Robert:

Here are some references for you. They will convince you that this topic

is a vigorous area of research.

1.

Can the dark energy equation - of - state parameter w be less than -1?

Sean M. Carroll, Mark Hoffman (Chicago U., EFI & Chicago U. & Chicago U., KICP), Mark Trodden (Syracuse U.).

Jan 2003. 29 pp.

Published in Phys.Rev. D68 (2003) 023509

2.

The State of the dark energy equation of state

Alessandro Melchiorri (Oxford U.), Laura Mersini-Houghton (Syracuse U.),

Carolina J. Odman (Cambridge U.), Mark Trodden (Syracuse U.).

Nov 2002. 6 pp.

Published in Phys.Rev. D68 (2003) 043509

3.

Transition of the dark energy equation of state in an interacting holographic dark energy model

Bin Wang (Fudan U.), Yun-gui Gong (CUPT, Chongqing),

Elcio Abdalla (Sao Paulo U.).

Jun 2005. 10 pp.

Published in Phys.Lett. B624 (2005) 141-146

4.

Cosmic Chronometers: Constraining the Equation of State of Dark Energy. I: H(z) Measurements

Daniel Stern, Raul Jimenez, Licia Verde, Marc Kamionkowski, S.Adam Stanford.

Jul 2009.

Published in JCAP 1002 (2010) 008

5.

A Single scalar field model of dark energy with equation of state crossing -1

Ming-zhe Li (Heidelberg U.), Bo Feng, Xin-min Zhang

(Beijing, Inst. High Energy Phys.).

Mar 2005. 4 pp.

Published in JCAP 0512 (2005) 002

6.

Probing dark energy perturbations: The Dark energy equation of state and speed of sound as measured by WMAP

Rachel Bean, Olivier Dore (Princeton U.).

Jul 2003.

Published in Phys.Rev. D69 (2004) 083503

7.

Probing the dark side: Constraints on the dark energy equation of state from CMB, large scale structure and Type Ia supernovae

Steen Hannestad (Nordita), Edvard Mortsell (Stockholm U.). May 2002. 6 pp.

Published in Phys.Rev. D66 (2002) 063508

No, ``dark energy'' does not, necessarily, mean ``non-zero cosmological constant''. The issue is, whether the expansion measured is consistent with a term in the Einstein equations, proportional to the metric (i.e. a cosmological constant) only, or, whether an, additional, contribution from an energy-momentum tensor is required, that *cannot* be reduced to the previous form: for it isn't, generically, true that the energy-momentum tensor is proportional, simply, to the metric. For the time being it seems that a cosmological constant contribution is sufficient to explain the data.

I have taken a quick look at the first reference Carroll et.al.,

It is said that .

1. Dark energy is smoothly distributed and slowly varying.

2. Simplest candidate is vacuum energy, which has a

equation of state w=P/rho = -1

3. w=-1 means it is a constant through space-time. This

is a reflection of the fact that vacuum energy is constant.

4. For varying w scenarios, w < -1/3 gives expansion.

A question naturally comes here is that what is the

experimental justification of assuming that vacuum energy

is a constant throughout space-time.

@ Charles

If one considers a single scalar field with potential V(phi) then the

vacuum energy at classical level is,

lambda_0 + V(phi)

Here lambda_0 is bare cosmological constant.

After including loop effects effective potential should

vary with time. For example it will develop minima

away from zero, required for SSB.

ref:astro-ph/0004075 section 1.3

Yes there is a serious problem with renormalization in quantum

field theories. Are there divergences in classical field theories

also ?

Since we're discussing Einstein's equations, we're in the context of classical gravity.

In these equations there are the two terms that come from the variation of the Ricci scalar and metric, in the Einstein-Hilbert action, and there's the term that comes from the contribution of the cosmological constant. All three terms are *always* present, since they are all consistent with the symmetries of the theory. These latter fix the relative coefficient between the first two terms-but don't constrain the value of the cosmological constant. Matter enters through the energy-momentum tensor. We may write this tensor as the sum of two terms: one, proportional to the metric, and one that is not. These statements refer tp the tensor structure. The energy-momentum tensor, of course, is that of some sort of matter and here's where the issue of the equation of state enters. If any such matter is present, there will be a contribution to the cosmological constant term. If not, then the question is, whether a cosmological constant term alone, i.e. a constant times the metric, can lead to a spacetime consistent with the observable universe. The answer seems to be Yes.

