Difference between revisions of "Homogeneous Universe"

From Universe in Problems
Jump to: navigation, search
Line 1: Line 1:
 
[[Category:Dynamics of the Expanding Universe|1]]
 
[[Category:Dynamics of the Expanding Universe|1]]
__NOTOC__
+
__TOC__
 
= Homogeneous and isotropic Universe, Hubble’s law =
 
= Homogeneous and isotropic Universe, Hubble’s law =
  
 
<div id="equ1"></div>
 
<div id="equ1"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1. ===
+
=== Problem 1: homogeniety vs isotropy ===
 
Most cosmological models are based on the assumption that the Universe is spatially homogeneous and isotropic. Give examples to show that the two properties do not automatically follow one from the other.
 
Most cosmological models are based on the assumption that the Universe is spatially homogeneous and isotropic. Give examples to show that the two properties do not automatically follow one from the other.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
Line 17: Line 17:
 
<div id="equ2"></div>
 
<div id="equ2"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 2. ===
+
=== Problem 2: global isotropy ===
 
Show that if some spatial distribution is everywhere isotropic then it is also homogeneous. Is the opposite true?.
 
Show that if some spatial distribution is everywhere isotropic then it is also homogeneous. Is the opposite true?.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
Line 29: Line 29:
 
<div id="equ3"></div>
 
<div id="equ3"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 3. ===
+
=== Problem 3: examples ===
 
What three-dimensional geometrical objects are both homogeneous and isotropic?
 
What three-dimensional geometrical objects are both homogeneous and isotropic?
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
Line 41: Line 41:
 
<div id="equ4"></div>
 
<div id="equ4"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 4. ===
+
=== Problem 4: on Big Bang ===
 
Why the notion of ''Big Bang'' regarding the early evolution of the Universe should not be treated too literally?
 
Why the notion of ''Big Bang'' regarding the early evolution of the Universe should not be treated too literally?
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
Line 53: Line 53:
 
<div id="equ5"></div>
 
<div id="equ5"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 5. ===
+
=== Problem 5: Hubble law Galilean invariance ===
 
Show that the Hubble's law is invariant with respect to Galilean transformations.
 
Show that the Hubble's law is invariant with respect to Galilean transformations.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
Line 68: Line 68:
  
 
<div id="equ6"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equ6"></div><div style="border: 1px solid #AAA; padding:5px;">
=== Problem 6. ===
+
=== Problem 6: Hubble law from homogeniety and isotropy ===
 
Show that the Hubble's law represents the only form of expansion compatible with homogeneity and isotropy of the Universe.
 
Show that the Hubble's law represents the only form of expansion compatible with homogeneity and isotropy of the Universe.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
Line 79: Line 79:
  
 
<div id="equ7"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equ7"></div><div style="border: 1px solid #AAA; padding:5px;">
=== Problem 7. ===
+
=== Problem 7: homogeniety conservation ===
 
Show that if expansion of the Universe obeys the Hubble's law then the initial homogeneity is conserved for all its subsequent evolution.
 
Show that if expansion of the Universe obeys the Hubble's law then the initial homogeneity is conserved for all its subsequent evolution.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
Line 97: Line 97:
  
 
<div id="equ9"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equ9"></div><div style="border: 1px solid #AAA; padding:5px;">
=== Problem 8. ===
+
=== Problem 8: stationary model of the Universe ===
 
In the 1940-ties Bondi, Gold and Hoyle proposed a stationary model of the Universe basing on the generalized cosmological principle, according to which there is no privileged position either in space or in time. The model describes a Universe, in which all global properties and characteristics (density, Hubble parameter and others) remain constant in time. Estimate the rate of matter creation in this model.
 
In the 1940-ties Bondi, Gold and Hoyle proposed a stationary model of the Universe basing on the generalized cosmological principle, according to which there is no privileged position either in space or in time. The model describes a Universe, in which all global properties and characteristics (density, Hubble parameter and others) remain constant in time. Estimate the rate of matter creation in this model.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
Line 123: Line 123:
  
 
<div id="equ10"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equ10"></div><div style="border: 1px solid #AAA; padding:5px;">
=== Problem 9. ===
+
=== Problem 9: Hubble flow and peculiar velocities ===
 
Galaxies typically have peculiar (individual) velocities of the order of $V_p \approx 100~\mbox{km/s}.$ Estimate how distant a galaxy should be for its peculiar velocity to be negligible compared to the velocity of Hubble flow $V_H=H_{0}R$.
 
Galaxies typically have peculiar (individual) velocities of the order of $V_p \approx 100~\mbox{km/s}.$ Estimate how distant a galaxy should be for its peculiar velocity to be negligible compared to the velocity of Hubble flow $V_H=H_{0}R$.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
Line 134: Line 134:
  
 
<div id="equ11"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equ11"></div><div style="border: 1px solid #AAA; padding:5px;">
=== Problem 10. ===
+
=== Problem 10: age of the Universe ===
 
Estimate the age of the Universe basing on the observed value of the Hubble's constant (the Hubble time $t_H$).
 
Estimate the age of the Universe basing on the observed value of the Hubble's constant (the Hubble time $t_H$).
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
Line 145: Line 145:
  
 
<div id="equ13"></div><div style="border: 1px solid #AAA; padding:5px;">
 
<div id="equ13"></div><div style="border: 1px solid #AAA; padding:5px;">
=== Problem 11. ===
+
=== Problem 11: Olbers' paradox resolution ===
 
Show that the model of the expanding Universe allows one to eliminate the Olbers' paradox.
 
Show that the model of the expanding Universe allows one to eliminate the Olbers' paradox.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">

Revision as of 11:44, 1 August 2012

Homogeneous and isotropic Universe, Hubble’s law

Problem 1: homogeniety vs isotropy

Most cosmological models are based on the assumption that the Universe is spatially homogeneous and isotropic. Give examples to show that the two properties do not automatically follow one from the other.


Problem 2: global isotropy

Show that if some spatial distribution is everywhere isotropic then it is also homogeneous. Is the opposite true?.


Problem 3: examples

What three-dimensional geometrical objects are both homogeneous and isotropic?


Problem 4: on Big Bang

Why the notion of Big Bang regarding the early evolution of the Universe should not be treated too literally?


Problem 5: Hubble law Galilean invariance

Show that the Hubble's law is invariant with respect to Galilean transformations.


Problem 6: Hubble law from homogeniety and isotropy

Show that the Hubble's law represents the only form of expansion compatible with homogeneity and isotropy of the Universe.


Problem 7: homogeniety conservation

Show that if expansion of the Universe obeys the Hubble's law then the initial homogeneity is conserved for all its subsequent evolution.


Problem 8: stationary model of the Universe

In the 1940-ties Bondi, Gold and Hoyle proposed a stationary model of the Universe basing on the generalized cosmological principle, according to which there is no privileged position either in space or in time. The model describes a Universe, in which all global properties and characteristics (density, Hubble parameter and others) remain constant in time. Estimate the rate of matter creation in this model.


Problem 9: Hubble flow and peculiar velocities

Galaxies typically have peculiar (individual) velocities of the order of $V_p \approx 100~\mbox{km/s}.$ Estimate how distant a galaxy should be for its peculiar velocity to be negligible compared to the velocity of Hubble flow $V_H=H_{0}R$.


Problem 10: age of the Universe

Estimate the age of the Universe basing on the observed value of the Hubble's constant (the Hubble time $t_H$).


Problem 11: Olbers' paradox resolution

Show that the model of the expanding Universe allows one to eliminate the Olbers' paradox.