Van Dokkum Research Articles

Cold Dark Matter and Self-interacting Dark Matter Interpretations of the Strong Gravitational Lensing Object JWST-ER1

van Dokkum et al. reported the discovery of JWST-ER1, a strong lensing object at redshift z ≈ 2, using data from the James Webb Space Telescope. The lens mass within the Einstein ring is 5.9 times higher than the expected stellar mass from a Chabrier initial mass function, indicating a high dark matter density. In this work, we show that a cold dark matter halo, influenced by gas-driven adiabatic contraction, can account for the observed lens mass. We interpret the measurement of JWST-ER1 in the self-interacting dark matter scenario and show that the cross section per particle mass σ/m ≈ 0.1 cm2 g−1 is generally favored. Intriguingly, σ/m ≈ 0.1 cm2 g−1 can also be consistent with the strong lensing observations of early-type galaxies at redshift z ≈ 0.2, where adiabatic contraction is not observed overall.

Flyby Galaxy Encounters with Multiple Black Holes Produce Star-forming Linear Features

We look for simulated star-forming linear features such as the one recently discovered by van Dokkum et al. in the cosmological hydrodynamical simulation ASTRID. Among the runaway black holes in ASTRID, none are able to produce clear star-forming wakes. Meanwhile, flyby encounters, typically involving a compact galaxy (with a central black hole) and a star-forming galaxy (with a duo of black holes), reproduce remarkably well many of the key properties (length and linearity, recent star formation, etc.) of the observed star-forming linear feature. We predict that the feature will persist for approximately 100 Myr in such a system and hence constitute a rare event. The feature contains a partly stripped galaxy (with M gal = 109–1010 M ⊙) and a dual black hole system (M BH = 105–107 M ⊙) in its brightest knot. The X-ray emission from AGN in the knot should be detectable in such systems. After 100–200 Myr from the first flyby, the galaxies merge, leaving behind a triple black hole system in a (still) actively star-forming early-type remnant of mass ∼5 × 1010 M ⊙. Follow-up JWST observations may be key for revealing the nature of these linear features by potentially detecting the older stellar populations constituting the bright knot. Confirmation of such detections may therefore help discriminate a flyby encounter from a massive black hole wake to reveal the origin of such features.

Creating a galaxy lacking dark matter in a dark matter-dominated universe

ABSTRACT We use hydrodynamical cosmological simulations to show that it is possible to create, via tidal interactions, galaxies lacking dark matter (DM) in a DM-dominated universe. We select dwarf galaxies from the NIHAO project, obtained in the standard cold dark matter model and use them as initial conditions for simulations of satellite–central interactions. After just one pericentric passage on an orbit with a strong radial component, NIHAO dwarf galaxies can lose up to 80 per cent of their DM content, but, most interestingly, their central (≈8 kpc) DM-to-stellar mass ratio changes from a value of ∼25, as expected from numerical simulations and abundance matching techniques, to roughly unity as reported for NGC 1052-DF2 and NGC 1054-DF4. The stellar velocity dispersion drops from ∼30 $\, \rm km\, s^{-1}$ before infall to values as low as 6 ± 2 $\, \rm km\, s^{-1}$. These, and the half-light radius around 3 kpc, are in good agreement with observations from van Dokkum and collaborators. Our study shows that it is possible to create a galaxy without DM starting from typical dwarf galaxies formed in a DM-dominated universe, provided they live in a dense environment.

