Comparative testing of dark matter models with 15 HSB and 15 LSB galaxies
^{1} Physics Institute, University of Szeged, Dóm tér 9, 6720 Szeged, Hungary
email: kun@titan.physx.uszeged.hu
^{2} Department of Computing Science, Umeå University, 901 87 Umeå, Sweden
^{3} Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary
Received: 8 April 2016
Accepted: 26 July 2017
Context. We assemble a database of 15 high surface brightness (HSB) and 15 low surface brightness (LSB) galaxies, for which surface brightness density and spectroscopic rotation curve data are both available and representative for various morphologies. We use this dataset to test the NavarroFrenkWhite, the Einasto, and the pseudoisothermal sphere dark matter models.
Aims. We investigate the compatibility of the pure baryonic model and baryonic plus one of the three dark matter models with observations on the assembled galaxy database. When a dark matter component improves the fit with the spectroscopic rotational curve, we rank the models according to the goodness of fit to the datasets.
Methods. We constructed the spatial luminosity density of the baryonic component based on the surface brightness profile of the galaxies. We estimated the masstolight (M/L) ratio of the stellar component through a previously proposed color–masstolight ratio relation (CMLR), which yields stellar masses independent of the photometric band. We assumed an axissymetric baryonic mass model with variable axis ratios together with one of the three dark matter models to provide the theoretical rotational velocity curves, and we compared them with the dataset. In a second attempt, we addressed the question whether the dark component could be replaced by a pure baryonic model with fitted M/L ratios, varied over ranges consistent with CMLR relations derived from the available stellar population models. We employed the Akaike information criterion to establish the performance of the bestfit models.
Results. For 7 galaxies (2 HSB and 5 LSB), neither model fits the dataset within the 1σ confidence level. For the other 23 cases, one of the models with dark matter explains the rotation curve data best. According to the Akaike information criterion, the pseudoisothermal sphere emerges as most favored in 14 cases, followed by the NavarroFrenkWhite (6 cases) and the Einasto (3 cases) dark matter models. We find that the pure baryonic model with fitted M/L ratios falls within the 1σ confidence level for 10 HSB and 2 LSB galaxies, at the price of growing the M/Ls on average by a factor of two, but the fits are inferior compared to the bestfitting dark matter model.
Key words: galaxies: halos / galaxies: structure / dark matter
© ESO, 2017
1. Introduction
Recent cosmological data suggest that 26.8 percent of the energy content of the Universe is in the form of dark matter (Planck Collaboration XIII 2016), but no compelling understanding of its nature has been possible so far. Galaxystructure studies provide an efficient test of various dark matter candidates, as dark matter plays a key role on this scale. The understanding of the kinematics of galaxy clusters (Zwicky 1937) and galactic rotation curves (e.g., Rubin et al. 1978, 1985; Mathewson et al. 1992; Prugniel et al. 1998) all require dark matter. When we compare different dark matter models with galactic rotation curves, it is crucial to estimate the mass of the baryonic (luminous) component accurately.
The baryonic mass density of the galaxy can be calculated from the luminosity distribution, assuming a certain masstolight (M/L) ratio. There are five basic techniques employed to estimate this ratio: a) using tabulated relations between color and M/L (e.g., Bell & de Jong 2001), b) modeling broadband photometry (e.g., Sawicki & Yee 1998), c) modeling moderateresolution spectra (e.g., Giallongo et al. 1998), d) the analysis of CMDs in nearby galaxies with resolved stellar populations (e.g. Dalcanton et al. 2012), and e) dynamical modeling via the Jeans equation in earlytype galaxies (e.g., Cappellari et al. 2013) or in intermediate and latetype disks (e.g., Bershady et al. 2010; Martinsson et al. 2013). In this paper we employ the first method.
The accuracy of the color–masstolight relations (CMLR, e.g., Bruzual & Charlot 2003; McGaugh & Schombert 2014) highly depends on the assumed initial mass function (IMF) of the employed stellar population synthesis model, the variations in the star formation histories of galaxies, the distribution of stellar metallicities, and the contribution of stars in bright but shortlived phases of evolution (e.g., TPAGB stars). According to McGaugh & Schombert (2014), the semiempirical stellar population synthesis model of Bell et al. (2003) provides the most identical stellar masses in different photometric bands. They modified the according CMLR relation to achieve the best match of the calculated stellar masses in different photometric bands.
When the Newtonian law of gravity is employed to deduce the rotational curve of the luminous component, the model curve is qualitatively different from the curve emerging from spectroscopic measurements (Doppler shift of the spectral lines). Either gravity is not well understood and needs refinement on galactic scales, or there is an invisible contribution to the mass of the galaxy, which interacts only gravitationally. In this paper we compare the compatibility with galactic rotation curves of three frequently used dark matter models and of the pure baryonic model.
The NavarroFrenkWhite (hereafter NFW) dark matter model is motivated by cold dark matter simulations. Navarro et al. (1997) used highresolution Nbody simulations to study the equilibrium density profiles of dark matter, and found that their halos have the same shape regardless of the halo mass, initial density fluctuations, and cosmological parameters. The NFW model has a divergent central density, and it is cuspy, the density scaling as r^{1} (r being the radial distance).
