The [NII] 205 micron Emission in Local Luminous Infrared Galaxies

In this paper, we present the measurements of the [NII] 205micron line ([NII]205) for a flux-limited sample of 122 (ultra-)luminous infrared galaxies [(U)LIRGs] and 20 additional normal galaxies, obtained with the Herschel Space Observatory. We explore the far-infrared (FIR) color dependence of the [NII]205 (L[NII]205) to the total infrared (LIR) luminosity ratio, and find that L[NII]205/LIR only depends modestly on the 70-to-160 micron flux density ratio (f70/f160) when f70/f160<~ 0.6, whereas such dependence becomes much steeper for f70/f160>0.6. We also investigate the relation between L[NII]205 and star formation rate (SFR), and show that L[NII]205 has a nearly linear correlation with SFR, albeit the intercept of such relation varies somewhat with f60/f100, consistent with our previous conclusion that \NIIab\ emission can serve as a SFR indicator with an accuracy of ~0.4 dex, or ~0.2 dex if f60/f100 is known independently. Furthermore, together with the ISO measurements of [NII] 122 micron emission we use a total of ~200 galaxies to derive the local [NII]205 luminosity function (LF) by tying it to the known IR LF with a bivariate method. As a practical application, we also compute the local SFR volume density ($\dot{\rho}_{\rm SFR}$) using the newly derived SFR calibrator and LF. The resulting $\log\,\dot{\rho}_{\rm SFR} = -1.96\pm0.11$ $M_\odot$\,yr$^{-1}$\,Mpc$^{-3}$ agrees well with previous studies. Finally, we determine the electron densities ($n_e$) of the ionized medium for a subsample of 12 (U)LIRGs with both [NII]205 and [NII]122 data, and find that $n_e$ is in the range of ~1-100 cm$^{-3}$, with a median value of 22 cm$^{-3}$

cooling the interstellar medium (ISM), and for providing critical diagnostic tools for the study of the star-forming ISM (e.g. Stacey et al. 1991;Lord et al. 1995;Malhotra et al. 2001;Farrah et al. 2013;De Looze et al. 2014;Fischer et al. 2014;Sargsyan et al. 2014). Among these lines, the [C ii] 158 µm emission is probably the most important and well studied, since it is the brightest single line in most galaxies and accounts for 0.1-1% of the total FIR luminosity (e.g. Stacey et al. 1991;Diaz-Santos et al. 2013).
The [N ii] 205 µm line is of particular interest for the following reasons. Firstly, this 3 P 1 → 3 P 0 transition (205.197 µm; hereafter [N ii] 205 µm) of singly ionized nitrogen is expected to be an excellent indicator of star formation rate (SFR) based on the following facts: 1) the ionization potential of nitrogen is only 14.53 eV. Thus the [N ii] 205 µm emission arises only from H ii regions, and essentially traces all warm ionized ISM. It can be utilized to estimate the ionizing photon rate (e.g. Bennett et al. 1994); 2) the [N ii] 205 µm transition can be easily collisionally excited because of its low critical density (44 cm −3 ; Oberst et al. 2006) and excitation energy (∼ 70 K); 3) this emission is usually optically thin and suffers much less dust extinction than optical and near infrared lines. Indeed, Zhao et al. (2013) have shown that the [N ii] 205 µm line can serve as a SFR indicator, which is especially useful for studying high-redshift galaxies for which the redshifted [N ii] 205 µm line is readily obtainable with modern submillimeter telescopes such as the Atacama Large Millimeter/submillimeter Array (ALMA).
Secondly, this line can provide complementary information on the origin of the [C ii] 158 µm emission (e.g. Oberst et al. 2006;Walter et al. 2009;Decarli et al. 2014;Parkin et al. 2013Parkin et al. , 2014Hughes et al. 2015). The [C ii] line can arise from both neutral and ionized gases since it takes only 11.3 eV to form C + , while the ionization potential of hydrogen is 13.60 eV. Therefore, it is important to know what fraction of the observed [C ii] line emission is from the ionized gas for the study of star-forming regions such as photodissociation region (PDR) modeling. Fortunately, the critical densities for the [N ii] 205 µm and [C ii] 158 µm lines in ionized regions are nearly identical (44 and 46 cm −3 at T = 8000 K, respectively; Oberst et al. 2006), and both require similar ionization potentials to further form N ++ (29.6 eV) and C ++ (24.4 eV). As a result, the [C ii]/[N ii] 205 µm line ratio from ionized gas is a function of only the assumed N + /C + abundances within the H ii region, and therefore the observed [C ii]/[N ii] 205 µm line ratio yields the fraction of the [C ii] emission that arises from the ionized gas (Oberst et al. 2006(Oberst et al. , 2011. Thirdly, the ratio of the [N ii] 122 µm to 205 µm lines (hereafter R 122/205 ) is an excellent density probe of lowdensity ionized gas due to their different critical densities (n crit ) required for the collisional excitations and being at the same ionization level. At an electron temperature of 8000 K, n crit are ∼ 293 and 44 cm −3 for the [N ii] 122 and 205 µm lines, respectively (Oberst et al. 2006). Therefore, R 122/205 is sensitive to gas densities of 10 n e 300 cm −3 (Oberst et al. 2006(Oberst et al. , 2011; also see §3. 3).
However, the [N ii] 205 µm line is generally inaccessible to ground-based facilities for local galaxies, and for extragalactic objects, only a handful were observed using satellite and airborne platforms (Petuchowski et al. 1994;Lord et al. 1995) prior to the advent of the Herschel Space Observatory (hereafter Herschel; Pilbratt et al. 2010). These studies have shown that the [N ii] 205 µm line is fairly bright, and the luminosity of [N ii] 205 µm line (L [N ii]205 µm ) may be up to ∼10 −3.5 times the total infrared luminosity (L IR ; 8 − 1000 µm; also see Zhao et al. 2013). With such a high luminosity, this line offers an excellent method for studying SFR and ionized gas properties in galaxies at high redshifts. The advantage of [N ii] 205 µm line over other FIR lines, such as [C ii] 158 µm, [N ii] 122 µm and [O iii] λ 88 µm, is that it starts to fall into atmospheric sub/millimeter windows that have higher transmission at lower-z, due to its longer wavelength. The detectability of the [N ii] 205 µm line and its potential for important astrophysical applications at high-z have already been demonstrated by a few experimental ALMA observing campaigns of a galaxy at z = 4.76 (Nagao et al 2012), and by the IRAM 30m telescope and Plateau de Bure Interferometer detection of distant, strongly lensed galaxies (Combes et al. 2012, z ∼ 5.2;Decarli et al. 2012, 2014, z ∼ 3.9 and4.7).
In Zhao et al. (2013), we reported our first results on the [N ii] 205 µm line emission for an initial set of 70 (Ultra-)Luminous infrared galaxies [(U)LIRGs; L IR ≥ 10 11(12) L ⊙ ] 15 , observed with the Fourier-transform spectrometer (FTS) of the Spectral and Photometric Imag-15 L IR is calculated by using the IRAS four-band fluxes and the equation given in Sanders & Mirabel (1996), i.e., L IR (8 − 1000 µm) = 4πD 2 L f IR , where D L is the luminosity distance, and ing Receiver (SPIRE; Griffin et al. 2010) on board Herschel. In that Letter we focused on the possibility of using the [N ii] 205 µm emission as a SFR indicator.
Here we expand our analysis to our full Herschel sample of 122 LIRGs, which is a flux limited subsample with f IR (8 − 1000 µm) > 6.5 × 10 −13 W m −2 from the Great Observatories All-Sky LIRGs Survey (GOALS; Armus et a. 2009). In addition, we include 20 additional nearby large galaxies for which SPIRE/FTS mapping observations covering the entire galaxy disk are available from the Herschel Science Archive (HSA). In this paper, besides further investigating the relation between L [N ii]205 µm and SFR, we also derive the luminosity function (LF) of the [N ii] 205 µm line and SFR density in the local Universe. It is important to constrain the LF of the [N ii] emission locally since now it becomes possible to build a large sample for studying the [N ii] LF at high-redshift using modern facilities such as ALMA. The local [N ii] LF can serve as a benchmark necessary for observational and theoretical (e.g. Orsi et al. 2014) studies on its evolution. Given the unprecedented sensitivity of Herschel at ∼200 µm, and the large number of galaxies in the local Universe already observed, for the first time we can derive the local [N ii] LF (see §3.2), using a bivariate method and by utilizing the local IR LF which has been studied extensively in the literature with IRAS observations (e.g. Soifer et al. 1986;Sanders et al. 2003).
In addition, we estimate the electron densities for a sub-sample of our (U)LIRGs by comparing the observed R 122/205 with theoretical predications. As shown in Rubin et al. (1994;also see Oberst et al. 2006), R 122/205 varies from 3 for n e ∼ 100 cm −3 to 10 for n e 10 3 cm −3 . Although Petuchowski et al. (1994) and Lord et al. (1995) observed both the [N ii] 205 µm and [N ii] 122 µm lines in M82, the use of R 122/205 for estimating n e was mostly limited to our own Galaxy (e.g. Wright et al. 1991;Bennett et al. 1994;Oberst et al. 2006Oberst et al. , 2011 prior to the advent of Herschel. Furthermore, so far there are only a handful of normal galaxies (e.g. M51 & Cen A: Parkin et al. 2013Parkin et al. , 2014NGC 891: Hughes et al. 2015) for which n e of the low-density gas has been derived using the ratio of these two lines. For (U)LIRGs, it is still unclear what is the typical n e for the low-density ionized gas.
The remainder of this paper is organized as follows: we give a brief introduction of the sample, observations and data reduction in Section 2, present the results and discussion in Section 3, and briefly summarize the main conclusions in the last section. Throughout the paper, we adopt a Hubble constant of H 0 = 70 km s −1 Mpc −1 , Ω M = 0.28, and Ω Λ = 0.72, which are based on the fiveyear WMAP results (Hinshaw et al. 2009), and are the same as those used by the GOALS project (Armus et al. 2009).

