The stellar initial mass function ( IMF ) is commonly assumed to be an invariant probability density distribution function of initial stellar masses , being generally represented by the canonical IMF which we define to be the result of one star formation event in an embedded cluster .
As a consequence the galaxy-wide IMF ( gwIMF ) , defined as the sum of the IMFs of all star forming regions in which embedded clusters form which spawn the galactic field population of the galaxy , should also be invariant and of the same form as the canonical IMF .
Recent observational and theoretical results challenge the hypothesis that the gwIMF is invariant .
In order to study the possible reasons for this variation , it is useful to relate the observed IMF to the gwIMF .
Starting with the IMF determined in resolved star clusters , we apply the IGIMF-theory to calculate a comprehensive grid of gwIMF models for metallicities , \mathrm { [ Fe / H ] } \in ( -3 , 1 ) and galaxy-wide star formation rates , \mathrm { SFR } \in ( 10 ^ { -5 } , 10 ^ { 5 } ) \mathrm { M _ { \odot } / yr } .
For a galaxy with metallicy [ Fe/H ] < 0 and SFR > 1 M _ { \odot } /yr , which is a common condition in the early Universe , we find that the gwIMF is both bottom-light ( relatively fewer low-mass stars ) and top-heavy ( more massive stars ) , when compared to the canonical IMF .
For a SFR < 1 \mathrm { M _ { \odot } / yr } the gwIMF becomes top-light regardless of the metallicity .
For metallicities \mathrm { [ Fe / H ] } > 0 the gwIMF can become bottom-heavy regardless of the SFR .
The IGIMF models predict that massive elliptical galaxies should have formed with a gwIMF that is top-heavy within the first few hundred Myr of the galaxy ’ s life and that it evolves into a bottom-heavy gwIMF in the metal-enriched galactic center .
Using the gwIMF grids , we study the SFR - H \alpha relation , its dependency on metallicity and the SFR , the correction factors to the Kennicutt SFR _ { K } - H \alpha relation , and provide new fitting functions .
Late-type dwarf galaxies show significantly higher SFRs with respect to Kennicutt SFRs , while star forming massive galaxies have significantly lower SFRs than hitherto thought .
This has implications for the gas-consumption time scales and for the main sequence of galaxies .
The Leo P and ultra-faint dwarf galaxies are discussed explicitly .