The Hubble Length and Mass of the Universe

[Ian]

What is the magnitude of the Hubble length and what does it signify?
Also, does anyone have any idea what the estimated mass of the universe is.

 

[marcus]

What is the magnitude of the Hubble length and what does it signify?
Also, does anyone have any idea what the estimated mass of the universe is.

it is about 13.8 billion LY

the radius of the observable universe is about 3 times the Hubble length
(the current distance to galaxies and stuff with we are now getting light from goes out to 40 some billion LY or roughly 3 times)

the Hubble length is a convenient length to work with
first off, it is good to know the definition
let H₀ = H(present) be the present value of the Hubble parameter

then by definition the Hubble length is simply
c/H₀

you can calculate it yourself from the measured value of H₀ which is 71 km/sec per Megaparsec

if you just take the speed of light and divide it by that value then you will get around 13.8 billion LY.

something to notice is that H(t) used to be hundreds of times larger and it has been decreasing as time goes along. H(t) decreasing makes c/H(t) increase. So the Hubble length has been growing---very fast in the early universe, really shooting out.

H(t) is still decreasing, but more slowly. H₀ is destined to decline some more. So the Hubble length still has a ways to increase.
IIRC according to one common version of the standard model it will increase some 30 percent to an assymptotic value. but that doesn't matter so much. the main thing is the Hubble length is growing only very slowly now.

 

[Ian]

Thanks Marcus, any idea on the estimated mass of the universe?
Ian.

 

[marcus]

If you know the Hubble law it says
v = H₀ D
where v is the current speed of recession and D is the current distance

(as measured by a chain of observers all at rest with respect to the Hubble flow, or if you prefer with respect to the CMB----Ned Wright goes into this "current distance" idea in his tutorial, it is the distance at this moment, not the light travel time over some long past history)

so the Hubble length is the current distance to things which are receding at speed c.

 

[marcus]

Thanks Marcus, any idea on the estimated mass of the universe?
Ian.

I can give you an estimate of the mass density. But the volume of the universe has not been estimated AFAIK

so I can give you an estimate of mass PER CUBIC KILOMETER
or per cubic lightyear

or in a cube which is 13.8 billion LY on an edge

which would you prefer? (per cubic kilometer is easiest)

but I do not know any estimate of how many cubic kilometers the universe is


of course there is the OBSERVABLE region containing things we have gotten light from, but that is presumably just a small piece of the universe

maybe you could be more specfic about what you are looking for?

 

[Ian]

That sounds odd Marcus, how can you estimate the density without knowing the mass or volume.
How many kg's of mass are there thought to be in the universe, (excluding dark matter)?

 

[marcus]

That sounds odd Marcus, how can you estimate the density without knowing the mass or volume.
How many kg's of mass are there thought to be in the universe, (excluding dark matter)?

hi Ian, if it sounds odd then you may need someone else to discuss with, because I have never heard an estimate of volume

of course there is the volume of the observable region from which light has already reached us
you could estimate the volume of the observable and the mass of the observable

but the easiest thing to do is estimate the density
I understand from your post that you want to exclude dark matter (and I assume the even more hypothetical dark energy) and just focus on matter that we can see: stars gas dust etc.

that comes to the equivalent of about 0.033 joules per cubic kilometer

roughly speaking 4E-19 kilogram per cubic km

a cube that is one million km on a side would therefore (being E18 cubic km) contain on average 0.4 kilogram.

Ian the reason it is so much easier to estimate the density is that one can just look. One takes a suitable large volume of space and does an inventory! It will turn out that you get about the same average density for one volume as another. It doesn't matter which direction you look, as long as you take a big enough sample that local clusters of galaxies don't bias.
it has to be big enough to include space between the clusters.
a lot of astronomers have worked on this kind of inventory for a lot of different volumes of space.

 

[yogi]

For a critical density Hubble sphere, its about 10^53 kgm. For those models of the universe where all observers view the same Hubble sphere, but from different vantage points, this may be all there is.

 

[moving finger]

something to notice is that H(t) used to be hundreds of times larger and it has been decreasing as time goes along. H(t) decreasing makes c/H(t) increase. So the Hubble length has been growing---very fast in the early universe, really shooting out.

H(t) is still decreasing, but more slowly. H₀ is destined to decline some more. So the Hubble length still has a ways to increase.
IIRC according to one common version of the standard model it will increase some 30 percent to an assymptotic value. but that doesn't matter so much. the main thing is the Hubble length is growing only very slowly now.

Has anyone noticed that (assuming the current lambda-CDM model) if you plot c/H(t)/R against R (where R is the scale size or "radius" of the universe), you get an interesting "peak" in the curve which corresponds to R approx equal to 10^10 light years?

For both low values of R (< 10^9 light years) and high values of R (>10^11 light years) this ratio c/H(t)/R is small; in the present epoch it goes through a maximum.

