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Chapter 18: Problem 29

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 $$ T=\left(\frac{3 c^{2}}{32 \pi G a_{\mathrm{s}}}\right)^{1+} t^{-1 / 2}=\frac{1.52}{\sqrt{t}} \mathrm{~K} \quad(t \text { in seconds }) $$

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.
04

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 \(T = \left(\frac{k}{a_3}\right)^{\frac{1}{4}}a = \left(\frac{k}{a_3}\right)^{\frac{1}{4}}n \cdot t^{\frac{1}{2}}\).
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

(a) For a perfect flud in general relat?uty, $$ T_{\mu v}=\left(\rho c^{2}+P\right) U_{\mu} U_{v}+P g_{\beta v} \quad\left(U^{\mu} U_{y}=-1\right) $$ show that the conservation identities \(T^{\mu v}, v=0\) imply \(\rho_{v} U^{v}+\left(\rho c^{2}+P\right) U_{v^{2}}^{*}\) \(\left(\rho c^{2}+P\right) U_{,}^{\mu} U^{v}+P_{v}\left(g^{\mu \prime}+U^{\mu} U^{\nu}\right)\) (c) In the Newtonian approximatuon where $$ U_{\mu}=\left(\frac{1}{c},-1\right)+O\left(\beta^{2}\right), \quad P=O\left(\beta^{2}\right) \rho c^{2}, \quad\left(\beta=\frac{v}{c}\right) $$ where \(|\beta| \ll 1\) and \(g_{\mu \nu}=\eta_{h^{x}}+\epsilon h_{\mu \nu}\) with \(\epsilon \ll 1\), show that $$ h_{+\mu} \approx-\frac{2 \phi}{c^{2}}, \quad h_{i j} \approx-\frac{2 \phi}{c^{2}} \delta_{j} \quad \text { where } \quad \nabla^{2} \phi=4 \pi G \rho $$ and \(h_{t 4}=O(\beta) h_{44}\) Show in this approxumation that the equations \(T^{\mu t},=0\) approvimate to $$ \frac{\partial \rho}{\partial t}+\nabla \cdot(\rho v)=0, \quad \rho \frac{d v}{d t}=-\nabla P-\rho \nabla \phi $$

Problem \(18.12\) (a) Show that in a pscudo-Riemannian space the action principle $$ \delta \int_{t_{1}}^{b_{2}} L, \mathrm{~d} t=0 $$ where \(L=g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}\) gives rise to geodesic equations with affine parameter \(t\). (b) For the sphere of radius \(a\) in polar coordinates, $$ \mathrm{ds}^{2}=a^{2}\left(\mathrm{~d} \theta^{2}+\sin ^{2} \theta d \phi^{2}\right) $$ use this variation principle to write out the equations of geodesics, and read off from them the Christoffel symbols \(\Gamma_{v \rho^{*}}^{\mu}\) where \(L=g_{k v} \dot{x}^{\mu} \dot{x}^{v}\) gives rise to geodesic equations with affine parameter \(t\). (b) For the sphere of radius \(a\) in polar coordinates, $$ \mathrm{ds}^{2}=a^{2}\left(\mathrm{~d} \theta^{2}+\sin ^{2} \theta \mathrm{d} \phi^{2}\right) $$ use this variation principle to write out the equations of geodesics, and read off from them the Christoffel symbols \(\Gamma_{i \rho^{*}}^{\mu}\) (c) Verify by direct substitution in the geodcsic equations, that \(L=\theta^{2}+\sin ^{2} \theta \dot{\phi}^{2}\) is a constant along the geodesics and use this to show that the general solution of the geodesic equations is grven by $$ b \cot \theta=-\cos \left(\phi-\phi_{0}\right) \text { where } b, \phi_{0}=\text { const. } $$ (d) Show that these curves are great circles on the sphere

(a) Compute the components of the Ricer tensor } R_{\mu v} \text { for a space-tume that has a }\end{array}\( metric of the form $$ \mathrm{d} s^{2}=\mathrm{dx}^{2}+\mathrm{d} v^{2}-2 \mathrm{~d} u \mathrm{~d} v+2 H \mathrm{~d} v^{2} \quad(H=H(\mathrm{x}, y, u, v)) $$(b) Show that the space-time is a vacuum if and only if \)H=\alpha(x, y, v)+f(v) u\( where \)f(v)\( is an arbitrary function and \)\alpha\( sat?sfies the two-dimensional Laplace equation $$ \frac{\partial^{2} \alpha}{\partial x^{2}}+\frac{\partial^{2} \alpha}{\partial y^{2}}=0 $$ and show that it is possible to set \)f(v)=0\( by a coordunate transformation \)u^{\prime}=u g(v), v^{\prime}=h(v)\(. (c) Show that \)R_{\text {taj } 4}=-H_{v}\( for \)i, j=1,2$.

Compute the Euler-Lagrange equations and energy-stress tensor for a scalar ficld Lagrangian in general relatwity given by $$ L_{5}=-\psi_{-\mu} \psi_{1} g^{n}-m^{2} \psi^{2} $$ Verify \(T^{n v},=0\)

If a Lagrangian depends on second and higher order derivatives of the fields, \(L=\) \(L\left(\Phi_{\lambda}, \Phi_{A \mu}, \Phi_{A, \mu \nu}, \ldots\right)\) derive the generalized Euler-Lagrange equations $$ \frac{\delta L \sqrt{-g}}{\delta \Phi_{A}} \equiv \frac{\partial L \sqrt{-g}}{\partial \Phi_{A}}-\frac{\partial}{\partial x^{\mu}}\left(\frac{\partial L \sqrt{-g}}{\partial \Phi_{A, \mu}}\right)+\frac{\partial^{2}}{\partial x^{\mu} \partial x^{\prime}}\left(\frac{\partial L \sqrt{-g}}{\partial \Phi_{A_{L} w}}\right)-\cdots=0 $$
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