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A Study of the Gas-Star Formation Relation over Cosmic Time

We use the first systematic data sets of CO molecular line emission in z~1-3 normal star forming galaxies for a comparison of the dependence of galaxy-averaged star formation rates on molecular gas masses at low and high redshifts, and in different galactic environments. Although the current high-z samples are still small and biased toward the luminous and massive tail of the actively star-forming 'main-sequence', a fairly clear picture is emerging. Independent of whether galaxy integrated quantities or surface densities are considered, low- and high-z SFG galaxy populations appear to follow similar molecular gas-star formation relations with slopes 1.1 to 1.2. The gas-depletion time scale in these SFGs grows from 0.5 Gyrs at z~2 to 1.5 Gyrs at z~0. Because star formation depletion times are significantly smaller than the Hubble time at all redshifts sampled, star formation rates and gas fractions are set by the balance between gas accretion from the halo and stellar feedback. In contrast, very luminous gas rich major mergers at both low-z and high-z produce on average 4 to10 times more far-infrared luminosity per unit gas mass. Only some fraction of this difference can be explained by uncertainties in gas-mass or luminosity estimators; much of it must be intrinsic. The most likely interpretation is that the star formation relation is driven by global dynamical effects. For a given mass, the more compact merger systems produce stars more rapidly because their gas clouds are more compressed with shorter dynamical times, so that they churn more quickly through the available gas reservoir than the typical normal disk galaxies. When the dependence on galactic dynamical time scale is explicitly included, disk galaxies and mergers appear to follow similar gas to star-formation relations. The mergers may be forming stars at slightly higher efficiencies than the disks.


Abstract
We use the first systematic data sets of CO molecular line emission in z~1-3 normal star forming galaxies (SFGs) for a comparison of the dependence of galaxy-averaged star formation rates on molecular gas masses at low and high redshifts, and in different galactic environments. Although the current high-z samples are still small and biased toward the luminous and massive tail of the actively star-forming 'main-sequence', a fairly clear picture is emerging. Independent of whether galaxy integrated quantities or surface densities are considered, low-and high-z SFG galaxy populations appear to follow similar molecular gas-star formation relations with slopes 1.1 to 1.2, over three orders of magnitude in gas mass or surface density. The gas-depletion time scale in these SFGs grows from 0.5 Gyrs at z~2 to 1.5 Gyrs at z~0. The average corresponds to a fairly low star formation efficiency of 2% per dynamical time. Because star formation depletion times are significantly smaller than the Hubble time at all redshifts sampled, star formation rates and gas fractions are set by the balance between gas accretion from the halo and stellar feedback.
In contrast, very luminous and ultra-luminous, gas rich major mergers at both low-z and high-z produce on average 4 to10 times more far-infrared luminosity per unit gas mass. We show that only some fraction of this difference can be explained by uncertainties in gas-mass or luminosity estimators; much of it must be intrinsic. A possible explanation is a top-heavy stellar mass function in the merging systems but the most likely interpretation is that the star formation relation is driven by global dynamical effects. For a given mass, the more compact merger systems produce stars more rapidly because their gas clouds are more compressed with shorter dynamical times, so that they churn more quickly through the available gas reservoir than the typical normal disk galaxies. When the dependence on galactic dynamical time scale is explicitly included, disk galaxies and mergers appear to follow similar gas to star-formation relations. The mergers may be forming stars at slightly higher efficiencies than the disks.

Introduction
Stars form from neutral interstellar gas. In the Milky Way and nearby galaxies most and arguably all star formation occurs in massive (10 4 …10 6.5 M  ), dense (n(H 2 )~10 2 …10 5 cm -3 ) and cold (T gas~1 0-30 K), gravitationally bound 'giant molecular clouds' (GMCs: Solomon et al. 1987, Bolatto et al. 2008, McKee & Ostriker 2007. But how does star formation proceed on a global galactic scale? How does it depend on environmental parameters, such as the mass, density, temperature and kinematics of the interstellar gas, on the mass and properties of the galaxy on large scales, and on cosmic epoch? Is the mass function of newly formed stars (the initial mass function (IMF)) universal, or does it vary depending on environment? In large part because the microphysics of stellar formation itself is still not fully understood, these questions, central to quantitative models of galaxy evolution, cannot yet be answered by ab-initio theory (McKee & Ostriker 2007).
Ever since the pioneering work of Schmidt (1959), empirical scaling relations have been fairly successful in describing galactic scale star formation in z=0 disk galaxies. The simplest hypothesis is to relate the star formation rate (SFR) integrated over a galaxy, SFR, and its total neutral gas mass M gas through d / (1), where d  is the gas consumption/depletion time scale , Bauermeister, Blitz & Ma 2010. The virtue of equation (1) is that it is easily accessible to global measurements of the tracers of star formation and gas (i.e. stellar luminosity, CO 1-0 line luminosity and HI mass) in a large number of galaxies (Young & Scoville 1991, Solomon & Sage 1988, Gao & Solomon 2004). Its disadvantage is that it probably is too simplistic.
A popular physical approach has been to assume that the star formation rate volume density,    , scales with gas volume density gas  and local free-fall time ff  as (Schmidt 1959, Kennicutt 1998a (2) with N=1.5 (K98a, Krumholz & McKee 2005. Brackets indicate spatial averages across the galaxy. Alternatively one can set (Elmegreen 1997, Silk 1997 (4), ( )  Leroy et al. (2008) have discussed other approaches that are variants of equations (1)-(4).
The basic empirical finding in the Milky Way and in local disk galaxies is that the global gas depletion times in z~0 disk galaxies are about 2 Gyrs , and the global star formation efficiency is 1-3% per free fall, or dynamical time (K98a, Leroy et al. 2008). In his seminal 1998 paper Kennicutt finds N=1.4±0.15 from galaxy integrated measurements of 97 z=0 galaxies. This sample consists of 61 normal spirals and 36 infrared-bright starburst galaxies, and includes 5 luminous (LIRG: 10 11 L  ) or ultra-luminous (ULIRG: 10 12 L  ) merging galaxies. Bouché et al. (2007) compared the K98a sample to the first spatially resolved data at z~1-3. They concluded that low-and high-z galaxies follow a similar star formation relation but find a somewhat steeper slope than K98a (N~1.7), mainly because of a more appropriate CO to gas mass conversion factor for z~0 ULIRGs and z~1-3 submillimetre galaxies (SMGs) (e.g. Downes & Solomon 1998). Krumholz & McKee (2005) have argued that the small star formation efficiencies are plausibly accounted for by the supersonic turbulent motions in GMCs (Mach numbers ~10-30) creating very broad (log-normal) gas density distributions. At any given time only a few percent of the gas is dense enough to collapse by self-gravity before it gets dispersed again by the turbulent motions. An analytic derivation (Krumholz, McKee & Tumlinson 2009) including internal stellar feedback and external pressure leads to a relation similar to equation (2) but with a power law index N that varies from ~0.8 at low surface densities to ~1.3 at high densities (Krumholz et al. 2009). In this model the transition occurs at below which the GMC pressure is dominated by internal star formation feedback, while external pressure dominates at higher surface densities. Recent spatially resolved studies of the star formation in nearby disk and irregular galaxies support the view that no single exponent N can account for the entire relationship ). At very low surface densities most of the gas is in atomic form and N is near 2(±0.5) if the relation between total gas (sum of atomic and molecular) and star formation is considered. Above a critical surface density of ~10 M  pc -2 most of the gas becomes molecular (Blitz & Rosolowsky 2006) and the relationship attains a slope near unity . A comparison of these newer spatially resolved data to the original integrated starburst sample of K98a suggests that above ~10 2 M  pc -2 the relation may steepen again, with a slope >1  A specific issue is the inferred gas depletion time scale for infrared luminous merging galaxies. Evidence has accumulated since the mid-1980s that the ratio of infrared luminosity to CO line luminosity is much larger in interacting galaxies (by a factor of ~4- 6, Young et al. 1986) and in z~0 ULIRG mergers (by a factor >10, Sanders, Scoville & Soifer 1991, Sanders & Mirabel 1996, Gao & Solomon 2004) than in normal spiral disks. In the interacting/merging systems gas surface densities are also much larger than in normal disk galaxies (Braine & Combes 1993, Downes & Solomon 1998, Tacconi et al. 2008. The interpretation of these findings is uncertain, however. Sanders et al. (1991) proposed that most z~0 ULIRGs are powered by AGN, which dominate the far-infrared (FIR) luminosity and thus lead to an overestimate of L FIR /L CO as far as star formation is concerned. Gao & Solomon (2004) find that in contrast to CO 1-0, there exists a linear relationship between the dense gas tracer HCN 1-0 and L FIR in z~0 ULIRGs, which favours a shorter gas depletion time scale in luminous mergers as the cause of the FIR excess. Gao & Solomon (2004) propose that the underlying reason for this boost in L IR /M mol-gas ratio is an increased fraction of very dense molecular gas in mergers.
Over the last decade CO line emission studies have become possible also for highredshift galaxies. Investigations concentrated initially on very luminous quasars (Omont et al. 1996, Walter et al. 2003, Greve et al. 2005) and submillimetre galaxies (Frayer et al. 1998, 1999, Genzel et al. 2003, Greve et al. 2005, Tacconi et al. 2006, Chapman et al. 2008. Because of significant advances in the sensitivity of the IRAM Plateau de Bure millimetre interferometer, integrated and spatially resolved CO measurements have very recently become feasible also in 'normal' z~1-2.5 SFGs (i.e. not predominantly major mergers and not extreme starbursts: Daddi et al. 2008, 2010, Dannerbauer et al. 2009. These high-z SFGs are situated on the 'main-sequence' in the star formation rate-stellar mass plane, and appear to have high-duty cycles (30-60%) indicative of near-continuous star formation and replenishment of new gas , Bouché et al. 2010. Less than 50% of high-z SFGs are major mergers (Shapiro et al. 2008, Förster Schreiber et al. 2009, Daddi et al. 2010. Although the high-z samples are still small and biased toward the massive tail of the main sequence (stellar masses 3-20x10 10 M  ), it is obviously of great interest to compare the star formation relation in these systems relative to their z=0 counterparts, as well as to the low-and high-z luminous mergers. This comparison is the subject of the present paper. All physical units used below are based on a concordance, flat ΛCDM cosmology with H 0 =70 km s -1 Mpc -1 , Ω m =0.28, Ω Λ =0.72. Stellar masses and star formation rates are based on a Chabrier (2003) initial stellar mass function (IMF).

