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 K. Xu & Y. Jing (2022), with the galaxy sample selected from the Baryon Oscillation Spectroscopic Survey (BOSS; C. P. Ahn et al. 2012; A. S. Bolton et al. 2012) CMASS, with an area of approximately 500 deg² overlapping with HSC. They identify galaxies that do not have more massive neighbors within a projected distance of 1 Mpc h⁻¹, considering these to be central galaxies. However, they did not account for fiber collisions or the stellar mass completeness of CMASS (A. Leauthaud et al. 2016; H. Guo et al. 2018; K. Xu et al. 2023), both of which can bias the selection of the most massive galaxies and lead to off-centering effects. In our lensing model, we incorporate these factors, along with the possibility that central galaxies may not be the most massive and that there may be offsets between the centers of halos and their central galaxies (C. Hikage et al. 2013; L. Wang et al. 2014; J. U. Lange et al. 2017). In their work, the stellar mass and rest-frame colors are measured by fitting the spectral energy distribution. The Sérsic indices are obtained by fitting the Sérsic profile to the corresponding HSC z-band images, as detailed in K. Xu & Y. Jing (2022). In this work, we use galaxies with stellar masses in the narrow range of , a redshift range of 0.45–0.7, and Sérsic indices ranging from 0.8 to 8.

The shear catalog we use is processed through the Fourier_Quad (FQ; J. 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 (C. Liu et al. 2024). The FQ method employs multipole moments of galaxy images in the Fourier space for shear measurements, including five shear estimators: G₁, G₂, N, U, and V. G_i is based on the galaxy quadrupole moments in the 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 probability density function-symmetrization (PDF-SYM) method, which is a novel way of estimating shear statistics. Their specific definitions can be found in J. Zhang et al. (2016).

The shear catalog records basic information about galaxies, such as the position, signal-to-noise ratio (ν_F; H. Li & J. Zhang 2021), magnitude, and shear estimators. Photometric redshift (photo-z) information is sourced from the DEmP method in A. J. 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, since the scatter photo-z for our sample is estimated to be about 0.02–0.04 (Table 2 in A. J. Nishizawa et al. 2020), to reduce the intrusion of foreground galaxies due to photo-z errors, we use the background galaxies whose photo-z are 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 a signal-to-noise ratio larger than 4 and an absolute field distortion (FD) ∣g_f∣ smaller than 0.02 (see Appendix A for more details on g_f). 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. Following C. 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 FD tests is at about 1–2% 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 Z. Shen et al. (2025) 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 FD information for each galaxy image/shear estimator, it therefore provides a convenient on-site calibration approach. Its details relevant to this work are 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 (A. Pillepich et al. 2017). We utilize the simulation with a box length of L_box = 205 Mpc h⁻¹, denoted as TNG300. The SIMBA simulation provides detailed modeling of black hole (BH) growth and feedback, successfully reproducing many observational results (R. Davé et al. 2019; N. Thomas et al. 2019; W. Cui et al. 2021). In particular, active galactic nucleus (AGN) feedback in SIMBA operates in three modes based on the accretion rate: the radiative mode at high Eddington ratios f_Edd, and the jet and X-ray modes at low f_Edd. For our primary analysis, we use the full physics simulation with L_box = 100 Mpc h⁻¹, denoted as S100. According to R. Weinberger et al. (2018), R. Davé et al. (2019), and F. 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_box = 50 Mpc h⁻¹ (S50) to study their effects.

We first select central galaxies with stellar masses in the range of from TNG300 and S100 to match the observation, and then choose the range 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 × 4 times the half-stellar-mass radius. Then, we convolve a small Gaussian PSF to reduce noise. We fit these images with a 2D Sérsic profile to obtain their Sérsic index n. Additionally, we fit the surface density of the host halos with an Navarro–Frenk–White (NFW) profile to obtain their halo mass M_h and concentration c. The definitions in the 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.

