a, Spectra in the vicinity of the λ = 8,183, λ = 8,195 Na i doublet for three stars from the IRTF library¹²: a K0 giant, which dominates the light of old stellar populations; an M6 dwarf, the (small) contribution of which to the integrated light is sensitive to the form of the IMF at low masses; and an M3 giant, which has potentially contaminating TiO spectral features in this wavelength range. b, Averaged Keck/LRIS spectra of NGC 4261, NGC 4374, NGC 4472 and NGC 4649 in the Virgo cluster (black line) and NGC 4840, NGC 4926, IC 3976 and NGC 4889 in the Coma cluster (grey line). Four exposures of 180 s were obtained for each galaxy. The one-dimensional spectra were extracted from the reduced two-dimensional data by summing the central 4″, which corresponds to about 0.4 kpc at the distance of Virgo and about 1.8 kpc at the distance of Coma. We found little or no dependence of the results on the choice of aperture. Coloured lines show stellar population synthesis models for a dwarf-deficient ‘bottom-light’ IMF¹⁴, a dwarf-rich ‘bottom-heavy’ IMF with x = −3, and an even more dwarf-rich IMF. The models are for an age of 10 Gyr and were smoothed to the average velocity dispersion of the galaxies. The x = −3 IMF fits the spectrum remarkably well. c, Spectra and models around the dwarf-sensitive Na i doublet. A Kroupa IMF, which is appropriate for the Milky Way, does not produce a sufficient number of low-mass stars to explain the strength of the absorption. An IMF steeper than Salpeter appears to be needed. d–f, Spectra and models near the λ = 9,916 Wing–Ford band. The observed Wing–Ford band also favours an IMF that is more abundant in low-mass stars than the Salpeter IMF. All spectra and models were normalized by fitting low-order polynomials (excluding the feature of interest). The polynomials were quadratic in a, b, d and e and linear in c and f.

a, Various stellar IMFs, ranging from a ‘bottom-light’ IMF with strongly suppressed dwarf formation¹⁴ (light blue) to an extremely ‘bottom-heavy’ IMF with a slope x = −3.5. The IMFs are normalized at 1M_⊙, because stars of approximately one solar mass dominate the light of elliptical galaxies. b, Comparison of predicted line Na i and Wing–Ford indices with the observed values. The indices were defined to be analogous to those in refs 4 and 8. The Na i index has central wavelength 0.8195 µm and side bands at 0.816 µm and 0.825 µm. The Wing–Ford index has central wavelength 0.992 µm and side bands at 0.985 µm and 0.998 µm. The central bands and side bands are all 20 Å wide. Both observed line indices are much stronger than expected for a Kroupa IMF. The best fits are obtained for IMFs that are slightly steeper than Salpeter.