Dense Gas in a Giant Molecular Filament

Recent surveys of the Galactic plane in the dust continuum and CO emission lines reveal that large ($\gtrsim 50$~pc) and massive ($\gtrsim 10^5$~$M_\odot$) filaments, know as giant molecular filaments (GMFs), may be linked to galactic dynamics and trace the mid-plane of the gravitational potential in the Milky Way. We have imaged one entire GMF located at $l\sim$52--54$^\circ$ longitude, GMF54 ($\sim$68~pc long), in the dense gas tracers using the HCN(1--0), HNC(1--0), HCO$^+$(1--0) lines, and their $^{13}$C isotopologue transitions, as well as the N$_2$H$^+$(1--0) line. We study the dense gas distribution, the column density probability density functions (N-PDFs) and the line ratios within the GMF. The dense gas molecular transitions follow the extended structure of the filament with area filling factors between 0.06 and 0.28 with respect to $^{13}$CO(1--0). We constructed the N-PDFs of H$_2$ for each of the dense gas tracers based on their column densities and assumed uniform abundance. The N-PDFs of the dense gas tracers appear curved in log-log representation, and the HCO$^+$ N-PDF has the largest log-normal width and flattest power-law slope index. Studying the N-PDFs for sub-regions of GMF54, we found an evolutionary trend in the N-PDFs that high-mass star forming and Photon-Dominate Regions (PDRs) have flatter power-law indices. The integrated intensity ratios of the molecular lines in GMF54 are comparable to those in nearby galaxies. In particular, the N$_2$H$^+$/$^{13}$CO ratio, which traces the dense gas fraction, has similar values in GMF54 and all nearby galaxies except ULIRGs.


Introduction
Studies towards nearby molecular clouds (MCs) show that the dense regions of MCs are permeated with filaments that contain sites of star formation (e.g, André et al. 2014). Recent observations identified a class of large ( 50 pc) and massive ( 10 5 M ) filaments, known as giant molecular filaments (GMFs) (Jackson et al. 2010;Goodman et al. 2014;Ragan et al. 2014;Zucker et al. 2015;Wang et al. 2015;Li et al. 2016;Wang et al. 2016;Abreu-Vicente et al. 2016;Wang et al. 2020). Many of these giant filaments lie along, or extremely close to spiral arms in the position-position-velocity space, suggesting they may trace the dense "spine" of the spiral arms and the mid-plane of the gravitational potential in the Milky Way (MW) (Goodman et al. 2014;Zucker et al. 2015). Detailed inter-comparison of a sample of long filaments from the literature suggests that there may be different classes of filaments in both their physical properties and their association with Galactic structure (Zucker et al. 2018). GMFs are the largest coherent cold gas structures in our Milky Way yet our physical understanding of GMFs is still poor, lim-ited to estimates of their occurrence, gas masses and lengths (Ragan et al. 2014;Abreu-Vicente et al. 2016;Zucker et al. 2018).
Observations of nearby galaxies have resolved giant MCs with masses and sizes similar to those of GMFs (10 to 200 pc, e.g., Hughes et al. 2013;Schinnerer et al. 2013;Leroy et al. 2016;Li et al. 2016;Faesi et al. 2018;Herrera et al. 2020), suggesting that GMFs could be analogous to these extragalactic giant MCs and could connect studies of star formation on Galactic and extragalactic scales.
So far, GMF studies have relied on Galactic plane surveys, either in dust tracers like (sub)mm emission (e.g., Hi-GAL 1 , Molinari et al. 2010; ATLASGAL 2 , Schuller et al. 2009) or in low-to intermediate-spatial resolution surveys of CO and its isotopologues (e.g., GRS 3 , Jackson et al. 2006;COHRS 4 , Dempsey 1 The Herschel infrared Galactic Plane Survey. 2 The APEX Telescope Large Area Survey of the Galaxy. 3 The Galactic Ring Survey. 4 Schuller et al. 2017). However, there exists little knowledge about the high-volume-density gas, which will eventually form stars.
To study this, we selected the giant molecular filament located at l ∼ 52 − 54 deg longitude (GMF54, Fig. 1), that was identified by Ragan et al. (2014) in infrared extinction and confirmed using the GRS 13 CO(1-0) data (Jackson et al. 2006) as a velocity-coherent filament at a distance of ∼ 2 kpc. With the 13 CO data, Ragan et al. (2014) estimated the total length and mass of GMF54 to be ∼ 68 pc and 6.8×10 4 M , respectively.
