# Cosmological Coincidences

Gold Member
One problem in the standard $\Lambda$CDM model is the near equality to an OOM of the densities of baryonic matter (4%), non-baryonic Dark Matter (23%) and Dark Energy (73%). The coincidence is more striking if it is realised that the proportion of DE, if due to the cosmological constant, will grow with the volume of the universe and the universe has expanded by something of the order of 1060 since the Planck era.

If DE is not a cosmological constant but some form of quintessence with $\omega$ <,>, -1, then its density proportion will also grow or possibly shrink with cosmic expansion.

In either case the question is why should this density be in the same 'ball-park' as that of the matter content?

A second coincidence is related to this.

If the presence of DE with negative pressure is accepted it allows the universe to accelerate. A purely accelerating universe can have any age from just over Hubble time (the inverse of Hubble's constant) upwards to infinity (Such as in the Steady State model).

Without any DE acceleration, but with the presence of ordinary matter and energy with a positive pressure, the universe purely decelerates. A decelerating universe has an age less than Hubble time.

The standard model first decelerates, then massively accelerates (inflation), then decelerates through the BBN era until the recent epoch (~1 < z < 0) when it accelerates again.

So what has been the result of this deceleration/acceleration process on the age of the universe?

The present best accepted values of cosmological parameters
(using the table at http://lambda.gsfc.nasa.gov/product/map/dr2/params/lcdm_all.cfm)
H0 = 70.4 km/sec/Mpsc
$Omega_{\Lambda}$ = 0.732
$Omega_{matter}$ = 0.268

Feeding these values into Ned Wright's Cosmology Calculator:
The age of the universe is = 13.81 Gyrs.
But with h100 = 0.704,
Hubble Time = 13.89 Gyrs.

Strange that the age of the universe should be equal to Hubble Time to within an error of 0.6%, almost as if the universe had been expanding linearly at the same rate all the way along!

Just food for thought.

Garth

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## Answers and Replies

sylas
Yes! I noted this myself previously, and started up a thread about it on "Bad Astronomy" some time ago. I also corresponded with Ned Wright directly. Ned considers it is probably just a co-incidence; but it is remarkable.

The thread I wrote at Bad Astronomy is Cosmic Coincidences (April 2005). I love it that we picked almost the same title!

Cheers -- Sylas

Gold Member
I raised the question of the Hubble time/Universe Age coincidence at last Friday's meeting of the RAS after a lecture on DE. Michael Rowan Robinson offered the explanation that the Age used to be always near Hubble time (TH) and so a coincidence should not be too significant.

However that was in the days of purely decelerating universes, when the age lay somewhere between TH and 2/3 TH depending on $\Omega$.

If we allow DE and acceleration into our model then the story changes.

Now it could be anything from 2/3 TH to infinity depending on the DE/Matter (DM and baryonic) ratio.

So why should the observed composition give an age that is so close to TH?

Garth

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hellfire
The expression for the age of the universe in a general cosmological model is:

$$T = \frac{1}{H_0} \int_0^1 \frac{da}{\sqrt{ \Omega_{k, 0} + \displaystyle \frac{\Omega_{m,0} }{a} + \displaystyle \frac{\Omega_{r,0} }{a^2}+ \Omega_{\Lambda,0} a^2 \right)}}}$$

Neglecting the radiation density, for the age to be equal to $T = 1/H_0$, it must hold that:

$$\mathcal{I}(\Omega_{m,0}, \Omega_{\Lambda,0}) = \int_0^1 \frac{da}{\sqrt{ \Omega_{k, 0} + \displaystyle \frac{\Omega_{m,0} }{a} + \Omega_{\Lambda,0} a^2 \right)}}} = 1$$

With $\Omega_{k, 0} = 1 - \Omega_{m,0} - \Omega_{\Lambda,0}$.

It would be nice to see graphically how the surface $$\mathcal{I}(\Omega_{m,0}, \Omega_{\Lambda,0})$$ behaves depending on different values of $\Omega_{m,0}$ and $\Omega_{\Lambda,0}$ (for example, between [0, 1]). Unfortunately I do not have the tools to do such graphics.

