Is Carlos Barcelo's Research Redefining Cosmology and Gravity?

[Chronos]

I was intrigued by this paper and the approach:

http://arxiv.org/abs/gr-qc/0611090
Cosmology as a search for overall equilibrium
Authors: Carlos Barcelo
Comments: 9 pages, 1 figure

"In this letter we will revise the steps followed by A. Einstein when he first wrote on cosmology from the point of view of the general theory of relativity. We will argue that his insightful line of thought leading to the introduction of the cosmological constant in the equations of motion has only one weakness: The constancy of the cosmological term, or what is the same, its independence of the matter content of the universe. Eliminating this feature, I will propose what I see as a simple and reasonable modification of the cosmological equations of motion. The solutions of the new cosmological equations give place to a cosmological model that tries to approach the Einstein static solution. This model shows very appealing features in terms of fitting current observations."

This builds upon earlier works in collaboration with Matt Visser on the concept of Analogue Gravity - an alternative to string and LQG approaches to quantum gravity. It is an interesting idea that, by all appearances, is more touchy feely [testable] than these approaches. It has some warts, mainly the inability to produce the expected quantum backreactions, but not yet mortally wounded so far as I can see.

The idea is substantially based on this also fascinating paper from 2001:

Einstein Gravity as an emergent phenomenon?
Authors: Carlos Barcelo, Matt Visser, Stefano Liberati
Comments: 8 pages, Essay awarded an honorable mention in the year 2001 Gravity Research Foundation essay competition
Journal-ref: Int.J.Mod.Phys. D10 (2001) 799-806

"In this essay we marshal evidence suggesting that Einstein gravity may be an emergent phenomenon, one that is not ``fundamental'' but rather is an almost automatic low-energy long-distance consequence of a wide class of theories. Specifically, the emergence of a curved spacetime ``effective Lorentzian geometry'' is a common generic result of linearizing a classical scalar field theory around some non-trivial background. This explains why so many different ``analog models'' of general relativity have recently been developed based on condensed matter physics; there is something more fundamental going on. Upon quantizing the linearized fluctuations around this background geometry, the one-loop effective action is guaranteed to contain a term proportional to the Einstein--Hilbert action of general relativity, suggesting that while classical physics is responsible for generating an ``effective geometry'', quantum physics can be argued to induce an ``effective dynamics''. This physical picture suggests that Einstein gravity is an emergent low-energy long-distance phenomenon that is insensitive to the details of the high-energy short-distance physics."

Some tantalizing related material has arisen in condensed matter physics along these lines. Perhaps there is an underlying 'apples to oranges' problem with existing approaches to quantum qravity.

 

[Garth]

Do these alternative approaches that asyptotically approach GR in the present low density limit also predict concordant BBN and CMB observables from the high density epoch?

Garth

 

[Chronos]

I don't think those questions are relevant to the model, Garth. It does not forbid or require BBN, or CMB anisotropy, so far as I perceive.

 

[Garth]

Thank you Chronos.

The model is an oscillating universe one.

From the OP paper:
The author seems to be saying that the SN Ia observations already contradict the model:

Taking the currently accepted values of \Omega_{\Lambda} ∼ 0.7 and \Omega_M ∼ 0.3 [21], our model predicts that the acceleration that we observe should be decreasing with time at present and so be bigger in the past. The fitting of the supernova data at high redshifts seems to provide an indication of the contrary [22]. The acceleration appears as nonexistent in the past, giving support to models of dark energy as the Chaplygin gas (see for example [23]). However, by making global comparisons between different cosmological models, other authors argue that it is still impossible to discern, for example, between a cosmological constant and varying dark energy [24].

However, the recent HST observations of SN Ia at high z may reverse this conclusion.

In the BBN high energy regime:

The “close to Big Bang” origin of the universe in our model allows for incorporating most of the predictive power of this paradigm. However one has to bear in mind that any realistic calculation within our model will have to take into account two new factors:
i) Since its entropic birth, the universe could have passed through a few entire cycles before entering in its current expansive phase;
ii) In each new cycle, the maximum value of the acceleration attained, proportional to (itex screwed up here), would be smaller.
For example, the diminishing of the duration of a single phase of nucleosynthesis, owing to the background acceleration predicted for that period, could be compensated with the plausible existence of a few cycles reaching large enough temperatures for nuclear reactions to take place.
Cosmology as a search for overall equilibrium 8
• In this model the time elapsed since the entropic birth of the universe (Big Banglike) would be much larger than in standard cosmological models. We have much more time to produce structures in the universe, something that has always been problematic in standard cosmology.
• This model does not have the so-called horizon problem of standard cosmology asthere is not a “beginning of time” event. Therefore, radiation could have enough time to thermalize in very large scales. If the current cycle would have started at a temperature smaller than T_{recombination}, then, the size of the inhomogeneities found in the cosmic microwave radiation would not directly constraint the size of the inhomogeneities in the barionic matter sector in recent times.
• Within this positive curvature model, the found values Omega_Lambda +Omega_M ~ 1 and H₀ ∼ 14.000 Gyears would be just telling us that the universe is very large, R₀ ≥ 100 Gpc with R₀ its current physical radius, so locally it would be almost flat.
Further checking of the compatibility of this cosmological model with actual observations will be the subject of future work.

So the model does affect BBN, but the author sees ways of compensating for this.

