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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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Most popular questions from this chapter

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 \(R_{1212}\).

(a) A particle falls radially inwards from rest at in finity in a Schwarzschild solution. Show that it will arrive at \(r=2 m\) m a finite proper time after crossing some fixed reference position \(r_{0}\), but that coordinate time \(t \rightarrow \infty\) as \(r \rightarrow 2 m\). (b) On an infalling extended body compute the tidal force in a radual direction, by parallel propagating a tetrad (only the radial spacelike unit vector need he considered) and calculating \(R_{1414}\). (c) Estimate the total tidal force on a person of height \(1.8 \mathrm{~m}\), weighing \(70 \mathrm{~kg}\), fallang head-first into a solar mass black hole \(\left(M_{3}=2 \times 10^{10} \mathrm{~kg}\right)\), as he crosses \(r=2 \mathrm{~m}\).

Consider an oscillator at \(r=r_{0}\) emitting a pulse of light (null geodesic) at \(t=t_{0}\). If this is received by an observer at \(r=r_{1}\) at \(t=t_{1}\), show that $$ t_{1}=t_{0}+\int_{r_{0}}^{r_{1}} \frac{d r}{c(I-2 m / r)} $$ By considering a signal emitted at \(t_{0}+\Delta t_{0}\), received at \(t_{1}+\Delta t_{1}\) (assuming the radial positions \(r_{0}\) and \(r_{1}\) to be constant), shou that \(t_{0}=t_{1}\) and the gravitational redsbift found by comparing proper times at cmission and reception is given by $$ 1+z=\frac{\Delta t_{1}}{\Delta \tau_{0}}=\sqrt{\frac{1-2 m / r_{1}}{1-2 m / r_{0}}} $$ Show that for two clocks at different heights \(h\) on the Earth's surface, this reduces to $$ z \approx \frac{2 G M}{c^{2}} \frac{h}{R} $$ where \(M\) and \(R\) are the mass and radius of the Earth.

Show that a space is locally flat if and only if there exists a local basis of vector ficlds \(\left\\{e_{t} \mid\right.\) that are absolutely parallel, \(D e_{1}=0\).

(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 $$ \frac{\partial \rho}{\partial t}+\nabla \cdot(\rho v)=0, \quad \rho \frac{d v}{d t}=-\nabla P-\rho \nabla \phi $$
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