Log out Go to App
Go to App

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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

Consider two radual light s?gnals (null geodesics) received at the spatial origin of coordinates at times \(t_{0}\) and \(t_{0}+\Delta t_{0}\), emitted from \(x=x_{1}\) (or \(r=r_{1}\) in the case of the flat models) at time \(t=t_{1}

(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$.

Let \((M, \varphi)\) be a surface of revolution defined as a submantfold of \(\mathbb{E}^{3}\) of the form $$ r=g(u) \cos \theta, \quad y=g(u) \sin \theta, \quad z=h(u) $$ Show that the induced metric (see Example 18.1) is $$ \mathrm{d} s^{2}=\left(g^{\prime}(u)^{2}+h^{\prime}(u)^{2}\right) \mathrm{d} u^{2}+g^{2}(u) \mathrm{d} \theta^{2} $$ Picking the parameter \(u\) such that \(g^{\prime}(u)^{2}+h^{\prime}(u)^{2}=1\) (interpret this choice!), and setting the basis 1-forms to be \(\varepsilon^{\prime}=\mathrm{d} u, \varepsilon^{2}=g d \theta\), calculate the connection I-forms \(\omega^{\prime}\), the curvature 1 -forms \(\rho_{\prime}^{\prime}\), and the curvature tensor component \(R_{1212}\).

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 $$ \frac{\delta L \sqrt{-g}}{\delta \Phi_{A}} \equiv \frac{\partial L \sqrt{-g}}{\partial \Phi_{A}}-\frac{\partial}{\partial x^{\mu}}\left(\frac{\partial L \sqrt{-g}}{\partial \Phi_{A, \mu}}\right)+\frac{\partial^{2}}{\partial x^{\mu} \partial x^{\prime}}\left(\frac{\partial L \sqrt{-g}}{\partial \Phi_{A_{L} w}}\right)-\cdots=0 $$
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free