Chapter 25, Problem 53Q

(a)

To determine

The value of $q_z$ at $z=0.5$, if the value of the deceleration parameter varies with time. It is given that the formula for the deceleration parameter at a redshift, z, for a flat universe, is $q_z=\frac{1}{2}-\frac{3}{2}\left[\frac{{\Omega }_{\Lambda }}{{\Omega }_{\Lambda }+\left(1-{\Omega }_{\Lambda }\right){\left(1+z\right)}^3}\right]$. Here, ${\Omega }_{\Lambda }$ is the dark energy density parameter. Use the value of ${\Omega }_{\Lambda }$ from table 25-2.

Answer to Problem 53Q

Solution:

$-0.22$

Explanation of Solution

Given data:

The value of the redshift is 0.5.

Using table 25-2, the value of ${\Omega }_{\Lambda }$ is found to be 0.76.

The formula for the deceleration parameter at redshift z is $q_z=\frac{1}{2}-\frac{3}{2}\left[\frac{{\Omega }_{\Lambda }}{{\Omega }_{\Lambda }+\left(1-{\Omega }_{\Lambda }\right){\left(1+z\right)}^3}\right]$.

Formula used:

The given formula is written as,

$q_z=\frac{1}{2}-\frac{3}{2}\left[\frac{{\Omega }_{\Lambda }}{{\Omega }_{\Lambda }+\left(1-{\Omega }_{\Lambda }\right){\left(1+z\right)}^3}\right]$

Here, ${\Omega }_{\Lambda }$ is the dark energy density parameter and $q_z$ is the deceleration parameter at the redshift, z.

Explanation:

Recall the provided formula.

$q_z=\frac{1}{2}-\frac{3}{2}\left[\frac{{\Omega }_{\Lambda }}{{\Omega }_{\Lambda }+\left(1-{\Omega }_{\Lambda }\right){\left(1+z\right)}^3}\right]$

Substitute 0.5 for z and 0.76 for ${\Omega }_{\Lambda }$.

$\begin{array}{c}{q}_z=\frac{1}{2}-\frac{3}{2}\left[\frac{.76}{.76+\left(1-.76\right){\left(1+.5\right)}^3}\right]\\ =\frac{1}{2}-\frac{3}{2}\cdot \left[\frac{.76}{1.57}\right]\\ =0.5-0.7261\\ =-0.22\end{array}$

Conclusion:

Therefore, the value of $q_z$ is $-0.22$.

(b)

To determine

The value of $q_z$ at $z=1$, if the value of the deceleration parameter varies with time. It is given that the formula for the deceleration parameter at the redshift, z, for a flat universe, is $q_z=\frac{1}{2}-\frac{3}{2}\left[\frac{{\Omega }_{\Lambda }}{{\Omega }_{\Lambda }+\left(1-{\Omega }_{\Lambda }\right){\left(1+z\right)}^3}\right]$. Here, ${\Omega }_{\Lambda }$ is the dark energy density parameter. Use the value of ${\Omega }_{\Lambda }$ from table 25-2.

Answer to Problem 53Q

Solution:

0.074

Explanation of Solution

Given data:

The value of the redshift is 1.

Using table 25-2, the value of ${\Omega }_{\Lambda }$ is found to be 0.76.

Formula used:

The provided formula is written as:

$q_z=\frac{1}{2}-\frac{3}{2}\left[\frac{{\Omega }_{\Lambda }}{{\Omega }_{\Lambda }+\left(1-{\Omega }_{\Lambda }\right){\left(1+z\right)}^3}\right]$

Here, ${\Omega }_{\Lambda }$ is the dark energy density parameter and $q_z$ is the deceleration parameter at the redshift, z.

Explanation:

Recall the provided formula.

$q_z=\frac{1}{2}-\frac{3}{2}\left[\frac{{\Omega }_{\Lambda }}{{\Omega }_{\Lambda }+\left(1-{\Omega }_{\Lambda }\right){\left(1+z\right)}^3}\right]$

Substitute 1 for z and 0.76 for ${\Omega }_{\Lambda }$.

$\begin{array}{c}{q}_z=\frac{1}{2}-\frac{3}{2}\left[\frac{.76}{.76+\left(1-.76\right){\left(1+1\right)}^3}\right]\\ =\frac{1}{2}-\frac{3}{2}\cdot \left[\frac{.76}{2.68}\right]\\ =0.5-0.4253\\ =0.074\end{array}$

Conclusion:

Therefore, the value of $q_z$ is 0.074.

(c)

To determine

An explanation for the result that shows that the expansion of the universe was decreasing at $z=1$. The expansion of the universe has changed from deceleration to acceleration between $z=1$ and $z=0.5$. It is given that the formula for the deceleration parameter at the redshift, z for a flat universe, is $q_z=\frac{1}{2}-\frac{3}{2}\left[\frac{{\Omega }_{\Lambda }}{{\Omega }_{\Lambda }+\left(1-{\Omega }_{\Lambda }\right){\left(1+z\right)}^3}\right]$. Here, ${\Omega }_{\Lambda }$ is the dark energy density parameter.

Explanation of Solution

Given data:

The values of the redshift are 1 and 0.5.

Using table 25-2, the value of ${\Omega }_{\Lambda }$ is found to be 0.76.

Formula used:

The provided formula is written as:

$q_z=\frac{1}{2}-\frac{3}{2}\left[\frac{{\Omega }_{\Lambda }}{{\Omega }_{\Lambda }+\left(1-{\Omega }_{\Lambda }\right){\left(1+z\right)}^3}\right]$

Here, ${\Omega }_{\Lambda }$ is the dark energy density parameter and $q_z$ is the deceleration parameter at the redshift, z.

Explanation:

The value of $q_z$ at $z=.5$ is $-0.22$, which is calculated in part (a) and value of $q_z$ at $z=1$ is 0.074, which is calculated in part (b). From this, it can be seen that the value at redshift .5 is negative and at redshift 1 is positive.

A positive value of $q_z$ indicates that the expansion of the universe is decelerated, which is at redshift 1. On the other hand, the value $q_z$ is negative at redshift 0.5, so the expansion of the universe is accelerated. So, it can be said that the universe must have accelerated between the redshifts, 1 and 0.5.

Conclusion:

Therefore, the universe accelerated between redshift 1 and 0.5.