Has this idea been explored? Why the universe expands

[mrspeedybob]

I lack the mathematical skills to determine if this idea is plausible. Maybe somebody has already been down this road.

The universe cools and the universe expands. At first this makes perfect sense because of the relationship between pressure, volume, and temperature. What if we have the cause and effect reversed?

Heat energy is the random motion of atoms and molecules. If All my atoms are moving randomly and all of your atoms are moving randomly then most of them are in motion relative to each other. This relative motion causes relativistic length contraction not only of your atoms relative to me and vice-versa but also of the distance between us. Now If I die and cool off there is less motion, less length contraction, more distance.

Could the universe be expanding because it is cooling off?

 

[bcrowell]

This relative motion causes relativistic length contraction not only of your atoms relative to me and vice-versa but also of the distance between us.

Heating an object doesn't change its distance to other objects. We don't measure the object's distance to other objects in a frame of reference tied to one of its moving atoms.

 

[StateOfTheEqn]

A universe consisting only of photons would cool by expanding. The energy of a photon varies inversely with its wavelength, that is, E=hc/(wavelength). As space in such a universe expanded the wavelengths would stretch and the energy of each photon would decline accordingly. The result would be a universe cooling itself through expansion. It can also be shown that the entropy S would change according to a time-independent version of the Hubble parameter. In the appropriate units, dS=da/a where a is the expansion scale factor.

The universe we live in contains energy in various forms other than photons so the situation is somewhat more complex. There is no current consensus on why the universe is expanding. One view is the expansion is kinematic. That means the expansion is due to the kinetic energy of the individual galaxies in the universe. Another view is that space itself is expanding, similar to the photon universe in the previous paragraph. But why is space itself expanding? It could be expanding due to a non-zero pressure term within the energy-momentum tensor or something like dark energy.

 

[bcrowell]

There is no current consensus on why the universe is expanding. One view is the expansion is kinematic. That means the expansion is due to the kinetic energy of the individual galaxies in the universe. Another view is that space itself is expanding, similar to the photon universe in the previous paragraph.

I don't think this is quite right. This is like saying that there is no consensus on why a rock falls when you release it: some people say it's because of F=ma, while others say it's because potential energy is getting converted to kinetic. The two interpretations you're giving for cosmological expansion are not contradictory. They're just two different verbal depictions of the same mathematics. There is no mystery about why the universe is expanding. GR explains it 100% in the Friedmann equations. There is some mystery in the equation of state, but the fact that you get expanding cosmologies that began with a big bang singularity is independent of details about the equation of state.

 

[StateOfTheEqn]

There is no mystery about why the universe is expanding. GR explains it 100% in the Friedmann equations. There is some mystery in the equation of state, but the fact that you get expanding cosmologies that began with a big bang singularity is independent of details about the equation of state.

The problem with the Friedmann models is that all three depend on the 'dust solution' where pressure=0. If the expansion is not kinematic, pressure is required. I agree there is some mystery. For the Friedmann models, 'pressure' must be brought in through the back door using the cosmological constant and/or 'dark energy'. That means the question of why the universe is expanding has not yet been answered in a really coherent manner.

 

[bcrowell]

The problem with the Friedmann models is that all three depend on the 'dust solution' where pressure=0.

FRW solutions can contain any mixture of dust, radiation, and cosmological constant. For example, the early universe was radiation-dominated, so the pressure was not zero, and the reason for the pressure is not mysterious.

If the expansion is not kinematic, pressure is required.

I assume "kinematic" refers to the distinction you were making in #3? That's a purely verbal distinction, not an empirically testable one. The existence of pressure is empirically testable. Therefore there can't be any logical link between the two.

I agree there is some mystery. For the Friedmann models, 'pressure' must be brought in through the back door using the cosmological constant and/or 'dark energy'. That means the question of why the universe is expanding has not yet been answered in a really coherent manner.

This is only necessary for some Friedmann models, not others. The very early universe is well described by a radiation-dominated FRW solution, with pressure from radiation. No mystery. The somewhat later universe was dominated by dust, with zero pressure. Expansion. No mystery.

 

[bapowell]

If the expansion is not kinematic, pressure is required. I agree there is some mystery. For the Friedmann models, 'pressure' must be brought in through the back door using the cosmological constant and/or 'dark energy'. That means the question of why the universe is expanding has not yet been answered in a really coherent manner.

