In Cosmology and in Fundamental Physics there is a crucial question like: where the elusive substance that we call Dark Matter is hidden in the Universe and what is it made of? that, even after 40 years from the Vera Rubin seminal discovery [1] does not have a proper answer. Actually, the more we have investigated, the more this issue has become strongly entangled with aspects that go beyond the established Quantum Physics, the Standard Model of Elementary particles and the General Relativity and related to processes like the Inflation, the accelerated expansion of the Universe and High Energy Phenomena around compact objects. Even Quantum Gravity and very exotic Dark Matter particle candidates may play a role in framing the Dark Matter mystery that seems to be accomplice of new unknown Physics. Observations and experiments have clearly indicated that the above phenomenon cannot be considered as already theoretically framed, as hoped for decades. The Special Topic to which this review belongs wants to penetrate this newly realized mystery from different angles, including that of a contamination of different fields of Physics apparently unrelated. We show with the works of this ST that this contamination is able to guide us into the required new Physics. This review wants to provide a good number of these “paths or contamination” beyond/among the three worlds above; in most of the cases, the results presented here open a direct link with the multi-scale dark matter phenomenon, enlightening some of its important aspects. Also in the remaining cases, possible interesting contacts emerges. Finally, a very complete and accurate bibliography is provided to help the reader in navigating all these issues.

In this paper we constrain the sizes of hypothetical new weak forces by making use the data coming from the precession of Planets. We consider the weak field approximation of Scalar–Tensor Fourth Order Gravity (STFOG), which include several models of modified gravity. The form of the corrections to the Newtonian potential if of the form of Yukawa-like potential (5th force), i.e. $$V(r)=\alpha \frac{e^{-\beta r}}{r}$$ V ( r ) = α e - β r r , where $$\alpha $$ α is the parameter related to the strength of the potential, and $$\beta $$ β to the range of the force. The present data on periastron advance allow to infer a constraint on the free parameter of the gravitational models. Moreover, the Non-commutative Spectral Gravity (NCSG) is also studied, being a particular case STFOG. Here we show that the precession shift of Planet allows to improve the bounds on parameter $$\beta $$ β by several orders of magnitude. Finally such an analysis is studied to the case of power-like potential, referring in particular to deformation of the Schwarzschild geometry induced by a quintessence field, responsible of the present accelerated phase of the Universe.

In the context of the periastron advance as a fundamental test for gravity theories, we show an approach to deduce constraints on the sizes of the new forces arising in the weak field limit of the Extended Theories of Gravity (ETG). It is also studied the force motivated by Quintessence Fields deforming the Schwarzschild geometry and associated to the Dark Energy, responsible of the acceleration of the Universe. As for the ETG, we consider the more general Scalar-Tensor Fourth Order Gravity (STFOG) and the NonCommutative Spectral Gravity (NCSG) as a special case. The solutions of the linearized field equations provide corrections to the Newtonian potential in the Yukawa-like form V ( r ) = α e − β r r , where α is the parameter related to the strength of the potential and β to the range of the force. Quintessence Fields lead to a power-law correction with parameters related to the Dark Energy. By analysing the periastron advance with the use of the current data, we find improvements on the parameters of the gravitational models as well as on the bounds of the parameter β by several orders of magnitude.