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petm1 said:
Sure. A photon is emitted from a billion year old atom a billion light years from here and thus by your reckoning is 2 billion years old when it gets here. It hits a newly formed H2O atom that is 1 second old.
Click For Summary
The discussion explores the concept of the universe's edge, presenting three main perspectives: the observable universe as a sphere with an edge beyond visibility, the expanding universe theory suggesting an inward perspective, and the idea of a universe without an edge. Participants argue that while the observable universe has a defined radius, it does not imply a physical edge, as the universe is expanding into itself rather than into an external space. The consensus leans towards the notion that the universe lacks an edge, emphasizing that this understanding challenges traditional concepts of space and time. Ultimately, the conversation highlights the complexities of defining boundaries in an infinite universe, suggesting that our definitions may be limited by human perception.
Sure. A photon is emitted from a billion year old atom a billion light years from here and thus by your reckoning is 2 billion years old when it gets here. It hits a newly formed H2O atom that is 1 second old.
Space news on Phys.org
HallsofIvy
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"The time it takes a letter to go from me to my nephew, added to my age, at the time I mail the letter, will always equal the age of my nephew." What is the error in my thinking?petm1 said:The duration of a photon from emission til reception added to the age of the emitting atom at emission will always equal the age of the atom that is receiving the photon in the present. Could you please show me the error in my thinking.
Age of an emitting atom at emission, added to any duration, will be the age of that atom at the end of the duration, just the time it takes a letter to go from me to my nephew add to my age when I send it will be my age when he receives it.
In other words "The duration of a photon from emission til reception added to the age of the emitting atom at emission" will be the age of the emitting atom at reception.
In asserting that it is also the age of the receiving atom, you appear to be thinking that all atoms are the same age. And that is not true.
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Adrian07
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mfb post 29
I think you may be right or perhaps, if this makes sense, time today is younger than yesterday, however I am older today than I was yesterday.
I think you may be right or perhaps, if this makes sense, time today is younger than yesterday, however I am older today than I was yesterday.
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Adrian07 said:mfb post 29
I think you may be right or perhaps, if this makes sense, time today is younger than yesterday, however I am older today than I was yesterday.
How do you define time in a way that makes it younger today than yesterday? Are you running time backwards?
petm1
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In asserting that it is also the age of the receiving atom, you appear to be thinking that all atoms are the same age. And that is not true.
I was under the impression that all matter, in the form of atoms, formed just after the decoupling of photons we see as the cmbr. I did not know new atoms were formed all the time I thought that new atoms were just a recombination of existing atoms into heaver ones not new ones. Thanks.
Thomas1989
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I didn't want to start yet another thread, so I'll post this here, it's quite closely related.
I've only recently started to comprehend that our universe could be flat and infinite, and has no boundary, this is really crazy. I still don't understand how the universe could have a finite age, a beginning in time and yet be infinite in spatial extent.
There are some truly brilliant minds here, I was hoping somebody might be good enough to explain exactly how this is mathematically possible.
I've only recently started to comprehend that our universe could be flat and infinite, and has no boundary, this is really crazy. I still don't understand how the universe could have a finite age, a beginning in time and yet be infinite in spatial extent.
There are some truly brilliant minds here, I was hoping somebody might be good enough to explain exactly how this is mathematically possible.
Adrian07
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I am not classed as brilliant but basically what you see in this thread is what you get regarding boundaries of the universe, got no idea about the maths though. As far as being edgeless goes imagine there is nothing outside the universe then you cannot have an edge as there is no outside you require an inside and outside to define the edge.
Thats space and you should be able to see the arguments caused when you include time.
Thats space and you should be able to see the arguments caused when you include time.
mfb
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It would have been infinite in its spatial size all the time.explain exactly how this is mathematically possible.
Boy@n
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Tu
So, based on what I read in this thread it seems Universe is considered to be infinite while Observable Universe is not.
I too would appreciate to see proper explanations.
It makes me also think that then we don't really know what's beyond that which we can observe (if anything), so, why do we need dark matter and dark energy theory to explain where the missing mass is? It could well be beyond that which we can observe, no?
Observable Universe is flat and has finite size... Google Universe and change results to images...Thomas1989 said:I didn't want to start yet another thread, so I'll post this here, it's quite closely related.
