The Dependence of Dark Matter Halo Properties on the Morphology of Their Central Galaxies from Weak Lensing

We use the galaxy morphology catalog measured by Xu & Jing (2022), with the galaxy sample selected from Baryon Oscillation Spectroscopic Survey (BOSS, Ahn et al. (2012); Bolton et al. (2012)) constant-mass (CMASS), with an area of approximately 500 deg² overlapping with HSC. We select central galaxies with stellar masses in the narrow range of $11.3<\log(M_{*}/{\rm M_{\odot}})<11.7$ and a redshift range of 0.45-0.7 as our lens sample. The Sérsic indices are obtained by fitting the Sérsic profile to the corresponding HSC z-band images, as detailed in Xu & Jing (2022). In this study, we focus on galaxies with Sérsic indices ranging from 0.8 to 8.

The shear catalog we use is processed through the Fourier_Quad (FQ; Zhang, 2008) pipeline, based on the third public data release of HSC, which covers an area of about 1400 deg² and includes about 100 million galaxies (Liu et al., 2024). The FQ method employs multipole moments of galaxy images in Fourier space for shear measurements, including five shear estimators: $G_{1}$, $G_{2}$, $N$, $U$, and $V$. $G_{i}$ is based on the galaxy quadrupole moments in Fourier space. $N$ is a normalization factor used to estimate and correct the impact of noise and the Point Spread Function (PSF). $U$ and $V$ are the additional terms taking care of the parity properties of the shear estimators in the PDF-SYM method, which is a novel way of estimating shear statistics. Their specific definitions can be found in Zhang et al. (2016).

The shear catalog records basic information about galaxies such as position, signal-to-noise ratio ($\nu_{F}$, Li & Zhang (2021)), magnitude, and shear estimators. Photometric redshift (photo-$z$) information is sourced from the DEmP method in Nishizawa et al. (2020), which demonstrates very low bias and scatter for photo-z less than 1.5. Therefore, we limit the maximum photo-$z$ of background galaxies to 1.5. Additionally, in the lensing analysis, we ensure that the photo-$z$ of background galaxies is larger than the lens redshift plus 0.1, i.e., $z_{s}>z_{l}+0.1$. To enhance the accuracy of shear measurements, we select background galaxies with $\nu_{F}>4$. HSC observes in five bands: $g$, $r$, $i$, $z$, and $y$, with the $i$ band offering the best imaging quality. In the FQ method, shear estimators are measured for each galaxy image without stacking images from different bands. According to Liu et al. (2024), we choose data from the $r$, $i$, and $z$ bands as the shear sample, as their multiplicative shear bias in the field distortion tests is at about 1-2 percent level at most, and their galaxy-galaxy lensing measurements show consistency with the results from the HSC year-one official shear catalog.

More recently, it is found in Shen et al. (2024) that additional shear multiplicative bias can be induced by the photo-z cut of the background sample, which is a selection effect unknown before. We therefore find it necessary to calibrate for the multiplicative bias that may arise due to the selection of a subset of the background sample in the measurement of, e.g., galaxy-galaxy lensing. Fortunately, the FQ shear catalog keeps the field distortion information for each galaxy image/shear estimator, it therefore provides a convenient on-site calibration approach. Its details relevant to this work is provided in Appendix A.

In our work, we study two hydrodynamic simulations, TNG and SIMBA, to compare our observational results and explore the origins of various relationships. TNG is a series of large cosmological gravity-magnetohydrodynamics simulations that incorporate comprehensive models of galaxy formation physics (Pillepich et al., 2017). We utilize the simulation with a box length of $L_{\rm box}=205{\rm Mpc}/h$, denoted as TNG300. The SIMBA simulation provides detailed modeling of black hole growth and feedback, successfully reproducing many observational results (Davé et al., 2019; Thomas et al., 2019; Cui et al., 2021). Specifically, AGN feedback in SIMBA operates in three modes based on the accretion rate: the radiative mode at high Eddington ratios $f_{\rm Edd}$, and the jet and X-ray modes at low $f_{\rm Edd}$. For our primary analysis, we use the full physics simulation with $L_{\rm box}=100\,{\rm Mpc}/h$, denoted as S100. According to Weinberger et al. (2018); Davé et al. (2019); Rodríguez Montero et al. (2019), kinetic feedback modes are primarily responsible for quenching massive central galaxies in both SIMBA and TNG, while major galaxy mergers are not the main drivers of most quenching events. Thus, we also utilize various AGN feedback simulations from SIMBA with $L_{\rm box}=50\,{\rm Mpc}/h$ (S50) to study their effects.

