The Cosmological Constant

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Problem 1

Derive the Einstein equations in the presence of the cosmological constant by variation of the gravitational field action with the additional term \[S_\Lambda=-\frac{\Lambda}{8\pi G} \int\sqrt{-g}\;d^4x.\]

Problem 2

Show that the $\Lambda$-term is a constant (the cosmological constant): $\partial_\mu\Lambda=0$.

Problem 3

Derive Friedman equations in the presence of the cosmological constant.

Problem 4

Consider the case of two-component Universe with arbitrary curvature filled by non-relativistic matter and dark energy in the form of cosmological constant. Show that in such case the first Friedman equation can be presented in terms of dimensionless variables $\bar a\equiv a/a_0$, $\Omega_m,\Omega_\Lambda$ and $\tau\equiv H_0 t)$ in the following way: \[\left(\frac{d\bar a}{d\tau}\right)^2=1+\Omega_{m0}\left(\frac{1}{\bar a}-1\right)+\Omega_{\Lambda0}({\bar a}^2-1).\]

Problem 5

Represent the first Friedman equation in terms of intrinsic Gaussian curvature of the three-space \[K(t)=K_0/a^2(t).\]

Problem 6

Analyze the contribution of different energy components to the intrinsic Gaussian curvature of the three-space.

Problem 7

Find equation for the scale factor in a two-component Universe filled with matter with equation of state $p=w\rho$ and cosmological constant.

Problem 8

Find the natural scale of length and time appearing due to introduction of cosmological constant into General Relativity.

Problem 9

Show that the relativity principle results in the state equation $p=-\rho$ for dark energy in form of cosmological constant if it is treated as the vacuum energy.

Problem 10

Show that the cosmological constant's equation of state $p=-\rho$ ensures Lorentz-invariance of the vacuum energy-momentum tensor.

Problem 11

Show that the equation of state $p=-\rho$ is the only form which ensures Lorentz-invariance of the vacuum energy-momentum tensor.

Problem 12

Show explicitly that the state equation $p=-\rho$ is Lorenz-invariant.

Problem 13

Consider the observer that moves with constant velocity $V$ in the Universe described by FLRW metrics and filled with substance with equation of state $p=w\rho$. Calculate the energy density, which the observer will register. Consider the cases of decelerated and accelerated expansion of the Universe.

Problem 14

Does the energy conservation law hold in the presence of dark energy in the form of cosmological constant?

Problem 15

Show that by assigning energy to vacuum we do not revive the notion of "ether", i.e. we do not violate the relativity principle or in other words we do not introduce the notions of absolute rest and motion relative to vacuum.

Problem 16

Suppose that density of the dark energy as cosmological constant is equal to the present critical density, $\rho_{\Lambda}=\rho_{cr}$. What is then the total amount of dark energy inside the Solar System? Compare this number with $M_\odot c^2$.

Problem 17

Estimate the upper limit for the cosmological constant. Can an upper or lower limit be derived from the observed rate of growth of cosmological structures?

Problem 18

Knowing the age of the oldest objects in the Universe, determine the lower physical limit of the physical vacuum density.

Problem 19

Find time dependence of the scale factor in the case of flat Universe filled by dark energy in the form of cosmological constant and non-relativistic matter with current relative densities $\Omega_{\Lambda0}$ and $\Omega_{m0}$ respectively (see Chapter~11 for more detailed analysis).

Problem 20

Consider flat Universe filled by matter and cosmological constant with $\Lambda<0$ and show that it collapses in time period \[t_{col}=\frac{2\pi}{\sqrt{3|\Lambda|}}.\]

Problem 21

Find the value of redshift in the cosmological constant dominated flat Universe, for which a source of linear size $d$ has the minimum visible angular dimension.

Problem 22

Find (in the Newton's approximation) a critical distance $r_0$ around a point mass $m$, embedded into medium which is the cosmological constant $\Lambda>0$, where the gravity vanishes: it is attractive if $r<r_0$ and repulsive if $r>r_0$.