Modeling biases from constant stellar mass-to-light ratio assumption in galaxy dynamics and strong lensing2021Modeling biases from constant stellar mass-to-light ratio assumption in galaxy dynamics and strong lensing–B

Within the dark matter cosmology framework, galaxies composed of stars and gas (baryons) live in dark matter halos. One primary goal of galaxy dynamical studies, often in combination with stellar population synthesis (SPS) and sometimes gravitational lensing, is to correctly disentangle the different contributions from the dark and the baryonic matter inside a galaxy. On the dark matter side, the mass fraction and density slope in the inner region of a galaxy (as well as their redshift evolution) may provide us with crucial information on the merger history of a galaxy (e.g., Boylan-Kolchin et al. 2005; Hilz et al. 2013; Tortora et al. 2014) as well as the dynamical influence of baryons (e.g., Blumenthal et al. 1986; Gnedin et al. 2004), in particular in the presence of various feedback mechanisms (e.g., Navarro et al. 1996; Duffy et al. 2010; Governato et al. 2012; Lovell et al. 2018). This is because during these processes, the dark matter halo will respond accordingly, altering its phase-space distribution. Observationally, however, the exact amount and shape of the inner dark matter halo are difficult to measure to high accuracy, largely because dark matter can only be probed indirectly, with the help of gravitational tracers and in combination of carefully deducting the baryonic contribution. This has substantially restrained our understanding on and solutions to the small-scale controversies of the standard cold dark matter theory, as inferred by observations (see Weinberg et al. 2015 and Bullock & Boylan-Kolchin 2017 for general review). On the baryonic side, although the stellar component can be directly probed by light, correctly converting luminosity $L$ to stellar mass $M_{\star}$ relies on knowing (the spatial distribution of) the mass-to-light ratio $M_{\star}/L$. The stellar mass-to-light ratio $M_{\star}/L$ depends on the age and metallicity of the underlying stellar population, as well as the initial mass function (IMF). In particular, the IMF describes the stellar mass distribution for a population at birth, and is an important fundamental property of a galaxy. A population IMF has been shown to depend on the turbulent environment of the interstellar medium (e.g., Padoan & Nordlund 2002; Hopkins 2013; Chabrier et al. 2014). While a galaxy-wide IMF has also been shown to correlate with properties such as the intensity of star formation, effective pressure and metallicity (e.g., Conroy & van Dokkum 2012; Jeřábková et al. 2018; Zhou et al. 2019), essentially reflecting the stellar assembly environment and history of galaxies.

Various modeling approaches have been developed and the exact implementation depends on the quality and richness of available observational data, which, under different circumstances, may suffer from different limitations and complications. For example, when high $S/N$ and high resolution spectroscopic data are available, IMF-sensitive absorption lines or features in a galaxy spectrum can be used to reveal the true IMF (as well as the stellar mass). Through this approach, many studies support that massive galaxies in the nearby Universe have Salpeter-like IMFs (Salpeter 1955), which are more bottom-heavy (i.e., a larger number of lower-mass stars with $M<1M_{\odot}$) than the Kroupa (Kroupa 2001) or Chabrier (Chabrier 2003) IMFs that are observed for the Milky-Way Galaxy (e.g., van Dokkum & Conroy 2010; Spiniello et al. 2012; Conroy & van Dokkum 2012; La Barbera et al. 2013; Spiniello et al. 2014; but note that some studies favor Chabrier IMFs for massive early-type galaxies, e.g., Smith et al. 2015; Sonnenfeld et al. 2019b). Meanwhile, spatially varying IMFs inside massive early-type galaxies have also been revealed by data such that the very central region tends to have a more bottom-heavy IMF than at outer parts (e.g., Martín-Navarro et al. 2015; La Barbera et al. 2016; van Dokkum et al. 2017; Parikh et al. 2018; Bernardi et al. 2023).

