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understanding the properties and behaviour of the interstellar medium ( ism ) of galaxies in the early universe is a cornerstone of modern galaxy evolution studies . galaxies at early epochs show significantly elevated gas fractions relative to their local analogues @xcite , and it is these massive gas reservoirs that drive the enhanced star formation rates characteristic of the high-@xmath11 universe ( i.e. , @xcite ; @xcite ; ) .    a variety of techniques have been used to observe the gas reservoirs in distant galaxies . traditionally , the detection of intergalactic molecular gas ( particularly at high-@xmath11 ) has relied on observations of various molecular emission lines of carbon monoxide ( @xmath6co ; @xcite ; @xcite ; @xcite ; @xcite ) . co exists in the centres of molecular clouds , and  with the aid of a co - to - h@xmath7 conversion factor  observations of the co line luminosity can be converted into a mass of molecular gas . this conversion is non - trivial however , with the abundance of co relative to h@xmath7 varying as a function of ism metallicity ( increased metal abundance results in larger quantities of dust to protect co from photodissociation ) and even galactic behaviour ( in merging systems the star - forming ism no longer consists of discrete molecular clouds , making co a more efficient tracer of molecular gas ) . moreover , recent results have suggested that co may also be destroyed by cosmic rays ( produced indirectly by star formation , via supernovae , as well as agn activity ) . the ism of intensely star - forming galaxies , which produce a large cosmic ray flux , may be a hostile environment for co molecules making it a less effective tracer of molecular gas than previously thought ( @xcite ; @xcite ) . another method for tracing molecular gas is via the long - wavelength dust emission . observations of the dust continuum can be converted into a total dust mass , which , with the aid of an assumed gas - to - dust ratio , can be used to calculate a gas mass ( see @xcite ; @xcite ; @xcite ) . this technique has some advantages over the use of co lines : dust continuum observations are generally less time intensive , allowing for larger samples to be assembled . the dust mass method is not without its disadvantages , however . the dust temperature must be constrained ( or assumed ) in order for a dust mass to be measured . in addition , the gas - to - dust ratio remains a relatively poorly studied quantity , which may vary by up to a factor of @xmath12 in bright star - forming galaxies @xcite , as well as potentially varying at high redshift ( @xcite ; @xcite ) . furthermore , observations of the dust continuum provide no kinematic information , which is available when observing co emission lines .    in recent years , the emission line of atomic carbon ( ( @xmath13 ) ; ( @xmath0 ) hereafter ) has been found to be an excellent alternate tracer of the cold molecular ism , being closely associated with low-@xmath5 co emission across a wide range of environments and redshifts ( @xcite ; @xcite ; @xcite ; @xcite ) . this conclusion is supported by both detailed studies of nearby galactic molecular clouds ( in which co and  are found to co - exist throughout the bulk of the cold molecular component ; @xcite ) , as well as hydrodynamic simulations @xcite .    using  as a molecular gas tracer offers a number of advantages , compared to observations of both co and the dust continuum . due to the simplicity of the quantum fine structure level of , many physical parameters ( including excitation temperature and total carbon mass ) can be calculated with minimal uncertainty . and while co becomes an increasingly poor tracer of molecular gas as metallicity decreases ( due to a lack of dust grains being available to shield co molecules from photodissociation ) ,  is affected far less severely  significantly reducing the uncertainty on the molecular gas mass introduced by unknown metal abundances . furthermore , @xcite found that a high cosmic - ray ionisation rate will destroy co molecules , dissociating them into , while leaving the underlying h@xmath7 unaffected . additionally , observations of the  line provide the same valuable kinematic information as co , offering a distinct advantage over dust - based methods . as a result can be a powerful and effective tracer of the molecular ism in distant galaxies .    