The expansion of the universe is a boundary condition imposed on the Friedmann equations. In a universe with only pressureless dust, there is still expansion, albeit a decelerated expansion. But this is true of radiation as well, which does have pressure. So I'm confused about why you are claiming that only stress-energy with pressure can lead to expansion -- it's not correct to think of the stress-energy as pushing the universe apart. A cosmological constant or dark energy actually has a negative pressure; both cause the universe to undergo accelerated expansion. So again, I'm confused about your logic.

 

[StateOfTheEqn]

The expansion of the universe is a boundary condition imposed on the Friedmann equations. In a universe with only pressureless dust, there is still expansion, albeit a decelerated expansion. But this is true of radiation as well, which does have pressure. So I'm confused about why you are claiming that only stress-energy with pressure can lead to expansion -- it's not correct to think of the stress-energy as pushing the universe apart. A cosmological constant or dark energy actually has a negative pressure; both cause the universe to undergo accelerated expansion. So again, I'm confused about your logic.

Let's start with the Friedmann models k=-1,0,+1. All three are dependent on the 'dust solution' (pressure=0). The one that seems most popular among cosmologists at the present is k=0. It predicts a relationship between density and the age of the universe. That prediction seems to be in error by a factor around 5. That is an error that cannot be ignored. So patchwork has been done by adding dark energy which may or may not solve the problem. In any case dark energy does not solve the density problem for the early universe since the density of dark energy is considered constant.

My fundamental suggestion is that we re-examine the assumptions that have gone into all three Friedmann models since none of the three seem entirely satisfactory and the most popular requires serious patchwork. The dust solution should be the first on our list of assumptions to question. My opinion is that a pressure term in the energy-,momentum tensor is required to explain the expansion. I would suggest something of the form T=(density)diag(1,-1/3,-1/3,-1/3) as the 'default' value of the energy-momentum tensor. The -1/3's are, of course, the pressure terms. The density is that of the total mass-energy of the universe. In any case it is a very small number. It would not significantly affect the value of T near gravitating bodies.

 

[bapowell]

Let's start with the Friedmann models k=-1,0,+1. All three are dependent on the 'dust solution' (pressure=0). The one that seems most popular among cosmologists at the present is k=0. It predicts a relationship between density and the age of the universe. That prediction seems to be in error by a factor around 5.
That is an error that cannot be ignored. So patchwork has been done by adding dark energy which may or may not solve the problem.

A flat matter dominated universe is also inconsistent with the CMB, and hasn't been taken seriously by cosmologists for quite some time. The motivation for dark energy was not in the form a fudge factor -- it was proposed to explain type 1a supernovae redshift data. The existence of dark energy is also supported by recent CMB measurements. And what do you mean by "dependent on the dust solution"?

In any case dark energy does not solve the density problem for the early universe since the density of dark energy is considered constant.

What do you mean by 'density problem'? Why do you think there's an early universe 'density problem'? Why would anyone suggest that dark energy, which is a recent phenomenon (past 5 billion years), have anything to do with your 'density problem'?

My fundamental suggestion is that we re-examine the assumptions that have gone into all three Friedmann models since none of the three seem entirely satisfactory and the most popular requires serious patchwork.

The assumptions that go into the FEs are 1) GR 2) spatial isotropy 3) spatial homogeneity. Which of these do you think should be reconsidered and why?

The dust solution should be the first on our list of assumptions to question.

This is not an assumption, but a case that can be studied. What is wrong -- physically -- with the Friedmann dust solution? You've not said. The concern you mentioned previously -- that it can't lead to expansion -- has been adequately addressed by bcrowell and myself above. If you disagree with our responses, please indicate which parts you have problems with.

My opinion is that a pressure term in the energy-,momentum tensor is required to explain the expansion. I would suggest something of the form T=(density)diag(1,-1/3,-1/3,-1/3) as the 'default' value of the energy-momentum tensor. The -1/3's are, of course, the pressure terms. The density is that of the total mass-energy of the universe. In any case it is a very small number. It would not significantly affect the value of T near gravitating bodies.

