Q50E Verify that if the universe were... [FREE SOLUTION]

Chapter 12: Q50E (page 558)

Verify that if the universe were critical $(惟_{M} = 1)$ . "flat鈥� $(K^{''} = 0)$ , and free of a cosmological constant $(惟_{蚂} = 0)$ , then equation ${(\frac{dR / dt}{R})}^{2} = \frac{1}{R^{3}}$ would be satisfied by a scale factor $R (t)$ of ${[\frac{3}{2} (t 鈭� \frac{1}{3})]}^{\frac{2}{3}} .$

Short Answer

Expert verified

The equation ${(\frac{dR / dt}{R})}^{2} = \frac{1}{R^{3}}$ is proved.

Step by step solution

Given data

The Friedmann equation can be expressed as ${(\frac{dR / dt}{R})}^{2} = \frac{惟_{M}}{R^{3}} + 惟_{A} 鈭� \frac{K^{'}}{R^{2}}$ .

Concept of Friedmann equation

If the universe were critical, flat and free of cosmological constant, the Friedmann equation is,

${(\frac{dR / dt}{R})}^{2} = \frac{1}{R^{3}}$ .

Differentiate the expression

The proposed solution is $R = {(\frac{3}{2} (t 鈭� \frac{1}{3}))}^{2 / 3}$ .

Differentiate the equation on both sides with respect to $t$ .

$\begin{array}{l} \frac{dR}{dt} = {(\frac{3}{2})}^{2 / 3} \frac{2}{3} {(t 鈭� \frac{1}{3})}^{鈭� 1 / 3} \\ \frac{dR}{dt} = {(\frac{3}{2} (t 鈭� \frac{1}{3}))}^{鈭� 1 / 3} \end{array}$

Substitute ${(\frac{3}{2} (t 鈭� \frac{1}{3}))}^{鈭� 1 / 3}$ for $\frac{dR}{dt}$ and ${(\frac{3}{2} (t 鈭� \frac{1}{3}))}^{2 / 3}$ for $R$ .

$\begin{matrix} {(\frac{d R / d t}{R})}^{2} = {[{(\frac{3}{2} (t 鈭� \frac{1}{3}))}^{鈭� 1 / 3} / {(\frac{3}{2} (t 鈭� \frac{1}{3}))}^{2 / 3}]]}^{2} \\ {(\frac{d R / d t}{R})}^{2} = {[{(\frac{3}{2} (t 鈭� \frac{1}{3}))}^{鈭� 1}]}^{2} \\ {(\frac{d R / d t}{R})}^{2} = {(\frac{3}{2} (t 鈭� \frac{1}{3}))}^{鈭� 2} \\ {(\frac{d R / d t}{R})}^{2} = {(\frac{3}{2} (t 鈭� \frac{1}{3}))}^{(\frac{2}{3}) (鈭� 3)} \end{matrix}$

Similarly, calculate further as shown below:

$\begin{array}{l} {(\frac{d R / d t}{R})}^{2} = R^{鈭� 3} \\ {(\frac{d R / d t}{R})}^{2} = \frac{1}{R^{3}} \end{array}$

Therefore, the equation ${(\frac{dR / dt}{R})}^{2} = \frac{1}{R^{3}}$ is proved.

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Step by step solution

Given data

Concept of Friedmann equation

Differentiate the expression

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91影视

Verify that if the universe were critical (惟M=1). "flat鈥�(K''=0), and free of a cosmological constant(惟蚂=0), then equation dR/dtR2=1R3would be satisfied by a scale factorR(t) of[32t鈭�13]23.

Short Answer

Step by step solution

Given data

Concept of Friedmann equation

Differentiate the expression

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Famous Physicists

Quantum Physics

Linear Momentum

Geometrical and Physical Optics

Electrostatics

Study anywhere. Anytime. Across all devices.