Sample Selection and Properties
In this paper we analyse data sets from a number of studies. Most importantly we discuss the recent data on z~1-2.5 normal, massive star forming galaxies reported in Tacconi et al. (2010), Daddi et al. (2010) and Tacconi et al. (in prep.), all obtained with the IRAM Plateau de Bure millimetre Interferometer. The Tacconi et al. (2010) study (with the addition of 3 galaxies in Tacconi et al. in prep.) reports on two mass-and star formation-matched samples of <z>=1.2 and <z>=2.3 SFGs, with currently 10 galaxies at <z>=1.2 and 11 galaxies at <z>=2.3, and with the same criteria of stellar mass (>4x10 10 M  ) and star formation (>70 M  yr -1 ) selection. Daddi et al. (2010) have observed a sample of 6 SFGs at <z>=1.5, with comparable selection criteria as in Tacconi et al. For a description of the observations and the data analysis we refer to the papers above. We compare these recent observations with measurements of z~1-3.5 submillimeter galaxies, and several data sets on z~0 normal, star bursting and merging galaxies from the literature. For all galaxies we adopt the K98b relation between star formation rates and infrared (8-1000μm luminosity), SFR (M  /yr)=10 -10 L IR (L  ).
In the following we discuss the properties of these different data sets.

<z>=1.2 sample
The z~1 SFGs discussed in this paper (10 galaxies) are drawn from AEGIS (Davis et al. 2007). The All-Wavelength Extended Groth Strip International Survey (AEGIS) provides deep imaging in all major wave bands from X-ray to radio (including Advanced Camera for Surveys (ACS) HST images), as well as optical spectroscopy (DEEP2/Keck) over a large area of sky (0.5 deg 2 ), with the aim of studying the panchromatic properties of galaxies over the last half of the Hubble time. The region studied is the Extended Groth Strip (EGS: RA=14 h 17 m , Dec= 52 0 30'). The AEGIS data provide the properties of a complete set of galaxies from 0.2z1.2 for the stellar mass range >10 10 M  . Extinction corrected star formation rates are derived from a combination of Spitzer MIPS 24μm fluxes, GALEX UV fluxes and Hα/[OII] fluxes (Noeske et al. 2007, Cooper et al. in prep.). Direct observations far-infrared spectral energy distributions (SEDs) of z~1 SFGs with luminosities comparable to our EGS-galaxies have now become possible with PACS on the Herschel 2 space telescope. The initial observations yield similar far-infrared luminosities as estimated from data at other wavelengths (Elbaz et al. 2010). From the AEGIS data set we selected SFGs at z~1.1 to1.3 without obvious major merger morphologies, with stellar masses 4x10 10 M  and with star-formation rates 70 M  yr -1 . The EGS galaxies thus represent the most luminous tail of the star forming <z>=1.2 galaxy population. Their star formation rates place them on the 'main sequence' in the M * -SFR plane ) and at the high mass tail of the overall population (left panel in Figure 1). Table 1 lists the inferred star formation rates and stellar masses, as well as half-light radii. The latter are obtained from Sersic fitting to the HST I/K-band images, with the exception of EGS13035123, EGS1207881 and EGS13003805, where the quoted radii are averages of these Sersic parameters and fits to the spatially resolved CO 3-2 distributions. In these three galaxies the half-light radii obtained from CO and stellar light are in reasonable agreement, similar to the finding for z=0 disk galaxies (Young & Scoville 1991).
Most of the <z>=1.2 SFGs in our sample are rotating disks. The CO kinematics in the 3 spatially resolved galaxies discussed above clearly indicate rotational motions in large disks with big local gas/star forming clumps (Tacconi et al. 2010, and in prep). In three additional galaxies velocity gradients are detected, which are indicative of rotational motions as well. The morphologies of the ACS images also are fully consistent with a clumpy disk interpretation. The values of the maximum disk circular velocity v d quoted in Table 1 were computed from the velocity difference in the two line profile emission peaks, or from the velocity width and corrected for inclinations obtained from the minor to major axis ratios in the I-band images. For details we refer to Tacconi et al. (2010).

<z>=2.3 sample
The z~2.3 SFGs (11 galaxies) were selected from the near-infrared long-slit Hα sample of Erb et al. (2006), which in turn was drawn from the larger survey of Steidel et al. (2004) and Reddy et al (2005), culled according to the so-called 'BX' criteria based on UV-colour (UGR) and R-magnitude (hereafter, simply BX sample). Our sub-sample was chosen from these surveys to cover the same stellar mass and star formation range as the z~1.2 AEGIS sample. The right panel of Figure 1 shows that within the measurement uncertainties and given the intrinsic scatter of the z~2 'main sequence', the 'BX' sample provides a fair census of the high mass end of the entire UV-/optically selected SFG population in this red-shift and mass range (Reddy et al. 2005, Förster Schreiber et al. 2009). As for the z~1 sample, many of the massive z~2 SFGs are turbulent, clumpy rotating disks (Cresci et al. 2009). Less than 50% of the massive z~2 SFGs galaxies studied with high-resolution Hα integral field spectroscopy exhibit kinematical properties that would be expected for major mergers (Shapiro et al. 2008, Förster Schreiber et al. 2009).
The star formation rates listed in Table 1 were derived from extinction corrected Hαluminosities , Förster Schreiber et al. 2009). For this purpose the conversions in Kennicutt (1998b (K98b)) were applied but the 'Salpeter IMF' star formation rates were divided by 1.7 to convert to a Chabrier IMF. Stellar masses are from the spectral energy distribution fits in , Förster Schreiber et al. 2009), and v d and R 1/2 values are from the data in the same references with the methods discussed in (Förster Schreiber et al. 2009). The typical uncertainties of the derived quantities are dominated in most sources by systematic errors in stellar masses, star formation rates and H 2 masses, all of which are at least 50%.

<z>=1.5 sample
In addition, we included in our analysis 6 <z>=1.5 SFGs from Daddi et al. (2008Daddi et al. ( , 2010. Four of these 6 sources have spatially resolved PdBI observations. All four of these resolved galaxies show strong velocity gradients with double-peaked profiles, indicative of ordered rotational motions in large disk galaxies, similar to our BX/EGS samples . The ACS images of these galaxies exhibit large clumps, again similar to what is observed at z~2. These BzK galaxies have comparable stellar masses (~5x10 10 M  ) and star formation rates (~150 M  yr -1 ). The star formation rates reported by Daddi et al. (2010) are based on several star formation indicators. They average estimates from the extinction corrected UV luminosities, from the radio luminosities, and from extrapolations of the observed 24μm Spitzer mid-infrared (restframe ~10μm) luminosities to far-infrared luminosities with the library of z=0 Chary & Elbaz (2001) spectral energy distributions. We estimate that the luminosities and implied star formation rates are uncertain by at least 50% (for a given IMF). Stellar masses were estimated from optical/UV SED fitting in the same way as at z~1 and 2, with similar uncertainties. Half-light radii are averages of the values obtained from CO and ACS.