As a common tool to study the physical properties of molecular clouds, column density probability density functions (N-PDFs) are widely used both in observational (e.g., Lombardi et al. 2008;Kainulainen et al. 2009;Alves de Oliveira et al. 2014;Sadavoy et al. 2014;Abreu-Vicente et al. 2015;Schneider et al. 2015a;Lin et al. 2017) and theoretical studies (e.g., Ostriker et al. 2001;Federrath et al. 2010;Federrath & Klessen 2013;Burkhart et al. 2017;Chen et al. 2018;Körtgen et al. 2019). The shape of the N-PDF depends on the physical processes dominating the cloud and can thus be used to study the evolution of the molecular clouds. From simulations, the early evolutionary stage of a molecular cloud is dominated by turbulence and the N-PDF shows a log-normal shape. The width of the log-normal N-PDF is determined by the turbulent motions (see e.g., Federrath et al. 2010;Ballesteros-Paredes et al. 2011;Kritsuk et al. 2011;Federrath & Klessen 2013;Burkhart et al. 2015;Bialy et al. 2017). As the cloud evolves to a gravity dominated system, the N-PDF develops a high column density power-law tail with a slope of around -2 (Klessen 2000;Girichidis et al. 2014). Observations indicate that star-forming clouds show such tails, providing support to this scenario (e.g., Kainulainen et al. 2009;Schneider et al. 2013). Furthermore, the slope of the power-law N-PDF can be related to evolutionary stages of the clouds as steeper slopes possibly indicating earlier quiescent stages (e.g., Kritsuk et al. 2011;Federrath & Klessen 2013;Ward et al. 2014), while flatter slopes or a second flatter power-law tail, often found in high-mass star-forming regions may indicate feedback effects (Tremblin et al. 2014;Schneider et al. 2015b).
N-PDFs are often constructed using dust extinction or continuum emission observations (e.g, Lombardi et al. 2008;Kainulainen et al. 2009;Sadavoy et al. 2014;Schneider et al. 2015b). However, since GMFs lie close to the Galactic mid-plane, heavy contamination from foreground and background emission makes it very difficult to study N-PDFs with continuum emission as it traces the entire line of sight through the Milky Way disk. With the velocity information of molecular line observations, we can easily distinguish different clouds along the line of sight. So far, a few studies investigated the molecular cloud properties with N-PDFs obtained from low J transitions of 12 CO or 13 CO observations (Goldsmith et al. 2008;Wong et al. 2008;Goodman et al. 2009;Carlhoff et al. 2013;Schneider et al. 2015b;Sun et al. 2020), which can not trace the column density of dense cores due to high optical depth. Schneider et al. (2016) found that compared to dust N-PDF, N-PDFs of the dense gas tracers N 2 H + (1-0) and CS(2-1) can recover the power-law tail well independent of the exact column-density. 5 Structure, excitation, and dynamics of the inner Galactic interstellar medium 2. Observation and data reduction 2.1. Dense gas tracers with IRAM 30 m The giant filament GMF54 was observed with the IRAM 30 m telescope in September 2017 and November 2017 with the EMIR receiver and the Fourier Transform Spectrometer (FTS) backends at 3 mm. The receiver was tuned to center at 91.4 GHz with dual-polarisation to cover the dense gas tracers HCO + (1-0), HCN(1-0), HNC(1-0) and their 13 C isotopologues, including the cold and dense gas tracer N 2 H + (1-0). The FTS backends were used to cover the 16 GHz bandwidth of the receiver with a uniform spectral resolution of 200 kHz, which results in a velocity resolution of ∼ 0.7 km s −1 at 3 mm. The observations were carried out in the on-the-fly (OTF) mode employing position switching to an OFF position sufficiently far away. The filament was scanned at least once along the Galactic longitude and latitude directions, respectively, in order to reduce scanning effects. The system temperature and other calibration parameters were measured every 10 to 15 min. Jupiter and standard mm calibrators were observed regularly to calibrate the pointing and focus. During the observation, the weather condition in general was good and the radiometer opacity τ at 225 was measured around 0.5, except for two OTF mapps observed on 9th September which have a precipitable water vapor (PWV) larger than 40 mm, the data was therefore dropped and the area was reobserved later. The spectra were calibrated with CLASS, which is part of the GILDAS software package 6 . We converted the data from antenna temperature (T A ) to main beam brightness temperature (T mb ) using a forward efficiency (F eff = 95% at 3 mm) and main beam efficiency (B eff = 81% at 3 mm) following equation T mb = T A × F eff /B eff . A linear function was fitted to the line free channels of the spectra to subtract the baseline. The final 3 mm data were all smoothed to a common spectral resolution of 0.7 km s −1 and a beam size of 32 (∼0.32 pc at 2 kpc distance), and the rms noise level of T mb measured at line free channels is around 0.05 K.