(Ultra-)Luminous Infrared Galaxies
The primary sample studied in this paper is from the Herschel open time project Herschel Spectroscopic Survey of Warm Molecular Gas in Local Luminous Infrared Galaxies (OT1 nlu 1; PI: N. Lu). This project aims primarily at studying the dense and warm molecular gas properties of 125 LIRGs (e.g. Lu et al. 2014Lu et al. , 2015, which comprise a flux limited subset of the GOALS sample (Armus et al. 2009). Lu et al. (2015;in preparation) will give the program details and complete set of spectra for individual galaxies. The [N ii] observations were available for 123 targets, one of which is a multiple-source system and our targeted object turned out to be a Galactic source according to its SPIRE/FTS spectrum, consequently excluded from our analysis. Here we present the [N ii] 205 µm data for the 122 galaxies (hereafter GOALS-FTS sample), including 111 LIRGS and 11 ULIRGs. Of these sources, 48 galaxies are point-like sources with respect to the ∼17 ′′ Herschel SPIRE/FTS beam at ∼210 µm, and 74 are extended sources, which were determined according to their flux fractions of the FIR continuum emissions at both 70 and 160 µm observed within the SPIRE/FTS beam (see the following subsection for details).
The observations were conducted with the SPIRE/FTS in its point source spectroscopy mode and high spectral resolution configuration, yielding a spectral resolution of 0.04 cm −1 (or 1.2 GHz) over the spectral coverage of 194-672 µm. The data were reduced using the default version of the standard SPIRE reduction and calibration pipeline for point source mode included in the Herschel Interactive Processing Environment (HIPE; Ott 2010) version 11.0.
In most cases the [N ii] 205 µm line is the brightest line in the SPIRE/FTS wavelength range (Lu et al. 2015, in preparation), and has high signal-to-noise ratios (S/N). As shown in Zhao et al. (2013), the line fluxes were obtained by fitting the observed profile using the instrumental Sinc function convolved with a free-width Gaussian profile. This is because the line width of most (U)LIRGs is 200 km s −1 (e.g., see Arribas et al. (2015) for ionized gas and Ronsenberg et al. (2015) for molecular gas). Given the instrumental resolution (∼300 km s −1 at 210 µm), the observed line might be marginally resolved and could not be modeled by a pure Sinc function. Therefore, we adopted the Sinc-convolved-Gaussian (SCG) profiles for the integrated [N ii] line flux measurements except for a few galaxies where a pure Sinc profile was a better choice due to the intrinsically narrow line width and/or relatively low S/N in the line. During the fitting process, the width of the Sinc function was fixed to be the SPIRE/FTS resolution (1.2 GHz), while the width of the Gaussian function was allowed to vary. The resulting full width at half maximum (FWHM) of the [N ii] 205 µm line, which was obtained from the Gaussian part of the SCG profile, is ∼ 100 − 600 km s −1 , with a median value of ∼ 300 km s −1 . Based on the 1σ statistical uncertainties, the lines in most (> 80%) sources are detected at better than 7σ, with the median at ∼14σ. The measured line fluxes are given in Table 1.