Looks like you've just reinvented inflation: $p = -\rho/3$. This will give an accelerated expansion when placed in the FE, as Guth discovered over 20 years ago. (I'm sure you're aware you've assigned a negative pressure to your cosmological fluid? This is an example of dark energy, which I thought you were trying to avoid.) But, as I stated above, one does not need pressure to have expansion -- this is a misconception. With respect to local effects (near gravitating bodies), the FE is not valid since the very existence of gravitating bodies violates the assumptions of homogeneity and isotropy.

 

[twofish-quant]

In any case dark energy does not solve the density problem for the early universe since the density of dark energy is considered constant.

It's not. You can put in all sort of different equations for dark energy. There are two directions. The theorists have been busy coming up with different equations for dark energy and then figuring out the consequences. The observers have been collecting more data so that we have more constraints on what dark energy could be.

My fundamental suggestion is that we re-examine the assumptions that have gone into all three Friedmann models since none of the three seem entirely satisfactory and the most popular requires serious patchwork.

I think you are using Friedmann model more restrictively than most cosmologists would use the term. If you put in some scalar pressure term then you still have a set of equations that work in the Friedman derivations. Setting pressure = zero is a solution of the Friedman equations, but you can put any sort of scalar pressure term that you want.

My opinion is that a pressure term in the energy-,momentum tensor is required to explain the expansion.

Which expansion?

You don't need a pressure term to create an expanding universe.

You do need a pressure term to create an expanding universe that looks like our universe. So you add one.

I would suggest something of the form T=(density)diag(1,-1/3,-1/3,-1/3) as the 'default' value of the energy-momentum tensor. The -1/3's are, of course, the pressure terms.

I think you've reinvented the cosmological constant here.

 

[twofish-quant]

The problem with the Friedmann models is that all three depend on the 'dust solution' where pressure=0. If the expansion is not kinematic, pressure is required. I agree there is some mystery. For the Friedmann models, 'pressure' must be brought in through the back door using the cosmological constant and/or 'dark energy'.

That's not true. You can have a perfectly good expanding universe with pressure = 0. One way of thinking about it is that if you throw a rock in the air, it will keep going without adding any extra energy to it. If you don't throw it hard enough it will arch back to the ground (i.e. big crunch). If you throw it hard enough then it goes past escape velocity and will keep going forever (i.e. open universe).

So if you see a rock nicely following an orbit, there is no need to assume any extra energy.

It happens that the universe is accelerating so you do need a pressure term.

 

[StateOfTheEqn]

What do you mean by 'density problem'? Why do you think there's an early universe 'density problem'? Why would anyone suggest that dark energy, which is a recent phenomenon (past 5 billion years), have anything to do with your 'density problem'?

The Friedmann k=0 model expresses a relationship between time (the age of the universe) and mass-energy density. The required density (usually called critical density) does not conform to observation for the current age of the universe. The predicted density is in error by around a factor of 5 (if we include dark matter). If we add dark energy derived from the cosmological constant with (density)=(LAMBDA)/8(pi)G, that might give the correct density for t=now but does not give the correct value for t not equal to now.

The assumptions that go into the FEs are 1) GR 2) spatial isotropy 3) spatial homogeneity. Which of these do you think should be reconsidered and why?

The 'dust solution' is also an assumption. That is what I suggest should be re-examined. I understand that the Friedmann k=0 model gives an expansion a=a(sub-zero)t^(2/3) but it also gives an incorrect age of the universe based on the currently observed density or looked at the other way, an incorrect density for the current age.

This is not an assumption, but a case that can be studied. What is wrong -- physically -- with the Friedmann dust solution?

Given the problems with the Friedmann k=0 model I think the most likely source of error is in the 'dust solution' assumption.

The concern you mentioned previously -- that it can't lead to expansion -- has been adequately addressed by bcrowell and myself above.

Like I said above I realize that a=a(sub-zero)t^(2/3) is a solution of the cosmological equations with k=0 and p=0 but it gives the wrong relationship between age and mass-energy density.

Looks like you've just reinvented inflation: $p = -\rho/3$. This will give an accelerated expansion when placed in the FE, as Guth discovered over 20 years ago.

There is a difference. p=-(1/3)(density) implies 3p+(density)=0 which, when introduced into the cosmological equations, gives a''=0. This implies a constant rate of expansion not an accelerated rate. I think that p=-(1/3)(density) can be derived from the expansion of all wavelengths, including De Broglie wavelengths (due to the expansion of space itself). That is what the 'dust solution' does not take into account.

