Cosmology is in exciting times. Over recent decades the use of space based observatories has resulted in a wealth of data that has allowed testing of various models with unprecedented precision; and this is a continuing trend.

However, cosmology is full of unsolved problems.

Put this together, and you get an exciting mix. My own feeling is that in future centuries people will look back on these times as the golden age of cosmology.

This also provides fertile ground for "bad astronomy"! It’s going to take some innovative work and new insights to resolve the various outstanding issues; and there are many folks anxious to put in an oar for how their own speculations may resolve the problems. Add to this that some of the solutions we have already are very counter intuitive, and you get bad astronomers who have weird "refutations" of fairly basic stuff.

I've been trying in recent months to learn more about relativity and the basic models for how space develops over time; and along the way I have learned some details of the various unsolved problems. This post is my own work, and the errors in it are my own. However, I am indebted to many sources for learning enough to even attempt to describe some of the issues. In particular, Sean Carroll’s paper The Cosmological Constant maintained at the fascinating livingreviews site was of enormous help to me. Readers should refer to that for some more details.

One key concept is that of the "scale factor" of the universe, which I will write as "a". This scale factor is a way of measuring expansion. The scale factor is at present "1", by definition. Over time, galaxies are receding from each other, and this is represented mathematically as an increase in "a". Galaxies that are "at rest" in the Hubble flow are called "comoving" observers. Their separation over time is proportional to the scale factor.

The question is, how does that scale factor develop over time? We have a couple of mathematical models for this, which we know are wrong, but at least they are easy to calculate. :wink:

For example, the "empty universe" model has "a" proportional to t, where t is the age of the universe. The "critical density" model has "a" proportional to t^(2/3). The "flat inflation" model has "a" proportional to e^t.

One way to calculate a scale factor is to assume that the dynamics of spacetime are based on such things as matter density, radiation density, vacuum energy density, and a curvature effect. Using general relativity, and using a’ as the time derivative of a, we end up with an equation as follows:

• (a'/a)^2 = F(4)/a^4 +F(3)/a^3 + F(2)/a^2 + F(1)/a + F(0)

The values F(n) are constants, corresponding to various contributing effects.

• F(0) is the vacuum energy contribution, or the “cosmological constant”.
• F(1) is not normally used. There is no energy that develops in this way, as far as we know.
• F(2) is a curvature term.
• F(3) is a term for energy contribution from matter density.
• F(4) is a term for energy contribution from radiation density.

The Hubble expansion “constant” is a measure of the rates of expansion. It is constant over space, but it changes over time. It is a’/a by definition. The Hubble constant quoted in papers, often written H0, is the value at the present time, establishing the present rate of expansion for the universe.

Therefore the first equation I provide is often written as

• H = H0 * sqrt(W(4)/a^2 + W(3)/a + W(2) + W(1)*a + W(0)*a^2)

The constants W(i) are actually the F(i), but scaled so that they sum to be one. These numbers are actually fractions of a “critical energy density” that would allow for a “flat” universe. W(3) is often written Omega_M, and W(0) is written Omega_Lambda. The other terms are often taken to be zero. W(1) is zero because we don’t know any energy for which the density scales with 1/a. W(2) is zero because the universe seems to be so close to being flat. W(4) is zero because radiation has a low contribution to the energy of the universe.

So much for the basic theory of expanding spaces. You’ll get this in a lot more detail in many papers. In the rest of this post I want to list some of the cosmic co-coincidences that match up with unresolved issues or questions in cosmology. You will also see frequently mentioned that the current best supported model of the universe corresponds to Omega_M ~ 0.3 and Omega_Lambda ~ 0.7.

Here are some cosmic co-incidences.

Flat universe. Why is the universe so close to being flat? That is, why is W(2) zero? The currently leading explanation is an inflationary epoch in the very early universe… that is, a time when W(0) was dominant. The questions are, how did it start, and how did it stop? Could the vacuum energy have “rolled down” with decay of an inflaton field, or did it just switch off with some kind of phase transition?

Omega_M / Omega_Lambda ~ 1. This is a cute problem. Astronomers are used to working with numbers that have huge or tiny magnitudes. The current best supported model is Omega_M ~ 0.3 and Omega_Lambda ~ 0.7. Those numbers are just about equal, at least to an order of magnitude. As time passes, they scale by different powers of “a”, which means that over time they will diverge enormously; that includes time into the past and into the future. So how come they are the same rough order of magnitude now? Are we just lucky to live at such a time? No-one knows.