We first select central galaxies with stellar masses in the range of $11.3<{\rm log}M_{*}/{\rm M_{\odot}}<11.7$ from TNG300 and S100 to match the observation, and then choose the range $11<{\rm log}M_{*}/{\rm M_{\odot}}<11.5$ for S50 series simulations due to their limited volume. The stellar mass is defined as the sum of the masses of stars within twice the stellar half-mass radius. Assuming a constant mass-to-light ratio, we create galaxy images for each galaxy’s stellar mass distribution, extending to $4\times 4$ times the half-stellar mass radius. Then we convolve a small gaussian PSF to reduce noises. We fit these images with a 2D ${\rm S\acute{e}rsic}$ profile to obtain their ${\rm S\acute{e}rsic}$ index $n$. Additionally, we fit the surface density of the host halos with an NFW profile to obtain their halo mass $M_{h}$ and concentration $c$. The definitions in NFW model are aligned with our lensing measurements. We use the position with the minimum gravitational potential energy as the real center of halos, hence the off-centering effect can be ignored in this part. Finally, since there is a slight correlation between $n$ and $M_{*}$, we resample the stellar mass distribution of galaxies with different $n$ to make their distributions similar, in order to eliminate the impact of stellar mass on the results.

Galaxy-galaxy lensing measures the correlation between the position of an object and the surrounding shear, which can reflect the excess surface density (ESD) around the object. For a spherically symmetric lensing potential, the relationship between shear and ESD is given by

where $\overline{\Sigma}_{0}(<r)$ refers to the average surface density within a radius $r$. $\Sigma_{c}$ is the comoving critical surface density, defined as

Here, $c$ is the speed of light, $G$ is the gravitational constant, and $D_{\rm s},D_{\rm l}$, and $D_{\rm ls}$ are the angular diameter distances for the lens, source, and lens-source systems, respectively.

In our work, we employ the PDF-symmetrization (PDF-SYM, Zhang et al. (2016)) method to measure the ESD of galaxies. The essence of this method lies in constructing the Probability Density Function (PDF) of the shear estimators, and adjusting the assumed underlying shear value to achieve the most symmetric state of the PDF, thereby obtaining the optimal shear estimate. For the ESD signal at a given radius $r$, we can obtain the PDF of the shear estimators $P(G_{t})$ for the background galaxies. By hypothesizing an ESD signal $\widehat{\Delta\Sigma}$, we adjust $P(G_{t})$ to $P(\hat{G}_{t})$ by modifying each value of $G_{t}$ via:

When $\widehat{\Delta\Sigma}$ equals the true ESD signal, the new distribution $P(\hat{G}_{t})$ will achieve the most symmetric state.

In this study, we uniformly divide the radius range from 0.05 to 20 Mpc/$h$ into 8 bins in logarithmic space for measurement and estimate the covariance matrix by dividing the sky into 86 sub-regions using Jackknife method. Additionally, to remove the effects of boundary from the observed region and some systematic bias, we subtract the signal contribution from random points, the number of which is 10 times the number of lens galaxies. Additionally, we correct the multiplicative biases introduced in shear measurements and details are shown in Appendix A.