<z>=1-3.5 SMG sample
Finally we also included 20 z=1-3.5 SMGs from the PdBI SMG survey presented in Greve et al. (2005), Tacconi et al. (2006Tacconi et al. ( , 2008, Engel et al. (2010) and Smail et al. (2010, in prep.). For 10 of these SMGs we have spatially resolved data. All but two of these SMGs are drawn from the luminous tail of the SMG population with S 850μm >4 mJy, corresponding to L FIR >4.4x10 12 L  and SFR>440 M  yr -1 when applying the Pope et al. (2006) relation between 850μm flux density and FIR luminosity, and the K98b relation between far-infrared luminosity and star formation rate (corrected to the Chabrier IMF). Direct far-infrared observations of SMGs in GOODS-N with Herschel/PACS (see footnote 2; Magnelli et al. 2010) confirm the Pope calibration. The stellar masses of the four SMGs with good quality SED fitting are >10 11 M  (Tacconi et al. 2008). The kinematics and structure of the SMGs are quite different from the z~1-2.5 SFGs. Only a few of the ~10 SMGs with high resolution PdBI CO imaging exhibit evidence for ordered motions and several consist of binary systems in close interaction. Their size distribution and 'messy' kinematic properties are best explained by a scenario where most of these luminous SMGs are powerful starbursts triggered by major gas rich binary mergers in different phases of the merging process, similar to z~0 ULIRGs (Engel et al. 2010). Table 1 summarises all relevant observed and derived properties of the spatially resolved high-z SFGs and SMGs. To compute surface densities we divided half of the star formation rate or gas mass by the area subtended by the half-light (or effective) radius (πR 2 ).

<z>~0 samples
We compare these high-redshift star forming galaxy samples to several z~0 star forming galaxy samples from the literature. We have taken normal disk galaxies from the original K98a paper and from Leroy et al. (2008Leroy et al. ( , 2009; henceforth 'z~0 normal') and Kuno et al. (2007).. We compiled a list of non-merging starburst galaxies from K98a, Gao & Solomon (2004), Kuno et al. (2007) and Gracia-Carpio (2009; henceforth z~0 'SFGs'). A sample of z~0 ULIRGs and interacting galaxies was constructed from the samples of K98a, Gracia-Carpio et al. (2008) and Gracia-Carpio (2009). For galaxies with multiple observations we either selected the measurement with the highest signal-tonoise ratio, or we selected observations sampling different spatial scales. We did not use observations of galaxy nuclei only, in an attempt to screen against the impact of AGN. We also only included galaxies with molecular surface densities above 3 M  pc -2 , where the molecular to atomic gas fractions become comparable or greater than 1 ). Our final z~0 data base has ~150 entries. All data were transformed to the same cosmology and conversion factors as given in the Introduction.

Conversion factors from CO luminosity to molecular gas mass
Observations in Milky Way GMCs have established that the integrated line flux of 12 CO millimetre rotational lines can be used to infer cold (molecular) gas masses, despite the fact that the CO molecule only makes up a small fraction of the entire gas mass and that the lower rotational lines (1-0, 2-1, 3-2) are almost always very optically thick (Dickman, Snell & Schloerb 1986, Solomon et al. 1987. This is because the CO emission in the Milky Way and nearby normal galaxies on average comes from moderately dense (volume averaged densities <n(H 2 )> ~200 cm -3 ), self-gravitating GMCs. In this regime the ratio of molecular hydrogen column density to line integrated CO intensity, or of molecular gas mass (including a 36% mass correction for helium) to CO luminosity L ' CO (L ' CO = source  line T R (v) dv dA [K km/s pc 2 ]) can be expressed as Here T R is the equivalent Rayleigh-Jeans brightness temperature of the (optically thick) CO line and c 1 and c 2 are appropriate numerical constants. In the 2.6mm CO (1-0) transition the typical gas temperature of GMCs ranges from 10 to 25 K. Several independent empirical techniques based on GeV -rays, optical extinction measurements, isotopomeric line ratios and excitation analysis have all shown that this 'virial' technique is appropriate and remarkably robust throughout the Milky Way (Dickman et al. 1986, Solomon et al. 1987, Strong & Mattox 1996, Dame, Hartmann & Thaddeus 2001. The best empirical Galactic conversion factor is X G~2 x10 20 [cm -2 (K km s -1 ) -1 ] and α G =3.2 [M  (K km/s pc 2 ) -1 ]. The virial approach can be shown to also apply to an ensemble of virialized clouds, instead of a single one, as long as again the factor n(H 2 ) 1/2 /T is approximately constant throughout the system and the CO line is optically thick (Dickman et al. 1986).
For the z~1-2 SFGs a Galactic conversion factor (α~3.2 M  /(K km s -1 pc 2 )) is appropriate since the CO emission in these systems, as in z~0 disk galaxies, probably arises in virialized giant molecular cloud systems (GMCs) and mean gas densities of <n(H 2 )>~10 2…3 cm -3 ). Based on far-infrared SEDs from PACS/SPIRE Herschel (see footnote 2; Elbaz et al. 2010) and an observation of several CO rotational lines in one galaxy (Dannerbauer et al. 2009) these SFGs may have gas/dust temperatures of 20-35 K. For the large column densities (and interstellar pressures) in the z>1 SFGs ( gas >>10 M  pc -2 ) most of the cold interstellar gas is probably in molecular form and the contribution of atomic hydrogen can be neglected (Blitz & Rosolowsky 2006).
The assumption of an ensemble of individual gas clouds in virial equilibrium with their own gravity breaks down, however, in galactic nuclei and mergers. In these cases gas velocities are dominated by disturbed large scale motions in the gravitational potential of gas, stars and dark matter, and the gas is often in a smoother, disk-like configuration, rather than in virialized individual clouds. In this limit the relationships above still hold but with a modified proportionality factor , Scoville, Yun & Bryant 1997, Downes & Solomon 1998, Sakamoto et al. 1999). For conditions appropriate for z~0 ULIRGs (n(H 2 )~10 3 -10 4 cm -3 and T R~2 0-60 K) with gas fractions of ~20%, the inferred empirical conversion factor ranges from X merger =5x10 19 (or α merger~0 . 8: Solomon et al. 1997, Downes & Solomon 1998 to 10 20 (α merger~1 . 6, Scoville, Yun & Bryant 1997). A Galactic conversion factor can be excluded for these systems as in that case the gas fractions would significantly exceed unity. Furthermore, the assumption of a Galactic conversion factor for these luminous mergers would imply that the two progenitor galaxies also had anomalously high molecular gas masses relative to normal spiral galaxies of the same stellar mass. An analysis of gas, stellar and dynamical masses suggests that the local ULIRG/merger conversion factor is also appropriate for very luminous z>1 SMGs (Tacconi et al. 2008). If a Galactic conversion is applied the ratio of gas masses to dynamical masses becomes much greater than unity.
In the z1 SFGs and SMGs rotationally excited (J>1) transitions are observed and corrections for the ratio of the intrinsic Rayleigh-Jeans brightness temperatures in the 1-0 line to that in the rotationally excited line may be necessary. Everything else being equal and all levels being thermalized, the difference between Planck (T Planck ) and Rayleigh-  ) leads to corrections between 1.15 (1.23, 1.33) and 1.33 (1.55, 1.82) for temperatures between 40 and 20 K for the J=2-1 (J=3-2, J=4-3) transitions. These are lower limits since the upper states may not be thermalized. CO line ratios in z~0 disk galaxies (Mauersberger et al. 1999) and in the z~1.5 SFG BzK21000 (Dannerbauer et al. 2009) suggest the following correction factors for SFGs: R 1J =L' CO 1-0 /L' CO J-(J-1)~1 .2 and 2 for J=2 and 3, respectively. For the SMGs in Table 1 we use R 1J =1.1, 1.3 and 1.6 for J=2,3 and 4 as motivated by observed line ratios in SMGs (Weiss et al. 2007) and z~0 ULIRGs (Iono et al. 2009).