We note that accurate estimates on the IMF are essential in order to correctly predict the stellar mass through stellar population synthesis (SPS). The stellar mass can then be used to derive the dark matter mass, when further combined with dynamical and/or lensing measurements that provide estimates on the total mass (see e.g., Newman et al. 2017). However, high-quality spectroscopic observations that can put stringent constraints on the IMF, via accurately identifying IMF-sensitive absorption features, are not cheaply available in particular for galaxies beyond the local Universe. The IMF is rather often taken as a basic assumption during SPS analyses. The inferred stellar mass may then be subject to a bias due to the difference between the referenced and the true IMFs. In this case, the true stellar mass can be derived through IMF-independent approaches. This is often done with the aid of stellar or gas kinematics, as well as gravitational lensing (e.g., Cappellari et al. 2013b; Sonnenfeld et al. 2018; Shajib et al. 2021). For example, one can choose to model the total density profile, either taking an observationally-motivated isothermal distribution or simply assuming “mass follows light” (e.g., Koopmans et al. 2006; Smith et al. 2015). Subtracting the best-modeled dark matter content from the total mass distribution then yields the stellar distribution. These gravitationally derived stellar masses are then treated as unbiased estimates, and then used to infer the true IMFs through comparisons to those obtained from the SPS approach. Up to now, studies in this regard found that the IMFs of spiral galaxies tend to have a normalization similar to a Chabrier/Kroupa IMF; assuming a Salpeter IMF normalization would simply results in a total mass that exceeds the one as required by kinematics or lensing measurements (Bell & de Jong 2001; Kassin et al. 2006; Brewer et al. 2012; Suyu et al. 2012). Conversely, studies of early-type galaxies typically found heavier Salpeter-like IMF (normalization) (e.g., Treu et al. 2010; Auger et al. 2010; Sonnenfeld et al. 2012; Li et al. 2017), including those studies that use quasar micro-lensing observations to directly constrain a stellar mass fraction (e.g., Oguri et al. 2014; Schechter et al. 2014; Jiménez-Vicente et al. 2015).

In these lensing- and dynamics-based studies, to disentangle the contribution between dark matter and stars, the dark matter density distribution is often modeled either using a standard NFW profile (Navarro et al. 1997) or by a generalized NFW profile (gNFW, Zhao 1996; Wyithe et al. 2001), or via some generally contracted or cored halo density profiles (e.g., Blumenthal et al. 1986; Burkert 1995; Gnedin et al. 2004). The latter cases allow for a variation in the inner slope of the dark matter halo, as a response to the central baryons (see Cappellari et al. 2013a for a variety of commonly assumed dark matter halo density profiles). On the stellar side, as the luminosity profile can be directly measured from spatially resolved images, the stellar mass distribution is often modeled by taking the best-fit luminosity profile and multiplying it with some assumed stellar mass-to-light ratio $M_{\star}/L$.

The combined two-component model is then used to fit observational data. Here, we shall make a further note that the exact implementation for the above-mentioned modeling approaches is also subject to whether or not one has access to spatially resolved kinematic measurements. For nearby galaxies where integral field spectroscopic data are available, e.g., those from the Atlas^3D (Cappellari et al. 2011) and the MaNGA (Bundy et al. 2015) surveys, spatially-resolved mass reconstructions can be achieved through, for example, Jeans anisotropic modeling (JAM) methods (Jeans 1922; Cappellari 2008; Li et al. 2017; Zhu et al. 2023; Lu et al. 2023) or via particle- or orbit-based modeling techniques (Syer & Tremaine 1996; Schwarzschild 1979; Zhu et al. 2018).

However, for galaxies at higher redshifts, in most cases the only available stellar kinematic measurements are single-aperture or long-slit velocity dispersions. Many studies have routinely combined such stellar kinematics with gravitational lensing, which put certain constraints on projected masses within some different radii. Specifically, the stellar velocity dispersion measured within a single aperture typically of a few kpc is a projected kinematic property of stars within the entire sight line through the stellar realm of a galaxy. Strong lensing phenomena often take place at a few kpc in projection from the galactic centre, the measurements can robustly provide a total (projected) mass normalization within the Einstein radius (see Koopmans et al. 2006; Treu 2010). While weak lensing phenomena take place at several hundreds of kpc, the measurements can put constraints on the total matter density profile at halo outskirts, where dark matter plays a dominant role (e.g., Auger et al. 2010; Schulz et al. 2010; Sonnenfeld et al. 2018; Sonnenfeld et al. 2019a; Shajib et al. 2021). Such jointly modeling approaches have been widely implemented for nearly a hundred strong lensing galaxies beyond the local Universe (see e.g., Bolton et al. 2012; Treu et al. 2010; Sonnenfeld et al. 2015; Shu et al. 2015; Oldham & Auger 2018b; Shajib et al. 2021). The majority of these lenses are obtained from the Sloan Lens ACS Survey (SLACS, Bolton et al. 2006; Shu et al. 2017), and the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) Strong Lensing Legacy Survey (SL2S, Cabanac et al. 2007; Gavazzi et al. 2012).