 

[bcrowell]

If we add dark energy derived from the cosmological constant with (density)=(LAMBDA)/8(pi)G, that might give the correct density for t=now but does not give the correct value for t not equal to now.

This sounds like you're saying there is a serious disagreement between ΛCDM and observation. This is incorrect.

The 'dust solution' is also an assumption. That is what I suggest should be re-examined. I understand that the Friedmann k=0 model gives an expansion a=a(sub-zero)t^(2/3) but it also gives an incorrect age of the universe based on the currently observed density or looked at the other way, an incorrect density for the current age.

There is no "'dust solution' assumption" in ΛCDM, so I don't understand what you're getting at. The model that works is ΛCDM.

Given the problems with the Friedmann k=0 model I think the most likely source of error is in the 'dust solution' assumption.

Again, there is no "'dust solution' assumption" in ΛCDM, so I don't understand what you're getting at.

 

[bapowell]

If we add dark energy derived from the cosmological constant with (density)=(LAMBDA)/8(pi)G, that might give the correct density for t=now but does not give the correct value for t not equal to now.

Why? The universe used to be dominated by radiation and matter. Why would we expect the addition of the CC to modify early universe densities? Nor does it need to: the LCDM model is (very) well supported by CMB data. What don't you think is in agreement here?

The 'dust solution' is also an assumption. That is what I suggest should be re-examined. I understand that the Friedmann k=0 model gives an expansion a=a(sub-zero)t^(2/3) but it also gives an incorrect age of the universe based on the currently observed density or looked at the other way, an incorrect density for the current age.
Given the problems with the Friedmann k=0 model I think the most likely source of error is in the 'dust solution' assumption.

How about this suggestion: since the k=0 dust model gives the wrong age of the universe, it's WRONG. So is the k=1 dust model. So is the Milne universe. And also the $\Omega_\Lambda = 1$ solution. I'm confused as to why you're hung up on the k=0 dust solution when lots of models give wrong cosmologies.

There is a difference. p=-(1/3)(density) implies 3p+(density)=0 which, when introduced into the cosmological equations, gives a''=0. This implies a constant rate of expansion not an accelerated rate. I think that p=-(1/3)(density) can be derived from the expansion of all wavelengths, including De Broglie wavelengths (due to the expansion of space itself). That is what the 'dust solution' does not take into account.

Ah, yes. I misread. So you are proposing a universe with nothing but curvature ($w =-1/3$). The next step is to test your model. Does your model fit SNIa data? CMB? LSS? Do you have dust and radiation in your universe too? You'll find that it won't fit current data.

 

[StateOfTheEqn]

The next step is to test your model. Does your model fit SNIa data? CMB? LSS? Do you have dust and radiation in your universe too? You'll find that it won't fit current data.

I have concerns about distance estimates using the inverse-square law which I attempted to address in another thread. https://www.physicsforums.com/showthread.php?t=488768

The model I am suggesting is correct gives a constant rate of expansion at the speed of light. For each observer the 'radius' of the universe is the length of his past world-line which is increasing at the speed of light. Within this model the Hubble relation holds rigorously: v=HD. However, the relation between v and cosmological redshift needs to be revised from v=cz to v=cz/(z+1) (for small z the former approximates the latter). Furthermore, D=cz/H needs to be revised to D=cz/H(z+1).

When you incorporate those revisions, does the model I'm proposing conflict with the observational data (subject to the above mentioned concern about the use of the inverse-square law)?

There is no dust. There are only particles whose wavelengths stretch as space expands which includes light and all known forms of matter.

 

[bapowell]

How are you defining H and z in terms of the scale factor a(t)?

 

[AstrophysicsX]

Space is expanding, not the matter. The cooling of our universe is due to the diffusion of heat throughout it. As our universe gets bigger and bigger, it will get colder and colder as well.

 

[bapowell]

There is no dust. There are only particles whose wavelengths stretch as space expands which includes light and all known forms of matter.

