From: goldenfieldquaternions@gmail.com   
      
   On Monday, July 15, 2019 at 12:53:14 AM UTC-5, Savin Beniwal wrote:   
   > Hi all   
   >   
   > I have questions regarding the Cosmological Principle that usually we   
   > study universe is SPATIALLY homogeneous and isotropic(around every   
   > point) at large scale (>150MPC).  Here homogenous means--> No special   
   > location and Isotropic means-->No special point. Also, this was   
   > confirmed by Hubble in 1929 that if distances are expanding (or   
   > contracting), the speed must be proportional to distance =E2=80=93 Hubble's   
   Law   
   > is inevitable.   
   >   
   > But my questions is that if there were a proportionality relation   
   > between velocity and square of distance rather than a linear relation   
   > between r and v. Even then can we understand the homogenous and   
   > isotropic concept from Hubble's law under this nonlinear relation?   
   >   
   > Thank you for your reply and discussion.   
   >   
   >   
   > With Regards!!!   
   > ----Savin(Darshan) Beniwal   
      
   The scale factor in FLRW cosmology expands as a(t) ~ a_0 exp(tH)   
   where H is the Hubble factor. Now take the derivative of this to   
   get   
      
   da/dt = Ha.   
      
   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.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)