Bulge $n$ and $B/T$ in High Mass Galaxies: Constraints on the Origin of Bulges in Hierarchical Models

Our dataset is derived from the 182 $H$-band images from the public data release of the Ohio State University Bright Spiral Galaxy Survey (OSUBSGS; Eskridge et al. 2002). These galaxies are a subset of the RC3 catalog that have $m_{B}\leq 12$, Hubble types $0\leq T\leq 9$ (S0/a to Sm), $D_{25}\leq 6^{\prime}.5$, and $-80^{\circ}<\delta<+50^{\circ}$. Imaging of OSUBSGS galaxies spans optical and near infrared (NIR) wavelengths with $BVRJHK$ images available for most galaxies. OSUBSGS images were acquired on a wide range of telescopes with apertures ranging from 0.9-2.4m. The $JHK$ data were acquired with a variety of telescopes and detectors, but mainly with the 1.8 m Perkins reflector at Lowell Observatory and the CTIO 1.5 m telescope with the OSIRIS detector, having 18.5 micron pixels (Depoy et al. 1993). Pixel scale is dependent on the telescope and for these observations ranged between $1-1.50^{\prime\prime}$/pix. Exposure times were heterogeneous, but the total observing time per object was typically between 10-15 minutes in $H$. The resulting limiting $H$-band surface brightness is $\sim 20$ mag arcsec^-2. The typical limiting surface brightnesses of the images $\sim 26$ mag arcsec^-2 in $B$-band and $\sim 20$ mag arcsec^-2 in $H$-band (Eskridge et al. 2002). The seeing depends on observing time and location. We find the $H$-band images have seeing of $\sim 3^{\prime\prime}$.

We choose to use the NIR images rather than optical ones for several reasons. Firstly, NIR images are better tracers of the stellar mass than optical images, and the mass-to-light ratio is less affected by age or dust gradients. Secondly, obscuration by dust and SF are minimized in the NIR, compared to the optical. As the $K$-band images are of poor quality, we settle on using the $H$-band images.

The OSUBSGS is widely used as the local reference sample for bright spirals by numerous studies (e.g. Eskridge et al. 2000; Block et al. 2002; Buta et al. 2005; MJ07; Laurikainen et al. 2004, 2007). Thus, there are numerous complementary results that we can use or compare with. In particular, MJ07 have identified bars in this sample using quantitative criteria based on ellipse fitting, and characterized their sizes, position angles, and ellipticities.

OSUBSGS is a magnitude-limited survey with objects whose distances range up to $\sim 60$ Mpc. Faint galaxies are inevitably missed at larger distances, resulting in the absolute magnitude distribution in Figure 1. We compare the $B$-band LF of this sample with a Schechter (Schechter 1976) LF (SLF) with $\Phi^{*}=5.488\times 10^{-3}$ Mpc^-3, $\alpha=-1.07$, and $M^{*}_{B}=-20.5$ (Efstathiou, Ellis & Peterson 1988) in Figure 2. The volume used to determine the number density in each magnitude bin is

where

is the maximum distance out to which a galaxy of absolute magnitude $M$ can be observed given the cutoff magnitude $m_{c}$. If the SLF is representative of the true LF, then Figure 2 suggests that the OSUBSGS sample is seriously incomplete at $M_{B}>-19.3$, while at the brighter end (-19.3 to -23) the shape of its LF matches fairly well the SLF. We thus conclude that the sample is reasonably complete for bright ($M_{B}\leq-19.3$ or $L_{B}>$ 0.33 $L^{*}$) galaxies.

We exclude highly inclined ($i>70^{\circ}$) galaxies for which structural decomposition does not yield accurate results. Thus, our final sample S1 consists of 143 bright ($M_{B}\leq-19.3$) low-to-moderately inclined ($i\leq 70^{\circ}$) spirals with Hubble types mainly in the range S0/a to Sc (Figure 1). Of the 126 for which we could derive stellar masses (see $\S$ 2.2, most have stellar masses $M_{\star}\geq 1.0\times 10^{10}M_{\odot}$ (Figure 3). Table 1 summarizes the morphologies, luminosities, and stellar masses of the sample. Note that there are few galaxies of late Hubble types (Scd or later) and we do not draw any conclusions on such systems from our study. In a future paper, we will tackle galaxies of lower mass and later Hubble types.

We derive global stellar masses for most of the OSUBSGS sample galaxies using the relation between stellar mass and rest-frame $B-V$ color from Bell et al. (2003). Using population synthesis models, the latter study calculates stellar $M/L$ ratio as a function of color using functions of the form $log_{10}(M/L)=a_{\lambda}+b_{\lambda}\times Color+C$, where $a_{\lambda}$ and $b_{\lambda}$ are bandpass dependent constants and C is a constant that depends on the stellar initial mass function (IMF). For the $V$ band Bell et al. (2003) find $a_{\lambda}=-0.628$ and $b_{\lambda}=1.305$; assuming a Kroupa (1993) IMF, they find $C=-0.10$. This yields an expression for the stellar mass in $M_{\odot}$ for a given $B-V$ color:

where

Here, $v_{lum}$ is the luminosity parametrized in terms of absolute $V$ magnitude.

How reliable are stellar masses determined from this procedure? Clearly, the above relationship between $M_{\star}$ and $B-V$ cannot apply to all galaxies, and must depend on the assumed stellar IMF, and range of ages, dust, and metallicity. However, it is encouraging to note that several studies (Bell et al. 2003; Drory et al. 2004; Salucci, Yegorova, & Drory 2008) find generally good agreement between masses based on broad-band colors and those from spectroscopic (e.g. Kauffmann et al. 2003) and dynamical (Drory et al. 2004) techniques. Typical errors are within a factor of two to three. Salucci et al. (2008) derive disk masses with both photometric and kinematic methods and find the two methods are equivalent on average. For a sample of 18, they find an rms scatter of 0.23 dex, while on an individual basis the deviation can be as high as 0.5 dex.

We used this relation to compute stellar masses for 126 of 143 (88%) objects. The remainder did not have $B-V$ colors available in the Hyperleda database or RC3. The mass distribution is summarized in Figure 3. Individual masses are listed in Table 1. This sample of 126 galaxies is referenced henceforth as sample S2.

Running GALFIT on an image requires initial preparation. The desired fitting region and sky background must be known, and the PSF image, bad pixel mask (if needed), and pixel noise map must be generated. We addressed these issues as follows: (1) The GALFIT fitting region must be large enough to include the outer galaxy disk, as well as some sky region. Since cutting out empty regions of sky can drastically reduce GALFIT run-time, a balance was sought between including the entire galaxy and some decent sky region, while excluding large extraneous blank sky areas. (2) It is possible for GALFIT to fit the sky background, but this is not recommended. When the sky is a free parameter, the wings of the bulge Sérsic profile can become inappropriately extended, resulting in a Sérsic index that is too high. Sky backgrounds were measured separately and designated as fixed parameters; (3) GALFIT requires a PSF image to correct for seeing effects. Statistics of many stars in each frame can be used to determine an average PSF. However, many of our images contain merely a few stars. Instead, a high S/N star from each frame was used as a PSF. (4) We used ordered lists of pixel coordinates to make bad pixel masks, which are useful for blocking out bright stars and other image artifacts. (5) We had GALFIT internally calculate pixel noise maps for an image from the noise associated with each pixel. Noise values are determined from image header information concerning gain, read noise, exposure time, and the number of combined exposures.

Figure 4 summarizes our method of decomposition, which we now detail. GALFIT requires initial guesses for each component it fits. It uses a Levenberg-Marquardt downhill-gradient algorithm to determine the minimum $\chi^{2}$ based on the input guesses. GALFIT continues iterating until the $\chi^{2}$ changes by less than 5e-04 for five iterations (Peng et al. 2002). We recognize that a drawback to any least-squares method is that a local minimum, rather than a global minimum, in $\chi^{2}$ space may be converged upon. We explore this possibility with multiple tests described in § 4. We adopted an iterative process, involving three separate invocations of GALFIT, to perform 1-component, 2-component, and 3-component decomposition:

1. 1.

   Stage 1 (single Sérsic): In Stage 1, a single Sérsic component is fitted to the galaxy. This serves the purpose of measuring the total luminosity, which is conserved in later Stages, and the centroid of the galaxy, which is invariant in later fits.

2. 2.

   Stage 2 (exponential plus Sérsic): In Stage 2, the image is fit with the sum of an exponential disk and a Sérsic component. During the Stage 2 fit, the disk $b/a$ and $PA$ are held constant at values, which we take from the published ellipse fits of MJ07, as well as ellipse fits of our own. This procedure reduces the number of free parameters in the fit by fixing the disk $b/a$ and $PA$, which are easily measurable parameters. It also prevents GALFIT from confusing the disk and bar, and artificially stretching the disk along the bar PA in an attempt to mimic the bar. As initial guesses for the Sérsic component in Stage 2, the output of Stage 1 is used. The Sérsic component in Stage 2 usually represents the bulge, in which case Stage 2 corresponds to a standard bulge-disk decomposition.

   However, in a few rare cases, where the galaxy has just a bar and a disk, the Sérsic component in Stage 2 represents a bar. The latter is recognizable by a low Sérsic index and large half-light radius.