Local Normal Galaxies
Almost all of our sample galaxies are (U)LIRGs, and hence have a rather limited dynamic range of several physical parameters such as luminosity, FIR color, etc. To increase the sample size and dynamic range of our study, we also include in our analysis 20 nearby normal galaxies having Herschel SPIRE/FTS mapping observations that cover the entire galaxy. As in Zhao et al. (2013), we further include another 53 unresolved galaxies (23 detections and 30 upper limits) observed by the Infrared Space Observatory (ISO; Kessler et al. 1996Kessler et al. , 2003 Long Wavelength Spectrometer (LWS), for which the [N ii] 122 µm fluxes were measured by Brauher et al. (2008; hereafter ISO sample).

Herschel SPIRE/FTS Mapping Observations
These observations were carried out by various Herschel projects [e.g.
Very Nearby Galaxies Survey (KPGT cwilso01 1; PI: C. D. Wilson; e.g. Spinoglio et al 2012b;Parkin et al. 2013;Schirm et al. 2014;Hughes et al. 2015) and the Beyond the Peak: Resolved Far-Infrared Spectral Mapping of Nearby Galaxies with SPIRE/FTS (OT1 jsmith01 1; PI: J. D. Smith)]. The level 0.5 raw data were obtained through HSA, and reduced with the default pipeline for mapping observations provided in HIPE 11. We then added up all pixels of the level 2 product which have valid data, and measured the integrated [N ii] 205 µm fluxes using the method described in Section 2.1 above. During this process, we have converted the units of the mapping data from MJy/sr to Jy/pixel using the area of an individual pixel.  · · · · · · · · · · · · · · · · · · · · · · · · Mapping Galaxy Sample NGC1266 03 · · · · · · · · · · · · · · · · · · · · · · · · ISO Sample NGC0520 01 · · · · · · · · · · · · · · · · · · · · · · · · Note. -Columns: (1) galaxy name; (2) and (3)  In Figure 1 we plotted R 122/205 against the IRAS FIR color, f 60 /f 100 . It seems that R 122/205 shows some dependence on f 60 /f 100 . Kewley et al. (2000) found that electron density tends to correlate with f 25 /f 60 . Therefore, the weak dependence of R 122/205 on f 60 /f 100 appears to be understandable. To further check whether there is a correlation between f 60 /f 100 and R 122/205 , we computed the Kendall's τ correlation coefficient using the cenken function in the NADA package within the public domain R statistical software environment 16 . For the whole dataset presented in Figure 1, we have τ = 0.18, with a p-value of 0.13, and thus we do not reject the null hypothesis that these two parameters are uncorrelated at the 0.05 significance level. However, we have τ = 0.35 with a p-value of 0.03 if we limit the sample to f 60 /f 100 < 0.9. Therefore, there exists a weak correlation between f 60 /f 100 and R 122/205 within this color range. Since almost all of the ISO galaxies fall within this color range, we adopt FIR color-dependent R 122/205 . Nevertheless, we caution that such an analysis is possibly limited by the small size of the sample. However, 16 http://www.R-project.org/ this (un)correlation will not affect our main conclusions since the two R 122/205 values adopted in the following only differ by ∼0.3, which is negligible compared to the overall uncertainties.

ISO LWS Observations
For sources with f 60 /f 100 < 0.7, we adopt R 122/205 = 1.2 ± 0.5, and for the other warmer galaxies (with f 60 /f 100 < 1.0), R 122/205 = 1.5 ± 0.7. These adopted R 122/205 values are the median of the corresponding detections, and are lower than the single value of 2.6 adopted in Zhao et al. (2013). The latter was based on the theoretical prediction for an electron density of n e = 80 cm −3 , i.e. the median value of H ii regions in late type galaxies (Ho et al. 1997). However, the adopted n e in Zhao et al. (2013) was measured from H ii regions in the centers of nearby galaxies, and thus might be an overestimate of the mean n e for the entire galaxy. As a result, the overall Zhao et al. (2013) was somewhat underestimated.
For these nearby galaxies, the redshift-independent distance was adopted if a direct primary measurement distance could be found in the NED 17 database, otherwise it was derived with the same method as used for our (U)LIRG sample (e.g. Armus et al. 2009), i.e. by correcting the heliocentric velocity for the 3-attractor flow model of Mould et al. (2000).