Luminosity-Luminosity Correlation
The first and most straightforward approach, independent of the choice(s) of conversion factor(s) from CO line luminosity to M mol-gas , is to investigate the relation between the observed (z=0) or inferred (z1) far-infrared luminosities and the observed (z=0) or inferred (z1) CO 1-0 luminosities. The results are shown in Figure 2. Compared to other plots discussed below, the total (statistical plus systematic) uncertainties are smallest in this plot. We estimate typical 1σ uncertainties of ±0.13 dex (30%) and ±0.17 dex (40%) for L CO and L FIR , respectively. In the case of the z1 SFGs and SMGs direct observations of the far-infrared luminosities with the Herschel satellite confirm the applied calibrations to get L FIR from the original star formation tracers (UV, Hα, midinfrared, submillimeter, radio: see footnote 2; Magnelli et al. 2010, Elbaz et al. 2010, or give corrections. Nordon et al. (in press.) find that at z~1.5-2.5 24μm based star formation rates typically overestimate the directly measured far-infrared luminosities by a factor of four.
The data and plots shown in Figure 2 are in excellent agreement with the results of all past studies on the subject (e.g. Sanders et al. 1991, Sanders & Mirabel 1996, Gao & Solomon 2004, Greve et al. 2005, Gracia-Carpio et al. 2008. They confirm that there is a close to linear (slope 1.15±0.12) relation between L CO and L FIR for SFGs (a constant gas depletion time scale) and that the luminous mergers lie on average a factor of about 4 above the relation for the 'normal' star forming galaxies. All uncertainties for slopes and offsets given henceforth and in the Figures are 3σ formal fit errors, which is a reasonable estimate of the total error, including systematic effects, calibration etc.
Our analysis adds two important new aspects. The first is that the gas depletion time for SFGs estimated from the right panel of Figure 2 depends only weakly on redshift. The redshift averaged ratio 1 0 FIR CO L L  is 27±5.6 L  /(K km/s pc 2 ) for SFGs. The correlation has a fairly substantial dispersion of 0.31 dex (see K98a). There are also several outliers. The CO luminosities in EGS12012083 and BX389 are surprisingly weak despite an estimated luminosity of >10 12 L  . To get from far-infrared to total infrared luminosity we use L IR (8-1000μm)/ L FIR (50-300μm)~1.3 (Gracia-Carpio et al. 2008). Taking the K98b conversion for SFR/L IR and α=α G =3.2 for SFGs (section 2.6) this results in an effective gas consumption time scale for SFGs of 1.2 (±0.3) Gyr averaged from z=0 to 2. If the sample is split by redshift, then for the z=0 SFGs the gas depletion time is 1.5 Gyrs and for the z1 SFGs it is 0.5 Gyrs. For comparison Leroy et al. (2008) find a value of 1.9±0.9 Gyrs for their z=0 disk sample. The difference between our z~0 time scale and that of Leroy et al. is the correction factor L IR /L FIR mentioned above. The gas depletion time scale for the z1 SFGs is comparable to their 'stellar ages', as estimated either from the ratio of stellar mass and star formation rate, or from synthesis modelling of their UVto near-infrared restframe SEDs (Förster Schreiber et al. 2009).
The second important point relates to the interpretation of the offset between SFGs and luminous mergers. In past studies there were no SFGs above ~10 10 K km/s pc 2 , such that the difference between mergers (z~0 ULIRGs and z1 SMGs) and normal z=0 disk galaxies was interpreted by most observers as a luminosity effect, with a change of slope occurring above ~10 10 K km/s pc 2 (or 10 11.5 L  ), in the merger regime (Bouché et al. 2007). With the addition of the z1 SFGs this interpretation may not be tenable anymore.
Instead it now appears that the locations of mergers and SFGs on average are well separated at the same CO and IR luminosities. Normal (disk) galaxies with star formation occurring over long time scales and with a high duty cycle constitute one extreme. Luminous major mergers, predominantly after the first encounter and later stages of the merging process (Veilleux, Kim & Sanders 2002, Engel et al. 2010) and in an extreme (maximum starbursts: Tacconi et al. 2006) and brief starburst phase (duty cycle ~10%: Tacconi et al. 2008) constitute the other. Luminous mergers have an average gas depletion time of 0.2±0.11 Gyrs, 2.5 to 7.5 shorter than the SFGs (right panel of Figure  2). Note that the somewhat lower luminosity interacting z~0 starbursts in our sample (filled cyan squares in Figure 2), which are not in a late stage of merging, are located closer to the isolated SFGs, and not near the extreme mergers.
SFGs and luminous mergers each adhere to a near-linear relation but the respective relations are offset by 0.6 dex. Such an offset has also been seen by other authors when comparing local ULIRGs (Gracia-Carpio et al. 2008) and z~2 SMGs (Bothwell et al. 2010) with local SFGs. For a fixed slope the probable uncertainty in this offset of the mean is ±0.15 and the offset is significant. However, the scatter of both merger and SFG distributions is large (~0.31 dex) at both low-and high-z and there is significant overlap between the two populations, as well as outliers. We come back to the interpretation of this offset in section 6.

SFGs
We now consider the distribution of the SFGs in the classical 'Kennicutt-Schmidt' surface density plane. Our best estimate for the typical 1σ total uncertainties (statistical plus systematic) are ±0.23 dex (52%) and ±0.27 dex (62%) for mol gas star form and   , respectively. Figure 3 summarises the distribution for the low-and high-z SFGs. As in the case of the pure luminosity plot in Figure 2 the correlation of all data has a standard deviation relative to the best fit of 0.32 dex. Again this scatter is somewhat larger than the typical total measurement uncertainties shown as a cross in the lower right of the left panel of Figure 3. The slope of the relation is 1.17 (±0.09), where the errors for slope-fits quoted here and below are formal 3σ fit errors. Note that in our fitting in Figure 3 (and also Figure 2) we did not attempt to assign individual errors (unlike K98a), since in our opinion essentially all uncertainties are systematic in nature and apply to all data equally. This slope is in very good agreement with the spatially resolved relation for nearby spirals in Bigiel et al. (2008, green/orange/red-shaded region in the left panel of Figure  3). The new data do not indicate a significant steepening of the slope at surface densities of >10 2 M  pc -2 , neither at z~0 nor at z1. Within the limited statistics of the currently available data, we do not find a break in the slope near 10 2 M  pc -2 , as proposed by Krumholz et al. (2009). The slope of 1.33 found by Krumholz et al. (2009) in the high density limit is marginally larger. A steeper slope in this regime (1.28 to 1.4) was suggested earlier by the K98a starburst sample, but that analysis included some mergers (see below) and the combined scatter of both data sets suggest a 1σ uncertainty of ~0.15, which makes the difference in slope of 0.1-0.23 only marginally significant.
Low-and high-z SFGs overlap completely, again with the obvious exception of EGS12012083 and BX389. The data in Figure 3 suggest that the KS-relation in normal star forming galaxies does not vary with redshift, in agreement with the conclusions of Bouché et al. (2007) and Daddi et al. (2010).
In the right panel of Figure 3 we analyse the data with the 'Elmegreen-Silk'-relation (see also K98a), which relates star formation rate surface density to the ratio of gas surface density and global galaxy dynamical timescale. There is a reasonably good correlation as well with a slope of slightly less than unity (0.84±0.09). The scatter in this relation (0.44 dex) is larger than in the surface density relation, which may in part be attributable to the larger total uncertainties in mol gas / dyn   , which we estimate to be ±0.32 dex (74%). Here and elsewhere we computed the dynamical time scale from the ratio of the radius to the circular velocity v c . For the z>1 SFGs and SMGs we took R=R 1/2 and applied a pressure correction to the inclination corrected rotation velocity v rot , v c =(v rot 2 + 2σ 2 ) 1/2 , where σ is the local 1d-velocity dispersion in the galaxy. This relation is applicable to rotation-dominated, as well as pressure dominated galaxies. The slope we find is close to that of K98a, who find a slope between 0.9 and 1. High-z SFGs have somewhat higher Σ star formation than low-z galaxies (by 0.71±0.21 dex) but the difference is probably only marginally significant. A fit with unity slope yields a star formation efficiency per dynamical time of 0.019 (±0.008). This is in excellent agreement with 0.01, the value found by K98a when corrected to a Chabrier IMF.

