Expanding universe

We wish to discuss the inventory of the mass (energy) of the universe. Assuming that for large scale the universe is homogeneous, we may use the Robertson-Walker metric

$$ds^2=dt^2-R^2(t)\left( \frac{dr^2}{1-kr^2}+r^2d\theta ^2+r^2\sin ^2\theta d\phi ^2\right) ,$$ (1)

where $k$ is a constant. It may be seen that the 3-dimensional space is closed, open or flat for $k>0,k<0$, or $k=0$, respectively. The Einstein equation relates the curvature of the metric to the energy-momentum tensor through the Newton constant. For a perfect fluid, characterized by energy density, $\rho$ and pressure $p$, the energy-momentum tensor is

$$T^{\mu \nu }=(\rho +p)u^\mu u^\nu -g^{\mu \nu }p,$$ (2)

where $u^\mu$ is the covariant velocity and $g^{\mu \nu }=\,(+1,-1,-1,-1)$ diagonal tensor.

The Einstein equation takes, therefore, the form of the Friedmann ones

$$\frac{\dot{R}^2}{R^2}+\frac k{R^2}=\frac{8\pi }3G_N\rho;$$ (3)

$$\frac{\ddot{R}}R=-\frac{4\pi }3G_N(\rho +3p).$$ (4)

The Hubble parameter is $H=\dot{R}/R$. Its value today is expressed in terms of a constant $h$ as

$$H_0=100h\frac{km}{sec\ Mpc}$$ (5)

and the critical density is that which comes from Eq.(3) for $k=0$, i.e.,

$$\rho _o=\frac{3H_o^2}{8\pi G_N}$$ (6)

Since $G_N=\frac 1{m_{pl}^2}$ with the Planck mass $m_{pl}=10^{19}GeV$ using the equalities of the Appendix

$$\rho _c=1.88\,h^210^{-29}gr\,cm^{-3}.$$ (7)

One important experimental result is the determination of the present value of the Hubble constant which confirms that the universe is still expanding. All current data are consistent with(2)

$$h=0.65\pm 0.05.$$ (8)

According to Eq.(3) if the actual universe density is larger, equal or smaller than the critical value of Eqs.(7, 8), the universe is closed, flat or open respectively.