Interestingly, studies in this regard so far have not reached consensus regarding the IMF normalization (and thus the central dark matter fraction and shape). The majority of the lensing galaxy population are relatively massive early-type galaxies (typically with stellar velocity dispersion of $\sigma\sim 200-350$ km/s). Treu et al. 2010 applied a joint model composed of two components to 56 SLACS lenses. The dark matter distribution therein was modeled by a standard NFW profile but with scale radius $r_{\rm s}$ fixed to 30 kpc. The light distribution was described by an Hernquist profile (Hernquist 1990). Under the assumption of a constant $M_{\star}/L$, a Salpeter IMF normalization was obtained. Auger et al. 2010 adopted a slightly different modeling technique to the same galaxy sample. In their study, an Hernquist profile was directly assumed for the stellar mass density distribution, with the scale radius proportional to the V-band effective radius. A Salpeter-like IMF normalization was also obtained when assuming a standard NFW halo. However, if a moderately contracted dark matter halo was adopted, the inferred IMF normalization became lighter than Salpeter but still heavier than Chabrier. Sonnenfeld et al. 2015 studied 81 early-type lensing galaxies from a combined sample of SL2S and SLACS. Through a hierarchical Bayesian modeling approach, population properties on central dark matter fraction and stellar IMF were inferred. Assuming a de Vaucouleurs light profile (de Vaucouleurs 1948) with a constant $M_{\star}/L$ and a generalized NFW halo with scale radius $r_{\rm s}$ fixed to 10 effective radius $R_{\rm eff}$, the strong lensing plus stellar kinematics data preferred a Salpeter IMF normalization (while the dark matter inner slope could not be well constrained). Oldham & Auger 2018b studied the IMF normalization in 12 early-type lensing galaxies at $z=0.6$. Adopting a two-component model composed of a gNFW profile for the dark matter halo and a Sérsic profile (Sérsic 1963) for the light distribution, the authors derived a Salpeter-like IMF normalization, as well as a clear bimodal distribution covering both cuspy and cored dark matter inner density slopes, under the constant $M_{\star}/L$ assumption.

However, some other studies obtained a different conclusion regarding the IMF normalization. For example, Sonnenfeld et al. 2019a studied 23 massive galaxies from the Baryon Oscillation Spectroscopic Survey (BOSS, Schlegel et al. 2009; Dawson et al. 2013) constant mass (CMASS) sample. The stellar mass was estimated by subtracting best-fit dark matter distribution (as constrained by weak lensing measurements at larger radii) from the total mass (as constrained by strong lensing at a few kpc). Then through a comparison to the SPS-inferred stellar mass, a normalization lighter than that of the Salpeter IMF was found within the strong lensing region. At more massive end ($\sigma>300$ km/s), Smith & Lucey 2013 and Smith et al. 2015 found Kroupa-like IMFs for three giant elliptical galaxies from the SINFONI Nearby Elliptical Lens Locator Survey (SNELLS). This was achieved through approximating the stellar mass by taking the difference between the lensing-derived total mass and a simulation-based dark-matter mass estimate (thus eliminating the necessity of making explicit assumptions on $M_{\star}/L$). Regarding the seemingly contradicting IMF trends between galaxies from SNELLS and SLACS/SL2S, Newman et al. 2017 carried out a detailed analysis using several different methods on the same SNELLS galaxies in order to derive the IMFs therein. One of them is to combine strong lensing modeling with the axisymmetric JAM method to fit stellar kinematics. Again, a lighter IMF normalization was reached for the three SNELLS lenses, consistent with the previous conclusion from Smith & Lucey 2013 and Smith et al. 2015. If this is true, this would indicate that a wide scatter in the IMF normalization (thus star assembly history) may exist among massive early-type galaxies due to unknown reasons¹¹ 1 Interestingly, while applying SPS modeling to high-quality spectroscopic data, Newman et al. 2017 allowed for variations in the power law slope as well as a lower-mass cut off of the IMF. They found that in order to reach consistent results between lensing, dynamical and spectroscopic constraints, a power-law IMF cannot extend to the canonical hydrogen-burning limit of $0.08\,M_{\odot}$; otherwise the derived stellar mass would be in conflict (too large) with the total-mass bound from lensing constraints. A lower-mass cutoff of $m_{\rm cut}\sim 0.3\,M_{\odot}$ was proposed therein, which is roughly consistent with $m_{\rm cut}\ga 0.1\,M_{\odot}$ as derived by Barnabè et al. 2013 and Spiniello et al. 2015. The uncertainty of the lower-mass cutoff simply further complicates the determination of galaxy IMFs..