Einasto (1965) proposed the density ρ ~ exp(−Ar^{α}) for a spherical stellar system, able to model both steep and shallow profiles. The Einasto model is formally similar to Sersic’s law, but it is fit to the space density as compared to the projected surface density for the latter. Merritt et al. (2006) pointed out that Sersic’s law is also an excellent description for Nbody dark matter halos (see references therein).
The pseudoisothermal sphere (hereafter PSE) halo has no cosmological motivation, but it often fits the rotational curves better than NFW (de Blok & Bosma 2002; Kuzio de Naray et al. 2008), as the PSE profile exhibits finite density at the center of the halo (e.g., Chemin et al. 2011).
For each density profile, the rotational velocity of the halo can be fit to the spectroscopically measured curves. We did this for either the pure baryonic model or for the three frequently used dark matter profiles (NFW, Einasto, and PSE). We have chosen 15 high surface brightness (HSB) and 15 low surface brightness (LSB) galaxies for this purpose.
In Sect. 2 we generate the spatial luminosity density of the baryonic component from the projected surface brightness profile of the galaxies. We also summarize the contributions of the baryonic and the dark matter components to the rotational velocity. In Sect. 3 we present the model fit results regarding the spatial luminosity density of the baryonic components. In Sect. 4 we compare the rotational velocity models with the spectroscopic rotational curves. In Sect. 5 we investigate the relevance of the dark matter models. In Sect. 6 we summarize the results.
The ΛCDM cosmological model is adopted throughout the paper, with the Hubble constant H_{0} = 67.8 km s^{1} Mpc^{1} and (baryonic + dark) matter density Ω_{m} = 0.308 (Planck Collaboration XIII 2016).
2. Galactic rotation curves
2.1. Contribution of baryonic matter
The baryonic rotational curves are derived based on the distribution of the luminous matter, which is deduced from the surface brightness of the galaxy.
The surface brightness S is the radiative flux F per solid angle ΔΩ of the image such that S ≈ F/ ΔΩ, which is a function of the redshift, and it is independent of the distance D of the emitting surface in a Friedmann universe. The observed surface brightness S_{obs} in units of L_{⊙}/kpc^{2} can alternatively be expressed as the quantity μ in units of mag/arcsec^{2}: (1)where R is the distance measured from the center of the galaxy in the galaxy plane, and ℳ_{⊙} is the absolute brightness of the Sun in units of mag. We translated μ(R) into S_{obs}(R) using Eq. (1), which is valid in the local Universe (z ≪ 1), having an (1 + z)^{3} factor suppressed.
We followed Tempel & Tenjes (2006) to derive the surface brightness, assuming that the spatial luminosity density distribution of each visible component is given by (2)Here is the central density, where a_{0} characterizes the harmonic mean radius of the respecting component, and k and h are scaling parameters. Furthermore, , where q is the axis ratio, and R and z are cylindrical coordinates. From the measurements the projection of l(a) onto the plane of the sky perpendicular to the line of sight is derived: (3)Here S(R) arises as a sum for n visible components, and we assumed constant axis ratios q_{i}. Equation (3) was fit to the surface brightness of the galaxies μ(R), in order to reveal the spatial luminosity density l(a).
We decomposed the baryonic model into two components, a bulge and a disk. Therefore the mass density is (4)where l_{b}(a) and l_{d}(a) are the spatial luminosity density of the bulge and of the disk, respectively, and σ and τ are the corresponding masstolight (M/L) ratios (both are given in solar units).
It follows from the Poisson equation that for spheroidal shape matter, the rotational velocity in the galactic plane induced by the ith baryonic component is given by (Tamm & Tenjes 2005) (5)where G is the gravitational constant, is the eccentricity of the ith component, and ρ_{i}(a) is its mass density. Then a summation of over all visible components gives the square of the rotational velocity of the baryonic model.
2.2. Contribution of the dark matter
For a spherically symmetric dark matter halo, the rotational velocity square is (6)with the spherical radial coordinate r, and cumulative mass within a sphere of r radius (7)The NFW dark matter density profile is (Navarro et al. 1997) (8)where ρ_{s} and r_{s} are the characteristic density and scale distance. The contribution of this dark matter to the rotational velocity squared at radial distance r is (9)The Einasto dark matter profile is described by (e.g., Merritt et al. 2006) (10)where n is a positive parameter. The term d_{n} is a function of n with the property that ρ_{e} is the density at r_{e} defining a halfmass radius. An empirical relation between d_{n} and n is (Merritt et al. 2006) (11)The total dark matter mass is (12)with the central density ρ_{E,0} = ρ_{e}e^{dn}, complete Gamma function Γ(3n), and (RetanaMontenegro et al. 2012). This dark matter contributes to the rotational velocity squared as (13)with the incomplete Gamma function Γ(3n,s^{1 /n}), and .