3. 3.

   Stage 3 (exponential plus two Sérsic components): In Stage 3, a three-component model consisting of an exponential disk, a Sérsic bulge, and a Sérsic bar is fit. As suggested by Peng et al. (2002), the bar can be well described by an elongated, low-index Sérsic ($n<1$) profile. As in Stage 2, the disk $b/a$ and $PA$ are held constant at values predetermined from ellipse fits. We provide initial guesses for the bar $b/a$ and $PA$, based on ellipse fits of the images from MJ07 or analysis of the images in DS9. We provide GALFIT with input guesses for the bulge parameters, based on the output from Stage 2. In principle, it is also possible to generate reasonable guess parameters for the bulge and disk from a bulge-disk decomposition on a $1D$ profile taken along a select PA. As described in $\S$ 4.3, we also experiment with initial guesses derived in this way, and find that the final convergence solution is the same. We also note that GALFIT fixes the bulge $b/a$ and does not allow it to vary with radius, while real bulges may have a varying $b/a$. We tested the impact of fixed and varying bulge $b/a$ on the derived $B/T$ ($\S$ 4.1) and find that there is no significant change in $B/T$.

For objects with central point sources, the bulge Sérsic index in the Stage 2 and Stage 3 models can be inadvertently overestimated unless an extra nuclear component is added to the model. Balcells et al. (2003) note that for galaxies imaged both from the ground and $HST$, the combination of unresolved central sources and seeing effects mimic high-index bulge Sérsic profiles in the ground images. Depending on sample and resolution, the frequency of central sources can range from 50% to 90% (Ravindranath et al. 2001; Böker et al. 2002; Balcells et al. 2003). Ravindranath et al. (2001) find a frequency of 50% in early type (E, S0, S0/a) galaxies, while Böker et al. (2002) measure a frequency of 75% for spirals with Hubble types Scd to Sm. Balcells et al. (2003) determine a frequency of 84% for S0-Sbc galaxies imaged with $HST$. Our dataset most closely resembles the latter sample, so we might expect that, as an upper limit, a similar fraction of our galaxies will need to be corrected with an extra compact component.

We added point sources as third or fourth components to the initial models. For those cases where the model successfully converged with the extra component, the images were visually inspected to verify the presence of a central bright source. Sometimes, the model converged to significantly different and unreasonable parameters for all components. Other times, the model would converge to a very dim point source without changing any of the other model parameters. Where new model parameters were not unreasonable and not identical to the case without the point source, the new model was adopted. This was the case for 111 of 143 (78%) of our sample. The point sources contribute less than 1%, 3%, and 5% of the total luminosity 55%, 86%, and 95% of the time, respectively. As the point sources take up such a small fraction of the light distribution, their contribution is folded back into $B/T$ in all cases where a point source was modeled. Inclusion of the point source reduced bulge index by $\sim 0.8$ for both barred and unbarred galaxies. Such a change is expected based on the above discussion. The decline in bulge index caused a minor decrease in $B/T$; on the mean, this change was 1.04% for barred galaxies and 0.32% for unbarred galaxies. For barred galaxies, this light most primarily added to $D/T$ rather than Bar/$T$.

It is important to recognize the physical significance of the added nuclear components. We began by determining which objects show evidence for AGN activity. The sample was checked against the catalog of Ho et al. (1997), the Véron Catalog of Quasars & AGN, 12th Edition (Véron-Cetty & Véron 2006), and NED. Of the 111 objects fit with point sources, 43 (39%) contain AGN. An additional 20 (18%) possess HII nuclei according to Ho et al. (1997) and visibly show bright compact nuclei. The remaining 48 (43%) probably contain neither AGN nor HII nuclei, but could house nuclear star clusters. For these objects, we visually inspected the images to ensure there was a bright compact source at the center. We are confident that the fitted point source components have physical counterparts in the data images.

GALFIT also allows a diskiness/boxiness parameter to be added to any Sérsic or exponential profile. We did not use this parameter for any bulge or disk profiles. Bars in general have boxy isophotes, and we could have included the diskiness/boxiness parameter in the bar profiles. However, it was found that adding boxiness to the bar did not change any model parameters, including fractional luminosities $B/T$, $D/T$, and Bar/$T$, by more than a small percentage, even though the appearance of the residual images improved in some cases due to the change in bar shape. Consequently, we chose to neglect bar boxiness altogether.

All objects in our sample were subjected to Stages 1, 2, and 3. Depending on whether a galaxy with a bulge is unbarred or barred, its best fit should be taken from the Stage 2 bulge-disk decomposition or the Stage 3 bulge-disk-bar decomposition, respectively. For objects with prominent bars, it is obvious that the Stage 3 model provides the best fit. However, it is more difficult to decide between Stage 2 versus Stage 3 fits in galaxies which host weak bars with no strong visual signature. In practice, we therefore applied the set of criteria below to each galaxy in order to select between the Stage 2 bulge-disk decomposition and Stage 3 bulge-disk-bar decomposition. Table 1 lists the model chosen for each galaxy. Table 2 summarizes the model parameters from the best fits.

For completeness, we note that for the few rare galaxies (see $\S$ 3.2) that have just a bar and a disk, the choice of a final solution is between the Stage 2 bar-disk decomposition and Stage 3 bulge-disk-bar decomposition. The same guidelines below can be used to identify the best model.

1. 1.

   GALFIT calculates a $\chi^{2}$ and $\chi^{2}_{\nu}$ for each model. It was found that $\chi^{2}$ almost universally declines between the Stage 2 and Stage 3 fits for a given object. This is because in the Stage 3 fit, five extra free parameters (bar luminosity, $r_{e}$, Sérsic index, $b/a$, and $PA$) are added with the Sérsic bar component, allowing GALFIT to almost always make a lower $\chi^{2}$ model during Stage 3. However, this does not necessarily mean that the solution in Stage 3 is more correct physically. Thus, an increasing $\chi^{2}$ was interpreted as a sign that the Stage 3 fit should not be adopted, but a decreasing $\chi^{2}$ was not considered as a sufficient condition to adopt Stage 3.

2. 2.

   In cases with prominent bars, a symmetric light distribution due to unsubtracted bar light was often found in the Stage 1 and Stage 2 bulge-disk residuals. This was strong evidence that the Stage 3 bulge-disk-bar fit be selected. NGC 4643 is shown in Figure 5 because it has a particularly striking bar residual; the corresponding fit parameters appear in Table 3. Note that in all figures and tables, we adopt the convention that PA values are positive/negative if they are measured from North counterclockwise/clockwise.

3. 3.

   The Stage 2 and Stage 3 models were selected only so long as the model parameters were all well behaved. In unbarred galaxies, the Stage 3 model parameters might be unphysically large or small, in which case the Stage 2 fit was favored. Conversely, in galaxies with prominent bars, the bulge component of the Stage 2 bulge-disk fit tends to grow too extended in size. Addition of a bar in the Stage 3 bulge-disk-bar fit removes this artifact, giving a more physical solution. An extreme example of this situation is the barred galaxy NGC 4548, which has a prominent bar and a faint disk. The Stage 2 fit, based on a Sérsic bulge and exponential disk, is highly inadequate to describe the bulge, disk, and the bar because it leads to an extremely extended bulge. The Stage 3 bulge-disk-bar fit, however, yields a believable fit with a prominent bar. The results of Stage 1, Stage 2, and Stage 3 are displayed in Figure 6 and Table 4.

4. 4.

   Not all barred galaxies had unphysical Stage 2 models. Instead, the bulge could be stretched along the $PA$ of the bar, giving the bulge a lower Sérsic index and larger effective radius. A Stage 3 model that returned the bulge to a size and shape more representative of the input image was favored over the Stage 2 fit. Figure 7 and Table 5 demonstrate this behavior in NGC 4902. We distinguish this effect from cases like NGC 4548 (Figure 6 and Table 4) where the Stage 2 fit is completely wrong.

5. 5.

   In cases where there was no bar, GALFIT can sometimes be enticed into fitting a bar to any existing spiral arms, rings, or the clumpy disks of late-type spirals. Stage 3 fits in these cases could be discarded by noting the resulting discrepancies in appearance between the galaxy images and the Stage 3 model images. Examples of false bars are shown in Figure 8.

After fitting the whole sample and picking the best fit from either the Stage 2 bulge-disk decomposition or the Stage 3 bulge-disk-bar decomposition, we also performed the following extra tests. For our sample S1 of 143 bright ($M_{B}\leq-19.3$) low-to-moderately inclined ($i\leq 70^{\circ}$) spirals in the OSUBSGS survey, we determine the fraction (75 of 143 or $\sim 52\%$) of spiral galaxies where a bulge-disk-bar decomposition was picked as the best fit for the $H$-band image. There are also eight galaxies with pure bar-disk fits. The $H$-band bar fraction, which is defined as the fraction of disk galaxies that are barred, is therefore $58.0\pm 4.13\%$ (83 of 143). We then compared our results ($58.0\%\pm 4.13\%$) with the $H$-band bar fraction (60%) determined from ellipse fits of the OSUBSGS sample by MJ07, with a slightly more conservative inclination cut ($i\leq 60^{\circ}$). The two numbers are in excellent agreement. As a further check to our fits, we compare the bar and unbarred classification for individual galaxies from our fits with those from MJ07, which were based on ellipse fits. Of the 73 galaxies that we classify as barred, and that are mutually fitted by MJ07, 54 (74%) are also classified as barred by MJ07. The remaining 19 (26%) galaxies are mainly weakly barred (with Bar/$T$ below 0.08). Their RC3 optical types are weakly barred AB (10), barred B (7), and unbarred A (2).