13 CO(1-0)
The 13 CO(1-0) data are from the Galactic Ring Survey (GRS, Jackson et al. 2006), with a spectral resolution of 0.21 km s −1 and a FWHM beam size of 46 . After converting the antenna temperature (T A ) to main beam brightness temperature (T mb ) using a main beam efficiency of 0.48 (Jackson et al. 2006), the line free channel rms noise level of T mb is around 0.22 K.

Dust continuum data
The dust temperature and column density maps were produced by Zucker et al. (2018) by performing pixel-by-pixel modified blackbody fits to the Hi-GAL 160, 250, 350, and 500 µm continuum map using the HiGal_SEDfitter 7 code (Wang et al. 2015) with a fixed dust opacity of β = 1.75 and a gas-to-dust ratio of 100. The 70 µm band was excluded from the fitting due to its high optical depth towards very dense region and to better tracing cold gas. We subtract a constant column density value (3.46×10 21 cm −2 ) measured from a 1 radius circular area centered at l = 53.69 • and b = −0.30 • to remove the line-of-sight contamination (Schneider et al. 2015b;Ossenkopf-Okada et al. 2016

Distribution of the dense gas
While the 13 CO(1-0) map ( Fig. 1) shows the extended nature of the giant filament GMF54, the integrated intensity maps of HCO + , HCN, HNC, and N 2 H + in Fig. 2 show that these dense gas spectral lines trace the extended structure of the filament although a much smaller area is detected compared to the 13 CO map. The integral velocity range for each tracer was determined to include all emissions associated with the filament. Since we want to include the emission of the all hyper-fine structures, a larger integrated velocity range (as shown in Fig. 2) is adopted for HCN(1-0) and N 2 H + (1-0), which also includes some emission at velocities 35 km s −1 or 10 km s −1 . These emissions that lie in regions marked with red ellipses in Fig. 2 are at different distance and unrelated to the filament, so we exclude them from the further discussion. As the spectra in Fig. 2 show, the filament is dominated by one component centered at 20 km s −1 . The contamination from the other two components (peaked at ∼ 5 km s −1 and ∼ 36 km s −1 , respectively) is negligible.
To quantify how much extended structure is traced by different molecular lines, we calculate the area filling factor of the dense gas integrated intensity map relative to the 13 CO(1-0) map. We measured the area of each molecular line map where S/N>3, and divided it by the 13 CO map area to get the filling fac- Notes. (a) E u /k are from the Cologne Database for Molecular Spectroscopy (CDMS, Müller et al. 2001Müller et al. , 2005Endres et al. 2016). (b) n crit are from Shirley (2015) assuming T k =20 K, except for 13 CO(1-0) and HN 13 C(1-0), which were calculated following the method described by Shirley (2015). The relevant data used for the calculation are from Schöier et al. (2005), and the Python script can be found at https: //github.com/ZhiyuZhang/critical_densities. (c) n eff assuming T k =20 K are from Shirley (2015).
tor, and the results are listed in Table 2. Comparing to the 13 CO emission, these typical dense gas tracers cover between 6% and 28% of the area of the 13 CO emission, depending on the tracer A&A proofs: manuscript no. gmf54 ( Table 2). If we smooth and regrid our 3 mm integrated intensity maps into the same angular resolution of 46 and pixel size of 22 as the 13 CO data, the filling factors of these dense gas tracers increases by ∼ 70% (see Table 2). The filling factors should be taken in to account for extragalactic studies. We list the upper energy level, the critical density (n crit ) under the simplifying assumptions of two-levels systems and optically thin emission, and the effective excitation density (n eff ) of the lines we used in Table 1. Compared to n crit , n eff gives a better estimate of the approximate density at which a modest (1 K km s −1 ) molecular line can be observed (Shirley 2015). Among these four dense gas tracers (HCO + , HCN, HNC, and N 2 H + ), we observe that HCO + (1-0) has the smallest n eff , and also traces the most extended structure, while N 2 H + (1-0) has the highest n eff and traces as expected the most compact structure. On the other hand, HCN(1-0) has both a higher n eff and a higher n crit than HNC(1-0), yet traces more extended structure than HNC(1-0). Kauffmann et al. (2017) also found that HCN(1-0) typically traces gas at a density of ∼ 870 cm −3 , while N 2 H + (1-0) traces a much higher density of ∼ 4 × 10 3 cm −3 .
