According to the cosmological principle, there is no preferred observer. There are two consequences of this 1) isotropy and 2) homogeneity. In a large enough scale (about 300 Mpc) the universe is homogeneous. Below that scale the matter distribution is not homogeneous. The universe is also isotropic: that is it looks similar in all directions. In a small scale, again,  there exists anisotropy. In the cosmic microwave background we have seen this anisotropy. Though CMBR has an unifirm  temperature of about 2.7 degree K, there exists anisotropy of the order of ΔT/T=10-5

If you ignore such small inhomogeneity and anisotropy, then you can conclude that there exists a large enough scale where there exists no preferred point in the  universe. Therefore it is not possible to choose an origin of space. In your question you state this as the given central point. According the the theory of inflation, in early universe indeed there was anisotropy and inhomogeneity in a primordial plasma. Therefore, space had undergone an exponential expansion which smoothed out initial inhomogeneity. However, the initial inhomogeneity served as  a seed of presently observed structures such as stars and galaxies. The anisotropy is still imprinted in the CMBR.

Notice that cosmological principle does not apply to space-time. There indeed exists a time which is regarded as t=0, that is the origin of time. The universe indeed looks different at different points of time.

For the other part of the question, by the term expansion rate we mean the Hubble expansion rate. According to the satellite based experiments such as PLANCK, the value of Hubble constant is, 67.80±0.77 (Km/sec)/Mpc. Prior to 1996 (when I was a Ph.D student) the hubble constant used to have large uncertainties in the range 50-90 (Km/sec)/Mpc.

Reference:  The First Three Minutes. A modern view of the origin of the universe, by Steven Weinberg.

In the standard relativistic models of the expanding Universe one cannot identify a particular point of expansion: it can be everywhere and nowhere. In a similar way, for an expanding spherical surface (balloon) the "center" of the sphere, where the "expansion started" is not any more a point of the 2D space of the sphere.  

According to the cosmological principle, there is no preferred observer. There are two consequences of this 1) isotropy and 2) homogeneity. In a large enough scale (about 300 Mpc) the universe is homogeneous. Below that scale the matter distribution is not homogeneous. The universe is also isotropic: that is it looks similar in all directions. In a small scale, again,  there exists anisotropy. In the cosmic microwave background we have seen this anisotropy. Though CMBR has an unifirm  temperature of about 2.7 degree K, there exists anisotropy of the order of ΔT/T=10-5

If you ignore such small inhomogeneity and anisotropy, then you can conclude that there exists a large enough scale where there exists no preferred point in the  universe. Therefore it is not possible to choose an origin of space. In your question you state this as the given central point. According the the theory of inflation, in early universe indeed there was anisotropy and inhomogeneity in a primordial plasma. Therefore, space had undergone an exponential expansion which smoothed out initial inhomogeneity. However, the initial inhomogeneity served as  a seed of presently observed structures such as stars and galaxies. The anisotropy is still imprinted in the CMBR.

Notice that cosmological principle does not apply to space-time. There indeed exists a time which is regarded as t=0, that is the origin of time. The universe indeed looks different at different points of time.

For the other part of the question, by the term expansion rate we mean the Hubble expansion rate. According to the satellite based experiments such as PLANCK, the value of Hubble constant is, 67.80±0.77 (Km/sec)/Mpc. Prior to 1996 (when I was a Ph.D student) the hubble constant used to have large uncertainties in the range 50-90 (Km/sec)/Mpc.

Reference:  The First Three Minutes. A modern view of the origin of the universe, by Steven Weinberg.

There is no meaning of Time until unless you specify the horizon. We are living on earth as sun is the horizon to us. Units and measurements of our time fixed with respect to sun. There is no meaning of expansion rate, why because according to life time of human being, we are unable to look and see the event occur supposed to every 1000 years practically, until unless our life time has to match the life time of what you looking for every 1000 years if it is the only variance.

In addition to that we don't have any evidence that how many end points exists for our universe. Suppose it is a multi verse theory and pocket universes, even though we don't have any evidence of particle which can pass through one pocket of universe to another make it into standard.

What we can observe is spacial expansion of  some constrained region only under the circumstances of light cone with time variance of sun as our horizon.

Still here a note to locally varying.... and so.. on.