In most previous bulge-disk and bulge-disk-bar $2D$ decomposition, the issue of parameter coupling and the systematic exploration of local versus global minima in $\chi^{2}$ have been ignored. Quantifying how the parameters are coupled is important in measuring error bars for the model parameters. With $2D$ models containing several free parameters, this is not an easy task. Although we also do not address this problem in rigorous detail, we describe in § 4.4 simple test that explores parameter coupling in our models.

Models generated with GALFIT do not allow the $b/a$ of the bulge, disk, or bar to vary with radius. Since real bulges may have a varying $b/a$, it is legitimate to investigate the impact of fixing the bulge $b/a$ on the estimated $B/T$. We therefore performed the following test on NGC 4548. To mimic a model bulge of varying $b/a$, we fitted the bulge light of NGC 4548 with ten concentric Sérsic profiles of increasing $r_{e}$ and varying $b/a$. The $r_{e}$ of the outermost profile comes from the original bulge model (see Table 4) where $b/a$ was kept constant with radius. The separation in $r_{e}$ between adjacent profiles is 0.5 pixels (0.75”). The luminosity, Sérsic index, $b/a$, and $PA$ of each profile were free parameters. The disk and bar components were fixed to the values in Table 4, as the emphasis was on the change in the bulge.

Figure 9 compares the $B/T$ obtained by fitting the bulge of NGC 4548 with a Sérsic model of constant $b/a$ as opposed to a Sérsic model varying $b/a$. The bulge $b/a$ (0.88), $PA$ (-66.5), and $B/T$ (13%) from the original Sérsic fit of constant $b/a$ (Table 4) are indicated with horizontal lines on the 3 panels. The top two panels show the run of $b/a$ (0.85 to 1.0) and $PA$ ($-90^{\circ}$ to $+90^{\circ}$) of the ten concentric Sérsic profiles. It can be seen that the Sérsic indices of the ten bulge models were generally higher toward the center and declined at larger $r_{e}$, indicating that the ‘fitted bulge is more concentrated at the center. The bottom panel shows the cumulative $B/T$ calculated by summing all models with $r\leq r_{e}$. The last point representing the summed $B/T$ from all ten components is 14.5%, in good agreement with the 13.0% value from the Sérsic fit of constant $b/a$. Thus, using a Sérsic model of constant $b/a$, does not have any significant adverse impact on our derived $B/T$ in NGC 4548.

An elementary test is to determine if GALFIT can recover the known parameters of artificially simulated noisy images. The images were simulated by taking parametric model images produced by GALFIT, and adding noise to the images with the PyFITS module for Python (Barrett & Bridgman 1999). Noise in each pixel was calculated by adding in quadrature the noise due to the source, sky, and read noise. The standard deviation of pixel noise in electrons was computed as

where $T_{source}$ is the number of electrons due to the source, $T_{sky}$ is the number of electrons due to the sky, and $T_{read}$ is the detector read noise. The contribution due to detector dark current was very small and therefore neglected. The offset added to each pixel was drawn from a normal distribution centered at zero with standard deviation $\sigma$.

Our test sample consisted of 40 models (20 bulge-disk and 20 bulge-disk-bar) with noise added as described. Thirty of the images included point sources as extra components. The range explored for each parameter in the model images is a fair representation of the parameter space covered by our full sample (e.g., $B/T$ ranges from $0.02-0.70$, the bulge index ranges from $\sim 0-5$, and the full range of possible bulge and bar PA was also tested ($-90^{\circ}$ to $+90^{\circ}$)). In terms of surface brightness, the models span five magnitudes in mean surface brightness inside the disk scalelength. Examples of the noise-added models are shown in Figure 10.

The noisy images were subjected to the $2D$ decomposition procedure outlined in Figure 4. GALFIT reproduced the model (bulge, disk, bar, and point source) parameters quite closely for the majority of the test cases. Figure 11 compares the recovered versus original model parameters. Except for some extreme cases where the images were highly distorted by noise, all parameters were recovered to within a few percent. Figure 12 plots the ratio of model-to-recovered parameter against mean surface brightness inside the disk scalelength; there is no strong trend in error with dimming surface brightness. This suggests our decompositions are effective across the parameter space spanned by our sample. The overall success of this test is evidence that GALFIT is able to converge to the absolute minimum in $\chi^{2}$ space for our bulge-disk and bulge-disk-bar decompositions when the input is the sum of parametric functions.

It is important to verify that GALFIT converges to the same solution even if the initial guesses for the bulge parameters in Stage 2 and 3 are different. Bulge-disk decomposition from $1D$ profiles provides an alternative means of generating initial guesses. While $1D$ bulge-disk decompositions of radial profiles along the bar major axis can be influenced by the bar, decomposition of cuts along the bar minor axis will not be influenced as heavily. The resulting bulge and disk parameters should be adequate guesses for Stage 3 of our $2D$ decomposition method.

We tested the robustness of our Stage 3 fits by extracting initial guesses for the bulge and disk using $1D$ decomposition along the bar minor axis. The nonlinear least-squares algorithm designed to perform the $1D$ decomposition simultaneously fits the sky-subtracted profiles with the sum of a Sérsic bulge and an exponential disk, while ignoring the PSF. The results from the $1D$ decomposition include a bulge magnitude, $r_{e}$, Sérsic index, disk magnitude, and disk scalelength.

The robustness of several bulge-disk-bar fits were tested by using the results of the $1D$ decomposition as input to Stage 3. The $1D$ decompositions do not provide information about the axis ratio ($b/a$) or $PA$, so these parameters for the bulge were estimated by eye; for the disk, the $b/a$ and $PA$ were fixed to the values determined by ellipse fitting, as described in $\S$ 3.2. The initial bar parameters were unchanged from the earlier Stage 3 fits. In all cases, the new models were identical to the Stage 3 models. As an example, Table 6 compares Stage 3 input derived from 1D decomposition and GALFIT for NGC 4548 and NGC 4643. In each case, both sets of input reproduced the same results.

Assessing the coupling between model parameters is complicated when models have a large number of free parameters. A standard approach is to calculate confidence regions using multi-dimensional ellipsoids for a given $\Delta\chi^{2}$ contour. As the errors in the GALFIT models are not normally distributed, but are instead dominated by the systematics of galaxy structure, this approach does not yield meaningful results because of the ambiguity in assessing which $\Delta\chi^{2}$ contour levels are statistically significant.

We carry out a simple test for representative galaxies to determine not only the effects of parameter coupling, but also the the effect parameter coupling has on model parameter errors, paying particularly close attention to $B/T$, $D/T$, and Bar/$T$, as they are of primary interest. We perform this test on four representative galaxies (NGC 3885, NGC 4151, NGC 4643, and NGC 7213). Two are barred (NGC 4151 and NGC 4643). Two have high ($n>2$) bulge indices (NGC 4643 and NGC 7213), and the other two have low ($n<2$) bulge indices.

We fit two and three-component bulge-disk and bulge-disk-bar models using fixed bulge indices of $n=1$ and $n=4$. The initial inputs to these fits were the same as those used to generate the model in which bulge $n$ is a free parameter. We then had GALFIT re-fit the models with bulge index as a free parameter using these output model parameters as input initial guesses. We compare the output of these two fits with the the best fit as selected in § 3.3, in which bulge $n$ is a free parameter. Ideally, the re-fits should converge to the same final parameters as the selected best fit.

Table 7 displays the outcome of this test for the four representative systems. For barred galaxies NGC 4643 and NGC 4151, the $n=1$ and $n=4$ re-fits converged to the same $\chi^{2}$ as when bulge $n$ is initially left a free parameter. $B/T$, $D/T$, and Bar/$T$ are precisely equal for NGC 4643, while for NGC 4151 there is a small dispersion of 0.1-0.2%. For unbarred galaxies NGC 3885 and NGC 7213, the $n=1$ re-fits again converged to the same $\chi^{2}$ and model parameters as when bulge $n$ is initially left a free parameter. This is not true for the $n=4$ re-fits. When the $n=4$ condition is enforced, the bulges in these cases become too extended and luminous while the $\chi^{2}$ drop below those when bulge $n$ is initially left a free parameter. During the $n=4$ re-fits, $B/T$ increases further at the expense of $D/T$ and the $\chi^{2}$ remain unchanged or decrease further. The $B/D$ ratio from the $n=4$ re-fit for NGC 3885 is 1.3, roughly 3.5 times higher than when $n$ is free. Given that NGC 3885 is an S0/a galaxy with a bulge embedded in a smooth extended disk, the latter $B/D$ is arguably too large. For NGC 7213, the $n=4$ re-fit yields starkly unphysical values. The $B/D$ ratio is 11.3, and $r_{e}/h$, the ratio between bulge effective radius and disk scalelength is 9.0. The lower $\chi^{2}$ in these cases cannot be taken seriously as the bulges are too luminous and the resulting $B/D$ ratios do not match the data images.