Figure 4 summarises our analysis of the luminous mergers at both low-and high
The left panel shows the case of applying the best single common conversion factor determined from the observations (α merger~1 , section 2.6), such that mergers and SFGs now have conversion factors that differ by a factor of 3.2. The slope of the merger relation (1.1±0.2) is consistent with that of the SFGs (1.17). Again low-and high-z mergers lie pl -z. ausibly on the same relation. Independent of whether the merger slope is fit or forced to be the sam t tor as =3.2), the mergers move by ~0.5 dex to higher gas mass surface densities (right panel of Figure 4), closer to but not fully overlapping with per slope of 1.27 e as that of the SFGs, the difference in star formation rate at a given gas surface density between the two branches is ~1.0 (±0.2) dex (see also Bothwell et al. 2010).
As we have argued in section 2.6, a Galactic conversion factor for all luminous lowand high-z mergers is almost certainly excluded (Downes & Solomon 1998, Sakamoto e al. 1999, Tacconi et al. 2008). If one, nevertheless, adopts the same conversion fac for 'normal' galaxies (α merger =α SFG the relation for SFGs. A fit to the entire data set then yields a stee (±0.075) for equal weighting.

Elmegreen-Silk relation for mergers and SFGs
In Figure 5 we added the low-and high-z luminous mergers in the ). However, the offset is less than in the gas surface density-star formation surface density plane. Given the larger intrinsic scatter, all data may thus be drawn from a single underlying distribution, albeit with a larger scatter than in the relation for either SFGs or mergers by themselves. The scatter may be driven by uncertainties, especially in the estimation of dynamical time, or by (a) additional (hidden) parameter(s). The best fit for all data combined has a slope consistent with unity (0.98, ±0.09), similar to K98a, and a zero point offset of -1.76 (±0.18). This offset corresponds to a star formation efficiency of 1.7% per dynamical time scale. The physical interpretation of this common underlyin relation then would be that the star formation relation is in part driven by the large scale dynamical time in a system. Spiral density waves and/or galaxy interactions drive density waves and bars, which in turn trigger cloud formation, enhanced cloud-cloud collisions and gas compression and global radial streaming (see Leroy et al. 2008). Mergers are more luminous because of a combination of their smaller sizes and dynamical time scale (Table 1, Downes & Solomon 1998) and an ad g s ditional larger efficiency of star formation per dynamical time scale. This is indeed the interpretation proposed by Bouché et al. (2007). Similar conclusions follow when considering galaxy integrated quantities (SFR as a function of M mol-gas / dyn , see section 6.3).

Comparison to K98a and Bouché et al. (2007)
We finish here with a brief discussion of the differences in slopes found in different studies. As discussed several times above, the slope determinations in the KS-surface density relation range between N~1.0 , section 4.1) and N=1.7 (Bouché et al. 2007), with K98a e ace dens and Kennicutt et al. (2007) in the middle (N=1.3 to 1.4). How can one understand the fairly large differences in slope, given that each of these papers quotes (1σ) uncertainties of ±0.15 or less?
A first important factor is the definition of gas surface densities. Throughout this paper we have discussed the 'molecular' gas-star formation relations and have restricted ourselves to 2 mol gas 3 M pc     where molecular gas should begin to dominate the ISM (Blitz & Rosolowsky 2006). The value of N=1.0 for Bigiel et al. (2008) is also for th molecular relation at ities. K98a and Bouché et al. (2007) consider the total cold gas column densities, including a contribution from HI. This contribution is plausibly negligible for z~0 starbursts, ULIRGs and z SFGs and SMGs but not for normal galaxies with he slope of N=1.17 der by a number of new normal and starburst galaxies we have included from the recent literature.These galaxies were not in the K98a sample and tend to increase the num r tota systematic uncertainty of slope determinations probably is ±0.2 to ±0.25. When ing all other assumptions the same as in K98a, we find N=1.33 instead of N=1.4. If the HI-columns are included for the normal galaxies in our SFG sample, the slope in Figure 3 would change from 1.17 to 1.28.
The second important factor is the choice of conversion factor. A actor for all galaxies, as assumed in K98a, yields a flatter distribution than s the bimodal value we favour. The molecular KS-relation for the K98a data reanalyzed in this way yields N=1.42 (compared to N=1.33 above).
Finally, the choice and weighting of data also significantly affects the resulting slopes. By adding a number of z1 SMGs at high surface densities, combined with the fairly extreme choice of a bimodal conversion factor (α merger =0.25α SFG ) and an inclusion of the HI contribution at the low surface density end, Bouché et al. (2007) arrived at the most extreme result in the literature of N=1.7 (see also Bothwell et al. 2010). K98 Kennicutt et al. (2007) show that by different weightings of data and by using data wit different spatial resolutions, the slope shifts by about ±0.1. Bigiel et al. (2008) average the spatially resolved molecular KS-relation for 18 z~0 normal galaxies and find N=1.0±0.2, while Kennicutt et al. (2007) find N=1.37±0.03 for the spatially resolved relation in M51 only. T ived for the SFGs in Figure 3 is in part driven ber of galaxies at 2.8...3.8 -2 mol gas~1 0 M pc   and push the fit to a somewhat shallower slope.
We conclude that the slope of the KS-surface density relation is sensitive to a numbe of subtle systematic effects, each of which can change the derived fit-value by ±0.1. The l com aring different results in the literature care needs to be taken to understand which assumptions were made to be able to assess the reality of the differences in quoted values.

Summary of the observations
or surface rmal star forming galaxies are correlated with a slope of 1.1 to ly mation ig has been low (1.7% per dynamical time), or the gas depletion time scale roduce 4 al SFGs. Apart from tions p In summary of the sections 3 and 4, we find  molecular gas and star formation rates (in terms of total quantities densities) in no 1.2 across the entire range observed (10 0.5 to 10 4 M  pc -2 ). The scatter is fair large (0.32 dex) but arguably significantly affected by systematic uncertainties;  low-and high-redshift samples follow similar relations. This means that in normal star forming galaxy populations with a semi-continuous star for history the star formation efficiency in SFGs ever since ~ 3 Gyrs after the B Bang long, ~60 times the dynamical time. The molecular gas depletion time throughout this period has increased slowly from 0.5 at z~2 to 1.5 Gyrs at z~0;  luminous starburst-mergers at both low and high redshift appear to p to 10 times more far-infrared luminosity per gas mass than norm a zero-point offset, the rela for merger driven systems are characterized by essentially the same slope as the "normal" SFGs;  the difference between mergers and SFGs is minimized in the /

AGN Contamination
For all galaxies in Figure 2 the total (infrared) luminosity is a measure of the star formation rate, unless there is a substantial contamination by an AGN. Most of the SFG are probably not strongly affected. This is by selection, since with one exception (NGC 1068) the z~0 SFG sample excludes bright AGN. In NGC1068, which is used here, the AGN contributes about half of the infrared luminosity (Telesco et al. 1984). In seve well-known z~0 ULIRGs, such as Mrk231 and Mrk273, the far-infrared emission of central AGN likely dominates the luminosity, in others it probably contributes a frac (50%) of the luminosity (Sander & Mirabel 1996, Genzel et al. 1998, Veilleux et al.2009). On average, the intrinsic z~0 ULIRG star formation luminosities could be overestimated in Figures 2 and 4 by a factor of 1.5 to 2. For most of the z1 SMGs, AGNs are present but the AGN contribution to L FIR is probably relatively small ( a few tens of percent: Lutz et al. 2005, Alexander et al. 2005, Valiante et al. 2007, Menendez-Delmestre et al. 2007, Pope et al. 2008. To minimize the possible contamination by AGNs we have taken the far-infrared luminosity and multiplied by 1.3 to estimate th total (8-1000μm) infrared luminosity associated with star formation, L IR . This approac avoids using the mid-infrared luminosity that is more prone to AGN contamination (Netzer et al. 2007). This correction facto The he high-z SFGs appears to be lower than comparably luminous low-z systems (Daddi et al. 200 ope 1.5, 2t al. 2010, Nordon et al. 2010). For 3 of the 6 z~1.5 BzK galaxies studied by Daddi et al. (2010), the PACS-FIR luminosities are comparable to those luminosities by a modest factor (1-1.5).