Structural parameters for the globular-cluster-like objects in NGC 1052-DF2

ABSTRACT Recently, van Dokkum et al. have found an ultra-diffuse galaxy NGC 1052-DF2 with little or no dark matter, based on a spectroscopic study of its 11 constituent globular-cluster-like objects. In this paper, we analyse these 11 objects using Hubble Space Telescope imaging. We derive the structural parameters for each object by fitting the surface brightness profiles to three different models. Our results indicate that these objects are better fitted by the Wilson model than by the King and Sérsic models. Using the distance of D = 20 Mpc to NGC 1052-DF2 obtained by van Dokkum et al., these 11 objects have half-light radii in the range from ∼11 to ∼16 pc. These values are much larger than for normal globular clusters in the Milky Way, and are comparable to the sizes of an extended star cluster and of a typical ultra-compact dwarf. The half-light radii obtained here are larger than those obtained by van Dokkum et al. and Trujillo et al. The offset of the differences between the half-light radii of van Dokkum et al. and ours is 6.17 pc, and the offset of the differences between the half-light radii of Trujillo et al. and ours is 4.99 pc. Our half-light radii are, on average, 11.74 pc, which corresponds to 53 and 43 per cent larger than those obtained by van Dokkum et al. and Trujillo et al., respectively. The Rh versus MV diagram shows that these objects occupy the same areas of extended star clusters and ultra-compact dwarfs. Using the distance of D = 13 Mpc obtained by Trujillo et al., there are still five objects that do not lie in the area of normal globular clusters in the Milky Way. So, we suggest that these globular-cluster-like objects in NGC 1052-DF2 are not normal globular clusters like those in the Milky Way.

The ultra-diffuse dwarf galaxies NGC 1052-DF2 and 1052-DF4 are in conflict with standard cosmology

ABSTRACT Recently van Dokkum et al. reported that the galaxy NGC 1052-DF2 (DF2) lacks dark matter if located at 20 Mpc from Earth. In contrast, DF2 is a dark-matter-dominated dwarf galaxy with a normal globular cluster population if it has a much shorter distance near 10 Mpc. However, DF2 then has a high peculiar velocity wrt. the cosmic microwave background of 886 $\rm {km\, s^{-1}}$, which differs from that of the Local Group (LG) velocity vector by 1298 $\rm {km\, s^{-1}}$ with an angle of $117 \, ^{\circ }$. Taking into account the dynamical M/L ratio, the stellar mass, half-light radius, peculiar velocity, motion relative to the LG, and the luminosities of the globular clusters, we show that the probability of finding DF2-like galaxies in the lambda cold dark matter (ΛCDM) TNG100-1 simulation is at most 1.0 × 10−4 at 11.5 Mpc and is 4.8 × 10−7 at 20.0 Mpc. At 11.5 Mpc, the peculiar velocity is in significant tension in the TNG100-1, TNG300-1, and Millennium simulations, but naturally in a Milgromian cosmology. At 20.0 Mpc, the unusual globular cluster population would challenge any cosmological model. Estimating that precise measurements of the internal velocity dispersion, stellar mass, and distance exist for 100 galaxies, DF2 is in 2.6σ (11.5 Mpc) and 4.1σ (20.0 Mpc) tension with standard cosmology. Adopting the former distance for DF2 and assuming that NGC 1052-DF4 is at 20.0 Mpc, the existence of both is in tension at ≥4.8σ with the ΛCDM model. If both galaxies are at 20.0 Mpc the ΛCDM cosmology has to be rejected by ≥5.8σ.

A new formulation of the external field effect in MOND and numerical simulations of ultra-diffuse dwarf galaxies – application to NGC 1052-DF2 and NGC 1052-DF4

