Chapter 12: Q50E (page 558)

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

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 ${(\frac{dR / dt}{R})}^{2} = \frac{Ω_{M}}{R^{3}} + Ω_{A} - \frac{K^{'}}{R^{2}}$ .

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 $R = {(\frac{3}{2} (t - \frac{1}{3}))}^{2 / 3}$ .

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

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

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

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

Similarly, calculate further as shown below:

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

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

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

Short Answer

Step by step solution

Given data

Concept of Friedmann equation

Differentiate the expression

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Verify that if the universe were critical (ΩM=1). "flat”(K''=0), and free of a cosmological constant(ΩΛ=0), then equation dR/dtR2=1R3would be satisfied by a scale factorR(t) of[32t−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

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