Garth

 

[Chaos' lil bro Order]

Contradicting type 1a observations seems a recipe for disaster. I thought type 1a's are our best standard candles.

 

[Garth]

Contradicting type 1a observations seems a recipe for disaster. I thought type 1a's are our best standard candles.

The best maybe, but that does not necessarily mean they are good!

There could be cosmological or systematic effects that result in their apparent magnitudes being fainter than expected.

The paper, as I read it, is simply claiming we cannot be sure about such a falsification.

Garth

 

[Chaos' lil bro Order]

What about Cepheid Variables? Does the author criticize their use as standard candles too? (I haven't read the paper)

 

[Garth]

They cannot be seen (yet) at such cosmological distances.

Garth

 

[Mike2]

I was intrigued by this paper and the approach:

http://arxiv.org/abs/gr-qc/0611090
Cosmology as a search for overall equilibrium
Authors: Carlos Barcelo
Comments: 9 pages, 1 figure

"In this letter we will revise the steps followed by A. Einstein when he first wrote on cosmology from the point of view of the general theory of relativity. We will argue that his insightful line of thought leading to the introduction of the cosmological constant in the equations of motion has only one weakness: The constancy of the cosmological term, or what is the same, its independence of the matter content of the universe..."

As I understand it, with the expansion of the universe comes an horizon, and horizons produce a bath of thermal particles. I take it that these particles can be permanent and can have mass. So it would seem that the matter/photon content of the universe is not constant as previously suspected. I wonder if this has been taken into account in the cosmological models.

 

[Chronos]

The universe has always had a horizon. Particles always emerged inside that horizon. Photons had already established the boundary conditions. But spacetime had the jump. Photons [and baryons] were trapped inside the surface of last scattering giving spacetime a huge, inflationary headstart.

 

[Mike2]

The universe has always had a horizon. Particles always emerged inside that horizon. Photons had already established the boundary conditions. But spacetime had the jump. Photons [and baryons] were trapped inside the surface of last scattering giving spacetime a huge, inflationary headstart.

I have to wonder how there could be an horizon before there were photons. If photons (and baryons) were formed because of the contraction of the horizon, how was there ever a horizon to begin with before photons? How can you establish the speed of photons without photons?

As R.M. Wald says, perhaps particles are not feasible in tightly curved and changing spacetimes. So I suspect that the reason Inflation was so fast was because it was so tightly curled that no particles could exist to prevent its expansion through gravity? In other words, all there was before particle was vacuum energy. All the particles came from this vacuum energy. But before this vacuum energy decayed into particle it caused a negative pressure of expansion on the universe causing it to inflate.

 

[hellfire]

I have to wonder how there could be an horizon before there were photons. If photons (and baryons) were formed because of the contraction of the horizon, how was there ever a horizon to begin with before photons? How can you establish the speed of photons without photons?

It is not necessary to have any particle to define a horizon. A horizon is determined by the behaviour of geodesics is space-time.

As R.M. Wald says, perhaps particles are not feasible in tightly curved and changing spacetimes. So I suspect that the reason Inflation was so fast was because it was so tightly curled that no particles could exist to prevent its expansion through gravity? In other words, all there was before particle was vacuum energy. All the particles came from this vacuum energy. But before this vacuum energy decayed into particle it caused a negative pressure of expansion on the universe causing it to inflate.

The question whether there existed particles or fields seams not relevant to me here. What is relevant is the energy density that determines the behaviour of the scale factor according to the Friedmann equations. During inflation the energy density of the inflaton with an equation of state p = - \rho was the dominant one. This lead to an accelerated expansion of space.

Moreover, the curvature of space-time seams also to be irrelevant. Consider a spatially flat universe (the universe became nearly flat soon after the start of inflation), with k = 0. The evolution of the scale factor during inflation is:

a = e^{Ht}

with H the constant Hubble parameter during inflation. The most simple quantity that tells you something about curvature of space-time is the Ricci scalar:

R = \frac{6}{a^2}(a \ddot a + \dot a^2 + k)

Inserting for the scale factor:

R = 6 H^2

As you can see, the curvature of space-time is the same before than after inflation (H remains constant in a de-Sitter space-time).

 

[Chronos]

Excellent post hellfire. I merely wish to add that if gravity was emergent in the early universe, inflation had no brakes until gravity emerged from the chaos. I'm not comfortable with an infinite rate of expansion [a quantum fluctuation thing], but, I can see how it might easily have run wild in the first few Planck ticks of time. I feel free to take poetic license here because causality has little meaning before matter freezes out of the primordial soup.

 

[Chaos' lil bro Order]

Excellent post hellfire. I merely wish to add that if gravity was emergent in the early universe, inflation had no brakes until gravity emerged from the chaos. I'm not comfortable with an infinite rate of expansion [a quantum fluctuation thing], but, I can see how it might easily have run wild in the first few Planck ticks of time. I feel free to take poetic license here because causality has little meaning before matter freezes out of the primordial soup.

I completely agree Chronos. Until space had grown sufficiently to thin out its Energy density, gravity was much to weak to influence any structural influence on the emerging universe.

 

[Mike2]

The question whether there existed particles or fields seams not relevant to me here. What is relevant is the energy density that determines the behaviour of the scale factor according to the Friedmann equations. During inflation the energy density of the inflaton with an equation of state p = - \rho was the dominant one. This lead to an accelerated expansion of space.

