Here is an example from cosmobiophysics. ))

Me: From the point of view of a microbiologist, the Hubble equation describes the expansion of the Universe as an exponential growth phase of a galaxy colony on a two-dimensional boundary surface of a dark nutrient medium. The colony grows by multiplying galaxies through binary fission. The specific growth rate of a galaxy colony is equal to the Hubble constant. The initial radius of the colony is equal to the radius of a Milky Way-type galaxy from which the colony began to grow. The number of galaxies currently observed is about 10^11.

Give a numerical estimate of the generation time (doubling time) of galaxies.

How will the speed of the colony front depend on the colony radius?

GPT-4: To estimate the generation time (doubling time) of galaxies, we need to use the relationship between the specific growth rate r and the doubling time Td in exponential growth. The specific growth rate is given by the Hubble constant H0.

Step 1: Relation Between Growth Rate and Doubling Time

The specific growth rate r is related to the doubling time Td by the following formula:

Td = ln(2) / r

Given that the specific growth rate r is equal to the Hubble constant H0, we can rewrite this as:

Td = ln(2) / H0

Step 2: Determine the Value of H0

The Hubble constant H0 is commonly expressed in units of km/s/Mpc. However, for this calculation, we need to convert it to units of s^{-1}.

A commonly accepted value for H0 is approximately: H0 ~ 70 km/s/Mpc

To convert H0 to s^{-1}:

1 Mpc (megaparsec) ~ 3.086*10^19 km

H0 ~ 70 km/s / 3.086*10^19 km ~ 2.27*10^{-18} s^{-1}

Step 3: Calculate the Doubling Time

Using the formula:

Td = ln(2) / H0

Td = ln(2) / 2.27*10^{-18} s}^{-1} ~ 3.05*10^17 s

Step 4: Convert the Doubling Time to More Convenient Units

Convert seconds to years:

year ~ 3.154*10^7 s

Td ~ 3.05*10^17 s / 3.154*10^7 s ~ 9.67*10^9 years

So, the generation time (doubling time) of galaxies is approximately 9.67*10^9 years.

Step 5: Speed of the Colony Front

The speed of the colony front v is related to the radius of the colony R and the specific growth rate r. For a colony growing exponentially, the speed at which the boundary expands is given by:

v = r*R

Since the specific growth rate r is equal to H0:

v = H0*R

Therefore, the speed of the colony front is directly proportional to the radius of the colony, with the proportionality constant being the Hubble constant.

In summary:

- The generation time (doubling time) of galaxies is approximately 9.67*10^9 years.

- The speed of the colony front depends linearly on the colony radius, with v = H0*R.