Let's see how the new board works GR-wise

[marcus]

General Relativity has been the main theme running through
the discussions I enjoyed in the previous PF version.

Cosmology issues, removal of the big bang singularity (loop quantum cosmology LQC), dark energy (cosmological constant), the limits of the observable universe, the CMB, redshift distance relations, the discovery that quantizing GR provides an explanation for inflation (incorporating inflation into LQC is another recent result),
playing around with the Friedmann equations, the recent consensus in cosmology based on COBE, HST, WMAP, and
discussion of black hole phenomena (holes are at home in GR)

The threads I've most enjoyed and the questions from other PF posters I've found most thought-provoking have all centered around GR (quantizing it, applying it to cosmology, thinking about the universe and its beginnings--- discussing what we can tell about it from observations, and so on.

Nothing else on the board is as good a fit, as far as I can see.
So for starters I guess I will hang out here. And see what happens.

The old board which had "Theoretical Physics" as a forum
and "Astronomy and Cosmology" as a forum seemed a
reasonable balance between generality and specialization----the present listing is a bit more like the college course catalog and
may be a bit numbing/sterilizing to some. But this may work out
all the same, who knows. There has to be a place for cross-fertilization and General Relativity Cum Cosmology might be a good venue

We'll give it a try

edit: "incorporating" added in line with Loop's point in the next post

 

[MathematicalPhysicist]

Originally posted by marcus
General Relativity has been the main theme running through
the discussions I enjoyed in the previous PF version.

Cosmology issues, removal of the big bang singularity (loop quantum cosmology LQC), dark energy (cosmological constant), the limits of the observable universe, the CMB, redshift distance relations, the discovery that quantizing GR provides an explanation for inflation (inflation is another recent LQC result),

are you sure inflation is a result of LQC if I am not mistaken inflation has been theorized by andrey linde and allan guth.

 

[marcus]

Originally posted by loop quantum gravity
are you sure inflation is a result of LQC if I am not mistaken inflation has been theorized by andrey linde and allan guth.

You are right that back in the 1980s Alan Guth, Linde and others conceived of inflation "scenarios"
and imagined various mechanisms that might cause inflation, dreaming up particles and fields called "inflatons".

Martin Bojowald recently derived inflation from the loop quantum cosmology model.

It doesn't have to be "put in by hand" but appears when the model is quantized-----the time zero singularity goes away and inflation appears, both as natural consequences of the non-perturbative background-independent quantization proceedure.

http://arxiv.org/gr-qc/0306008
and additional references Bojowald gives here

Some recent results (Alexander et al) fit inflation into context of a quantum description of the early universe:

http://arxiv.org/hep-th/0309045

 

[marcus]

http://arxiv.org/hep-th/0309045
[/B]

Loop, it would be interesting to delve into the question

"How does inflation fit into quantizing General Relativity?"

We could see if anybody wants to go thru the Alexander et al paper "Quantum Gravity and Inflation"

I gave the link above. It is 18 pages.

Page 4: "Thus, there appears to be no longer any reason to restrict the study of quantum cosmology to the semiclassical approximation. In this paper we provide more evidence of this, by finding exact solutions to the equations of quantum cosmology that provide exact quantum mechanical descriptions of inflation..."

Page 16: "The results of this paper represent a step towards a detailed study of the very early universe beyond the semiclassical approximation, in which quantum gravitational effects are treated in a non-perturbative and background independent manner. For each potential V(f) and classical slow-roll solution u(T) consistent with inflation, we have found a quantum state given by (56) which is an exact solution to the quantum equations of motion, but has a classical limit given by the classical solution. Furthermore we can construct normalizable states which are wavepackets around the initial conditions that generate that classical solution.

Thus, inflation is here described in terms of exact quantum states."

Stephon Alexander is a member of the high energy theory group at SLAC. It is an interesting paper and not very long, we could try to go through it and get some understanding of how inflation fits into the ongoing effort to quantize GR.

 

[marcus]

"what we believe is the first direct derivation..."

The initial paper in this line of investigation was by Martin Bojowald (over a year ago)

"Inflation from Quantum Geometry"

http://arxiv.org/gr-qc/0206054

His concluding paragraph on page 4 begins:

"In this paper we presented what we believe is the first direct derivation of inflation from a candidate for a quantum theory of gravity...Inflation with a graceful exit into a standard Friedmann phase can be regarded as a natural prediction of quantum geometry."

The italics are Bojowald's emphasis.
By "candidates for a quantum theory of gravity" I assume he means primarily efforts to quantize general relativity, which is a background-independent theory, but it could refer more generally to perturbative, fixed-background theories as well.

 

[marcus]

The abstract of the June 2002 Bojowald paper just linked is interesting:

"Inflation from Quantum Geometry

Quantum geometry predicts that a universe evolves through an inflationary phase at small volume before exiting gracefully into a standard Friedmann phase. This does not require the introduction of additional matter fields with ad hoc potentials; rather, it occurs because of a quantum gravity modification of the kinetic part of ordinary matter Hamiltonians.

An application of the same mechanism can explain why the present-day cosmological acceleration is so tiny."

What seems to be the key equation in this short (4 page) paper
is equation (8) on page 3. It is a quantized form of the usual Friedmann equation which gives the square of the Hubble parameter H-----the expansion rate, i.e. a key player.

What this equation says, in essence, is that as long as the scale factor a(t) is small, the square of the Hubble parameter H² is proportional to a⁹
So the expansion rate H is proportional to a^9/2
By elementary differential calculus one can indeed solve
for a(t) to see how it behaves at very early times while it is still small and it turns out to behave as if it had a "pole" at some early time t₀ and were proportional to
1/(t₀ - t)^2/9

this is is what is known as "super-inflationary" expansion with an equation of state parameter that starts out at w = -4 which translates into radically extreme negative pressure and super-fast accelerated expansion
(dark energy accelerating contemporary expansion is usually assumed to have w = -1, considerably less extreme than -4) .

But after the scale factor a(t) has grown some then the character of the quantized Friedmann equation automatically changes into a "large a" regime where
H² is no longer proportional to a⁹ but is instead proportional to a^-3-----the equation of state w goes from -4 to +1 as the scale grows and the system makes a smooth transition over to non-inflation.

This automatic civilized behavior is contained in Bojowald's equation (8) together with his "small a" and "large a" approximations. But you can see from looking at page 3 of this short paper that it is a bit hairy.

There are a couple of computer generated curves plotted, to help understand, but all the same it isn't immediately clear why this works. It seems to ultimately depend on the discreteness of volume (!) in quantum general relativity---which goes back to the basic background-independence. Having volume discrete----the volume and density operators have discrete spectra---means that when things get really really dense near the beginning of expansion some non-classical behavior takes over. This is referred to in the abstract as "a quantum gravity modification of the kinetic part of the ordinary matter Hamiltonian"

It is the matter Hamiltonian that appears in the quantized Friedmann equation (8) which I referred to earlier and which seems to be the key equation in this paper's approach to deriving inflation.

Well here is an interesting place where ordinary GR (the Friedmann equations) and classical cosmology meet up with inflation scenarios and with the work in progress to quantize GR.


There are some followup papers by Bojowald and others that I should get links to.