Cosmology

Abstract

The Lemaître–Friedmann–Robertson–Walker (LFRW) universe models are deduced as solutions of Einstein’s field equations, and the Hubble–Lemaître expansion law is found as a general property of these models. It is shown that the cosmic redshift due to the expansion of the space contains both the kinematic Doppler effect due to the velocity of the emitter relative to the observer and the gravitational shift of wavelength for light moving vertically in a gravitational field. Several observational properties of a flat LFRW universe model with dust and Lorentz invariant vacuum energy (LIVE) are deduced. Also anisotropic and inhomogeneous universe models are considered. Finally some inflationary universe models are discussed, and their predictions for some observational properties are confronted with observed data.

Exercises

• 12.1. Gravitational collapse

In this problem we shall find a solution to Einstein’s field equations describing a spherically symmetric gravitational collapse. The solution shall describe the spacetime both exterior and interior to the star. To connect the exterior and interior solutions, the metrics must be expressed in the same coordinate system. We will assume that the interior solution has the same form as a Friedmann solution. The Friedmann solutions are expressed in co moving coordinates, thus freely falling particles have constant spatial coordinates.

Let $\left( {\rho ,\tau } \right)$ be the infalling coordinates, i. co-moving coordinates with freely falling particles. τ is the proper time of a freely falling particle starting at infinity with zero velocity. These coordinates are connected to the curvature coordinates via the requirements

$$\begin{array}{*{20}l} {\rho = r,} \hfill & {{\text{for}}\quad \tau = 0,} \hfill \\ {\tau = t,} \hfill & {{\text{for}}\quad r = 0.} \hfill \\ \end{array}$$

(12.376)

• (a) Show that the transformation between the in falling coordinates and the curvature coordinates is given by

$$\begin{aligned} \tau & = \frac{2}{{3c\sqrt {R_{\text{S}} } }}\left( {\rho^{{\frac{3}{2}}} - r^{{\frac{3}{2}}} } \right), \\ t & = \tau - \frac{2}{c}\left( {R_{\text{S}} r} \right)^{{\frac{1}{2}}} + \frac{{R_{\text{S}} }}{c}\ln \left[ {\frac{{\left( {\frac{r}{{R_{\text{S}} }}} \right)^{{\frac{1}{2}}} + 1}}{{\left( {\frac{r}{{R_{\text{S}} }}} \right)^{{\frac{1}{2}}} - 1}}} \right], \\ \end{aligned}$$

(12.377)

where $R_{\text{S}}$ is the Schwarzschild radius of the star.

• (b) Show that the Schwarzschild metric in these coordinates takes the form

$$\begin{aligned} {\text{d}}s^{2} & = - c^{2} {\text{d}}\tau^{2} + \left[ {1 - \frac{3}{2}\left( {R_{\text{S}} } \right)^{{\frac{1}{2}}} c\tau \rho^{{ - \frac{3}{2}}} } \right]^{{ - \frac{2}{3}}} {\text{d}}\rho^{2} \\ & \quad + \left[ {1 - \frac{3}{2}\left( {R_{\text{S}} } \right)^{{\frac{1}{2}}} c\tau \rho^{{ - \frac{3}{2}}} } \right]^{{\frac{4}{3}}} \rho^{2} \left( {{\text{d}}\theta^{2} + \sin^{2} \theta {\text{d}}\phi^{2} } \right). \\ \end{aligned}$$

(12.378)

Assume the star has a position-dependent energy density $\rho (\tau )$, and that the pressure is zero. Assume further that the interior spacetime can be described with a Friedmann solution with Euclidean geometry (k = 0).

• (c) Find the solution when the radius of the star is R₀ at τ = 0.

• 12.2. The volume of a closed Robertson–Walker universe

Show that the volume of the region contained inside a radius $r = a\chi = a\,{ \arcsin }\,r$ is

$$V = 2\pi a^{3} \left( {\chi - \frac{1}{2}\sin 2\chi } \right).$$

(12.379)

Find the maximal volume. Find also an approximate expression for V when $\chi \ll R$.