The PSE density profile is given by (e.g., Jimenez et al. 2003) (14)where ρ_{P,0} is the central density, and r_{c} scales the size of the core. For this model the contribution to the square of the rotational velocity reads (15)Taking into account both visible and dark matter, the rotational velocity squared in the galactic plane becomes: (16)This was fit to the spectroscopic rotational curves for each of the dark matter models (NFW, Einasto, and PSE).
Bestfit parameters describing the luminosity density distribution of the baryonic matter of 15 HSB spiral and 15 LSB galaxies.
3. Bestfit surface brightness profile of the galaxies
We calculated the S_{obs}(R) of the galaxies from μ(R), given in the literature. The ESO HSB galaxies were imaged in I band (Palunas & Williams 2000). Eight of the 15 LSB galaxies were detected in R band (F5611, UGC1230; van der Hulst et al. 1993), (F5631, F579V1, F5831; de Blok et al. 1996), and (F5683, UGC128, UGC5750; de Blok et al. 1995), and 7 in V band (F730V1, UGC10310, UGC11454, UGC11616, UGC11748, UGC11819, and UGC6614; Kim 2007). The absolute brightness of the Sun (ℳ_{⊙}) was substituted into Eq. (1) according to the observing photometric filter, assuming the values of Binney & Merrifield (1998; Table 2.1) as M_{I, ⊙} = 4.08 mag, M_{R, ⊙} = 4.42 mag, and M_{V, ⊙} = 4.83 mag.
Then by using Eq. (3) and varying the parameters q, l(0), ka_{0}, and N, the spatial luminosity densities were fit to the surface brightness of the galaxies by nonlinear leastsquares fitting methods. We set the initial value of the axial ratio of the components as q = 0.7 for the bulge and q = 0.1 for the disk, which were shown to be the most frequently used values for the nearby galaxies (Tamm & Tenjes 2005). We set a lower limit of 0.4 for the bulge and an upper limit of 0.3 for the disk based on SDSS results (Padilla & Strauss 2008; Rodríguez & Padilla 2013). Using the photometry of the 15 HSB and 15 LSB galaxies, we decomposed the surface brightness profile into bulge and disk components. The fit results are given in Table 1 and shown in Figs. A.4 and A.5. In Table 1 we also list the redshift z of the galaxies from the NASA/IPAC Extragalactic Database, confirming that all galaxies belong to the local Universe. In the case of the LSB galaxies F5631 and F5831, the surface brightness distribution indicates only a bulge component. Knowing the spatial distribution of the luminous matter l(a), we are able to construct the mass model, and consequently the rotational velocity curve of the baryonic component. In the next subsections we present the fitting results with the baryonic matter and three different dark matter density profiles.
4. Bestfit galaxy rotational curves
We proceeded in a similar way for the HSB and LSB galaxies. First we fit the surface brightness distributions with the model described in Sect. 2, inferring the luminosity distribution of the baryonic component. When we fit the dark matter models, we included the baryonic component, with fixed M/L ratios as described in the next subsection. We applied the nonlinear leastsquares method to perform the fit with 1/error^{2} weights, minimizing the residual sum of squares (χ^{2}) between the data and the model.
4.1. Estimated masstolight ratio of the bulge and the disk
Coefficients of the color–masstolight ratios employed in this paper.
Color indices of the chosen galaxies, their gas mass fraction (f_{g}), total luminosity in i photometric band (L_{i}), stellar (M_{⋆}) and gas masses (M_{gas}), and estimated M/L ratios for the bulge (Υ_{b}), and for the disk (Υ_{d}).
To estimate the M/L ratios, we employed colortomasstolight ratio relations (CMLR). The relation between the stellar M/L ratio and the color index is log Υ_{⋆} = α_{i} + β_{i}(m_{i}−m_{j}), where Υ_{⋆} is the stellar M/L ratio, and m_{i}−m_{j} is the color index calculated from i and j photometric bands.
McGaugh & Schombert (2014) combined Spitzer 3.6 μm infrared observations of a sample of disk galaxies with optical luminosities to test four different population synthesis prescriptions for computing stellar mass. In their analysis the bulge and disk were not distinguished. They found that the semiempirical stellar population synthesis model of Bell et al. (2003) is selfconsistent, and the revised CMLR based on the model of Bell et al. (2003) has the least scatter (the selfconsistency satisfies that different photometric bands provide the same stellar mass). This model is one of the two population synthesis models that required the smallest corrections to the Spitzer data. Therefore we calculated Υ_{⋆} considering the revised CMLR of McGaugh & Schombert (2014), which explores the color index B−V. This model was tested by circular arguments and assumes that the scatter in the baryonic TullyFisher relation should be minimized for all galaxies. For the HSB galaxies with known color index B−R, we used the original CMLR of Bell et al. (2003). We summarize the coefficients of these CMLRs in Table 2.
The color indices were collected from the literature as indicated in Table 3. When transforming the SDSS colors to B−V color index, we applied the relation B−V = 0.98(g−r) + 0.22 given by Jester et al. (2005). Where the color indices were not corrected for extinction (AdelmanMcCarthy et al. 2008; Kim 2007), we have done it using Landolt standardfields.