From star formation tracers to SFRs
Different observers have used different star formation tracers, with each method having its own strength and weakness (see e.g. K98b, Kennicutt et al. 2009). The Hα recombination line luminosity is an excellent measure of the instantaneous formation rat of the most massive (O) stars ( Figure 5), but requires a good knowledge of the extinct correction. In practice, the Hα indicator is very well suited for low extinction star forming galaxies, including most normal z~0 disk galaxies and probably all but the dustiest z1 SFGs. It tends to fail in the most extreme, dusty SFGs and z~0 ULIRGs/ z1 SMGs because of the very large extinctions (e.g. Goldader et al. 1995, Genzel et al. 1998). far-UV luminosity and the far-infrared luminosity (a proxy of bolometric luminosity in dusty SFGs) measure the integral of star formation over a fairly long period of time and thus require a good knowledge of the star formation history ( Figure 5). The far-UV luminosity is even more sensitive to extinction than Hα, and for this reason it fails in t case of very dusty star forming galaxies (Goldader et al. 2002). Since the extinction of 7, Reddy & Steidel 2009), the applicability of the UV-estimator (like that of Hα) appears well justified for most z1 SFGs.
With the advent of very sensitive measurements from the Spitzer Space Telesc for studying high-z SFGs, it has become increasingly popular to use the observed 24μm luminosity (corresponding to 6-12μm in the rest frame), in conjunction with a library of z=0 SFG SEDs (e.g. Chary & Elbaz 2001), for an extrapolation to the far-infrared luminosity. Direct observations of FIR emission of z1 SFGs and SMGs with PACS on the Herschel Observatory indicate that the 24μm-extrapolation method is robust at z but overestimates FIR luminosities in z>1.5, L >10 12 L  SFGs and SMGs by factors of 4 (footnote 2; Elbaz e  Figure 4 (L IR =10 SFR) applies only for ages >>10 years. Hα as an estimator of instantaneous star formation does better but is bviously strongly dependent on the exact shape of the IMF in the O-star mass range (>20 M ).

ple ic sion
Galactic conversion appears to apply in the dense star forming clumps themselves (Bolatto et al. 2008). Our data ived from the 24μm data, and on average no corrections to the SFRs are necessary (Nordon et al. 2010) Several authors are using combinations of the various tracers to overcome their respective disadvantages , 2009. The luminosities in our <z>~1.2 sample are based on such combinations, and thus should be fairly secure. For the <z>=2.3 SFGs we use the observed Hα luminosities from Erb et al. (2006) and Förster Schreiber et al. (2009) and apply the Calzetti et al. (2000) prescription, with the proposed extra nebular attenuation relative to stellar light by 2.3, to estimate the Hα-extinction from the E(B-V) values in Erb et al. (2006). A comparison of these Hα-luminosities/star formation rates, extinctio corrected far-UV luminosities/star formation rates and direct PACS-Herschel far-infrared luminosities/star formation rates yields encouraging agreement (Förster Schreibe 2009, Nordon et al. 2010). This agreement instils some trust in the appl for the ratio of nebular to continuum extinction, for high redshift SFGs (see discussion in Förster Schreiber et al. 2009, but see Reddy et al. 2010).
Once one has obtained a best estimate intrinsic far-infrared/UV-/Hα-luminosity and has ascertained that it is powered by star formation, the next step is to go from lumin to star formation rate. This step involves a star formation history and an IMF. Figure 56 shows the dependence of the various tracers on star formation history. It is clear that the various standard conversions from these tracer luminosities to SFR require that star formation does not vary rapidly and preferably is constant (see K98b for a discussio Hα, far-IR and far-UV luminosities all tend to underestimate the peak SFR for a sta past its maximum, in some cases by large factors, unless the star formation history explicitly included in the modelling. Even for a constant star formation history, the infrared to S 10 7 o 

From L CO 1-0 to M mol-gas
We have discussed in section 2.6 the basic assumptions and reasoning for being able to convert integrated CO 1-0 line luminosities to total molecular gas masses with a sim proportionality factor. This conversion is fundamentally fraught with several systemat uncertainties. Yet studies with a range of methods have come up with similar conver factors in the Milky Way and nearby near-solar metallicity disk galaxies, to within a factor of 2 (see the Appendix in Tacconi et al. 2008 for a more detailed discussion). Because of photo-dissociation of the molecular gas, the situation in low metallicity (<50% solar) dwarf systems is more uncertain, although even there a base does not include these lower metallicity systems, and most of the more massive z1 SFGs probably have near-solar metallicities (Erb et al. 2006b).
We have emphasized in section 2.6 the important assumption that the molecular line emission originate in self-gravitating clouds where gas dominates the mass balance. This is because the basic mass information is contained in the observed line width (virial estimate) and not in the intensity (equation 6, Dickman e et a This assumption is reasonably well justified in isolated disk galaxies close to thermal an dynamical equilibrium. It probably also holds in many star bursting systems, as long as the integrity of the GMCs is maintained, and the factor t al. 1986, Solomon l. 1987). d 2 ( ) / gas n H T     is approximately the same as in the Milky Way (0. 5-1.5). It arguably is also applicable to the z~1-2 SFGs since in the few cases studied so far, their properties are close enough to the Milky Way (presence of giant clouds, gas densities and temperatures) that a similar con l.
mpletely ral, z~0 ULIRGs and z1 SMGs are highly disturbed systems with large velocity dispersions, and the application of dyn ctor e of the overall gas-star formation rate relation at >10 M  pc , with a slope ~1.4, consistent with the interpretation of Bouché et al. (2007). This does not change the basic conclusion, however, that luminous mergers on average have a larger ed version factor is plausible , Dannerbauer et a 2009).
The situation is quite different in the merging galaxies. While there still may be a reasonably well defined conversion factor 3 to 4 times smaller than the Galactic value, its value is probably uncertain by a factor of at least 2. In some instances, such as the gaseous bridge between the two nuclei of NGC 6240, the gas motions may be co out of equilibrium and the concept of a virial mass estimator based on line width may not be applicable (c.f. Tacconi et al. 1999). In gene amical tracers, such as virial estimators or rotation curves, may yield uncertain (fa 2) or even questionable results (Mihos 1999).
We have chosen the simplest approach of assigning a single constant conversion factor each for mergers and SFGs. The reality undoubtedly is more complex. The conversion factor may be a function of physical parameters, such as surface density of gas and star formation, metallicity etc. (Obreschkow & Rawlings 2009). There may b some support for such a smooth conversion function as a function of gas surface in Figure 10 of Tacconi et al. (2008) but the trend is not well enough defined to apply it here. The effect of introducing such a function would be for luminous (high surface density) SFGs to have lower conversion factors, more akin to mergers. The gap between mergers and SFGs in Figures 2 and 4 would then be less pronounced and the overall distribution would be more continuous, and not as bimodal as Figure 4 suggests. There would then be a steepening 3 -2 L FIR /M gas ratio than SFGs.

Spatially resolved vs. integrated measurements
If star formation in a given galaxy is a strong function of position because of the presence of a radial gradient in the Toomre Q-parameter, spiral arms, bar-induced resonances, nuclear concentrations and large star forming complexes in the disk, galaxy integrated studies may yield a different gas-star formation relation than spatially resolv studies. Because galaxy integrated measurements smear out such structures one would expect galaxy integrated relations to be shallower than spatially resolved data. This is in fact observed in M51, where Kennicutt et al. (2007) find N=1.56±0.04 in 520 pc data, N=1.37±0.03 in 1850 pc resolution data, and N=1.4±0.15 for the global average data (when compared to other galaxies). Undoubtedly such effects will be also present in the 1 SFGs, which have a very clumpy and irregular star formation and gas distributions . Future spatially resolved studies of the gas-star formation relations in the z1 SFGs will shed light on this issue.

cosmic evolution of the gas-star formation relation in SFGs
n gas dep e n s are el et rnal fference in tion, rather than or in addition to small-scale processes in star forming clumps (e.g. Struck et al. 2005). Alternatively the star formation relation could, in p z

