1. the Friedmann Equation Is Essentially a Statement of Conservation of Energy

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1. The Friedmann equation is essentially a statement of conservation of energy. Starting with a sphere of radius R ﬁlled with galaxies, write a statement of the total energy of a test galaxy of mass m that sits on the edge of the sphere. Solution: There is nothing unusual in the conservation of the total energy for isolated system, it comes to be a sum of the kinetic and potential components:

E = V + T,

dr 2 ( ) GMm where T = m dt is the kinetic eenrgy of trhe system, V = - is gravitational 2 r potential energy, thus dr 2 ( ) GMm E = m dt − . (1) 2 R 2. The scale factor is a time-dependent measure of cosmological distances. The Hubble rate, given by the following equation, describes how the scale factor changes as the universe evolves.

dr/dt H(t) = r Rewrite the conservation of energy equation of the sphere of galaxies to describe a homogeneous and isotropic expansion (or contraction) of the sphere over time. dr Solution: Deriving the ration dr/dt from the Hubble rate, we obtain = H(t) · r. Con- dt necting this obtained relation with the 1, we get the total energy expressed with the Hubble rate: H(t)2R2 GMm E = m − . (2) 2 R For the convenience we can also introduce in the 2 the total matter density of the universe as M ρ = 4 πR3 3 , thus the 2 becomes

H2(t)R2 4 mR2 8π mR2 (dR/dt)2 8π E = m − πGR2m = (H2(t) − Gρ) = ( − Gρ). (3) 2 3 2 3 2 R2 3

3. Rewrite the equation to consider a curvature parameter k? What is the physical meaning of k? Solution: The curvature parameter is deﬁned as −2E k = m and using the 3: 8π (dR/dt)2 k = R2( Gρ − ), (4) 3 R2

1 thus 3 can be expressed with the curvature parameter as:

(dR/dt)2 k 8π + = Gρ (5) R2 R 3

The physical meaning of this curvature parameter: k = +1, 0 or 1 depending on whether the shape of the universe is a closed 3-sphere, ﬂat (i.e. Euclidean space) or an open 3-hyperboloid, respectively. If k = +1, then R a is the radius of curvature of the universe. If k = 0, then R a may be ﬁxed to any arbitrary positive number at one particular time. If k = 1, then one can say that i·R a is the radius of curvature of the universe.

4. How does this equation relate to the Friedmann equation as given in class? In this case, what is the critical density, c? What is the physical meaning of the critical density? 8πG ρ − ρ H2(t) = [ρ(t) − cr 0 ] (6) 3 a2(t)

Solution: It is quite obvious: the last term stays caries the same purpose as the curvature related term in the 5. The critical density deﬁnes the spatial geometry of the Universe, it can be also illustrated with the relation: 8π ρ − ρ k = G cr 0 , (7) 3 R hence if the matter density equals to the critical, we have to deal with the ﬂat Universe. 5. What assumptions allow us to apply this conservation of energy statement to the universe? Solution: Homogeneous and isotropic, which is in general relevant to the large scale universe (100 Mpc or more).