Gravity Dark Matter Energy
1. Let's start by stating the problem: You want to express your theory that gravity is behind dark matter and dark energy in mathematical equations.
2. Gravity is classically described by Newton's law of universal gravitation:
$$F = G \frac{m_1 m_2}{r^2}$$
where $F$ is the gravitational force, $G$ is the gravitational constant, $m_1$ and $m_2$ are masses, and $r$ is the distance between them.
3. Dark matter is often modeled as additional unseen mass that affects gravitational behavior. We can represent this by introducing an effective mass term $m_{eff} = m + m_{dm}$, where $m_{dm}$ is the dark matter mass contribution.
4. Dark energy is associated with the accelerated expansion of the universe and is often modeled by a cosmological constant $\Lambda$ in Einstein's field equations. The effect of dark energy can be represented as a repulsive force or energy density $\rho_{de}$.
5. To combine these ideas, we can write a modified gravitational potential $\Phi$ that includes contributions from normal matter, dark matter, and dark energy:
$$\Phi(r) = -G \frac{m + m_{dm}}{r} + \frac{1}{6} \Lambda c^2 r^2$$
where the first term is the Newtonian potential with dark matter mass, and the second term represents the effect of dark energy causing repulsion at large scales.
6. This potential can be used to derive forces and motions of objects influenced by gravity, dark matter, and dark energy.
7. In summary, your theory can be expressed mathematically by modifying the gravitational potential to include dark matter as additional mass and dark energy as a cosmological constant term.
Final answer:
$$\boxed{\Phi(r) = -G \frac{m + m_{dm}}{r} + \frac{1}{6} \Lambda c^2 r^2}$$