I've only recently started to comprehend that our universe could be flat and infinite, and has no boundary, this is really crazy. I still don't understand how the universe could have a finite age, a beginning in time and yet be infinite in spatial extent.
There are some truly brilliant minds here, I was hoping somebody might be good enough to explain exactly how this is mathematically possible.
So, based on what I read in this thread it seems Universe is considered to be infinite while Observable Universe is not.
I too would appreciate to see proper explanations.
It makes me also think that then we don't really know what's beyond that which we can observe (if anything), so, why do we need dark matter and dark energy theory to explain where the missing mass is? It could well be beyond that which we can observe, no?
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Boy@n said:Observable Universe is flat and has finite size... Google Universe and change results to images...
Based on what I read in this thread it seems Universe is considered to be infinite while Observable Universe is not.
I'd appreciate too to see proper explanations.
The universe is undefined in shape and size. Might be infinite, might be finite and unbounded. Believed to NOT be finite and bounded as that causes a myriad of problems in physics and seems unreasonable.
The observable universe is a well-defined sphere based on you and extending out for about 50billion light years in all directions.
Mordred
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Boy@n said:Tu
It makes me also think that then we don't really know what's beyond that which we can observe (if anything), so, why do we need dark matter and dark energy theory to explain where the missing mass is? It could well be beyond that which we can observe, no?
Dark matter and dark energy is not an explanation in regards to shape or edge of the universe.
dark matter explains the missing mass that attributes to the rate of galaxy spins, by Newtonian physics the outer edge of a galaxy would spin slower than the outer edge. This does not occur, it was found that having dark matter the spin works out correctly. this explanation does not cover all the evidence of dark matter which is numerous. However it covers what led scientists to infer its existence.
dark energy is an energy term used to explain the rate of expansion. It was thought at one time that the universe was static. However Einsteins GR predicted either a contracting or expanding universe. Hubble later showed that it was expanding. Dark energy is said to be the cause of that expansion. There is still debate on the cause of expansion mechanism. Could be false vacuum as stated by A. Guths false vacuum model. Could be a bounce as LQG states (hopefully I have that part right LQC is not my familiarity).
In an infinite Universe the only things we can confirm is the observable portions only, no matter how far back in time we look we will never see an edge or border to the universe not that one exists,. The flat geometry was once considered indicative of an open universe (infinite) however later research showed that other flat shapes such as a torus can be flat and finite.
other classical geometries of a closed universe are circle or sphere, and saddle both these geometries are finite geometries.
here is a straightforward link on Universe geometry
http://abyss.uoregon.edu/~js/cosmo/lectures/lec15.html
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mfb
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Dark energy is the cause for the acceleration of the expansion. The universe could expand without dark energy as well.Mordred said:Dark energy is said to be the cause of that expansion.
Mordred
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Yeah I should have worded that better lol
Boy@n
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That seems to me to be the most certain thing we can say on this topic.phinds said:The universe is undefined in shape and size.
All else are just various possible scenarios/theories with higher or lower degree of probability to describe truth of nature.
Though, I find all of them fascinating, because we infer so much from so little (all from light and particles hitting our planet). Amazing.
Adrian07
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Another question.
Draw a simple time-line for the universe from BB to present day.
1 Where on that time-line has the universe expanded to.
2 Where on that line do we sit.
The answer to both appears to be the same place. This appears to make no sense at least as far as the observable universe goes. Any explanations. Am assuming time and space expand at the same rate.
Draw a simple time-line for the universe from BB to present day.
1 Where on that time-line has the universe expanded to.
2 Where on that line do we sit.
The answer to both appears to be the same place. This appears to make no sense at least as far as the observable universe goes. Any explanations. Am assuming time and space expand at the same rate.
Mordred
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Time does not have a spatial component. It neither expands or contracts. Rather its observed rate changes as described by GR and SR.
If your referring to the relation of light speed of travel and the observable universe.
Yes the universe is 13.78 billion years old.
The observable universe today has a radius from our locale of 47 billion light years.
As light travels towards us space is expanding.
Thats how we see further than the age of the universe.
See the FAQ sub forum for more detail.
During the first second of the universe. The universe exponentially expanded. Then expansion slowed down during the matter dominated era. This matter dominated era lasted till roughly 7.3 billion years after the BB. during the matter dominated era gravity was dominant and caused a contraction. However enough vacuum or dark energy was created referred to as the cosmological constant or Lambda \Lambda in the lambdaCDM. The cosmological constant became dominant leading to a lambda dominant era where expansion is increasing.