I think you have a misconception of what is meant by 'dust' in cosmology. This is not the stuff you clean off your bookshelf. It is a generic name for pressureless matter. Pressureless matter is all matter that is sufficiently non-relativistic to lack radiation pressure. For example, interstellar hydrogen gas is dust, as are non-relativistic electrons and cold dark matter. Anything with a density that scales as $\rho \sim 1/a^3(t)$ is pressureless dust. Surely your universe contains these things.

 

[twofish-quant]

I have concerns about distance estimates using the inverse-square law which I attempted to address in another thread.

The inverse-square law doesn't work at large cosmological distances. At large distances, there are several different and perfectly good definitions for "distance" so people don't talk about distances, they talk about redshifts, and then convert to a particular definition of distance as necessary.

You can *define* a brightness distance as the number that you get assuming the inverse square law is true, but that "distance" may be (and at large scales is) different from other reasonable definitions of "distance."

Within this model the Hubble relation holds rigorously: v=HD. However, the relation between v and cosmological redshift needs to be revised from v=cz to v=cz/(z+1) (for small z the former approximates the latter).

You really can't do that without throwing away GR. If you throw away GR, then you need to propose an alternative theory of gravity that gives you those revisions.

Also any model in which the Hubble relation holds rigorously in which you do use GR is excluded by experiment. If you throw away GR, then you need to come up with some other theory of gravity.

There is no dust. There are only particles whose wavelengths stretch as space expands which includes light and all known forms of matter.

In cosmology, "dust" means anything in which doesn't have any pressure. It can be jelly beans, neutrinos, or black holes.

One problem with "non-standard" theories is their vagueness. The good thing about lambda-CDM is that by putting in seven or eight numbers, you can curve fit the expanding universe. The problem with what you are proposing is that it doesn't get that far.

 

[twofish-quant]

Here is an article on the issue of "distance"

http://en.wikipedia.org/wiki/Distance_measures_(cosmology)

The weird thing is that when people say "space is curved" they really mean it. You take a ruler and none of things you expect at short distances work any more.

 

[StateOfTheEqn]

I think you have a misconception of what is meant by 'dust' in cosmology. This is not the stuff you clean off your bookshelf. It is a generic name for pressureless matter. Pressureless matter is all matter that is sufficiently non-relativistic to lack radiation pressure. For example, interstellar hydrogen gas is dust, as are non-relativistic electrons and cold dark matter. Anything with a density that scales as $\rho \sim 1/a^3(t)$ is pressureless dust. Surely your universe contains these things.

By 'dust' I mean pressureless matter. I do not think such a thing exists in an expanding universe. As space expands, the deBroglie wavelength of all particles expands causing a mass-energy flux we call pressure.

 

[StateOfTheEqn]

You really can't do that without throwing away GR. If you throw away GR, then you need to propose an alternative theory of gravity that gives you those revisions.

Also any model in which the Hubble relation holds rigorously in which you do use GR is excluded by experiment.

What are the assumptions? You always need to look at the assumptions.

If you throw away GR, then you need to come up with some other theory of gravity.

There is no throwing away GR in my model. Do you agree that if 3p+(density)=0 then the only solutions of the cosmological equations give a'=constant? The topology I am using is U X 3-sphere where U={t:t=0 or t>0}. Using such a topology and a constant rate of expansion at the speed of light you get v=c(1-e^-A). A is the primary angular quantity in the R-W metric. A is related to z by e^A=(absorption wavelength)/(emission wavelength)=z+1. H=a'/a. v=cz/(z+1) and D=cz/H(z+1).

When I say the universe is expanding at the speed of light I mean that for each observer the length of his past world-line is increasing at the speed of light.

 

[bapowell]

By 'dust' I mean pressureless matter. I do not think such a thing exists in an expanding universe. As space expands, the deBroglie wavelength of all particles expands causing a mass-energy flux we call pressure.

Yes, of course all particles have a de Broglie wavelength -- that's not the issue. This issue is with the fact that non-relativistic matter obeys a Maxwell-Boltzmann distribution very closely, and, hence, has a density that scales inversely to volume: $\rho \sim V^{-1}$. See the first few chapters in Kolb and Turner for a good discussion of this.

But you still haven't answered my earlier question: what are H and z in terms of the scale factor, a?

 

[bcrowell]

By 'dust' I mean pressureless matter. I do not think such a thing exists in an expanding universe. As space expands, the deBroglie wavelength of all particles expands causing a mass-energy flux we call pressure.