You seem to be assuming that other fields coexisted with the Inflaton field. But as I understand it, other fields developed from the Inflaton field. So if the ONLY field is the Inflaton field causing Inflation, then there is nothing to stop Inflation.

As you can see, the curvature of space-time is the same before than after inflation (H remains constant in a de-Sitter space-time).

Hold on there. I though one reason that motivated Inflation to begin with is to explain how the universe has become flat when it was curved to begin with. What are you talking about?

 

[hellfire]

You seem to be assuming that other fields coexisted with the Inflaton field. But as I understand it, other fields developed from the Inflaton field. So if the ONLY field is the Inflaton field causing Inflation, then there is nothing to stop Inflation.

Other fields existed already, however, their energy density was negligible. After inflation the energy density of the inflaton was transferred to the other fields.

Hold on there. I though one reason that motivated Inflation to begin with is to explain how the universe has become flat when it was curved to begin with. What are you talking about?

You are right that the theory of inflation was motivated to explain the flatness problem, providing a plausible mechanism to make the universe spatially flat. Flatness in cosmology means zero curvature of space, not space-time. However, the question whether particles may or not exist in very curved space-times relates to the curvature of space-time.

 

[Mike2]

Other fields existed already, however, their energy density was negligible. After inflation the energy density of the inflaton was transferred to the other fields.

Can you give me an idea of what epoc this was? Was this during the time when the universe consisted of a quark soup? Or was it before that when particles received their mass from the Inflaton field? I'm thinking it would have to be before the Standard Model since those fields are of particles which already have mass, etc. And if it is before the SM, then we really don't have a good idea what's going on in that mass receiving interation, do we?

You are right that the theory of inflation was motivated to explain the flatness problem, providing a plausible mechanism to make the universe spatially flat. Flatness in cosmology means zero curvature of space, not space-time. However, the question whether particles may or not exist in very curved space-times relates to the curvature of space-time.

So what are examples of curved spacetime if not near singularity such as black holes and big bangs? I thought Wald was trying to explain the problems with the particle picture associated with these "singularities". But if these situations are not what he is talking about, then I don't know what situations he is referring to.

 

[hellfire]

Can you give me an idea of what epoc this was? Was this during the time when the universe consisted of a quark soup? Or was it before that when particles received their mass from the Inflaton field?

The start and end of inflation is not known. I assume it can start at any time after the Planck time. The quark soup existed after inflation. See this for more information.

I'm thinking it would have to be before the Standard Model since those fields are of particles which already have mass, etc. And if it is before the SM, then we really don't have a good idea what's going on in that mass receiving interation, do we?

I am sorry but I do not understand what you mean here. Particles or fields is not an issue here. Fields have also mass.

So what are examples of curved spacetime if not near singularity such as black holes and big bangs? I thought Wald was trying to explain the problems with the particle picture associated with these "singularities". But if these situations are not what he is talking about, then I don't know what situations he is referring to.

You are right, these are the situations: when the length scale at which space-time curvature is noticeable is comparable to the de-Broglie wavelength of the particles. My intention was just to point out that the fact that energy was transferred to other fields from the inflaton has nothing to do with curvature of space-time.

 

[Mike2]

You are right, these are the situations: when the length scale at which space-time curvature is noticeable is comparable to the de-Broglie wavelength of the particles. My intention was just to point out that the fact that energy was transferred to other fields from the inflaton has nothing to do with curvature of space-time.

Can you give me a little story of what's going on in the Lagrangian or Action Integral, or path integral that causes energy transfer from one field to the next? Would it be some way in which coupling constants are used in flat spacetime verses curved spacetime. Any references would be appeciated too. Thanks.

 

[Chaos' lil bro Order]

Hellfire.
The Inflationary Period is 10-37 to 10-32 seconds.

I'm not saying that by any means known, but that's what the Standard Model says.

 

[Mike2]

Moreover, the curvature of space-time seams also to be irrelevant. Consider a spatially flat universe (the universe became nearly flat soon after the start of inflation), with k = 0. The evolution of the scale factor during inflation is:

a = e^{Ht}

with H the constant Hubble parameter during inflation. The most simple quantity that tells you something about curvature of space-time is the Ricci scalar:...

I suspect a contradiction in your argument here. If the dominant field was the Inflaton field which exhibits a negative pressure during Inflation, then wouldn't this cause an acceleration in expansion so that the Hubble parameter would not remain constant but would increase with the acceleration in expansion?

 

[SpaceTiger]

I suspect a contradiction in your argument here. If the dominant field was the Inflaton field which exhibits a negative pressure during Inflation, then wouldn't this cause an acceleration in expansion so that the Hubble parameter would not remain constant but would increase with the acceleration in expansion?

If the Hubble parameter is constant, the expansion is accelerating. Try taking the second time derivative of the expression hellfire gave.

 

[Mike2]

If the Hubble parameter is constant, the expansion is accelerating. Try taking the second time derivative of the expression hellfire gave.

Now I'm confused. I thought that accelerated expansion was equivalent to an increasing Hubble constant. At present the Hubble constant is 72km/(sec*Mpc) which means at a given distance space is receding away by a certain emount. But recently supernavae observation show that this rate is increasing with time. So in the future the 72km/(sec*Mpc) value will increase. Isn't that what is meant by "accelerating expansion"?