Possible Interpretations
The near-unity slope of the gas-star formation relations and the slow variatio of the letion time scale as a function of cosmic time in normal SFGs are at first glanc surprising. Bigiel et al. (2008) have interpreted the near-linear relation between    and mol gas   in low-z disks in terms of the galaxy-scale filling factor of the basic building blocks, namely star forming GMCs. Since GMC properties in normal z~0 galaxies are probably quite comparable (Bolatto et al. 2008), a similar linear relation in different galaxies is plausible. The interstellar medium in z1 SFGs is quite different from that i the local Universe, however. The turbulent velocities and Mach numbers in z1 SFG far greater than at z~0 (σ gas~2 0-80 km/s instead of 5-10 km/s, Förster Schreiber et al. 2009, Cresci et al. 2009. Gas column densities and pressures are also much larger (Elmegreen 2009). As a result the Toomre-mass, that is the largest gravitational unstable mode of gas structure formation, is about 10 2 greater, and feedback probably less destructive (e.g. Elmegreen 2009, Dekel, Sari & Ceverino 2009, Genz al. 2008). If the global star formation relation were strongly sensitive to these inte properties of the star forming complexes, one might have expected to see a di slope or an offset for the high-z SFGs. The fact that the gas-star formation relation apparently is not strongly altered by these rather different internal properties thus suggests to us that the global gas depletion time scale (or efficiency per free fall/dynamical time scale) is set or strongly influenced at large scales, as assumed in the Elmegreen-Silk rela rinciple, be sensitive to the internal properties but their effects may be compensated by self-regulation.
Another important feature is the slow change in gas depletion time from 0.5 to 1.5 Gyrs over 10 Gyrs, which is smaller than the Hubble time for all redshifts observed here. At the same time the SFGs sampled by the current observations have large star format rates (e.g. Noeske et al. 2007, Franx et al. 2008) and high gas fraction (30-60%, Tacconi et al. 2010) throughout the z~1-2.5 redshift range. Taken together these facts require that the gas reservoir is replenished semi-continuously over cosmic time , Bouché et al. 2010, Bauermeister et al. 2010. Bauermeister et al. (2010) have shown that the observed change in depletion time scale with cosmic time can be understood as the consequence of the variations in gas ac and star formation. This conclusion is consistent with other observational (e.g. Erb 2008, Förster Schreiber et al. 2006, 2009) and theoretical findings (Dekel & Birnboim 2006, Kereŝ et al. 2005, Ocvirk, Pichon & Teyssier 2008), which all suggest a semi-continuous supply o ion s cretion f fresh gas in 'cold flows' from the inte alactic cosmic web, below a critical halo mass of ~10 12 M  . Since the gas depletion tim cosmic epoch, star formation rates and gas fractions as a function of red-shift are likely set by the balance between accretion from the halo and feedback (Bouché et al. 2010).
as ons have made a compelling case that most z1 SMGs and the majority of z~0 ULIRGs are powered predominantly (but by no means exclusively) by star AGN er tially Robinson 2003). The results emerging from PACS on Herschel probably exclude that possibility. Magnelli et al. ( . rg e scale in SFGs is not a strong function of

what accounts for the excess in L FIR /L CO in mergers?
What is the interpretation of the excess in L FIR /L CO and mol gas       in z~0 ULIRGs and z1 SMGs? Sanders et al. (1991) proposed that AGNs strongly dominate the energy output of ULIRG mergers. In the meantime ISO and Spitzer spectroscopy, as well Chandra X-ray observati formation (Genzel et al. 1998, Alexander et al. 2005, Veilleux et al. 2009). The contribution may on average reduce the excess of 4-10 by a factor of 2 or less but unlikely will remove it.
As long as only submillimetre data were available for z1 SMGs, one possible explanation for the strong emission of SMGs was to postulate that they are much cold than expected, thereby lowering the required far-infrared luminosities substan (Kaviani, Haehnelt & Kauffmann 2003, Efstathiou & Rowan-2010 find that the SEDs of luminous z1 SMGs are very similar to the standard interpretations , Pope et al. 2006, with 30 to 40 K dust temperatures. z1 SMGs are indeed very luminous systems.
The uncertainties in going from star formation tracers to star formation rates (given an IMF) are substantial but have now been mitigated by combining different star formation tracers and having available direct far-infrared luminosities from Herschel for the luminous high-z populations (SFGs as well as SMGs, see footnote 2; Magnelli et al 2010, Nordon et al. 2010, Elbaz et al. 2010. A definite concern is the step from L FIR to SFR, which requires the assumption of a star formation history. Most SF 7 R calibrations assume steady star formation histories over >>10 years, which are a poor description for the in the ergers. n ion e) is in all likelihood excluded in most mergers because of mass balance. We thus conclude that improved estimates of the conversion factor, perhaps in form of a con move ntly ound under physical conditions similar to major mergers (Bastian, Covey & Meyer, 2010). Studies of super-massive star clusters in star burst galaxies and nea d ever, major mergers are also more turbulent than SFGs (Mach numbers in ULIRGs are ~200, compared to ~25 in Milky Way GMCs). Turbulent dissipation time scales thus are correspondingly smaller as well. The star formation efficiency per free & Mc ale hs bursting merger systems. However, we have shown that in this case the correction would go in the direction of increasing the intrinsic excess in the ratio of SFR/M gas merging systems relative to the SFG populations.
The conversion factor from CO luminosity to gas mass is quite uncertain in m However, the excess in the merger data already is strongly present in the L FIR /L CO relation, which does not make any assumptions about a conversion factor. The discussio in section 4.2 has shown that any reasonable application of an improved convers factor applicable for mergers makes the excess stronger, not weaker. A Galactic conversion factor (which would remove the discrepancy at least in the surface density plan version function dependent on a number of physical parameters, may perhaps lessen and smear out the difference between mergers and SFGs, but will not completely re it.
This leaves in our opinion two options. If the IMF in mergers would be significa more top-heavy than in normal disks, the L FIR /M mol-gas ratio would increase proportionally. This option has already been proposed for other reasons by Baugh et al. (2005), Davé (2008) andvan Dokkum (2008). The trouble with that explanation is that there is no convincing local Universe evidence (yet ?), other than in the immediate vicinity of the black hole in the Galactic Centre (Bartko et al. 2010), that significantly top-heavy IMFs are f rby mergers are perhaps the most promising hunting ground, but the observational support for a top-heavy in these cases is marginal at best (de Grijs & Parmentier 2007, Bastian et al. 2010. The most likely explanation thus is that luminous mergers are forming stars faster (shorter depletion time scale) and/or more efficiently than normal disks at any cosmic epoch. Plausible qualitative explanations might appeal to large scale shocks for compressing the gas and increasing the fraction of dense gas that can then collapse an form stars. How fall/dynamical time may even decrease with increasing Mach number (Krumholz Kee 2005).

the global dynamical time scale is an important parameter
The discussions in section 4.3 and in 6.1 point to the large scale, dynamical time sc as a critical additional element in the global star formation relations at low-and hig redshifts and in different mass, gas fraction and dynamical time regimes. Figure 7 show what happens if the dynamical time scale is added as an explicit parameter in the 3dspace spanned by  dyn -SFR-M mol-gas . In the M mol-gas -SFR (similar to Figure 2) and  dyn -SFR projections the four different galaxy types (z~0 and z1 SFGs, and luminous mergers) clearly separate, as discussed throughout this paper. However, there exists a rojection of the three-dimensional distribution, in which mergers and SFGs at both lowpe s 0 to K98a and the theoretical work of Krumholz & McKee (2005) and Krumholz et al. (2009). The slope in the  dyn -SFR projection is only mar is real an s, us physical differences in their star forming clouds. If the global star formation efficiency per free fall/dynamical time scale is driven by large scale dynamical effects, such as spiral arms, global disk fr ale compression, then a single gas-star formation relatio th the mergers and SFGs in our sample. egrated measurements of molecular gas and star formation rates in galaxies spanning a wide range of properties. We have included z~0 p and high-redshift follow a well-defined relation of modest scatter (±0.47 dex) given by 1 log( (M )) 0.78( 0.23) log( ( )) 1.37( 0.16) log( ( )) 6.9( dyn mol gas The uncertainties given in equation (8) again are 3σ fit errors, Note that the -0.8-slo dependence of SFR on  dyn can be easily seen in the sequence between SMGs and z1 SFGs, which plausibly sample galaxies of similar gas, stellar and dynamical masse (Tacconi et al. 2008. The detailed values of the slopes determined from the data naturally depend strongly on the weighting of the different types of galaxies in the extremes of the distribution. The values given in equation (8) are for equal weights to zS FGs, z1 SFGs, z~0 mergers (ULIRGs) and z1 mergers (SMGs). Equation (8) describes a 'fundamental plane' of star formation. For the relation above the slope in the M mol-gas -SFR projection is similar 1.9) (8).  ginally different from a simple SFR~M mol-gas / dyn relation, which was discussed in section 3. Future work is needed to determine whether this subtle tilt of the relation and what it might be caused by.
The basic interpretation is that the more compact z1 SMGs and z~0 ULIRG mergers form stars more rapidly because their gas clouds are more compressed with shorter dynamical times, so that they churn more quickly through the available gas reservoir th the typical normal disk galaxies. When the dependence on galactic dynamical time scale is explicitly included, disk galaxies and mergers appear to follow similar gas to star formation relations. The mergers may be forming stars at slightly higher efficiencies than the disks. This interpretation, also favoured by Bouché et al. (2007), is consistent with our remarks earlier in this section about the remarkable uniformity of z~0 and z1 SFG despite obvio agmentation and galaxy sc n may broadly explain bo

