., in preparation). The best-fit z ∼ 2 FMR on the Denicoló et al. (2002) scale has β = 0.31 (±0.06 1σ) for data stacked in stellar mass bins (all for γ fixed at 0.32). The rms scatter of the z ∼ 2 data is ±0.11 dex.

We list in Table 1 the molecular depletion rates and SFRs computed from Equation (5) for the four most extreme SFGs in Figure 4, and compare these values to the corresponding values computed from Equations (3) and (4), for n = 1.1 and 1.3, and for β = 0.31, 0.25, and 0.37 (reflecting the 1σ uncertainty in the z ∼ 2 FMR relation). The value of β = 0.37 on the Denicoló et al. (2002) scale is equivalent to β = 0.47 on the Maiolino et al. (2008) scale deduced by Mannucci et al. (2010) to be the best fit for the z ∼ 0 SFGs. With this range of values Table 1 shows that the effect of the FMR cannot simultaneously quantitatively account for the observed SFRs and depletion rates (for constant α_G = 4.36) in ZC406690, cB58, and BX389, for any combination of the parameters n and β. Either the predicted depletion rate is too small, the SFR is too large, or both. Obviously no effect is predicted for a KS-slope of n = 1, favored by the work of Bigiel, Leroy, Schruba, and collaborators. The largest impact of the FMR naturally is for steeper KS-slopes (n = 1.3), as favored by Kennicutt et al. (2007) and Daddi et al. (2010b). In the case of BX482 a match is possible for small β ∼ 0.2–0.3 and n > 1.1.

The unrealistically large SFRs found for many of the cases in Table 1 immediately show that the strict application of an ''inverse'' FMR (as in Equation (5)) is not a good tool for individual galaxies. This is because the dependence on star formation in Equation (4) is very shallow (β × γ ∼ 0.1) and the scatter of the FMR is large. This means that the inverse solution for SFR depends very strongly on the location of a source in the Z_O–M_*-plane and the assumption of a scatter-free relation.

We conclude that including the effect of the FMR may indeed somewhat decrease the amplitude of the effect seen in Figure 4. However, the FMR cannot account for the observed increase of the depletion rate with decreasing metallicity.

Theoretical work on UV-illuminated molecular clouds reaching back 25 years has consistently predicted a strong dependence of (t_depletion(CO))⁻¹ on metallicity, especially in somewhat diffuse molecular gas (van Dishoeck & Black 1986, 1988; Maloney & Black 1988; Sternberg & Dalgarno 1995; Hollenbach & Tielens 1999; Wolfire et al. 1990, 2010; Bolatto et al. 1999; Papadopoulos & Pelupessy 2010; Shetty et al. 2011a; Glover & Mac Low 2011; Glover & Clark 2011). The short-dashed blue curve in Figure 4 (from Krumholz et al. 2011) is the result of calculating (t_depletion(CO, Z_O))⁻¹ for a spherical molecular cloud with constant column density (Σ_gas ∼ 80 M_☉ pc⁻² ∼ 〈Σ_gas(GMC, MW)〉), and exposed to a diffuse UV radiation field with a characteristic ratio of UV energy density G_UV to gas density n_H similar to that in the solar neighborhood (G_sn = 2.7 × 10⁻³ erg cm⁻² s⁻¹, n_sn = 23 cm⁻³; Wolfire et al. 2010; Krumholz et al. 2011). For this curve the volume filling factor f_V (the inverse of the clumping factor c as defined by Krumholz et al. 2011) of molecular gas is assumed to be 0.2. This curve represents the ratio of the total H₂ column through the cloud to the H₂ column in which CO remains molecular, given the adopted UV radiation field and its density, clumpiness, and total column (Wolfire et al. 2010). The depth of the CO-photodissociation zone is controlled by dust (A_UV ∼ 1) and thus scales directly with metallicity (e.g., van Dishoeck & Black 1986, 1988; Maloney & Black 1988; Sternberg & Dalgarno 1995; Wolfire et al. 2010; Glover & Mac Low 2011; Feldmann et al. 2011). The work of Krumholz et al. (2011) and Glover & Clark (2011) implies that star formation does not require the formation/presence of CO but can also occur in ''CO-dark'' gas. Krumholz et al. (2011) find that this star-forming ''CO-dark'' gas is molecular (in terms of hydrogen), while Glover & Clark (2011) conclude that star formation can even occur in atomic hydrogen gas as long as dust shielding is efficient (A_V > 3).

Simple estimates based on SFRs and galaxy sizes suggest that z ⩾ 1 SFGs in Figure 1 have UV radiation field densities ∼10² to 10³ times that in the solar neighborhood. Average ISM densities may also be greater, such that the ratio is probably similar to that in the solar neighborhood, or in nearby starburst galaxies. The onset of the up-turn in the blue-dashed curve in Figure 4 depends strongly on f_V. Model curves with larger f_V values than chosen by Krumholz et al. (2011) have up-turns at higher metallicity. For a homogeneous cloud (f_V = 1) the up-turn occurs at solar metallicity.