We estimate the mass of the gaseous disk M_{gas} employing the gas mass fraction where M_{⋆} is the stellar mass of the galaxy. There are empirical relations between the gas mass fraction, the brightness, and the color index of the galaxies. McGaugh & de Blok (1997) found that in general, f_{g} computed from Bband and Iband data of spiral galaxies correspond closely. More precisely, for the Iband HSB galaxies f_{g} = 0.12(M_{B} + 23) holds for the gas mass fraction and the absolute magnitude of the galaxy in B band (McGaugh & de Blok 1997). The absolute B magnitudes are derived based on the galaxy distances collected from Palunas & Williams (2000) and their apparent B magnitudes collected from the NASA/IPAC Extragalactic Database (Lauberts & Valentijn 1989). McGaugh & de Blok (1997) gave a relation between f_{g} and the B−V color index, f_{g} = −1.4 [(B−V)−0.95]. We used this equation to calculate f_{g} for the LSB galaxies with available B−V indices: F730V1, UGC10310, UGC11454, UGC11616, UGC11748, and UGC11819. For the other galaxies, f_{g} is available directly from the literature: F5611, F5683, UGC128, UGC1230, UGC6614 (McGaugh & de Blok 1997), F579V1, and UGC5750 (Schombert & McGaugh 2014).
The stellar masstolight ratio Υ_{⋆} was taken as the bulge M/L, Υ_{b}. Then the stellar mass, encompassed by the bulge and the disk, is M_{⋆} = Υ_{⋆}L, where L = L_{b} + L_{d} is the total luminosity of the galaxy, calculated from the bestfit central luminosity density of the bulge (L_{b}) and the disk (L_{d}). The corresponding scaling parameters, k = Γ(2N)/Γ(3N) and h = Γ^{2}(3N)/ [NΓ^{3}(2N)] (Tamm et al. 2012) were calculated based on the bestfit N and ka_{0} of the bulge and disk components. The gas mass M_{gas} was derived based on f_{g} and M_{⋆}. Then the M/L ratio of the disk is Υ_{d} = (M_{⋆} + M_{gas}) /L.
We summarize the photometric informations, M/L ratios, and total luminosities and masses in Table 3. These M/L ratios are assumed constant when fitting the baryonic + dark matter models to the rotation curve data.
4.2. HSB galaxies
We collected the rotational velocity data of the HSB galaxy sample from Palunas & Williams (2000), who presented maximum disk models for a sample of 74 field and cluster spiral galaxies located in the vicinity of the HydraCentaurus cluster. For each galaxy they had Iband CCD images and twodimensional (2D) Hα velocity maps, from which the surface brightness distribution and rotational velocity curve of each galaxy were produced. From this sample 15 galaxies were selected such that these galaxies did not show bars, or rings, which might contradict the assumption of axisymmetry. We summarize the bestfit parameters in Table 4. We present the bestfit rotational velocity models and the galaxy velocity curves in Fig. A.1. We note that the error bars on the individual data points are quite large compared to the scatter in the mean data values, suggesting that the error bars are slightly overestimated.
4.3. LSB galaxies
We have explored a database of LSB galaxies taken from the literature as follows: the smoothed hybrid HαHI rotational velocity curves of the LSB galaxies F5611, F5631, F5683, F5831, F730V1, UGC5750, UGC10310, UGC 11454, UGC11616, UGC11748, UGC11819, and UGC6614 from de Blok et al. (2001), the HI rotational velocity curve of F579V1 from de Blok et al. (1996), the HI rotational velocity curves of UGC128 and UGC1230 from van der Hulst et al. (1993). de Blok et al. (2001) calculated the errors on the smoothed rotational curves as follows. The error bars consist of two components: (1) observational errors that are due to the measurement uncertainties in the individual raw data points, and (2) the differences between approaching and receding sides and noncircular motions. For the final error estimate, these two uncertainties were added quadratically. The original data for the approaching and receding sides are available in McGaugh et al. (2001)^{1}, showing only slight asymmetries. In the other four cases, no errors were published with the data. As the radial distribution of the velocities shows a quite regular pattern, we assumed for them 10 percent errors. We have determined the bestfit parameters for each of the baryonic + NFW, baryonic + Einasto, baryonic + PSE models, and they are listed in Table 5. We present the rotational velocity curves of the bestfit models and the galaxy velocity curves in Fig. A.2.
Parameters describing the bestfit pure baryonic, baryonic + NFW, baryonic + Einasto, and baryonic + PSE models of 15 HSB galaxies.
Parameters describing the bestfit pure baryonic, baryonic + NFW, baryonic + Einasto, and baryonic + PSE models of 15 LSB galaxies.