This supernovae data was used to prove the existence of "dark energy" as the cause of the accelerating expansion. So if the negative pressure of a vacuum energy (dark energy) is what causes accelerating expansion defined by increasing Hubble constant, then how can you have during Inflation both a very high vacuum energy causing acceleration and a constant Hubble constant? Thanks.

 

[hellfire]

Can you give me a little story of what's going on in the Lagrangian or Action Integral, or path integral that causes energy transfer from one field to the next? Would it be some way in which coupling constants are used in flat spacetime verses curved spacetime. Any references would be appeciated too. Thanks.

Reheating is a rather difficult subject, at least for my level of understanding. What I can do is provide you some basic concepts that may help you to understand some papers.

Lets start with the general expression for the action of a scalar field with some potential in curved space-time:

S = \int \sqrt{-g} d^4x \left( \frac{1}{2} g^{\mu \nu} \partial^{\mu} \phi \partial_{\mu} \phi + V(\phi) \right)

For a massive field one may take:

V(\phi) = \frac{1}{2} m_{\phi}^2 \phi^2 + \mathcal{O}(\phi^4)

Note that the Lagrangian takes over the square of the determinant of the metric for the volume element:

\mathcal{L} = \sqrt{-g} \left( \frac{1}{2} g^{\mu \nu} \partial^{\mu} \phi \partial_{\mu} \phi - V(\phi) \right)

To calculate the Euler-Lagrange equation for the scalar field you should note first that the only relevant derivative is the time derivative and \partial_1 \phi = \partial_2 \phi = \partial_3 \phi = 0 due to homogeneity and isotropy. This means that the Lagrangian reduces to:

\mathcal{L} = \sqrt{-g} \left( \frac{1}{2} \dot \phi^2 - V(\phi) \right)

Furthermore, for an FRW space-time \sqrt{-g} = a^3. You will get:

\ddot \phi + 3 \frac{\dot a}{a} \dot \phi + m^2 \phi = 0

Inserting the definition of the Hubble parameter:

\ddot \phi + 3 H \dot \phi + m^2 \phi = 0

For the time dependence of the Hubble parameter the first Friedmann equation in a nearly flat space (k = 0) can be used:

H^2 = \frac{8 \pi G}{3 c} \rho

For a scalar field:

\rho = \frac{1}{2} \dot \phi^2 + V

If you consider this scalar field to be the inflaton, then these two equations are the starting point to analize reheating. They describe the behaviour of the scalar field in the regime of inflation (negligible kinetic energy) and in the regime of coherent oscillations (similar kinetic and potential energies). In http://arxiv.org/hep-ph/9407247 you can see how the two solutions are obtained.

Furthermore, you can consider two coupled scalar fields:

\mathcal{L} = \sqrt{-g} \left( \frac{1}{2} \dot \phi^2 - V(\phi) + \frac{1}{2} \dot \chi^2 - V(\chi) - g \phi^2 \chi^2 \right)

And, starting from or inserting the oscilatory behaviour of \phi, analize how \chi behaves and energy transfer or particle production takes place. The formalism to analize this is rather complex. Take a look for example to this paper: http://arxiv.org/hep-ph/9704452

 

[hellfire]

Now I'm confused. I thought that accelerated expansion was equivalent to an increasing Hubble constant.

Consider:

\dot H = \frac{\ddot a}{a} - \left( \frac{\dot a}{a} \right)^2

and insert there the two Friedmann equations for a flat universe k = 0. If you impose the condition

\dot H > 0

you will see that this implies an equation of state of phantom energy different from the inflationary one.

 

[SpaceTiger]

Now I'm confused. I thought that accelerated expansion was equivalent to an increasing Hubble constant. At present the Hubble constant is 72km/(sec*Mpc) which means at a given distance space is receding away by a certain emount. But recently supernavae observation show that this rate is increasing with time. So in the future the 72km/(sec*Mpc) value will increase. Isn't that what is meant by "accelerating expansion"?

"Acceleration" refers to the behavior of objects moving with the Hubble expansion. If the physical distance of an objects is given by

d=ax

where x is the comoving distance and a is the scale factor, it will be accelerating (dd/dt²) if

\ddot{a}>0

The Hubble parameter, however, is

H=\frac{\dot{a}}{a}

Try plugging in a variety of time dependences for the scale factor and see how this works out.

This supernovae data was used to prove the existence of "dark energy" as the cause of the accelerating expansion. So if the negative pressure of a vacuum energy (dark energy) is what causes accelerating expansion defined by increasing Hubble constant

A cosmological constant will also lead to a constant Hubble constant. Consider the Friedmann eq. in a flat, pure lambda universe:

H^2=\frac{\Lambda}{3}

If \Lambda[/tex] is a constant, so is H.

 

[Mike2]

"Acceleration" refers to the behavior of objects moving with the Hubble expansion. If the physical distance of an objects is given by

d=ax

where x is the comoving distance and a is the scale factor, it will be accelerating (dd/dt²) if

\ddot{a}>0

The Hubble parameter, however, is

H=\frac{\dot{a}}{a}

Try plugging in a variety of time dependences for the scale factor and see how this works out.




A cosmological constant will also lead to a constant Hubble constant. Consider the Friedmann eq. in a flat, pure lambda universe:

H^2=\frac{\Lambda}{3}

If \Lambda[/tex] is a constant, so is H.

