# Increasing energy scale in an expanding Universe?

1. May 24, 2014

### johne1618

A homogeneous and isotropic Universe is described by the FLRW metric:
$$ds^2 = c^2dt^2 + a^2(t)\ d\Sigma^2,$$
where $a(t)$ is the scale factor and $d\Sigma$ is an interval of uniformly curved co-ordinate 3-space which is independent of cosmic time $t$.

If we set $dt=0$ then we find that the interval of proper distance, $ds$, between two points with co-ordinate space separation, $d\Sigma$, is given by:
$$ds = a(t)\ d\Sigma.$$
Now at the present time $t_0$ we can define the scale factor $a(t_0)=1$. Therefore, at time $t_0$, the co-ordinate separation, $d\Sigma$, is equal to an interval of proper distance, $ds_0$, given by:
$$ds_0 = d\Sigma.$$
Therefore we can eliminate the co-ordinate interval $d\Sigma$ in the two equations above to give:
$$ds = a(t)\ ds_0,$$
where $ds$ is a proper length at time $t$ and $ds_0$ is the corresponding proper length at the present time $t_0$.

Imagine that we have a rigid ruler of length one meter at the present time $t_0$.

Let us transport that ruler into some future time $t$. Since the ruler is rigid it remains one meter in length. But the corresponding length of the future ruler, at our present time $t_0$, $l_0$, is given by the above formula with $ds=1$ and $l_0=ds_0$ so that we have:
$$l_0 = \frac{1}{a(t)}\ \hbox{meters}.$$
Energy and length are related by the Compton wavelength:
$$E = \frac{\hbar c}{\lambda}.$$
Thus the energy associated with one meter measured at time $t$ by a contemporary observer is:
$$E = \hbar c\ \hbox{joules}.$$
However the energy of one meter measured at time $t$, when described in the co-ordinate system of an observer at the present time $t_0$, is given by:
$$E_0 = \frac{\hbar c}{l_0}\\ E_0 = a(t)\hbar c\ \hbox{joules}.$$
When solving the FLRW equations one refers distance scales back to the present time by using $a(t_0)=1$. Thus everything should be in terms of an observer at the present time $t_0$.

For example the energy density of a co-moving volume of dust is conventionally taken to be given by:
$$\rho \propto \frac{1}{a^3}.$$
Given the above discussion I would say that the energy density of a co-moving volume of dust (whose atoms always have a fixed size), from the perspective of an observer at the present time $t_0$, should be given by:
$$\rho \propto \frac{a}{a^3}\\ \rho \propto \frac{1}{a^2}.$$
Rather than assuming that energy density is increasing, I would explain this effect as an increase in Universal energy scale from the perspective of an observer at the present time.

The Planck mass, $M_{Pl}$, is the fundamental energy scale.

Newton's constant $G$ is related to the Planck mass by definition:
$$G \propto \frac{1}{M_{Pl}^2}.$$
Instead of the Planck mass being a constant let us assume that it is proportional to $a(t)$.

Thus, from the perspective of an observer fixed at the present time, Newton's constant is no longer constant but given by the expression:
$$G = \frac{G_0}{a^2(t)}.$$
The Friedmann equation for a spatially flat Universe would then be given by:
$$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G_0 \rho}{3\ a^2(t)}.$$
Instead of the density $\rho$ varying with time let us assume that it is constant $\rho=\rho_0$ and instead $G$ varies with time.

The Hubble constant at the present time, $H_0$, is given by:
$$H^2_0 = \frac{8 \pi G_0 \rho_0}{3}.$$
Therefore we have:
$$\left(\frac{\dot{a}}{a}\right)^2 = \frac{H^2_0}{a^2(t)}.$$
This equation has the simple linear solution:
$$a(t) = H_0\ t\\ a(t) = \frac{t}{t_0},$$
where $t_0$ is the current age of the Universe.

Therefore we obtain a remarkably elegant cosmology with the following features:
• The density $\rho$ is constant (in accord with the perfect cosmological principle)
• The scale factor $a(t) \propto t$
• The Planck mass $M_{Pl} \propto t$
All from the perspective of an observer at the present time $t_0$.

I realize that current observations favor an accelerating universal expansion. But even so this model is a lot closer to observations than a conventional matter-dominated Einstein-de Sitter Universe.

Last edited: May 24, 2014
2. May 24, 2014