Does anyone have an explanation for this?

Thanks

MF

 

[marcus]

Has anyone noticed that (assuming the current lambda-CDM model) if you plot c/H(t)/R against R (where R is the scale size or "radius" of the universe), you get an interesting "peak" in the curve which corresponds to R approx equal to 10^10 light years?

For both low values of R (< 10^9 light years) and high values of R (>10^11 light years) this ratio c/H(t)/R is small; in the present epoch it goes through a maximum.

Does anyone have an explanation for this?

Thanks

MF

hi MF, no I hadnt noticed this, and the notation c/H(t)/R is at least superficially ambiguous.

but I understand that you mean (c/H(t))/R(t) which would be a dimensionless number.

so you are plotting

$$\frac{\frac{c}{H(t)}}{R(t)}= \frac{\text{Hubble length at time t}}{\text{scale factor at time t}}$$

I do not remeber ever seeing a plot of that. I wish you would post a dozen or so sample values in the form of a list, or a table, so then i could see what you are talking about

 

[marcus]

For a critical density Hubble sphere, its about 10^53 kgm. For those models of the universe where all observers view the same Hubble sphere, but from different vantage points, this may be all there is.

pleased to meet someone who regularly calculates. I expect you are right and I will try to confirm, to see if my number is consistent with yours:
"a cube that is one million km on a side would therefore (being E18 cubic km) contain on average 0.4 kilogram."

LY = 9.46E12 km, 9.46E6 million km
cubic LY contains 3.4E20 kg

hubble sphere, I guess would a sphere with radius 13.8 billion LY
1.1E31 cubic LY

so a Hubble sphere wd contain 1.1E31 x 3.4E20 kilogram
3.7E51 kilogram

now I was only counting baryonic matter, which is said to be about 4 percent of the total

so if you meant to include every component of the critical density, in your critical density sphere,
then our figures agree!

because my 3.7E51 kg is about 4 percent of your E53 kg.

===========
BTW I was excluding dark energy and dark matter because that is what Ian asked about back in post #6 where he asked
"How many kg's of mass are there thought to be in the universe, (excluding dark matter)?"

It seemed he was only interested in knowing the mass of ordinary familiar visible or baryonic matter and not these more hypothetical things.

 

[moving finger]

hi MF, no I hadnt noticed this, and the notation c/H(t)/R is at least superficially ambiguous.

but I understand that you mean (c/H(t))/R(t) which would be a dimensionless number.

so you are plotting

$$\frac{\frac{c}{H(t)}}{R(t)}= \frac{\text{Hubble length at time t}}{\text{scale factor at time t}}$$

I do not remeber ever seeing a plot of that. I wish you would post a dozen or so sample values in the form of a list, or a table, so then i could see what you are talking about

Hi

Yes, I did mean $$\frac{\frac{c}{H(t)}}{R(t)}= \frac{\text{Hubble length at time t}}{\text{scale factor at time t}}$$

The curve is attached, assuming :

normal Friedman equations
rho(lambda) (energy density of vacuum) is 6.9 x 10^-27 kg/m^3
equation of state for vacuum energy = -1 (ie density = - pressure)
Omega(tot) = 1
present-day value for Omega(matter) = 0.27
k (curvature of space) = 0

I get the attached curve (with R(t) in light-years).

MF

 

[SpaceTiger]

Has anyone noticed that (assuming the current lambda-CDM model) if you plot c/H(t)/R against R (where R is the scale size or "radius" of the universe), you get an interesting "peak" in the curve which corresponds to R approx equal to 10^10 light years?

It looks like you're just plotting:

$$\frac{c}{\dot{R}}$$ vs $$R$$

The critical point for this curve will just be the point at which $$\ddot{R}=0$$, which is the point at which the acceleration starts and lambda takes over (so to speak).

 

[moving finger]

It looks like you're just plotting:

$$\frac{c}{\dot{R}}$$ vs $$R$$

The critical point for this curve will just be the point at which $$\ddot{R}=0$$, which is the point at which the acceleration starts and lambda takes over (so to speak).

Thank you Space Tiger!
Yes you are correct.
I guess this is the source of the "coincidence" problem, that Omega(mass) is of the same order of Omega(vacuum energy) at the present epoch (ie the vacuum energy density is JUST about the right value that exponential expansion starts to take over just about now).

What does everyone think - is this purely a coincidence, or is there something deeper?

MF

 

[Chronos]

I don't believe it's a coincidence. I think omega has, and will always be exactly 1. I haven't figured out the why part. But I refuse to believe we live at just the right time to see it pass from not 1 through 1 and continue on in whatever direction it had in mind.

 

[moving finger]

I don't believe it's a coincidence. I think omega has, and will always be exactly 1. I haven't figured out the why part. But I refuse to believe we live at just the right time to see it pass from not 1 through 1 and continue on in whatever direction it had in mind.