The agreement of the theoretical predictions with the observations suggests that the trends seen in Figure 4, at both low and high redshifts, may mainly be the consequence of the ratio of the volume and mass of molecular gas traced by CO (relative to that in H₂) decreasing rapidly at low metallicity due to photodissociation by the ambient UV field (see Pelupessy & Papadopoulos 2009; Krumholz et al. 2011; Glover & Mac Low 2011; Shetty et al. 2011b).

We now turn the results in Figure 4 around and derive an empirical dependence of α_{CO 1–0} on metallicity, based on the assumption of a universal KS-scaling relation. Using the KS-relation in reverse, the molecular gas mass can be obtained from the SFR (e.g., Erb et al. 2006b),

$$\begin{equation} M_{{\rm mol\; gas}} (M_ \odot) = \xi {\rm SFR}\;(M_ \odot \;{\rm yr}^{ - 1})^{1/n} {\rm }R_{1/2}\; ({\rm kpc})^{(2n - 2)/n} {\rm.} \end{equation} \tag{ 6 }$$

R_1/2 is the half-light radius of the gas/star-forming disk. For the z > 1 data with Z_O > 8.7 in Figure 3 and the best-fit slope of n = 1.1 we find ξ = 1.02 × 10⁹. For a slope fixed to n = 1.3 (Daddi et al. 2010b), the data in Figure 3 yield ξ = 1.2 × 10⁹. In comparison to Equation (6) the simpler assumption of a constant depletion rate of 1 Gyr⁻¹ (n = 1) at z > 1 yields typically 5% larger gas masses for all but the most compact galaxies. The relation given in Equation (8) of Genzel et al. (2010), which depends on the ratio of R_1/2 to the circular velocity (i.e., the global dynamical time), instead of R_1/2 alone, on average yields half the gas masses. The relation in Kennicutt et al. (2007; n = 1.37) yields 20% smaller gas masses. These differences are all within the scatter of the empirical KS-relation.

We apply Equation (6) to each galaxy and divide the inferred gas mass by L'_CO. The result is a direct measure of the conversion factor, and Figure 5 shows the derived CO 1–0 conversion factor as a function of metallicity for our z ⩾ 1 sample. Figure 5 has two insets. The left inset applies for our favored KS-slope of n = 1.1 (Figure 3). The right inset is for n = 1.3 favored by Daddi et al. (2010b) and Kennicutt et al. (2007). For comparison, we show estimates of α_{CO 1–0} of z ∼ 0 SFGs by Leroy et al. (2011). The Leroy et al. (2011) estimates of α_{CO 1–0} come from simultaneous parametric fitting of spatially resolved H i, CO, and dust emission. They do not rely on the KS-relation. The three sets of dashed lines denote previous CO conversion factor scaling relations for z ∼ 0 SFGs (Wilson 1995; Arimoto et al. 1996; Israel 1997, 2000; Boselli et al. 2002). This earlier work was partially based on similar galaxies as in the Leroy et al. (2011) sample but employed different methods.

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In agreement with the theoretical expectations, the CO → H₂ conversion factor increases with decreasing metallicity for all these samples, with similar trends at both low- and high-z, albeit with a large scatter. If the z ∼ 0 points of Leroy et al. (2011) and the z ⩾ 1 SFGs are combined (treating 3σ upper limits as detections), the best linear fit yields the relation

$$\begin{equation} \log (\alpha _{{\rm CO 1}\hbox{--}{\rm 0}}) = - 1.3{\rm (} \pm {\rm 0}{\rm.25) (12} + {\rm log(O/H))}_{{\rm Denicol\scriptsize\hbox{\'{o}}\;02}} + {\rm 12 (} \pm {\rm 2),} \end{equation} \tag{ 7 }$$

where the quoted uncertainties are 2σ fit uncertainties for equal weights to all data points (because of the dominance of the systematic errors discussed above). The result in Equation (7) is the same for n = 1.1 and n = 1.3. For n = 1.1 a fit to only the z ⩾ 1 data yields a slope of −1.8 (±0.6) and zero point of 17 (±6), which is somewhat steeper than, but not significantly different from the combined fit. This slope may get somewhat flatter, however, depending on the FMR effects and the slope of the KS-relation, as discussed in Section 3.4. Our method constrains the relative conversion-factor–metallicity relation to no better than ±50%, since the rms scatter of the data points around any of the two relations given above is ±0.23 dex. Future observations are needed to clarify whether the outliers in Figure 4 are due to observational uncertainties, or whether they might point to intrinsic variations in the conversion factor, perhaps as a result of variations in the clumping factor. Absolute uncertainties are larger because of the inherent uncertainties in the metallicity calibrations (and their applicability at high-z), stellar masses, and SFRs, and the excitation of the CO rotational ladder. These uncertainties reflect a combination of the systematic uncertainties and plausible physical variations.