Aperture Corrections
Around 205 µm the SPIRE/FTS beam can be well represented by a symmetrical Gaussian profile with a FWHM of 17 ′′ (Makiwa et al. 2013;Swinyard et al. 2014). However, this beam can not fully cover the entire [N ii] emission region in most of our targets assuming their FIR sizes indicate the extent of the [N ii] emission. To define a source as "extended" compared to the SPIRE/FTS beam, we calculated the fractional 70 and 160 µm fluxes within a Gaussian beam of FWHM of 17 ′′ (see below). An extended source will have both fractions less than 90%. Based on this definition, 74 galaxies are classified as extended. Therefore, to achieve our ultimate goals of deriving the LF of the [N ii] 205 µm emission, as well as of further exploring the applicability of L [N ii]205 µm as a SFR indicator, we need to apply an aperture correction to the observed [N ii] 205 µm fluxes for most sources. Zhao et al. (2013) found that L [N ii]205 µm correlates almost linearly with L IR . Hence, the aperture correction can be done by utilizing PACS photometry images and estimating the L IR measured within the region outside the SPIRE/FTS beam. However, as already shown in Zhao et al. (2013), the L [N ii]205 µm /L IR ratio also depends somewhat on the FIR color. To account for this dependence and to minimize the uncertainty in the final, total L [N ii]205 µm , we used the FIR color-dependent L [N ii]205 µm -L IR relation (see below) to correct the L [N ii]205 µm which was measured directly from the SPIRE/FTS spectra.
To measure the L IR and FIR color within the 17 ′′ SPIRE/FTS beam near 205 µm for our sample of (U)LIRGs, we applied the following steps. Firstly, in order to have the same resolution as the SPIRE/FTS, we convolved the 70 and 160 µm images (e.g. Chu et al. 2015, in preparation), which were obtained with the Photodetector Array Camera and Spectrometer (PACS; Poglitsch et al. 2010), with kernels computed through the algorithm described in Aniano et al. (2011). These convolution kernels were generated by comparing the PACS PSFs at 70 µm and 160 µm with a Gaussian profile of FWHM of 17 ′′ (as a representative of the SPIRE/FTS beam around 210 µm). Then we converted the units of the convolved images from Jy/arcsec 2 to Jy/beam by multiplying the area of the Gaussian profile. The fluxes within the SPIRE/FTS beam at 70 and 160 µm (hereafter f 70, beam and f 160, beam , respectively) were measured from the convolved PACS images at the SPIRE/FTS pointing position. The total fluxes (f 70, tot and f 160, tot ), were also measured from the convolved images by doing aperture photometry. Therefore, the fluxes outside the SPIRE/FTS beam are f 70, out = f 70, tot − f 70, beam and f 160, out = f 160, tot − f 160, beam , for the 70 and 160 µm respectively. Note that in this subsection the IR luminosity (L IR, PACS ) is calculated using the f 70 and f 160 fluxes and the formula (L IR, PACS = 1.010νL(70 µm) + 1.218νL(160 µm)) presented in Galametz et al. (2013) since the PACS data have much higher angular resolution than the IRAS data, which is necessary for our purpose. However, the L IR used for the remainder of our analysis is derived from the IRAS four-band fluxes and the well known equation given in Sanders & Mirabel (1996).
Since the galaxies in our sample are (U)LIRGs, and our SPIRE observations usually were targeted at the center of each object, the measured FIR color within the beam is very warm, whereas the part missed by the SPIRE beam, which needs to be corrected for, is gen-erally much colder. This is illustrated in Figure 2a, in which we plotted the distributions of the FIR color inside ((f 70 /f 160 ) beam ; dotted histogram) and outside ((f 70 /f 160 ) out ; solid histogram) the SPIRE/FTS beam. We can see that (f 70 /f 160 ) beam and (f 70 /f 160 ) out peak at ∼1.41 and ∼0.45 respectively. Therefore, it is necessary to include more fiduciary data with cooler FIR colors to better establish the L [N ii]205 µm /L IR -FIR color relationship.
For this purpose, we include in our analysis a dozen nearby, spatially resolved galaxies, which have SPIRE/FTS mapping observations in the HSA and are mainly from the same projects listed in section 2.2.1. These SPIRE/FTS observations (3 of them are in the sample of the 20 galaxies mentioned in §2.2) were reduced with the same method as described in §2.2.1. The PACS imaging data of these nearby, very extended galaxies were reduced using the Scanamorphos technique (Roussel 2013) provided in HIPE 12.1, and then were convolved from their native resolutions to the 17 ′′ resolution of SPIRE at ∼210 µm using the same method as for our GOALS-FTS sample. The convolved images were rebinned to maps with pixel size corresponding to the SPIRE/FTS mapping observations. In order to increase the S/N for the SPIRE/FTS mapping observations, we stacked spectra from regions having similar f 70 /f 160 colors. Also, to reduce the uncertainties in the stacked spectrum and IR flux for each color bin, only pixels with S/N > 3 both at 70 and 160 µm were used. The [N ii] 205 µm fluxes of the stacked spectrum from these galaxies were measured using the same method as for the GOALS-FTS sample described in §2.1 above.
The final L [N ii]205 µm /L IR, PACS -FIR color relation is shown in Figure 3, where L [N ii]205 µm , L IR, PACS and FIR color were measured within the SPIRE/FTS beam for all of our GOALS-FTS (U)LIRGs, and for other galaxies these were measured within the stacked spaxels. This relation is rather flat for log(f 70 /f 160 ) −0.2 (equivalent to f 60 /f 100 0.46; after Dale et al. 2001), but becomes much steeper when the FIR color is getting warmer, and has the largest scatter at the warmest end. To investigate this relation, we used the Kaplan-Meier estimate (Kaplan & Meier 1958)  We also fitted the relation estimated by locfit.censor (i.e. the solid line in Figure 3) with a third-order polynomial function because an analytical form could be more convenient for future studies. The best-fit gives with x = log (f 70 /f 160 ), and a scatter of 0.01 dex, and it is plotted as a dash line in Figure 3. Please note this relation is only valid for the color range we have inves-18 Implemented in the locfit.censor function in the locfit package in R To further examine whether the color-dependent aperture correction is essential, we also plotted the L IR, PACS /L IR, PACS, beam distribution (dotted line) in Figure 2b.