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 refers to the average surface density within a radius r. Σ_c is the comoving critical surface density, defined as

Here, c is the speed of light, G is the gravitational constant, and D_s, D_l, and D_ls are the angular diameter distances for the lens, source, and lens–source systems, respectively.

In our work, we employ the PDF-SYM (J. Zhang et al. 2016) method to measure the ESD of galaxies. The essence of this method lies in constructing the 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 , we adjust P(G_t) to by modifying each value of G_t via:

where N and U_t are the normalization factor and rotated correction factor introduced in Section 2.2. When equals the true ESD signal, the new distribution will achieve the most symmetric state.

In this study, we uniformly divide the radius range from 0.05 to 50 Mpc h⁻¹ into nine bins in logarithmic space for measurement and estimate the covariance matrix by dividing the sky into 86 subregions using the Jackknife method. Additionally, to remove the effects of the 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 the effects of stellar mass of galaxies, individual halos, off-centering effects, and the two-halo term into account.

In this work, we assume that the galaxy luminosity distribution follows a Sérsic profile. However, relative to halos, the mass of galaxies is concentrated within a very small radius, allowing it to be treated as a point source for its ESD and the ESD of galaxy is

where M_* is the stellar mass of the central galaxy, as shown in Table 2 for all subsamples. We briefly verify the point-source model in Appendix B, showing that its ESD is consistent with that of the Sérsic profile within our radius range.

Table 2. Fitting Results for Different Subsamples

Samples	n/Color	Nlens	logM*		c	σs	fc	bh	χ2/ν
	0.8 < n < 2.6	1453	11.46						2.41
	2.6 < n < 3.6	1744	11.48						0.94
All	3.6 < n < 4.6	1985	11.48						2.17
	4.6 < n < 5.6	1883	11.48						1.72
	5.6 < n < 8	1783	11.46						0.40
	0.8 < n < 3	826	11.50						1.37
Fixed color	3 < n < 5	2560	11.49						2.35
(2.3 < u − r < 2.7)	5 < n < 8	2102	11.47						1.29
Fixed n	2 < u − r < 2.25	1195	11.42						2.73
(1 < n < 6)	2.25 < u − r < 2.6	1202	11.43						1.53
Fixed Mh	0.8 < n < 4	1361	11.49						1.74
	4 < n < 8	1923	11.49						0.77

Note. Halo mass logM_h, concentration c, scatter in off-centering σ_s, proportion of well-centered galaxies f_c, and halo bias b_h. The results in the parentheses are the ratios of the measured bias b_h to the theoretical value b_h(M_h, z) modeled by J. L. Tinker et al. (2010), with the measured halo mass M_h and mean redshift of the samples. “All” refers to samples controlled only by constant stellar mass. “Fixed color” refers to samples where galaxy color is also controlled, and “Fixed n” refers to samples controlled by Sérsic index. “Fixed M_h” is to control the halo mass of subsamples by applying the selection on M_G, where M_G is the halo mass assigned by abundance matching in the group catalog of X. Yang et al. (2021). The middle panel shows conditions of subsamples and the information, including the number of galaxies N_lens and their average stellar mass logM_*/M_⊙.

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We adopt NFW model (J. F. Navarro et al. 1997) to describe an individual halo profile, whose density is expressed as

with ρ₀ = ρ_mΔ_vir/(3I), where . r_vir is the virial radius, r_s is the scale radius, and the concentration c of halos is defined by c = r_vir/r_s. In this paper, we define a halo as a sphere with an overdensity of Δ_vir = 180 times the mean matter density ρ_m, with a halo mass . The unit of M_h in this paper is M_⊙/h. We employ the analytical expression for the surface density ΔΣ_NFW as presented in X. 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 one-halo term ESD model typically divides the halo profile into well-centered and off-centered components,

where f_c is the proportion of well-centered halos. Σ_off refers to the mean surface density of miscentered halos, which can be derived from

where is the distribution of offset centers on the radius, and it is generally assumed to be a Rayleigh distribution (D. E. Johnston et al. 2007)

σ_s represents the scatter of off-centered positions. In this work, we set it as a free parameter and use the halo radius, r_vir, as the unit.