As illustrated by the above discussion and Table 7, this test shows that in some cases (e.g., NGC 4643 and NGC 4151) GALFIT converges to the similar model parameters and $B/T$, $D/T$, and Bar/$T$ while starting from highly different initial input guesses (e.g. bulge $n=1$, bulge $n=4$, and bulge $n$ based on the Stage 2 fits) in different regions of parameter space. For NGC 3885 and NGC 7213, however, during the $n=4$ re-fits GALFIT converged to models that were unphysical and different compared with the reasonable models generated with input guesses corresponding to the bulge fixed at $n=1$ or the bulge $n$ based on output from Stage 1. In effect, when the initial input guesses were very different from the data images, the resulting models were found, in spite of the lower $\chi^{2}$, to be unphysical through comparison with the input data images. We emphasize that for all sources analyzed in the paper, the data, converged model output, and residuals are always inspected before adopting the best final fit (see § 3.3).

Table 7 also provides hints as to how the model parameters are coupled to bulge index. As suggested in the above discussion, fixing the bulge index to $n=4$ leads to a more extended and luminous bulge, causing bulge $r_{e}$ and $B/T$ to rise without fail for increasing bulge index. The disk is coupled with the bulge such that increasing bulge index, bulge $r_{e}$, and $B/T$ yields a reduction in $D/T$. At the same time, disk scalelength either increases (NGC 3885 and NGC 4643) or decreases (NGC 3885 and NGC 4151); in the latter two cases, the disk becomes very compact and the bulge quite extended. The behavior of the bar is coupled with both the bulge and disk. In the case of NGC 4643, as bulge index is raised to $n=4$, bar $r_{e}$ becomes slightly larger, but Bar/$T$ falls by a factor of 2.6 as light is redistributed from the bar and disk to the bulge. Bar index also declines as the bar assumes a flatter profile. For NGC 4151, bar $r_{e}$ again increases slightly, but this time Bar/$T$ rises by a factor of 1.7 as light is transferred from the disk to the bulge and bar.

Based on the above test, we stress that GALFIT was able to overcome this parameter coupling in the cases where the input guess parameters well-reflected the data images.

From the Stage 2 bulge-disk decomposition and Stage 3 bulge-disk-bar decompositions, which we performed on all objects ($\S$ 3.2) we saw firsthand the effects of adding a bar to the fit of a barred galaxy. We summarize below some of these effects in order to underscore the importance of including a bar component in the $2D$ luminosity decomposition of barred galaxies

1. 1.

   During the Stage 2 bulge-disk decomposition of a barred galaxy, the luminosity which comes from the galaxy’s disk, bulge, and bar gets distributed only between two model components: the model bulge and disk. Since the disk $b/a$ and $PA$ are measured independently and held constant during the fits, the Stage 2 model tends to distort the bulge in order to fit the bar. Thus, the bulge in the Stage 2 bulge-disk decomposition of a barred galaxy can be artificially long or too bright and extended. When a model bar component is added in the Stage 3 bulge-disk-bar decomposition of a barred galaxy, it forces a reshuffling of the luminosity between the three components. Generally, the bulge declines in luminosity, whereas light can be either taken from, or added back, to the disk.

2. 2.

   We find that the inclusion of a bar component in the Stage 3 bulge-disk-bar decomposition of a barred galaxy reduces the bulge fractional luminosity $B/T$, compared with the Stage 2 bulge-disk decomposition. For our 75 barred galaxies, the reductions correspond to factors of less than two, 2 to 4, and above 4, in 36%, 25%, and 39% of barred galaxies, respectively. The larger changes in $B/T$ occur in very strongly barred galaxies, where a prominent bar cause the Stage 2 bulge-disk decomposition to overestimate the bulge. For instance, $B/T$ declines in both of NGC 4643 (Figure 5 and Table 3) and NGC 4548 (Figure 6 and Table 4). In the latter case, $B/T$ is reduced by a factor of $\sim 5$ between Stage 2 and Stage 3. These examples underscore the importance of including bars in $2D$ luminosity decomposition of very strongly barred galaxies.

3. 3.

   The scalelength of the disk is generally unchanged by including the bar. NGC 4548 (Figure 6 and Table 4) is a good example. Sometimes, however, the disk from the Stage 2 bulge-disk decomposition of a barred galaxy is erroneous due to a poor fit. The disk parameters from the Stage 3 bulge-disk-bar decomposition are quite different in such cases. NGC 4643 (Figure 5 and Table 3) illustrates this behavior.

We find that for our sample S1 of 143 bright ($M_{B}\leq-19.3$) low-to-moderately inclined ($i\leq 70^{\circ}$) spirals (Figure 1) in the OSUBSGS survey, 75 of 143 or $\sim 52\%$ are better fit with a Stage 3 bulge-disk-bar decomposition than a Stage 2 bulge-disk decomposition. There are also eight galaxies with pure bar-disk fits. As stated in $\S$ 3.3, the resulting $H$-band bar fraction ($58.0\%\pm 4.13\%$ or 83 of 143) is in excellent agreement with the $H$-band bar fraction of 60% reported by MJ07 based on ellipse fits of the OSUBSGS sample, with a slightly more conservative inclination cut ($i\leq 60^{\circ}$).

The fractional $H$-band luminosities in the bulge, disk, and bar ($B/T$, $D/T$, Bar/$T$) of each galaxy can be considered as a fractional mass if we assume that the same mass-to-light ($M/L$) ratio can be used to convert the $H$-band luminosities of both the numerator ($B$, $D$, or Bar) and the denominator ($T$) terms into a stellar mass. This is not an unreasonable assumption as the $H$-band $M/L$ ratio is not very sensitive to differences in dust or age that might exist between the bulge, disk, and bar. The uncertainties in $M/L$ can be estimated by looking at population synthesis models. Charlot, Worthey, & Bressan (1996) find that for idealized galaxies with a single generation of stars, the uncertainties in $M/L$ ratio due to different input stellar models and spectra are roughly $\pm 35\%$ for a fixed metallicity and IMF. Furthermore, as the age of a stellar population varies from $\sim 0.5$ Gyr to 10 Gyr, the $K$-band $M/L$ ratio rises by a factor of $\sim$ 2 to 3 (Charlot 1996). Asymptotic giant branch (AGB) stars dominate the NIR light for ages between 0.1 and 1 Gyr, while red giant branch (RGB) and super-giant branch (SGB) stars dominate between 1 Gyr and 10 Gyr.

In this paper, we convert the $B/T$ light ratio determined from $H$-band images to a $B/T$ mass ratio by assuming a constant mass-to-light ($M/L$) in the $H$-band for both the bulge and the rest of the galaxy. However, Moorthy & Holtzman (2006) present line strengths of bulges and inner disks for 38 spirals with Hubble types S0 to Sc. They show 76% of spirals have negative metallicity gradients. $B-K$ color gradients are shown to largely match metallicity gradients and are likewise negative outward, indicating bulge $M/L$ is higher than in the disk and bar. If bulges are much older than the disks, then our prescription would underestimate the true $B/T$ mass ratio. If we assume an extreme case where bulges are $\sim 12$ Gyr and the disk light is dominated by a young 3 Gyr population, our assumption of a constant $H$-band $M/L$ ratio would underestimate the true $B/T$ by a factor of $\leq 2$ (Charlot 1996). In several sections of the paper (e.g., $\S$ 5.3, $\S$ 5.8), we illustrate how our main results would change if the true $B/T$ was higher by up to a factor of two. On the other hand, central regions of galaxies may harbor intense episodes of star formation. If the bulge is younger than the disk and happens to have star formation and a significant young population of massive stars, then our prescription could overestimate the true $B/T$ mass ratio. This would make our current results on the high fraction of low $B/T$ bulges even stronger.

Using the total galaxy stellar mass from $\S$ 2.2, the fractional masses can be converted into absolute masses. (We do not convert the $H$-band luminosity directly into a mass as the $H$-band images do not have photometric calibration). The results are shown in Table 8. For our sample S1 of 143 bright ($M_{B}\leq-19.3$) low-to-moderately inclined ($i\leq 70^{\circ}$) spirals with a mass-weighted mean Hubble type of Sab-Sb, we find that $\sim 70\%$ of the stellar mass is in disks, $\sim$ 10% is in stellar bars and $\sim 20\%$ is in bulges (with $\sim 11\%$ in $n>2$ bulges and $\sim 9\%$ in $n\leq 2$ bulges). Thus while bulges with $n\leq 2$ are highly ubiquitous (see next section), they only account for a small fraction of the total stellar mass budget.

Figure 13 shows the stellar mass for bulges, disks, and bars along the Hubble sequence.