If your referring to the relation of light speed of travel and the observable universe.
Yes the universe is 13.78 billion years old.
The observable universe today has a radius from our locale of 47 billion light years.
As light travels towards us space is expanding.
Thats how we see further than the age of the universe.
See the FAQ sub forum for more detail.
During the first second of the universe. The universe exponentially expanded. Then expansion slowed down during the matter dominated era. This matter dominated era lasted till roughly 7.3 billion years after the BB. during the matter dominated era gravity was dominant and caused a contraction. However enough vacuum or dark energy was created referred to as the cosmological constant or Lambda \Lambda in the lambdaCDM. The cosmological constant became dominant leading to a lambda dominant era where expansion is increasing.
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Boy@n
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Well, I guess current science says that space and time are not separate, but one 'thing' as spacetime.Adrian07 said:Am assuming time and space expand at the same rate.
Barbour has different opinion though, that time doesn't really exist, that time is merely our perception of motion of all things.
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Mordred
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{\scriptsize \begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)&a'R_{0}\\ \hline 1090.000&0.000917&0.000378&0.000637&45.731&0.042&0.056&0.000&20.164\\ \hline 849.077&0.001178&0.000574&0.000953&45.543&0.054&0.072&0.001&17.299\\ \hline 661.405&0.001512&0.000866&0.001419&45.325&0.069&0.092&0.002&14.917\\ \hline 515.214&0.001941&0.001301&0.002103&45.073&0.087&0.118&0.003&12.919\\ \hline 401.336&0.002492&0.001943&0.003106&44.782&0.112&0.151&0.005&11.230\\ \hline 312.629&0.003199&0.002891&0.004574&44.448&0.142&0.193&0.007&9.791\\ \hline 243.528&0.004106&0.004284&0.006717&44.066&0.181&0.246&0.011&8.558\\ \hline 189.701&0.005271&0.006328&0.009846&43.629&0.230&0.313&0.016&7.496\\ \hline 147.771&0.006767&0.009322&0.014408&43.131&0.292&0.399&0.024&6.576\\ \hline 115.109&0.008687&0.013699&0.021057&42.563&0.370&0.507&0.036&5.776\\ \hline 89.667&0.011152&0.020093&0.030744&41.917&0.467&0.644&0.053&5.079\\ \hline 69.848&0.014317&0.029424&0.044849&41.182&0.590&0.816&0.079&4.469\\ \hline 54.409&0.018379&0.043031&0.065385&40.348&0.742&1.032&0.116&3.935\\ \hline 42.383&0.023594&0.062863&0.095274&39.400&0.930&1.303&0.172&3.467\\ \hline 33.015&0.030289&0.091754&0.138771&38.325&1.161&1.640&0.253&3.056\\ \hline 25.718&0.038884&0.133830&0.202060&37.105&1.443&2.057&0.372&2.694\\ \hline 20.033&0.049917&0.195082&0.294128&35.722&1.783&2.572&0.547&2.376\\ \hline 15.605&0.064080&0.284235&0.428027&34.154&2.189&3.202&0.802&2.096\\ \hline 12.156&0.082263&0.413945&0.622664&32.377&2.663&3.964&1.176&1.850\\ \hline 9.469&0.105605&0.602597&0.905329&30.363&3.207&4.876&1.722&1.633\\ \hline 7.376&0.135570&0.876751&1.315024&28.084&3.807&5.950&2.520&1.443\\ \hline 5.746&0.174038&1.274606&1.906448&25.507&4.439&7.190&3.683&1.278\\ \hline 4.476&0.223420&1.850354&2.753067&22.602&5.050&8.581&5.378&1.136\\ \hline 3.487&0.286815&2.678630&3.944165&19.346&5.549&10.082&7.837&1.018\\ \hline 2.716&0.368198&3.856519&5.562897&15.738&5.795&11.615&11.390&0.927\\ \hline 2.116&0.472674&5.494908&7.622994&11.826&5.590&13.062&16.471&0.868\\ \hline 1.648&0.606794&7.688893&9.965115&7.742&4.698&14.289&23.622&0.852\\ \hline 1.284&0.778970&10.465993&12.220152&3.709&2.889&15.203&33.467&0.892\\ \hline 1.000&1.000000&13.753303&13.999929&0.000&0.000&15.793&46.686&1.000\\ \hline 0.848&1.178769&16.127949&14.832270&-2.203&-2.596&16.037&57.611&1.113\\ \hline 0.720&1.389495&18.618735&15.419791&-4.150&-5.766&16.198&70.616&1.262\\ \hline 0.611&1.637894&21.189539&15.814752&-5.855&-9.590&16.300&86.034&1.450\\ \hline 0.518&1.930698&23.813226&16.071344&-7.332&-14.156&16.363&104.265&1.682\\ \hline 0.439&2.275846&26.470879&16.234284&-8.601&-19.575&16.400&125.793&1.963\\ \hline 0.373&2.682696&29.149943&16.336240&-9.687&-25.986&16.420&151.192&2.299\\ \hline 0.316&3.162278&31.842341&16.399447&-10.612&-33.558&16.429&181.147&2.700\\ \hline 0.268&3.727594&34.543144&16.438244&-11.399&-42.492&16.438&216.465&3.175\\ \hline 0.228&4.393971&37.248853&16.462174&-12.069&-53.030&16.462&258.104&3.737\\ \hline 0.193&5.179475&39.957670&16.476843&-12.637&-65.454&16.477&307.189&4.401\\ \hline 0.164&6.105402&42.668392&16.485825&-13.120&-80.101&16.486&365.051&5.185\\ \hline 0.139&7.196857&45.380278&16.491324&-13.529&-97.369&16.491&433.258&6.110\\ \hline 0.118&8.483429&48.092876&16.494692&-13.877&-117.723&16.495&513.659&7.200\\ \hline 0.100&10.000000&50.805908&16.496757&-14.172&-141.718&16.497&608.434&8.487\\ \hline 0.085&11.787686&53.519206&16.498025&-14.422&-170.001&16.498&720.152&10.003\\ \hline 0.072&13.894955&56.232831&16.498643&-14.634&-203.342&16.499&851.842&11.791\\ \hline 0.061&16.378937&58.946390&16.499128&-14.814&-242.642&16.499&1007.074&13.898\\ \hline 0.052&19.306977&61.660009&16.499432&-14.967&-288.969&16.499&1190.057&16.382\\ \hline 0.044&22.758459&64.373666&16.499625&-15.097&-343.577&16.500&1405.752&19.311\\ \hline 0.037&26.826958&67.087345&16.499751&-15.207&-407.947&16.500&1660.006&22.763\\ \hline 0.032&31.622777&69.801037&16.499835&-15.300&-483.825&16.500&1959.712&26.832\\ \hline 0.027&37.275937&72.514739&16.499894&-15.379&-573.267&16.500&2312.997&31.628\\ \hline 0.023&43.939706&75.228445&16.499938&-15.446&-678.699&16.500&2729.438&37.282\\ \hline 0.019&51.794747&77.942154&16.499972&-15.503&-802.978&16.500&3220.325&43.947\\ \hline 0.016&61.054023&80.656031&16.499836&-15.551&-949.475&16.500&3798.968&51.804\\ \hline 0.014&71.968567&83.369743&16.499861&-15.592&-1122.161&16.500&4481.054&61.065\\ \hline 0.012&84.834290&86.083456&16.499884&-15.627&-1325.718&16.500&5285.075&71.981\\ \hline 0.010&100.000000&88.797170&16.499905&-15.657&-1565.665&16.500&6232.831&84.849\\ \hline \end{array}}
Time now (at S=1) or present age in billion years: 13.753301 'T' in billion years (Gy) and 'D' in billion light years (Gly)
Look at the 7.6 billion year timeline you can see where the change over occcurs in the last columm.
this graph is for a theoretical galaxy at the edge of the observable universe today. or stretch 1090 which is the first row. the S row 1.000 is when the BB occurred at 10-43 seconds.
the a'Ro column shows the distance increase to the hypothetical galaxy. you can see the change in contraction and expansion related to its recessive velocity in S=1.648 or first column
the calculator is developed by forum members and is in the sticky or pinned thread above "look 88 years into the .." thread above.
