The document discusses key concepts in cosmology, including: 1) The universe appears homogeneous and isotropic at large scales, with only small fluctuations in density. 2) Olbers' paradox explains why the night sky is dark if the universe is assumed to be infinite and unchanging, which may be incorrect assumptions. 3) The expanding universe model provides a solution to Olbers' paradox, as the universe had a hot, dense beginning and a finite age, limiting the number of observable stars. 4) The fate of the universe depends on its total density - if density exceeds a critical threshold, the universe will eventually collapse, while a lower density means continued expansion.

1. Introduction to modern astronomy 26
島袋隼⼠(Hayato Shimabukuro)（云南⼤学、
SWIFAR）
©GETTYIMAGES

2. 16.Cosmology（宇宙学）

3. The Cosmic principle（宇宙原理）
The properties of the universe
•Homogeneous（均匀的）
•Isotropic（各向同性的）
The cosmic principle(宇宙原理)
“Our universe does not have any special region”
Really??
•Our solar system seems to be special. The earth is a
special place!
“Our universe does not have any special region
at large scales”
The fluctuations in number density
are small.

4. 9 persons 8 persons 8 persons
8 persons 8 persons 8 persons
•The average number density is roughly ~8 persons. In many regions, the number density
is 8. But, a region has 9 people. This is fluctuation.
•Same situation happens at large scales in the universe. The number density in the universe is
almost the same as the average value everywhere at large scales.

5. The Cosmic principle（宇宙原理）
•If we observe the universe at small scales, we can find galaxies, galaxy clusters, and large-scale
structures of the universe. Thus, the universe is not isotropic and homogeneous.
•But, when we average the number density at large scales (~10Mpc), the number density
fluctuations are very small (the number density everywhere has almost average value).
The universe is very smooth.
The universe has no center!

6. Olbers’s paradox（佯谬）
•In addition that the universe is homogeneous and isotropic, we assume the universe is
infinite(⽆限) in spatial extent and unchanging in time.
r
L
F =
L
4πr2
:Brightness depends on the distance to the stars
•The number of stars ( ) also depends on
n r2
•Thus, the total brightness in the sky can be expressed by
Ftotal ∝
∫
∞
0
L
4πr2
⋅ r2
dr = ∞

7. Olbers’s paradox（佯谬）
Ftotal ∝
∫
∞
0
L
4πr2
⋅ r2
dr = ∞ The sky should be too bright!
•But, we know the actual sky is not bright.
Olbers’s paradox
Why is it dark in the night sky?
Our assumption
The universe is…
•Infinite in spatial extent.
•Unchanged in time.
These assumptions may be incorrect

8. Olbers’s paradox（佯谬）
Ftotal ∝
∫
∞
0
L
4πr2
⋅ r2
dr = ∞ The sky should be too bright!
•But, we know the actual sky is not bright.
Olbers’s paradox
Why is it dark in the night sky?
Our assumption
The universe is…
•Infinite in spatial extent.
•Unchanged in time.
These assumptions may be incorrect

9. The expanding universe（膨胀宇宙）
•In order to solve Olbers’s paradox, we first start Hubble’s law.
t =
v
r
=
1
H
(Velocity)=(constant) (distance)
×
time =
distance
velocity
•The inverse of the Hubble constant roughly
provides the age of the universe.
•Hubble constant is roughly 70km/s/Mpc.
This indicates the age of the universe is
roughly 14 billion years(140亿年)
The age of the universe is finite!
Age of the universe

10. The expanding universe（膨胀宇宙）
The age of the universe is finite!
The number of stars (galaxies) is also finite.
The age of the universe is finite, which means that the number of
galaxies that can be observed is finite and their ages are shorter
than the age of the universe. There are not enough stars
(galaxies) in the universe to brighten the night sky.
This is an answer to Olbers’s paradox.
The night sky is not too bright.

11. The expanding universe（膨胀宇宙）
t =
v
r
=
1
H
•The age of the universe is 14 billion years.
•The universe is expanding.
These facts suggest that our universe was small in the past.
•Smaller universe is hotter.
•The beginning of the universe was a
hot and dense fireball! This universe
is called Big Bang
•Big Bang happened everywhere. Big
Bang was not an explosion of the
universe.

12. Big Bang is not an explosion!

13. Big Bang happens everywhere
and
Big Bang does not have a center.
The universe was hot and dense everywhere

14. The expanding universe（膨胀宇宙）
•The behavior of the expansion is like a balloon and galaxies are located on the balloon.
Galaxies themselves do not expand.

15. The expanding universe（膨胀宇宙）
•The behavior of the expansion is like a balloon and galaxies are located on the balloon.
Galaxies themselves do not expand.

16. •We often use “redshift(红移)” to express distant or past（过去） universe.
Redshift
Redshift
Due to the expansion of the universe, the wavelength of the distant or past universe extends.
z =
0
0
Original wavelength
Extended wavelength
z=0: present universe
z=0.1: 1.4 Gyr ago
z=1: 7.8 Gyr ago
z=10: 13.2 Gyr ago
z=20: 13.7 Gyr ago
G:Giga, 10亿年

17. The fate of the cosmos
What happened in the universe's future? Will the universe expand forever?
(Big question in cosmology)
Let’s begin with another analogy.
Rocket escaping from the earth Expanding universe
Physics is similar!

18. The fate of the cosmos
Scenario 1
•A rocket leaving the Earth with a speed greater than the
Earth’s escape speed escapes from the earth.

19. The fate of the cosmos
•If the launch speed of the rocket is less than the escape
speed, the rocket eventually goes back to the earth and
falls.
Scenario 2

20. The fate of the cosmos
•The expansion of the universe also behaves like a rocket.
•The distance between two galaxies
as a function of time.
•The distance between two galaxies
expands forever or eventually
collapses.
Expand forever
Eventually
collapse

21. The fate of the cosmos
•In the case of a rocket, the speed of the rocket determines the fate of the rocket.
What determines the fate of the universe?
Remember that gravity pulls material.
The universe
•According to Einstein’s theory, gravity
is caused by material (energy, mass).
Thus, the fate of the universe depends on
the density of mass in the universe.
Critical density
ρcr =
3H0
8πG
= 9 × 10−27
kg/m3

22. The fate of the cosmos
•In the case of a rocket, the speed of the rocket determines the fate of the rocket.
What determines the fate of the universe?
Remember that gravity pulls material.
Gravity also pulls the universe itself.
The universe
•According to Einstein’s theory, gravity
is caused by material (energy, mass).
Thus, the fate of the universe depends on
the density of mass in the universe.
Critical density
ρcr =
3H0
8πG
= 9 × 10−27
kg/m3

23. The fate of the cosmos
ρ > ρcr
ρ < ρcr
(a)The universe turns to shrink and is collapsed (Big Crunch). We do not know what happened.
(b)The universe expands forever.