For the ESD model, we take into account the effects of stellar mass of galaxies, individual halos, off-centering effects, and the 2-halo term. The mass of the galaxy primarily affects the ESD at small radii, typically regarding the central galaxy as a point source and

where $M_{*}$ is the stellar mass of central galaxy, as shown in Table 1 for all subsamples. We adopt NFW model (Navarro et al., 1997) to describe an individual halo profile, whose density is expressed as

with $\rho_{0}=\rho_{m}\Delta_{\rm vir}/(3I)$, where $I=[\ln(1+c)-c/(1+c)]/c^{3}$. $r_{\rm vir}$ is the virial radius, $r_{s}$ is the scale radius and the concentration $c$ of halos is defined by $c=r_{\rm vir}/r_{s}$. In this paper, we define a halo as a sphere with an overdensity of $\Delta_{\rm vir}=180$ times the mean matter density $\rho_{m}$, with a halo mass $M_{h}=180(4\pi/3)\rho_{m}r^{3}_{\rm vir}$. The unit of $M_{h}$ in this paper is ${\rm M_{\odot}}/h$. We employ the analytical expression for the surface density $\Delta\Sigma_{\rm NFW}$ as presented in Yang et al. (2006).

The off-centering effect is a significant consideration within the virial radius of halos, as it can lead to an underestimation of the ESD signal at smaller radii. Ignoring this effect might result in an underestimation of the halo’s mass and concentration. Thus, the 1-halo term ESD model typically divides the halo profile into well-centered and off-centered components,

where $f_{\rm c}$ is the proportion of well-centered halos. $\Sigma_{\rm off}$ refers to mean surface density of mis-centered halos, which can be derived from

where $P\left(R_{\rm off}\right)$ is the distribution of offset centers on radius and it is generally assumed to be Rayleigh distribution (Johnston et al., 2007)

Given that our measurement radii significantly exceed the common virial radii of halos, the 2-halo term exerts a substantial influence on the ESD. According to Yang et al. (2006), the surface density of the 2-halo term is correlated with the galaxy-matter cross-correlation,

where $\xi_{\rm gm}(r)=b_{h}(M_{h},z)\xi_{\rm mm}(r)$. $b_{h}(M_{h},z)$ is the halo bias, which is modeled by Tinker et al. (2010). $\xi_{\rm mm}(r)$ is calculated by COLOSSUS (Diemer, 2018) using a semi-analytical power spectrum from Eisenstein & Hu (1998). Finally, our ESD model can be summerized as

To enhance the accuracy of our model, we divide the redshift range of 0.45 to 0.7 into five bins and perform an integration based on the distribution of lens galaxies. Our model comprises four free parameters, halo mass $M_{h}$, concentration $c$, the scatter of off-centering $\sigma_{s}$ and the proportion of well-centered halos $f_{c}$, which we fit using Markov Chain Monte Carlo techniques (Christensen & Meyer, 2000).

We divide the galaxy samples with stellar masses in the range of $11.3<{\rm log}M_{*}/{\rm M_{\odot}}<11.7$ into five subsamples. The range of their Sérsic indices $n$, the number of galaxies, and the average stellar mass are presented in Table 1. Figure 1 shows the measurements of ESD, with their corresponding best-fit curves and the different components of the model. It is evident that the ESD increases with higher $n$. The “All” section in Table 1 presents the fitting results for all parameters. To clearly show the trend of the parameters, we depict the relationship between halo mass $M_{h}$ and halo concentration $c$ as blue dots in Figure 2, with the horizontal axis representing the average $n$ for each sample. We find that halo mass increases significantly with the increase of the ${\rm S\acute{e}rsic}$ index, which is consistent with the conclusions from Xu & Jing (2022). Additionally, we observe a possible evidence that more concentrated galaxies (high $n$) may reside in more concentrated halos (high $c$), although the uncertainties are relatively large with current data. We fit the blue dots in the figure using linear relationships , yielding:

In terms of galaxy properties, higher Sérsic indices are typically associated with early type galaxies with red colors, while lower Sérsic indices refer to late type galaxies with blue colors (see, e.g., Figure 1 of Xu & Jing, 2022). It has long been recognized that red and blue galaxies exhibit distinct SHMR (e.g. More et al., 2011; Peng et al., 2012; Wang & White, 2012; Rodríguez-Puebla et al., 2015; Mandelbaum et al., 2016; Wang et al., 2021; Posti & Fall, 2021; Xu & Jing, 2022; Alonso et al., 2023), with red galaxies hosted by more massive dark matter halos than blue ones at fixed stellar mass. Given the strong correlation between morphology and color, it is crucial to determine which of these secondary dependencies is more fundamental. Does the halo mass dependence on $n$ arise due to color, or is it the other way around?