One of the major uncertainties in these lensing- and dynamics-based studies lies in the assumption of constant $M_{\star}/L$. Mounting evidence has suggested the existence of a radial variation in $M_{\star}/L$ across galaxy populations (e.g., Tortora et al. 2011; García-Benito et al. 2019; Ge et al. 2021; Lu et al. 2023). A radial gradient in $M_{\star}/L$ has also been gradually implemented in dynamical and lensing studies. For example, Oldham & Auger 2018a applied a Jeans analysis to multiple dynamical tracers in M87, a nearby massive brightest cluster galaxy in Virgo. The data strongly favors a radial $M_{\star}/L$ variation. As such, a Salpeter IMF was found to only dominate the central 0.5 kpc, and a Chabrier IMF was needed in regions beyond. Sonnenfeld et al. 2018 combined the strong lensing and stellar kinematic measurements of 45 SLACS lenses, plus weak lensing measurements made by the Hyper-Suprime-Cam (HSC, Aihara et al. 2018) survey for 1700 massive quiescent galaxies from the Sloan Digital Sky Survey (SDSS). The authors found that a radial gradient in $M_{\star}/L$ of 0.2 was needed, and the data favored a Chabrier IMF normalization plus a standard NFW dark halo. Interestingly, Shajib et al. 2021 took a similar approach, combining strong lensing, weak lensing and stellar kinematics measurements of 23 SLACS lenses. Adopting double-Sérsic profile to fit galaxy light and allowing for an anisotropic velocity profile as well as a radial variation in $M_{\star}/L$, they however, obtained a Salpeter-like IMF normalization together with a mildly expanded NFW halo. In addition, the radial slope in $M_{\star}/L$ as required by data was only as small as $0.02$, consistent with zero gradient in the mass-to-light ratio.

It is worth noting that even though pioneering studies (as mentioned above) have gradually moved on using power-law $M_{\star}/L$ models, a consensus towards the inference on the IMF and/or dark matter halo shape is still not in place. Meanwhile many dynamical studies that target modern IFU-kinematic galaxy surveys still take constant $M_{\star}/L$ as a basic assumption (e.g., Zhu et al. 2023; Zhu et al. 2024; Lu et al. 2023). The goal of this study is, using a realistic galaxy population from state-of-the-art cosmological simulations, to address following questions: (1) How well can a constant value of $M_{\star}/L$ be used to approximate the profile in early-type galaxies? (2) Assuming there are sufficient observables about the total matter distribution and the light distribution, how well can we constrain the average $M_{\star}/L$, as well as the central dark matter fraction $f_{\rm DM}$ and the inner dark matter slope $\gamma_{\rm DM}$, under the assumption of a constant $M_{\star}/L$? (3) How do the systematic biases differ among different models that are adopted to describe the dark matter and stellar distributions? (4) What are the reasons behind model failures in correctly predicting the above-mentioned dark matter and stellar properties under the assumption?

To do so, we take an early-type galaxy sample from the cosmological hydrodynamic TNG100 simulation (Springel et al. 2018; Nelson et al. 2018; Pillepich et al. 2018b; Naiman et al. 2018; Marinacci et al. 2018). A two-component model fitting is carried out within a Bayesian inference framework. Specifically, we adopt a gNFW model to describe the dark matter density distribution. The model for stellar mass distribution is made to closely follow the light distribution (up to a constant $M_{\star}/L$ normalization factor), which is modeled using two different luminosity density profiles: a single-Sérsic profile and double-Sérsic profile (Sérsic 1963).