The ultra-diffuse dwarf galaxy NGC 1052-DF2 (DF2) has ten (eleven) measured globular clusters (GCs) with a line-of-sight velocity dispersion of $\sigma=7.8^{+5.2}_{-2.2}\,$km/s ($\sigma=10.6^{+3.9}_{-2.3}\,$km/s). Our conventional statistical analysis of the original ten GCs gives $\sigma=8.0^{+4.3}_{-3.0}\,$km/s. The overall distribution of velocities agrees well with a Gaussian of this width. Due to the non-linear Poisson equation in MOND, a dwarf galaxy has weaker self-gravity when in close proximity to a massive host. This external field effect is investigated using a new analytic formulation and fully self-consistent live $N$-body models in MOND. Our formulation agrees well with that of Famaey and McGaugh (2012). These new simulations confirm our analytic results and suggest that DF2 may be in a deep-freeze state unique to MOND. The correctly calculated MOND velocity dispersion agrees with our inferred dispersion and that of van Dokkum et al. (2018b) if DF2 is within 150 kpc of NGC 1052 and both are 20 Mpc away. The GCs of DF2 are however significantly brighter and larger than normal GCs, a problem which disappears if DF2 is significantly closer to us. A distance of 10-13 Mpc makes DF2 a normal dwarf galaxy even more consistent with MOND and the 13 Mpc distance reported by Trujillo et. al. (2019). We discuss the similar dwarf DF4, finding good agreement with MOND. We also discuss possible massive galaxies near DF2 and DF4 along with their distances and peculiar velocities, noting that NGC 1052 may lie at a distance near 10 Mpc.

Effects of mass models on dynamical mass estimate: the case of ultradiffuse galaxy NGC 1052-DF2

ABSTRACT NGC 1052-DF2 was recently discovered as the dark-matter-deficient galaxy claimed by van Dokkum et al. However, large uncertainties on its dynamical mass estimate have been pointed out, concerning the paucity of sample, statistical methods, and distance measurements. In this work, we discuss the effects of the difference in modelling of the tracer profile of this galaxy on the dynamical mass estimate. To do this, we assume that the tracer densities are modelled with power-law and Sérsic profiles, and then we solve the spherical Jeans equation to estimate the dynamical mass. Applying these models to kinematic data of globular clusters in NGC 1052-DF2, we compare 90 per cent upper limits of dynamical mass-to-light ratios estimated from this analysis and from van Dokkum et al. We find that the upper limit obtained by the power-law is virtually the same as the result from van Dokkum et al, whilst this limit estimated by the Sérsic is significantly greater than that from van Dokkum et al, thereby suggesting that NGC 1052-DF2 can still be a dark-matter-dominated system. Consequently, we propose that dynamical mass estimate of a galaxy is largely affected by not only the small kinematic sample but the choice of tracer distributions, and thus the estimated mass still remains quite uncertain.

Does the galaxy NGC1052-DF2 falsify Milgromian dynamics?

Galaxies normally have far more dark matter than normal matter, but the dynamics of objects within the ultra-diffuse galaxy NGC1052–DF2 suggest that it has a very little dark matter component. Most galaxies studied need a component of dark matter associated with them in order to explain their observed properties. Normally there is far more dark matter than normal matter, with an average of 30 times more for galaxies about the mass of the Milky Way. Dwarf galaxies are thought to have even higher ratios of dark to normal matter (more than 400 times as much). Pieter van Dokkum and colleagues report that the ultra-diffuse galaxy NGC1052–DF2 has a dynamical mass, determined by the motions of globular-cluster-like objects, that is essentially the same as the mass in stars, meaning that it does not have a dark matter component. Studies of galaxy surveys in the context of the cold dark matter paradigm have shown that the mass of the dark matter halo and the total stellar mass are coupled through a function that varies smoothly with mass. Their average ratio Mhalo/Mstars has a minimum of about 30 for galaxies with stellar masses near that of the Milky Way (approximately 5 × 1010 solar masses) and increases both towards lower masses and towards higher masses1,2. The scatter in this relation is not well known; it is generally thought to be less than a factor of two for massive galaxies but much larger for dwarf galaxies3,4. Here we report the radial velocities of ten luminous globular-cluster-like objects in the ultra-diffuse galaxy5 NGC1052–DF2, which has a stellar mass of approximately 2 × 108 solar masses. We infer that its velocity dispersion is less than 10.5 kilometres per second with 90 per cent confidence, and we determine from this that its total mass within a radius of 7.6 kiloparsecs is less than 3.4 × 108 solar masses. This implies that the ratio Mhalo/Mstars is of order unity (and consistent with zero), a factor of at least 400 lower than expected2. NGC1052–DF2 demonstrates that dark matter is not always coupled with baryonic matter on galactic scales.

