From: goldenfieldquaternions@gmail.com   
      
   Technically you are right, and I was referring to de Sitter spacetime. I   
   was blurring the two sort of intentionally to show how Hubble law as a   
   linear law is "deformed" into a nonlinear one.   
      
   As for below, the nonlinear situation is what gives the departure from   
   13.8 billion years in time and 46 billion years in radius out to the CMB   
   with z = 1100.   
      
   > > Now take the derivative of this to   
   > > get   
   > >   
   > > da/dt = Ha.   
   >   
   > Yes, but the velocity is ALWAYS EXACTLY proportional to the distance in   
   > ANY FLRW model.   
   >   
   > > The actual distance is the scale factor times the "ruler" with some   
   > > unit distance x so the distance d is d = xa and with v = x dx/dt   
   > > we have v = Hd. That is the standard Hubble rule. However, in this   
   > > case d is based on an expanding scale and this lacks linearity, so   
   > > for d_0 = xa_0 we have   
   > >   
   > > v = Hd_0exp(tH).   
   > >   
   > > The time t = d_0/c and now Taylor expand   
   > >   
   > > v = Hd_0 + (Hd)^2/c + 1/2(Hd)^3/c^2 + ... .   
   > >   
   > > The rule v = Hd_0 is the linear rule that Hubble found. This is how   
   > > the expansion for sufficiently large distances, usually with z > 1,   
   > > is nonlinear.   
   >   
   > No.  This is not even wrong.  The velocity is always exactly   
   > proportional to the distance, but this regards the proper distance and   
   > its derivative.  Edward Harrison devoted an entire chapter in his   
   > excellent cosmology textbook to this:   
   >   
   > @BOOK         { EHarrison81a ,   
   >                AUTHOR       = "Edward R. Harrison",   
   >                TITLE        = "Cosmology, the Science of the Universe",   
   >                PUBLISHER    = CUP,   
   >                  YEAR       = "1981",   
   >                    ADDRESS    = "Cambridge (UK)"   
   >               }   
   >   
   > (Note that there is also a second edition, from 2000 I believe.)  He   
   > also wrote a paper detailing this:   
   >   
   > @ARTICLE      { EHarrison93a ,   
   >                AUTHOR       = "Edward R. Harrison",   
   >                TITLE        = "The Redshift-Distance and Velocity-Distance   
   >                                Laws",   
   >                JOURNAL      = APJ,   
   >                YEAR         = "1993",   
   >                  VOLUME       = "403",   
   >                  NUMBER       = "1",   
   >                  PAGES        = "28",   
   >                  MONTH        = jan   
   >               }   
   >   
   > Even professional astronomers get it wrong, as I pointed out here:   
   >   
   >    http://www.astro.multivax.de:8000/helbig/research/publicati   
   ns/info/a_formula_for_confusion.html   
      
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