When we describe dust as pressureless, we don't mean exactly pressureless. It's just an approximation. If you want to get unstuck from the confusion you're currently stuck in, a good exercise would be to calculate the density of a liter of helium gas at standard temperature and pressure, then compare with its pressure, doing this all in units where c=1.

 

[bapowell]

When I say the universe is expanding at the speed of light I mean that for each observer the length of his past world-line is increasing at the speed of light.

Even for an observer at rest with respect to the expansion? How is that possible?

 

[StateOfTheEqn]

When we describe dust as pressureless, we don't mean exactly pressureless. It's just an approximation. If you want to get unstuck from the confusion you're currently stuck in, a good exercise would be to calculate the density of a liter of helium gas at standard temperature and pressure, then compare with its pressure, doing this all in units where c=1.

The mass-energy flux is due to the stretching of the DeBroglie wavelength as space expands. p=dM/dV. Traditionally this would be considered zero. However, The energy and hence the mass of a particle varies inversely with the wavelength and therefore inversely with the expansion factor. I have used the topology U x 3-sphere where U={t : t=0 or t>0}. So p=dM/dV=(dM/dR)(dR/dV). Since the wavelength of a particle varies as 1/R, d(MR)=0. Then p=dM/dV=(-M/R)(dR/dV)=(-M/R)(1/(3*constant*R^2)) and density=M/(constant*R^3). So, 3p+(density)=0.

Applying that final equation to the cosmological equations gives a family of constant solutions. I use the fact that the past world-line of an observer increases at the speed of light due to SR. This allows us to define the radius as the length of the past world-line of an observer. This gives the rate of expansion of the universe = c.

 

[bapowell]

The mass-energy flux is due to the stretching of the DeBroglie wavelength as space expands. p=dM/dV. Traditionally this would be considered zero. However, The energy and hence the mass of a particle varies inversely with the wavelength and therefore inversely with the expansion factor.

OK, but what we are trying to tell you is that this is not correct -- non-relativistic matter is more accurately approximated as pressureless dust than radiation -- period. It is not a matter of opinion: see any introductory cosmology text (Kolb & Turner, Liddle, Peebles) for a good discussion of the thermodynamics of the early universe and how different species of matter/energy evolve in time. Your prediction that the mass of particles decreases with the expansion of the universe is 1) not supported by evidence 2) not predicted by any accepted theory.

I have used the topology U x 3-sphere where U={t : t=0 or t>0}. So p=dM/dV=(dM/dR)(dR/dV). Since the wavelength of a particle varies as 1/R, d(MR)=0. Then p=dM/dV=(-M/R)(dR/dV)=(-M/R)(1/(3*constant*R^2)) and density=M/(constant*R^3). So, 3p+(density)=0.

What do your symbols mean? I'd like to try to find out where you are going wrong. Your equation of state, p = -density/3, is that of curvature. I mentioned this in a previous email that appears to have gone ignored. This means that you are modeling an empty universe with only curvature. This is not the only Friedmann solution, and so you must have made an overly restrictive or incorrect assumption at some point in your calculation that is leading you down this path. In an earlier post, you found expressions for redshift that were inconsistent with standard GR, and I'm curious about what the Hubble parameter, H, and the redshift, z, are as functions of the scale factor a(t). If you can write these down, we'll have a better idea of where you might be going wrong. Lastly, this model is not in agreement with current CMB or LSS data. Or SN data.

I use the fact that the past world-line of an observer increases at the speed of light due to SR.

You should not be using special relativity at all here. The motion of all observers should be fully determined within GR. Your terminology is also confusing: what's the difference between an observer moving at the speed of light and their worldline "increasing at the speed of light." For example, comoving observers are at rest with respect to the expansion. Given that we are approximately comoving observers, how does your solution apply to us?

 

[bapowell]

This gives the rate of expansion of the universe = c.

But the rate of expansion of the universe is not a speed. And the rate of expansion has nothing to do with the speed of various observers -- it's an observer-independent quantity. The rate of expansion is given by the Hubble parameter in terms of the scale factor:
$$H = \frac{\dot{a}}{a}$$
In your model, with $p = -\rho/3$, you find $a(t) \sim t$ and therefore $H \sim 1/t$.