The ratios of the molecular line integrated intensities are often used to trace different physical properties of the gas. The HCN/CO integrated intensity ratio is commonly used to trace the dense gas fraction in galactic and extragalactic studies (e.g., Gao & Solomon 2004;Lada et al. 2012), and HCN/HNC ratio is considered to be able to trace the evolutionary stages of the molecular cloud (Schilke et al. 1992;Graninger et al. 2014;Hacar et al. 2019). To compare the intensity of different lines properly, we first integrated all four dense gas tracers and the 13 CO line in the velocity range 10 < v LSR < 36 km s −1 . We then convolved and regridded all dense gas maps to the same resolution as the 13 CO one. We divided the HCN integrated intensity map by the HCO + , HNC, N 2 H + and 13 CO map to obtain the line ratio maps shown in Fig. 3.
To study the dense gas properties in different evolutionary stages, we divided the filament into four regions. As shown in Figs. 2 and 3, the four sub-regions are: Region A, infrared dark filament dominated; Region B, infrared bright H ii regions with photon dissociation regions (PDR) (Urquhart et al. 2011;Anderson et al. 2014); Region C, infrared dark filament dominated; Region D, high-mass star forming region with a hypercompact (HC) H ii region, methanol maser, and water maser (Pandian & Goldsmith 2007;Pandian et al. 2009;Sánchez-Monge et al. 2011;Urquhart et al. 2011;Anderson et al. 2014). Therefore, we propose an evolutionary sequence among these four sub-regions, that Region A and Region C are the youngest, Region D is more evolved, Region B with PDRs is the most evolved region.
As shown in Fig. 3 where we overlaid the HCN integrated intensity contours on line ratio maps and HI-GAL dust temper-ature map (Marsh et al. 2017), compared to HCO + and HNC, the relative intensity of HCN increases dramatically in regions with high temperature (i.e., Regions B and D). For instance, the HCN/HNC ratio remains relative constant ( 2) in Regions A and C, where the dust temperature is T dust 18 K, while in Regions B and D the dust temperature rises to T dust 20 K and the HCN/HNC ratio also increases to 3. Similar trends can also be seen in the HCN/HCO + ratio map, where the HCN/HCO + ratio is approximately a factor of 3 higher in Region B than in Region C.

Column density
Following the standard method described in other works (e.g., Eq. 15.37 in Wilson et al. 2013, appendix in Caselli et al. 2002and Feng et al. 2016, we estimated the column density of the molecular cloud from the 13 CO emission as well as the dense gas tracers. Assuming the optically thick for 12 CO(1-0) emission lines, Zhang et al. (2019) estimated the excitation temperature T ex for 13 GMFs from T mb ( 12 CO), and they obtained the mean value and standard deviation are ∼ 10 K and 2.5 K, respectively. Since we do not have 12 CO(1-0) observation with compatible angular resolution with GRS for GMF54, we assume 13 CO emission to be optically thin and adopt the mean excitation temperature T ex =10 K from Zhang et al. (2019). We discuss the uncertainty of this in Sect. 4.5. For N 2 H + (1-0), we applied a hyperfine structure (HFS) fit pixel-by-pixel using PySpecKit (Ginsburg & Mirocha 2011) to derive the optical depth τ and the excitation temperature T ex .
Since our observations also cover the H 13 CO + (1-0), H 13 CN(1-0), and HN 13 C(1-0) lines, we assume these lines to be optically thin and estimated the optical depth for HCO + (1-0), HCN(1-0), and HNC(1-0). To do this, we estimated the 12 C to 13 C abundance ratio following the relation [ 12 C]/[ 13 C] = 6.2D GC [kpc] + 9.0 (Giannetti et al. 2014). For a Galactocentric distance of D GC = 7.3 kpc (Ragan et al. 2014), [ 12 C]/[ 13 C] ∼ 54. For regions where the 13 C line integrated intensity is higher than 5σ, we calculated τ for the 12 C lines following Eq. 6 in Giannetti et al. (2014), and assuming the remaining regions are optically thin. Since the derived τ maps are calculated pixelby-pixel, due to limited signal-to-noise ratio of the data, the τ value could vary significantly between two neighboring pixels, which is not physically plausible. To avoid this pixelization problem, we further convolved the τ map with a gaussian FWHM beam of 32 (gaussian kernel ∼13.59 ), which is the beam of our 3 mm data. We then took into account the optical depth and estimated the column densities of these dense gas tracers assuming T ex =10 K. Eventually, only a small percentage (8%, 1%, and 2% for HCO + , HCN, and HNC, respectively) of the total emission area above 3σ (Table 2) were corrected for optical depth with the factor τ/(1 − exp(−τ)) > 1.01.
Assuming an abundance of these molecules (Gerner et al. 2014;Giannetti et al. 2014), we can derive the column density of the molecular gas traced by different tracers shown in Fig. 4. The abundances we used are listed in Table 3. For comparison, we also show the column density map derived from Hi-GAL dust continuum emission (Zucker et al. 2018) in Fig. 4.