We point out that except for galaxy matter and light profiles, we do not create any specific mock observations, such as galaxy spectra, multi-band photometries, or gravitationally lensed images. Instead we make the model to directly fit the total matter density distributions of simulated galaxies, which are not directly observable. However, by doing so, we can evaluate model degeneracy and bias that arise as a consequence of a constant $M_{\star}/L$ assumption, without being affected by the assumptions adopted in the dynamical and/or lensing modeling processes, such as an isotropic velocity dispersion distribution.

The rest of this work is organized as follows. In section 2, we will introduce the simulation we used and the preliminary data selections(section 2.1), profile descriptions (section 2.2) and our two-component fitting models (section 2.3). Secondly, in section 3, we will present the investigation on the mass-to-light ratio radial variation in our simulated sample. In section 4.1, we will present the results of using the two-component model to fit our mock data, the total density profile, without other constraints. After that, we explore the results of alternative models with constraints on the central mass-to-light ratio and dark matter component in the section 4.2. The further discussions related to our results are provided in section 5, including the limitation of numerical resolution (section 5.1) and the stellar mass-to-light ratio model (section 5.2). Finally, we will summarize our work in section 6. In this work, we adopt the cosmology as Planck Collaboration et al. 2016 which is used in IllustrisTNG: $\Omega_{\rm M}=0.3089$, $\Omega_{\Lambda}=0.6911$ and $h=0.6774$.

Before presenting the composite model fitting results and the systematic biases rooted in assuming a constant mass-to-light ratio $M_{\star}/L$ for the sub-sample of galaxies in section 4, we first, in this section, display the actual spatial distribution of $M_{\star}/L$ for our full massive early-type galaxy sample.

In order to quantitatively describe the $M_{\star}/L$ radial variation, we adopt a simple power-law to fit the $M_{\star}/L$ profile within $R_{\rm eff}$:

where $\Upsilon_{\star}(R)$ represents the value stellar mass-to-light ratio around the given projected radius $R$. And the power-law index $\alpha$ is logarithmic gradient of $M_{\star}/L$. Without loss of generality, we simply divide our massive early-type galaxies into two categories. The galaxies with $\alpha$ exceeding 0.1 are classified as non-constant $M_{\star}/L$ systems while the remaining galaxies are still considered to exhibit a constant $M_{\star}/L$. Among our 129 massive early-type galaxies, there are totally 68 galaxies identified as a non-constant $M_{\star}/L$ system. In Fig. 1, such two categories are also labelled by blue and red, respectively. As can be seen already, the non-constant $M_{\star}/L$ systems are in general more massive and larger in size, as well as more actively star-forming, than their constant $M_{\star}/L$ counterparts. This can be seen again in Fig. 2, where the $M_{\star}/L$ gradient $\alpha$ is presented as the function of the total stellar mass and color-coded by the specific star formation rate within $2R_{\rm eff}$. We note that the distribution of $\alpha$ has a peak around $\sim 0.2$, which is consistent with observations, e.g., Sonnenfeld et al. 2018; García-Benito et al. 2019; Ge et al. 2021.

We present the stacked radial profiles of the two categories of galaxies in Fig. 3. As can be seen, constant $M_{\star}/L$ galaxies (red) on average have a flatter radial gradient than the non-constant $M_{\star}/L$ galaxies (blue). To provide a more comprehensive understanding of the properties of galaxies exhibiting non-constant $M_{\star}/L$, we have selected several representative examples to illustrate their characteristics.

The top panels of Fig. 4 provide the comparisons between the stellar mass density and luminosity profiles of four example non-constant $M_{\star}/L$ galaxies. The stellar mass (blue circles) and luminosity (yellow stars) density profiles are roughly compatible with each other around $R_{\rm eff}$, but the former is markedly more centrally concentrated than the latter (for the galaxies with stellar masses in a range as investigated in this study). This corresponds to generally increased $M_{\star}/L$ profiles towards galaxy centres, as can be seen from the bottom panels.