Therefore, we select galaxies with rest-frame $u-r$ colors in a narrow range of 2.3 to 2.7. We divide them into three subsamples according to the Sérsic index, $n$, and resample their color distributions to be the same to eliminate the correlation between color and $n$. The left panel of Figure 3 shows their ESD measurements. The red empty squares with errorbars in Figure 2 show the best-fit $M_{h}$ and $c$ versus $n$, and the “Fixed color” row in Table 1 presents the best-fit parameters and associated errors. We find that after the color is fixed, the dependencies of both $M_{h}$ and $c$ on $n$ are still consistent with the results of the blue dots. The best-fit relations are

This implies that the correlation between $n$ and halo properties is not predominantly driven by differences in galaxy color. Conversely, to explore whether the color dependence is driven by morphology, we also select galaxies with $1<n<6$ and divide them into two color subsamples: blue galaxies with $2<u-r<2.25$ and red galaxies with $2.25<u-r<2.6$. For fair comparison, these two sub-samples are resampled to have the same distribution of $n$, with average $n$ of $2.98\pm 1.07$ and $3.10\pm 1.09$, respectively. The middle panel of Figure 3 shows their ESD measurements. The orange and blue rings in Figure 2 represent the results for the red and blue galaxy samples, with their sizes indicating the 1$\sigma$ error range. The “Fixed $n$” section in Table 1 shows the best-fit results. We find that even when $n$ is fixed, halo mass still exhibits a color bimodality, with red galaxies residing in more massive halos than blue galaxies, while their halo concentrations remain consistent. This suggests that a better secondary proxy for halo mass should involve a combination of both color and morphology. Due to data limitations, we are unable to determine the optimal combination, but this presents an intriguing topic for future research. On the other hand, halo concentration appears to be primarily linked to morphology, although more precise measurements are needed to confirm this conclusion.

Galaxy properties and evolution are typically shaped by their surrounding environment, including local matter density and the evolutionary history of their host halos. We further investigate the properties of galaxy groups hosting the central galaxies using the group catalog from Yang et al. (2021), which provides details on group richness and member galaxies.

The left panel of Figure 4 shows the richness distributions of groups with different $n$ of central galaxies in our sample. The vertical dashed lines indicate the mean richness for each sample. We find that central galaxies with larger $n$ are more likely to be located in groups with higher richness. Also, we calculate the projected number density profiles of satellite galaxies at various projected radii, displayed in the right panel of Figure 4. It shows that the surrounding density of companion satellite galaxies increases with the $n$ of central galaxies, consistent with the results from Xu & Jing (2022). In fact, many previous studies have pointed out the tight correlation between the host halo mass and cluster richness or satellite abundance (e.g. Rozo et al., 2009; Andreon & Hurn, 2010; Wang & White, 2012; Simet et al., 2017; Murata et al., 2018, 2019; Phriksee et al., 2020; Chiu et al., 2020; Wang et al., 2021; Tinker et al., 2021; Chen et al., 2024), and thus our results here are related to the fact that galaxies with larger Sérsic indices are hosted by more massive dark matter halos.

In Figure 2, we find evidence of a positive correlation between the Sérsic index and halo concentration. A similar pattern is observed in the satellite distribution in Figure 4. While more precise measurements are needed to fully confirm this relation, it would be intriguing to explore its theoretical implications.

It is generally believed that more massive halos form later and have lower concentrations (e.g. Maccio et al., 2007; Duffy et al., 2008; Mandelbaum et al., 2008; Correa et al., 2015; Ludlow et al., 2014, 2016), as material continues to flow in from outer regions (e.g. Fong & Han, 2021; Gao et al., 2023; Diemer, 2022; He et al., 2024). However, in our results, we find that both the host halo mass and halo concentration are positively correlated with the Sérsic index of galaxies (showing positive $n-M_{h}$ and $n-c$ relations). If the $n-c$ relation were solely driven by the negative halo mass-concentration relation for dark matter halos, we would expect a negative $n-c$ relation instead. Therefore, the positive correlation we observe between halo concentration and the Sérsic index is likely driven by more intrinsic factors beyond the typical mass-concentration relation of dark matter halos.