Column density probability density function
We regridded all the column density maps of the dense gas tracers (Fig. 4)   the column density probability density functions (N-PDFs) of the filament. Figure 5 presents the N-PDFs for the entire filament traced by different molecular lines. All column density values lower than 5σ (N Threshold in Table 4 corresponding to the thick closed contours in Fig. 4) were ignored when calculating the N-PDFs. We also normalized all N-PDFs by the mean column density of each tracer ( N(H 2 ) in Table 4).
As shown in Fig. 5, although all the N-PDFs from different tracers are curved, the shapes are different from each other. Qualitatively, the 13 CO and N 2 H + N-PDFs resemble a log-normal shape, while the N-PDFs of HCO + , HCN, and HNC are closer to a broken power-law shape. It is obvious that one single powerlaw cannot describe the N-PDFs well. We fit the N-PDFs with different functions to quantify their shapes. We first fit a sin-gle log-normal function 8 to each N-PDF. The errors of the lognormal fitting are estimated with bootstrap method, that we ran the fitting procedure 100 times while removing 10% of the data points randomly in each run, and we take the difference between the maximum and minimum width as the error of the fit. We cannot identify a clear peak in N-PDFs, thus the width we derived could have large uncertainties. Among all the N-PDFs, the ones from 13 CO and N 2 H + are fitted best by a log-normal function, and all the remaining N-PDFs show strong excess over the fit at the high column density side. The widths we derived from the log-normal fitting are listed in Table 4, in which 13 CO has the smallest width of 0.52, and HCO + has the largest width of 0.77.
We further fit the N-PDFs with a power-law (p(x) ∝ x −α ) from an optimal column density threshold (N min in Table 4, and    Fig. 4. Molecular hydrogen column density maps derived from HCO + , HCN, HNC, and N 2 H + . All maps are plotted with the same column density range (0 to 1.5×10 22 cm −2 ) except for HNC, which is plotted with a range of 0 to 3.75×10 21 cm −2 to show the emission structure better. The thick white contours in each panel outline the column densities we used to construct the N-PDFs, and the thin white contours mark the optimal column density threshold (N min for the power-law fit for the whole filament. The red contours in the dense gas molecular line panels mark N min for the power-law for the sub-regions (see Sect   Giannetti et al. (2014), where D GC = 7.3 kpc (Ragan et al. 2014), X12 C 16 O [8.5 kpc] = 9.5 × 10 −5 . (b) We take the observed median abundances for IRDCs for T = 15 K reported by Gerner et al. (2014). For HCO + and HNC, we take the abundances for H 13 CO + and HN 13 C reported by Gerner et al. (2014), and divide them by the 12 C to 13 C abundance ratio. the thin white contours in Fig. 4), which results in the minimum Kolmogorov-Smirnov (K-S) distance between the fit and the N-PDF. The fitting procedure was performed with the python package Powerlaw 9 (Alstott et al. 2014). The obtained N min and power-law index α are listed in Table 4. Among different tracers, the N-PDF of HCO + has the lowest power-law index of 2.52, and N 2 H + has the highest index of 4.15. For comparison we also fit a power-law function which includes all data points to each N-PDF, and all the derived power-law indices are quite similar to each other (between 2.0 and 2.36), which are lower than the ones using the optimal column density threshold N min .
To study the dense gas properties in different evolutionary stages, we constructed the dense gas N-PDFs for each sub-region (Fig. 2). Since we only compare between the dense gas tracers, the N-PDFs were constructed with the column density maps at the original resolution (8 pixel size and 32 beam size). The derived N-PDFs of the dense gas tracers for each sub-region are shown in Fig. 6.
Qualitatively, the N-PDFs of each dense gas tracer show clear differences from region to region. As we proposed in Sect. 3.1, Region A and Region B are in earlier relative quiescent stage, Region D is more evolved with HCH ii region, Region B with PDRs is the most evolved region. An evolutionary trend can be seen in some molecular lines, for instance, the HCN and HNC N-PDFs evolve from a clear log-normal shape in the younger Region A to an almost straight power-law shape in the more evolved Region D. To quantify this, we fit a log-normal function to each N-PDF, we also fit a power-law function to each N-PDF with the python package Powerlaw (Alstott et al. 2014). The fitting results are also listed in Table 4. Since we do not recover the log-normal peaks of the N-PDFs, and some N-PDFs, such as the ones of Region D, show a clear power-law shape, the derived log-normal widths not reliable, we focus on the powerlaw slope in the following discussion. As shown in the top panel in Fig. 7, from quiescent regions to HCH ii region, the power-law slope becomes flatter systematically, then from HCH ii region to more evolved PDRs the power-law slope steepens again.