<br /> <br /> Yes, that helps alot. Thanks ST. So you&#039;re saying that the value of 72km/(sec*Mpc) for the Hubble constant will not change whether there is a dark energy/ cosmological constant or not, is this right? I thought this Hubble constant was measured as 72km/(sec*Mpc) even before the supernovae data when the cosomolgical constant was considered to be zero.

 

[SpaceTiger]

Yes, that helps alot. Thanks ST. So you're saying that the value of 72km/(sec*Mpc) for the Hubble constant will not change whether there is a dark energy/ cosmological constant or not, is this right?

The universe we live in isn't purely \Lambda-dominated, so the Hubble constant will continue to decrease with time, but eventually asymptote to a constant value (if \Lambda CDM is correct).

I thought this Hubble constant was measured as 72km/(sec*Mpc) even before the supernovae data when the cosomolgical constant was considered to be zero.

The Hubble constant was approximately measured prior to the SNe results, but I'm not sure why this would matter.

 

[Mike2]

The universe we live in isn't purely \Lambda-dominated, so the Hubble constant will continue to decrease with time, but eventually asymptote to a constant value (if \Lambda CDM is correct).

I'm starting to get confused again. I was thinking that the Hubble parameter of 72km/(sec*Mpc) could remain constant and there could still be acceleration in expansion because if the scale factor increase by two, then the rate of change of the scale factor would also increase by two so that the fraction would remain constant. This would mean an increase in the expansion rate with time as the scale factor grows (an acceleration). However, now that I've thought about it, if the Hubble perameter remains constant, then by definition that means that the recession rate is the same for a specific proper distance, even as time increases. And that's not what I understand acceleration in expansion to mean. If the expansion rate does not change and stays at 72km/(sec*Mpc), then the proper distance to the Hubble sphere (where the expansion rate is the speed of light) does not change either. But I understand acceleration in expansion to mean that for a given proper distance the recession rate is increasing with time, and this is equal to saying that the recession of 72km/(sec*Mpc) must increase with time. And this also means that the proper distance to the Hubble sphere would be shrinking with time.

The Hubble constant was approximately measured prior to the SNe results, but I'm not sure why this would matter.

Well, I don't mean to be a pest, but you just got through saying that H^2=\frac{\Lambda}{3}. And I equated a positive cosmological constant to equal acceleration in expansion as I understand it in the above paragraph. If all this is true, then it would seem that the Hubble parameter in the above equation should be zero if the cosmological constant were zero. But this seems inconsistent with a non-zero Hubble parameter measured to be 72km/(sec*Mpc) before the supernovae data when we thought the cosmological constant was zero with no dark energy to drive an acceleration in expansion. Thus my confusion. Any help is appreciated.

 

[hellfire]

This may be a matter of definitions. As SpaceTiger wrote, with accelerated expansion it is usually meant \ddot a &gt; 0.

You may define "accelerated expansion" to be \dot H &gt; 0, but then you are not using the standard definition for it, and furthermore, you must be aware that both are not equivalent.

You can understand this also with the deceleration parameter q = - a \ddot a / \dot a^2. Accelerated expansion means q &lt; 0 whereas \dot H &gt; 0 means q &lt; -1.

Accelerated expansion requires dark energy and \dot H &gt; 0 requires, at least in flat space, a phantom energy.

 

[Mike2]

This may be a matter of definitions. As SpaceTiger wrote, with accelerated expansion it is usually meant \ddot a &gt; 0.

You may define "accelerated expansion" to be \dot H &gt; 0, but then you are not using the standard definition for it, and furthermore, you must be aware that both are not equivalent.

You can understand this also with the deceleration parameter q = - a \ddot a / \dot a^2. Accelerated expansion means q &lt; 0 whereas \dot H &gt; 0 means q &lt; -1.

Accelerated expansion requires dark energy and \dot H &gt; 0 requires, at least in flat space, a phantom energy.

You might be right. I'd have to see the math worked out in full. But doesn't the supernovae data say that it is the Hubble parameter, H= 72km/(sec*Mpc), that is increasing in more recent times?

 

[SpaceTiger]

If the expansion rate does not change and stays at 72km/(sec*Mpc), then the proper distance to the Hubble sphere (where the expansion rate is the speed of light) does not change either.

That's correct. In a pure \Lambda phase, the Hubble sphere has a constant physical size, but a decreasing comoving size. This means that it will enclose fewer objects as time goes on, but not recede in proper distance. Again, acceleration generally refers to the behavior of objects at a constant comoving distance, not a constant physical distance.

Well, I don't mean to be a pest, but you just got through saying that H^2=\frac{\Lambda}{3}.

I said that expression would be valid in a pure \Lambda phase. Surely you know that \Lambda CDM is not pure \Lambda? Prior to the SNe results, many assumed that the universe was matter-dominated (and flat), which would give the following for the Friedmann equation:

H^2=\frac{8\pi G\rho}{3}

 

[Mike2]

That's correct. In a pure \Lambda phase, the Hubble sphere has a constant physical size, but a decreasing comoving size. This means that it will enclose fewer objects as time goes on, but not recede in proper distance. Again, acceleration generally refers to the behavior of objects at a constant comoving distance, not a constant physical distance.

