# Homogeneous universe solutions

An important difference between GR and GLET is the difference in their notions of a homogeneous universe. A homogeneous GR universe may be curved. Instead, a homogeneous GLET universe must be flat — the symmetry of the solution has to be a symmetry of everything physical, thus, also for the fields expressed in the preferred coordinates. This difference remains even in the GR limit of GLET, which defines an ether interpretation of the Einstein equations of GR. In this limit, the curved homogeneous universe solutions will be also solutions of the GLET equations — but, as GLET solutions, they will no longer be homogeneous. The point is that a homogeneous GLET solution must be homogeneous in the preferred (harmonic) coordinates.

Therefore in GLET only the flat universe may be homogeneous. Curved Friedman solutions describe inhomogeneous ether configurations. This justifies the start with the flat Robertson-Walker universe as the ansatz for the homogeneous GLET universe: $ds^2 = d\tau^2 - a(\tau)^2(dx^2+dy^2+dz^2)$ This ansatz is a nice starting point, because the three spatial coordinates are already harmonic functions, thus, can be identified with the preferred coordinates. It remains to find the harmonic time coordinate. Fortunately, this is easy too, and the ansatz which is completely harmonic is $ds^2 = a(\mathfrak{t})^6 d\mathfrak{t}^2 - a(\mathfrak{t})^2(dx^2+dy^2+dz^2)$ where we can find an easy connection between proper time and absolute time defined by $d\tau = a(\mathfrak{t})^3 d \mathfrak{t}.$

Note that we have, as in any physical theory, the full freedom of choice of coordinates. In the previous ansatz we have used this freedom — instead of the preferred coordinates, we have used the preferred spatial coordinates and proper time instead of preferred time. This makes it easier to compare the equations with the usual GR equations: $\begin{eqnarray} 3 \left(\frac{a'}{a}\right)^2 &=& -\Upsilon a^{-6} + 3 \Xi a^{-2} + \Lambda + \varepsilon,\\ 2 \frac{a''}{a} + \left(\frac{a'}{a}\right)^2 &=& +\Upsilon a^{-6} + \Xi a^{-2} + \Lambda - k \varepsilon. \end{eqnarray}$

## $$\Upsilon > 0$$: Big Bounce instead of Big Bang

In the case of $$\Upsilon > 0$$ the left hand side of the first equation is nonnegative, thus, the right hand side has to be nonnegative too. That means, $$a(\tau)$$ cannot become arbitrary small, but there will be, instead, a minimal value $$a_0$$ defined by $\Upsilon a_0^{-6} = 3 \Xi a_0^{-2} + \Lambda + \varepsilon.$ So, there will be a big bounce instead of the big bang — a big crunch until the minimal value $$a_0$$ is reached, followed by a big bang.

## $$\Upsilon < 0$$: Big Bang starting without singularity

In the case of $$\Upsilon < 0$$ we have, for very small $$a(\tau)$$, to care only about the $$\Upsilon$$-term. This gives approximately $a' \sim \sqrt{\frac{-\Upsilon}{3}} a^{-2}, \qquad a \sim \tau^{\frac13}$ Then, as a consequence, we obtain $d \mathfrak{t} = a^{-3} d\tau = \tau^{-1} d\tau = d \ln \tau, \qquad \mathfrak{t} \sim \ln \tau,$ which gives a remarkable picture: While in proper time, the universe starts with a big bang, and this has happened only a finite amount of proper time before now, absolute time reaches $$-\infty$$ in the limit $$\tau\to 0$$. Thus, from point of view of the ether interpretation, there is no big bang singularity in finite true time — the singularity only seems to be a singularity in proper time, but proper time is only clock time, thus, time measured by clocks distorted by gravity.

## $$\Xi > 0$$: Expansion shifts toward coasting

If in some region the term with $$\Xi$$ would be the most important one in comparison with all others, the evolution equation would be $3 \left(\frac{a'}{a}\right)^2 \approx 3 \Xi a^{-2}$ or, in other words, $a' \approx \sqrt{\Xi}$ so that $$a(\tau) \approx \sqrt{\Xi} \tau$$. Such a linear expansion of the universe is named "coasting".

There is, now, some empirical evidence that the evolution during the early universe was evolving much closer to coasting than predicted by the standard $$\Lambda$$CDM evolution.