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Gold Member
The expression for the age of the universe in a general cosmological model is:

$$T = \frac{1}{H_0} \int_0^1 \frac{da}{\sqrt{ \Omega_{k, 0} + \displaystyle \frac{\Omega_{m,0} }{a} + \displaystyle \frac{\Omega_{r,0} }{a^2}+ \Omega_{\Lambda,0} a^2 \right)}}}$$

Neglecting the radiation density, for the age to be equal to $T = 1/H_0$, it must hold that:

$$\mathcal{I}(\Omega_{m,0}, \Omega_{\Lambda,0}) = \int_0^1 \frac{da}{\sqrt{ \Omega_{k, 0} + \displaystyle \frac{\Omega_{m,0} }{a} + \Omega_{\Lambda,0} a^2 \right)}}} = 1$$

With $\Omega_{k, 0} = 1 - \Omega_{m,0} - \Omega_{\Lambda,0}$.

It would be nice to see graphically how the surface $$\mathcal{I}(\Omega_{m,0}, \Omega_{\Lambda,0})$$ behaves depending on different values of $\Omega_{m,0}$ and $\Omega_{\Lambda,0}$ (for example, between [0, 1]). Unfortunately I do not have the tools to do such graphics.
As $$\int_0^1 da = 1$$

Then one possible solution would be

$$\Omega_{k, 0} + \displaystyle \frac{\Omega_{m,0} }{a} + \Omega_{\Lambda,0} a^2 \right) = 1$$

As $\Omega_{k, 0} = 1 - \Omega_{m,0} - \Omega_{\Lambda,0}$ then this would require

$$\Omega_{m,0}\left(1- \frac{1}{a}\right) = \Omega_{\Lambda,0} \left(a^2 -1\right)$$

or $$\Omega_{m,0}= a\left(a + 1\right)\Omega_{\Lambda,0}$$

Garth

Chronos
Gold Member
Nice exchange Garth, I perceive you did not take the bait. It proves nothing, but the exercise is refreshing. A key variable is still missing, IMO. I am reluctant to call it a hidden variable, but something is clearly missing.

Gold Member
Nice exchange Garth, I perceive you did not take the bait. It proves nothing, but the exercise is refreshing. A key variable is still missing, IMO. I am reluctant to call it a hidden variable, but something is clearly missing.

What I did not point out before was that in a (non-standard) static universe model, where cosmological red shift would not be explained by expansion but instead by a mass field effect, then

$$a(t) = a_0 = 1$$

and if

$$\Omega_{m,0}= a\left(a + 1\right)\Omega_{\Lambda,0}$$

then

$$\Omega_{m,0}= 2 \Omega_{\Lambda,0}$$

And it just so happens that in http://en.wikipedia.org/wiki/Self_creation_cosmology [Broken] we find $\Omega_{m,0} = 0.22$ and $\Omega_{\Lambda,0} = 0.11$

Of course in my post #5 if a = 1 the solution becomes degenerate so you would have to say:

$$\Omega_{k, 0} + \displaystyle \frac{\Omega_{m,0} }{a} + \Omega_{\Lambda,0} a^2 \right) = X$$

if a = 1 then

$$\Omega_{k, 0} + \Omega_{m,0} + \Omega_{\Lambda,0} = X$$

but as $\Omega_{k, 0} = 1 - \Omega_{m,0} - \Omega_{\Lambda,0}$ then

X = 1 as required.

Of course in the static model case the concept of Hubble time and an 'Age of the universe' has to be reinterpreted....

But just food for more thought.

Garth

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Gold Member
For those interested in other coincidences.....