Orbital Decay of Globular Clusters in the Galaxy with Little Dark Matter

Abstract Recently, van Dokkum et al. presented an important discovery of an ultra-diffuse galaxy, NGC1052-DF2, with a dark matter content significantly less than predicted from its stellar mass alone. The analysis relies on measured radial velocities of 10 globular clusters (GCs), of estimated individual masses of a few ×106 M ⊙. This is about 1% of the inferred mass of NGC1052-DF2 of 2 × 108 M ⊙ within a half-light radius, R e = 2.2 kpc. The large relative mass and the old age of these objects imply that they might be susceptible to orbital decay by dynamical friction. Using analytic estimates and N-body simulations of an isolated system matching the inferred mass profile of NGC1052-DF2, we show that the orbits of the most massive GCs should already have decayed on a timescale of a few Gyrs. These findings should help in constraining mass profile and formation scenarios of NGC1052-DF2.

The Initial Mass Function in the Nearest Strong Lenses from SNELLS: Assessing the Consistency of Lensing, Dynamical, and Spectroscopic Constraints

Abstract We present new observations of the three nearest early-type galaxy (ETG) strong lenses discovered in the SINFONI Nearby Elliptical Lens Locator Survey (SNELLS). Based on their lensing masses, these ETGs were inferred to have a stellar initial mass function (IMF) consistent with that of the Milky Way, not the bottom-heavy IMF that has been reported as typical for high-σ ETGs based on lensing, dynamical, and stellar population synthesis techniques. We use these unique systems to test the consistency of IMF estimates derived from different methods. We first estimate the stellar M */L using lensing and stellar dynamics. We then fit high-quality optical spectra of the lenses using an updated version of the stellar population synthesis models developed by Conroy & van Dokkum. When examined individually, we find good agreement among these methods for one galaxy. The other two galaxies show 2–3σ tension with lensing estimates, depending on the dark matter contribution, when considering IMFs that extend to 0.08 M ⊙. Allowing a variable low-mass cutoff or a nonparametric form of the IMF reduces the tension among the IMF estimates to <2σ. There is moderate evidence for a reduced number of low-mass stars in the SNELLS spectra, but no such evidence in a composite spectrum of matched-σ ETGs drawn from the SDSS. Such variation in the form of the IMF at low stellar masses (m ≲ 0.3 M ⊙), if present, could reconcile lensing/dynamical and spectroscopic IMF estimates for the SNELLS lenses and account for their lighter M */L relative to the mean matched-σ ETG. We provide the spectra used in this study to facilitate future comparisons.

CATALOG OF ULTRA-DIFFUSE GALAXIES IN THE COMA CLUSTERS FROM SUBARU IMAGING DATA*

ABSTRACT We present a catalog of ultra-diffuse galaxies (UDGs) in the Coma cluster. UDGs are a subset of low surface brightness (SB) galaxies with very large effective radii defined by van Dokkum et al. We surveyed the Subaru data archive for deep Suprime-Cam/Subaru R-band images, and used data covering the 1.°7 × 2.°7 region of the Coma cluster. The data are ∼1 magnitude deeper than the data of van Dokkum et al (2015a) in limiting SB. This paper explains the details of our sample selection procedure. This UDG catalog includes positions, magnitudes, effective radii, mean and central SBs, and colors (when available). Comparisons with previous galaxy catalogs in the literature are performed, and we show that the current catalog is the largest for UDGs. We also discuss that most of the UDGs are members of the Coma cluster, and the major axis of the UDGs tends to align toward the cluster center (radial alignment).

