# Why ratios of the scale factor?

Posted by on Apr 6, 2013 in Blog, Science | 0 comments

I introduced the notion of a scale factor as a concept separate from a fixed and comoving cosmological lattice in a previous post. Developing that idea lead to the Hubble non-constant, $H = \dot{a}(t)/a(t)$.

Later, in developing a couple of simple forms of the Friedmann equations, we encountered something of the form $\ddot{a}(t)/a(t)$. This raises a question.

## Why all these ratios of a(t)?

The answer is simple. The scale factor depends on the scale of the comoving cosmological grid that’s being used. Since proper distances depend upon the product of the scale factor and the size of the grid, $D = a(t) \sqrt{x^2 + y^2 + z^2}$, changing the size of the lattice must have an inverse effect on the corresponding scale factor assuming that the proper distance, $D$, is a given.

Hence, it is only ratios of the scale factor that can be invariant. For example, independent of the choice of grid, the value of $a(t_1)/a(t_0)$; that is, the ratio of the scale factor at two different times, $t_1$ and $t_0$, would be an invariant as to the choice of the scale of the lattice. Likewise, in the Hubble constant, we divide the time derivative of the scale factor by the scale factor itself in order to get an invariant measure of the rate of change of the scale factor. The same is true of the acceleration of $a(t)$ is expressed in the Friedmann equations.

If we expected fluctuations in $a(t)$, we would write these in the form $\delta a(t) / a(t)$ as well. This is not surprising either; fluctuations in some parameter are often expressed in this way. Then, we can consider a fluctuation of, say, 1%, as being greater than one of, say, 0.01%, or alternatively, 1 part per million, by some invariant factor.

### Scaling density

A similar question arises with regard to scaling the density of, say, matter, in our comoving cosmological grid. We introduced a density that was invariant to the grid, call it $\nu$; but this value will again depend upon the scale of the lattice. We could not call this a proper density, if such a term were used. Instead, we employ $\rho$ as a measure of the density of matter relative to a proper volume comprised of a sphere of a radius of given proper length, $D$.

In this way, when we wrote a simple form of the Friedmann equations for a universe with a net $0$ of energy

$(\frac{ \dot{a}(t)}{a(t)})^2 = \frac{8 \pi G}{3} \rho$

we had invariants on both sides of the equation. This equation expresses an exact balance between a kinetic energy term on the LHS and a potential energy term on the RHS. Not too surprisingly, they have the form of Einstein’s field equations; and the kinetic energy term on the LHS also has the character of a metric of space that is in balance with a 00-component of a tensor on the RHS.

However, this form of the equation contains an implicit occurrence of the scale factor in the term $\rho$ that we have to make explicit in order to solve the equation, $\rho = \nu /a^3$. Assuming that we are working with a density of ordinary matter, made up of protons, neutrons, and electrons, then we can just about assume that $\nu$ is some constant. Protons are stable and slow, and we can hang their numbers to our cosmological grid of some grand scale. Neutrons are not quite so stable though, but decay into protons and electrons on their own with a half-life of about 14 – 15 minutes or so. The problem is not so much getting out another proton from a neutron decay, the quantity of matter is roughly equivalent, it is getting out the occasional photon.

This puts us in mind of the fact that we have so far been considering only the density of ordinary matter in our equation for density, and we are well aware that we should really be adding up all of the energy present on the RHS of our last equation. This is true even if we want an exact null balance between that kinetic energy term on the LHS and a matter-energy term on the RHS.

In subsequent posts, we’ll begin to account for these other terms that have to be considered in a complete treatment …