A Relationship between the Comoving Particle Number and the Effective Cosmological Constant

Abstract

In order to discuss and obtain the remaining inflaton potential, we introduced an idea called "effective static friction" in our last paper to balance the "force", $\frac{dV}{d\phi}|_{\phi=\phi(t_{rest})}$, of inflaton. According to this idea, we now discover that, after the course of particle creation, there will be a relationship between the final relativistic particle number inside an arbitrary chosen comoving volume $N_r(t_{rest})$ and the effective cosmological constant $\Lambda$ in our Universe. This relationship can be expressed as $N_r(t_{rest})=(\frac{\ell^2m_\phi^2\Lambda}{4\pi G\sigma^2(t_{rest})})^{3/2}$ when we employ the classical chaotic model, $V(\phi)=1/2m_{\phi}^2\phi^2$, and consider that $\phi$ comes to rest at $t_{rest}$. Moreover, we obtain an evolution equation for the particle number ($N_r(t)$) inside the comoving volume. Meanwhile, a new inflaton field equation which contains parameters of $N_r(t)$ and "particle creation coefficients" can also be found. Importantly, the results illustrate the fact that $\Lambda$ and $N_r$ are the results of probability.