Navarro et al. (1997) propose that older halos, which formed in higher cosmic densities, tend to have larger characteristic densities and concentrations. Using high-resolution N-body simulations, Zhao et al. (2009) find that the halo concentration is tightly correlated with the universe age when its progenitor first reaches 4% of their current mass, i.e. $t/t_{0.04}$ in their work. Therefore, we speculate that two main factors may influence the $n-c$ relation in our results. One is introduced by the halo mass-concentration relation, that the positive $n-M_{h}$ relation would introduce negative correlations between $n$ and $c$. The other is that more concentrated galaxies with larger Sérsic indices form earlier at fixed stellar mass, with their host halos form earlier as well, and dark matter halos with earlier formation times have larger concentrations. Among these two factors, the latter may play a dominant role for our galaxy samples.

To further test our hypothesis, we control halo mass in the sample by using the halo mass from the group catalog $M_{G}$, which is assigned using the abundance matching technique (Yang et al., 2021). We select galaxies with $10^{13.3}<M_{G}<10^{13.8}{\rm M_{\odot}}/h$ and divide them into two subsamples based on their Sérsic index ($n$) to measure the ESD. The right panel of Figure 3 shows the ESD measurements, Table 1 presents sample information and best-fit parameters , which are also shown by the green square points in Figure 2. We find that the halo masses from the group catalog of the two subsamples are very consistent, but halo concentrations for galaxies with larger Sérsic indices are higher, still consistent with the $n-c$ relation shown and discussed earlier. Furthermore, compared to the $n-c$ relation without controlling halo mass, the relation becomes steeper. This implies that by reducing the negative $M_{h}-c$ relationship caused by accretion and merging, the positive correlation between $n$ and $c$ becomes slightly strengthened. This also supports our speculation that the host halos of galaxies with higher Sérsic indices are formed earlier, leading to higher halo concentrations, and this effect is stronger than the influence of later-time halo mergers to decrease the concentration at least for the massive CMASS galaxies at $z\sim 0.57$.

Figure 5 compares our main results (blue dots) with those obtained using the PAC method in the work of Xu & Jing (2022) (orange squares). To ensure a consistent basis for comparison, we convert the halo masses in our results to their definition, where $\Delta_{\rm vir}=18\pi^{2}+82x-39x^{2}$ and $x=\Omega_{m}(z)-1$. We observe that our results systematically yield higher halo masses. This discrepancy may stem from the assumption in Xu & Jing (2022) that massive central galaxies are perfectly situated at the halo centers, neglecting the off-centering effects in observations. By setting $f_{c}=1$ in our model, disregarding the central offsets, we show the fitting results as green hollow points in the figure. In this test, our results align well with theirs.

Before proceeding, we need to clarify some problems when measuring the galaxy morphology in SIMBA: the galaxies in these simulations generally exhibit very low galaxy concentrations, with $n$ not exceeding 2. According to Davé et al. (2019), the size of low-mass quenched galaxies in SIMBA is a notable failure of the current model, as their half-light radii are larger than those of star-forming galaxies, which contradicts observational trends. A larger radius usually implies a lower concentration, leading to elliptical galaxies that should have higher $n$ but obtaining very low $n$ values in the simulation. Nevertheless, we endeavor to incorporate the results from SIMBA in our analysis within the limited range of $n$ it produces.

Firstly, we use snapshots from TNG300 and SIMBA at two redshifts for the main analysis, $z=0$ and $z=0.55$ respectively¹¹1The latter is actually $z=0.556839$ for SIMBA, but denoted as 0.55.. Figure 6 displays the relationships between the ${\rm S\acute{e}rsic}$ index, $n$, and the halo mass, $M_{h}$, halo concentration, $c$, and richness, $\lambda$, for central galaxies in simulations. The properties of galaxies and halos are represented by scatter plots, with the linear fits represented by lines, and the 16-84 percentiles displayed as shaded areas. Regarding the relationship between halo mass and $n$, we find that both TNG300 and SIMBA exhibit correlations, even though the correlation in TNG300 is weaker than our observational results (Eq.12). While the S100 simulation shows a stronger correlation, considering its incorrect galaxy morphology, we only qualitatively analyze the trends. Both simulations demonstrate a positive correlation between $n$ and halo abundance $\lambda$, which is qualitatively consistent with our results (Figure 4). Moreover, we note that the correlations between $n$ and other galaxy and halo properties at higher redshift are weaker than those at lower redshift.