In Fig. 5, we present the face-on projected brightness map of the same four galaxies. Notably, all four galaxies have star formation occurring during recent $1\rm\,Gyr$ out to $3R_{\rm eff}$ (see bottom panel). Despite the insignificant mass fraction of newly formed stars, they markedly decrease $M_{\star}/L$ at the outskirts. It is interesting to observe that there are galaxies (subfind ID: 88592, 167636) with a dented feature in the $M_{\star}/L$ profile as a consequence of a disk-like star formation in the past $1\rm\,Gyr$ within the region around $1-2R_{\rm eff}$.

It is noteworthy that the mass-to-light ratio of the galaxy depends on many physical properties including the stellar IMF, the metallicity and the star forming history (as have been mentioned in the introduction). In observations, the radial gradient of the mass-to-light ratio is consistently recognized as an indicator of the radial variation in the stellar IMF (Oldham & Auger 2018a; Oldham & Auger 2018b; Sonnenfeld et al. 2018). In those works, researchers proposed that the stellar IMF might be heavier in the centre but keep the MW-like IMF in the outskirts, and such a centrally heavier IMF could exacerbate the declining trend of $M_{\star}/L$ profile. However, in current simulations, the luminosity of stellar particles is essentially generated under a uniform IMF (the Chabrier IMF in the TNG100 simulation), whereas the gradient in the mass-to-light ratio is primarily attributed to stellar particles that form in distinct epochs. As depicted in Fig. 3, the simulated $M_{\star}/L$ also exhibits a declining radial profile, which is similar with the $M_{\star}/L$ inferred by the centrally heavier IMF scenario in those previous works. Therefore, although the $M_{\star}/L$ variation in the real universe is attributed to more intricate factors, these simulated non-constant $M_{\star}/L$ galaxies can serve as a tool for discussing the potential biases that arise from the assumption of a constant $M_{\star}/L$.

In this section, we report the inference results of the four composite models introduced in section 2.3, and discuss reasons behind the statistical bias under the constant stellar mass-to-light ratio assumption. We primarily present our baseline model in section 4.1, and summarize the results of three alternative models in section 4.2.

In this work, we study the modeling bias from constant stellar mass-to-light ratio assumption, which has been widely used in galaxy dynamics and lensing studies (e.g., Treu & Koopmans 2002; Koopmans et al. 2006; Sonnenfeld et al. 2012; Sonnenfeld et al. 2015). We take a sample of over one hundred of simulated galaxies from the TNG100 project (Weinberger et al. 2017; Pillepich et al. 2018a; Pillepich et al. 2018b; Nelson et al. 2018; Springel et al. 2018; Naiman et al. 2018; Marinacci et al. 2018) and fit their total density profiles with two-component models within a Bayesian framework, following procedures widely adopted in observations (e.g., Sonnenfeld et al. 2015; Oldham & Auger 2018a; Oldham & Auger 2018b; Sonnenfeld et al. 2018). In particular, we use a generalized NFW model (Zhao 1996; Wyithe et al. 2001) to describe the dark matter halo density profile. We fit the light surface brightness profile with both a single- and double-Sérsic model (Sérsic 1963). We then multiply the best-fit light model with a constant stellar-mass to light ratio $\Upsilon_{\star}$, which is then used to describe the stellar mass density model. We fit the two-component model to the total density profiles of simulated galaxies, and examined the modeling performances using the density profile data within $R_{200}$ in an all-free model (with flat prior for all model parameters, as our baseline model, see section 2.3.1 for details) and in a model that assumes a Gaussian prior on $\Upsilon_{\star}$ (centred on the true value and with a scatter of $\sim 10\%$, see section 2.3.2). The other two models restrict the range to $4R_{\rm eff}$, emulating the study without constraints at large radii. In addition, some priors on the model parameters are set in models with partially constrained gNFW profiles where either the dark matter inner slope was fixed to 1 (see section 2.3.3), or the halo concentration was fixed by empirical relations (see section 2.3.4).

