Stringballs and Planckballs for dark matter

As a follow up of the seminal work by Guiot, Borquez, Deur, and Werner on "Graviballs and Dark Matter", we explicitly show that in string theory, local and nonlocal higher derivative theories, as well as general asymptotically-free or finite theories, gravitationally interacting bound states can form when the energy is larger than the Planck energy. On the other hand, in higher derivative or nonlocal theories with interaction governed by a dimensionless or a dimensionful coupling constant, the bound states form when the energy is smaller than the Planck energy. Such bound states are allowed because of the softness of the scattering amplitudes in the ultraviolet region. Indeed, in such theories, the potential is finite while the force is zero or constant in r = 0. Finally, since the bound states that form in the early Universe may have an energy that ranges from the Planck mass to any arbitrarily large or small value, we argue that they can serve as dark matter candidates and/or as the seeds for the structure's formation at large scale in the Cosmos.