I'm attempting to solve an equation to find the critical density (Pc) of the universe; Pc=(3*H squared)/(8pi*G) This is found through the Friedmann equation which is derived from Einsteins' field equations. In order to solve this I need to determine a value for something called the Hubble parameter (H). This parameter is also the solution to the Friedmann equation and can be found using something called a scale factor (a(t)); a.(t) / a(t) = H where the dot represents a time derivative. For whatever reason I'm having a hard time comprehending the value for H. The definition states that a scale factor relates the comoving distances for an expanding universe with the distances at a reference time arbitrarily taken to be the present. Here is how I interpret this, please tell me if I am correct or not. If we measured the recession velocity of space at a distance of one mega parsec (Mpc) to be 70820.53 m/sec (universe being 13.8Gyr old) and then divide that by the distance of 1Mpc (3.262Mly which must be expressed as 3.083977973e22 meters), this will equal a value for H. Using this value we find the critical density of a 13.8Gyr old universe should be around 6.288e-27 kg per cubed meter This would mean that the Hubble parameter is a ratio between the recession velocity of space at a known distance (a value that changes with time) and the fixed distance of space (in this case 1 Mpc)? Just to be clear, H is not the ratio between the recession velocity measured today to that of the past? An example being that the recession velocity once equaled 70.8372 km/sec over 1 Mpc 3.262Myr ago but now equals 70.8205 km/sec over 1 Mpc giving the ratio of .9998 or -1.0002 between the present and 3.262Myr ago. Unless I'm mistaken this ratio is related to the cosmological red-shift and has little to do with the critical density of the universe. So, do I have the general concept correct or not?
You have a view of 13.7 billion light years. Maximum recessional speed is light speed. So you divide the maximum speed 186,282 mps by 13,700 (million light year lengths) between us and the "edge" which gives you a figure of just under 13.6 mps which is the believed rate of expansion for each million light year length of that area. Or another way of putting it. The universe is expanding as fast as it can and no more.
Man. I really know how to make something overly complicated. Let me try this again. The Hubble parameter is not the same thing as the Hubble constant yet you need to know one in order to find the other. What I am interested in is finding the critical density of the universe. To do this I must first find a value for the Hubble parameter by way of the Friedmann equation. One form of this equation requires the knowledge of the cosmological constant, the normalized spatial curvature of the universe and the actual density of the universe. Another form for this is called a scale factor which is how I chose to find the value for H. It is my understanding that a scale factor is not limited to this single concept. What all does a scale factor represent? If I understand the equation correctly, in this case all I need to do is divide the Hubble constant (time derivative) by the distance in which it is measured. Does that sound right?
