# A Cusp-Core like challenge for Modified Newtonian Dynamics

## TL;DR

This paper examines how Modified Newtonian Dynamics (MOND) predicts specific geometric patterns in galaxy acceleration data, revealing a cusp-core challenge that conflicts with observed galaxy rotation curves, especially in cuspy galaxies.

## Contribution

It introduces a new geometric classification in $g2$-space to distinguish between MOND modified gravity and inertia, highlighting a cusp-core challenge for MOND based on galaxy data.

## Key findings

- MOND predicts cored geometries for isolated galaxies under modified gravity.
- MOND modified inertia yields neutral geometries, neither cuspy nor cored.
- Observed cuspy galaxies deviate from MOND predictions, posing a challenge.

## Abstract

We show that Modified Newtonian Dynamics (MOND) predict distinct galactic acceleration curve geometries in $g2$-space - the space of total observed centripetal accelerations $g_{\rm tot}$ vs the inferred Newtonian acceleration from baryonic matter $g_{\rm N}$ - and corresponding rotation speed curves: MOND modified gravity predicts cored geometries for isolated galaxies while MOND modified inertia yields neutral geometries, ie. neither cuspy or cored, based on a cusp-core classification of galaxy rotation curve geometry in $g2$-space - rather than on inferred DM density profiles. The classification can be applied both to DM and modified gravity models as well as data and implies a {\it cusp-core} challenge for MOND from observations, for example of cuspy galaxies, which is different from the so-called cusp-core problem of dark matter (DM). We illustrate this challenge by a number of cuspy and also cored galaxies from the SPARC rotation curve database, which deviate significantly from the MOND modified gravity and MOND modified inertia predictions.