 

[StateOfTheEqn]

But the rate of expansion of the universe is not a speed. And the rate of expansion has nothing to do with the speed of various observers -- it's an observer-independent quantity.

That is correct. In the R-W metric for the topology U x 3-sphere we have ds^2=c^2dt^2 - R^2d(sigma)^2. The c^2dt^2 can be interpreted as a distance term. It can be thought of as dR^2.

The rate of expansion is given by the Hubble parameter in terms of the scale factor:$$H = \frac{\dot{a}}{a}$$ In your model, with $p = -\rho/3$, you find $a(t) \sim t$ and therefore $H \sim 1/t$.

That is correct. I am saying R=ct and R'=c. Therefore H=R'/R=c/ct=1/t.

 

[StateOfTheEqn]

Your equation of state, p = -density/3, is that of curvature.[...]This means that you are modeling an empty universe with only curvature. This is not the only Friedmann solution, and so you must have made an overly restrictive or incorrect assumption at some point in your calculation that is leading you down this path.

I'm rejecting all the Friedmann solutions because they all depend on p=0.

In an earlier post, you found expressions for redshift that were inconsistent with standard GR, and I'm curious about what the Hubble parameter, H, and the redshift, z, are as functions of the scale factor a(t).

As I stated earlier, in the R-W metric for the topology U x 3-sphere we have ds^2=c^2dt^2 - R^2d(sigma)^2 where d(sigma)^2=dA^2 + sin^2(A)(d(theta)^2 + sin^2(theta)d(phi)^2).

[URL]http://s1109.photobucket.com/albums/h421/StateOfTheEqn/R-W-metric.jpg[/URL]

The c^2dt^2 can be interpreted as a distance term. It can be thought of as dR^2. I use R=ct and then R'=c so H=R'/R=1/t.

I have derived 3p+(density)=0 and applied that to the cosmological equations derived from GR. That gives a family of constant solutions. To select the correct one apply SR. That gives R'=c because c is the rate observers move along their respective world-lines.

What's the difference between an observer moving at the speed of light and their worldline "increasing at the speed of light."?

World-lines are increasing at the speed of light in the direction of time as in the R-W metric: ds^2=c^2dt^2 - R^2d(sigma)^2. But c^2dt^2 can be interpreted as a distance measure and equated to dR as long as R is interpreted as the length of the past world-line. The co-moving coordinates are all contained in (sigma).

Our past light cone, being a collection of null geodesics, intersects world-lines at angle (pi)/4 in space-time. Within the R-W metric (theta) and (phi) can be suppressed. That is equivalent to observing along a single line of sight which leaves dR=-RdA at each point of intersection of our past light cone with other world-lines. From dR=-RdA along our past light cone we can derive recession v=c(1-e^-A) and z+1=(wavelength observed)/(wavelength emitted)=e^A. Therefore v=cz/(z+1) and D=cz/H(z+1). Plotting the last equation gives:
[URL]http://s1109.photobucket.com/albums/h421/StateOfTheEqn/z-D-Plot.jpg[/URL]

 

[bapowell]

OK. Good luck with your model.

 

[StateOfTheEqn]

What do your symbols mean?

This might explain the derivation of 3p+(density)=0 a little better:

 

[twofish-quant]

I'm rejecting all the Friedmann solutions because they all depend on p=0.

No they don't.

What you do is to take the FRW equation, put in your favorite expression for P, and then you get expansion rates. Or you go backward and put in observed expansion rates, and that will give you P.

You can do this with your equation for P, and I am 99% certain that you'll get expansion rates that look nothing like what the universe looks like.

You can fix this by tossing FRW and coming up with a new theory of gravity. People *have* been doing this, and there are hundreds maybe thousands of papers that try to explain observations by assuming new gravity. The general technique is to use an f(R) model in which they new theory of gravity is like GR for short distances (since we can see how gravity behaves at short distances) but different for long distances.

The problem with what you are doing is that if you start with a specific theory about what is causing the universe to expand and you work through the numbers, you end up with something that looks nothing at all like the universe. What people are doing is starting with the observations, then working out what *could* cause those observations, and then hopefully we will be able to pin down, what it is or isn't.