Conclusions
We have analysed in this paper galaxy-int normal star forming galaxies, very luminous and gas rich, but otherwise normal star forming disks at z~1-2.5, z~0 ULIRG mergers, and z1 submillimetre galaxies, many of which are also gas rich, dissipative mergers.
To extract from the basic data (CO line luminosities and far-infrared/submillime UV luminosities) the underlying physical parameters, namely mole tre or cular gas masses and star formation rates, we have used the best available consensus conversion and cali nd much . The gas depletion time (~1.5 Gyr at z~0 and ~0.5 Gyr at z~1-2.5) is ~50 times greater than average local free-fall, or galaxy-scale dynamical times. However, the gas dep an ween iently f the more rapid driving of large scale gas compression and the higher fractions of dense gas. Our second major conclusion thus is that global star formation in normal disk galaxies as well as the most extreme galactic environments can be largely captured in a 'universal' gas-star formation relation explicitly including the dynamical time.
bration factors. While these calibrations are undoubtedly approximate and have substantial uncertainties, we find that our main findings are quite robust and do not change qualitatively over the plausible range of these calibrations.
Our first main conclusion is that the gas-star formation relation does not depe on redshift. Star formation in both z~0 and z~1-2.5 normal star forming disks is quite inefficient letion time is also much shorter than the Hubble time at all redshifts. A semicontinuous replenishment of gas through smooth accretion or minor mergers is thus required.
The ratio of star formation rate to available molecular gas mass in ULIRGs and SMGs is four to ten times greater than in SFGs at all redshifts. Since z~1-2.5 SFGs are as luminous as z~0 ULIRGs and almost as luminous as z1 SMGs, this L/M-excess cannot be a pure luminosity effect. If the global galactic dynamical time scale is introduced as explicit third parameter, all galaxies appear to lie in a plane, with a scatter that is only somewhat greater than the systematic measurement uncertainties. The difference bet SFGs, ULIRGs and SMGs disappears almost completely in the edge-on projection of this plane, which may be somewhat tilted with respect to 45 0 (which would correspond to SFRM mol-gas / dyn ). We propose that major gas rich mergers form stars more effic and rapidly because of the combination o  Bouché et al. 2010). The grey crosses are the SFR-M * data from Noeske et al. (2007) and Daddi et al. (2007) scaled to the same mean redshift as the CO observations with the (1+z) 2.7 dependence given above. The dashed vertical and horizontal lines mark the common matched selection criteria for both red-shift ranges (M * >4x10 10 M  , SFR>40 M  yr -1 ).  Figure 2. Correlation between observed/inferred CO 1-0 luminosity and observed/inferred FIR luminosity for different galaxy samples in our data base. Open grey circles denote isolated normal (and starburst) galaxies at z~0, from K98a, Gao & Solomon (2004), Kuno et al.(2007), Gracia-Carpio et al. (2008), Gracia-Carpio (2009) and Leroy et al. (2008Leroy et al. ( , 2009; filled cyan squares denote interacting z~0 galaxies from the same references; filled blue circles are <z>=2.3 SFGs (BX) and filled black circles <z>=1.2 SFGs (EGS), both from Tacconi et al. (2010, in prep.); and filled green triangles are <z>=1.5 SFGs (BzK) from Daddi et al. (2010). Crossed magenta squares are z~0 LIRG/ULIRG mergers from K98a, Gracia-Carpio et al. (2008) and Gracia-Carpio (2009 (2010) and Smail et al. (in prep). In cases where rotationally excited CO lines were observed correction factors discussed in section 2.6 have been applied to the observed luminosities (see also Table 1). The typical total (statistical plus systematic) 1σ uncertainty is shown as a large black cross in the lower right of the panels. Left panel: luminosity-luminosity correlation. Dotted grey and red lines give the results of the fits to the SFGs and luminous mergers, respectively, including in each case all red-shifts. The fits assign equal weight to all data points. Right panel: L FIR /L CO as a function of L CO . Dotted grey and red lines give the results of average values to the SFGs and luminous mergers, respectively, including in each case all red-shifts. The fits assign equal weight to all data points, and uncertainties in brackets are 3σ formal fit errors. The specific star formation rate (SFR/M mol-gas ) computed from L FIR /L CO is given on the right vertical axis.  Figure 2 but there are fewer galaxies than in Figure 2 because of the lack of availability of size measurements. For the high-z SFGs surface densities are calculated by dividing 50% of the integrated star formation rate and molecular gas mass (including a 36% correction for helium) by the effective area of the half-light radius (πR 1/2 2 ). Left panel: Σ molgas -Σ starform relation (' Kennicutt-Schmidt'-relation). In addition to our data base, we have also included in green contours with green/orange/red shading the average distribution of the spatially resolved data in normal galaxies given in Bigiel et al. (2008, green/orange colours). The grey shaded region denotes the relation found from spatially resolved observations in M51 ). Right panel: Σ molgas / dyn -Σ starform relation ('Elmegreen-Silk' relation). The fits assign equal weight to all data points, and uncertainties in brackets are 3σ formal fit errors. The crosses in the bottom right denote the typical total (statistical + systematic) 1σ uncertainties. The black dash-dotted, blue continuous and magenta long-dashed lines mark lines of (constant) 0.1%, 1% and 10% efficiency per dynamical time scale   (equation 4).  Figure 3 if the a priori best conversion factors for SFGs (α=α G ) and mergers (α=α G /3.2) are chosen. The right panel shows the same plot for the choice of a universal conversion factor of α=α G for all galaxies in the data base. This was the choice in the K98a paper but leads to a significant overestimate of gas fractions in almost all major mergers. The fits assign equal weight to all data points and uncertainties in brackets are 3σ formal fit errors. The crosses in the lower right denote the typical total (statistical + systematic) 1σ uncertainty. Here the a priori best conversion factors for SFGs (α=α G ) and mergers (α=α G /3.2) are chosen. The short-dashed red line is the best fit to all data. The fits assign equal weight to all data points and the uncertainties (in brackets) are 3σ formal fit errors. The cross at the bottom denotes the typical total (statistical + systematic) 1σ uncertainty. As in Figure 3, the black dash-dotted, blue continuous and magenta longdashed lines mark lines of (constant) 0.1%, 1% and 10% efficiency per dynamical time scale   (equation 4).  Figure 6. Dependence of different star formation tracers on star formation histories. Left panel: Ratio of luminosity in a given tracer (Hα, far-infrared (=total bolometric), far-UV and near-infrared luminosities, see legend) to the star formation rate for a constant star formation rate, as a function of time. Right panel: the same quantities for a moderately short duration (δt=10 7 years) burst model (SFR(t)=SFR 0 exp(-t/δt)). In both panels we have used the STARS stellar population synthesis code (Sternberg et al. 2003), with solar metallicity templates and a Kroupa (~Chabrier) IMF. In both panels the dashed blue and red horizontal lines mark the values of the K98a calibrations of the far-IR and Hα indicators (corrected to a Chabrier/Kroupa IMF). These plots show that these calibrations are applicable for the equilibrium (t>>10 7 yrs) in constant star formation histories but not for smaller ages for the far-IR indicator and not for any age for short duration bursts post the peak of the burst. In all these cases the indicators underestimate the intrinsic peak star formation rate (SFR 0 ). SFGs, crossed open magenta squares are z~0 LIRG/ULIRG mergers and filled red squares are z1 SMGs. The bottom right panel (the M mol-gas -SFR projection) is essentially the same as Figure 2 and shows the clear separation of mergers and SFGs (in all three panels we use α merger =1 and α SFG =3.2). The top right panel shows the  dyn -SFR projection, which again shows a clear separation between compact mergers on the left and extended SFGs on the right. For the high-z SFGs and mergers (which are of comparable dynamical, stellar mass and gas fraction: Tacconi et al. 2008Tacconi et al. , 2010 there appears to be a clear sequence of star formation rate on dynamical time (slope -0.8). The central left panel shows that there exists a three-dimensional projection of the data in which differences between different types of galaxies are minimal. The best fitting scaling relation (log(SFR)=-0.78 (±0.23) log( dyn ) +1.37 (±0.16) log(M mol-gas ) -6.9(±1.9)) has a scatter of ±0.47 dex.