Then what's this talk about a possible big rip if the universe keeps on accelerating in its expansion? In such a scenario, even galaxies are torn apart by the space between them accelerating away faster than gravity can hold it together. I'm not aware of the mention of phantom energy as a cause of the big rip, they only presume a cosmological constant. In such a case, space would be accelerating ever faster so that the distance at which it recedes at at the speed of light gets closer.[/QUOTE]

 

[SpaceTiger]

Then what's this talk about a possible big rip if the universe keeps on accelerating in its expansion? In such a scenario, even galaxies are torn apart by the space between them accelerating away faster than gravity can hold it together. I'm not aware of the mention of phantom energy as a cause of the big rip, they only presume a cosmological constant. In such a case, space would be accelerating ever faster so that the distance at which it recedes at at the speed of light gets closer.

Nah, the big rip won't occur with a cosmological constant, it only occurs with phantom energy, as hellfire said. Basically, it happens because the dark energy density is increasing with time (rather than being constant, as with \Lambda), eventually becoming larger than the binding energy of the universe's constituents (galaxies, atoms, etc.) and tearing them apart.

 

[George Jones]

Nah, the big rip won't occur with a cosmological constant, it only occurs with phantom energy, as hellfire said. Basically, it happens because the dark energy density is increasing with time (rather than being constant, as with \Lambda), eventually becoming larger than the binding energy of the universe's constituents (galaxies, atoms, etc.) and tearing them apart.

According to current ideas in physics, John Baez writes in an http://math.ucr.edu/home/baez/end.html" that galaxies and atoms will eventually come apart even without a big rip. Of course, future discoveries could change this scenario.

John Baez[/URL]]Indeed, this whole discussion should be taken with a grain of salt: future discoveries in physics could drastically change the end of this story.

 

[Mike2]

Nah, the big rip won't occur with a cosmological constant, it only occurs with phantom energy, as hellfire said. Basically, it happens because the dark energy density is increasing with time (rather than being constant, as with \Lambda), eventually becoming larger than the binding energy of the universe's constituents (galaxies, atoms, etc.) and tearing them apart.

This issue is a stumbbling block for me. I must be missing something fundamental, and I can't proceed until it's resolved. I don't seem to be making my question very clear. So let me try again.

Even before the Friedmann-Robertson-Walker metric or the application of GR, the expansion rate was measured by the red shift of distant galaxies. Eventually, this recession rate was pinned down to H = 72km/(sec*Mpc) - the result of direct measurement. This means that objects that are 1Megaparsec away are receding at 72km/sec. Objects that are 2Mpc away are receding at 144km/sec. Ultimately there is a distance that is receding at the speed of light - the Hubble sphere. Am I right so far?

Then came the observation that the rate of expansion is not constant, that more distance supernovae were dimmer than expected because they had receded faster than expected of a linear Hubble law. This means that points of space had a rate of recesion that changed. Is this right so far?

OK, then doesn't that mean the Hubble rate of 72km/(sec*Mpc) is not constant but has changed?

 

[hellfire]

The Hubble law relating proper distance (on a simultaneity hypersurface) to recession speed is always linear. For the current time:

d_p = \frac{1}{H_0} v

For small redshifts this can be approximated to:

d_p = \frac{1}{H_0} z

However, what you have probably read is about the recession speed vs. luminosity distance (instead of proper distance). As you can see in Ned Wright's tutorial on Supernova Cosmology the relation depends on the cosmological model.

For small redshifts, the luminosity distance can be written as:

d_L = \frac{1}{H_0} \left(z + \frac{1}{2} (1 - q_0) z^2 + ...\right)

where q_0 is the current deceleration parameter. You can see that the more q_0 &lt; 0 the more deviates this from the linear relation.

 

[SpaceTiger]

According to current ideas in physics, John Baez writes in an http://math.ucr.edu/home/baez/end.html" that galaxies and atoms will eventually come apart even without a big rip. Of course, future discoveries could change this scenario.

The heat death is really quite different from the big rip. In the scenario Baez describes, atoms are gradually ionized as their density decreases and galaxies slowly boil away. In the big rip, on the other hand, both are suddenly torn apart by the increasing energy density of the phantom energy.

 

[SpaceTiger]

OK, then doesn't that mean the Hubble rate of 72km/(sec*Mpc) is not constant but has changed?

It has indeed changed, but I don't think that explains the confusion you're having. Remember that we're looking back in time when we look at high redshifts, so they weren't expecting a linear relation (nor did they get one). In both \Lambda CDM and in the matter-dominated universe, the Hubble constant will decrease with time, but it will decrease much more quickly in the latter.

 

[George Jones]

The heat death is really quite different from the big rip.

I didn't mean to imply that it wasn't, I just meant to highlight that a big rip isn't the only way that galaxies and atoms can come apart.

 

[Mike2]

It has indeed changed, but I don't think that explains the confusion you're having. Remember that we're looking back in time when we look at high redshifts, so they weren't expecting a linear relation (nor did they get one). In both \Lambda CDM and in the matter-dominated universe, the Hubble constant will decrease with time, but it will decrease much more quickly in the latter.

Then what can an accelerating universe means except that distances between points are receding away from each other with ever greater velocity. I suppose it does complicate things to have to guess what the simultaneous points are doing when we can only look down our light cone to the past. But I would think the we would eventually witness what is now simultaneous. Or are you saying that the distance to the Hubble sphere will appear to be getting farther away because we are accelerating away from it? That the light emitted there is still within our past light cone, but since we are accelerating, it will take longer to get here, and so it appears further away? That the path length to reach an accelerating object is longer than for a inertial body?

