From: helbig@asclothestro.multivax.de   
      
   In article <88138831-7ffc-462d-900e-b69eac1ecbb4@googlegroups.com>,   
   Lawrence Crowell  writes:   
      
   > The scale factor in FLRW cosmology expands as a(t) ~ a_0 exp(tH)   
   > where H is the Hubble factor.   
      
   No, not in general.  FLRW means Friedmann-Lemaitre-Robertson-Walker.   
   Actually, Robertson-Walker is enough to answer the question: by   
   definition these are homogeneous and isotropic models, for which the   
   only possbible velocity-distance relation is a linear one.  The   
   Friedmann-Lemaitre models are based on general relativity and have   
   ordinary matter and the cosmological constant as components (whereby the   
   possibility that one or both of these is zero is also covered).  The   
   expansion law depends on the amounts of these components; in general,   
   the relative amounts also change with time.  Exponential expansion is   
   the case ONLY for no matter and a cosmological constant (and holds for   
   all times).  It is true that this is APPROXIMATELY true in OUR universe   
   NOW (and in the future will become more and more true, since as the   
   matter thins out due to expansion and the cosmological constant is,   
   errm, constant, asymptotically our universe will approach the so-called   
   de Sitter model of exponential expansion with (just) a cosmological   
   constant).   
      
   > 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/publication   
   /info/a_formula_for_confusion.html   
      
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