# GLET is compatible with $$\Lambda$$CDM

For the flat homogeneous universe the FLRW ansatz is $ds^2 = d\tau^2 - a(\tau)^2(dx^2+dy^2+dz^2).$

The GLET equations for this ansatz are $\begin{eqnarray} 3 \left(\frac{\dot{a}}{a}\right)^2 &=& -\Upsilon a^{-6} + 3 \Xi a^{-2} + \Lambda + \varepsilon,\\ 2 \frac{\ddot{a}}{a} + \left(\frac{\dot{a}}{a}\right)^2 &=& +\Upsilon a^{-6} + \Xi a^{-2} + \Lambda - k \varepsilon. \end{eqnarray}$

So GLET gives here for $$\Xi,\Upsilon \to 0$$ the GR limit for the spatially flat universe, together with the $$\Lambda$$ term. So, there is no problem of compatibility with the usual $$\Lambda$$CDM cosmology.

## Why $$\Lambda < 0$$ seems preferable

On the other hand, I would prefer a negative cosmological constant. There are several reasons:

• A vacuum solution: For $$\varepsilon=0$$ one would hope for some vacuum solution. But if $$\Xi,\Upsilon>0$$, there would be such a vacuum solution only for $$\Lambda < 0$$.
• An oscillating universe: For $$\Xi,\Upsilon>0, \Lambda < 0$$ the general solution would be an oscillating universe. If $$a(\tau)$$ increases, on the right hand side all terms except the $$\Lambda$$ term will decrease. Thus, after some time, the $$\Lambda$$ term becomes more and more important. Nonetheless, once the left hand side of the first equation cannot be negative, there will be an upper bound for $$a(\tau)$$. Similarly, for $$\Upsilon>0$$ there will also be a lower bound, so that we get a big bounce instead of a big bang. So, there will be a periodic, oscillating universe.
• No singularity: A positive $$\Lambda$$ would even lead to a singularity. In this case, for large $$a(\tau)$$ we would have $$\dot{a}(\tau) \sim \sqrt{\frac13\Lambda} a(\tau)$$, which gives $$a(\tau) \sim \exp(\sqrt{\frac13\Lambda}\tau)$$. But then the equation for harmonic time gives $\frac{d \mathfrak{t}}{d\tau} = a(\tau)^{-3} \sim \exp(-\sqrt{3\Lambda}\tau).$

That means that for $$\tau\to\infty$$ absolute time reaches only a finite limit, or, the equation in absolute time would reach infinity in finite absolute time. $\mathfrak{t}(\tau) \sim \left. -\frac{1}{\sqrt{3\Lambda}} \exp(-\sqrt{3\Lambda}\tau') \right|^{\tau}_0 = \frac{1}{\sqrt{3\Lambda}} \left(1- \exp(-\sqrt{3\Lambda}\tau')\right) \xrightarrow[\tau\to\infty]{} \frac{1}{\sqrt{3\Lambda}}.$ This singularity would not exist for $$\Lambda < 0$$.

## Why the singularity is harmless

Given that there is a singularity for $$\Lambda > 0$$, one could think that this would be fatal for GLE, so that there would not be only some weak preference based on metaphysical ideas, but a strong requirement for $$\Lambda < 0$$. But this is not correct. First of all, the competitor is GR, and GR has singularities too. If one considers the GR singularities as not being a decisive argument against GR, then it would be double standard if one would reject GLET because of this. Then, the singularity would be harmless in comparison with the GR singularities, because it will be reached only in infinite proper time, so that the solution would be complete from point of view of GR, and we would not have observable differences.

But the more important argument is a different one, namely that the GLE is, from the start, assumed to be only a continuous approximation, which has to be replaced by an atomic ether theory below some critical distance. The expansion of the universe is explained not as an expansion, but as an effect of shrinking of local rulers. At the singularity, the size of such local rulers would reach zero. But rulers cannot shrink indefinitely, they can shrink only as long as they reach the critical distance. Thus, before the singularity would be reached, we would have to switch anyway to atomic ether theory.

So, even if $$\Lambda > 0$$ would lead to a singularity, it would be a harmless singularity, which will happen inside a region where the GLE has to be replaced by an atomic ether theory anyway.

That means that the arguments in favor of $$\Lambda < 0$$ are of a metaphysical nature, and in no way decisive.

## How strong is the evidence for $$\Lambda > 0$$?

In fact, what seemed to be quite certain and strong evidence in favor of $$\Lambda > 0$$ has been recently questioned: First, observations have found a correlation between the luminosity of the SN 1a and the probable age of the star which collapses. Then, the seeming acceleration appears to be anisotropic. Last but not least, there is Wiltshire's "timescape cosmology" where the acceleration appear to be an effect of the local inhomogeneity. Every one of these three possible corrections alone could be sufficient to make $$\Lambda \le 0$$ possible.

Moreover, in GLE we have the additional $$\Xi$$-term. This term would not allow to lead to an acceleration of the expansion rate, but would shift the expansion rate toward a constant. But such a term could become important because without $$\Lambda > 0$$, with cold dark matter alone, there would be deceleration, and that deceleration could be too strong to be compatible with observation.