Perhaps if we were to rescale these numbers to distances where the time lag was negligable and assume we could accurately measure the recession velocity at these close distances, then the situation might become more obvious. 75km/sec/Mpc = 2.4311183144246E-12m/sec/m. Assume we can measure such small distances as accurately as we do a mile. On this scale what does it mean that the universe is accelerating in its expansion?

 

[SpaceTiger]

Then what can an accelerating universe means except that distances between points are receding away from each other with ever greater velocity.

Take an object that's expanding away from us at a constant comoving distance. As time goes on, how does its proper distance from us change? If it accelerates away, then we say the universe is accelerating. Note that this is not the same as asking what will happen to a set of hypothetical objects at the same proper distance at a variety of different times. In a pure \Lambda universe, the Hubble constant remains constant and this set of objects all recedes from us at the same rate, but a single object will accelerate away from us if followed through time. This is because its proper distance is increasing with time, and a constant Hubble constant means that its speed should increase as it moves away from us.

I suppose it does complicate things to have to guess what the simultaneous points are doing when we can only look down our light cone to the past.

With the cosmological principle, it's not so difficult, we just assume that on average, the entire universe behaves the same way at a given time in its history.

But I would think the we would eventually witness what is now simultaneous. Or are you saying that the distance to the Hubble sphere will appear to be getting farther away because we are accelerating away from it?

In a pure \Lambda (which would be accelerating) universe the Hubble sphere stays at a constant proper distance. In our current universe it will slowly recede from us until things become completely \Lambda-dominated. You shouldn't think of the Hubble sphere as an object, just a scale distance that depends upon the expansion rate.

That the light emitted there is still within our past light cone, but since we are accelerating, it will take longer to get here, and so it appears further away?

We don't directly observe the Hubble sphere, it's just a scale radius.

That the path length to reach an accelerating object is longer than for a inertial body?

It's not acceleration in the Newtonian sense. In a \Lambda-dominated universe, distant galaxies will remain on geodesics despite "accelerating" away from us.

Perhaps if we were to rescale these numbers to distances where the time lag was negligable and assume we could accurately measure the recession velocity at these close distances, then the situation might become more obvious.

If the time lag were negligible, then by the cosmological principle, the Hubble constant would be a constant and we would see a linear distance-redshift relationship, regardless of cosmological model.

 

[Mike2]

This is because its proper distance is increasing with time, and a constant Hubble constant means that its speed should increase as it moves away from us.

Well, this much we knew before the supernovae data. What do the researchers of the supernovae data mean by acceleration of expansion? Does it not mean that space at a specified proper distance will receed with greater velocity with time?

 

[SpaceTiger]

Well, this much we knew before the supernovae data. What do the researchers of the supernovae data mean by acceleration of expansion? Does it not mean that space at a specified proper distance will receed with greater velocity with time?

No, it doesn't, read the first paragraph again. That's where I explain what it means.

 

[Mike2]

The Hubble law relating proper distance (on a simultaneity hypersurface) to recession speed is always linear.

We can only get the proper distance by use of a model that we're not quite sure of yet (e.g. what is dark energy)

When I look at the first figure in your reference at:
http://www.astro.ucla.edu/~wright/sne_cosmology.html

I notice how the supernovae line is concave downward. Taking the slope of this curve gives the Hubble parameter for a particular time, right? And I notice how that slope is increasing as we approach the present. I take it this is what is meant by an "accelerating expansion".

But you note how the proper distance recession rate is always the same. I'm not able to understand this yet, and I could use some help. I'm hoping there is a way to get from this to a linear proper distance relation without resorting to a model that we're not sure of yet. For example, is it possible to integrate a curve connecting the dots to get the linear relation? Or is there a non-linear correction factor for say, the interference of dust, that linearized this curve?

If the Hubble diagram were linear, there would be no issue. But since we are dealing with a curved Hubble diagram, this is relatively new (to me), and I don't understand how you get from this to a linear relation with proper distance. Any help you could give would be appreciated.

 

[hellfire]

I don't understand how you get from this to a linear relation with proper distance. Any help you could give would be appreciated.

The linear Hubble law for the proper distance follows from the properties of space-time. Start with the FRW line element with a scale factor a (we do not need the angular coordinates):

ds^2 = -c^2 dt^2 + a^2 dr^2

Take a hypersurface of constant time dt = 0:

ds = a dr

Note that on a hypersurface of constant time the scale factor is the same everywhere and it is possible to integrate to get the proper distance (which is defined on a hypersurface of constant time), let's call it D:

d_{proper} = D = a r

Now, consider a galaxy located at a proper distance D at a specific time on a specific hypersurface. ¿How does its proper distance vary with time?

\frac{dD}{dt} = \frac{da}{dr} r + a \frac{dr}{dt}

The second term of the rhs is the peculiar speed which is zero for a comoving galaxy dr/dt = 0. It follows then that the proper distance changes according to:

\frac{dD}{dt} = \frac{da}{dt} r

The definition of the Hubble parameter is:

H = \frac{1}{a} \frac{da}{dt}

Inserted in the previous equation:

\frac{dD}{dt} = H a r

Moreover, considering the relation between proper distance and radial coordinate we got above:

\frac{dD}{dt} = H D

Which is actually the Hubble law:

v = H D

This is a linear relation on a hypersuface of constant time.

Note that the definition of luminosity distance is a different one.

I hope this helps. Otherwise keep asking and I will do my best.

