96. Quintessence with w less than -1
Save This Post as PDFIn another very interesting recent paper on quintessence the Italian Team (Creminelli, D’Amico, Norena, Vernizzi – and warmest regards from Helsinki if you read it, Filippo 
) is trying to construct an reliable example of QFT that behaves like the quintessence with ?ghost-like? effective equation of state 
.
(If you want first two know what quintessence is, kindly read my recent post on quintessence realized on string theory landscape, where I try to explain the basics.)
Using the word ?reliable? in the first sentence I mean the following. If quintessence is a QFT, this QFT should naturally behave as hydrodynamics at very large scales ? the only scales where its behavior really interests us (since it mimics dark energy). This ideal hydrodynamics (viscosity is irrelevant at cosmological scales) can be characterized by two quantities: one is the effective equation of state (EOS) 
and the other is the speed of sound 
. One can say that the first quantity governs the background, homogeneous, dynamics of the quintessence field, while the second one shows how perturbations in the quintessence field behave.
Note that the case of cosmological constant 
corresponds to an imaginary speed of sound, since 
and 
, and corresponding perturbations should grow without bound due to the gradient instability.
Generally, according to a very old very well known statement, if the EOS parameter and the speed of sound is real (or speed of sound is imaginary and EOS is larger than -1), the corresponding theory has a ghost-like instability and is therefore badly defined (see the Fig. below). The Italian Team is arguing that this is not actually so!
They ask: what actually prevents us to introduce higher derivative terms in the QFT Lagrangian that will stop developing gradient instability (actually, the Team has earlier constructed a formalism for analyzing cosmological perturbations in theories with higher order gradient terms), so the QFT will be free of gradient and ghost-like instabilities in the end.
On the other hand, if we consider cosmological scales, higher order gradient terms are not important (they are suppressed by additional powers of 
), but of course due to their presence the gradient instability is cured at all scales. This leads to the conclusion that long wave length part of the QFT actually behaves as k-essence, i.e., theory with and zero speed of sound for long wave length modes we are interested in.
Higher derivative terms become important also at cosmological scales if one gets really close to the bound corresponding to exact cosmological constant. In this limit, QFT lagrangian reduces to the ghost condensate theory (in turn equivalent to a modification of gravity).
What remains to be understood (and what is actually interesting, I think) is how rapidly does the quintessence cluster in the model used by the authors. What we know for sure is that dark energy does not cluster at all at scales smaller than the cosmological horizon. However, gradient instability (even in the presence of higher order gradient terms curing it) may lead, I expect, to some clustering at smaller scales as well.
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