It's actually a fun thing to do. If you want to join in on the hunt, we could use people. But people have tried what you are doing, and it hasn't worked, and if you aren't willing to listen to why it doesn't work, then I don't know what to do.

 

[twofish-quant]

One last thing...

There is astrophysics the "tooth fairy rule." Which is to say that in any paper, you are allowed to invoke the tooth fairy once. You can assume *one* crazy thing about the universe and if that one thing explains everything, you can publish.

If you assume some weird thing about the property of matter, and by invoking the tooth fairy *once* you fit all of the observations, then that's publishable. If you assume that the expansion of the universe inherently reduces energy, and this happens to fit observations, you win, and that one tooth fairy honestly is not any worse than some other the other tooth fairies that have been invoked.

However, what you should to is to plot your predictions of universe expansion with observations. My guess is that they won't match, and if you have to come up with some reason why they don't match, you've already invoked the tooth fairy, and so you don't get a second wave of the wand.

One other thing about scientific publishing is to be *interesting* and *original*. For example pointing out that putting P=0 into the Friedman equations won't work is not publishable because we knew that in 1999. In fact if you can get P=0 to work by invoking some other tooth fairy, *that* would be interesting.

 

[StateOfTheEqn]

No they don't.

The following are equivalent:
1) p=0
2) M=constant.
3) The Friedmann equation holds true.

The proof is in Semi-Riemannian Geometry by Barrett O'Neill (1983) p.351

Let's take this step by step.

1)Is the p=0 assumption in the three Friedmann models necessary?
2)By looking at the influence of wavelength stretching coinciding with the expansion of space, we can see there is a mass-energy flux which gives a non-zero value for p. The idea that DeBroglie wavelengths stretch with the expansion of space is not a "tooth fairy" idea. It was introduced by Peebles in Principles of Physical Cosmology (1993) p.96.
3)We can derive 3p+(density)=0 from (2). I have used the topology U x 3-sphere. U is cosmic time. That is not controversial since it is the required topology for quantum cosmo-genesis. See Atkatz and Pagels (1982) and/or Vilenkin (1982).
4)What solutions does that give us for the cosmological equations derived from GR?
5)It gives us a family of constant solutions.
6)How do we decide among them? We look at the R-W metric and ask whether there is a canonical (natural) constant rate of expansion associated with that metric. There is. Set dR^2=c^2dt^2. That gives us R=ct where R is to be considered the length of the past world-line of an observer.

What you do is to take the FRW equation, put in your favorite expression for P, and then you get expansion rates. Or you go backward and put in observed expansion rates, and that will give you P.

You can do this with your equation for P, and I am 99% certain that you'll get expansion rates that look nothing like what the universe looks like.

You can fix this by tossing FRW and coming up with a new theory of gravity. People *have* been doing this, and there are hundreds maybe thousands of papers that try to explain observations by assuming new gravity. The general technique is to use an f(R) model in which they new theory of gravity is like GR for short distances (since we can see how gravity behaves at short distances) but different for long distances.

There is no discarding or modification of the equations of GR nor any new theory of gravity. The equation of state in (3) does require a default value of (density)diag(1,-1/3,-1/3,-1/3) for the energy-momentum tensor but (density) is so small that the modification would not appreciably affect the gravitational fields of gravitating bodies. So, I would not call it a significant modification of the GR field equations. It is certainly less of a modification than adding a cosmological constant.

The problem with what you are doing is that if you start with a specific theory about what is causing the universe to expand and you work through the numbers, you end up with something that looks nothing at all like the universe.

That may be true but there is one thing to consider. Any interpretation of data involves assumptions, especially in cosmology. I think the theory I am proposing explains the redshift anomaly without resorting to 'dark energy'. I have tried to follow Occam's Razor - do not add unnecessary entities! I have proposed a theory that has no parameters but from which you can derive the Hubble Relation.
I admit it does require reinterpreting redshift somewhat. Instead of v=cz , I have v=cz/(z+1) and D=cz/H(z+1).

It's been fun and I'll leave you with the last word.

What people are doing is starting with the observations, then working out what *could* cause those observations, and then hopefully we will be able to pin down, what it is or isn't.

It's actually a fun thing to do. If you want to join in on the hunt, we could use people. But people have tried what you are doing, and it hasn't worked, and if you aren't willing to listen to why it doesn't work, then I don't know what to do.