Steady-State Universe and Matter Creation

[ajl1989]

Homework Statement

Estimate the rate of continuous creation required to keep the density of the universe constant at 10^-29 g/cm³
Assume Hubble's constant H_o=70 km/s/Mpc

Homework Equations

I'm not sure if these are relevant, but for spherical shells: dV=4*(pi)*r²dr and N=4*(pi)*r²ndr, where N is the total number of particles in the shell and n is the density (#/unit volume)
Hubble's law: v=Hd or dr/dt=H*r

The Attempt at a Solution

So, given the mass of a proton, the density is approximately 6*10⁹ protons/km³.
I also converted Hubble's constant to 2.2*10⁹ km/year/Mpc.
I'm kind of stuck after that... I feel like there should be some easy way to figure out how much a volume changes per year, but that depends on how many Mpc away we are.
I feel like there's some simple equation that I'm missing...

 

[anathema]

I would first start by identifying the key information and variables given in the problem:

1. Density of the universe (ρ): 10^-29 g/cm^3
2. Hubble's constant (H0): 70 km/s/Mpc

Next, I would consider the concept of critical density (ρc), which is the density required for the universe to be flat. This can be calculated using the following equation:

ρc = (3H0^2)/(8πG)

Where G is the gravitational constant.

Using the given value of H0, we can calculate ρc to be approximately 9.2 x 10^-30 g/cm^3.

Since the density of the universe is given as 10^-29 g/cm^3, we can see that it is slightly higher than the critical density. This means that the universe is currently in a state of expansion.

To maintain a constant density, the rate of creation of matter must be equal to the rate of expansion. This can be calculated using the following equation:

ρ = (3H^2)/(8πG)

Where H is the Hubble parameter, which is equal to H0 at the present time.

Substituting the given values, we get:

10^-29 = (3(70)^2)/(8πG)

Solving for G, we get a value of 6.7 x 10^-11 m^3/kg*s^2.

This value of G can then be used in the equation for critical density to calculate the rate of continuous creation required to maintain the density of the universe at 10^-29 g/cm^3:

dρ/dt = (3H^2)/(8πG)

Substituting the values, we get a rate of approximately 1.3 x 10^-28 g/cm^3/year.

So, to summarize, the rate of continuous creation required to keep the density of the universe constant at 10^-29 g/cm^3 is approximately 1.3 x 10^-28 g/cm^3/year.

 

[nlightner]

I would approach this problem by first clarifying the concept of a steady-state universe and matter creation. In a steady-state universe, the density of matter remains constant over time, meaning that new matter is continuously created to replace the matter that is lost through the expansion of the universe. This idea is in contrast to the Big Bang theory, which suggests that the universe began with a singularity and has been expanding and cooling ever since.

In order to estimate the rate of continuous creation required to maintain a constant density of 10^-29 g/cm^3, we can use the Hubble's law equation: v=Hd or dr/dt=H*r. This equation relates the expansion rate of the universe (v) to the distance (d) and the Hubble's constant (H). We can rearrange this equation to solve for the rate of expansion (dr/dt) at a given distance.

Using the given value for Hubble's constant (Ho=70 km/s/Mpc), we can calculate the expansion rate at a distance of 1 Mpc (1 Mpc = 3.09*10^22 km) as follows:
dr/dt = 70 km/s/Mpc * 3.09*10^22 km = 2.17*10^24 km/year

Next, we can calculate the volume of a spherical shell with a radius of 1 Mpc using the equation dV=4*(pi)*r^2dr. This will give us the volume of new matter that needs to be created per year to maintain a constant density of 10^-29 g/cm^3.

dV = 4*(pi)*(3.09*10^22 km)^2 * 2.17*10^24 km/year = 2.13*10^97 km^3/year

To convert this volume into the number of protons, we can use the density formula n=N/V, where n is the density (#/unit volume), N is the total number of particles, and V is the volume. Rearranging this equation, we get N=n*V.

Using the given density of 6*10^9 protons/km^3 and the calculated volume, we can estimate the number of protons that need to be created per year to maintain a constant density of 10^-29 g/cm^3:

N = 6*10^9 protons/km