A large-scale heuristic modification of Newtonian gravity as an alternative approach to dark energy and dark matter

Abstract

The peculiarities of the inverse square law of Newtonian gravity in standard big bang cosmology are discussed. It is shown that the incorporation of an additive term to Newtonian gravitation, as the inverse Yukawa-like field, allows removing the incompatibility between the flatness of the Universe and the density of matter in the Friedmann equation, provides a new approach for dark energy, and enables theoretical deduction of the Hubble–Lemaïtre law. The source of this inverse Yukawa-like field is the ordinary baryonic matter and represents the large-scale contribution of gravity in accordance with the Mach principle. It is heuristically built from a specular reflection of the Yukawa potential, in agreement with astronomical and laboratory observables, resulting null in the inner solar system, weakly attractive in ranges of interstellar distances, very attractive in distance ranges comparable to the clusters of galaxies, and repulsive in cosmic scales. Its implications in the missing mass of Zwicky, virial theorem, Kepler’s third law in globular clusters, rotation curves of galaxies, gravitational redshift, and Jean’s mass are discussed. The inclusion of the inverse Yukawa-like field in Newtonian gravitation predicts a graviton mass of at least 10⁻⁶⁴ kg and could be an alternative to the paradigm of non-baryonic dark matter concomitant with the observables of the big bang.

Appendix

1.1 Additional note about the virial theorem

Beginning with the Clasius’s virial expression, we have:

$$ \begin{aligned} \frac{1}{\tau }\int_{0}^{\tau } {\frac{{{\text{d}}G}}{{{\text{d}}t}}\,} {\text{d}}t^{\prime} = & \frac{1}{\tau }\int_{0}^{\tau } {\sum\limits_{i} { - \vec{\nabla }U_{i} \cdot \vec{r}_{i} } } \,{\text{d}}t^{\prime} + \frac{1}{\tau }\int_{0}^{\tau } {\sum\limits_{i} {\frac{{p_{i}^{2} }}{{2m_{i} }}} } \,{\text{d}}t^{\prime}, \\ \frac{G(\tau ) - G(0)}{\tau } = & - \frac{1}{\tau }\int_{0}^{\tau } {\sum\limits_{i} {\left( {\frac{{GMm_{i} }}{{r_{i} }} + m_{i} \vec{F}_{{{\text{IY}}}} (r) \cdot \vec{r}_{i} } \right)} } \,{\text{d}}t^{\prime} \\ & + \frac{1}{\tau }\int_{0}^{\tau } {\sum\limits_{i} {T_{i} } } \,{\text{d}}t. \\ \end{aligned} $$

$$ \begin{aligned} 0 = & - \frac{1}{\tau }\int_{0}^{\tau } {\sum\limits_{i} {\left( {\frac{{GMm_{i} }}{{r_{i} }} + \frac{{m_{i} U_{0} (M)}}{r}\;{\text{e}}^{ - \alpha /r} [ {r^{2} + \alpha ( {r - r_{0} } )} ]} \right)} \,} {\text{d}}t^{\prime} + 2\langle T \rangle , \\ 0 = & - \Big \langle {\sum\limits_{i} {\frac{{GMm_{i} }}{{r_{i} }}} + \sum\limits_{i} {m_{i} U_{0} (M)} \,{\text{e}}^{ - \alpha /r} ( {r - r_{0} } )} \Big \rangle \\ & + 2\langle T \rangle + \frac{1}{\tau }\int_{0}^{\tau } {\sum\limits_{i} {m_{i} {\kern 1pt} U_{0} (M)\,{\text{e}}^{ - \alpha /r} \left( {\alpha + r_{0} - \frac{{\alpha r_{0} }}{r}} \right)\,} } {\text{d}}t^{\prime}. \\ \end{aligned} $$

The term within the integral is a continuous function, whose domains are all real numbers, differentiable on the interval (0, τ) and bounded by 1 and f(α), then follow with (38).