Are Hubble's Sphere and the Universe's Age Coincidentally Similar?

[zeromodz]

Is it just a coincidence that Hubble's Sphere which is c/H

and the age of the universe being almost if not exactly the same

 

[Garth]

You may find this thread interesting: Cosmological Coincidences.

Garth

 

[nicksauce]

I don't see it as a coincidence. If you admit only c and Ho as the only constants with dimensions in your model, then you would expect them to give the characteristic length scale c/Ho and the characteristic time scale 1/Ho. In other words, the age of the universe must be of order 1/Ho, because there are no other numbers you can play around with to give a time.

 

[Garth]

I don't see it as a coincidence. If you admit only c and Ho as the only constants with dimensions in your model, then you would expect them to give the characteristic length scale c/Ho and the characteristic time scale 1/Ho. In other words, the age of the universe must be of order 1/Ho, because there are no other numbers you can play around with to give a time.

The Age of the universe A is not necessarily equal to Hubble Time T_H = H^-1.

In GR without DE
$$T_H \geqslant A \geqslant \frac{2}{3} T_H$$
depending on a cosmological density of $0 \geqslant \Omega \geqslant 1$.

The actual density was generally thought (pre 1998) not to be greater than the critical density because i) such density was not observed and ii) that would introduce an Age Problem to the model.

However, after 1998, in the LCDM model with DE: A can be $( \frac{2}{3} \rightarrow \infty)T_H$ depending on the amount of density and DE.

Why then should there be just enough DE and total density so that A is so close to $T_H$ that it is exactly equal to it to within observational errors?

You will find a discussion on this topic on the thread I linked to above.

Garth

 

[Chalnoth]

I don't see it as a coincidence. If you admit only c and Ho as the only constants with dimensions in your model, then you would expect them to give the characteristic length scale c/Ho and the characteristic time scale 1/Ho. In other words, the age of the universe must be of order 1/Ho, because there are no other numbers you can play around with to give a time.

Well, the only difficulty with this analysis is that if you compared the age of our universe to $$1/H_0$$ much earlier than now or much later, you'd end up with a very very different result than the one we get.

I'm sure I could calculate this more explicitly, but my suspicion is that this is another feature of the cosmological coincidence problem: that the matter and dark energy density are both within an order of magnitude of one another right now.

 

[Garth]

Well, the only difficulty with this analysis is that if you compared the age of our universe to $$1/H_0$$ much earlier than now or much later, you'd end up with a very very different result than the one we get.

I'm sure I could calculate this more explicitly, but my suspicion is that this is another feature of the cosmological coincidence problem: that the matter and dark energy density are both within an order of magnitude of one another right now.

Well as hellfire said in this post:

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.

There you have the relationship between the matter and DE densities, but why should the integral pretty well equal unity?

I cannot help but see it as more than just a coincidence.

Garth

 

[nicksauce]

Ah, I see what you mean now. Then yes, it is an interesting coincidence I suppose.

 

[twofish-quant]

Ah, I see what you mean now. Then yes, it is an interesting coincidence I suppose.

Or you can go anthropic and say that the universe needs about this much time for intelligent life to develop. One curious thing about this is that this would solve the Fermi paradox.

One thing that would be interested is to do a calculation in which you estimate the earliest point in the universe in which intelligent life could develop and then the latest point, and the compare that with Hubble times.

 

[Garth]

Or you can go anthropic and say that the universe needs about this much time for intelligent life to develop. One curious thing about this is that this would solve the Fermi paradox.

One thing that would be interested is to do a calculation in which you estimate the earliest point in the universe in which intelligent life could develop and then the latest point, and the compare that with Hubble times.

If you look at the integral:
$$\mathcal{I}(\Omega_{m,0}, \Omega_{\Lambda,0}) = \int_0^1 \frac{da}{\sqrt{ \Omega_{k, 0} + \displaystyle \frac{\Omega_{m,0} }{a} + \Omega_{\Lambda,0} a^2 \right)}}} = 1$$

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

You see both Hubble Time and the Age of the universe are nowhere to be seen.

In order to get the 'coincidence' you need the correct balance of Matter, Dark Matter and Dark Energy.

Then, with that relative abundance ratio, you would get the coincidence at all ages of the universe.

Garth

 

[Chalnoth]

Well as hellfire said in this post:

If we take the flat case, that integral is within about 10% of 1 between around $$0.18 < \Omega_m < 0.38$$.

 

[Garth]

If we take the flat case, that integral is within about 10% of 1 between around $$0.18 < \Omega_m < 0.38$$.

The present best accepted values of cosmological parameters
(using the table at WMAP Cosmological Parameters)
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.

So the integral is equal to A/T_H = 0.994, i.e. to within 0.6% of unity.

Garth

 

[Chalnoth]

The present best accepted values of cosmological parameters
(using the table at WMAP Cosmological Parameters)
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.

So the integral is equal to A/T_H = 0.994, i.e. to within 0.6% of unity.

Garth

Given that the errors on those parameters is at the 2% level, this is probably a statistical fluke.

 

[Garth]

Given that the errors on those parameters is at the 2% level, this is probably a statistical fluke.

However, note in precision cosmology the values of H and A are given to 3 decimal places, so rounding to 2 places, A and T_H are equal to within observational errors.