We find that the $M_{\star}/L$ profile of simulated massive early-type sample generally declines with radius, i.e., the true stellar mass density distributions more centrally concentrated than the luminosity density distributions (see Fig. 4 and section 3). In particular, we define the galaxies that profiles possess comparatively shallower logarithmic $M_{\star}/L$ gradient ($\alpha<0.1$, see Fig. 3) as the constant $M_{\star}/L$ system. However, a considerable fraction of galaxies exhibit markedly non-constant $M_{\star}/L$ profiles ($\alpha>0.1$ as the non-constant $M_{\star}/L$ galaxy). Almost half of our simulated massive early-type galaxies have a significant $M_{\star}/L$ gradient, indicating that assuming constant $M_{\star}/L$ is not a good approximation. The $M_{\star}/L$ variation of these simulated galaxies is typically around the region, where on-going star formation exists within a quenched stellar halo (see Fig. 3 and Fig. 5). This often causes $M_{\star}/L$ to decrease dramatically at the associated radii, and leads to seemingly steeper $M_{\star}/L$ gradient.

In previous works, the lensing and stellar kinematic data can effectively constrain the total mass distribution. But the dark matter distribution is still degenerate with baryons in the galaxy centre. We find that the assumption of a constant stellar mass-to-light ratio $M_{\star}/L$ would in general artificially break such a degeneracy and causes non-negligible model biases for those non-constant $M_{\star}/L$ galaxies (i.e., the model prefers under-estimating one specific component). In composite model fitting where an all-free model (baseline model) has been adopted (see section 4.1), due to the similarity of the logarithmic slope between central light distribution and the total density profile, the model would artificially enhance $\Upsilon_{\star}$ and thus result in under-estimated $f_{\rm DM}$ and flatter $\gamma_{\rm DM}$. When we take the single-Sérsic model to describe the stellar component for those galaxies with a non-constant stellar mass-to-light ratio, the mean bias of the stellar mass-to-light ratio $\log{\Upsilon^{(\rm mod)}_{\star}/\Upsilon^{(\rm true)}_{\star}}$ is $0.19\pm 0.07\,\rm dex$, and the mean bias of the dark matter fraction $f^{(\rm mod)}_{\rm DM}-f^{(\rm true)}_{\rm DM}$ is $-0.18\pm 0.08$. As a comparison, for those galaxies with a constant stellar mass-to-light ratio, the mean biases as well as their errors of $\log\Upsilon_{\star}/\Upsilon_{\odot}$ and $f_{\rm DM}$ are much smaller, $-0.04\pm 0.14$ and $0.02\pm 0.12$, respectively. When we consider the double-Sérsic model for the stellar component, the general trend between the $M_{\star}/L$ and modeling biases is not significantly changed.

It is then interesting to ask what if we have some knowledge about the average stellar mass-to-light ratio $\Upsilon_{\star}$? To test this, we have then added a prior on $\Upsilon_{\star}$ centered around the truth. The model in general can better reproduce the central dark matter fraction $f_{\rm DM}$. However, as the model now cannot freely enhance $\Upsilon_{\star}$ to compensate for a more compact stellar mass density distribution, it is only left with the choice of systematically overestimating the dark matter inner slope $\gamma_{\rm DM}$ for those non-constant $M_{\star}/L$ systems (see section 4.2.1). We have also partially constrained the gNFW model to describe the dark matter component only with the data from the central region. These fixed models, such as NFW (see section 4.2.2) or assuming $r_{\rm DM}=10R_{\rm eff}$ (section 4.2.3) also introduce additional biases. In particular, the application of NFW model results in a systematically under-predicted dark matter component since the simulated massive galaxies always keep a $\gamma_{\rm DM}$ exceeding 1.

As can be seen, without a good description to $M_{\star}/L$, different models (on both dark matter and light) would artificially break the degeneracy in different manners, therefore resulting in different inferences. This study, investigating the consequence of the constant $M_{\star}/L$ assumption, simply builds up the first stage of a series of investigation along this direction.

We thank Dr. Shengdong Lu for helpful and constructive discussions about this paper. We thank the anonymous referees whose comments have helped to significantly improve the quality of the paper. This work is supported by the China Manned Space Project (No. CMS-CSST-2021-A07). LY and XDD also acknowledge the General Program from National Natural Science Foundation of China (No. 12073013). We would like to thank the high-performance computing cluster at the Department of Astronomy, Tsinghua University for providing computational and data storage resources for this study.

The data of simulated galaxies in this work come from the TNG100 cosmological simulation project. All of the data are public and can be obtained on the website of the TNG project (https://www.tng-project.org). We will also share the galaxy IDs and other underlying data for reasonable requests.