Ultra-diffuse cluster galaxies as key to the MOND cluster conundrum

MOND reduces greatly the mass discrepancy in clusters of galaxies, but does leave a consistent global discrepancy of about a factor of two. It has been proposed, within the minimalist and purist MOND, that clusters harbor some indigenous, yet-undetected, cluster baryonic (dark) matter (CBDM). Its total amount has to be comparable with that of the observed hot gas. Following an initial discovery by van Dokkum & al. (2015a), Koda & al. (2015) have recently identified more than a thousand ultra-diffuse galaxy-like objects (UDGs) in the Coma cluster. Robustness of the UDGs to tidal disruption seems to require, within Newtonian dynamics, that they are much more massive than their observed stellar component. Here, I propound that a considerable fraction of the CBDM is internal to UDGs, which endows them with robustness. The rest of the CBDM objects formed in now-disrupted kin of the UDGs, and is dispersed in the intracluster medium. While the discovery of cluster UDGs is not in itself a resolution of the MOND cluster conundrum, it lends greater qualitative plausibility to CBDM as its resolution, for reasons I discuss. Alternatively, if the UDGs are only now falling into Coma, their large size and very low surface brightness could result from the adiabatic inflation due to the MOND external-field effect, as described in Brada & Milgrom (2000). I also consider briefly solutions to the conundrum that invoke more elaborate extensions of purist MOND, e.g., that in clusters, the MOND constant takes up larger-than-canonical values of the MOND constant.

OUTFLOWS IN SODIUM EXCESS OBJECTS

van Dokkum and Conroy revisited the unexpectedly strong Na I lines at 8200 A found in some giant elliptical galaxies and interpreted it as evidence for unusually bottom-heavy initial mass function. Jeong et al. later found a large population of galaxies showing equally-extraordinary Na D doublet absorption lines at 5900 A (Na D excess objects: NEOs) and showed that their origins can be different for different types of galaxies. While a Na D excess seems to be related with the interstellar medium (ISM) in late-type galaxies, smooth-looking early-type NEOs show little or no dust extinction and hence no compelling sign of ISM contributions. To further test this finding, we measured the doppler components in the Na D lines. We hypothesized that ISM would have a better (albeit not definite) chance of showing a blueshift doppler departure from the bulk of the stellar population due to outflow caused by either star formation or AGN activities. Many of the late-type NEOs clearly show blueshift in their Na D lines, which is consistent with the former interpretation that the Na D excess found in them is related with star formation-caused gas outflow. On the contrary, smooth-looking early-type NEOs do not show any notable doppler component, which is also consistent with the interpretation of Jeong et al. that the Na D excess in early-type NEOs is likely not related with ISM activities but is purely stellar in origin.

The IMF-SFH connection in massive early-type galaxies

The stellar initial mass function (IMF) is one of the fundamental pillars in studies of stellar populations. It is the mass distribution of stars at birth, and it is traditionally assumed to be universal, adopting generic functions constrained by resolved (i.e. nearby) stellar populations (e.g., Salpeter 1955; Kroupa 2001; Chabrier 2003). However, for the vast majority of cases, stars are not resolved in galaxies. Therefore, the interpretation of the photo-spectroscopic observables is complicated by the many degeneracies present between the properties of the unresolved stellar populations, including IMF, age distribution, and chemical composition. The overall good match of the photometric and spectroscopic observations of galaxies with population synthesis models, adopting standard IMF choices, made this issue a relatively unimportant one for a number of years. However, improved models and observations have opened the door to constraints on the IMF in unresolved stellar populations via gravity-sensitive spectral features. At present, there is significant evidence of a non-universal IMF in early-type galaxies (ETGs), with a trend towards a dwarf-enriched distribution in the most massive systems (see, e.g., van Dokkum & Conroy 2010; Ferreras et al. 2013; La Barbera et al. 2013). Dynamical and strong-lensing constraints of the stellar M/L in similar systems give similar results, with heavier M/L in the most massive ETGs (see, e.g., Cappellari et al. 2012; Posacki et al. 2015). Although the interpretation of the results is still open to discussion (e.g., Smith 2014; La Barbera 2015), one should consider the consequences of such a bottom-heavy IMF in massive galaxies.

