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# Stellar initial mass function varies with metallicity and time

## Abstract

Most structural and evolutionary properties of galaxies strongly rely on the stellar initial mass function (IMF), namely the distribution of the stellar mass formed in each episode of star formation1,2,3,4. The IMF shapes the stellar population in all stellar systems, and so has become one of the most fundamental concepts of modern astronomy. Both constant and variable IMFs across different environments have been claimed despite a large number of theoretical5,6,7 and observational efforts8,9,10,11,12,13,14,15. However, the measurement of the IMF in Galactic stellar populations has been limited by the relatively small number of photometrically observed stars, leading to high uncertainties12,13,14,15,16. Here we report a star-counting result based on approximately 93,000 spectroscopically observed M-dwarf stars, an order of magnitude more than previous studies, in the 100–300 parsec solar neighbourhood. We find unambiguous evidence of a variable IMF that depends on both metallicity and stellar age. Specifically, the stellar population formed at early times contains fewer low-mass stars compared with the canonical IMF, independent of stellar metallicities. In more recent times, however, the proportion of low-mass stars increases with stellar metallicity. The variable abundance of low-mass stars in our Milky Way establishes a powerful benchmark for models of star formation and can heavily affect results in Galactic chemical-enrichment modelling, mass estimation of galaxies and planet-formation efficiency.

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## Data availability

The raw dataset that supports the findings of this study is publicly available at the National Astronomical Data Center (https://doi.org/10.12149/101070). The data generated and/or analysed during the study are available at the National Astronomical Data Center (http://paperdata.china-vo.org/jordan/jdli22_imf.csv).

## Code availability

The code used to determine the stellar mass of M-dwarf stars and model fitting is publicly available on GitHub at https://github.com/jiadonglee/MDwarfMachine.

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## Acknowledgements

We thank L. Deng and R. de Grijs for their contributions in the very early stage of the project. We thank J. Liu for discussions. This work is supported by the National Key R&D Program of China no. 2019YFA0405500, the China Manned Space Project with no. CMS-CSST-2021-A07 and CMS-CSST-2021-A08. C.L. thanks the National Natural Science Foundation of China (NSFC) for grant no. 11835057. Z.-Y.Z. and Z.-Q.Y. acknowledge the support of NSFC grants no. 12041305 and no. 12173016, and the Program for Innovative Talents, Entrepreneur in Jiangsu. H.T. acknowledges the support of NSFC grant no. 12103062. X.F. acknowledges the support of China Postdoctoral Science Foundation no. 2020M670023, the NSFC grants no. 12203100, no. 11973001 and no. 12090044, and the National Key R&D Program of China no. 2019YFA0405504. Jiao Li acknowledges the science research grants from the China Manned Space Project with no. CMS-CSST-2021-A10 and no. CMS-CSST-2021-B05, and the NSFC grants no. 12090043 and no. 11873016. Z.-Q.Y. acknowledges support from NSFC grants no. 12203021, the Jiangsu Funding Program for Excellent Postdoctoral Talent under grant no. 2022ZB54, the Fundamental Research Funds for the Central Universities under grant no. 0201/14380049. Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. The National Development and Reform Commission has provided funding for the project. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC; https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This work also benefited from the International Space Science Institute (ISSI/ISSI-BJ) in Bern and Beijing, thanks to the funding of the team ‘Chemical abundances in the ISM: the litmus test of stellar IMF variations in galaxies across cosmic time’.

## Author information

Authors

### Contributions

Jiadong Li contributed most of the modelling and calculations and wrote the initial manuscript. C.L. provided the ideas to initialize the project, supervised Jiadong Li on the modelling, and revised the manuscript. Z.-Y.Z., X.F. and Z.-Q.Y. compared the results with other theoretical and observational work and helped write the manuscript. H.T. ran the calculations to derive vertical actions. Jiao Li helped with discussions on the effect of the binary stars. All authors discussed and commented on the manuscript.