5. Relevance of dark matter models
5.1. Comparison of dark matter models
Since the dark matter models are not submodels of each other, comparing them by exact statistical tests, such as the likelihood ratio test, is not conclusive. Instead we applied the Akaike information criterion AIC = 2N + χ^{2} (Akaike 1974) that is usually employed in the literature (e.g., Chemin et al. 2011). The AIC is a measure of the fit of a given model to the statistical data based on both the residual sum of squares χ^{2} and the number of parameters N. A lower AIC value represents a better model performance. In Table A.1 the columns AIC_{NFW}, AIC_{E}, and AIC_{P} are the calculated AIC values for the baryonic + NFW, the baryonic + Einasto, and the baryonic + PSE model, respectively, for those galaxies for which at least one model fits the dataset within the 1σ confidence level. The lowest values are marked in boldface.
A model is worse than the bestfit model when the difference Δ of the corresponding AIC values is higher. These differences are presented in the last three columns of Table A.1. We establish the following thresholds: Δ ≤ 2 refers to approximately equal performances, 4 ≤ Δ ≤ 7 represents a measurable difference in the fit of the two models, while Δ > 10 clearly favors one of the models over the other. We stress, however, that the imposition of these limits is somewhat arbitrary. For related considerations see Chapter 2 of Burnham & Anderson (2003).
Bestfit (BF) dark matter models in Col. 2, where N/E/P marks hold for the NavarroFrankWhite/Einasto/pseudoisothermal sphere models, and the estimated M/L ratios of the baryonic + dark matter models (Υ_{b} and Υ_{d} for the bulge and disk, respectively) in Cols. 3 and 4.
For 7 galaxies (2 HSB, and 5 LSB) neither the pure baryonic nor any of the dark matter models fit the dataset within the 1σ confidence level. In order to identify the performances of the bestfit models in the case of the other 23 galaxies, the Δvalues are listed in the last three columns of Table A.1, and their interpretation is summarized in Table A.2. The bestfit PSE model is favored in Σ_{++} + Σ_{+} = 23 cases (14 HSB, and 9 LSB) and cannot be ruled out in either case. The bestfit NFW is favored in Σ_{++} + Σ_{+} = 13 cases (10 HSB, and 3 LSB), and is ruled out in Σ_{−−} = 2 cases (1 HSB, and 1 LSB). The bestfit Einasto model is favored in Σ_{++} + Σ_{+} = 10 cases (3 HSB, and 7 LSB), and is ruled out in Σ_{−−} = 1 cases (1 HSB).
5.2. Pure baryonic model with M/L ratio fit
The rotational curves cannot be explained by the pure baryonic matter with the M/L ratios derived in Sect. 4.1. However, the estimation of M/L depends on the applied stellar population model and the IMF. Previously, we used the selfconsistent model of Bell et al. (2003).
Now we compare the pure baryonic model with fitted M/L to the models that contain baryonic matter with fixed M/L and dark matter. For this purpose we employ lower and upper limits on the M/L ratios based on the nine different models of Bell & de Jong (2001) and the corresponding CMLRs, tabulated for different color indices, and for cosmological redshifts z = 0.008 and z = 0.02. We used the CMLRs evaluated at z = 0.02, as this redshift applies for our galaxy sample. In all cases the CMLR based on the Bruzal & Charlot population synthesis model with a modified Salpeter IMF gave the lower limits, and the CMLR based on the PEGASE model with x = −1.85 IMF gave the upper limits to the M/L ratios. These M/L ratios (separately for the bulge and the disk) are summarized in Table 6, from Cols. 5 to 8. The bestfit values of the M/L ratios are presented in Cols. 9 and 10 of the same table. In Table 6 we also present the M/L ratios of the baryonic+dark matter models (in Cols. 3 and 4) in order to compare the bestfit M/L ratios of the pure baryonic model to them.
Columns 9 and 10 of Table 6 show the bestfit M/L ratios for the bulge and the disk, respectively. In Fig. A.3 we represent the pure baryonic model fits with rotational curve data within the 1σ confidence level. We found goods fits like this for ten HSB and two LSB galaxies. Although the fitting is within 1σ for the HSB galaxies ESO323G25, ESO374G02, ESO445G19, and ESO446G01, the pure baryonic model still does not reproduce the plateau as well as the bestfit dark matter models, which are also indicated in Fig. A.3. This appears most clearly for ESO446G01.
Columns 2 and 6 of Table A.1 show the AIC_{B} and Δvalue (i.e., the difference of AIC_{B} to the bestfit AIC), respectively. The second column of Table A.2 indicates the relevance (based on the AIC) of the pure baryonic model that is favored in Σ_{++} + Σ_{+} = 2 cases (1 HSB, and 1 LSB), which is ruled out in Σ_{−−} = 13 cases (7 HSB, and 6 LSB).
6. Summary and concluding remarks
In this paper we have assembled a database consisting of 15 HSB and 15 LSB galaxies that are representative for the various possible galaxy morphologies of both types. In particular, the HSB galaxy set contained spiral galaxies of various brightness profiles, while the LSB galaxy set contained both disk and dwarf galaxies. For the selected galaxies, both surface brightness density data and spectroscopic rotation curve data were available in the literature.