The median values of f corr and L IR, PACS /L IR, PACS, beam are 1.66 and 1.44, respectively, which indicates that the overall L [N ii]205 µm would be underestimated by about 12% if we used a constant L [N ii]205 µm /L IR ratio to do the aperture correction. This is insignificant compared to the scatter of the L [N ii]205 µm /L IR -FIR color relation. However, the underestimation will reach 30% for sources with log(f 70 /f 160 ) out > −0.2. Therefore, it still worth using a color-dependent L [N ii]205 µm /L IR relation to do the aperture correction since the underestimation is systematic. Here L IR was calculated with the IRAS four-band data. Without taking into account the dependence of SFR on the assumed initial mass function, the uncertainty in SFR from this composite calibrator is dominated by the uncertainty in the coefficient of the first term on the right side of the equation. Here we adopted an uncertainty of 40% (e.g. Kennicutt & Evans 2012), and it was propagated to the final SFR after taking a quadratic sum of all errors. For the GOALS-FTS sample, the FUV data were adopted from Howell et al. (2010), while for the normal galaxy and ISO samples, the FUV data were compiled from the literature (mainly from, e.g. Dale et al. 2007;Gil de Paz et al. 2007) and the GALEX data release GR7 19 . Since our SFR estimate relies on the availability of UV observations, we restricted our sample to 121 galaxies with available UV photometric data (hereafter SFR sample).
Before further analysis, however, it is instructive to check whether our dataset is capable of exploring a relation between L [N ii]205 µm and SFR. This is due to the fact that (1) for the GOALS-FTS sample, the aperture correction is essential the conversion of L IR into L [N ii]205 µm ; (2) The SFRs for the GOALS-FTS galaxies are dominated by L IR . Therefore, L [N ii]205 µm and SFR may artificially correlate with each other even if they do not have an intrinsic relationship. As shown in Figures 4a  and 4b, sources with f corr ≤ 1.5 and f corr > 1.5 for our GOALS-FTS sample reside in similar phase space of L [N ii]205 µm and L IR . In addition, very extended sources with f corr > 2.0 only occupy a small fraction (∼17%) of the SFR sample. Therefore, we conclude that our dataset will not artificially make a correlation between L [N ii]205 µm and SFR.
In Figure 5, we plot SFR against L [N ii]205 µm for both (U)LIRGs and normal galaxies. Squares represent the (U)LIRGs in the GOALS-FTS sample. Circles show normal galaxies observed by Herschel, whereas diamonds are the ISO sources from Brauher et al. (2008). The solid symbols in Figure 5 indicate that the fractional contribution from a possible active galactic nucleus (AGN) to the bolometric luminosity, f AGN , is greater than 0.35. Here From Figure 5 we can see that the scatter in L [N ii]205 µm −SFR relation becomes larger with the increase of SFR, consistent with Zhao et al. (2013). In this figure we demonstrate that the increase in scatter is traced to the individual galaxy colors (f 60 /f 100 ). To isolate the color-dependence (and thus reduce the scatter) of the L [N ii]205 µm −SFR relation, we divide our sample galaxies into three sub-samples according to their f 60 /f 100 , i.e. a "cold" one with 0.2 ≤ f 60 /f 100 < 0.6 (i.e. a blackbody temperature of 30 T 50 K); a "warm" one with 0.6 ≤ f 60 /f 100 < 0.9 (50 T 60 K), and a "hot" one with 0.9 ≤ f 60 /f 100 < 1.4 (60 T 90 K). These color bins were chosen according to the FIR color distribution (three peaks in Figure 4c) of our sample galaxies. Additional considerations for the separation of cold and warm/hot samples are that (1) Starburst galaxies usually have f 60 /f 100 > 0.55 (Buat & Burgarella 1998); (2) The turnover of the L [N ii]205 µm /L IR -f 60 /f 100 relation happens at f 60 /f 100 ∼ 0.5.
To investigate the relationship between L [N ii]205 µm and SFR, we fitted each sub-sample using a least-squares, geometrical mean functional relationship (Isobe et al. (2) to all galaxies except those having f AGN > 0.35. Using the same method, we also fitted the whole galaxy sample. To take into account the uncertainties both in L [N ii]205 µm and SFR, we used two independent approaches, which allow us to evaluate their reliabilities, to estimate the final coefficients (a, b) and associated errors in equation 2. The first method (M1) was carried out by using a Monte Carlo simulation described as follows. Firstly, we generated a simulated sample by assuming a Gaussian error using the measured data points and uncertainties. Secondly, we fitted this sample using equation 2 and the geometrical mean method. Thirdly, we repeated the previous two steps 10000 times. The distributions of the fitted results from this process are shown in Figure 6. The second method (M2) is that the observed data points were fitted by using a weighted least-squares, geometrical mean regression. The weighting is defined after Williams et al. (2010) and σ 2 SFR are the errors in L [N ii]205 µm and SFR, respectively. Table 2 lists the number of objects in each sample, the fitting coefficients, 1σ errors and scatters from both methods, for the L [N ii]205 µm −SFR relation. From the table we can see that M1 and M2 give consistent results. Hence, we only discuss the results from M2 hereafter. In Table 2 we also show the Spearman's rank correlation coefficient (ρ, assessing how well an arbitrary monotonic function could describe the relationship between two variables), and the level of significance (Sig), which was computed from the p-value using a Student's t distribution. These ρ and Sig indicate that there exists a very strong correlation between L [N ii]205 µm and SFR.
On The nearly linear relation between L [N ii]205 µm and SFR indicates that the power source of the [N ii] 205 µm emission may be related to the details of the star formation processes that take place in each galaxy. Given such a strong correlation, and to reduce any systemic uncertainties caused by the sample itself (such as sample size, dynamic range, etc), we also fitted the L [N ii]205 µm −SFR relation with a fixed slope of 1. These results are also given in Table 2, and plotted in Figure 5 as a dashed line for each sample. The reduced χ 2 from both fixed and varying slopes, as listed in 2, agree with each other within < 15%, and thus the fitted results with a fixed slope of 1 are recommended to be used for computing SFRs.
Are the fitted relations sensitive to R 122/205 for our (sub-)samples? To further check this, we fitted the L [N ii]205 µm −SFR relations for the (sub-)samples by excluding the ISO galaxies and using the method M2. We found that the resultant slopes and intercepts only have tiny changes, as shown in Table 2. Therefore, we conclude that our results are not affected substantially by including the ISO galaxies. Table 2 shows that the scatter of each sub-sample is a factor of ∼1.5 smaller compared to that of the full sample. This suggests that the color-dependence of the [N ii] 205 µm emission contributes significantly to the total scatter of the L [N ii]205 µm −SFR relation. This is further confirmed by the following two checks: (1) A principal component analysis indicates that the FIR color accounts for 41% of the total variance of the entire sample;