If I assume that the expansion would originate from a location in deep space far from galactic influences, I don't know why you fix and choose the coordinates (shape, size) is spherical geometry to generate a function under the limits of light !

http://www.nature.com/nature/journal/v216/n5111/pdf/216148a0.pdf

http://www.nature.com/news/physics-bell-s-theorem-still-reverberates-1.15435#/impossible

As said above, there is no central point from which the rate of expansion of the universe could be described by a simple mathematical formula. However Hubble law (v=Ho.D) is a simple formula giving the recession velocities of galaxies "v" in function of their distance from earth "D". "Ho" is the Hubble constant nowadays. As mentioned by Robert J. Low and given in the wikipedia reference in his answer, the value of Ho is not known exactly, but, according to a few last measurements, it is thought to be of the order of 67-70 km/s.MPc. Maybe you would also be interested to read the (40) answers to the question of Nitin Jha on "Is Hubble constant really constant ? Or just a parameter ?"

https://www.researchgate.net/post/Is_Hubbles_Constant_really_a_constant_Or_just_a_parameter10

 Thanks all for your answers. The links were helpful, especially this one:

https://www.researchgate.net/post/I...ly_a_constant_Or_just_a_parameter10

which confronted the subject from numerous perspectives.

I don't know, but I accept that the Hubble-parameter is a parameter, It is not a constant.

Can you help me to understand that ?

Hi Charles, that link did not work.

here is an alternate view that i venture was not considered in that link :)

that the rate of expansion is zero.

the expansion rate will depend on the size of the universe and the speed of the expansion.

let us define the universe as the visible universe.

let us define the visible universe as the current radius distance to stars from which we can see light from.

per the standard model we see light from stars that were 13 B LY away 13 B years ago that are now 45B LY away, +/-

so the universe has been expanded 32 B LY in 13 B years for an average expansion speed of 2.46 the speed of light. 

Assume the speed of the expansion is steady, that means the expansion rate of the visible universe slows as the size of the visible universe increases. 

the formula for that solution is the answer i assume you were asking in the OP?

that calculation is moot if one holds by the Pearlman SPIRALL cosmological redshift hypothesis which concludes the universe has been somewhat static since the end of the inflation epoch.

 Roger, here's the corrected link as previously listed above by Guibert:

https://www.researchgate.net/post/Is_Hubbles_Constant_really_a_constant_Or_just_a_parameter10 

Hi Alapati,

The present Hubble parameter Ho is a constant in space. It is the proportionality factor between the recession speed of the observed galaxies and their distance. According to recent measurements, its value is 67-70 km/s.MPc. This value is the same no matter the direction where you are looking in space for your measurements nor the (far) distance of the galaxy. The proportionality factor between recession speed of galaxies and their distance is not constant in time. It could have been of the order of 300 km/s.MPc 12 billions years ago.   

Dear Sir (Guibert Crevecoeur),

The critical density for a flat universe depends only on Hubble's parameter H, which is not accurately known. Concepts of Physics, Sixth Edition, Eleventh Print 2005, McGraw-Hill, Arthur Beiser, Page No. 501 to 503, and also page No. 13-14. A reasonable value for H is 21 km/s per million light-years, which gives  8.9 x 10 -27 kg/m3.

                                    ρc =(3H2)/(8πG) 

Also a link  (PNAS | March 17, 2015 | vol. 112 | no. 11 | 3175)

t=0  condition -------------> Hubble Constant

t ≠ 0 condition ------------> Hubble's parameter. 

http://www.pnas.org/content/112/11/3173.full

Dear Alapati, I indeed suggest you to read the first answer by Gordon Fleming to the question "Is Hubble's constant really a constant or just a parameter ?" : see link above last given by Charles Wohl.  

79

I don't see a problem with calling it Hubble Constant - that is historical.  As mentioned, it is constant in space (same everywhere), but not necessarily in time.  I was surprised to see a few years ago, the announcement of a conference on "The Variations of the Solar Constant".  {Solar Constant is the flux received at earth from the sun}

The latest value for the Hubble constant is  73.8 ± 2.4 km/s/Mpc (pdg.lbl.gov.).

There should be relativistic corrections to

v=Hr , a=HHr,  but I dont know any.

"constant" vs "parameter"

Hubble parameter H=(da(t)/dt)/a(t). Where a(t) is the cosmological scale factor depending on time, defined by the relation d(t)=a(t) d0. Here d0 is the distance between two pints at a reference time t0, a(t) is the scale factor, and d(t) is the distance at time t.

Current experimental results suggest that the Hubble parameter is decreasing with time, whereas the universe is accelerating meaning that the second time derivative of a(t) is positive.

If I compare this Hubble Parameter with Rydburg Constant, What objections you made to answer to the discussion of Hubble Constant ?