Eventually we have published on the ArXiv Brandon Carter's previously unpublished paper The significance of numerical coincidences in nature
Postscript, October 2007.
As the preceeding notes were too long for the journals (such as Nature) that seemed suitable from the point of view of subject metter, I prepared an abbreviated version that was ultimately published in 1973 [1].
It had been my intention that this “Part I”, dealing with the numbers characterising local cosmogony, would be followed by a separate “Part II”, dealing with the numbers characterising lage scale cosmology. Instead, however, the two parts were finally merged in a relatively short (13 page) set of lecture notes entitled “Large numbers in astrophysics and cosmology” that was presented at a Princeton meeting (organised by John Wheeler for the Clfford Centennial) on 21 February, 1970, developing the (for some purposes indispensible [2]) notions that – in the version I subsequently published [3, 4] – were designated as (strong and weak versions of) the anthropic principle.
The original 1967 notes – namely the “Part I” reproduced here – were circulated in crude stencil printed form (cf figures) just before the observational confirmation (following the discovery of pulsars) that neutron stars actually do exist. The epoch making introduction, at about the same time, of the term “black hole” was helpful for the presentation of the simpler published version [1] . The later did however (for the sake of wider readability) omit many of the details that were thought noteworthy by subsequent authors. It is for this reason, as well as for its purely historical interest, that the original version reproduced here has been directly cited in a variety of relatively ancient [5, 6, 7, 8, 9], and also more recent [10, 11, 12, 13, 14, 15, 16] publications. The purpose of this belated transcription is therefore to make the omitted details more generally accessible
via electronic archiving. It is pertinent to add some comments about how much of the contents of this restoration remains effectively applicable today.

Better late than never!!

Garth

SpaceTiger
Staff Emeritus
Gold Member
I'm not a big fan of cosmology by coincidence. It worked well for the flatness and horizon problems because they were so extreme and required a great deal of fine-tuning in the absence of inflation.

These "coincidences", however, are less striking. The trouble is that, because of randomness in nature, there will be apparently unusual things in any astronomical data set even if nothing is unusual going on. Unless the coincidence is overwhelming, there's no way to know whether it's just an artifact of selective attention or a real indication of new physics.

Wallace
The 'age' co-incidence is of little consequence. The reason for this is that the things that matter, e.g. what can be observed, generally do not depend on the total age of the universe. This includes formation and evolution of structure, distance measures and observed CMB fluctuations. The age of the Universe is a quantity that you get by integrating a model that has been determined by fitting its parameters to what has been observed.

What you claim is that there is significance in the fact that two models, one of which is well fit by observations and the other at odds with observational data have a similar value for a quantity that is intrinsically unobservable.

What matters is what we can see, not numerology!

Chronos
Gold Member
I concur, Wallace. The issue must be decided by observational evidence. WMAP3 is a powerful tool, but not inscrutable. The 'axis of evil hypothesis' will eventually be proven to be a selection effect, IMO.

Gold Member
The 'age' co-incidence is of little consequence. The reason for this is that the things that matter, e.g. what can be observed, generally do not depend on the total age of the universe. This includes formation and evolution of structure, distance measures and observed CMB fluctuations. The age of the Universe is a quantity that you get by integrating a model that has been determined by fitting its parameters to what has been observed.

What you claim is that there is significance in the fact that two models, one of which is well fit by observations and the other at odds with observational data have a similar value for a quantity that is intrinsically unobservable.

What matters is what we can see, not numerology!
Agreed on the last point Wallace, but do you think that the near equality (to an OOM) of the densities of baryonic matter (4%), non-baryonic Dark Matter (23%) and Dark Energy (73%) is also an insignificant coincidence?

As I said, this coincidence is more striking if it is realised that the proportion of DE, if due to the cosmological constant, will grow with the volume of the universe and the universe has expanded by something of the order of 1060 since the Planck era.

The two coincidences are linked because it is the mix of DE/Matter that determines the age of the universe in terms of Hubble's constant.

Garth

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Wallace
The energy density co-incidence is interesting, as is the related relisation that 'dark energy' comes to dominate at the point at which the universe becomes void dominated.

Both of these are interesting because a) they are the result of things that can be measured and b) the quantities in the co-incidence (energy densities, the state of structure growth) have a bearing on the evolution of the universe. The age of the universe in and of itself does not which is why the age co-incidence is pure numerology and nothing more.