in this work , we present observations of the ( @xmath0 ) emission line in a sample of strongly lensed dusty star forming galaxies ( dsfgs ) which were identified via the south pole telescope ( spt ; @xcite ) wide - field survey ( @xcite ) . dsfgs are extremely luminous star - forming galaxies ( typical sfrs @xmath14 m@xmath4/yr ) , which are thought to be the high-@xmath11 progenitors of the most massive galaxies in the @xmath15 universe . the spt has proven to be an efficient machine for finding the brightest ( strongly - lensed ) dsfgs in the universe ( @xcite ; @xcite ) . the dsfgs in this work were all taken from the 26-galaxy sample targeted for spectroscopic redshift identification by the atacama large millimeter array ( alma ) in cycle 0 @xcite , which confirmed redshifts via the identification of a number of atomic and molecular emission lines . thirteen dsfgs were found to lie at redshifts @xmath16 , shifting the ( @xmath0 ) into alma band 3 . in this work we present an analysis of the  properties of these 13 sources . this paper falls broadly into two halves . in the first half , we present an analysis of the  properties of our sample . we describe the sample and the various ancillary data in  2 , and in  3 we present our analysis of the ( @xmath0 ) data ( including calculations of the mass and cooling contribution of atomic carbon , analysis of the kinematic properties of the sample , and discussions of the use of ( @xmath0 ) as a tracer of molecular gas and star formation mode ) . moving to the second half of the paper , in 4 we combine the ( @xmath0 ) line with a variety of atomic and molecular emission lines in order to constrain the physical conditions in the ism of our galaxies using a modern photodissociation region ( pdr ) modelling code , 3d - pdr . we present a discussion of our results in  5 , and we present our conclusions in  6 . throughout this work we adopt a standard @xmath17-cdm cosmology with parameters taken from @xcite ; @xmath18 , @xmath19 , and @xmath20 . the 13 sources presented in this work were originally targeted as part of our alma blind redshift search programme @xcite , in which 26 spt dsfgs were observed across the entirety of alma band 3 ( @xmath21 ghz ) as part of the cycle 0 ` early compact array ' setup . each observation consisted of 5 distinct 7.5 ghz tunings , spaced to cover the band . each source was observed for 120s in each tuning configuration . further details of the observing programme and data reduction can be found in @xcite . 13 of the 26 dsfgs lie in the redshift range @xmath16 , causing the ( @xmath0 ) emission line ( @xmath22 492.161 ghz ) to be redshifted into band 3 . of these 13 sources , just one ( spt0345 - 47 ) was not detected in ( @xmath0 ) . a further two sources ( spt0300 - 46 and spt2103 - 60 ) are tentatively detected at the @xmath23 level ( i.e. , 3 times the rms channel noise , which we measure to be 2.1 mjy and 2.6 mjy per 50 km / s channel respectively ) . the remaining 10 sources all show clearly detected ( @xmath0 ) emission at @xmath24 significance .    for spt0345 - 47 , we calculate a @xmath25 upper limit on the intensity of the ( @xmath0 ) emission line :    @xmath26    where @xmath27 is the rms channel noise in the spectrum of spt0345 - 47 ( which we measure to be 2.4 mjy per 50 km / s channel ) , @xmath28 is the mean linewidth of the detected sample ( = 410 km / s ) , and @xmath29 is the bin size in kms@xmath30 ( = 50 km / s ) . we calculate an upper limit on the  line intensity for spt0345 - 47 of @xmath31(1 - 0 ) } < 1.03 $ ] jy km / s . figure [ fig : prop1 ] shows spectral cutouts at the position of the ( @xmath0 ) line for the 13 spt dsfgs analysed in this work . for reference , we have overlaid ( where available ) the co(@xmath32 ) emission line ( scaled arbitrarily in flux for ease of comparison ) .    due to a lack of x - ray data for our sample we can not rule out a possible agn component in any of our 13 dsfgs . however it is unlikely that any of our sources contains a significant agn . dsfgs as a class are star - formation dominated ( i.e. , the bolometric output of the galaxy originates predominantly from young massive stars , rather than accretion onto a central compact object ) ; even dsfgs with some measurable agn activity tend to be primarily star - formation - driven objects @xcite . furthermore , chandra x - ray observations of the most compact and ir - luminous spt - dsfg , spt0346 - 52 , found no sign of agn activity ( ma et al . , 2016 ) . we proceed with the assumption that our the dsfgs in our sample are star - formation dominated objects . all 13 galaxies presented in this work are strongly gravitationally lensed , and have detailed lens models based on alma @xmath33 m observations , which allow their lensing magnifications to be calculated @xcite . we discuss the use of these lens models to remove the effects of gravitational lensing in  3.1 below . the spt - dsfg sample has been the target of several followup programmes designed to survey a number of ism diagnostics . in particular , in addition to the ( @xmath0 ) emission lines presented in this work , 9/13 sources have observations covering the co(@xmath32 ) emission line , 10/13 have observations covering the co(@xmath34 ) emission line ( the remaining 3 have observed co(@xmath35 ) emission , which can be converted to ( @xmath34 ) with the aid of an assumed co spectral line energy distribution ( @xcite ; @xcite ) ) . in addition , 10/13 have the  emission line observed . the mid-@xmath5 lines of co come from the alma spectra used in this work . the  observations were obtained from the apex first light apex submillimetre heterodyne receiver ( flash ) , and the herschel ` spire ' fts spectrometer . observations were made at 345 ghz ( for sources with redshifts @xmath36 ) and 460ghz ( @xmath37 ) , with system temperatures of 230 k and 170 k respectively . further details of the observations and data reduction can be found in @xcite . the low-@xmath5 co observations were taken by the australia telescope compact array ( atca ) , as part of a targeted program aimed at obtaining co(@xmath0 ) or co(@xmath32 ) for spt dsfgs with secure redshifts . atca was used in its h214 hybrid array configuration , with the compact array broadband backend in wide bandwidth mode . the mean rms noise for the co(@xmath32 ) observations used in this work was 0.5 mjy per 50 km / s channel . further details of the observations and data reduction can be found in @xcite . this suite of molecular emission lines enables a more detailed treatment of the conditions of the ism , as multiple line ratios can be used to independently constrain various parameters of interest . throughout this work , we calculate emission line luminosities using the formulae below . luminosities in solar units ( l@xmath4 ) , which represent the true energy output carried by the emission line ( used , for example , for calculating cooling contributions ) are calculated following @xcite :      where @xmath39 is the velocity - integrated line flux in jy km s@xmath30 , @xmath40 is the rest frequency in ghz , and @xmath41 is the luminosity distance in mpc . alternatively , line luminosities in units of k km s@xmath30 pc@xmath42 are calculated using        table [ tab : lum ] lists line luminosities ( in solar units ) for the dsfgs used in this work . in addition , we also list in table [ tab : lum ] the bolometric far - ir luminosity , calculated using greybody sed fits to far - ir / millimetre photometry ( greve et al . 2012 ) . note that all values in table [ tab : lum ] are observed quantities , which have not been corrected for the effects of gravitational lensing . .line luminosities , far - ir luminosity , /fir ratios , and lensing magnifications ( @xmath44 ) for the dsfgs studied in this work . all luminosities are given in solar units , and have not been corrected for gravitational lensing . where @xmath6co(@xmath34 ) transition line luminosities are not directly measured , they are inferred from the @xmath6co(@xmath35 ) line luminosity using the conversions derived by @xcite , and are shown in parentheses . the far - infrared luminosities are derived by integrating under a modified blackbody curve from 8 - 1000@xmath44 m . lensing magnifications are derived from visibility - based lens models fit to alma 870@xmath44 m observations ( spilker et al . data are taken from gullberg et al . co@xmath45 data are taken from aravena et al . ( 2016 )  other co lines are taken from the program described in weiss et al . ( 2013 ) . [ cols="<,^,^,^,^,^,^,^,^,^,^,^,^,^,^",options="header " , ]
we present alma ( @xmath0 ) ( rest frequency 492 ghz ) observations for a sample of 13 strongly - lensed dusty star - forming galaxies originally discovered at 1.4 mm in a blank - field survey by the south pole telescope . 
 we compare these new data with available  observations from the literature , allowing a study of the ism properties of @xmath1 extreme dusty star - forming galaxies spanning a redshift range @xmath2 . using the  line as a tracer of the molecular ism 
 , we find a mean molecular gas mass for spt - dsfgs of @xmath3 m@xmath4 . 
 this is in tension with gas masses derived via low-@xmath5 @xmath6co and dust masses ; bringing the estimates into accordance requires either ( a ) an elevated co - to - h@xmath7 conversion factor for our sample of @xmath8 and a gas - to - dust ratio @xmath9 , or ( b ) an high carbon abundance @xmath10 . using observations of a range of additional atomic and molecular lines ( including , , and multiple transitions of co ) 
 , we use a modern photodissociation region code ( 3d - pdr ) to assess the physical conditions ( including the density , uv radiation field strength , and gas temperature ) within the ism of the dsfgs in our sample . 
 we find that the ism within our dsfgs is characterised by dense gas permeated by strong uv fields . 
 we note that previous efforts to characterise pdr regions in dsfgs may have significantly under - estimated the density of the ism . 
 combined , our analysis suggests that the ism of extreme dusty starbursts at high redshift consists of dense , carbon - rich gas not directly comparable to the ism of starbursts in the local universe . 
 galaxies : high - redshift  galaxies : ism  gravitational lensing : strong  galaxies : evolution  galaxies : formation