Omega_Lambda small. When particle physics is used to get a rough estimate for the order of magnitude of the vacuum energy density, we get numbers that are something like a 120 orders of magnitude more than what we observe at present. That is quite a large difference! It could be different aspects “balancing” to cancel out the energy contributions, but it is pretty effective balancing to get rid of 120 orders of magnitude!

H0 ~ 1/t0. This co-incidence is what has prompted me to write this post. I would very much welcome any comment on this particular issue, which I have not seen published elsewhere. It is one I found for myself.

Basically, I have attempted to explore these ideas by solving the equations personally. I used Runge Kutta numeric solutions, and looked at the results for various models. And I noticed something rather odd! The age of the universe, written t0, can be deduced for various models from the present value of the Hubble constant H0. For example, the age of the universe in the empty model is 1/H0. In the critical mass density model it is (2/3)/H0. So what is the age in the present (0.3, 0.7) model? It seems to be 0.975/H0, by my calculations. Here are some comparisons of models, and of the ages my calculations have given.

• ( 0.40, 0.60 ) gives 0.9/H0
• ( 0.30, 0.70 ) gives 0.975/H0
• ( 0.28, 0.72 ) gives 0.992/H0
• ( 0.27, 0.73 ) gives 1.002/H0
• ( 0.26, 0.74 ) gives 1.013/H0
• ( 0.20, 0.80 ) gives 1.085/H0

Here’s the kicker; the WMAP papers which proposed the (0.3,0.7) model actually propose a higher precision of (0.27,0.73). These seems tuned to give t0 = 1/H0, which strikes me as a co-incidence. I’m not aware of any deep reason it should be so, and the wide use of coarser approximations like (0.3, 0.7) seems to back this up.

What’s going on here? I don’t know. Can anyone tell me whether this really is a co-incidence? Is anyone out there with their own model calculators who could confirm my figures?

Thanks -- Sylas

What evidence do we have that the Hubble constant changes over time?
Cheers
lyndon

Well, if expansion in space is uniform, the expansion rate must change with time. First, we know that expansion is space is uniform because the Hubble curve is a straight line (at least, close enough to us). An object X times as far away as another object has X times the cosmological redshift, isotropically, so the expansion of space is uniform. The rate of expansion is governed by gravity and the cosmological constant. (Remember that radiation can act like matter in a grav field.) One component, the self-gravitational acceleration, changes with time as the universe expands since it depends on the distance between objects.

In other words, if any F(i) in the equation
----- H^2 = (a'/a)^2 = F(4)/a^4 +F(3)/a^3 + F(2)/a^2 + F(1)/a + F(0)
is time-dependant, then H is as well. As I said above, we *know* that F(3) and F(4) are time-dependant in an expanding universe, so we know that H is time-dependant, too.

I know what you mean, but for clarity I need to interject a quibble.

None of the F(i) is time dependent in this equation. They are boundary conditions defined by the present instant. The time dependent term is a; and hence F(i)/a^i is time dependent as long as i is non-zero.

F(3), for example, is the factor for normal matter, and relates to matter density in the current epoch. Of course, as the scale factor increases the density drops as the cube of the scale factor; which is why we have F(3)/a^3. That is, "a" is the time dependent variable, and the F(i) are constants. Each of terms F(i)/a^i is thus time dependent due to the dependence on "a"... except F(0).

F(0) is the vacuum density. The simplest inflation model is one where F(0) is positive, and all other F(i) factors are zero. In this case the Hubble constant really is constant. Objects at a given distance will be receding at a given velocity. Hence any object, as it moves further away, will be moving exponentially faster, and the scale factor is proportional to e^t.


Consequently, in answer to Lyndon's question, evidence that the Hubble constant is not constant over time is the same as evidence that the current rate of expansion is not exponential. Observations of distant galaxies is also an observation back in time; inflation would have a dramatic effect on the observations of high redshift objects, and this is not seen. A Hubble "constant" that is constant over time is thus not a serious contender in the light of redshift vs luminosity relations for high redshift objects; and well as being theoretically impossible given the fact of a significant matter density in the current universe.

