## Universe size

thank you very much for your kind answer George. In fact, I was not particularly afraid by recession velocities higher than c given that this limit does not apply to space expansion, but I just wondered (for other reasons) if Vrec > c were really observed because H has been proven to be lower than H0 in the past (expansion accelerates)
I have an other question with respect to your answer: I wonder why SR is not sufficient for studying the geometry of expanding universe models, supposed to be isotropic and homogeneous as empty. General relativity deals with gravity?
 Recognitions: Gold Member Science Advisor SR does not take gravity into account. That was the whole point of GR.
 Yes. But precisely, all what I read about universe expansion and horizons, begins with a redefinition of the line element of SR (ds2=(cdt)2+a(t)2dl2)

Mentor
 Quote by denism Yes. But precisely, all what I read about universe expansion and horizons, begins with a redefinition of the line element of SR (ds2=(cdt)2+a(t)2dl2)
I am not sure what you mean. How is this necessarily a redefinition of the line element of SR?
 Indeed I was not clear. I just wanted to say that authors talking about light connection rates in the expanding universe never make use of GR, but only derive their conclusions from the simple relationship cdt=a(t)*dl (note there is a typing error of sign in my previous equation). SR seems to be sufficient. Gravity and GR are generally not involved in the universe model used in these studies. Furthermore, even SR seems to be not observed: calculations using speed substractions such as c-Vrec, rather resemble to classical mechanics ... even if I understood that Vrec is not a genuine speed.

 Quote by denism I would be very pleased seeing any supporting data! I did not want to be contrarian. Considering my poor knowledge in this field, I am sure that data (more convincing than the theoretical extension of Vrec = HD) should exist. This was precisely the purpose of my question
There's no evidence - yet - that there's anything far beyond the present horizon. An infinite flat space-time is just the simplest topology to assume. Alternative theories exist for a range of different space-time structures which are consistent with the data, some of which are finite and some are actually smaller than the current apparent horizon. The problem with assuming the limits of what we currently see are the actual physical limits is the implication we exist in a privileged moment of history - a possibility that needs a theoretical explanation that'd convince other cosmologists or they'll stick to their current assumptions.

The limits of what we can see are about x3 times the current Hubble light-travel-time. A good place to learn about all the different cosmological distances is Professor Ned Wright's Cosmology Tutorial...
Ned Wright's Cosmology Tutorial
...which features a handy Java calculator of all sorts of cosmic parameters. So handy that people have referenced it in their papers.

Many thanks for these informations

 Quote by qraal An infinite flat space-time is just the simplest topology to assume. .
does "infinite flat" means euclidean 3D?

Recognitions:
 Quote by denism does "infinite flat" means euclidean 3D?
"Flat" means Euclidean 3D. It need not be infinite.
 But how do you conceive a finite euclidean 3D space? with a peripheral boundary? looks strange

 Quote by denism But how do you conceive a finite euclidean 3D space? with a peripheral boundary? looks strange
That requires some tricky topology I'm guessing :-)

Recognitions:
 Quote by qraal That requires some tricky topology I'm guessing :-)
A torus.
 4D torus?

Recognitions:
 Quote by denism 4D torus?
You can think of it that way, yes. The best way to visualize it, though, is as a cube with opposite faces identified (by identified, I mean topologically connected -- for example, a circle is a line segment with the end points identified.)
 I have a naive perception of topology: the number of dimensions of any shape is that of the minimal euclidean space capable of embedding it. Your torus is 4D to me, even for something confined in its surface the circle is a good example: unidimensional if envisioned from the interior, but in fact genuinely bidimensionnal

Mentor
 Quote by denism Indeed I was not clear. I just wanted to say that authors talking about light connection rates in the expanding universe never make use of GR, but only derive their conclusions from the simple relationship cdt=a(t)*dl (note there is a typing error of sign in my previous equation).
$ds^2 = 0$ for a lightlike worldline in both special and general relativity.
 Quote by denism SR seems to be sufficient. Gravity and GR are generally not involved in the universe model used in these studies.
This is just plain wrong. In order to use the equation in your post above, the dependence of $a\left(t\right)$ on $t$ is needed. This is given by the solution of the differential equation

$$\left( \frac{da}{dt} \left(t\right) \right)^2 = H_0^2 \left( \Omega_{m0} a\left(t\right)^{-1} + \Omega_{r0} a\left(t\right)^{-2} + \Omega_{\Lambda 0} a\left(t\right)^2 + 1 - \Omega_{m0} - \Omega_{r0} - \Omega_{\Lambda 0} \right),$$
where the constants $\Omega_{m0}$, $\Omega_{r0}$, $\Omega_{\Lambda 0}$ are the current densities (relative to critical density) of matter, radiation, and dark energy, respectively. This equation comes from Einstein's equation of general relativity, i.e., it come form Einstein's theory of gravity.
 Quote by denism Furthermore, even SR seems to be not observed: calculations using speed substractions such as c-Vrec, rather resemble to classical mechanics ... even if I understood that Vrec is not a genuine speed.
With appropriate definitions of time and distance, c - V_rec is true in special relativity, and in the FRW cosmological models of general relativity

 Quote by George Jones $ds^2 = 0$ for a lightlike worldline in both special and general relativity.
sure, SR no way contradicts GR

 Quote by George Jones This is just plain wrong. In order to use the equation in your post above, the dependence of $a\left(t\right)$ on $t$ is needed. This is given by the solution of the differential equation $$\left( \frac{da}{dt} \left(t\right) \right)^2 = H_0^2 \left( \Omega_{m0} a\left(t\right)^{-1} + \Omega_{r0} a\left(t\right)^{-2} + \Omega_{\Lambda 0} a\left(t\right)^2 + 1 - \Omega_{m0} - \Omega_{r0} - \Omega_{\Lambda 0} \right),$$ where the constants $\Omega_{m0}$, $\Omega_{r0}$, $\Omega_{\Lambda 0}$ are the current densities (relative to critical density) of matter, radiation, and dark energy, respectively. This equation comes from Einstein's equation of general relativity, i.e., it come form Einstein's theory of gravity.

I disagree with you, the scale factor has first been naturally postulated because of the observation of Hubble. You describe one of the multiple a-posteriori attempts to calculate the expansion rate(s) from the universe constituents: (matter/energy and now the more exotic dark energy). These attempts are very interesting from a physical viewpoint but please do not inverse the string. a(t) did not emerge from matter/energy density calculations but was just postulated a-priori. To my knowledge its time-dependence has not been firmly established yet and it is likely to be underlain by different successive functions in the course of cosmic time

 Quote by George Jones With appropriate definitions of time and distance, c - V_rec is true in special relativity, and in the FRW cosmological models of general relativity
you are certainly right but this typically looks a Newtonian approach in Galilean coordinates: you know the celebrated thought experiment of Einstein, this approach would lead to the absurd conclusion that the speed of light emitted by a lamp in a train, depends on the speed of the train. Ironically, this is erroneous for the train but true for galaxies