However, neither of the two simulations reproduce significant positive correlations between halo concentration and $n$ as have been observed in the real data. A strong negative correlation even shows in SIMBA. As previously discussed, the $n-c$ relation at fixed stellar mass is affected by: 1) the mass-concentration ($M_{h}-c$) relation of dark matter halos; and 2) the intrinsic $n-c$ relation at fixed halo mass. To validate our idea that more concentrated galaxies are hosted by more concentrated halos, we control both halo mass and stellar mass in a subsample from the TNG300 at $z=0$. The left panel of Figure 7 shows the correlation between halo mass and stellar mass for the red points in Figure 6, color coded by $n$. We select galaxies within the red box in the panel and replot the correlations between $n$ and other properties for them in right panels of Figure 7. We find that after controlling both the stellar mass and halo mass to a narrow range, the halo concentration shows a slight positive correlation with $n$. This might imply that, for central galaxies with the same stellar mass and halo mass, more concentrated central galaxies are more likely to reside in more concentrated halos, which is consistent with our previous hypothesis and results. However, the correlation is still weak and may depend on the details of galaxy formation models adopted in the simulations and how realistic are these models.

We further examine the relationship between $n$ and the black hole (BH) mass $M_{\rm BH}$ in the galaxies for the two simulations at $z=0$, as shown in Figure 8. We find a weak positive correlation between $n$ and $M_{\rm BH}$ in both TNG300 and S100. We consider this to be natural because a larger $n$ typically implies that a galaxy has a more dominant bulge component, and many studies reported a positive correlation between bulge and black hole mass (McLure & Dunlop, 2002; Wandel, 2002; McConnell & Ma, 2013; Davis et al., 2019). We also display the relationship between $M_{\rm BH}$ and $M_{h}$, revealing strong positive correlations. We can derive the relationship between ${\rm log}M_{h}$ and $n$ by combining the best-fit equations in ${\rm log}M_{\rm BH}-n$ and ${\rm log}M_{\rm BH}-{\rm log}M_{h}$ panels, and we find that the results are consistent with those obtained from the direct fitting shown in Figure 6. This implies that the correlation between the ${\rm S\acute{e}rsic}$ index and halo mass is probably derived from the different black hole activities.

According to Thomas et al. (2019), the Eddington ratio, $f_{\rm Edd}$, of black holes is inversely correlated with their mass. When $f_{\rm Edd}<0.2$, it would trigger jet-mode AGN feedback and subsequent X-ray AGN feedback. Therefore, to validate the $n-M_{h}$ relationship influenced by black hole mass, we use simulations from the SIMBA series with different modes of AGN feedback, including simulations that incorporate all physical processes (“S50”), that exclude X-ray feedback (“S50, no-X”), and that exclude both X-ray feedback and jet-mode AGN feedback (“S50, no-jet”). Figure 9 shows the correlations between $n$ of central galaxies and other halo properties for three simulations, where the shaded regions represent the 16-84 percentiles. We observe distinctly different trends of $n$ with other properties in the three simulations. As shown in the leftmost panel, the average stellar mass of galaxies without X-ray and jet-mode AGN feedback (orange) is higher than those with jet-mode AGN feedback (red and blue). This indicates that jet-mode feedback primarily reduces the star formation rate in galaxies, as stated in Davé et al. (2019). Furthermore, as AGN feedback modes are progressively turned off, the slope between halo mass and $n$ gradually decreases. Especially in the “S50, no-jet” simulation, the correlation between $n$ and $M_{h}$ almost disappears, and the correlations with $c$ and $\lambda$ are also significantly weakened. These results indicate that the relationship between $n$ and halo properties might be strongly influenced by the strength of AGN feedback, especially the jet mode, which can rapidly and effectively expel gas from galaxies, leading to the cessation of star formation.