Cheers -- Sylas

I research a model of the Rotating Universe. In this model the expansion is governed by gravity and repulsive force (Dark Energy ?) of rotating Galaxy Clusters (Cosmic Voids creation).
A neighbouring rotating Clusters drives an intergalaxy gas (Dark Matter ?) and it creates arms of spiral galaxies moving very fast around galaxy bulge (like a gear transmission).

In this Model of Rotating Observable Universe is an energy supplied from outside from existing Total space because of the entropy increase and energy conservations law.

The expansion may be determine by observations of the radiation spectrum redshift caused by Doppler effect, gravity effect or others. Probably it is not very high.

The expansion may be balanced by new matter supply as well (creation of the matter-antimatter and separation according spin polarization .

Being pedantic here but for good reason, there are too many assumptions in these posts regarding the Hubble constant not being a 'constant' but time dependent (isn't this one of Arp's proposals that our 'constants' change with time?).
The evidence is that redshift, z, increases with distance where the redshift is delta lambda divided by lambda.
The Hubble relation in terms of evidence is z = (H/c)d
Now Tobin Dax tells me that gravity etc means that the 'expansion' Must be slowed down - but he has assumed expansion and that redshifts represent 'velocity'. 'If' and 'must' is not evidence.
The evidence for an accelerating universe is that distant galaxies are dimmer than they should be if we calcluate their distances according to Hubble's relation. To interpret this evidence as acceleration one has to interpret redshifts as velocities.
So maybe the Big Bang theory is interpreting the evidence wrongly.
I don't want to take over Sylas' thread with this and am happy to drop this point here and now, but to say that the Hubble constant is 'time dependent' is maybe taking too much for granted - that is all.
Cheers
lyndon

Thanks Lyndon; I appreciate your willingness to allow a few assumptions here. You are right; my post works on an assumption that cosmological refshifts are a consequence of recession velocities from expanding space.

Sometime, no doubt, we will debate/discuss other hypotheses, including perhaps your own. For this thread, I prefer not to debate or defend the notion of expanding spacetime as the basis for cosmological redshifts. I may do so in other threads. Here I am hoping to learn if anyone has noticed the co-incidence of H0 = 1/t0 in the present (0.27, 0.73) model, or if there is any hypothesis within the conventional models which explains this close match.

Cheers -- Sylas

Thank you, Sylas. That's what I get for responding so late. I did mean for the F(i)'s to include the scale-factor terms.

I have taken this up directly with Ned Wright.

On the coincidence of 1/H0 being about the age of the universe in the current model, Ned confirms that this has been noticed by the working cosmologists; but did not give a reference. If anyone else knows of a reference, please let me know.

The most exciting words in science are not “Eureka!”, but “That’s funny…”. A coincidence like this might just be a coincidence, and it might be an indicator of something deeper going on that we don’t yet understand. There’s scope here for bad astronomy wackos to come up with a wild new model. The real trick is to find an explanation that is consistent with all the other evidence already known and explained within the existing models; if an explanation is actually required.

On the age calculations, I was able to confirm my figures approximately with Ned Wright's cosmology calculator, and a few simple arithmetic conversions. Using a Hubble flow value of 977.792354 in the calculator gives 1/H0 as 1 GigaYear, and this helps conversions. The numbers I get are in the same ball park, but far enough away to indicate a problem. This was what prompted my emails to Ned.

Here is a side by side comparison of the results I quoted previously and the results from Ned’s calculator. I’m using “code” tags as the best way to get a table.
It turns out that the errors in my figures seem to have at least two parts. One is the crude calculation method. The other is that Ned’s calculator includes a small factor for the radiation energy of the universe.


As for the radiation energy, a large part of this is due to neutrinos, and Ned directed me to his page Neutrinos as Dark Matter for a discussion. I have put the appropriate term into my models, and the results are a lot closer. The radiation pressure corresponds to a W_R term of 0.4165/H0^2. You can find this term in Ned’s calculator by entering H0 as 1, and W_M as 1, and calling up a flat universe model. The W_V term becomes -0.417. With H0 as 0.1, it becomes -41.650

I’m also reworking my integration code to use a more accurate method.