Given that our measurement radii significantly exceed the common virial radii of halos, the two-halo term exerts a substantial influence on the ESD. Following F. C. van den Bosch et al. (2013) and J. Wang et al. (2022), the surface density of the two-halo term can be expressed as

where . b_h is the halo bias, which is a free parameter in our model. ζ(r, z) is radial bias function obtained in F. C. van den Bosch et al. (2013)

where

and

ξ_mm(r) is calculated by COLOSSUS (B. Diemer 2018) using a semianalytical power spectrum from D. J. Eisenstein & W. Hu (1998f). f_exc(r) describes the halo exclusion effect, where it equals 0 when r < r_vir and equals 1 otherwise.

Combining Equations (5)–(10) and Equation (2), our ESD model can eventually be summarized as

To enhance the accuracy of our model, we divide the redshift range of 0.45–0.7 into five bins and perform an integration based on the distribution of lens galaxies. Although more sophisticated modeling approaches can be adopted to model the scatter of halo mass or others (e.g., A. Leauthaud et al. 2012; J. Han et al. 2015), we leave more sophisticated modeling to future studies. Also, we test our model in simulations shown later.

In total, our model comprises five free parameters: halo mass M_h, concentration c, halo bias b_h, the scatter of off-centering σ_s, and the proportion of well-centered halos f_c, which we fit using Markov Chain Monte Carlo techniques (N. Christensen et al. 2001). The flat priors of the parameters in the fitting process are listed in Table 1.

We divide the galaxy samples with stellar masses in the range of into five subsamples. The range of their Sérsic indices n, the number of galaxies, and the average stellar mass are presented in Table 2. Figure 1 shows the measurements of ESD, with their corresponding best-fit curves and the different components of the model, and Figure 2 shows their 68% and 95% confidence level contour plots. It is evident that the ESD increases with higher n. The “All” section in Table 2 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 3, with the horizontal axis representing the average n for each sample. We find that halo mass increases significantly with the increase of the Sérsic index, which is consistent with the conclusions from K. Xu & Y. Jing (2022).

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Additionally, we observe 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:

Nevertheless, due to the large uncertainties and the strong degeneracy between the concentration parameter c and the off-centering fraction f_c, as shown in Figure 2, higher-quality data are required to validate our findings. Our best-fit results suggest that only about 50% of galaxies in our sample reside at the true centers of their halos, consistent with previous weak lensing studies using analyses (e.g., W. Luo et al. 2018; J. Wang et al. 2022). In contrast, some theoretical studies based on semianalytical and hydrodynamical models report lower off-centering fractions (e.g., Planck Collaboration et al. 2013; W. Wang et al. 2019). Further investigation is necessary to reconcile these differences and confirm the observational results.

Assembly bias refers to the phenomenon where the clustering of dark matter halos is influenced not only by their mass but also by their formation history and surrounding environment (R. H. Wechsler et al. 2006; N. Dalal et al. 2008). This means that halos of the same mass, but with different formation processes or environments, could exhibit different halo bias. We also set the halo bias b_h as a free parameter in the hope of detecting assembly bias, and show the ratios of the measured bias b_h to the theoretical value b_h(M_h, z) modeled by J. L. Tinker et al. (2010) in Table 2. But for subsamples of “All,” we do not observe a clear trend of b_h or assembly bias varying with n.

