Pho.rs: Quintessence

A1 ^?? Find the total energy $E$ of the universe as a function of its radius $R$ in the form
\[E = -\frac{\alpha}{R} + \frac{\beta}{R^2} + E_K,\]
where $E_K$ is the kinetic energy of the universe.

H1 Use Guass's law for the gravitational field
\[\int_\Gamma \vec{g} \cdot d\vec{S} = -4\pi Gm,\] where $m$ is the mass enclosed within the closed surface $\Gamma$ to find a gravitational field and a gravitational potential in the universe.

H2 The gravitational field $g(r)$ in the massive ball with radius $R$ and mass $M$
\[ g(r) = GM \frac{r}{R^3}\]

H3 The gravitational potential $\varphi(r)$ in the massive ball with radius $R$ and mass $M$
\[ \varphi(r) = -\frac{3GM}{2R} + \frac{GMr^2}{2R^3}.\]

A2 ^?? For given $M_s$, $M_q$, $A$ and $E$ of the universe find a range of radii which it can have. Write the expression in terms of $\alpha$, $\beta$ and $E$.

A3 ^?? Could the universe described by TCM be stable, i.e., have a constant radius? If so, describe it quantitatively.

A4 ^?? How long will the universe continue to expand at a positive acceleration?

H1 To calculate the integral
\[\int\frac{x \, dx}{\sqrt{x-1}}\]
you can use the substitution $u = x - 1$.

Quintessence

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