Difference between revisions of "Homogeneous Universe"

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= Homogeneous and isotropic Universe, Hubble’s law =
 
= Homogeneous and isotropic Universe, Hubble’s law =
  
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=== Problem 1. ===
 
=== Problem 1. ===
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A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous.
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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.
 
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=== Problem 2. ===
 
=== Problem 2. ===
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Due to isotropy the distribution must coincide in any two points, because they can be transformed to each other by rotation by $180^\circ$ around the middle of the interval connecting them. The opposite is not true (see [[#Problem_1. | problem #1]]).
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Show that if some spatial distribution is everywhere isotropic then it is also homogeneous. Is the opposite true?.
 
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=== Problem 3. ===
 
=== Problem 3. ===
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There are three cases: three-dimensional plane (zero curvature), sphere (positive curvature) and hyperboloid (negative curvature).
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What three-dimensional geometrical objects are both homogeneous and isotropic?
 
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=== Problem 4. ===
 
=== Problem 4. ===
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The main qualitative difference from the usual explosion lies in the fact that the explosive charge is usually surrounded by atmospheric air. The expansion is then caused by the difference between the huge pressure of the gaseous products of the explosion and comparatively small pressure of the surrounding air. But when considering the expanding Universe, one assumes that the pressure (according to the cosmological principle) is uniformly distributed too. Therefore there are neither pressure gradients nor forces that could cause or even affect the expansion. The expansion of the Universe itself is the result of initial velocity distribution.
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Why the notion of ''Big Bang'' regarding the early evolution of the Universe should not be treated too literally?
 
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=== Problem 5. ===
 
=== Problem 5. ===
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Suppose the Hubble's law holds in the reference frame $K$, in which matter is at rest at the origin. Consider another frame $K'$, with origin at some point $A$, moving with velocity $\vec{v}_{A}$ relative to $K$. The quantities measured in the latter frame are denoted by primes. Then in $K'$ one has <math>\vec{r}' = \vec{r} - {\vec{r}_A}</math> and
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Show that the Hubble's law is invariant with respect to Galilean transformations.
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\[\vec{v}' = \vec{v} - {\vec{v}_A}
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= H\vec r' = H\vec r - H{\vec r_A}
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= H\left( {\vec r - {{\vec r}_A}} \right) = H\vec r'.\]
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Therefore the distribution law for velocities in the new frame has the same form with the same value of Hubble's parameter as in the original one.
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=== Problem 6. ===
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Show that the Hubble's law represents the only form of expansion compatible with homogeneity and isotropy of the Universe.
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"> A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous. </p>
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  </div>
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</div>
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<div id="equ7"></div>
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=== Problem 7. ===
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Show that if expansion of the Universe obeys the Hubble's law then the initial homogeneity is conserved for all its subsequent evolution.
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"> A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous. </p>
 +
  </div>
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</div>
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 +
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<div id="equ9"></div>
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=== Problem 8. ===
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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.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"> A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous. </p>
 +
  </div>
 +
</div>
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<div id="equ10"></div>
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=== Problem 9. ===
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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 $V_H=H_{0}R$.
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<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"> A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous. </p>
 +
  </div>
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</div>
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<div id="equ11"></div>
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=== Problem 10. ===
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Estimate the age of the Universe basing on the observed value of the Hubble's constant (the Hubble time $t_H$).
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<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"> A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous. </p>
 +
  </div>
 +
</div>
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<div id="equ13"></div>
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=== Problem 11. ===
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Show that the model of the expanding Universe allows one to eliminate the Olbers' paradox.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"> A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous. </p>
 +
  </div>
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</div>
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 +
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Revision as of 19:37, 28 May 2012


Homogeneous and isotropic Universe, Hubble’s law

Problem 1.

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.

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


Problem 3.

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


Problem 4.

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


Problem 5.

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


Problem 6.

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


Problem 7.

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.

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.

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.

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


Problem 11.

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



















Problem 4. The main qualitative difference from the usual explosion lies in the fact that the explosive charge is usually surrounded by atmospheric air. The expansion is then caused by the difference between the huge pressure of the gaseous products of the explosion and comparatively small pressure of the surrounding air. But when considering the expanding Universe, one assumes that the pressure (according to the cosmological principle) is uniformly distributed too. Therefore there are neither pressure gradients nor forces that could cause or even affect the expansion. The expansion of the Universe itself is the result of initial velocity distribution.


Problem 5. Show that the Hubble's law is invariant with respect to Galilean transformations.


Problem 6. Show that the Hubble's law represents the only form of expansion compatible with homogeneity and isotropy of the Universe.


Problem 7. 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. 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. Galaxies typically have peculiar (individual) velocities of the order of $V_p \approx 100~\mbox{\it km/s}.$ 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. Estimate the age of the Universe basing on the observed value of the Hubble's constant (the Hubble time $t_H$).


Problem 11. Show that the model of the expanding Universe allows one to eliminate the Olbers' paradox.
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