I am adding some links to experimental searches.

SNAP - the SuperNova Acceleration Probe - is a proposed

space observatory designed to measure the expansion of the

Universe and to determine the nature of the mysterious

Dark Energy that is accelerating this expansion. If selected it

will be launched before 2020.

http://snap.lbl.gov/

Euclid is an ESA mission to map the geometry of the dark Universe.

The mission will investigate the distance-redshift relationship and

the evolution of cosmic structures by measuring shapes and

redshifts of galaxies and clusters of galaxies

http://sci.esa.int/euclid/

SUBARU telescope

http://www.naoj.org/

I am also attaching a figure on the currently allowed

values of w. The source of the figure is cosmology page

by Andy Howell. I thank Prof. Probir Roy for names of the

experiments looking for dark energy.

The statement ``the cosmological constant may not be constant'' *means* that dark energy requires the contribution of a new form of matter, described by an energy-momentum tensor. And such matter must be ``exotic'', since it must have negative pressure. However, in contrast to the Hubble ``constant'', there *does* exist a reason, namely the symmetries of the Einstein-Hilbert action, that lead to the appearence of the cosmological constant-so it's there, anyway.

No-we do know what the cosmological constant means and its effects in the solutions to the Einstein equations. We do not know of any matter that can exert negative pressure, so the question, whether with the cosmological constant alone it is possible to quantitatively describe spacetime is perfectly well-defined-and it does seem to work, as the supernova measurements indicate.

Farthest supernova is about 7-8 billion light years away. Accelerated

expansion started roughly at that time of the history

of the universe. Cosmological constant does not change in

space and time. That means it's effect is the same throughout

the history of the universe which is about 14 billion years old.

How to reconcile this two things. If cosmological

constant is what is driving an accelerated expansion then

it would have started right at the beginning. If things were

flying apart from the beginning structure formation would not

have happened. I am attaching a picture of the history of

universe souce:

http://www.nasa.gov/missions/deepspace/f_dark-energy.html

@Charles.

Sorry to ask for one more very basic clarification from you:

Gravitational attraction (in Newtonian sense) depends on the

mass. But how in Einstein's gravity it depends on density

not the mass ? Are these two notions conflicting.

Thanks in advance.

In Einstein's gravity gravitational interaction is expressed through the coupling of the energy-momentum tensor of matter to the metric. This realizes the equivalence principle-and, in the Newtonian limit, becomes the equality between gravitational and inertial mass.

Regarding supernova measurements, the expansion of the Universe and the cosmological constant, it might be useful to read the Nobel Prize lectures of 2011

http://www.nobelprize.org/nobel_prizes/physics/laureates/2011/

The cosmological constant doesn't appear in the Einstein equations through arbitrary choice-even though it was, originally, introduced that way: It appears because the corresponding term in the Einstein-Hilbert action enjoys all of the symmetries of this action and thus may not be left out: The Ricci scalar alone is *not* the most general term, containing at most two derivatives, that is invariant under general coordinate transformations-the cosmological term (containing zero derivatives), is also, admissible and, thus, must be added-it's a dynamical issue what its value is. This action, then describes empty spacetime, in the absence of any energy-momentum tensor. It, also, can describe an expanding Universe, as de Sitter Freidmann and Lemaïtre showed. (Schrödinger later showed that *ordinary* matter can be created in such a Universe.) So the question is, whether such an expanding Universe is consistent with *our* Universe-or is an additional energy-momentum tensor (describing the putative exotic matter content of ``dark energy'') necessary to describe the expansion. The answer seems to be that such an energy-momentum tensor is not necessary.

No: once more, the *most* general action, in four spacetime dimensions, that contains, at most, two derivatives of the metric is given by the Ricci scalar and the cosmological constant term. That's a theorem that can be easily proved. Varying this action, the gives rise to the Einstein equations with cosmological constant. If this constant is given a positive value, the solutions to the Einstein equations describe an expanding Universe-in particular, consistent with what we observe.

People have, also, shown the following: that, if one, only, imposes the condition that the Einstein equations do not contain terms with more than two derivatives, then it's possible to add more terms to the Einstein-Hilbert action. Such terms are called ``Galileon'' terms and have been studied as possible alternatives. But they face significant constraints from solar system measurements.