It maybe that Omega always has and always will be 1, but the cosmological dynamics right now depend on how this is split between Omega(mass) and Omega(vacuum energy) (assuming Omega(radiation) is negligible).

In the not so distant past Omega(mass) was ~1 and Omega(vacuum energy) was ~0

In the not so distant future Omega(mass) will be ~0 and Omega(vacuum energy) will be ~1

We seem to live at a very strange epoch where the transition is taking place, where deceleration due to Omega(mass) has stopped, and acceleration due to Omega(vacuum energy) has just taken over.

This is the coincidence I was referring to.

MF

 

[SpaceTiger]

I guess this is the source of the "coincidence" problem, that Omega(mass) is of the same order of Omega(vacuum energy) at the present epoch (ie the vacuum energy density is JUST about the right value that exponential expansion starts to take over just about now).

What does everyone think - is this purely a coincidence, or is there something deeper?

You always seem to ask the best questions.

Yeah, I dunno, but I suspect that it's nothing particularly deep. We were talking about this in the lounge a few weeks ago and I was drawing a logarithmic plot, marking the major events in cosmic history (decoupling, M-R equality, nucleosynthesis, etc.). I basically asked the question, if I pick a random point on this plot (i.e. a random proper time), what are the chances that I'll land near (in logarithmic interval) some major "event" in the universe's evolution?

It's obviously an extremely ambiguous question, as it depends on what one defines to be a "major" event, but I think it was worth the qualitative look, at least. My general conclusion was that it wasn't obviously strange that we landed this near the lambda transition, but is still worth looking into. Perhaps it's in some way related to the anthropic principle and that the growth of structure in the universe somehow naturally led to us existing near the transition. It's probably not worth thinking too hard about it until we have a better idea of what the dark energy is.

 

[moving finger]

You always seem to ask the best questions. .

Advances are made by answering questions. Discoveries are made by questioning answers. (Bernard Haisch, Astrophysicist and past Editor of the Journal of Scientific Exploration)

Yeah, I dunno, but I suspect that it's nothing particularly deep. We were talking about this in the lounge a few weeks ago and I was drawing a logarithmic plot, marking the major events in cosmic history (decoupling, M-R equality, nucleosynthesis, etc.). I basically asked the question, if I pick a random point on this plot (i.e. a random proper time), what are the chances that I'll land near (in logarithmic interval) some major "event" in the universe's evolution?

It's obviously an extremely ambiguous question, as it depends on what one defines to be a "major" event, but I think it was worth the qualitative look, at least. My general conclusion was that it wasn't obviously strange that we landed this near the lambda transition, but is still worth looking into. Perhaps it's in some way related to the anthropic principle and that the growth of structure in the universe somehow naturally led to us existing near the transition. It's probably not worth thinking too hard about it until we have a better idea of what the dark energy is.

Doesn't look so impressive on a linear plot though, does it?

MF

"And if you take one from three hundred and sixty-five what remains?"
"Three hundred and sixty-four, of course."
Humpty Dumpty looked doubtful, "I'd rather see that done on paper," he said.
(Lewis Carroll)​

 

[Ian]

Thanks everyone for the info, I made an estimate of the mass of the universe of ~10^50 kg. Comparing the gravitational length (GM/c^2) of this mass to the Hubble length tells me something is out order with expansionist big bang ideas.

 

[moving finger]

Thanks everyone for the info, I made an estimate of the mass of the universe of ~10^50 kg. Comparing the gravitational length (GM/c^2) of this mass to the Hubble length tells me something is out order with expansionist big bang ideas.

actually I reckon the total mass contained within the present Hubble radius (1.3 x 10^26m), assuming a mean mass-density of 2.55 x 10^-27 kg/m^3, is about 2.4 x 10^52 kg.

This would give a value of ~1.76 x 10^25m for the ratio GM/c^2; contrasted with 1.3 x 10^26m for the present Hubble radius.

Why does this show something is out of order with expansionist Big Bang ideas?

MF

 

[Garth]

In the Freely Coasting model R(t) = ct

$$\frac{\frac{c}{H(t)}}{R(t)}= \frac{\text{Hubble length at time t}}{\text{scale factor at time t}} = 1$$ and $$\ddot{R}=0$$ at all times..

May this be the explanation you are looking for moving finger?

Garth

 

[moving finger]

In the Freely Coasting model R(t) = ct

$$\frac{\frac{c}{H(t)}}{R(t)}= \frac{\text{Hubble length at time t}}{\text{scale factor at time t}} = 1$$ and $$\ddot{R}=0$$ at all times..

May this be the explanation you are looking for moving finger?

Garth

Interesting, but I was actually querying (in my last post) why Ian seemed to think there was a problem with the numbers?

MF