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The central regions of nearby elliptical galaxies are dominated by baryons(stars) and provide interesting laboratories for studying the radialacceleration relation (RAR). We carry out exploratory analyses and discuss thepossibility of constraining the RAR in the super-critical acceleration range$(10^{-9.5},\hspace{1ex}10^{-8})$~${\rm m}~{\rm s}^{-2}$ using a sample ofnearly round pure-bulge (spheroidal, dispersion-dominated) galaxies including24 ATLAS$^{\rm 3D}$ galaxies and 4201 SDSS galaxies covering a wide range ofmasses, sizes and luminosity density profiles. We consider a range of currentpossibilities for the stellar mass-to-light ratio ($M_\star/L$), its gradientand dark or phantom matter (DM/PM) halo profiles. We obtain the probabilitydensity functions (PDFs) of the parameters of the considered models viaBayesian inference based on spherical Jeans Monte Carlo modeling of theobserved velocity dispersions. We then constrain the DM/PM-to-baryonacceleration ratio $a_{\rm X}/a_{\rm B}$ from the PDFs. Unless we ignoreobserved radial gradients in $M_\star/L$, or assume unreasonably stronggradients, marginalization over nuisance factors suggests $a_{\rm X}/a_{\rm B}= 10^{p} (a_{\rm B}/a_{+1})^q$ with $p = -1.00 \pm 0.03$ (stat)$^{+0.11}_{-0.06}$ (sys) and $q=-1.02 \pm 0.09$ (stat) $^{+0.16}_{-0.00}$ (sys)around a super-critical acceleration $a_{+1}\equiv 1.2\times 10^{-9}~{\rmm}~{\rm s}^{-2}$. In the context of the $\Lambda$CDM paradigm, this RARsuggests that the NFW DM halo profile is a reasonable description of galactichalos even after the processes of galaxy formation and evolution. In thecontext of the MOND paradigm, this RAR favors the Simple interpolating functionbut is inconsistent with the vast majority of other theoretical proposals andfitting functions motivated mainly from sub-critical acceleration data.

Radial Acceleration Relation between Baryons and Dark or Phantom Matter in the Super-critical Acceleration Regime of Nearly Spherical Galaxies