The Scatter in the L [N ii]205 µm −SFR Relation
(2) We simply normalized the L [N ii]205 µm by the galaxy FIR color, i.e., log L [N ii]205 µm, norm = log L [N ii]205 µm + log (f 60 /f 100 ), and then fitted the log L [N ii]205 µm, norm − SFR relations with method M2 (varying slope) and calculated the scatters, which are 0.15, 0.23, 0.20 and 0.26 dex for the "Cold", "Warm", "Hot" and "All" samples, respectively. For the former three samples, these values  (Rubin 1985) since the levels emit these two lines have their critical densities differing by a factor of >10. Therefore, here we used the [O iii] 88 µm/[N ii] 122 µm ratio, which is insensitive to density, as a hardness indicator (Ferkinhoff et al. 2011) to further check the hardness effect.
However, we note that the [O iii] 88 µm/[N ii] 122 µm ratio is only a good hardness indicator for a fixed

U .
It is correlated with U at a given hardness (Cormier et al. 2015). Therefore, for the sample hav-   (1): Sample defined according to their f60/f100, i.e. 0.2 ≤ f60/f100 < 0.6, 0.6 ≤ f60/f100 < 0.9, and 0.9 ≤ f60/f100 < 1.4 for "Cold", "Warm" and "Hot", respectively. "All" represents that the sample includes all galaxies. Col. (2): Number of sources in each sample. Cols. (3) and (4) 8) and (9): The Spearman's rank correlation coefficient (ρ) and the level of significance (Sig; computed from the p-value using a Student's t distribution, which approaches the normal distribution as the sample size increases), respectively. Col. (10): Whether the ISO galaxies were included in the sample. None of the ISO galaxies has 0.9 ≤ f60/f100 < 1.4. The bold rows show the results that we used to calculate the 3σ range of the slope and discussed in more details in the main text.
scatter of the OIII sample is reduced from 0.28 dex to 0.20 dex after further correcting for this hardnessdependence. Therefore, the variations of ionization parameter and hardness can be mainly responsible for the scatter in the [N ii] 205 µm-SFR relation. Nevertheless, the OIII sample is small and biased towards higher log (L [N ii]205 µm /SFR) at a given FIR color (see Figure  7a). A larger and more unbiased sample is needed in future studies.

Comparisons with Other FIR Fine-structure Line-based SFR Indicators
Other FIR atomic and ionic fine-structure lines have been studied for their application as a SFR tracer (e.g. Farrah et al. 2013;De Looze et al. 2014). In comparison, our [N ii] 205 µm line-based SFR tracer fares relatively well in terms of the overall uncertainty and the systematic dependence on the FIR color. Farrah et al. (2013)  Many studies (i.e., Malhotra et al. 2001;Brauher et al. 2008;Graciá-Carpio et al. 2011;De Looze et al. 2014;Cormier et al. 2015) have demonstrated that most of the FIR fine-structure lines show some dependence on the dust temperature or FIR color. From ISM modeling the [O i] 63 and 145 µm lines may be the most FIR colorinsensitive lines at a given density, and the line-to-IR continuum luminosity ratios only vary about 0.3-0.5 dex for 0.3 ≤ f 60 /f 100 2.2 (Abel et al. 2009;Cormier et al. 2015). However, De Looze et al. (2014) showed that the ratio between [O i] 63 µm-based and FUV+IRbased SFRs has a span of ∼1.3 dex for f 100 /f 160 ∼ 0.5 − 2.1 (equal to f 60 /f 100 ∼ 0.1 − 1 for a graybody SED assuming a dust emissivity index β = 1.5), and the span is even larger for [C ii]-based SFRs. For the [N ii] 205 µm line, the L [N ii]205 µm /SFR ratios vary about 1.7 dex (see Figure 7a) for 0.2 ≤ f 60 /f 100 ≤ 1.4. Therefore, it seems that the [N ii] 205 µm emission is not more sensitive to the FIR color than other lines, and we can conclude that the [N ii] 205 µm emission is one of the most reliable SFR tracers among these fine-structure lines, in terms of its relatively small uncertainty.   (5) and (4), respectively.
The quantitative correlation between the [N ii] 205 µm line and SFR for the local LIRGs implies that the [N ii] 205 µm can be particularly a useful SFR tracer at high redshifts, because of the following reasons. Firstly, the [N ii] 205 µm emission has an important advantage over the usual SFR calibrator, L IR , for high-redshift galaxies: It might be relatively easier to be detected, since (a) it is resource intensive to construct a FIR SED to measure L IR at high redshift (especially when λ rest ≤ 100 µm, it needs very good weather to observe a source at z = 6 even with ALMA, and gets harder for a lower redshift), but L IR derived with a single-band flux is very uncertain (see the review of Carilli & Walter (2013)), and (b) for sources having very little dust, the IR continuum is very weak, but the FIR fine-structure lines might still be strong (e.g. Decarli et al. 2014;Capak et al. 2015;Ferkinhoff et al. 2015).
Secondly, as already mentioned in §1, the [N ii] 205 µm line has an important advantage over other shorterwavelength FIR lines: it starts to fall into atmospheric sub/millimeter windows that have higher transmission at lower-z. Furthermore, as shown in Lu et al. (2015), the combination of the [C ii] 158 µm [or CO(7-6)] and [N ii] 205 µm emissions can be used to estimate the f 60 /f 100 (see their equations 4, 5 and 6), and hence we are able to obtain a more accurate SFR for those highredshift galaxies if both lines are measured.
However, for the application to high-z galaxies, an issue that should be explored further is whether our SFR indicator depends on metallicity as some sources at higher redshifts might have significantly lower metallicities. Indeed, low-metallicity systems (∼0.25Z⊙) at high-z have already been observed (Capak et al. 2015). In our sample, there is one galaxy, ESO 350-IG 038 (Haro 11), which has a relatively low gaseous metallicity value of ∼0.48 Z ⊙ (Guseva et al. 2012)  , where the SFR was calculated using the FIR and UV fluxes adopted from Leitherer et al. (2011). These two SFR/L [N ii]205 µm ratios are both within 1σ of the fitted relation at their colors f 60 /f 100 > 0.9. Hence, it seems that the [N ii] 205 µm emission is not very sensitive to metallicity. It might also be simply because the UV radiation field has a stronger influence on the [N ii] emission than metallicity does, given the fact that a lower metallicity will result in a stronger UV radiation field (e.g. Binney & Merrifield 1998).
In addition, Cormier et al. (2015) found that the average L [N ii]122 µm /L IR ratios for their dwarf galaxy sample with 7.0 < 12 + (O/H) < 8.5 and for the metal rich sample in Brauher et al. (2008) only differ by a factor of ∼2, which is very small compared to the effects on the [O iii] 88 µm line. Hence, the [N ii] 205 µm line is not expected to strongly depend on metallicity down to Z = 1/15Z ⊙ if we assume that [N ii] 205 µm/[N ii] 122 µm is not a strong function of metallicity, which is reasonable. Therefore, the [N ii] 205 µm emission is one of the FIR tracers least affected by metallicity effects, and can serve as a useful SFR indicator for high-redshift galaxies.

