Cosmology (Big Bang Theory, Dark Matter, Dark Energy)

Introduction to Cosmology
Cosmology is the study of the origin, evolution, and ultimate fate of the universe. It explores the largest-scale structures and dynamics, including fundamental questions about the universe’s beginnings, its current state, and its eventual destiny. Central topics in cosmology include the Big Bang Theory, dark matter, and dark energy, which together form the foundation of our understanding of the universe.

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The Big Bang Theory

The Big Bang Theory is the prevailing cosmological model that explains the early development of the universe. According to this theory, the universe began as a singularity approximately 13.8 billion years ago. This singularity rapidly expanded in an event known as the Big Bang, leading to the formation of all matter, energy, space, and time.

Key Concepts in the Big Bang Theory

1. Singularity: A point of infinite density and temperature where the laws of physics as we know them break down.
2. Cosmic Inflation: A brief period immediately following the Big Bang, during which the universe expanded exponentially.
3. Expansion of the Universe: The universe has been expanding ever since the Big Bang. The discovery that distant galaxies are moving away from us supports this theory.
4. Cosmic Microwave Background (CMB): The faint radiation left over from the Big Bang, providing evidence of the universe’s early hot and dense state.

Example:

Question: How do we calculate the temperature of the cosmic microwave background radiation using the age of the universe?

Answer:

Step 1: Given Data:

• Age of the universe $t=13.8,\text{billion years}$.
• The temperature of the universe at the Big Bang was around ${10}^{32},\text{K}$.

Step 2: Formula:
The temperature decreases as the universe expands:

$T=\frac{T_0}{\sqrt{t}}$

Where:

• $T$ is the current temperature,
• $T_0$ is the temperature at the Big Bang,
• $t$ is the age of the universe in billions of years.

Step 3: Solution:

$T=\frac{{10}^{32}}{\sqrt{13.8}}$

$T\approx \frac{{10}^{32}}{3.71}$

$T\approx 2.69\times {10}^{31},\text{K}$

Step 4: Final Answer:
The temperature of the cosmic microwave background is approximately $2.69\times {10}^{31},\text{K}$.

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Dark Matter

Dark matter is a form of matter that does not emit, absorb, or reflect light, making it invisible to current detection methods. However, its existence is inferred from its gravitational effects on visible matter, such as galaxies and galaxy clusters. Dark matter constitutes about 27% of the universe, vastly outnumbering regular matter, which makes up only about 5%.

Evidence for Dark Matter

1. Galaxy Rotation Curves: The observed rotation speeds of galaxies suggest that there is much more mass in galaxies than can be accounted for by visible matter alone.
2. Gravitational Lensing: The bending of light from distant objects by massive galaxy clusters indicates the presence of unseen matter.
3. Cosmic Microwave Background: The detailed patterns in the CMB provide indirect evidence of dark matter.

Example:

Question: How do we calculate the mass of dark matter in a galaxy using its rotational velocity?

Answer:

Step 1: Given Data:

• Rotational velocity $v=200,\text{km/s}$,
• Distance from the center $r=10,\text{kpc}$,
• Gravitational constant $G=6.674\times {10}^{-11},{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$.

Step 2: Formula:
The mass of dark matter $M$ can be calculated using:

$M=\frac{v^{2}r}{G}$

Step 3: Solution:

$M=\frac{(200\times {10}^3{)}^2\times (10\times 3.086\times {10}^{19})}{6.674\times {10}^{-11}}$

$M=\frac{4.00\times {10}^{10}\times 3.086\times {10}^{20}}{6.674\times {10}^{-11}}$

$M=1.85\times {10}^{41},\text{kg}$

Step 4: Final Answer:
The mass of dark matter in the galaxy is approximately $1.85\times {10}^{41},\text{kg}$.

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Dark Energy

Dark energy is a mysterious force that is driving the accelerated expansion of the universe. It constitutes about 68% of the universe and is uniformly distributed throughout space. Unlike dark matter, which pulls objects together through gravity, dark energy works in the opposite way by pushing them apart.

Evidence for Dark Energy

1. Accelerated Expansion: Observations of distant supernovae suggest that the universe’s expansion rate is increasing, which requires an unknown form of energy—dark energy.
2. Cosmic Microwave Background: The distribution of fluctuations in the CMB supports the presence of dark energy.
3. Large Scale Structure: The growth of large-scale structures in the universe is slower than expected, which can be explained by the repulsive force of dark energy.

Example:

Question: How do we calculate the effect of dark energy on the expansion of the universe using the cosmological constant?

Answer:

Step 1: Given Data:

• Cosmological constant $\mathrm{\Lambda }=1.1\times {10}^{-52},{\text{m}}^{-2}$,
• Energy density $\rho =7.6\times {10}^{-27},{\text{kg/m}}^3$,
• Gravitational constant $G=6.674\times {10}^{-11},{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$.

Step 2: Formula:
The effect of dark energy on the expansion rate can be calculated using:

$H^2=\frac{8\pi G\rho }{3}+\frac{\mathrm{\Lambda }}{3}$

Where $H$ is the Hubble constant, $\rho$ is the energy density, and $\mathrm{\Lambda }$ is the cosmological constant.

Step 3: Solution:

$H^2=\frac{8\times 3.14159\times 6.674\times {10}^{-11}\times 7.6\times {10}^{-27}}{3}+\frac{1.1\times {10}^{-52}}{3}$

$H^2=\frac{1.27\times {10}^{-36}}{3}+\frac{1.1\times {10}^{-52}}{3}$

$H^2=4.23\times {10}^{-37}+3.67\times {10}^{-53}$

$H\approx \sqrt{4.23\times {10}^{-37}}$

$H\approx 6.5\times {10}^{-19},{\text{s}}^{-1}$

Step 4: Final Answer:
The Hubble constant due to dark energy is approximately $6.5\times {10}^{-19},{\text{s}}^{-1}$.

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Conclusion

Cosmology seeks to answer some of the biggest questions in science, particularly those concerning the universe’s origin, its expansion, and its ultimate fate. The Big Bang Theory explains how the universe began and evolved, while dark matter and dark energy remain two of the most mysterious forces shaping its large-scale structure. Dark matter’s gravitational effects help explain galaxy formation, while dark energy drives the accelerated expansion of the universe. Together, these concepts form the core of modern cosmological studies, providing insight into the structure and future of the universe.

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Frequently Asked Questions (FAQs)

1. What is dark matter?
   Dark matter is a form of matter that does not emit light but is detected through its gravitational effects.
2. What is the Big Bang Theory?
   The Big Bang Theory explains the origin of the universe, describing its expansion from a hot, dense state about 13.8 billion years ago.
3. What is dark energy?
   Dark energy is a mysterious force that is causing the accelerated expansion of the universe.
4. What evidence supports the Big Bang Theory?
   Key evidence includes the cosmic microwave background radiation, the expansion of the universe, and the abundance of light elements.
5. What role does dark matter play in the universe?
   Dark matter provides the gravitational “glue” that holds galaxies and galaxy clusters together.