Gold Member
The age of the universe in and of itself does not which is why the age co-incidence is pure numerology and nothing more.
We should add: "in the standard GR $\Lambda$CDM model"....

Actually the age of the universe is the result of

$$T = \frac{1}{H_0} \int_0^1 \frac{da}{\sqrt{ \Omega_{k, 0} + \displaystyle \frac{\Omega_{k, 0} }{a} + \displaystyle \frac{\Omega_{r,0} }{a^2}+ \Omega_{\Lambda,0} a^2 \right)}}}.$$

I still find it interesting that, ignoring radiation density, our present day values for $\Omega_{k, 0}$, $\Omega_{k, 0}$, $\Omega_{\Lambda,0}$ yield
TH0 = 0.994 and wonder whether that $\pm$0.6% near equality with unity might be telling us something about that standard GR $\Lambda$CDM model....

Garth

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Wallace
Right, the age of the universe is the result of the integral, but the age doesn't appear in any calculations in cosmology, it is the result of other calculations and observations.

What is it do you think this co-incidence 'is telling us'?

I'm not a big fan of cosmology by coincidence.....

...because of randomness in nature, there will be apparently unusual things in any astronomical data set even if nothing is unusual going on. Unless the coincidence is overwhelming, there's no way to know whether it's just an artifact of selective attention or a real indication of new physics.

Ah yes. Or we might be concocting new physics out of the fact that we can just observe total eclipses, because of the near-equality of the angles subtended here by the sun and moon.

That way lies anthropic reasoning and much else ....

Gold Member
Right, the age of the universe is the result of the integral, but the age doesn't appear in any calculations in cosmology, it is the result of other calculations and observations.

What is it do you think this co-incidence 'is telling us'?

Well, either it is just a coincidence or that for some as yet unknown reason

$$1 = \int_0^1 \frac{da}{\sqrt{ \Omega_{k, 0} + \displaystyle \frac{\Omega_{k, 0} }{a} + \displaystyle \frac{\Omega_{r,0} }{a^2}+ \Omega_{\Lambda,0} a^2 \right)}}}.$$

This would place a constraint on the DE equation of state so that, for example, it resulted in
$$\Omega_{\Lambda,0}= \frac{1}{a\left(a + 1\right)}\Omega_{m,0}$$,

(ignoring radiation density) as discussed above in post#5.

Garth

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Wallace
Placing a constraint means that a parameter of a model is examined by seeing the effects of different values of that parameter on observable and confronting that with observation data.

What you are suggesting it that we first constrain the parameters by looking at available data, then given that model the data prefers calculate the unmeasurable age of the Universe that the model implies. Once we have that value of the age, you suggest that we then ignore the data and claim that the region of allowed models is constrained to those which give the same age. This is an incredibly silly way to do cosmology, since most of the region of allowed models under your suggesting will be ruled out by the very data that arrived at the age of unity in the first place!

To be clear about this, the process you suggest is somewhat analogous to the following: We have an unknown function describing the universe. We can measure this function at some points to some accuracy. Based on these measurements and known physics we come up with a plausible physical model with some free parameters that we use the measured points to constrain. Having arrive at our model, which describes the unknown function we then can integrate the function from time zero (or close to it) to today. Having done that integral you suggest therefore that any set of values of the parameters which give you the same integrated value between the end points is therefore allowed and suggest that this is a 'constraint'. It is not. The constraint comes from whether the curve goes through the measured points, not the total integrated value since that is what we can observe!

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Gold Member
All I am pointing out is that in the present epoch, with the observed values determined by the 'precision cosmology' mainstream model,

$$\int_0^1 \frac{da}{\sqrt{ \Omega_{k, 0} + \displaystyle \frac{\Omega_{m, 0} }{a} + \displaystyle \frac{\Omega_{r,0} }{a^2}+ \Omega_{\Lambda,0} a^2 \right)}}}\approx 1 .$$

We note that according to that mainstream $\Lambda$CDM model this near identity with unity ($\pm$0.6%) did not hold at various earlier epochs and indeed it need not hold in the present epoch. Depending on the mix of DE and matter, and the equation of state of DE, the integral could have a value between 0.66 and $\infty$. It therefore seems a suspicious coincidence that the present value of the integral should be so near to unity.