It is useful to compare our results with those of Driver et al. (2006), who performed bulge-disk decomposition of $B$-band images with GIM2D on 10,095 galaxies from the Millennium Galaxy Catalog (Liske et al. 2003; Driver et al. 2005). They found 68.6% of the stellar mass to be in disks, and 32.6% in bulges (with 30.8% in high $n$ bulges, and 1.8% in low $n\leq 2$ bulges). Their study thus finds a higher stellar mass fraction in all bulges (32.6% vs our 18.9%), and in high $n$ bulges (30.8% vs our 10.4%), and a lower fraction in low $n\leq 2$ bulges (1.8% vs our 8.4%), and disks+bars (68.6% vs our 71.6% + 9.6%). This difference can be attributed to the fact that the Driver et al. (2006) study did not perform bulge-disk-bar fits and thus, their $B/T$ ratios may be skewed to higher values.

Figure 14 shows the individual and mean $B/T$ and bulge Sérsic index, plotted, as a function of Hubble type and galaxy stellar mass. Barred and unbarred galaxies are shown separately. Figure 15 shows the relationship between bulge index and $B/T$.

We first consider the $B/T$ values in Figure 14. The mean $B/T$ in barred galaxies is lower than in unbarred galaxies, but there is a large overlap in the individual values. The offset in the mean $B/T$ of barred and unbarred galaxies reported here, agrees with the result of Laurikainen et al. (2007; see $\S$ 5.4) on the same sample. We also note that $B/T$ does not correlate with Bar/$T$ (Fig. 16): aside from the six galaxies with large Bar/$T$ ($>0.3$), most galaxies have low-to-moderate Bar/$T$ and a wide range of $B/T$ is seen at each Bar/$T$. This is reassuring and suggests that the bar fit is not arbitrarily biasing the $B/T$ values. The distribution of Bar/$T$ is further discussed in $\S$ 5.5.

How does the $B/T$ vary as a function of Hubble type and galaxy stellar mass? Bulges with very high $B/T$ ($>0.4$) exist primarily in galaxies with high mass ($M_{\star}>6\times 10^{10}M_{\odot}$) and early types (S0/a to Sab). Bulges with very low $B/T$ ($<0.1$) lie primarily in lower mass galaxies with later morphologies (Sb to Sc). It is striking that $\sim 69\%$ of bright ($M_{B}\leq-19.3$) low-to-moderately inclined ($i\leq 70^{\circ}$) spirals have $B/T\leq 0.2$: these bulges are pervasive and exist across the whole spectrum of S0/a to Sd. The results are summarized in Table 9. We shall return to this point in § 5.8. We note again that these $B/T$ mass ratios were calculated assuming a constant $M/L$ ratio in the $H$-band for the bulge and disk components. As noted in $\S$ 5.2, if the bulge in these high mass spirals is much younger (older) than the disk and bar, then the $B/T$ can be overestimated (underestimated) by up to a factor of two, and the limiting value of 0.2 for the $B/T$ cited in the above fraction, would have to be modified in the extreme case to 0.1 and 0.4, respectively.

Some of the low $B/T\leq 0.2$ values for six barred S0/a and Sa galaxies in Figure 14 may at first look suspicious. Balcells et al. (2007) report the mean $B/T$ for S0 galaxies to be 0.25, so much smaller $B/T$ are potentially worrisome. OSUBSGS $H$-band images of these objects in Figure 17 show a smooth extended disk around the bulge. It should be noted that Hubble types were originally assigned on a combination of criteria including disk smoothness and spiral arm topology in addition to the prominence of the bulge. It is likely that these galaxies were assigned early Hubble types due to their smooth extended disks, in spite of their low bulge-to-disk ratio.

Similarly, some of the high $B/T\sim 0.4$ bulges in three of the Sc galaxies may at first seem odd. However, again, visual inspection of their image (Figure 17) reveals prominent spiral arms and clumpiness, which may explain why they were assigned late Hubble types.

How does the bulge Sérsic index $n$ vary as a function of Hubble type, and galaxy stellar mass (Figure 14), as well as $B/T$ (Figure 15)? The results are summarized in Table 9. Only a small fraction ($\sim 1\%$) of bright spirals have high $n\geq 4$ bulges; such bulges lie primarily in S0/a to Sab, and have a large $B/T>0.2$. A moderate fraction ($\sim 22\%$) have intermediate $2<n<4$ bulges; these exist in barred and unbarred S0/a to Sd, and their $B/T$ spans a wide range (0.03 to 0.5) with a mean of 0.29. A strikingly large fraction ($\sim 76\%$) of bright spirals have low $n\leq 2$ bulges; such bulges exist in barred and unbarred galaxies across a wide range of Hubble types, and their $B/T$ varies from 0.01 to 0.4, with most having $B/T\leq 0.2$.

As an independent check of our decomposition method, we compare our results with independently published decompositions.

We find our mean $H$-band $B/D$ (Figure 18) ratios are comparable to the $K$-band $B/D$ derived with the $1D$ bulge-disk decompositions of Graham (2001) and Trujillo et al. (2002). Like Graham (2001) and Trujillo et al. (2002), we find $B/D$ is widely variable with Hubble type and that mean $B/D$ steadily declines from Sa through Scd galaxies. Graham (2001) and Trujillo et al. (2002) find bulge indices are widely variable within a Hubble type, but they are in general $>1$ for early types and $<1$ for late types. We likewise find wide scatter in bulge index with $n<1$ bulges existing in both early and late types. Figure 18 is also in good agreement with the more recent results of Graham & Worley (2008), who have calculated, as a function of morphology, inclination and dust-corrected $B/D$ and bulge Sérsic indices. They find $B/D$ values are typically $<1/3$.

Another meaningful comparison can be made with Laurikainen et al. (2007) who, using their own $2D$ decomposition code, fit a hybrid sample containing some OSUBSGS galaxies combined with additional S0 galaxies. One difference between their work and ours is that they typically model bars with a Ferrers function, but may sometimes use a Sérsic profile, while we use only the latter. Also, they include additional components to model secondary bars or inner disks. They report a distinct offset in the mean $B/T$ between barred and unbarred galaxies, which we confirm in Figure 14. Their mean $B/T$ are similar to ours, and they conclude that pseudobulges exist throughout the Hubble sequence. The Sérsic indices derived by Laurikainen et al. (2007) are likewise similar, on the mean, to ours for both barred and unbarred systems. They likewise find $n\leq 2$ bulges across early and late Hubble type galaxies.

Balcells et al. (2003) emphasized that bulges typically have indices $\sim 3$ or lower. Our results in Figure 14 are consistent. We find a low frequency ($\sim 1\%$) of high $n\geq 4$ bulges, with most bulges having $n\leq 3$.

Stellar bars exert gravitational torques on the gaseous component and are particularly efficient in driving gas from the outer disk into the inner kiloparsec (see $\S$ 1). Thus, it would be natural to have a measure of bar strength that is sensitive to the strength of the gravitational torque and hence measures both the shape and mass of the bar.

Many measures of bar strength have been formulated. The $Q_{b}$ method of (Block et al. 2002; Buta et al. 2003; Buta et al. 2005) measures directly the gravitational torque at a single point along the bar. This method requires a value of scaleheight for the disk and a model of the potential to be made from the image. In the bar/inter-bar contrast method of Elmegreen & Elmegreen (1985) and Elmegreen et al. (1996), bar strength is parameterized as the ratio between peak surface brightness in the bar region and the minimum surface brightness in the inter-bar region. Elmegreen & Elmegreen (1985) and Elmegreen et al. (1996) also characterize bar strength with the maximum amplitude of the $m=2$ mode from Fourier decomposition. When ellipse fitting is applied, the maximum ellipticity of the bar, $e_{bar}$, can be used to characterize bar strength (e.g. MJ07). This constitutes a partial measure of bar strength only, however, as it offers no information about mass of the bar.

Bulge-disk-bar decomposition in the $H$-band provides another partial measure of bar strength through the $H$-band Bar/$T$ light ratio, which is a measure of the Bar/$T$ mass ratio under the assumption that the $H$-band $M/L$ ratio is the same for the bar and the rest of the galaxy, as discussed in $\S$ 5.2. Figures 19 and 20 explore the derived bar properties.

The upper left panel of Figure 19 plots the individual and mean Bar/$T$ against Hubble type. There is a wide range ($\sim 0.03$ to $\sim 0.47$) in the individual Bar/$T$ at a given Hubble type. The mean Bar/$T$ remains fairly constant with Hubble type from Sa to Sb, but shows a possible weak decline by about 0.1 from Sb to Sc. Their number statistics are too small to make any robust statement for later Hubble types. We also note that six systems have high Bar/$T$ above $0.3$; these are displayed in Figure 21.

Bar Sérsic indices are mostly below unity. Neither the individual, nor the mean bar Sérsic index, show any trend with Hubble type or with stellar mass, for Sa to Sc galaxies (Fig. 19). Thus, the steepness of the bar profile does not seem to depend on the Hubble type. Is the bar mass ratio and its mass profile related? There is a wide range in the individual Bar/$T$ at a given bar Sérsic index (Fig. 20). The mean Bar/$T$ rises with bar index out to a bar index of $\sim 0.6$, and then flattens out. This suggests that on the mean, bars of lower Bar/$T$ have flatter profiles.

