The Dynamical Equations of Cosmology

Abstract

The Einstein equations applied to the FLRW metric give basic dynamical equations for cosmology, specifically for the scale factor. The dynamical equations depend on the physical properties of the constituents of the cosmic fluid, which we take to be vacuum or dark energy, cold matter, radiation, and an effective curvature. Together with the behavior of the constituents the dynamical equations lead to what we here call the Friedmann master equation for the scale factor of the universe; it is remarkably useful.

Appendix 1: The Einstein Tensor for the FLRW Metric

The Riemann and Ricci tensors were defined and discussed in Chaps. 8 and 10, and in particular the Ricci tensor is given in (8.27). The Einstein tensor which occurs on the geometric left side of the field equations was defined in terms of the Ricci tensor in (8.31). For the diagonal FLRW metric it is straight-forward to calculate the Ricci tensor and the Ricci scalar; the result for the nonzero components is (Schutz 2009)

$$ \begin{aligned} & R_{{00}} = \frac{{3a^{{\prime \prime }} }}{{ac^{2} }},\quad R_{{11}} = \frac{{ - 1}}{{1 - kr^{2} }}\left[ {2k + \frac{{aa^{{\prime \prime }} }}{{c^{2} }} + \frac{{2a^{{{\prime }2}}}}{{c^{2} }}} \right], \\ & R_{{22}} = - r^{2} \left[ {2k + \frac{{aa^{{\prime \prime }} }}{{c^{2} }} + \frac{{2a^{{{\prime }2}}}}{{c^{2} }}} \right], \quad R_{{33}} = - R_{{22}} \sin ^{2} \theta , \\ & R = {R^{\beta }}_{\beta } = \frac{6}{{a^{2} }}\left[ {k + \frac{{aa^{{\prime \prime }} }}{{c^{2} }} + \frac{{a^{{{\prime }2}}}}{{c^{2} }}} \right]. \\ \end{aligned} $$

(14.20)

From the Ricci tensor and Ricci scalar the 0,0 and 1,1 components of the Einstein tensor are

$$ G_{00} = - \frac{{3a^{{{\prime}2}} }}{{a^{2} c^{2} }} + \frac{3k}{{a^{2} }},\quad G_{11} = \frac{ 1}{{1 - kr^{2} }}\left[ {k + \frac{{2a a^{{\prime \prime }} }}{{c^{2} }} + \frac{{a^{{{\prime }{2}}}} }{{c^{2} }}} \right]. $$

(14.21)

Finally we raise an index and obtain the mixed index forms

$$ {G^{0}}_{0} = - 3\left[ {\frac{{a^{{{\prime }2}} }}{{a^{2} c^{2} }} + \frac{k}{{a^{2} }}} \right],\quad {G^{1}}_{1} = - \left[ {\frac{k}{{a^{2} }} + \frac{{2a a^{\prime\prime}}}{{a^{2} c^{2} }} + \frac{{a^{{{\prime }{2}}} }}{{a^{2} c^{2} }}} \right]. $$

(14.22)

This verifies the Einstein tensor (14.4) in the text. Note that in the mixed index form there is no explicit spatial dependence in the Einstein tensor, which is a convenient feature. The other components of the field equations are either identically zero or the same as the above. See Exercise 14.2.

Exercises

• 14.1 Calculate the affine connections for the FLRW metric. Alternatively see Misner (1973) and Schutz (2009).

• 14.2 Verify the calculation of the Ricci and Einstein tensors for the FLRW metric in the Appendix or see Schutz (2009) and Misner (1973). Show that the other components of the field equations are either identically zero or redundant. In particular show ${G^{1}}_{ 1} = {G^{2}}_{ 2} = {G^{3}}_{ 3}$.

• 14.3 Consider a star orbiting near the edge of a galaxy in a circular orbit, with the mass of the galaxy concentrated in the central bulge. What is the relation between the orbital velocity and radius of the orbit?

• 14.5 Now suppose that the galaxy is dominated by dark matter distributed spherically symmetrically as in Fig. 14.1 with density $\rho \left( r \right)$. What is the relation between the orbital velocity and radius of the orbit? In the special case that the velocity is constant show that the density distribution is proportional to $1/r^{2}$. What is the total mass of the dark matter in the galaxy? Is this a problem?

• 14.6 Look up a reference on the work of Zwicky, then work out the way that the random motion in a cluster of galaxies (velocity dispersion) can determine their mass (Zwicky 1933; Wiki DM).