We explored this dataset for a comparative testing of frequently applied and wellestablished dark matter models (NFW, Einasto, and PSE). We investigated the compatibility of the pure baryonic model and baryonic plus various dark matter models with observations on the galaxy database. The mass distribution of the baryonic component of the galaxies was derived from the spatial luminosity distribution by estimating the M/L ratios through colortomasstolight relations and gas mass fractions. For our analysis we constrained the axial ratio of the galaxies based on SDSS results as 0.4 <q_{b} < 1, and 0 <q_{d} < 0.3.
We calculated the Akaike information criterion to characterize the goodness of the bestfit models. In case of the pure baryonic model, the M/L ratios were varied between reasonable limits in the fitting to the rotation curves, while in case of baryonic + dark matter combined models, the baryonic component was inferred using M/L ratios derived based on the CMLR of Bell et al. (2003) and McGaugh & Schombert (2014). In case of 7 galaxies (2 HSB and 5 LSB), neither model fits the dataset within the 1σ confidence level. According to the Akaike information criterion, the pseudoisothermal sphere emerges as most favored in 14 cases out of the remaining 23 galaxies, followed by the NavarroFrenkWhite (6 cases) and the Einasto (3 cases) dark matter models.
The pure baryonic model with an M/L ratio fit did not provide the best model performance in any of the cases, and it was ruled out in case of 7 HSB and 6 LSB galaxies based on the AIC. On the other hand, the pure baryonic model fit the dataset within the 1σ confidence level in case of 10 HSB and 2 LSB galaxies. From these 12 galaxies, the pure baryonic model provided a poor fit compared to the bestfit DM model in 5 cases, giving Δ > 10. This clearly disfavors the baryonic model.
By crosscorrelating the results of the fits by the two methods, we found that the following seven galaxies could not be described with any of the considered models: ESO322G82, ESO376G02, UGC6614, UGC10310, UGC11454, UGC11616, and UGC11748. The proper modeling of these galaxies needs more sophisticated descriptions. Employing massive datasets of 2D velocity fields from integral field unit surveys, such as SAMI (Allen et al. 2015) and MaNGA in SDSSIV (Bundy et al. 2015), will help to further develop the method we presented and test it for nonaxisymmetry on highquality velocity fields with welldefined errors.
The remaining 23 galaxies from the dataset, which require a dark matter component to be modeled, might be useful in the comparative testing of spherically symmetric dark matter substitutes emerging in either alternative gravity theories or from the study of nonstandard dark matter candidates, like BoseEinstein condensates.
Acknowledgments
The authors are grateful to the anonymous referee for the valuable suggestions provided throughout the refereeing process, which have contributed to significant improvement of the paper. E.K., Z.K., and L.Á.G. acknowledge the support of the Hungarian National Research, Development and Innovation Office (NKFI) in the form of the grant 123996. Z.K. and L.Á.G. were further supported by COST Action CA15117 “Cosmology and Astrophysics Network for Theoretical Advances and Training Actions” (CANTATA).
References
 AdelmanMcCarthy, J. K., Agüeros, M. A., Allam, S. S., et al. 2008, ApJS, 175, 297 [NASA ADS] [CrossRef] [Google Scholar]
 Akaike, H. 1974, IEEE Transactions on Automatic Control, 19, 716 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Allen, J. T., Croom, S. M., Konstantopoulos, I. S., et al. 2015, MNRAS, 446, 1567 [NASA ADS] [CrossRef] [Google Scholar]
 Bell, E. F., & de Jong, R. S. 2001, ApJ, 550, 212 [NASA ADS] [CrossRef] [Google Scholar]
 Bell, E. F., McIntosh, D. H., Katz, N., & Weinberg, M. D. 2003, ApJS, 149, 289 [NASA ADS] [CrossRef] [Google Scholar]
 Bershady, M. A., Verheijen, M. A. W., Swaters, R. A., et al. 2010, ApJ, 716, 198 [NASA ADS] [CrossRef] [Google Scholar]
 Binney, J., & Merrifield, M. 1998, Galactic Astronomy (NJ: Princeton University Press) [Google Scholar]
 Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000 [NASA ADS] [CrossRef] [Google Scholar]
 Bundy, K., Bershady, M. A., Law, D. R., et al. 2015, ApJ, 798, 7 [NASA ADS] [CrossRef] [Google Scholar]
 Burnham, K., & Anderson, D. 2003, Model Selection and Multimodel Inference: A Practical InformationTheoretic Approach (New York: Springer) [Google Scholar]