### Corresponding author

Correspondence to Chao Liu.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

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Nature thanks Charlie Conroy, Antonio Sollima and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Extended data figures and tables

### Extended Data Fig. 1 The Sigmoid function fitting result of IMF power-law slope as a function of [M/H].

The yellow line is the best-fit result. The red squares are results in our study using to fit. The blue lines are 100 fitting results selected randomly from the MCMC chains. Error bars represent 1σ uncertainty.

### Extended Data Fig. 2 The derived stellar mass by PARSEC model as a function of absolute magnitude of Ks-band ($${M}_{{K}_{s}}$$).

The colours of each pixel represent the median [M/H] in each colour-magnitude bin.

### Extended Data Fig. 3 The IMF power-law index as a function of [M/H] measured with different methods and different stellar models.

a, The red filled rectangles are derived by all stars with the hierarchical Bayesian model considering the exponential Galactic disk profile. The filled stars denote the values of α derived directly from the observed densities of stars in the |z| range from 100 to 250 pc. b, The IMF power-law slope as a function of [M/H] based on various stellar models and empirical MLR, respectively. The red filled rectangles are derived by PARSEC, and the black rectangles are derived by Dartmouth56. The blue rectangles denote the values of α from the empirical mass–luminosity relation48. Error bars represent 1σ uncertainty.

### Extended Data Fig. 4 Red giant branch stars65 reveals vertical actions Jz increase with stellar age.

a, Age–metallicity distributions of red giant branch stars within 500 pc in the solar vicinity. The red and blue contours display the age–metallicity distribution of dynamically hot (Jz > 20 kpc km s−1) and dynamically cold (Jz < 10 kpc km s−1) stars, respectively. Their contour densities are smoothed by the kernel density estimation method. The dotted lines indicate the separation of [M/H] slices used in the M dwarf samples. b, Stellar ages as a function of Jz in logarithmic form. The left side of the vertical dashed line denotes the dynamically cold stars, and the right side of the vertical dash-dotted line represents the dynamically hot stars. c, The ratio of the normalized number of dynamically hot stars to that of the dynamically cold stars as a function of [M/H]. fdyn.hot is the proportion of the number of dynamically hot stars in each [M/H] bin to all dynamically hot stars of M dwarf star sample. fdyn.cold denotes the similar proportion, but for dynamically cold stars. fdyn.hot/(fdyn.hot + fdyn.cold) indicate the normalized number ratio between dynamically hot and the sum of dynamically hot and dynamically cold ratios. The vertical shaded region represents the metallicity range of −0.5 < [M/H] < 0.2, corresponding to the area of α variation in Fig. 2b and Fig. 3. Error bars represent 1σ uncertainty.

### Extended Data Fig. 5 The test results of the effect of the binary stars on the IMF.

a, The results of simulations to verify the effect of the binary stars in the IMF. The results show the difference of the estimated α from the true values versus binary fraction with the numbers of mock stars equal to 1,000, 10,000 and 100,000. The vertical dotted line represents the binary fraction is 30%, which is approximately the observed mean binary fraction for solar metallicity stars. The vertical dashed line denotes the binary fraction is 60%. b, The test results by setting different slopes of binary fraction as a function of [M/H]. The blue, orange, green and red solid lines show the trend of α with [M/H] by adopting dfb/[M/H] ≈ −0.12, −0.20, −0.50 and −1.00, respectively. The blue dashed line indicates the IMF formula in Yan et al.24. The annotations on the right of the vertical line denote the binary fractions of [M/H] = −0.8. Error bars represent 1σ uncertainty.

### Extended Data Fig. 6 The ΔRVmax distribution of the dynamically hot M-dwarf stars and dynamically cold stars.

The red solid line and blue dashed line represented the dynamically hot and dynamically blue stars, respectively. The vertical dotted line denotes the typical 1σ uncertainty of radial velocity uncertainty of LAMOST M-dwarf stars.

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Li, J., Liu, C., Zhang, ZY. et al. Stellar initial mass function varies with metallicity and time. Nature 613, 460–462 (2023). https://doi.org/10.1038/s41586-022-05488-1

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• DOI: https://doi.org/10.1038/s41586-022-05488-1