Evolution of the Milky Way with radial motions of stars and gas

We study the role of radial migration of stars on the chemical evolution of the Milky Way disk. In particular, we are interested in the impact of that process on the local properties of the disk (age-metallicity relation and its dispersion, metallicity distribution, evolution of abundance ratios) and on the morphological properties of the resulting thick and thin disks.We use a model with several new or up-dated ingredients: atomic and molecular gas phases, star formation depending on molecular gas, yields from the recent homogeneous grid provided by Nomoto et al. (2013), observationally inferred SNIa rates. We describe radial migration with parametrised time- and radius-dependent diffusion coefficients, based on the analysis of a N-body+SPH simulation. We also consider parametrised radial gas flows, induced by the action of the Galactic bar. Our model reproduces well the present day values of most of the main global observables of the MW disk and bulge, and also the observed "stacked" evolution of MW-type galaxies from van Dokkum et al. (2013). The azimuthally averaged radial velocity of gas inflow is constrained to less than a few tenths of km/s. Radial migration is constrained by the observed dispersion in the age-metallicity relation. Assuming that the thick disk is the oldest (>9 Gyr) part of the disk, we find that the adopted radial migration scheme can reproduce quantitatively the main local properties of the thin and thick disk. The thick disk extends up to ~11 kpc and has a scale length of 1.8 kpc, considerably shorter than the thin disk, because of the inside-out formation scheme. We also show how, in this framework, current and forthcoming spectroscopic observations can constrain the nucleosynthesis yields of massive stars for the metallicity range of 0.1 solar to 2-3 solar.

FLUCTUATION SPECTROSCOPY: A NEW PROBE OF OLD STELLAR POPULATIONS

We introduce a new method to determine the relative contributions of different types of stars to the integrated light of nearby early-type galaxies. As is well known, the surface brightness of these galaxies shows pixel-to-pixel fluctuations due to Poisson variations in the number of giant stars. Differential spectroscopy of pixels as a function of fluctuation strength ("fluctuation spectroscopy") effectively measures the spectral variation of stars as a function of their luminosity, information that is otherwise difficult to obtain for individual stars outside of the Local Group. We apply this technique to the elliptical galaxy NGC 4472, using HST/ACS imaging in six narrow-band ramp filters tuned to spectral features in the range 0.8-1.0 micron. Pixels with +- 5% broad-band variations show differential color variations of 0.1% - 1.0% in the narrow-band filters. These variations are primarily due to the systematic increase in TiO absorption strength with increasing luminosity on the upper giant branch. The data are very well reproduced by the same Conroy & van Dokkum (2012) stellar population synthesis model that is the best fit to the integrated light, with residuals in the range 0.03% - 0.09%. Models with ages or metallicities that are significantly different from the integrated-light values do not yield good fits. We can also rule out several modifications to the underlying model, including the presence of a significant (>3% of the light) population of late M giants. The current observations constitute a powerful test of the expected luminosities and temperatures of metal-rich giants in massive early-type galaxies. Studies of pixels with much larger (negative) fluctuations will provide unique information on main sequence stars and the stellar initial mass function.

Variations in the initial mass function in early-type galaxies: a critical comparison between dynamical and spectroscopic results

Abstract I present a comparison between published dynamical (ATLAS3D) and spectroscopic (Conroy & van Dokkum) constraints on the stellar initial mass function (IMF) in early-type galaxies, using the 34 galaxies in common between the two works. Both studies infer an average IMF mass factor α (the stellar mass relative to a Kroupa-IMF population of similar age and metallicity) greater than unity, i.e. both methods favour an IMF which is heavier than that of the Milky Way, on average over the sample. However, on a galaxy-by-galaxy basis, there is no correlation between α inferred from the two approaches. I investigate how the two estimates of α are correlated systematically with the galaxy velocity dispersion, σ, and with the Mg/Fe abundance ratio. The spectroscopic method, based on the strengths of metal absorption lines, yields a correlation only with metal abundance ratios: at fixed Mg/Fe, there is no residual correlation with σ. The dynamical method, applied to exactly the same galaxy sample, yields the opposite result: the IMF variation correlates only with dynamics, with no residual correlation with Mg/Fe after controlling for σ. Hence, although both methods indicate a heavy IMF on average in ellipticals, they lead to incompatible results for the systematic trends, when applied to the same set of galaxies. The sense of the disagreement could suggest that one (or both) of the methods has not accounted fully for the main confounding factors, i.e. element abundance ratios or dark matter contributions. Alternatively, the poor agreement might indicate additional variation in the detailed shape of the IMF, beyond what can currently be inferred from the spectroscopic features.