 Cappellari, M., McDermid, R. M., Alatalo, K., et al. 2013, MNRAS, 432, 1862 [NASA ADS] [CrossRef] [Google Scholar]
 Chemin, L., de Blok, W. J. G., & Mamon, G. A. 2011, AJ, 142, 109 [NASA ADS] [CrossRef] [Google Scholar]
 Dalcanton, J. J., Williams, B. F., Melbourne, J. L., et al. 2012, ApJS, 198, 6 [NASA ADS] [CrossRef] [Google Scholar]
 de Blok, W. J. G., & Bosma, A. 2002, A&A, 385, 816 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 de Blok, W. J. G., van der Hulst, J. M., & Bothun, G. D. 1995, MNRAS, 274, 235 [NASA ADS] [CrossRef] [Google Scholar]
 de Blok, W. J. G., McGaugh, S. S., & van der Hulst, J. M. 1996, MNRAS, 283, 18 [NASA ADS] [CrossRef] [Google Scholar]
 de Blok, W. J. G., McGaugh, S. S., & Rubin, V. C. 2001, AJ, 122, 2396 [NASA ADS] [CrossRef] [Google Scholar]
 de Vaucouleurs, G., de Vaucouleurs, A., Corwin, Jr., H. G., et al. 1992, VizieR Online Data Catalog, 7137 [Google Scholar]
 Einasto, J. 1965, Trudy Astrofizicheskogo Instituta AlmaAta, 5, 87 [NASA ADS] [Google Scholar]
 Giallongo, E., D’Odorico, S., Fontana, A., et al. 1998, AJ, 115, 2169 [NASA ADS] [CrossRef] [Google Scholar]
 Jester, S., Schneider, D. P., Richards, G. T., et al. 2005, AJ, 130, 873 [NASA ADS] [CrossRef] [Google Scholar]
 Jimenez, R., Verde, L., & Oh, S. P. 2003, MNRAS, 339, 243 [NASA ADS] [CrossRef] [Google Scholar]
 Kim, J. H. 2007, Ph.D. Thesis, University of Maryland, College Park [Google Scholar]
 Kuzio de Naray, R., McGaugh, S. S., & de Blok, W. J. G. 2008, ApJ, 676, 920 [NASA ADS] [CrossRef] [Google Scholar]
 Lauberts, A., & Valentijn, E. A. 1989, The surface photometry catalogue of the ESOUppsala galaxies [Google Scholar]
 Martinsson, T. P. K., Verheijen, M. A. W., Westfall, K. B., et al. 2013, A&A, 557, A131 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Mathewson, D. S., Ford, V. L., & Buchhorn, M. 1992, ApJS, 81, 413 [NASA ADS] [CrossRef] [Google Scholar]
 McGaugh, S. S., & de Blok, W. J. G. 1997, ApJ, 481, 689 [NASA ADS] [CrossRef] [Google Scholar]
 McGaugh, S. S., & Schombert, J. M. 2014, AJ, 148, 77 [NASA ADS] [CrossRef] [Google Scholar]
 McGaugh, S. S., Rubin, V. C., & de Blok, W. J. G. 2001, AJ, 122, 2381 [NASA ADS] [CrossRef] [Google Scholar]
 Merritt, D., Graham, A. W., Moore, B., Diemand, J., & Terzić, B. 2006, AJ, 132, 2685 [NASA ADS] [CrossRef] [Google Scholar]
 Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, AJ, 490, 493 [NASA ADS] [CrossRef] [Google Scholar]
 Padilla, N. D., & Strauss, M. A. 2008, MNRAS, 388, 1321 [NASA ADS] [Google Scholar]
 Palunas, P., & Williams, T. B. 2000, AJ, 120, 2884 [NASA ADS] [CrossRef] [Google Scholar]
 Planck Collaboration XIII. 2016, A&A, 594, A13 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Prugniel, P., Zasov, A., Busarello, G., & Simien, F. 1998, A&AS, 127, 117 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 RetanaMontenegro, E., van Hese, E., Gentile, G., Baes, M., & FrutosAlfaro, F. 2012, A&A, 540, A70 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Rodríguez, S., & Padilla, N. D. 2013, MNRAS, 434, 2153 [NASA ADS] [CrossRef] [Google Scholar]
 Rubin, V. C., Thonnard, N., & Ford, Jr., W. K. 1978, ApJ, 225, L107 [NASA ADS] [CrossRef] [Google Scholar]
 Rubin, V. C., Burstein, D., Ford, Jr., W. K., & Thonnard, N. 1985, ApJ, 289, 81 [NASA ADS] [CrossRef] [Google Scholar]
 Sawicki, M., & Yee, H. K. C. 1998, AJ, 115, 1329 [NASA ADS] [CrossRef] [Google Scholar]
 Schombert, J. M., & McGaugh, S. 2014, PASA, 31, e011 [NASA ADS] [CrossRef] [Google Scholar]
 Stark, D. V., McGaugh, S. S., & Swaters, R. A. 2009, AJ, 138, 392 [NASA ADS] [CrossRef] [Google Scholar]
 Tamm, A., & Tenjes, P. 2005, A&A, 433, 31 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Tamm, A., Tempel, E., Tenjes, P., Tihhonova, O., & Tuvikene, T. 2012, A&A, 546, A4 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Tempel, E., & Tenjes, P. 2006, MNRAS, 371, 1269 [NASA ADS] [CrossRef] [Google Scholar]
 van der Hulst, J. M., Skillman, E. D., Smith, T. R., et al. 1993, AJ, 106, 548 [NASA ADS] [CrossRef] [Google Scholar]
 Zwicky, F. 1937, ApJ, 86, 217 [NASA ADS] [CrossRef] [Google Scholar]
Appendix A:
Akaike information criterion of the pure baryonic model with the M/L ratio fit (AIC_{B}), baryonic model with estimated M/L ratio combined with the NFW (AIC_{NFW}), the Einasto (AIC_{E}), and the PSE dark matter models (AIC_{P}).