Garth

 

[Wallace]

It doesn't matter how close to unity this integral is, the whole co-incidence is a red herring. Since the age of the Universe is a derived quantity, not a model parameter, it doesn't matter what value it has (beyond any 'age problems' which don't seem overwhelming at this point, and not relevant to any co-incidence in any case).

Everytime this comes up, people want to try and suggest that this is somehow a problem for LCDM or that is points to some alternate model, but you can't make model selection decisions based on derived quantities after you have considered the data. If some other model exists that would say enforce this co-incidence to be true at all times (and hence not be a co-incidence) then the only way to test that model is against the original data, not against the derived age from the fit of the LCDM model to the data.

To put it another way, an infinite number of incompatible models could give you the same age, so looking at the age alone tells you nothing.

 

[Garth]

So A/T_H ~ 1 is just a coincidence in the LCDM model.

As the drived age A is dependent on $\Omega_m$ and $\Omega_{\Lambda}$, it is obviously related by the integral in post 9 to the Cosmic Coincidence Problem in that model which is the DM (and baryonic matter) and DE densities are roughly equal in the present epoch, when they could be very disparate in value.

Garth

 

[Chalnoth]

However, note in precision cosmology the values of H and A are given to 3 decimal places, so rounding to 2 places, A and T_H are equal to within observational errors.

Garth

But as I said, if those parameters are only measured to within 2%, any measurement that they are closer than that is likely just a fluke. Did you try looking at other combinations of data, for instance?

 

[Wallace]

DE densities are roughly equal in the present epoch, when they could be very disparate in value.

They could be, but they don't appear to be so. Just as I could have made this post at any moment, but against all odds happened to make it at precisely 11:36 am (CET) on the 5th of January 2010!

Less facetiously, there really isn't a big deal about this co-incidence in energy densities. If you start from the assumption that you are in a Lambda /= 0 universe, and can actually observe this cosmologically, then it is not so surprising to see this confluence. If we existed to early we'd just see matter domination and if we exist too late we'd see nothing outside the local group.

There are a lot of more pressing problems in cosmology that one could get your knickers in a knot about than a few curious numerological co-incidences.

 

[Garth]

But as I said, if those parameters are only measured to within 2%, any measurement that they are closer than that is likely just a fluke. Did you try looking at other combinations of data, for instance?

I agree that the fact that A and T_H are within 0.6% of each other is a fluke and by choosing slighty different values (within the error bars) for the densities the answer would come out less close, but the fact that they are equal within observational error (i.e. closer than 2%) might be telling us something about the relationship between matter, DE and DM.

Garth

 

[Garth]

They could be, but they don't appear to be so. Just as I could have made this post at any moment, but against all odds happened to make it at precisely 11:36 am (CET) on the 5th of January 2010!

Less facetiously, there really isn't a big deal about this co-incidence in energy densities. If you start from the assumption that you are in a Lambda /= 0 universe, and can actually observe this cosmologically, then it is not so surprising to see this confluence. If we existed to early we'd just see matter domination and if we exist too late we'd see nothing outside the local group.

There are a lot of more pressing problems in cosmology that one could get your knickers in a knot about than a few curious numerological co-incidences.

Well, I'm not getting my knickers in an twist, it might just be a coincidence, I was simply responding to the OP question!

However, others do think about the Cosmic Coincidence Problem such as here and as Funkhouser says:

The fact that the large number coincidence occurs only in the same epoch in which other coincidences among cosmic parameters occur could be considered a distinct coincidence problem suggesting an underlying physical connection.

.

In may prove interesting not to dismiss such coincidences as 'just coincidences' in order to explore a possible "underlying physical connection".

Garth

 

[Chalnoth]

I agree that the fact that A and T_H are within 0.6% of each other is a fluke and by choosing slighty different values (within the error bars) for the densities the answer would come out less close, but the fact that they are equal within observational error (i.e. closer than 2%) might be telling us something about the relationship between matter, DE and DM.

Garth

Possible, but unlikely. The thing is, it's just not statistically significant enough to really tell us anything, at least not yet.

If we started to measure these parameters to within, say, 0.05% or so, and the age was still, within the error bars, equal to the inverse of the Hubble constant, then we might have something that really needs explaining. But within 2%? That's not really special.

 

[Wallace]

But that's the point, you can't find an underlying physical connection from these, because they are not physically relevant parameters, that is to say, they don't represent a physical theory, they are just some numbers. It's like saying if a piece of wood and a piece of metal are the same length it might imply a similarity in their internal chemistry.

The only way to look for physics is to construct physical models and compare them to data, not read the tea leaves that you get after doing a fit to data.

 

[Garth]

It's like saying if a piece of wood and a piece of metal are the same length it might imply a similarity in their internal chemistry.

It might imply that you had unknowingly stumbled across a collection of wooden and steel rulers, we don't know.

you can't find an underlying physical connection from these, because they are not physically relevant parameters, that is to say, they don't represent a physical theory, they are just some numbers.

We also don't know much about DM and DE, other than how they behave gravitationally, in particular we don't know how they relate to each other, but the A/T_H coincidence AND the Cosmic Coincidence between $\Omega_{DM}$ and $\Omega_{\Lambda}$ might indicate that they are physically relevant parameters connected to each other.

Garth