Is there a relation between the bar strength and the bulge present in a galaxy? There is a wide range in the individual Bar/$T$ at a given $B/T$, and at a given bulge Sérsic index (Fig. 20). The mean Bar/$T$ shows a weak decline for bulge Sérsic indices above two. Similarly the mean Bar/$T$ shows a weak rise from 0.1 to 0.25 as $B/T$ rises out to 0.15, after which the trend flattens or reverses.

Both Bar/$T$ and maximum bar ellipticity $e_{bar}$ are partial measures of bar strength. Figure 19 shows mean Bar/$T$ may scale weakly with Hubble type. The bars with highest $e_{bar}$ (i.e, thin bars) are often termed strong bars, and $e_{bar}$ has been shown to correlate with $Q_{b}$. Total bar strength should scale with both bar mass and bar ellipticity. Does bar strength have a dependence on Hubble type? The upper left panel of Figure 20 plots the product of Bar/$T$ and $e_{bar}$, as determined by MJ07 for galaxies mutually classified as barred, against Hubble type. There is a wide range in Bar/$T\times e_{bar}$ in each bin, and mean bar strength shows no definite trend with Hubble type. We note that bars with high Bar/$T$ and high $e_{bar}$ should exert the largest gravitational torque and be most effective at driving gas inflows. A nice example is the oval or lens galaxy NGC 1317 (Figure 21); the bar has a low ellipticity, but its $B/T$ is large as it is extended and massive. Such bars or lenses may exert significant gravitational torques although they are not very elongated.

As outlined in $\S$ 5.3, we found that as many as $\sim 76\%$ of bright spirals have bulges with $n\leq 2$; such bulges exist in barred and unbarred galaxies across a wide range of Hubble types, and their spread in $B/T$ is from 0.01 to 0.4, with most having $B/T\leq$ 0.2. The variation of the bar fraction as a function of $B/T$ and bulge $n$ can provide important constraints on bulge formation scenarios ($\S$ 5.8). Table 10 shows our results. The bar fraction declines with bulge index; $\sim 65\%$ of the spirals with low $n\leq 2$ bulges host bars while intermediate $2<n<4$ bulges have a lower bar fraction ($\sim 38\%$). The high $n\geq 4$ bulges in the sample are unbarred, so the bar fraction is $0\%$. Systems with low $B/T$ are more likely to be barred. For $B/T\leq 0.2$, the bar fraction is high ($\sim 68\%$). Systems with $0.2<B/T<0.4$ and $B/T\geq 0.4$ have lower bar fraction ($\sim 42\%$ and $\sim 17\%$).

Overall, Table 10 shows bulges with low $n\leq 2$ and low $B/T\leq 0.2$ preferentially exist in barred galaxies. This is consistent with earlier work (Odewahn 1996; Barazza et al. 2008; Marinova et al. 2008; Aguerri et al. 2008, in prep.) where an enhanced optical bar fraction is seen is galaxies with late Hubble types or low $B/D$. It may be tempting to infer this result to mean the formation pathway of two-thirds of low-$B/T$ bulges is related to bars in that spontaneous or/and tidally induced bars play a role in bulge formation (with the remaining one-third of such bulges may have been formed either by mechanisms like retrograde minor mergers or short-lived bars). We caution that this type of cause-effect relationship is not the only scenario consistent with this result. It may also be possible that bar instabilities are favored in galaxies with low $B/T$ and no inner Lindblad resonances (ILR). Under these conditions, the swing amplifier model with a feedback loop (Julian & Toomre 1966; Toomre 1981; Binney & Tremaine 1987) may be responsible for bar formation and partly account for the high bar fraction in galaxies of low $B/T$.

Our observational results provide some interesting challenges for models of galaxy evolution that try to address the origin of present-day bulges. Any successful model must be able to account for the observed distribution of bulge $B/T$ and $n$ in high mass ($M_{\star}\geq 1.0\times 10^{10}M_{\odot}$) spirals, as shown in Table 9 and Table 10. In particular, the following results must be reproduced:

1. 1.

   In terms of the overall distribution of bulge $n$, as much as ($\sim 74\%$) of high mass spirals have bulge $n\leq 2$: such bulges exist in barred and unbarred galaxies and their $B/T$ ranges from 0.01 to 0.4, with most having $B/T\leq 0.2$ (Table 9). A moderate fraction ($\sim 24\%$) of high mass spirals have $2<n<4$, and just ($\sim 2\%$) have $n\geq 4$.

2. 2.

   Theoretical models often make more robust predictions on the bulge-to-total mass ratio $B/T$ than on the bulge index $n$, so we consider the empirical $B/T$ distribution in detail. We note that as much as $\sim 66\%$ of high mass spirals have bulges with $B/T\leq 0.2$ (Table 9). In terms of bar fraction, $\sim 68\%$ are barred (Table 10).

3. 3.

   The fraction of bars rises among spirals with low bulge index $n$. About 63% of spirals with low $n\leq 2$ bulges host bars, while the bar fraction in spirals with $2<n<4$ bulges (44%) is two-thirds as large (Table 10).

In a hierarchical Universe, there are several physical processes that contribute to the assembly of bulges: major mergers, minor mergers, and secular evolution. We briefly describe these, expanding on our introduction in $\S$ 1.

Major mergers, defined as those with mass ratio $M_{1}/M_{2}\geq 1/4$, typically destroy the extended outer stellar disks during violent relaxation, leaving behind a classical bulge. Such bulges are associated with modest-to-high bulge Sérsic indices, in the range $2<n<6$ (Hopkins et al. 2008; Springel et al. 2005; Robertson et al. 2006; $\S$ 5.8) in simulations. This trend is also consistent with the fact that among ellipticals, high luminosity ones tend to have a Sérsic index $n>4$, while low luminosity ones tend to have $2\leq n\leq 3.5$ (Caon et al. 1993; Kormendy et al. 2008). The final Sérsic index depends on the amount of residual gas the settles into a somewhat disky component. Simulations by Hopkins et al. (2008) find that the Sérsic indices of remnants from 1:1 gas-rich major mergers lie in the range of $2<n<4$, with most above 2.5 (see Fig. 22). This body of evidence strongly suggests that many bulges with $n>2$ might have a major merger origin.

Minor mergers, typically defined as those with mass ratio $1/10<M_{1}/M_{2}<1/4$, do not destroy the stellar disk of the primary system, but can contribute to building bulges via three pathways. Firstly, a fraction $F_{sat}$ of the satellite’s stellar mass can end up in the central region of the primary galaxy. The value of $F_{sat}$ depends on how centrally concentrated the in-falling satellite is. Typically, the more diffuse outer stellar body is tidally stripped, while the central core sinks by dynamical friction to the central region (e.g., Quinn et al. 1993; Walker et al. 1996). Secondly, a non-axisymmetric feature (e.g., a stellar bar or bar-like feature) can be induced in the main disk, and gravitational torques exerted by the feature can drive gas into the inner kpc (e.g., Hernquist & Mihos 1995; Jogee 2006 and references therein), where subsequent SF forms a compact high $v/\sigma$ stellar component, or disky pseudobulge. Most of the gas inflow happens during the merger phase and large gas inflow rates (e.g., $\gg 1$ $M_{\odot}$ per year) may be generated. Thirdly, gas inflow can also be caused by direct tidal torques from the companion (e.g., Hernquist & Mihos 1995; Eliche-Moral et al. 2006). It is to be noted that in the simulations by Hernquist & Mihos (1995), the gas inflow generated by non-axisymmetric features (e.g., bar-like features) in the inner part of the disk is much larger than that caused by direct tidal torques from the satellite. In the recent work of Eliche-Moral et al. (2006), N-body simulations of minor mergers followed by fits of 1D Sérsic+exponential models to the remnants, suggest that the bulge Sérsic index and $B/D$ ratio can grow as a result of the central re-concentration of stellar disk material in the primary system by tidal forces. Minor mergers are frequent under $\Lambda$CDM, and the likelihood of multiple successive minor mergers occurring during the formation of a galaxy is high. Bournaud, Jog, & Combes (2007) study the effects of repeated minor mergers on galaxy structure. They show that a disk galaxy undergoing successive minor mergers will eventually transform into an elliptical galaxy with an $r^{1/4}$-law profile and high $V/\sigma$. However, galaxy growth is not completely merger-driven and the efficiency of minor mergers at creating ellipticals must be regulated by other mechanisms (e.g., cold gas accretion).

In addition, the process of secular evolution can build a disky bulge (pseudobulge) between merger events. Here a stellar bar or globally oval structure in a non-interacting galaxy drives gas inflow into the inner kpc, where subsequent star formation forms a compact high $v/\sigma$ stellar component (e.g., Kormendy 1993; Jogee 1999; KK04; Jogee, Scoville, & Kenney 2005; Athanassoula 2005; Kormendy & Fisher 2005; Kormendy 2008). This process is different from that of minor mergers in the sense that it happens in the quiescent phase of the galaxy, between minor or major merger events. The prevalence of pseudobulges in galaxies of different Hubble types is discussed in KK04, and select examples of S0 galaxies with pseudobulges are also shown in Kormendy & Cornell (2004) and KK04.

