# Don't Be Fooled by w!

Before our right-wing commenters get too excited by the title, I should point out that the “w” is lower-case, and most certainly is not to be pronounced “*dubya*“. Rather, this post is all about dark energy.

Tomorrow morning, my Ph.D. student, Antonio De Felice, defends his thesis. Assuming all goes well, he will become Dr. De Felice, some time shortly after noon, and I will then take him out to a slap-up lunch at Mrs. Miggins’ Pie Shop, or some reasonably nice restaurant at any rate (if the preceding sentence is puzzling to you, immediately go out and rent the entire Blackaddder series).

The Cosmology group here in Syracuse has already pre-celebrated with Antonio, because he’s leaving town pretty soon (so long as we pass him tomorrow) to take a short vacation and then head off to his postdoc at Sussex in England. We went out for a few beers and snacks the other night (no expense spared here) and a good time was had by all.

Antonio’s thesis contains a number of different pieces of work that he has carried out during his time at Syracuse. I thought it might be interesting to sketch one of the ideas here.

As has been discussed a number of times both by Sean and by me, the observed acceleration of the universe is one of the most puzzling problems facing physicists today. Many different cosmological observations, including those of the cosmic microwave background and large scale structure, imply that, within the context of General Relativity, the majority of mass-energy in the universe is in the form of a smoothly-distributed, negligibly-clustering component referred to as *dark energy*. If the results obtained from observing the light-curves of type-Ia supernovae are included, one is forced, at a high level of confidence, to also conclude that this dark energy is causing the expansion of the universe to speed up at present times – *cosmic acceleration*. This is a weird effect to be sure, and presents a formidable obstacle for theorists. Nevertheless, putting aside the ridiculous fine-tunings required, there are some candidates for what this dark energy might be; the best know of which is Einstein’s cosmological constant.

Hard as it is to imagine, the situation could be even worse for theorists. It is sometimes convenient to parameterize dark energy by what is called *the equation of state parameter*, w (not “dubya” remember). This parameter tells us how, if we treat dark energy as a perfect fluid, the pressure is related to the energy density of the fluid. Current observational bounds place w in the range (very roughly, I’m not trying to quote precise numbers here) -0.8 to -1.2 (it’s important to note that acceleration will occur for any w<-1/3). What I want to focus on for the rest of this post is the possibility that w might be less than -1.

This possibility makes physicists shudder, because various stability proofs in General Relativity require so-called *energy conditions*, all of which are violated by a source of energy-momentum with w<-1. Thus, this type of dark energy runs the risk of being unstable. In fact, as (among others) Sean, his student Mark Hoffman and I showed, specific models satisfying w<-1 exhibit a catastrophic decay under most circumstances.

Given this state of affairs, Sean, Antonio and I asked the question; is there a way that observers could infer w<-1 in a theory within which all energy-momentum sources obey sensible energy conditions (i.e. no matter has an equation of state parameter that is less than -1)? Our idea was to investigate what happens to how one interprets the data in a theory that is not quite Einstein’s gravity.

The concept is extremely simple. Einstein’s equations for cosmology relate the expansion rate of the universe, and its time derivatives, to the amounts and equations of state of the various components of the cosmic energy budget. If, for simplicity, we discuss only constant equations of state, and approximately instantaneous measurements, then those equations can be rewritten to express the equation of state of dark energy as a function of the various amounts of different types of matter (e.g. dark matter and dark energy) and derivatives of the expansion rate of the universe. This is how one infers w from the data.

Now imagine that gravity at large distances is a little different from Einstein’s theory. One would still write down the same relationship to extract w from the data but, because the way this relationship was originally derived is no longer valid, the value extracted is now, in general, just some function of the expansion rate and other variables and, most importantly, does not represent the equation of state parameter of *any* matter content. Thus, under certain circumstances, one might infer w<-1 from this relationship without any unstable matter being present!

Thus far this is just a general statement. What Antonio and his Obi-Wan-like mentors did, was to consider how this might work within a simple model of non-Einstein gravity – *Brans-Dicke theories* with a potential. This isn’t the place to go into the details of such theories; rather for now it’s enough to think of them as theories in which Newton’s constant is replaced by a field, so that, on cosmological scales, it might vary with time.

The dynamics of this field become important in the relationship between w and the various observable cosmological parameters. Part of our work therefore involved us playing around with possible potentials for the field, and identifying what is required to make the relationship yield w<-1 today. Obviously, this isn’t too hard to do if satisfying the data on the equation of state parameter is all one has to achieve. Unfortunately, nature isn’t so kind.

Whenever one modifies gravity, there are all kinds of trouble one can get into. There are strong cosmological constraints on such theories from measurements within the solar system, for example from careful measurements of the time delay of communications with spacecraft far from Earth. In fact, in the last couple of years these constraints have been tightened by an order of magnitude due to measurements made on communications with the Cassini mission (and you thought it was only useful for looking at Saturn and Titan). Applying these constraints to the kind of potentials we constructed yields some pretty odd-looking results.

The upshot is that any potential one constructs to achieve a value for w that is measurably less than -1 and which simultaneously satisfies all other constraints, is exceptionally fine tuned. There are two well-known fine tunings associated with dark energy. The first is the ridiculously small energy scale associated with this component (120 orders of magnitude less than the Planck scale). The second is the coincidence problem – why has dark energy begun to dominate only recently in cosmic history. What Sean, Antonio and I showed was that, if Brans-Dicke theories (with a potential) are to explain an observed w<-1 then there is a *third* fine-tuning – a significant feature in the potential (essentially a local maximum), occurring precisely where the field is at the present cosmological epoch. Obviously, we find this less than compelling and a strike against such theories. Nevertheless, it is important to underscore that our interpretations of cosmological datasets can be complicated by the possibility of a modification of gravity at cosmological scales.

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