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 K. Xu & Y. Jing 2022). It has long been recognized that red and blue galaxies exhibit distinct SHMR (e.g., S. More et al. 2011; Y.-j. Peng et al. 2012; W. Wang & S. D. M. White 2012; A. Rodríguez-Puebla et al. 2015; R. Mandelbaum et al. 2016; L. Posti & S. M. Fall 2021; W. Wang et al. 2021; K. Xu & Y. Jing 2022; P. 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 2.3–2.7. We divide them into three subsamples according to the Sérsic index, 0.8 < n < 3, 3 < n < 5, and 5 < n < 8. To eliminate the correlation between color and n, we resample their color distributions to make them similar by randomly removing some galaxies. The left panel of Figure 4 shows their ESD measurements. The red empty squares with error bars in Figure 3 show the best-fit M_h and c versus n, and the “Fixed color” row in Table 2 presents the best-fit parameters and associated errors. We find that, after fixing the color, the dependencies of both M_h and c on n are still consistent with, and even stronger than, the results for 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 the galaxy color.

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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 subsamples are resampled by randomly selecting galaxies to ensure a similar distribution of n, with average n of 2.98 ± 1.07 and 3.10 ± 1.09, respectively, where the errors are the standard deviations of n. The middle panel of Figure 4 shows their ESD measurements. The orange and blue rings in Figure 3 represent the results for the red and blue galaxy samples, with their sizes indicating the 1σ error range. The “Fixed n” section in Table 2 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.

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 X. Yang et al. (2021), which provides details on group richness λ and member galaxies. However, galaxy/group samples from observations are generally flux-limited. As shown in the left panel of Figure 5, which plots the luminosity versus photo-z diagram of all member galaxies in groups, the member galaxies and their richness are influenced by different absolute magnitude limits at various redshifts. To mitigate the effects of redshift dependence, we define a volume-limited sample by setting a minimum luminosity threshold for the member galaxies. Consequently, the richness here is defined as the number of member galaxies in a group with .

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The middle panel of Figure 5 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 5. It shows that the surrounding density of companion satellite galaxies increases with the n of central galaxies, consistent with the results from K. Xu & Y. 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., E. Rozo et al. 2009; S. Andreon & M. A. Hurn 2010; W. Wang & S. D. M. White 2012; M. Simet et al. 2017; R. Murata et al. 2018, 2019; I. N. Chiu et al. 2020; A. Phriksee et al. 2020; J. L. Tinker et al. 2021; W. Wang et al. 2021; M. 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 3, we find a slight positive correlation between the Sérsic index and halo concentration. Also, the right panel of Figure 5 shows that for galaxy clusters with smaller n, the distribution of satellite galaxies is slightly more extended. While more precise measurements are needed to fully confirm this relation, it would be intriguing to explore its theoretical implications.

Numerical simulations predict the trend of lower concentrations in more massive halos (e.g., A. V. Maccio et al. 2007; A. R. Duffy et al. 2008; R. Mandelbaum et al. 2008; A. D. Ludlow et al. 2014, 2016; C. A. Correa et al. 2015; B. Diemer & A. V. Kravtsov 2015), which can be explained by that these halos form later, allowing more material and subhalos to continuously accrete from the surrounding regions, resulting in a more diffuse distribution (e.g., M. Fong & J. Han 2021; B. Diemer 2022; H. Gao et al. 2023; J. 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 other factors beyond the typical mass–concentration relation influenced by the environment.

J. F. 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, D. 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 forming 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 (X. Yang et al. 2021). Considering the positive correlation between the Sérsic index n and halo mass M_h, we could adjust the selection range of M_G to ensure the halo mass is the same for different galaxy samples. In particular, we divide the sample into two groups based on Sérsic index: for galaxies with 0.8 < n < 4, we select samples satisfying 10^13.3 < M_G < 10^13.8M_⊙ h⁻¹; for galaxies with 4 < n < 8, we use samples with 10^13.3 < M_G < 10^13.7M_⊙ h⁻¹. The right panel of Figure 4 shows the ESD measurements, Table 2 presents sample information and best-fit parameters, which are also shown by the green squares in Figure 3. 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 ∼ 0.57.