 

[hellfire]

Ned Wright has a great page explaining this and the relation to other distances here: http://www.astro.ucla.edu/~wright/cosmo_02.htm

 

[Mike2]

The linear Hubble law for the proper distance follows from the properties of space-time. Start with the FRW line element with a scale factor a (we do not need the angular coordinates):

ds^2 = -c^2 dt^2 + a^2 dr^2

Take a hypersurface of constant time dt = 0:

ds = a dr...

I hope this helps. Otherwise keep asking and I will do my best.

Thanks, hellfire, for your help. It was gracious of you to take the time to do the math, and I appreciate it.

Yes, the Hubble constant is spatially invariant, the same at every point in space at a specific time. That's implied by the cosmological principle, and now you've proven it from the metric.

However, my concern is with how the Hubble constant may vary with time. Ned Wright's website that I link to shows recession velocity plotted against luminousity distance. The slope of this curve would seem to give the Hubble constant at various distances. This slope increases for closer distance. And since closer distance means more recent time, the curve would seem to indicate an increase of the Hubble parameter with more recent time. And with increased Hubble parameter comes an increase in the rate at which the universe is expanding. I thought this is what was meant by accelerating expansion.

But it would seem that luminosity distance needs to be corrected for things like redshift. So now I'm not sure that the graph does indicate an increase of the Hubble parameter with time.

And why do I keep asking?... I simply took the word of those who said that the universe was accelerating in its expansion. I thought this meant that the recession velocity for a fixed physical distance was increasing with time. This would mean that the distance out to which the recession velocity was the speed of light was getting closer. I had hoped that this might mean that there would be an entropy associated with this horizon that would then be seen as shrinking. And if horizon entropy constrained the entropy inside it, then a shrinking horizon might be a force of creation inside. But now I'm hearing that the GR predicts never an increase in the Hubble parameter, only decreases to a final value, and that even the graph against luminosity distance when corrected for redshift, etc, does not indicate a change in the Hubble constant, that the acceleration in expansion is only appearent, not actual.

 

[George Jones]

As an example of what hellfire and Space Tiger have said, consider a toy universe and galaxies A, B, C, D at three different instants of cosmological times, t = 1, t = 2, and t = 3.

At times t = 1, t = 2, and t = 3, the proper distances to galaxies A, B, C, D are given by the table:

<br /> \begin{matrix}<br /> &amp; | &amp; A &amp; B &amp; C &amp; D \\<br /> -- &amp; | &amp; - &amp; - &amp; - &amp; - \\<br /> t = 1 &amp; | &amp; 1 &amp; 2 &amp; 3 &amp; 4 \\<br /> t = 2 &amp; | &amp; 4 &amp; 8 &amp; 12 &amp; 16 \\<br /> t = 3 &amp; | &amp; 9 &amp; 18 &amp; 27 &amp; 36<br /> \end{matrix}<br />

At times t = 1, t = 2, and t = 3, the recessional speed of galaxies A, B, C, D are given by the table:

<br /> \begin{matrix}<br /> &amp; | &amp; A &amp; B &amp; C &amp; D \\<br /> -- &amp; | &amp; - &amp; - &amp; - &amp; - \\<br /> t = 1 &amp; | &amp; 2 &amp; 4 &amp; 6 &amp; 8 \\<br /> t = 2 &amp; | &amp; 4 &amp; 8 &amp; 12 &amp; 16 \\<br /> t = 3 &amp; | &amp; 6 &amp; 12 &amp; 18 &amp; 24<br /> \end{matrix}<br />

What are the values of the Hubble constant H at the three times? Since v = H d, the Hubble constant is given by H = v/d. This give that H equals 2, 1, and 2/3 at times 1, 2, and 3.

Note: 1) at each instant in time, the Hubble constant is constant, i.e., independent of the galaxy used to calculate it; 2) the Hubble constant decreases with time.

What about acceleration or deceleration of the expansion of this universe? During the time interval from t = 1 to t = 2, Galaxy A "moves" a distance \Delta d = 4 - 1 = 3. During the later but equal-length interval from t = 2 to t = 3, the same galaxy, Galaxy A, "moves" a greater distance, \Delta d = 9 - 4 = 5. This is an indication that the expansion of the universe is accelerating. The fact that this universe is accelerating is independent of which galaxy is used.

This toy model is a Freidman-Robertson-Walker universe that has its scale factor given by a(t) = t^2.

 

[Mike2]

What about accleration or deceleration of the expansion of this universe? During the time interval from t = 1 to t = 2, Galaxy A "moves" a distance \Delta D = 4 - 1 = 3. During an equal intreval of time, but now from t = 2 to t = 3, Galaxy A "moves" a distance \Delta D = 9 - 4 = 5. Thiis is an indication that the universe is accelerating. The fact that this universe is accelerating is independent of which galaxy is used.

Yes, of course. There is no argument there. Even if the Hubble constant were to decrease with time, the simple fact that space is expanding with time means that galaxies pick up speed with time which is an acceleration. This much was know when the Hubble law was first discovered. So am I now to believe that the supernova data just now discovered this effect? I don't think so. I'm sure they were pointing to an apparent increase in the Hubble constant when they say the word "acceleration". For they say things like, "the expansion rate of the universe is increasing", etc. The problem I have is that now I'm not so sure that the "expansion rate of the universe is accelerating" after corrections for redshift are taken into account. I would think that such experts would make the distinction clear as to an appearent acceleration and an actual acceleration. Or did they?