NaD excess objects and its implications on initial mass function

Abstractvan Dokkum and Conroy (2010) have heralded the possible implication of NaI 8200A lines on extra bottom-heavy IMF of bright elliptical galaxies. A flurry of papers have followed it and revisited the half a century old issue all over again from various angles. For a sanity check, we investigate on the issue this time using the better measured NaD 5900A line instead. We found from the SDSS database that a large (~ 10%) fraction of galaxies (both early and late types) exhibit strong NaD lines and that the same bottom-heavy models of van Dokkum and Conroy that reproduced the strong NaI 8200 lines of elliptical galaxies are not capable of reproducing NaD line strengths at all. The mystery prevails. This article is mainly a review of our recent paper (Jeong et al. 2013).

The stellar IMF in early-type galaxies from a non-degenerate set of optical line indices

We investigate the optical spectral region of spectra of ∼1000 stars searching for initial mass function (IMF)-sensitive features to constrain the low-mass end of the IMF slope in elliptical galaxies. The use of indicators bluer than near-infrared features (NaI, CaT, Wing-Ford FeH) is crucial if we want to compare our observations to optical simple stellar population (SSP) models. We use the MILES stellar library (Sánchez-Blázquez et al.) in the wavelength range 3500–7500 Å to select indices that are sensitive to cool dwarf stars and that do not or only weakly depend on age and metallicity. We find several promising indices of molecular TiO and CaH lines. In this wavelength range, the response of a change in the effective temperature of the cool red giant (RGB) population is similar to the response of a change in the number of dwarf stars in the galaxy. We therefore investigate the degeneracy between IMF variation and ΔTeff, RGB, and show that it is possible to break this degeneracy with the new IMF indicators defined here. In particular, we define a CaH1 index around λ6380 Å that arises purely from cool dwarfs, does not strongly depend on age and is anticorrelated with [α/Fe]. This index allows the determination of the low-mass end of the IMF slope from integrated-light measurements when combined with different TiO lines and age- and metallicity-dependent features such as Hβ, Mgb, Fe5270 and Fe5335. The use of several indicators is crucial to break degeneracies between IMF variations, age, abundance pattern and effective temperature of the cool red giant (RGB) population. We measure line-index strengths of our new optical IMF indicators in the Conroy & van Dokkum SSP models and compare these with index strengths of the same spectral features in a sample of stacked Sloan Digital Sky Survey early-type galaxy spectra with varying velocity dispersions. Using different indicators, we find a clear trend of a steepening IMF with increasing velocity dispersion from 150 to 310 km s−1 described by the linear equation x = (2.3 ± 0.1) log σ200 + (2.13 ± 0.15), where x is the IMF slope and σ200 is the central stellar velocity dispersion measured in units of 200 km s−1. We test the robustness of this relation by repeating the analysis with 10 different sets of indicators. We found that the NaD feature has the largest impact on the IMF slope, if we assume solar [Na/Fe] abundance. By including NaD, the slope of the linear relation increases by 0.3 (2.6 ± 0.2). We compute the ‘IMF mismatch’ parameter as the ratio of stellar mass-to-light ratio predicted from the x-σ200 relation to that inferred from SSP models assuming a Salpeter IMF and find good agreement with independent published results.

Recovery of bimanual coordination is delayed after left hemispheric and/or purely cortical stroke lesions

Recovery of bimanual coordination is delayed after left hemispheric and/or purely cortical stroke lesions