Furthermore there is a related coincidence in the near equality of density of the components of that DE/DM/Matter mix in the present epoch.

It is not "an incredibly silly way to do cosmology", but a complementary observation that may indicate we are missing something in that standard model.

Garth

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Gold Member
Another example in today's physics arXiv of somebody (Zhao) using an apparent coincidence as a hint that the standard model may need modification, the coincidence may or may not be significant: Coincidences of Dark Energy with Dark Matter - Clues for a Simple Alternative?
A rare coincidence of scales in standard particle physics is needed to explain why $\Lambda$ or the negative pressure of cosmological dark energy (DE) coincides with the positive pressure P0 of random motion of dark matter (DM) in bright galaxies.
Recently Zlosnik et al. (2007) propose to modify the Einsteinian curvature by adding a non-linear pressure from a medium flowing with a four-velocity vector field Uμ. We propose to check whether a smooth extension of GR with a simple kinetic Lagrangian of Uμ can be constructed, and whether the pressure can bend space-time sufficiently to replace the roles of DE, Cold DM and heavy neutrinos in explaining anomalous accelerations at all scales.

Garth

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At the risk of incurring everyone's wrath,

I suggest that the universe is fundamentally, inherently flat geometrically. That it has no capability, through any (true) law of physics, of departing from precisely perfect flatness. (Notwithstanding the various formulas which calculate theoretical curvature values.) Flatness requires nothing other than that the universe expand at a rate equal to the escape velocity of the total mass/energy of its contents. If one starts from this perspective, one would expect various calculations to lead to similar results.

This is not a personal theory, it's just a simpleminded suggestion that some apparent "coincidences" might not be coincidental.

Jon

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Wallace
We note that according to that mainstream $\Lambda$CDM model this near identity with unity ($\pm$0.6%) did not hold at various earlier epochs and indeed it need not hold in the present epoch. Depending on the mix of DE and matter, and the equation of state of DE, the integral could have a value between 0.66 and $\infty$. It therefore seems a suspicious coincidence that the present value of the integral should be so near to unity.

Why it this suspicious? The integral could have had a wide range of values but happens to have a value near 1 given current observations. I see nothing suspicious in this being similar to the result that a different model, one that is at odds with observations, gives. There is an infinite range of possible values of the cosmological parameters and therefore every possible universe that we found ourselves in would have the same age as an infinite number of other universes that are in principle possible, but ruled out by observations. Why is it that you choose a particular discordant model that gives the same age as the observationally favored model? What do you think is special about that particular incorrect model as opposed to the rest of the set of incorrect models that give the same age as observations?

This is why this is pure numerology, since it requires a subjective input of a pet model that is counter to observations.

Some of the other co-incidences you mention are indeed more interesting as I've said and discussion of them is nothing new. But the age co-incidence is not something that people discuss for the reasons I've explained.

Gold Member
At the risk of incurring everyone's wrath,

I suggest that the universe is fundamentally, inherently flat geometrically. That it has no capability, through any (true) law of physics, of departing from precisely perfect flatness. (Notwithstanding the various formulas which calculate theoretical curvature values.) Flatness requires nothing other than that the universe expand at a rate equal to the escape velocity of the total mass/energy of its contents. If one starts from this perspective, one would expect various calculations to lead to similar results.

This is not a personal theory, it's just a simpleminded suggestion that some apparent "coincidences" might not be coincidental.

Jon
Hi Jon,

In the standard model it is Inflation that drives the geometrodynamics of the universe onto near spatial flatness. To say that the universe has no capacity to depart from perfect flatness is contrary to the understanding of GR and would introduce a "personal theory". Do you have a reason why the universe should behave like this?

Actually such a suggestion does nothing to explain the near equality of the calculated Age of the universe and Hubble Time.