http://www.einsteins-theory-of-relativity-4engineers.com/TabCosmo9.html
here is a quote from wiki that's appropriate
The metric expansion of space is the increase of the distance between two distant parts of the universe with time. It is an intrinsic expansion whereby the scale of space itself is changed. That is, a metric expansion is defined by an increase in distance between parts of the universe even without those parts "moving" anywhere. This is not the same as any usual concept of motion, or any kind of expansion of objects "outward" into other "preexisting" space, or any kind of explosion of matter which is commonly experienced on earth.
the keyword is scale of space itself is expanding
Time now (at S=1) or present age in billion years: 13.753301 'T' in billion years (Gy) and 'D' in billion light years (Gly)
Look at the 7.6 billion year timeline you can see where the change over occcurs in the last columm.
this graph is for a theoretical galaxy at the edge of the observable universe today. or stretch 1090 which is the first row. the S row 1.000 is when the BB occurred at 10-43 seconds.
the a'Ro column shows the distance increase to the hypothetical galaxy. you can see the change in contraction and expansion related to its recessive velocity in S=1.648 or first column
the calculator is developed by forum members and is in the sticky or pinned thread above "look 88 years into the .." thread above.
http://www.einsteins-theory-of-relativity-4engineers.com/TabCosmo9.html
here is a quote from wiki that's appropriate
The metric expansion of space is the increase of the distance between two distant parts of the universe with time. It is an intrinsic expansion whereby the scale of space itself is changed. That is, a metric expansion is defined by an increase in distance between parts of the universe even without those parts "moving" anywhere. This is not the same as any usual concept of motion, or any kind of expansion of objects "outward" into other "preexisting" space, or any kind of explosion of matter which is commonly experienced on earth.
the keyword is scale of space itself is expanding
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Mordred
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This wiki page has some decent coverage which I don't often say for wiki pages lol.
http://en.wikipedia.org/wiki/Metric_expansion_of_space
the images unfortunately do not show great detail in the slight contraction mentioned above.
http://en.wikipedia.org/wiki/Metric_expansion_of_space
the images unfortunately do not show great detail in the slight contraction mentioned above.
RByrne6722
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Try to imagine 3 orthogonal (mutually perpendicular) planes.
That’s pretty easy; the corner of any cube shows how 3 orthogonal planes meet. It defines the 3-dimensional (3-D) space we live in. Look at the corner of your room, 2 walls and the ceiling are 3 mutually perpendicular planes (in most normal rooms).
Only, we don’t live in a 3-D space. Time is a dimension. We appear to live in a 4-dimensional (4-D) space.
We can’t actually visualize 4 orthogonal planes. The 4 planes of ordinary space and time are mutually perpendicular, but we have 3-D eyes and can’t see or draw 4-D space. We can draw 2-D spaces and build 3-D spaces, but images beyond that are not possible (yet).
However, we can describe it mathematically. It’s not too difficult to do, just add a 4th dimension to the X, Y & Z Cartesian coordinates, call it T if you will.
Our 3-D (X,Y, Z) space is within this 4-D (X, Y, Z, T) space.
Wouldn’t our 3-D space be expanding within 4-D space? What are the boundaries of this 4-D space? What would constitute an edge in 4-D space?
What if the universe is actually 9, or 11 or 12 dimensional?
The expansion of 3-D (and 4-D etc.) space would be subsumed by these higher dimensions, wouldn’t it?
There's an edge to a sheet of paper, but I'm not so sure there's an edge to the Universe.
You might enjoy reading Flatland by Edwin Abbott
That’s pretty easy; the corner of any cube shows how 3 orthogonal planes meet. It defines the 3-dimensional (3-D) space we live in. Look at the corner of your room, 2 walls and the ceiling are 3 mutually perpendicular planes (in most normal rooms).
Only, we don’t live in a 3-D space. Time is a dimension. We appear to live in a 4-dimensional (4-D) space.
We can’t actually visualize 4 orthogonal planes. The 4 planes of ordinary space and time are mutually perpendicular, but we have 3-D eyes and can’t see or draw 4-D space. We can draw 2-D spaces and build 3-D spaces, but images beyond that are not possible (yet).
However, we can describe it mathematically. It’s not too difficult to do, just add a 4th dimension to the X, Y & Z Cartesian coordinates, call it T if you will.
Our 3-D (X,Y, Z) space is within this 4-D (X, Y, Z, T) space.
Wouldn’t our 3-D space be expanding within 4-D space? What are the boundaries of this 4-D space? What would constitute an edge in 4-D space?