The present-day bulge mass can be written as the sum of mass contributed from each process:

where

• •

  $f_{\rm maj}$ is the percentage of the bulge stellar mass, which is built by major mergers,

• •

  $f_{\rm min1}$ is the percentage of the bulge stellar mass, which is built during minor mergers from stars of the satellite. This depends on the fraction $F_{sat}$ of the satellite’s stellar mass, which ends up in the central region of the primary galaxy during each minor merger.

• •

  $f_{\rm min2}$ is the percentage of the bulge stellar mass, which is built during minor mergers from gas inflow caused by a tidally induced bar.

• •

  $f_{\rm min3}$ is the percentage of the stellar mass, which is built during minor mergers from gas inflow caused by tidal torques from the companion.

• •

  $f_{\rm sec}$ is the percentage of the stellar mass, which is built secularly between merger events from gas inflow caused by bars or ovals

In § 5.8, we compare our derived distribution of bulge $n$ and $B/T$ with hierarchical models that model major and minor mergers, but not secular evolution. The main goal of the model is to see whether bulges built via major mergers can account for the large fraction of high mass spirals with bulges of low $B/T$ or/and low $n$. A secondary goal is to see if a first order simplified prescription for minor mergers can broadly account for the observations. We stress here that bulge-building during minor mergers is modeled in a very simple way: all the stars in the satellite are assumed to contribute to the bulge of the larger galaxy (i.e., $F_{sat}=100\%$), and bulge-building via gas inflow driven through tidal torques and via gravitational torques from induced bars are ignored (i.e., $f_{\rm min2}$ = 0, and $f_{\rm min3}$ = 0). Furthermore, the models entirely ignore secular evolution between mergers. In a future paper, these extra terms will be addressed and a comprehensive picture built of the relative importance of minor mergers and secular processes in making present-day bulges.

We compare our data with the predictions from cosmological semi-analytical models based on Khochfar & Burkert (2005) and Khochfar & Silk (2006). We briefly describe the models first. The merger trees of dark matter (DM) halos are derived by using the extended Press-Schechter formalism (Press & Schechter 1974), as in Somerville & Kolatt (1999). When two DM halos merge, the merger time scale of the galaxies is calculated by considering the timescale it would take the satellite galaxies to reach the central galaxy at the center of the halo via dynamical friction (e.g., Kauffmann et al. 1999; Springel et al. 2001). The baryonic physics, which includes radiative cooling, star formation, and feedback from supernovae, is treated via semi-analytic prescriptions (see Khochfar & Silk (2006) and references therein). Baryonic mass inside the dark matter halos is divided between hot gas, cold gas, and stars. The hot gas is initially shock-heated to the halo virial temperature. As the gas radiatively cools, it settles down into a rotationally supported disk at the halo center. Cold disk gas is allowed to fragment and subsequently form stars according to the Schmidt-Kennicutt law (Kennicutt 1998). Star formation is regulated by feedback from supernovae using the prescription in Kauffmann et al. (1999).

Major mergers are typically considered as those with stellar mass ratio $M_{1}/M_{2}\geq 1/4$. In the simulations, one assumes that during a major merger any existing stellar disk is destroyed, gas is converted to stars with some star formation efficiency (SFE), and all stars present undergo violent relaxation to form a bulge. Therefore, the bulge-to-total stellar mass ratio ($B/T$) of a bulge immediately after a major merger is always one. Note that the SFE during a major merger is not assumed to be 100% as there is mounting evidence from SPH simulations (Springel & Hernquist 2005; Cox et al. 2008) that not all cold gas is converted to stars. Instead, the burst efficiency defined by Cox et al. (2008) is applied to control the fraction of stars formed due to the interaction. This efficiency is dependent on the relative masses of merging galaxies and is expressed as

where $e_{1:1}$ is the burst efficiency for a 1:1 merger and $\gamma$ fixes the dependence on mass ratio; Cox et al. (2008) find $e_{1:1}=0.55$ and $\gamma=0.69$. The remaining fraction (1-$e$) of gas is added to the gaseous disk and can start making stars.

As stated above, immediately after a major merger, the remnant is a bulge with a $B/T$ equal to one. As time proceeds, $B/T$ falls because a stellar disk grows around the bulge as hot gas in the halo cools, settles into a disk, and forms stars. The formation of stars by any residual cold gas left at the end of the major merger also helps to grow the disk. Thus $B/T$ falls until the next major merger happens, at which point $B/T$ is reset to one in the models.

The bulge may also grow in stellar mass due to minor mergers. Minor mergers are defined as mergers with mass ratio $1/10<M_{1}/M_{2}<1/4$, and the stellar disk of the large companion is not destroyed during such mergers. The models assume that during minor mergers, all the stars in the satellite are added to the bulge of the host, while the gas settles in the disk. When DM halos grow by accretion or minor mergers, the hot gas that comes in with a satellite is immediately stripped and added to the hot gas component of the host. The cold gas in the disk of the satellite is only added to the cold gas of the host if they merge. Until they merge the satellite is using up its own cold gas to make stars.

Fig. 23 shows the relationship between the present-day $B/T$ of a a high mass ($M_{\star}\geq 1.0\times 10^{10}M_{\odot}$) spiral and the redshift $z_{\rm last}$ of its last major merger. As expected, systems where the last major merger occurred at earlier times, have had more time to grow a disk and have a lower $B/T$. The dispersion in the present-day $B/T$ at a given $z_{\rm last}$ is due to the different times spent by a galaxy in terms of being a satellite versus a central galaxy in a DM halo, since the cooling of gas and the growth of a disk is stopped when a galaxy becomes a satellite. Thus, galaxies that became a satellite galaxy shortly after their last major merger stayed at high $B/T$. Conversely, galaxies that continued to be a central galaxy for a long time after their last major merger will have low $B/T$.

The present-day $B/T$ of a high mass ($M_{\star}\geq 1.0\times 10^{10}M_{\odot}$) spiral depends on its major merger history. In particular, we note from Fig. 23 that a high mass ($M_{\star}\geq 1.0\times 10^{10}M_{\odot}$) galaxy, which has undergone a past major merger since $z\leq 2$ will end up hosting a present-day $B/T>0.2$. In effect, a high mass spiral can have a present-day $B/T\leq 0.2$ only if its last major merger occurred at $z>2$ (lookback times $>10$ Gyr).

The predicted distribution of present-day $B/T$ depends on the galaxy merger history in the models and it is relevant to ask how well the latter is constrained observationally. Over the redshift range $z\sim 0.24$ to 0.80 (lookback times of 3 to 7 Gyr), recent studies by Jogee et al. (2008, 2009) find that among high mass ($M_{\star}\geq 2.5\times 10^{10}M_{\odot}$) galaxies, $\sim$ 10% of galaxies are undergoing mergers of mass ratio $>1/10$, and $\sim$ 3% are undergoing major mergers of mass ratio $>1/4$. These findings agree within a factor of less than $\sim 2$ with the merger rates from the models of Khochfar & Burkert (2001) over $z\sim$ 0.24 to 0.80. At higher redshifts, the empirical merger rate/fraction is uncertain due to relatively modest volumes and bandpass shifting effects, but there is a general trend towards higher merger fractions at higher redshifts (e.g., Conselice et al. 2003). The models used here (Khochfar & Burkert 2001) agree with this trend and predict that $\sim$ 13.5% and $\sim$ 20% of high mass ($M_{\star}\geq 1.0\times 10^{10}M_{\odot}$) spirals have undergone major mergers since $z\leq 2$ and $z\leq 4$, respectively (see Table 11).

The contribution of galaxies with different merger histories to the present-day $B/T$ distribution are shown in Table 11. The top and middle parts of the table describe systems with and without major mergers since $z\leq 2$ and $z\leq 4$, respectively. In the model, $\sim$ 13.5% of present-day high mass ($M_{\star}\geq 1.0\times 10^{10}M_{\odot}$) spirals, experienced a major merger since $z\leq 2$, causing most of them ($\sim$ 11.2%) to have a present-day high $B/T>0.4$ and a negligible fraction ($\sim$ 0.1%) to have a low present-day $B/T\leq 0.2$. In contrast, the remaining $\sim$ 86.5% spirals experienced no major merger at $z\leq$ 2, and most (67.2%) of them have a present-day low $B/T\leq 0.2$. If the comparisons are extended to systems without a major merger since $z\leq 4$, the numbers are very similar (see middle part of Table 11).

Table 11 shows that there is good agreement between the model and data for the fraction of high mass spirals with present-day low $B/T\leq 0.2$ ($\sim$ 67% in the model versus $\sim$ 66% in the data). The model contribution to low $B/T\leq 0.2$ comes almost entirely from galaxies, which have not had a major merger since $z\leq 2$ (see column 4 in Table 11). In fact, most of these galaxies have not even had a major merger since $z\leq 4$, as illustrated by the bottom part of Table 11. In the model, the fraction ($\sim 1.6\%$; column 3 of Table 11) of high mass spirals, which have undergone a major merger since $z\leq 4$ and host a bulge with a present-day $B/T\leq 0.2$, is a factor of over thirty smaller than the observed fraction ($\sim 66\%$) of high mass spirals with $B/T\leq 0.2$. Thus, bulges built via major mergers since $z\leq 4$ seriously fail to account for the bulges present in $\sim 66\%$ of high mass spirals. These results are also illustrated in Fig. 24, which shows the comparison between data and models for the cumulative fraction of high mass spirals as a function of present-day $B/T$.

