As you know, the universe is pervaded by 3 K black body...

As you know, the universe is pervaded by 3 K black body radiation. In a simple view, this radiation arose from the adiabatic expansion of a much hotter photon cloud which was produced during the big bang. (a) Why is the recent expansion adiabatic rather than, for example, isothermal? (b) If in the next $10^{10}$ years the volume of the universe increases by a factor of two, what then will be the temperature of the black body radiation? (Show your work.) (c) Write down an integral which determines how much energy per cubic meter is contained in this cloud of radiation. Estimate the result within an order of magnitude in joules per (meter) ${ }^3$.

As you know, the universe is pervaded by 3 K black body radiation. In a simple view, this radiation arose from the adiabatic expansion of a much hotter photon cloud which was produced during the big bang.
(a) Why is the recent expansion adiabatic rather than, for example, isothermal?
(b) If in the next $10^{10}$ years the volume of the universe increases by a factor of two, what then will be the temperature of the black body radiation? (Show your work.)
(c) Write down an integral which determines how much energy per cubic meter is contained in this cloud of radiation. Estimate the result within an order of magnitude in joules per (meter) ${ }^3$.

Key Concepts

Adiabatic Expansion

In cosmology, adiabatic expansion refers to a process during which the universe expands without exchanging heat with any external system. Because the universe is effectively isolated, its radiation cools due to the expansion; the photon energies get redshifted, which decreases the overall temperature, and no external thermal energy is added to maintain isothermal conditions.

Temperature Scaling in an Expanding Universe

For an adiabatically expanding photon gas, the temperature is inversely proportional to the scale factor (T ? 1/a), and since the volume of the universe scales as the cube of the scale factor (V ? a³), the temperature changes according to T ? V^(-1/3). This relationship allows one to compute the temperature change when the volume increases, such as finding the new temperature when the volume doubles.

Blackbody Radiation and Planck's Law

Blackbody radiation describes an idealized spectrum of electromagnetic radiation which is characterized solely by its temperature. Planck's law gives the spectral energy density as a function of frequency (or wavelength), and by integrating this function over all frequencies, one obtains the total energy per unit volume contained in the radiation.

Stefan-Boltzmann Law and Energy Density

The Stefan-Boltzmann law relates the total energy radiated per unit area of a blackbody to the fourth power of its temperature. In the context of energy density in the universe, integrating the Planck spectrum yields an energy density that is also proportional to T?, which is essential for estimating the magnitude of energy per cubic meter in the radiation field.

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