Model performances based the Δvalues.
Fig. A.1 Bestfit rotational curves for the HSB galaxy sample. The dots with error bars denote archive rotational velocity curves derived from spectroscopic data. The fitted models are pure baryonic (black shortdashed curve), baryonic + NFW (red continuous curve), baryonic + Einasto (purple dashed curve), and baryonic + PSE (green continuous curve). 

Open with DEXTER 
Fig. A.2 Bestfit rotational curves for the LSB galaxy sample. The dots with error bars denote archive rotational velocity curves derived from spectroscopic data. The fitted models are pure baryonic (black shortdashed curve), baryonic + NFW (red continuous curve), baryonic + Einasto (purple dashed curve), and baryonic + PSE (green continuous curve). 

Open with DEXTER 
Fig. A.3 Bestfit rotational curves for the HSB and LSB galaxies for which the χ^{2} values with the fit M/L ratios are within the 1σ confidence level of the fitting. The bestfit pure baryonic models are indicated by solid red curves, and the bestfit models composed of baryonic matter with fixed M/L and dark matter are indicated by dashed green curves. 

Open with DEXTER 
Fig. A.4 Bestfit 1D surface brightness models of the 15 HSB galaxies (upper panels) and their residuals (lower panels). The measurements are represented as larger black dots, the residuals by points, while the model predictions are shown as black continuous lines. 

Open with DEXTER 
Fig. A.5 Bestfit 1D surface brightness models of the 15 LSB galaxies (upper panels) and their residuals (lower panels). The measurements are represented as larger black dots, the residuals by points, while the model predictions are shown as black continuous lines. 

Open with DEXTER 
All Tables
Bestfit parameters describing the luminosity density distribution of the baryonic matter of 15 HSB spiral and 15 LSB galaxies.
Color indices of the chosen galaxies, their gas mass fraction (f_{g}), total luminosity in i photometric band (L_{i}), stellar (M_{⋆}) and gas masses (M_{gas}), and estimated M/L ratios for the bulge (Υ_{b}), and for the disk (Υ_{d}).
Parameters describing the bestfit pure baryonic, baryonic + NFW, baryonic + Einasto, and baryonic + PSE models of 15 HSB galaxies.
Parameters describing the bestfit pure baryonic, baryonic + NFW, baryonic + Einasto, and baryonic + PSE models of 15 LSB galaxies.
Bestfit (BF) dark matter models in Col. 2, where N/E/P marks hold for the NavarroFrankWhite/Einasto/pseudoisothermal sphere models, and the estimated M/L ratios of the baryonic + dark matter models (Υ_{b} and Υ_{d} for the bulge and disk, respectively) in Cols. 3 and 4.
Akaike information criterion of the pure baryonic model with the M/L ratio fit (AIC_{B}), baryonic model with estimated M/L ratio combined with the NFW (AIC_{NFW}), the Einasto (AIC_{E}), and the PSE dark matter models (AIC_{P}).
All Figures
Fig. A.1 Bestfit rotational curves for the HSB galaxy sample. The dots with error bars denote archive rotational velocity curves derived from spectroscopic data. The fitted models are pure baryonic (black shortdashed curve), baryonic + NFW (red continuous curve), baryonic + Einasto (purple dashed curve), and baryonic + PSE (green continuous curve). 

Open with DEXTER  
In the text 
Fig. A.2 Bestfit rotational curves for the LSB galaxy sample. The dots with error bars denote archive rotational velocity curves derived from spectroscopic data. The fitted models are pure baryonic (black shortdashed curve), baryonic + NFW (red continuous curve), baryonic + Einasto (purple dashed curve), and baryonic + PSE (green continuous curve). 

Open with DEXTER  
In the text 
Fig. A.3 Bestfit rotational curves for the HSB and LSB galaxies for which the χ^{2} values with the fit M/L ratios are within the 1σ confidence level of the fitting. The bestfit pure baryonic models are indicated by solid red curves, and the bestfit models composed of baryonic matter with fixed M/L and dark matter are indicated by dashed green curves. 

Open with DEXTER  
In the text 
Fig. A.4 Bestfit 1D surface brightness models of the 15 HSB galaxies (upper panels) and their residuals (lower panels). The measurements are represented as larger black dots, the residuals by points, while the model predictions are shown as black continuous lines. 

Open with DEXTER  
In the text 
Fig. A.5 Bestfit 1D surface brightness models of the 15 LSB galaxies (upper panels) and their residuals (lower panels). The measurements are represented as larger black dots, the residuals by points, while the model predictions are shown as black continuous lines. 

Open with DEXTER  
In the text 