To investigate the evolution of the N-PDFs, we estimated the virial parameter α virial of each sub-region following the method described in Bertoldi & McKee (1992). To eliminate the uncertainty brought in by the abundances of the dense gas tracers, we estimated the mass of each sub-region with the Hi-GAL column density map (Fig. 4). For HCO + and HNC, we fit the spectra with a Gaussian profile to derive the velocity dispersion. For HCN and 9 https://github.com/jeffalstott/powerlaw N 2 H + we fit the hyperfine structures to derive the velocity dispersion. All fittings were performed with PySpecKit (Ginsburg & Mirocha 2011). Assuming a circular shape, we estimated the effective radius from the projected pixel area of each sub-region. As shown in Fig. 7 where the power-law indices are plotted as a function of the virial parameters, the virial parameters of all regions and tracers are around or smaller than two.
There is not clear correlation between the virial parameters and the power-law slope (Fig. 7). Among all four molecular lines, N 2 H + always has the lowest virial parameter, which makes sense, since N 2 H + traces the dense cores that are gravitational bond. Virial parameters of Regions A and C measured with all four tracers are all 1, indicating the gas traced by these four tracers in these two relative quiescent regions are all at gravitational virial equilibrium and could be undergoing gravitational collapsing (e.g., Bertoldi & McKee 1992;McKee & Zweibel 1992). For the more evolved Regions B and D, while the gas traced by N 2 H + is still gravitational bond (α virial 1), the gas traced by HCO + show much larger α virial and could be confined by pressure (Bertoldi & McKee 1992).

N-PDFs of the whole filament
As mentioned in Sect. 1, there is only one prior dense gas N-PDF study. Schneider et al. (2016) studied the N-PDFs of Cygnus X with Herschel dust continuum, 12 CO(1-0), 13 CO(1-0), C 18 O(1-0) and the dense gas tracers N 2 H + (1-0) and CS(2-1). They found that compared to the dust N-PDF, the N-PDFs derived from CO observations can recover the log-normal shape in the lower column density regime, whereas, the N-PDFs derived from dense gas tracers only recover the power-law tail in the high column density regime. Although the absolute column density depends on the abundance and excitation temperature adopted, the power-law index of the dense gas is robust (-1.41 for N 2 H + , and -1.56 for CS). Compared to Schneider et al. (2016), our observation is deeper; for instance, the 3σ level of the N 2 H + (1-0) observation from Schneider et al. (2016) corresponds to an N(H 2 ) of ∼ 1 × 10 22 cm −2 , while applying the same abundance, the 3σ level of our N 2 H + (1-0) observation corresponds to an N(H 2 ) of 3.6 × 10 21 cm −2 . However, we still could not recover the lognormal peak of the N-PDFs.
As we show in Fig. 5, the N-PDFs of the dense gas tracers appear curved in log-log representation, even if we only include the data above the completeness levels (5σ). Among the N-PDFs of the four dense gas tracers, the N 2 H + one is more close to a lognormal shape, while the HCO + N-PDF is more close to a powerlaw shape. This is counter-intuitive at first glance, as HCO + has the lowest n crit and n eff and N 2 H + has the highest ones (Table 1). However, since we used the H 13 CO + (1-0) line to correct the optical depth, and H 13 CO + (1-0) traces effective density two orders of magnitude higher than that of HCO + (Table 1). Thus the HCO + N-PDF in Fig. 5 is not limited by the optical depth and n eff of HCO + (1-0), which is the reason that the HCO + N-PDF is probing high column density and shows a power-law shape. We also constructed the N-PDFs of the dense gas tracer without the opticaly depth correction shown in Fig. A.1. Wang et al. (2020) studied the N-PDFs of the atomic and molecular gas in the giant molecular filament GMF38a, and found that the N-PDF of H i emission has the smallest width, followed by the cold neutral media traced by H i self absorption (HISA) and the 13 CO N-PDF has the largest width. Furthermore, the log-normal widths of the dense gas for the whole filament are A&A proofs: manuscript no. gmf54 all larger than the 13 CO N-PDF as listed in Table 4. It seems that the width of the N-PDF is correlated with the density it traces. The power-law indices derived by fitting all data points in the N-PDFs are quite similar for all molecular lines, between 2.00 and 2.36 (α * in Table 4). On the other hand, the optimal powerlaw indices (α in Table 4) are different among different lines. Following the method described in the appendix of Schneider et al. (2016), we can estimate the radial density profile of the filament. Assuming a singular polytropic cylinder, ρ(r) ∝ r −α f , then α f = 1 + 1/α. Similar calculations are also discussed by Federrath & Klessen (2013), Fischera (2014), and Myers (2015). For the power-law indices we derived from the dense gas, the radial density profile α c is estimated to be between 1.2 and 1.5, which is consistent with the self-gravitating filament model (e.g., Myers 2015).
