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Chapter 12: Q52E (page 558)

Show that the presence of a cosmological constant I friedmann equation must be , as R become very large, lead to an exponential expansion of the universe.

Short Answer

Expert verified

The universe will expand exponentially if it is dominated by the cosmological constant.

Step by step solution

01

Given data

The Friedmann equation can be expressed as:

${(\frac{d R / d t}{R})}^{2} = \frac{惟_{M}}{R^{3}} + 惟_{A} 鈭� \frac{K^{'}}{R^{2}}$

02

Concept of Friedmann equation

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

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

03

Calculate the derivative of the expression

The Friedmann equation is given by,

${(\frac{dR / dt}{R})}^{2} = \frac{惟_{M}}{R^{3}} + 惟_{A} 鈭� \frac{K^{'}}{R^{2}}$

As the $R$ is very large,

$\frac{惟_{M}}{R^{3}} 鈫� 0$ and $\frac{K^{'}}{R^{2}} 鈫� 0$

So, the equation becomes,

${(\frac{dR / dt}{R})}^{2} = 惟_{A}$

This means:

$\begin{matrix} \frac{dR / dt}{R} = const \\ \frac{dR}{dt} = (const) R \end{matrix}$

The solution to the above differential equation is:

$R 鈭� e^{(const) t}$

Thus, the universe will expand exponentially if it is dominated by the cosmological constant.

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Most popular questions from this chapter

Although not truly fundamental, the residual strong force shared by nucleons can also be represented by a feynmandiagram, with pion filling the role of mediating boson.The accompanying diagram represents a force between a neutron and a proton mediated by a $蟺^{0}$ boson.Suggest a diagram in which they exchange a $蟺^{-}$ particle.

Sketch the Feynman diagram if the proposed decay is possible.

$萤^{* -} 鈫� 危^{0} + e^{-} + v_{e}$

Sketch the Feynman diagram if the proposed decay is possible.

$K^{+} 鈫� 渭^{+} + {\overset{炉}{v}}_{渭}$

Sketch the Feynman diagram if the proposed decay is possible.

$惟^{-} 鈫� 萤^{0} + 蟺^{-}$

Suppose a force between two particles decreases distance according to $F = k / r^{b}$ . What is the limit on b if the energy required to separate the particlesInfinitely far is not to be infinite?

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