# Questions on Hubble's law and Friedmann equation

1. Jun 8, 2014

### kdlsw

The Hubble's Law v=H0D
LHS of The Friedmann equation H2/H02

I am a bit confused with the following:

1: The Hubble constant H0 is the current universe velocity/distance ratio, it is basically a constant, so galaxies in the universe with a further distance, move away from each other at a faster speed? so the universe is accelerating expending? is this statement correct?

2: Does the Hubble constant H0 represent the current velocity/distance ratio? if it is, then the meaning of Friedmann equation is to compare a different expanding rate H (for past or further or any other imaging situations) to our current H0, is this correct?

3:Rotation curve suggests that the circular velocity of galaxies are the same for any large value of distance r, doesn't that mean velocity/distance H0 is not constant?
Thanks

2. Jun 8, 2014

### Mordred

first off Hubble's constant isn't a constant,(it is a time dependant constant) v = H0D, with H0 the constant of proportionality (the Hubble constant) between the "proper distance" D to a galaxy (which can change over time, unlike the comoving distance) and its velocity v (i.e. the derivative of proper distance with respect to cosmological time coordinate.

for the evolution of Hubble's constant you need to also consider the scale factor a

$$H(t)=\frac{\stackrel{.}{a}(t)}{a(t)}$$ H0 is Hubble's constant and corresponds to the value of H (often termed the Hubble parameter which is a value that is time dependent and which can be expressed in terms of the scale factor) in the Friedmann equations taken at the time of observation denoted by the subscript 0. This value is the same throughout the universe for a given comoving time.

now as far as the velocity relations, think of it this way.

Hubble's law states the greater the distance, the greater the recessive velocity in other words the measured recessive velocity depends on the separation distance.

this article has a good coverage of the related misconceptions it will also better explain some of the relations your having difficulty with (its one of the best articles I've read in a straight easy to understand format, it by far beats my poor attempt lol http://cosmology101.wikidot.com/redshift-and-expansion)

far better article is this one
http://tangentspace.info/docs/horizon.pdf :Inflation and the Cosmological Horizon by Brian Powell

here is also a couple of textbook style articles with one now free textbook to help you
http://arxiv.org/pdf/hep-ph/0004188v1.pdf :"ASTROPHYSICS AND COSMOLOGY"- A compilation of cosmology by Juan Garcıa-Bellido
http://arxiv.org/abs/astro-ph/0409426 An overview of Cosmology Julien Lesgourgues
http://arxiv.org/pdf/hep-th/0503203.pdf "Particle Physics and Inflationary Cosmology" by Andrei Linde

Last edited: Jun 8, 2014
3. Jun 8, 2014

### timmdeeg

No, the Hubble law shows the proportionality between distance and recession velocity. Accelerated expansion means that the scale factor $a$ increases accelerated, which is the case if the second derivative of $a$ with respect to time is positive. According to the Friedmann acceleration equation

$$H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3}\rho - \frac{kc^2}{a^2}\dot{H} + H^2 = \frac{\ddot{a}}{a} = - \frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right)$$

the second derivative of $a$ is proportional to
$$- \left(\rho + \frac{3p}{c^2}\right)$$.
That shows how the amount of energy density and pressure determines the expansion of the universe.

Last edited: Jun 8, 2014