Chapter 26, Problem 46Q

(a)

To determine

The mass of neutrinos in the unit of kilograms and also as a fraction of the mass of the electron, if the Hubble constant is about $73\text{km/s/Mpc}$, critical density ${\rho }_c$ is $1\times {10}^{-26}{\text{kg/m}}^3$ and its average density is about $0.20{\rho }_c$. The average density of a neutrino in space is about $100{\text{neutrino/cm}}^3$.

Answer to Problem 46Q

Solution:

$2\times {10}^{-35}\text{kg, 2}\text{.1}\times {\text{10}}^{-5}$

Explanation of Solution

Given data:

The Hubble constant has a value of $73\text{km/s/Mpc}$.

The critical density has a value of $1\times {10}^{-26}{\text{kg/m}}^3$.

The average density of dark matter is $0.20{\rho }_c$.

The average density of neutrino throughout the space is $100{\text{neutrino/cm}}^3$.

Formula used:

The relation between the Hubble law and critical density is presented below.

${\rho }_c$= $\frac{3H_0{}^2}{8\pi G}$

In the equation, G is the universal constant whose value is $6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^{\text{2}}\text{/kg}$, $H_0$ is the Hubble constant and ${\rho }_c$ is the critical density.

Explanation:

Convert the value of $H_0$ into seconds.

$\begin{array}{c}{H}_0=\text{73}\text{km/s/Mpc}\\ =73\frac{\text{ km}}{\text{s}\cdot \text{Mpc}}\left(\frac{1\text{ Mpc}}{3.08\times {10}^{19}\text{ km}}\right)\\ =\text{2}\text{.37}\times {\text{10}}^{-18}{\text{s}}^{-1}\end{array}$

Recall, the expression for critical density.

${\rho }_c$= $\frac{3H_0{}^2}{8\pi G}$

Substitute $2.37\times {10}^{-18}{\text{ s}}^{-1}$ for $H_0$ and $6.67\times {10}^{-11}\text{ N}\cdot {\text{m}}^{\text{2}}\text{/kg}$ for G.

$\begin{array}{c}{\rho }_c=\frac{3{\left(2.37\times {10}^{-18}{\text{ s}}^{-1}\right)}^2}{8\pi \left(6.67\times {10}^{-11}\text{ N}\cdot {\text{m}}^{\text{2}}\text{/kg}\right)}\\ =\text{1}\times {\text{10}}^{-26}\text{}{\text{ kg/m}}^3\end{array}$

Average density of dark matter is $0.20{\rho }_c$.

${\rho }_{dm}=0.20{\rho }_c$

In the relation above, ${\rho }_{dm}$ is the average density of dark matter.

Substitute $\text{1}\times {\text{10}}^{-26}\text{}{\text{ kg/m}}^3$ for ${\rho }_c$.

$\begin{array}{c}{\rho }_{dm}=0.20\left(1\times {10}^{-26}{\text{ kg/m}}^{\text{3}}\right)\\ =\text{2}\times {\text{10}}^{-27}{\text{ kg/m}}^3\end{array}$

The average density of neutrino throughout the space is $100{\text{neutrino/cm}}^3$. So, the number of the neutrino in a cubic meter of space is obtained as follows.

$\begin{array}{c}n=100{\text{ cm}}^{-3}\left(\frac{{10}^6{\text{cm}}^3}{{\text{1 m}}^3}\right)\\ ={\text{10}}^8{\text{ m}}^{-3}\end{array}$

Therefore, the mass of the neutrino is obtained from the following calculation.

$m=\frac{{\rho }_{dm}}{n}$

Substitute $2\times {10}^{-27}{\text{kg/m}}^3$ for ${\rho }_{dm}$ and ${10}^8{\text{ m}}^{-3}$ for n.

$\begin{array}{c}m=\frac{2\times {10}^{-27}{\text{kg/m}}^3}{{10}^8{\text{ m}}^{-3}}\\ =2\times {10}^{-35}\text{ kg}\end{array}$

The mass in terms of mass of electron $\left(m_e\right)$ which has a value of $9.1\times {10}^{-31}\text{ kg}$. Thus, the above value calculated in terms of electron mass as presented below.

$\begin{array}{c}m=2\times {10}^{-35}\text{ kg}\left(\frac{1m_e}{9.1\times {10}^{-31}\text{ kg}}\right)\\ =2.2\times {10}^{-5}{\text{ m}}_e\end{array}$

Conclusion:

The mass of the neutrino in kilograms is $2\times {10}^{-35}\text{ kg}$ and as a fraction of mass of electron is $2.2\times {10}^{-5}$.

(b)

To determine

The reason due to which astronomers do not consider massive neutrinos as the prominent type of dark matter.

Answer to Problem 46Q

Solution:

The cold matter is the dominant form of dark matter. This suggests that massive neutrinos, which are hot, are not the dominant type of dark matter in the universe.

Explanation of Solution

Introduction:

Dark matter: Dark matter does not emit electromagnetic radiations and thus cannot be detected directly. It constitutes 80% of the universe. It is thought to be non-baryonic in nature which means it is composed of some undiscovered subatomic particles.

Explanation:

Example of hot dark matters are Neutrinos because they comprise of lightweight particles which travel at high speed. Cold matter, on the other hand, comprises of massive particles at low speed.

Massless neutrinos, like photons, always travel at the speed of light. However, neutrinos possess a small mass and the gravitational pull on surrounding matter. This graviational pull could lead to the formation of galaxies.

In cold matter, the formation of the galaxy takes place from bottom to up means the intersection between filaments occurs. These filaments merge together into galaxies. These galaxies are converted into clusters of galaxies, after which they become superclusters.

On the other hand, hot matter galaxies follow a top to bottom sequence of formation. Observation from the remote galaxies suggests that the formation of matter occurs by “bottom-up” scenario. Thus, hot matter such as massive neutrino is not responsible for formation of galaxies and hence is not the dominant form of dark matter.

Conclusion:

From this observation, it is concluded that the prominent form of dark matter is cold. This suggests that massive neutrinos (which are hot) do not constitute prominent type of dark matter present in the universe.