It is also interesting to note from Table 11 that although the models reproduce well the frequency of bulges with present-day low $B/T\leq 0.2$, they tend to over-produce the frequency of present-day high $B/T>0.4$ systems by nearly a factor of two ($\sim$ 14% in the model versus $\sim$ 8% in the data; see columns 6 and 2 in middle part of Table 11). Most of this overproduction stems from major mergers at $z\leq 4$ ($\sim$ 13%; column 4 in middle part of Table 11). This suggests that major mergers, as currently modeled here, are building bulges too efficiently.

One possible solution to this problem might relate to the suggestion by Hopkins et al. (2009) that the efficiency of bulge-building during the major merger of two spirals depends not only on the mass ratio $M_{\rm 1}/M_{\rm 2}$ of the merger, but also depends on the cold gas mass fraction $f_{\rm gas}$ in the disk. In their semi-analytic models, the entire stellar mass of the satellite violently relaxes, but the fraction of stellar mass in the primary disk that violently relaxes and adds to the bulge is $M_{1}/M_{2}$. This differs with our models where the entire primary stellar disk is always destroyed in a major merger. Furthermore, the fraction $F$ of the total gas mass, which loses angular momentum, falls to the nucleus, and is transformed into stellar mass in a nuclear starburst, is $\sim(1-f_{\rm gas})\times(M_{1}/M_{2})$. In particular, $F$ is lower for more gas-rich disks, causing a suppression of the burst efficiency in gas-rich systems, and a reduction in the stellar mass that ends up in the bulge built during the major merger. The predictions for the distribution of $B/T$ from the Hopkins et al. (2009) models are shown in the bottom of Table 11, with major mergers considered as those with baryonic galaxy-galaxy mass ratio $>$ /1/3. Due to the reduced stellar mass that ends up in the bulge after a major merger, the Hopkins et al. (2009) models tend to yield lower $B/T$ after such a merger. Thus, the models produce overall fewer high $B/T>0.4$ systems, and more intermediate 0.2 $<B/T<$ 0.4 and low $B/T\leq 0.2$ systems (see column 6 in Table 11). Nonetheless, the predictions are not very different from the models by Khochfar & Burkert (2005) and Khochfar & Silk (2006), which we use in this paper. In particular, the conclusion that the large frequency of high mass ($M_{\star}\geq 1.0\times 10^{10}M_{\odot}$) spirals with low $B/T\leq 0.2$ bulges can only be accounted for by spirals without a major merger since $z\leq 2$ also holds with the Hopkins et al. (2009) models.

Thus, we conclude that the observed large frequency ($\sim$ 66%) of high mass ($M_{\star}\geq 1.0\times 10^{10}M_{\odot}$) spirals with low present-day $B/T\leq 0.2$ can be accounted for in our and other hierarchical models by high mass spirals, which have not undergone a major merger since $z\leq 2$, and most of which have not even experienced a major merger over the last 12 Gyr since $z\leq 4$. In the models, most of these present-day low $B/T\leq 0.2$ bulges are built by minor mergers since $z\leq 4$. As noted earlier, our models explore bulge-building via minor and major mergers, but do not explicitly incorporate secular processes (see $\S$ 5.7). In practice, secular processes may contribute to the building of present-day low $B/T\leq 0.2$ bulges, and are particularly relevant at $z\leq 2$, where major mergers cannot build such bulges.

For completeness, we further explore how sensitive are our results to other assumptions made in the data and models of Khochfar & Burkert (2005) and Khochfar & Silk (2006):

• •

  How sensitive are the results to the mass ratio used to separate major and minor mergers? Fig. 25 is similar to Fig. 24 except that the model now defines major mergers as those with mass ratio $M_{1}/M_{2}\geq 1/6$. In this case, about 30% of the model spirals undergo major mergers since $z\leq 4$ rather than $\sim 20\%$. The overall model $F$ (black dashed line) now under-predicts the data $F$ by about 10% for $B/T>$ 0.2. However, the main conclusion that bulges built by major mergers since $z\leq 4$ cannot account for most of the low $B/T\leq 0.2$ bulges, present in a large percentage ($\sim 66\%$) of spirals still holds.

• •

  How sensitive are the results to the $B/T$ cut used to define spirals? Fig. 26 is similar to Fig. 24 except that here spirals are considered to be systems with a $B/T\leq 0.55$ rather than 0.75 in the models, and a corresponding cut is applied to the data points. The results are similar to Fig. 24

• •

  How sensitive are the results to our assumed constant $H$-band mass-to-light ($M/L$) for the bulge, disk, and bar? Fig. 27 is similar to Fig. 24 except that the $B/T$ of all the observed galaxies has been multiplied by a factor of two, in order to test what would happen in the case where the $M/L$ ratio of the bulge in $H$-band is twice as high as that of the disk and bar. This could happen in an extreme example where the dominant bulge stellar population was much older (e.g. 12 Gyr) than the age of the dominant disk stellar population (e.g., 3 Gyr). In such a case, the fraction of high mass spirals with $B/T\leq$ 0.2 would change from $\sim 66\%$ in Fig. 24 to $\sim 55\%$, and deviate from the model overall model $F$ (black dashed line) by $\sim 20\%$. However, the main conclusion that bulges built by major mergers since $z\leq 4$ cannot account for most of the low $B/T\leq 0.2$ bulges, present in a large percentage ($\sim 55\%$) of spirals still holds.

Finally, it is important to note that so far we have compared the data and model only in terms of bulge $B/T$, but not in terms of bulge index $n$ or in terms of bar fraction. In effect, we have only shown that the models reproduce a subset of the results outlined in points (1)-(3) of $\S$ 5.7. Since the semi-analytic models do not produce a distribution of bulge index $n$, we resort to presenting only an indirect comparison in Table 12. We assume that bulges, which form in major mergers have a bulge $n>2.5$. This assumption is based the evidence presented in $\S$ 5.7. Thus, in Test 1 of Table 12, we compare the fraction ($\sim$ 66%) of galaxies in the semi-analytic models having $B/T\leq 0.2$ and no major merger since $z\leq 4$, to the observed fraction ($\sim$ 65%) of galaxies with $B/T\leq 0.2$ and bulge $n\leq 2.5$. There is close agreement between the two values. In Test 2 of Table 12, the fraction ($\sim$ 12.7%) of model galaxies with $B/T>0.4$ and no major merger since $z\leq 4$ is a factor of $\sim$ 3 higher than the fraction ($\sim$ 3.5%) of high mass spirals with $B/T>0.4$ and bulge $n>2.5$. Similarly, the fraction ($\sim$ 1.6%) of model galaxies with $B/T\leq 0.2$ and a past major merger since $z\leq 4$ is also a factor of $\sim$ 2 higher than the observed fraction ($\sim$ 0.9%) of high mass spirals with $B/T\leq 0.2$ and bulge $n>2.5$ (Test 3, Table 12). Thus, in terms of bulge $B/T$ and $n$, there is good agreement between data and model for Test 1 (involving model galaxies with no major mergers since $z\leq 4$). This suggests that the vast majority of bulges with $B/T\leq 0.2$ and $n\leq 2.5$ likely formed in galaxies having had no major merger since $z\leq 4$. However, the agreement is less good for Tests 2 and 3 (involving model galaxies with major mergers since $z\leq 4$) and this suggests that the models may be building bulges a little bit too efficiently during a major merger, in agreement with the conclusion reached earlier.

What about the role of bars in the formation of these bulges of low $B/T$ and low $n$? A detailed direct comparison with the semi-analytic models is not possible as the role of bars is not yet modeled, but related comparisons are possible. First, it is important to note that bar-driven gas inflow into the inner kpc and the subsequent building of disky stellar components or ‘pseudobulges’ (see $\S$ 1) can happen in both isolated galaxies and in minor mergers ($\S$ 5.7), since bars can be spontaneously induced in an isolated disk or tidally induced during an interaction or minor merger. The triggering of a bar is favored in a prograde interaction or minor merger. Thus, bulge-building via induced bars is more likely to happen in prograde rather than retrograde minor mergers. Statistically about half of minor mergers might be prograde or prograde-like, and half retrograde. Thus, one would expect bars to be induced in only half of the minor mergers. If this assumption is correct and if most of the mass in bulges with present-day $B/T\leq 0.2$ is formed in minor mergers, then one would expect only about half of these bulges to host bars. This is close to what is observed, as shown by Table 10. We see that $\sim 63\%$ of high mass spirals with low $n\leq 2$ bulges and $\sim 68\%$ of spirals with low $B/T\leq 0.2$ bulges host bars. This suggests that in high mass spirals, spontaneous and/or tidally induced bars may play a part in forming up to two-thirds of $B/T\leq 0.2$ or $n\leq 2$ bulges. The remaining one-third of such bulges may have been formed either by mechanisms that do not involve bars (e.g., retrograde minor mergers) or by bars that are not long-lived.