• 12.3. Conformal time

Find the form of the line-element (12.1) if the cosmic time t is replaced by conformal time $\eta$ defined by Eq. (12.126).

What is the equation of light moving radially when we use conformal time?

• 12.4. Lookback time and the age of the universe

The lookback time of an object is the time required for light to travel from an emitting object to the receiver. Hence, it is $t_{L} \equiv t_{0} - t_{e}$, where t₀ is the point of time the object was observed and t_e is the point of time the light was emitted.

• (a) Show that the lookback time is given by

$$t_{L} = \int\limits_{0}^{2} {\frac{{{\text{d}}y}}{(1 + y)H(y)}} ,$$

(12.380)

where z is the redshift of the object and the Hubble parameter $H\left( y \right)$ is given in Eq. (12.111).

• (b) Show that the lookback time in the Milne universe model with $a(t) = (t/t_{0} ),\,\,\,\,k < 0$, is

$$t_{L} = \frac{1}{{H_{0} }}\frac{z}{1 + z}.$$

(12.381)

and find the age of this universe.

• (c) Show that $t_{L} = t_{0} \left[ {1 - (1 + z)^{ - 3/2} } \right]$, where ${\text{t}}_{ 0} { = 2}/( 3 {\text{H}}_{ 0} )$, in a flat, matter-dominated universe.

What is the age of this universe.?

• (d) Find the lookback time of an object with redshift z and the age of the matter-dominated universe models with positive and negative spatial curvature.

• (e) Find the age of a matter-dominated universe from the parametric solutions (12.140)–(12.144).

Are the resulting expressions in agreement with those found in d)?

• (f) Find the lookback time–redshift relation and the age of a flat universe with dust and LIVE.

• (g) Find the lookback time—redshift relation for a flat, LIVE-dominated universe.

• 12.5. The LFRW universe models with a perfect fluid

In this problem we will investigate FRW models with a perfect fluid. We will assume that the perfect fluid obeys the equation of state

$$p = w\rho ,$$

(12.382)

where $- 1 \le w \le 1$.

• (a) Write down the Friedmann equations for a LFRW universe model with a w-law perfect fluid. Express the equations in terms of the scale factor a only.

• (b) Assume that a(0) = 0. Show that when $- 1/3 < w \le 1$, the closed model will recollapse. Explain why this does not happen in the flat and open models.

• (c) Solve the Friedmann equation for a general $w \ne - 1$ in the flat case. What is the Hubble parameter and the deceleration parameter? Also write down the time evolution for the matter density.

• (d) Find the particle horizon distance in terms of H₀, w and z.

• (e) Specialize the above to the dust-dominated, radiation-dominated and LIVE-dominated universe models.

• (f) Find a general formula in terms of the density parameters for the present value of the deceleration parameter of a LFRW universe model.

• 12.6. Age—density relation for a radiation-dominated universe

Show that the age of a radiation-dominated universe model is given by

$$t_{0} = \frac{{t_{\text{H}} }}{{1 + \sqrt {\Omega _{{{\text{rad}}0}} } }}$$

(12.383)

for all values of k.

• 12.7. Redshift–luminosity relation for matter-dominated universe: Mattig’s formula

The luminosity distance of an object with redshift z is

$${\text{d}}_{L} = \frac{1 + z}{{H_{0} \sqrt {\left| {\Omega _{{{\text{k}}0}} } \right|} }}S_{k} \left[ {H_{0} \sqrt {\left| {\Omega _{{{\text{k}}0}} } \right|} \,\int\limits_{0}^{z} {\frac{{\text{d}}z}{H\left( z \right)}} } \right],$$

(12.384)

where the function $S_{\text{k}}$ is defined in Eq. (12.17) and $H\left( z \right)$ is given in Eq. (12.111).