Local Luminosity Function and Star Formation
Rate Density 3.2.1. Local LF of the [N ii] 205 µm Emission As shown in §3.1, the [N ii] emission can be used as a SFR calibrator, which is particularly useful for highredshift galaxies observable with submillimeter spectroscopic facilities such as ALMA. Therefore, besides its own evolution, studying the [N ii] LFs at different redshifts can probe the evolution of SFR density. In the following we derive the local [N ii] LF using the data ob-  (solid), which was used to construct the local infrared LF in Sanders et al. (2003), and our sample (dashed), which is used to derive the local LF of the [N ii] 205 µm emission. In the right panel, the labels give the L IR bins, and the number of sources in each bin (in the format of N=our sample, RBG sample). Only sources with log L IR < 11 L ⊙ are shown in the plots. Although our GOALS-FTS sample is IR flux-limited, it is not a [N ii] flux-limited sample. Also, we have added more normal galaxies into the study to improve the extent of our GOALS-FTS sample. Therefore, considering the breadth of our sample, we are able to use the socalled bivariate method (see, e.g., Xu et al. 1998) Figure 3 , the L [N ii]205 µm /L IR ratio is dependent on the FIR color. Therefore, we further checked whether ours can be a representative sample of the IRAS RBGS. To this end, we plot the fractional distribution of L IR , and the cumulative fraction of f 60 /f 100 in three L IR bins, in the left and right panels of Figure 8 , respectively. Due to the fact that our GOALS-FTS sample is complete and unbiased for galaxies of L IR > 10 11 L ⊙ , we plot only the sources with log L IR < 11 L ⊙ in Figure 8.
In Figure 8 we can see that, in general, our sources have similar FIR color distributions to the IRAS RBGS galaxies, albeit the discrepancy is relatively larger in the low luminosity bins, which could be attributed to the small number statistics involved. Using the ad.test task in the kSamples package within R, we performed an Anderson-Darling test (Adderson & Darling 1952) to these two samples, and got a p-value of 0.17. Therefore, we do not reject the null hypothesis that these two samples arise from a common distribution. Although the discrepancy between the cumulative fractions of the f 60 /f 100 distribution for the least luminous L IR bin with f 60 /f 100 <∼ 0.6 appears large, the L [N ii]205 µm /L IR ratio is much less sensitive to the FIR color for these sources (see Figure 3). Nevertheless, one should keep this caveat in mind when using the local [N ii] LF derived in the current work.
The algorithm and the formulation used in this work are the same as those presented in §3 of Xu et al. (1998) 21 , which were based on the Kaplan-Meier estimator (Kaplan & Meier 1958) and therefore can take into account the information contained in upper limits. Here we adopted the local infrared bolometric (i.e. 21 The correct versions of equations (14) and (15) of Xu et al. (1998) and Var [log(L 15 µm L IR (8 − 1000 µm)) LF derived by Sanders et al. (2003).
In Table 3 we list the derived local [N ii] LF with bin width δ log (L [N ii] ) = 0.5 L ⊙ , and the associated uncertainties. In Figure 9 we plot this new local [N ii] LF, as well as the best fitted Schechter (1976) function, i.e., where L is the galaxy luminosity, and φ(L)dL is the number density of galaxies in luminosity range L + dL.
The parameters, L * , φ * and α, determined from the fit, describe the characteristic luminosity, the normalization factor at L * , and the slope of the LF at faint luminosities, respectively. The fitted parameters are given in Table 3.
As described earlier, we included the ISO sources into our analysis to make our sample larger and more representative in the low L IR bins. To check how our results will change, we also derived the [N ii] 205 µm LF by excluding the ISO galaxies (see Table 3), and found that the differences are pretty large (∼0.2 − 0.5 dex) for log (L [N ii] /L ⊙ ) < 6.5. This is because only 13 galaxies with L IR < 10 10.5 L ⊙ have [N ii] 205 µm data. Furthermore, L IR of these sources do not cover the full range of the RBG sample. However, the inclusion of the ISO galaxies in our analysis is not unreasonable since the uncertainty of ∼0.2 dex in the resultant L [N ii]205 µm (converted from L [N ii]122 µm ) is much less than the luminosity bin we used to derived the LF. Nevertheless, we caution the reader to keep this caveat in mind when interpreting/using our results.

Local Star Formation Rate Density
It appears that the Schechter function provides a good fit to our data, with a reduced Chi-square of χ 2 = 0.31 (see also Figure 9). We can further use this LF to compute the volume-averaged SFR density (SFRD) for the local Universe, by integrating the fitted Schechter function and multiplying by the SFR calibration provided in §3.1, i.e.
where the adopted L [N ii]205 µm −SFR relation is based on the total sample with a fixed-slope (see the last line in Table 2). The quoted uncertainty inρ SFR is about 0.11 dex when only the fitted errors of the local LF are taken into account. However, in the worst case scenario, i.e., the full amount of uncertainty in the SFR calibrator propagates to the final error, the resulting uncertainty inρ SFR could be as large as 0.37 dex.
The local SFRD obtained here is in good agreement with those of Gallego et al. (1995) and Yun et al. (2001), which give logρ SFR of −2.02 ± 0.2 and −1.86 ± 0.14 M ⊙ yr −1 Mpc −3 , respectively. These values were derived by assuming the same cosmological model as in this paper, and using the Kennicutt  Our results provide a local benchmark for studying the evolution of the [N ii] 205 µm emission andρ SFR . Here we further predict the [N ii] 205 µm LFs at higher redshifts using the method described below. We will also compare our results with those from simulations given by Orsi et al. (2014), and those derived with the method presented in Spinoglio et al. (2012a).
While nearby (U)LIRGs show a [N ii] deficit (relative to IR continuum emission) as compared to local normal galaxies (e.g. Graciá-Carpio et al. (2011), observations at high-z indicate that not all of the highredshift (U)LIRGs show this similar deficit (Ferkinhoff et al. 2011(Ferkinhoff et al. , 2015Combes et al. 2012;Decarli et al. 2012Decarli et al. , 2014Nagao et al. 2012). This is expected if these (U)LIRGs at high redshifts have lower FIR colors than the local galaxies of comparable luminosities. Indeed, Casey et al. (2014 and references therein) shows that, for a fixed L IR , the average dust temperature of a galaxy decreases as redshift increases. This implies that the "turning" point in Figure 3 might occur at a higher characteristic L IR if the trend shown in that figure remains independent of redshift. Inspired by these observations, here we have derived the [N ii] 205 µm LFs at different redshifts from the IR LFs given by Gruppioni et al. (2013), by making the following three assumptions: 1. The L [N ii]205 µm /L IR ratio has the same T dustdependence (cf. Eq. (1)) at any redshift; 2. The L IR −λ peak, dust relation has the same power law (i.e., λ peak, dust = KL −0.06