What if the universe is actually 9, or 11 or 12 dimensional?
The expansion of 3-D (and 4-D etc.) space would be subsumed by these higher dimensions, wouldn’t it?
There's an edge to a sheet of paper, but I'm not so sure there's an edge to the Universe.
You might enjoy reading Flatland by Edwin Abbott
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Mordred
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welcome to the forum.
Your example while good on the 4 dimensions, lacks further explanation in regards to a finite, flat geometry. Take that piece of paper you just described and fold it into a Klein bottle. Now it has no edge.
http://www.math.osu.edu/~fiedorowicz.1/math655/Klein2.html
there are other possible flat shapes that do not have an edge.
torus
Moebius strip. are two other examples.
images and write up on universe geometry is discussed here.
http://abyss.uoregon.edu/~js/cosmo/lectures/lec15.html
as far as the universe being more than 4 dimensions is a string theory view point. Unfortunately string theory does not have a lot of successes. Its still a relatively new theory though.
Your example while good on the 4 dimensions, lacks further explanation in regards to a finite, flat geometry. Take that piece of paper you just described and fold it into a Klein bottle. Now it has no edge.
http://www.math.osu.edu/~fiedorowicz.1/math655/Klein2.html
there are other possible flat shapes that do not have an edge.
torus
Moebius strip. are two other examples.
images and write up on universe geometry is discussed here.
http://abyss.uoregon.edu/~js/cosmo/lectures/lec15.html
as far as the universe being more than 4 dimensions is a string theory view point. Unfortunately string theory does not have a lot of successes. Its still a relatively new theory though.
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Mordred, I believe the Moebius strip DOES have an edge, it just has a single surface so you don't to cross the edge to get to the other side as you would on an unfolded piece of paper which has two sides
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phinds said:Mordred, I believe the Moebius strip DOES have an edge, it just has a single surface so you don't to cross the edge to get to the other side as you would on an unfolded piece of paper which has two sides
It depends on how you define the Mobius strip. If you define it as a compact surface, then it will indeed have an edge (=manifold with boundary). But if you remove the edge, then you have some kind of Mobius strip which does not have an edge. If you want to work with manifolds without boundary, then you don't want this edge.
Mordred
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Wiki has a couple of examples
of mobius strips without boundaries.
http://en.m.wikipedia.org/wiki/Möbius_strip
scroll down to mobius band round section.
of mobius strips without boundaries.
http://en.m.wikipedia.org/wiki/Möbius_strip
scroll down to mobius band round section.
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Mordred said:Wiki has a couple of examples
of mobius strips without boundaries.
http://en.m.wikipedia.org/wiki/Möbius_strip
scroll down to mobius band round section.
Very interesting. Thanks.
Adrian07
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I am getting the impression that people don't believe time is real.
How can a universe expanding out from a singularity become anything other than at least roughly, spherical in shape, as far as I can see any other shape would require weird starting conditions.
How can a universe expanding out from a singularity become anything other than at least roughly, spherical in shape, as far as I can see any other shape would require weird starting conditions.
Boy@n
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From images it looks like observable Universe is far from expected 3D sphere, it looks more like 2D ellipse.Adrian07 said:I am getting the impression that people don't believe time is real.
How can a universe expanding out from a singularity become anything other than at least roughly, spherical in shape, as far as I can see any other shape would require weird starting conditions.
So, why is it flat?
mfb
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What you see is a projection of the whole sky on a surface (paper, or a computer screen). This does not work perfectly, and the problem is similar to the issue how to make a map of earth: you need a globe to do it right. An ellipse is a way to graph the data without too much deformation.
It is not a 3D-map.
It is not a 3D-map.
I don't think a torus or an infinite universe would be weird.How can a universe expanding out from a singularity become anything other than at least roughly, spherical in shape, as far as I can see any other shape would require weird starting conditions.
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Adrian07 said:I am getting the impression that people don't believe time is real.
.
HUH? Why do you say that?
Mordred
Science Advisor
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when we say the universe is flat, curved etc we are talking about a relation between the ratio of the universes energy-mass density \rho compared to the Hubble constant
this article shows the math involved.
http://www.ams.org/notices/199811/cornish.pdf
this article shows the math involved.
http://www.ams.org/notices/199811/cornish.pdf
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