Chapter 25, Problem 52Q

To determine

(a)

The proof that in the absence of a cosmological constant, the expansion of the universe must slow down.

Answer to Problem 52Q

The proof that the expansion of the universe must slow down in the absence of a cosmological constant is as stated below.

Explanation of Solution

Given:

The deceleration parameter $q_0$ is used to check whether the expansion of the universe is accelerating or decelerating.

The expansion of the universe is decelerating when $q_0$ has a positive value, the expansion of the universe is accelerating when $q_0$ has a negative value and the expansion rate is constant for $q_0=0$.

Formula Used:

The expression for the deceleration parameter is given by,

$q_0=\frac{1}{2}{\Omega }_0-\frac{3}{2}{\Omega }_{\text{A}}$

Here, ${\Omega }_0$ is the density parameter and ${\Omega }_{\text{A}}$ is the dark energy density parameter.

Calculation:

For the case when there is no cosmological constant, the dark energy density parameter is zero.

Consider table 25-2 of the book “Universe, Stars and Galaxies 6^th Edition”. The density parameter is ${\Omega }_0=1$.

The deceleration parameter is calculated as,

$\begin{array}{c}{q}_0=\frac{1}{2}{\Omega }_0-\frac{3}{2}{\Omega }_{\text{A}}\\ =\frac{1}{2}\left(1\right)-\frac{3}{2}\left(0\right)\\ =0.5\end{array}$

Since $q_0$ has a positive value, the universe expansion is slowing down.

Conclusion:

Therefore, it is proved that the universe expansion is slowing down in the absence of a cosmological constant.

To determine

(b)

The value of the deceleration parameter for the universe at present and to check whether the universe expansion is speeding up or slowing down.

Answer to Problem 52Q

The value of the deceleration parameter for the universe at present is $-0.64$, and thus, the universe expansion is speeding up.

Explanation of Solution

Given:

Consider table 25-2 of the book “Universe, Stars and Galaxies 6^th Edition”.

The matter density parameter is, ${\Omega }_{\text{m}}=0.24$.

The dark energy density parameter is, ${\Omega }_{\text{A}}=0.76$.

Formula Used:

The density parameter is given by,

${\Omega }_0={\Omega }_{\text{m}}+{\Omega }_{\text{A}}$

The expression for the deceleration parameter is given by,

$q_0=\frac{1}{2}\left({\Omega }_{\text{m}}+{\Omega }_{\text{A}}\right)-\frac{3}{2}{\Omega }_{\text{A}}$

Here, ${\Omega }_{\text{m}}$ is the matter density matter.

Calculation:

The deceleration parameter is calculated as,

$\begin{array}{c}{q}_0=\frac{1}{2}\left({\Omega }_{\text{m}}+{\Omega }_{\text{A}}\right)-\frac{3}{2}{\Omega }_{\text{A}}\\ =\frac{1}{2}\left(0.24+0.76\right)-\frac{3}{2}\left(0.76\right)\\ =-0.64\end{array}$

Since $q_0$ has a negative value, the universe expansion is .speeding up.

Conclusion:

The value of the deceleration parameter for the universe at present is $-0.64$, and thus, the universe expansion is speeding up.

To determine

(c)

The value of dark energy density parameter for a universe that has the same value of matter density parameter as our universe but the expansion of the universe is neither speeding up nor slowing down and to check whether matter or dark energy will be dominant in such a universe.

Answer to Problem 52Q

The value of the dark energy density parameter is $0.12$ and the matter will be dominant in such a universe.

Explanation of Solution

Given:

Consider table 25-2 of the book “Universe, Stars and Galaxies 6^th Edition”.

The matter density parameter is, ${\Omega }_{\text{m}}=0.24$.

The other universe is expanding at a constant rate; that is, $q_0=0$.

Formula Used:

The density parameter is given by,

${\Omega }_0={\Omega }_{\text{m}}+{\Omega }_{\text{A}}$

The expression for the deceleration parameter is given by,

$q_0=\frac{1}{2}\left({\Omega }_{\text{m}}+{\Omega }_{\text{A}}\right)-\frac{3}{2}{\Omega }_{\text{A}}$

Calculation:

The deceleration parameter is calculated as,

$\begin{array}{c}{q}_0=\frac{1}{2}\left({\Omega }_{\text{m}}+{\Omega }_{\text{A}}\right)-\frac{3}{2}{\Omega }_{\text{A}}\\ 0=\frac{1}{2}\left(0.24+{\Omega }_{\text{A}}\right)-\frac{3}{2}\left({\Omega }_{\text{A}}\right)\\ 0=0.12-{\Omega }_{\text{A}}\\ {\Omega }_{\text{A}}=0.12\end{array}$

Conclusion:

The value of the dark energy density parameter is $0.12$ and the matter will be dominant in such a universe.