IR
, where λ peak, dust is the SED peak wavelength and is inversely proportional to T dust ) at any redshift, but with the scaling factor K depending on redshift (Casey et al. 2014); 3. In order to determine the scaling factor K in (2), we used a fixed T dust at L IR = L * , where L * is the characteristic luminosity in the luminosity function defined by Gruppioni et al. (2013) and is an increasing function of redshift. Here we related T dust to λ peak, dust using a greybody with β = 1.5 and τ = 1 at 200 µm (e.g. Casey 2012). Generally, the measured T dust at L * for z = 0 and higher redshifts (Casey et al. 2014) are consistent with our assumption here.
The LFs predicted by our model calculation at various redshifts are shown in Figure 9 with solid colored lines. Our measured [N ii] 205 µm LF (z = 0; solid black line) from Table 3 generally agrees (within 1 − 3σ) with the one at 0 < z < 0.3 predicted by our method. The difference may be caused by different IR LFs (Sanders et al. LF vs Gruppioni et al. 2013 LF), and/or methods employed. For comparison, we also plotted the results  difference is expected since our L [N ii]205 µm /L IR ratios for warmer (and more luminous) galaxies are smaller than Spinoglio et al. (2012a).
In Figure 9, the dashed lines show the model predictions for z = 0 − 5 from Orsi et al. (2014) with n e = 10 cm −3 , which were obtained by using a semianalytical model of galaxy formation and photoionization code. Except for one or two points, their model predicted LF at z = 0 is consistent with our results within 1 − 3σ. However, the Orsi et al. (2014) LF at z = 0 is systematically larger than ours. The difference might be reduced by adopting a larger n e and/or varying n e . From the figure we can see that, for higher redshifts, although the Orsi et al. (2014) results in the low-L [N ii]205 µm bins are much different from our predicted LFs, they show better agreement in the high-luminosity bins. The most probable reason for this disagreement is that too few points were used to construct the IR LFs at higher redshifts (Gruppioni et al. 2013), which could result in large uncertainties in the IR ([N ii] 205 µm) LFs in the low-luminosity bins.

Electron Density of the Ionized Gas in (U)LIRGs
In Figure 10, the solid curve is the theoretically expected R 122/205 as a function of electron density. This was derived by assuming collisional excitation by electrons from the ground-state levels of N + , and using the collision strengths calculated with the fitted results (Draine 2011b) of Hudson & Bell (2005) with T = 8000 K. For galaxies in our GOALS-FTS sample, sixteen have [N ii] 122 µm data (12 detections and 4 upper limits), as listed in Table 2. Among these 16 galaxies, Mrk 231 has PACS observations and the flux has been measured by Fischer et al. (2010), whereas the other sources were ob-served by ISO and the fluxes are adopted from Brauher et al. (2008). To derive n e , we match our observed R 122/205 (circles in Figure 10) to the theoretically expected curve. The derived n e ranges from < 1 to tens cm −3 (also see Table 4), and has a median value of 22 cm −3 (corresponding R 122/205 = 1.27) for the 12 galaxies having [N ii] 122 µm detections. This median R 122/205 for our sample is a bit larger than those of the Milky Way (0.9; Wright et al. 1991;Bennett et al. 1994) and the nearby spiral galaxy M51 (0.8 − 0.9; Parkin et al. 2013) as well as Cen A (0.8; Parkin et al. 2014), which might suggest that the density of the ionized medium traced by the [N ii] emission is higher in these (U)LIRGs. However, more data are needed to reach a solid conclusion since the current sample is too small and not representative of the (U)LIRG population.
Our results indicate that the [N ii] 205 µm emission is dominated by the low-density ionized gas even in (U)LIRGs. This might be due to the fact that, for dusty H ii regions, the ionization parameter is so high in the high-density regime (U is almost proportional to density; see, e.g. Draine 2011a), such as (Ultra-)compact H ii regions, that (1) the UV photons are preferentially absorbed by dust (Voit 1992;Bottorff et al. 1998;Draine 2011a), and (2) Most N + become N ++ ; producing much less [N ii] emissions (e.g. Abel et a. 2009;Fischer et al. 2014). Hence, most of the [N ii] emissions should come from the low-density surrounding halos of these dense H ii regions in (U)LIRGs. Indeed, high angular resolution ALMA observations of (U)LIRGs have revealed that compact regions of high gas density (∼ 10 4 cm −3 ) are surrounded by more extended low density regions . This scenario is also consistent with our finding that sources with warmer f 60 /f 100 colors gener-  Brauher et al. (2008). The data for Mrk 231 are adopted from Fischer et al. (2010). b The quoted error is obtained from the Monte Carlo simulation by assuming a Gaussian noise distribution.
c Aperture corrections have been applied. d The observed ratios are smaller than (or approach) the theoretical lower limit. ally have lower L [N ii]205 µm -to-SFR ratios.

SUMMARY
We present Herschel SPIRE/FTS observations of the [N ii] 205 µm emission for an IR flux-limited sample of 122 (U)LIRGs. Combining them with SPIRE/FTS mapping observations of nearby normal galaxies, along with GALEX UV observations, PACS and IRAS photometry, and ISO [N ii] 122 µm spectra, we have studied the dependence of the L [N ii]205 µm /L IR ratio on the FIR color and L [N ii]205 µm -SFR correlation. We also derived the [N ii] 205 µm luminosity function and SFR density for the local Universe, as well as investigated the electron densities for a sub-sample of (U)LIRGs. Our main results are summarized in the following. is from <∼ 1 cm −3 to ∼66 cm −3 , with a median value of 22 cm −3 .