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  1. It is sometimes stated that absence of a symmetry to make cosmological constant zero is a problem. But observed value of dark energy is very small and non-zero. So why is it desirable to have a symmetry to make cosmological constant zero?

  2. Second, why is the energy density due to vacuum fluctuations expected to be Lorentz invariant? What happens if it is not so? (In other words, why should the vacuum expectation value of T_{ij} be of the form \Lambda g_{ij}?)

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Related: physics.stackexchange.com/q/22847/2451 –  Qmechanic Nov 27 '12 at 14:31
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1) The cosmological constant in the context of quantum field theory is a set of calculations and leads to a sum that will look something schematically like this cc = 3-5+22-120+3042-50242+... +O(M^4) where M is some heavy mass scale

The problem is we don't know the exact numbers in the sum, we can only calculate a few of them and the best we can do is guess their average magnitude, as there are known constants that set scales.

Naively if you weren't quite sure of your calculation, you would 'guess' the value of any such set of cancellations is going to be basically on the order of the biggest number there (eg M^4). The problem is that the measured value of the cosmological constant is about 10^-47 Gev^4 and if you set M to be the Planck scale (a natural cutoff scale where new physics ought to be important) you will find something like ~10^70 Gev^4. This implies that the sum above must cancel to one part in 10^120 or so, which is a fantastic coincidence, and highly unnatural unless there was some mechanism there that forced the cancellations to be close.

Now, if instead there was a symmetry that made the sum identically vanish (for instance, unbroken supersymmetry), you might be able to cook up a theory that for instance has an undetermined integration constant left over, which you could arbitrarily set to be whatever you wanted. Its just a number, big deal.

However trying to tailor a set of physical laws, between quantities that naively shouldn't know anything about one another (why should the vacuum energy of electron loops, very precisely cancel vacuum energy loops by another set of particles) is a daunting task that requires either a miracle, or an as yet set of new principles (symmetries or other) that force mathematical relations inside of the sum.

2) Lorentz invariance of the energy momentum tensor guarantees that the only term available that satisfies the tensor structure is in fact of the form (Pvac Guv). You can of course choose another type of theory that is not Lorentz invariant (and consequently are no longer dealing with General Relativity) but then you are actually generically presented with new finetuning problem. A quantum vacuum that is not Lorentz invariant will typically generate relevant operators at our own scale that are constrained by experiment to be incredibly small. So you haven't really helped matters much, but then some people try to do this with varying degrees of success. See eg Lifshitz gravity.

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