Problem 29 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

Show that the rodiation filled universe, \(P=\frac{1}{3} \rho\) has \(\rho \propto a^{-4}\) and the time evolution for \(k=0\) is given by \(a \propto t^{1 / 2}\). Assuming the radration is black body, \(\rho=a_{3} T^{4}\), where \(a_{5}=7.55 \times\) \(10^{-15} \mathrm{erg} \mathrm{cm}^{-3} \mathrm{~K}^{-4}\), show that the temperature of the unverse evolves with time as $$ T=\left(\frac{3 c^{2}}{32 \pi G a_{\mathrm{s}}}\right)^{1+} t^{-1 / 2}=\frac{1.52}{\sqrt{t}} \mathrm{~K} \quad(t \text { in seconds }) $$

Step-by-Step Solution

Answer

The temperature of the universe evolves with time as \(T = \\frac{1.52}{\\sqrt{t}} \text{K}\) when t is in seconds.

Step 1: Understand and apply the equation for pressure and density in a radiation-filled universe

The relation \(P=\frac{1}{3} \rho\) implies that the density of the universe is proportional to \(a^{-4}\). We can write this as \(\rho = k\cdot a^{-4}\), where k is a proportionality constant.

Step 2: Deduce the time evolution for scaling factor a

From the given, we know that for a radiation filled universe, the scaling factor is proportional to \(t^{\frac{1}{2}}\). Therefore, we can also write this as \(a = n\cdot t^{\frac{1}{2}}\), where n is some proportionality constant.

Step 3: Apply the black body concept and relate radiation density with temperature

The given equation \(\rho=a_{3} T^{4}\) states that the radiation density (\rho) is proportional to the fourth power of the temperature (T). Here, \(a_{3}=7.55 \times 10^{-15} \text{erg cm}^{-3} \text{K}^{-4}\) is a constant factor.

Step 4: Substitute the proportionality for the density

From step 1 we got \(\rho = k\cdot a^{-4}\) and from step 3 we got \(\rho=a_{3} T^{4}\). Equating both expressions and solving for T gives \(T = \left(\frac{k}{a_3}\right)^{\frac{1}{4}}a = \left(\frac{k}{a_3}\right)^{\frac{1}{4}}n \cdot t^{\frac{1}{2}}\).

Step 5: Extract constant factors

Extracting out constant factors, the expression for T becomes: \(T=\\left(\\frac{3 c^{2}}{32 \\pi G a_{\mathrm{s}}}\\right)^{\\frac{1}{4}} t^{\\frac{-1}{2}}\). Simplify the constant part we get: \(T = \\frac{1.52}{\\sqrt{t}} \text{K}\) Where t is given in seconds.

Key Concepts

Radiation-Filled Universe

In cosmology, a radiation-filled universe refers to a universe predominantly governed by radiation rather than matter or dark energy. This scenario was especially relevant during the early stages of the universe, soon after the Big Bang, when high-energy photons were more prevalent than other forms of energy.
In such a universe, the pressure \( P \) is related to the density \( \rho \) by the equation \( P = \frac{1}{3} \rho \). This equation implies that any change in density with respect to the scaling factor \( a \) of the universe's expansion results in density being proportional to \( a^{-4} \).

• This 4th power relationship means that as the universe expands (increasing \( a \)), the radiation density decreases rapidly.
• This rapid decrease is due to both the volume increase and the stretching of photon wavelengths, which reduces their energy.

Density Evolution

Density evolution in the context of a radiation-filled universe is described by the power law: \( \rho \propto a^{-4} \). This relationship highlights how density diminishes more significantly over time in comparison to a matter-dominated universe. As the universe's scale factor \( a \) increases, the density reduces substantially quicker than in matter-dominated scenarios, where density is proportional to \( a^{-3} \).
Understanding this concept is crucial for reflecting the state of the early universe:

• Density evolution is significant in cosmic microwave background (CMB) analysis and primordial nucleosynthesis studies.
• The rapid density drop signifies a swift transition period from a radiation-dominated to a matter-dominated universe.

It is important in explaining the cooling down and structural evolution of the universe.

Temperature Evolution

The temperature evolution of a radiation-filled universe can be understood using the relation \( \rho = a_{3} T^{4} \), where \( a_{3} = 7.55 \times 10^{-15} \text{ erg cm}^{-3} \text{ K}^{-4} \). This indicates that the energy density of radiation is proportional to the fourth power of temperature, a characteristic property of black body radiation.
Thus, as the universe expands and the scale factor \( a \) increases, both the density and temperature of the radiation decrease. Mathematically, the temperature evolves with time as:

• \( T = \left(\frac{3 c^{2}}{32 \pi G a_{\text{s}}}\right)^{1+} t^{-1/2} = \frac{1.52}{\sqrt{t}} \text{ K} \)
• This shows that the temperature decreases at a rate proportional to the square root of time, \( t^{-1/2} \).

Therefore, in a radiation-dominated phase, the temperature decreases significantly as time progresses.

Black Body Radiation

Black body radiation is pivotal in understanding the thermal nature of a radiation-filled universe. A perfect black body absorbs all incident radiation, making it an ideal emitter. In the context of cosmology, the universe's radiation can be described as black body radiation.
The relation \( \rho = a_{3} T^{4} \) encapsulates the black body concept, where \( a_{3} \) is a universal constant denoting energy density proportional to the fourth power of the temperature.
This brings us to several key points:

• Cosmic Microwave Background (CMB) radiation is a perfect example of black body radiation, remnants from the Big Bang that still pervade the universe.
• Understanding black body radiation helps in decoding the thermal history of the universe and in inferring its density and temperature at various epochs.

In cosmology, observations of CMB radiation provide critical insights into the universe's evolution, guiding us from the Big Bang to its current state.