Question

How does Hubble’s Law change in an accelerating universe?

Answer 1

Hint: Hubble’s Law is considered the first observational basis for the explanation of the universe. Hubble Parameters directly proportional to density of the universe. Hubble’s Law is used to find the speed of galaxies that are moving away from earth and their speed is always proportional to their distance from the earth.

Complete answer:
Hubble’s Law is the observation in physics that galaxies are moving away from the earth at their speed which is proportional to their distances.
So According to Hubble's law, objects further away are moving faster than objects closer to us.
And velocity of the galaxy is determined by their red shift. So, Technically we can say that the redshifts of distant galaxies are proportional to their distances from us.
Mathematically Hubble’s Law can be written as-$$v=H_{0}D$$ $$-(Eq1)$$
Where,
v = Current recession velocity of the galaxy.
D = Current distance from galaxy.
H0 = Hubble Constant.
Mathematical expression for Hubble parameter and density of universe can be written as-
 $$H^2={\displaystyle \frac{8\pi G\rho }{3}}$$
Where,
H = Hubble Parameter.
ρ = Density of Universe.
Rest all other parameters are constant. So we can write this equation as
$$H^2=k\rho$$$$-(Eq2)$$
Where k is a constant whose value is$$k={\displaystyle \frac{8\pi G}{3}}$$.
So we can write that the Square of the Hubble Parameter is directly proportional to the density of the universe.
As objects move away from each other then the density of the universe will decrease, according to equation 2, if density decreases then Hubble parameter also decreases.
According to equation 1 when Hubble parameter decreases then velocity also decreases and we know that change in velocity will create acceleration.
So we can justify that expanding the universe will change the density of the universe. So Hubble law changes with the accelerating universe.

Note:
The Hubble Law is a consequence of the physics cosmological principle, i.e. that the universe on large scales is isotropic and homogeneous. This means that the expansion of the universe can be described by a single function of time, the so-called scale factor.