The cosmological constant term does receive corrections, in the presence of matter. So the question arises, whether its effects can be described by some exotic form of matter. Such matter would have to show negative pressure and is phenomenologically described by a fluid with equation of state p = w rho, with w = -1. Normal matter has w>0. So there has been a lot of work on whether exotic matter with -1< w < 0 could be detected. For ordinary matter, with w>0, leads, of course, to contraction and not expansion.

A lot is semantics, so I've been careful to stress it. Matter *is* described by the energy-momentum tensor; but its presence *also* affects the cosmological constant, inasmuch as it isn't obvious how, *in the presence of matter*, to attribute the spacetime average of the part of the energy-momentum tensor to the cosmological constant or the energy-momentum of matter itself. If one *assumes* the *absence* of matter, then the cosmological constant term is unambiguously identified. If one *assumes* the existence of matter, one may ask what kind of equation of state could give rise to accelerated expansion-and one is led to exotic matter-which, for the moment, cannot be distinguished from a cosmological constant. The rest is a matter of opinion and taste.

The statements about Omega and Omega_Lambda are what this discussion is about, namely: what are the assumptions that underlie identifying Omega with what kind of ``matter'' and identifying Omega_Lambda with the contribution of the cosmological constant term. The issue is, in what way is it possible to test *these* assumptions, that can be experimentally realized, given what is known-and what might be the consequences. This means, of course, working out the content of the terms, *before* one ends up with the Friedmann equations.

The statement of what Omega_Lambda ``is'' or the role it plays is an assertion, whose assumptions and consequences can be tested. The statement that Omega_Lambda is or isn't ``matter'' or ``energy'', similarly, can be tested. There's no a priori reason to declare its role.

The phrase ``varying Lambda'' is not compatible with the phrase ``cosmological constant''. When people talk about ``varying Lambda'' what they *mean* is the part of the energy-momentum tensor, that's proportional to the metric and describes the putative exotic matter that *could* contribute to the accelerated expansion, in addition to the cosmological constant.

Well, as said before, one should pay more attention to what's meant, not what things are called. ``Exotic'' matter, that could contribute to an accelerated expansion is *called* dark energy; ``ordinary'' matter, that has, therefore, attractive gravitational interactions, but, otherwise, interacts very weakly (if at all? that's still not clear) with ordinary matter, is *called* dark matter. The former can be, apparently, described by the cosmological constant contribution; the latter requires its own energy-momentum tensor, about which we currently know very little, save that it should satisfy the same conditions, gravitationally, as ordinary matter. Therefore, the effort is in trying to understand, in what way, new particles, predicted by extensions of the Standard Model, could play the role of dark matter.

No, ``exotic'' matter is not part of Omega, because it doesn't satisfy the energy condition-that's the important point. That's how it's distinguished from dark matter.

Of course dark matter is different from dark energy: the difference isn't that one is ``part of T'' while the other is ``part of Omega_Lambda''. The difference is that the former can contribute to the accelerated expansion of the universe, while the other can't. That's the important point. It then turns out that the cosmological constant, that's inevitably present, can describe the physical properties of ``dark energy'', consistently with experiment.

In any case, once more, what the categories are *called* is a metaphysical question, what their properties are is of physical interest.

We do need measurements to know their ratio (their relative contribution). And we need to know how, from the measurements, we can separate the contributions. Therefore we need models of how such contributions can arise.

Einstein's equations do *not* specify the energy-momentum tensor. That has to be provided separately.

And, in the presence of an energy-momentum tensor, the cosmological constant part isn't easy to identify.

Not that bizarre: unless it's known beforehand that the energy-momentum tensor is traceless, it's not that easy to say what part of the trace of the RHS to Eisntein's equations is part of the cosmological constant and what part belongs to the trace of the energy-momentum tensor itself, for example. There are quite some subtleties.

Charles and Stam, It seems to me that in the Newtonian notion

that says gravitational attraction depends on mass is a result

of `action at a distance' nature of this law. Whereas in Einstein

gravity we have local interactions taking part. So, density as a

function of space time coordinates is coming into play.

There does exist a gravitational potential in the Newtonian approximation, so there isn't any ``action at a distance''. Howver the equality between the mass that enters Newton's law of gravitation and the mass that enters Newton's second law of motion must be imposed separately. Einstein's theory contains Newton's law as a non-relativistic limit and, through the equivalence principle, contains the equality of gravitational and inertial masses, as well as the generalization thereof. However this doesn't have anything to do with dark energy.

Theoretically, for an accelerating universe, \ddot{a}/ a >0, which means \rho+3p