Cheers -- Sylas

Sylas wrote
I know I said we would drop this Sylas, but since you have reintroduced it, and I seem to be regarded as a ‘Wacko’…

When one gets ‘coincidences’ like this then one has to go back and check your original assumptions.
As I posted earlier, you have assumed that the cosmological redshifts are due to expansion. There is no basis for this, it is just an assumption. What these coincidences are telling you is that this assumption is wrong. Especially when one also includes the ‘coincidence’ that H0 is equal in magnitude to hr/m for the electron. There is an old page on my website that looks at all the nonsense one has to believe if one believes in the expanding universe. In Tired Light we expect this sort of coincidence since we get H = 2nhr/m. True, H = hr/m is still a coincidence but not as surprising when one realises that n has values between 0.1 and 10.
I will keep quiet again now! (maybe)
Cheers,
Lyndon.
P.S. When I e mailed Ned Wright I didn’t get a reply. I wonder why?

Sure, and also check your original working. A genuine co-incidence is something to be looked at very carefully for some possibility of a new insight, or for an error, or for whether it is just a co-incidence. This is precisely why I started this thread! I found that Age(0.27,0.73) to be very close to 1/H0, and I do think this is something well worth looking at!

This week I’ve been checking my maths; since this involves a numeric integration. I corrected some discrepancies with Ned's calculator, with his help, and I have also upgraded to use a higher order integration method. I now get results identical to Ned's calculator except in the last digit in a few places. I am basically doing a numerical solution to the differential equations for the scale factor formula. I'm comparing models with and without the radiation energy; since the cause and significance of the co-incidence is not known.

It turns out that the radiation energy does not make much difference to age in a model with mass about 0.3 of critical; which is as I would have expected. Here is a side by side comparison; and this time I am confident that the values reflect the conventional relativistic equations. The units for 1/H0 are GigaYears; and I use a dimensionless ratio to get the unit value. The ratio is also independent of H0, except that the model with radiation has the Omega_R value based on an observed density, and so is scaled with H0^2.

In this table, Ho is shown in km/s/MPc; and 1/Ho is shown in GigaYear. The table entries are dimensionless, and are obtained as Age of universe divided by 1/H0.
In this case, if the co-incidence has any significance, it will tend to be support for something fundamentally valid about expanding spaces, since the value is obtained by an integration of a formula for the rate of change of scale factor. If you have some other model of the universe, there is no way to obtain the co-incidence at all.

I've ignored your "co-incidence". We've done this before, and I am happy to discuss your alleged coincidence in its own thread. Not here.

Cheers -- Sylas

PS. Ned's calculator uses a less sophisticated integration method, but much more sophisticated models that may (I think) diverge from the simple polynomial I have given, by taking into account such things as particle annihilation and so on. It makes little difference to the results for age, given the (0.3,0.7) model with Ho around 70 or so.

I have found that if we restrict ourselves to two terms, the dark energy term (f(0)) and the matter density term (f(3)) we can actually perform the integral with elementary functions and I obtain:

For H^2 = H0^2 * (A + B(a0/a)^-3), where a0 is the value of the scale parameter now, A and B are dimensionless and A+B=1, we find

(2/3)/sqrt(A)*ln(sqrt(A/B)+1/sqrt(B)) = H0*t0,

where t0 is the age of the universe now.

It turns out that the function on the left has a limit of 2/3 as A-->0, B-->1 (use l'Hospital's rule) and is very nearly one if we choose A=0.73, B=0.27. I don't think anything special can be inferred from this, however.

As for your general expression, I suspect that it can be done but it will be quite messy and will probably involve elliptic integrals of the third type and possibly of the other elliptic integrals as well. I am searching through Gradshteyn and Ryzhik, but I haven't found it yet. I probably have to solve the quartic equation for the roots first. Nasty!

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I now officially condemn CM's skits as smartaleck, ignorant, sophomoric, inflammatory and and a poor reflection on the level of discussion in BAUT. -- Bob Angstrom

The first sentence, which describes the whole first paragraph of Sean Carroll’s paper The Cosmological Constant reads.

“Science is rarely tidy..”

This is my oppinion, if it is untidy, be suspisious. Look for beuty, look for form and process. (If the expansion of space time were truly uniform, there is no cosmological constant. If you have time, look over a recent post I made to ExpErdMan on Gravity and E. ).

Snowflake

How do we know that?

We have only been observing space for an short time. Space could currently be on an arc at such a spot where it all lines up from our point of view.

example:

if you draw an S
and you were a molecule on that curvy line wouldn't everything look like it was a straight line. Until you could observe the entire S you would not know you were on a curve.

Ok back to your regularly scheduled program...

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