The N-PDFs of molecular clouds in the nearby galaxy M33 derived from 12 CO observations (Druard et al. 2014;Corbelli et al. 2018) peak at lower column density, but the shape and width is similar to the 13 CO one in Fig. 5. Our study provides the foundation to interpret future high angular resolution dense gas tracer observations towards nearby galaxies with ALMA.

N-PDFs and evolutionary stages
Previous observations show that clouds without star formation have log-normal shape N-PDF with little or no excess in the high column density tail, while active star-forming clouds present prominent power-law-like wings (e.g., Kainulainen et al. 2009;Schneider et al. 2013). Furthermore, the slope index of the power-law tail is correlated with star formation activities, as clumps with flatter power-law tails are more efficient at form-ing Class 0 protostars (Sadavoy et al. 2014;. Louvet et al. (2014) demonstrated that the star formation rate per free fall time steeply decreases with the virial parameter. The dense gas N-PDFs for the sub-regions further confirm this in a way that more quiescent regions have flatter N-PDFs (Table 4). Although with large scatter, we also found a correlation between the evolutionary stages of subregions and N-PDF power-law indices (Fig. 7). Sources at early quiescent stages have a lower fraction of high density gas which would result in a steeper power-law slope, while in high-mass star forming region with HCH ii region the fraction of high density gas is higher which results in a flatter power-law slope. When the source evolves into the PDR stage, the feedback processes from high-mass stars could reduce the dense gas fraction especially the one traced by HCN, HNC, HCO + , and the power-law slope of the N-PDF is also steeper again.

Compare to dust continuum N-PDF
To compare with the N-PDFs we derived from dense gas tracers, we also constructed the N-PDF from the Herschel dust continuum column density map. We convolved and regridded the dust column density map to the same angular resolution and pixel size as the GRS 13 CO(1-0) map, and constructed the N-PDF shown in Fig. 8. Similar to the N-PDFs of the dense gas tracers, we also fitted a log-normal function and a power-law function (with a closed contour N(H 2 )=6.0×10 21 cm −2 ) to the dust N-PDF. Since there is no closed contour in the low column density regime of the dust N-PDF, and the log-normal width is not reliable (Alves et al. 2017), we only compare the power-law slope and the column density range to the ones traced by molecular lines. Compared to N-PDFs from dense gas tracers, the dust N-PDF has a similar power-law slope to N 2 H + . Indeed, Figs. 2 and 4 also show that the N 2 H + emission follows the dust column density very well, and traces the dense cores (see also Kauffmann et al. 2017). Similar to Schneider et al. (2016), we can also shift the N 2 H + N-PDF to"calibrate" the N 2 H + abundance with the dust N-PDF, and the "calibrated" abundance of N 2 H + is 1.6×10 −9 . This "calibrated" abundance is similar to the abundance Gerner et al. (2014) estimated for high-mass protostellar objects.

Line ratios
Integrated intensity ratios of molecular lines are often used to trace molecular gas properties, which can be linked to star formation activities. Numerous surveys have been carried out towards nearby galaxies in the 3 mm band (Watanabe et al. 2014;Aladro et al. 2015;Meier et al. 2015;Nishimura et al. 2016a,b;Jiménez-Donaire et al. 2019). We would like to compare our observations of GMF54 with the observed line ratios in such nearby galaxies. Figure 3 shows the line ratio map of GMF54, and we list the averaged line ratios of GMF54 and in the nearby galaxies in Table 5. Most of the observations of dense gas in the nearby galaxies in Table 5 were carried out with single-dish telescopes, where the spatial resolution varies from ∼ 10 pc towards the Large Magellanic Cloud (LMC) to ∼24 kpc towards some active galactic nucleus (AGNs). In comparison to this, GMF54 has a size of 68 pc and our observations have a spatial resolution of 0.32 pc. Despite the huge spatial resolution differences, the line ratios we derived in GMF54 are in general comparable to those seen in the nearby galaxies.
The HCN/CO ratio is used to trace the dense gas fraction, which is directly related to star formation activity (Gao & Solomon 2004;Usero et al. 2015;Bigiel et al. 2016). Since we do not have CO observations, we use the HCN/ 13 CO ratio. Compared to nearby galaxies, the HCN/ 13 CO ratio in GMF54 is much lower than in nearby galaxies, while ULIRGs present the highest HCN/ 13 CO ratio. As discussed by Shirley (2015); Kauffmann et al. (2017); Pety et al. (2017), HCN traces much lower density than its critical density n crit , and typically traces densities around ∼ 10 2 − 10 3 cm −3 . The HCN flux is also influenced by far ultra-violet (UV) radiation (Pety et al. 2017). Similarly, HCO + (1-0) also traces only moderate dense gas, and the flux can be affected by far-UV radiation (Shirley 2015;Pety et al. 2017). Thus ULIRGs and AGNs have the highest HCO + / 13 CO ratio, while GMF54 has the lowest ratio.
