Our universe is filled with black body radiation (photons)...

Our universe is filled with black body radiation (photons) at a temperature $T=3 \mathrm{~K}$. This is thought to be a relic, of early developments following the "big bang". (a) Express the photon number density $n$ analytically in terms of $T$ and universal constants. Your answer should explicitly show the dependence on $T$ and on the universal constants. However, a certain numerical cofactor may be left in the form of a dimensionless integral which need not be evaluated at this stage. (b) Now estimate the integral roughly, use your knowledge of the universal constants, and determine $n$ roughly, to within about two orders of magnitude, for $T=3 \mathrm{~K}$.

Our universe is filled with black body radiation (photons) at a temperature $T=3 \mathrm{~K}$. This is thought to be a relic, of early developments following the "big bang".
(a) Express the photon number density $n$ analytically in terms of $T$ and universal constants. Your answer should explicitly show the dependence on $T$ and on the universal constants. However, a certain numerical cofactor may be left in the form of a dimensionless integral which need not be evaluated at this stage.
(b) Now estimate the integral roughly, use your knowledge of the universal constants, and determine $n$ roughly, to within about two orders of magnitude, for $T=3 \mathrm{~K}$.

Key Concepts

Cosmic Microwave Background (CMB)

The Cosmic Microwave Background is the relic radiation from the early universe, observable today as a nearly perfect blackbody spectrum at about 3 K. It provides a snapshot of the universe approximately 380,000 years after the Big Bang and is a cornerstone for modern cosmology, linking theoretical predictions from the Big Bang model with observational data.

Blackbody Radiation

Blackbody radiation describes the electromagnetic radiation emitted by an idealized object that absorbs all incident radiation, regardless of frequency or angle. Its spectrum depends solely on the temperature of the body and is characterized by a specific distribution function, which is fundamental for understanding thermal radiation processes in both laboratory settings and astrophysical phenomena.

Planck Distribution

The Planck distribution provides the mathematical formulation for the spectral energy density of blackbody radiation. It describes the number of photons present at each frequency for a given temperature, incorporating universal constants such as Planck's constant, the speed of light, and Boltzmann's constant. This distribution is essential for deriving quantities like energy density and photon number density by integrating over all frequencies.

Photon Number Density

Photon number density refers to the total number of photons per unit volume in a system and is obtained by integrating the Planck distribution over all photon frequencies. This concept connects statistical mechanics and quantum theory in the context of radiation fields and is critical in describing the properties of a photon gas, particularly in cosmological settings.

Dimensional Analysis and Universal Constants

Dimensional analysis involves using universal constants—such as the speed of light, Planck’s constant, and Boltzmann’s constant—to determine the scaling behavior and units of physical quantities. In the context of blackbody radiation and photon number density, these constants help express the temperature dependence and provide the correct physical dimensions for derived quantities, often leaving numerical factors in the form of dimensionless integrals.

Transcript

00:01 So in this problem, we're asked to find the ratio of the number of baryons to the number of photons in today's universe.

00:09 So we'll call the number of baryons today be for beryon, not for today, and the number of photons and gamma for photons, not for today.

00:19 So this is what we're looking for.

00:22 And we're given in the problem, the baryon energy density, will be not today.

00:30 And we're also given an expression for the energy density of black body photons.

00:45 So from these two things we need to find the number of baryons today and the number of photons today.

00:51 So we'll start with the number of baryons.

00:54 The baryon energy density can be written in terms of the number of baryons and the mass of the baryons.

01:05 Right, so this number of baryons today has units like number per volume.

01:13 And of course the mass units of mass energy so this becomes like mass per volume or energy per volume can energy density now we're also told in this problem that we can assume for convenience that the universe is made solely of hydrogen so what this means is that the mass of a baryon is just the mass of hydrogen which is about the mass of a proton right and we can look up the mass of the proton, it's about 1 .67 times 10 to the negative 24 grams.

01:52 So from here, we can just solve for the number of barions, the barion number density today.

02:02 That's just the barion energy density today divided by the mass of proton.

02:10 Of course, the barion energy density can also be written in terms of dimensionless parameter omega the barons today, multiplied by the critical energy density of the universe today.

02:29 So a reminder, this is just three times the hubble parameter squared over 8 pi g.

02:35 G is the gravitational constant.

02:38 You can just look this up in the textbook.

02:41 It's about .879 times 10 to negative 23, h squared, reduced hubble constant grams per meter cubed.

02:58 Alright, so now we have a nice expression for the number density of the barions in terms of a bunch of values that are known, or we can look up in the textbook.

03:19 So this is about 0 .04.

03:24 All right, so this is the numerator of the this ratio that we're trying to find.

03:31 The second thing we need to find is the number density in the photons.

03:36 So we start with the number density of blackbody photons it's given to us.

03:40 So i'll rewrite this down here and explain the expression a bit more.

03:52 So here, k is just the boltman constant.

03:58 So this is a value.

04:01 Also look up in the textbook.

04:02 If we want the photon energy density today, we need today's temperature photons or the cmb temperature...