From the perspective of assembly bias, we find that the halo bias of the galaxy sample with a higher Sérsic index is lower, which is also reflected in the lower large-scale ESD shown in Figure 4. R. H. Wechsler et al. (2006) proposed that for massive halos with fixed halo mass, those that form earlier (with higher concentration) have weaker clustering, resulting in a smaller halo bias. From this standpoint, our findings further support the inference that halos hosting galaxies with higher Sérsic indices formed earlier. This also suggests that the Sérsic index could serve as a proxy for halo concentration or formation time, which could be used in the future to probe the assembly bias of galaxy clusters.

Figure 6 compares our main results (blue dots) with those obtained using the PAC method in the work of K. Xu & Y. Jing (2022; orange squares). To ensure a consistent basis for comparison, we convert the halo masses in our results to their definition, where Δ_vir = 18π² + 82x − 39x² and x = Ω_m(z) − 1. We observe that our results systematically yield higher halo masses. This discrepancy may stem from the assumption in K. Xu & Y. 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.

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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 R. 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.

First, we use snapshots from TNG300 and SIMBA at two redshifts for the main analysis, z = 0 and z = 0.55, respectively.⁶ Figure 7 displays the relationships between the Sérsic index, n, and the halo mass, M_h, halo concentration, c, and richness, λ, 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 16–84th percentiles displayed as shaded areas. Note that, since there is a slight correlation between n and M_*, we randomly remove galaxies at different n to decorrelate these two parameters. As a result, the slope of the relationship in the leftmost panel of Figure 7 is zero, thereby avoiding the impact of stellar mass on other correlations between n and halo properties.

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We also bin the galaxies into three groups in TNG300 and two groups in S100, and stacked their ESDs as in the observations. The results are presented in Figure 8 with their best-fit curves. Similar to fitting individual halos, we assume that the central galaxies in the simulations are well-centered and treat M_h, c, and b_h as free parameters for fitting. The fitting results for M_h and c are shown as green and yellow squares in Figure 7. Note that the error in the stacked fitting results represents the uncertainty of the average properties of halos, which is so small that it is almost invisible in the figure. The stacked fitting results are close to the best-fit solid lines of scatter points obtained from fitting individual halos, demonstrating the robustness of the stacking lensing measurements.

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Regarding the relationship between halo mass and n, we find that both TNG300 and SIMBA exhibit correlations, also shown by the rising ESD in Figure 8. However, compared to observational results, halo masses from TNG300 are higher, and their correlation with n is weaker (Equation (15)). 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 λ, which is qualitatively consistent with our results (Figure 5). 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 reproduces 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 check whether the TNG simulation reveals a similar trend between n and c as what we have inferred from the data, we control both halo mass and stellar mass in a subsample from the TNG300 at z = 0. The left panel of Figure 9 shows the correlation between halo mass and stellar mass for the red points in Figure 7, 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 the right panels of Figure 9. 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 these models are.

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We further examine the relationship between n and the BH mass M_BH in the galaxies for the two simulations at z = 0, as shown in Figure 10. We find a weak positive correlation between n and M_BH in both TNG300 and S100. It might be because a larger n typically implies that a galaxy has a more dominant bulge component, and many studies reported a positive correlation between the bulge and BH mass (R. J. McLure & J. Dunlop 2002; A. Wandel 2002; N. J. McConnell & C.-P. Ma 2013; B. L. Davis et al. 2019). We also display the relationship between M_BH and M_h, revealing strong positive correlations. We can derive the relationship between and n by combining the best-fit equations in the and panels, and we find that the results are consistent with those obtained from the direct fitting shown in Figure 7. This implies that the correlation between the Sérsic index and halo mass might be related to the different BH activities.

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According to N. Thomas et al. (2019), the Eddington ratio, f_Edd, of BHs is inversely correlated with their mass. When f_Edd < 0.2, it would trigger jet-mode AGN feedback and subsequent X-ray AGN feedback. Therefore, to validate the n–M_h relationship in SIMBA influenced by BH 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 11 shows the correlations between n of central galaxies and other halo properties for three simulations, where the shaded regions represent the 16–84th 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 R. 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 λ are also significantly weakened. These results indicate that the relationship between n and halo properties might be 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.

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