On the other hand, N 2 H + traces the real dense gas fraction with an effective density n eff 5.5 × 10 3 cm −3 (T k =20 K). The N 2 H + / 13 CO ratio, which traces the high density gas, shows similar values in GMF54 and nearby galaxies, except in ULIRGs which have a much higher N 2 H + / 13 CO ratio. Considering ULIRGs are forming many stars, it is reasonable that they have a higher fraction of high density gas. It is surprising that GMF54 has a similar or even higher N 2 H + / 13 CO ratio than the starbursts in Table 5. Considering the filling factor of N 2 H + is 10% (with respect to 13 CO), it is likely that the extragalactic studies are underestimating the true N 2 H + / 13 CO ratio.
Another useful line ratio is HCN/HNC, which is proposed as a tracer of evolutionary stages of the molecular cloud because at temperature 30 K HNC starts to convert to HCN (Schilke et al. 1992;Herbst et al. 2000;Graninger et al. 2014;Hacar et al. 2019). Therefore, the HCN/HNC ratio would increase as temperature increases. As we demonstrate in Fig. 3, the HCN/HNC ratio remains relative constant 2 in infrared dark regions, and increases to 3 in the PDR region and around infrared bright HCH ii region. To further demonstrate this, we plotted the HCN/HNC ratio as a function of the dust temperature derived from Hi-GAL (Zucker et al. 2018). A clear correlation between the HCN/HNC ratio and T dust can be seen in   Fig. 9, with a Pearson correlation coefficient of 0.77 and p-value < 0.001. The mean HCN/HNC ratio across the whole GMF54 is also comparable to nearby galaxies.

Uncertainties
Besides the rms noise of the data, the main factors that introduce uncertainties to the column densities and N-PDFs are excitation temperature and abundances. Assuming a distance of 2 kpc, we estimate the mass of the filament with the GRS 13 CO(1-0) to be 2.0×10 4 M , which agrees with the mass derived by Zhang Using the same GRS 13 CO(1-0) data and the same distance of 2 kpc. they estimated the mass of the filament to be ∼1.4 to 4.3×10 4 M . Zhang et al. (2019) studied column density of a sample of GMFs and further discussed in detail the uncertainties brought in by T ex and 13 CO abundance, and they conclude that the 1σ uncertainty of the column density estimated from 13 CO is ∼ 50%. Another uncertainty source for 13 CO(1-0) is optical depth. Since we do not have 12 CO or C 18 O observation with compatible angular resolution, we cannot estimate the 13 CO(1-0) optical depth directly . With a similar angular resolution, Schneider et al. (2016) estimated the 13 CO(1-0) optical depth for the Cygnus X North and found out the optical depth is negligible and does not influence the N-PDF significantly. Riener et al. (2020) estimated the 13 CO(1-0) GMF24, and found a median τ value of ∼ 0.5. Wang et al. (2020) found the GRS 13 CO(1-0) optical depth for another filament GMF38a (l ∼ 33 − 37 • ) is mostly smaller than 1 with an median value of ∼0.4. If we apply this 0.4 to GMF54, the column density increases by ∼21%.
For HCO + (1-0), HCN(1-0), and HNC(1-0), adopting an excitation temperature of T ex =20 K increases the column density HiGal α = 4.06 ± 0.13 Fig. 8. Probability density functions of the molecular hydrogen column density derived from Hi-GAL (Fig. 2). The dashed vertical line marks the column density threshold, and the solid vertical line marks the mean column density. We list in the figure the mean column density N(H 2 ) , the log-normal (green curve) width σ LN , the power-law index α (solid line). Error bars are calculated from Poisson statistics. by ∼50%. We obtain the uncertainties of T ex and τ for N 2 H+ from the HFS fitting, which have a median value of 2 K and 0.5, respectively. Therefore, the column density uncertainty of N 2 H+ column density brought in by HFS fitting is ∼ 50%. The largest uncertainty source for these molecular line column density is the abundance. As studied by Gerner et al. (2014), the typical standard deviation for the abundances of these molecules is ∼50 to 75%. However, from IRDC to ultracompact (UC) H ii region, the abundance can change by an order of magnitude. In summery, the H 2 column density traced by these molecular lines have an uncertainty of a factor of a few (see also, Schneider et al. 2016).