However Inflation itself is being questioned by some people, so you could ask:"Given that the universe is observed to be flat, and if Inflation is not 'true', then, as an alternative, is there some as yet unknown other reason making the universe flat?"

It was such a question that gave rise to Inflation theory in the first place!

Garth

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Gold Member
Why it this suspicious?
You may not find the near equality suspicious, even though the accuracy of the equality is to within the error margin of the relevant measured entities ($h_{100}$, $\Omega_{m, 0}$, $\Omega_{r, 0}$, $\Omega_{\Lambda, 0}$), but I do.

On the other hand it could be a complete coincidence, I have never denied that.

As far as the "pet model" is concerned, the present (published 2002) theory of Self Creation Cosmology has been robustly falsified by the first results of the Gravity Probe B satellite, as I have posted here and here on these Forums.
Even though the accuracy is not too great the data clearly shows 6.6"/yr, which is fatal to SCC. http://einstein.stanford.edu/cgi-bin/highlights/showpic.cgi?name=gyro_drift_plot.png [Broken]

Therefore there is now no extant published "pet theory" of mine.

Nevertheless, I find the above near equality very intriguing and still wonder whether we are missing something...

Garth

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Wallace
I'm not referring to you in particular with the pet theory comment. The issue is that for any possible universe there are an infinite number of other sets of parameter values that give the same age. Therefore there is no co-incidence, since this is a generic feature of any possible universe not a quirk of the particular values we find for ours, unless of course there is a special significance to the constant expansion rate universe, which there is not.

Gold Member
The issue is that for any possible universe there are an infinite number of other sets of parameter values that give the same age. Therefore there is no co-incidence, since this is a generic feature of any possible universe not a quirk of the particular values we find for ours.
Wallace, I do think I understand you, are you saying that any of these other sets of parameter values also give the same equality of Calculated Age and Hubble Time?

I understand that the set of parameter values that give the same age is not unique, but the fact that that age is also the Hubble Time in those universes is surely unusual, as I have said, could those Calculated Ages not be any proportion of Hubble Time (with a flat universe) from 2/3 to $\infty$?

Garth

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Wallace
The assumption underpinning this is that there is something special about the Hubble time! The Hubble time (as you well know) is simply the inverse of the present value of the Hubble parameter. In one particular model, in which H is constant for the entire history of the universe, the age of the universe is (obviously) unity in units of the Hubble time. There is nothing special about that particular model so therefore there is no significance to finding that in our current model we find a very similar age.

I am well aware that is equality is only valid right now in our universe and this might be of interest if there was any significance to the constant expansion model, which there is not. In other words the fact that the age is unity in units of the Hubble age doesn't mean anything, since as you say the age could be any value multiplied by the Hubble time. The age being unity in these units is just one option.

To make this really clear imagine the age of the universe was 1.24 in units of the Hubble time (to pick a random number), as implied by some different set of cosmological parameters measured in some other universe. We could then say "Oh my, that's the same age as model X, and these models only give the same age right now, what a co-incidence, surely that is telling us something!". Sure in that case X is a different model from the constant expansion model but why does that matter? There is nothing special about the constant expansion model so we could pick any suitable model for any universe and convince ourselves of some co-incidence, if we were so inclined.

Gold Member
Thank you,
I now understand what you are saying.

The value of the integral in #19 is a pure number.

It appears from present observations of the parameter set that that pure number is unity, or near unity.

Whether this is considered to be interesting or not depends on whether the value is exactly unity and if so whether it is thought '1' is special number or not.

Garth

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One problem in the standard $\Lambda$CDM model is the near equality to an OOM of the densities of baryonic matter (4%), non-baryonic Dark Matter (23%) and Dark Energy (73%). The coincidence is more striking if it is realised that the proportion of DE, if due to the cosmological constant, will grow with the volume of the universe and the universe has expanded by something of the order of 1060 since the Planck era.
Hello Garth,

This is indeed much interesting. If we combine it with the anomalous Pioneer acceleration $a_p$ which is almost equal to $H_0 c$, all of this could indicate that GR is not the more appropriate tool to understand cosmological matters.

Paul