Dear Biswajoy,

If "H" has two terms, one "c" constant both in time and space and the other one "b/t" only constant in space, i.e. if we have "H=c+b/t", one gets for an universe accelerating from 7 Gyrs after the Big Bang (second time derivative of a(t) positive) and with "b=2/3" (Einstein-de Sitter flat matter-dominated universe) : c ~ 21 km.s-1.Mpc-1. So the universe is indeed accelerating and the Hubble parameter is decreasing with time. One finds, e.g. : 

• at present time t0 ~ 14 Gyrs -> H0 ~ 67.5 km.s-1.Mpc-1 (in line with Planck mission results : 67.8+-0.77 km.s-1.Mpc-1)
• at time t ~ 16 Gyrs -> H ~ 61.7 km.s-1.Mpc-1

Kind regards

Hubble parameter gives the expansion rate which is constant for a given epoch. But it was associated with a very high energy at the beginning of the universe which rolled down during inflation to give the very low value of the  Hubble parameter at present epoch. In all other theories, there exists no relation between the Hubble parameter, the value of high initial energy at the beginning and the particle frequencies. The theory given below relates these values with each other for the first time and provides a single formula which yields precise values of masses of all the standard model fundamental particles. This theory is based on the periodic nature of time as opposed to linear time assumed by general relativity. So the rate of expansion of the universe continuously vary from the beginning of the universe to the present epoch. During inflation it is the highest.

Article Periodic quantum gravity and cosmology

The Hubble Constant is in terms of velocity divided by distance: (km/s)/Mpc.  dividing out distance gives 1/seconds.  Inverting, converting seconds to years, gives an accurate value for the age of the universe.  Thus the Hubble Constant seems a reasonable answer to the question.

The statement "space expands" is incomplete; the complete statement is "space expands in standard units"; or, in other words, space expands in relation to us. And this expansion is a scaling - it disappears if, instead or standard length unit, one uses another unit, the comoving unit. This means that the observed space expansion can be the consequence of we being decreasing - the phenomenon being a size decrease of matter instead of an expansion of space. This reminds us of the ancient heliocentric case - instead of a rotation of the universe around us, now an expansion of the universe in relation to us. Is it possible that also now it is we that vary, not the universe?

The answer is found here: http://vixra.org/abs/1107.0016.

It is rather clear, simply no one has ever analyzed the problem. Which raises an interesting problem for Psychology: a hypothesis that is against our empirical knowledge can never appear in our mind? The answer is: no! The possibility of the matter being decreasing should have been presented as soon as the scalar characteristic of space expansion was identified; however, no one presented it. This is a very important characteristic of human mind. Descartes defined the proper methodology to deal with it. And Descartes methodology is the key to solve all current problems of Physics, in Cosmology and other fields. However, it implies an all new organization of the scientific work.

Dear Charles,

The only answer is “uncertainty”.

e.g.: If someone claims that the age of the Universe is 13 or 15 billion years, every I&C engineer might ask: “Where in the Universe the unit of time which you use are defined?”

We can hardly imagine that, for example, the neutron stars have strong magnetic fields, between 108 and 1015 times that of Earth’s and the gravitational field on the surface of the neutron star is stronger 1011 times than that of the Earth. Obviously, the duration of one “second” in the extremely strong gravitational field on the neutron stars surface can be equal to the duration of years on the Earth’s surface. Therefore, the above-mentioned claim about the age of the Universe is meaningless... and corresponds to uncertainty.

We can discuss and other thinks, but the starting point I suggest to be the project “Collecting proofs about invalidity of the special theory relativity” started at ResearchGate.net.

Marry Christmas & Happy New Year!!!

The Hubble Constant uses time as defined here.  Incidentally, I use the special theory in radiation transport, and it produces very accurate results.

 @ Gocho

I can appreciate your examples about the duration of time as my thinking has evolved since I asked my original question. 

I'm sympathetic to your suggestion about the thread "Collecting proofs about invalidity of the special theory relativity" and I've gone down this route and continue to do so. I haven't settled upon a concrete belief one way or the other. Though, for me, evaluating so-called invalidity is certainly a way to study and gain a better understanding of the concepts. I spend a lot of time spawning and disproving my own little pet theories, and as soon as I put one to bed, another invariably pops up. And it appears that I have little control over the dynamics of this process; it's as though it just happens to me. Maybe I should seek professional help ;-)

The Special Theory is clearly valid over energies up to 10^15 eV..  I've used it for cosmic-ray transport up to that point.

Red shift of distant objects suggest the expansion is about half fast as light speed, and accelerating. You can get a lot of different opinions about the details and precision.

In all of the cases the universe is about 8 times the volume we observe due to the time of light travel to us.

A better estimate could be made if we could get some experimental data on a fast moving vehicle in deep space under prolonged acceleration testing the physical limits of general relativity and the high energy remedies that are offered. Then the cosmic models could be improved.

@ Jerry

Thanks, almost forgot I'd asked that question. How deep are we talking about? Would anybody alive now be around to read the returning data?

Put the question in light years, and the question answers itself.