Show that the luminosity distance of an object with redshift z in a matter-dominated universe with relative density $\Omega _{{{\text{m}}0}}$ and Hubble constant H₀ is

$${\text{d}}_{L} = \frac{2c}{{H_{0}\Omega _{{{\text{m}}0}}^{2} }}\left[ {\Omega _{{{\text{m}}0}} z + (\Omega _{{{\text{m}}0}} - 2)(\sqrt {1 +\Omega _{{{\text{m}}0}} z} - 1)} \right].$$

(12.385)

This is called Mattig’s formula. Find the corresponding formula for the Einstein–de Sitter universe, with $\Omega _{{{\text{m}}0}} = 1$.

• 12.8. Newtonian approximation with vacuum energy

• (a) Show that Einstein’ linearized field equation for a static spacetime containing dust with density ρ and vacuum energy with density $\rho_{\Lambda }$ takes the form of a modified Poisson equation

$$\nabla^{2} \phi = 4\pi G(\rho - 2\rho_{\Lambda } ).$$

(12.386)

• (b) Assume there is a particle with mass m at the origin. Solve Eq. (12.386) in the space outside the particle, and find the acceleration of gravity at a function of the distance from the origin.

Find the radius of a spherical surface where the acceleration of gravity vanishes.

How large is the mass inside this surface compared to the mass of the particle at the origin?

Evaluate the importance of LIVE for gravitational phenomena in the solar system.

• 12.9. Universe models with constant deceleration parameter

  • (a) Show that the universe with constant deceleration parameter q has expansion factor

$$a = \left( {\frac{t}{{t_{0} }}} \right)^{{\frac{1}{1 + q}}} ,\quad q \ne - 1,\;{\text{and}}\,a \propto e^{Ht} ,\quad q = - 1.$$

(12.387)

This is the scale factor of a flat universe with a perfect fluid. Find the equation of state of the fluid.

• 12.10. Density parameters as functions of the redshift

Show that the relative densities of LIVE and matter as functions of a are

$$\begin{aligned}\Omega _{\text{L}} & = \frac{{\Omega _{{{\text{L}}0}} a^{3} }}{{\Omega _{L0} a^{3} + \left( {1 -\Omega _{{{\text{L}}0}} -\Omega _{m0} } \right)a +\Omega _{m0} }}, \\\Omega _{m} \, & = \frac{{\Omega _{m0} }}{{\Omega _{\text{L0}} a^{3} + \left( {1 -\Omega _{{{\text{L}}0}} -\Omega _{m0} } \right)a +\Omega _{m0} }}. \\ \end{aligned}$$

(12.388)

What can you conclude from these expressions concerning the universe at early and late times?

• 12.11. FRW universe with radiation and matter

Show that the scale factor and the cosmic time as functions of conformal time of a universe with radiation and matter are [28].

$$k > 0:\left\{ {\begin{array}{*{20}l} {a = a_{0} \left[ {\alpha (1 - \cos \eta ) + \beta \sin \eta } \right]} \hfill \\ {t = a_{0} \left[ {\alpha (\eta - \sin \eta ) + \beta (1 - \cos \eta )} \right],} \hfill \\ \end{array} } \right.$$

(12.389)

$$k = 0:\left\{ {\begin{array}{*{20}l} {a = a_{0} \left[ {\frac{1}{2}\alpha \eta^{2} + \beta \eta } \right]} \hfill \\ {t = a_{0} \left[ {\frac{1}{6}\alpha \eta^{3} + \frac{1}{2}\beta \eta^{2} } \right],} \hfill \\ \end{array} } \right.$$

(12.390)

$$k < 0:\left\{ {\begin{array}{*{20}l} {a = a_{0} \left[ {\alpha (\cosh \eta - 1) + \beta \sinh \eta } \right]} \hfill \\ {t = a_{0} \left[ {\alpha (\sinh \eta - \eta ) + \beta (\cosh \eta - 1)} \right],} \hfill \\ \end{array} } \right.$$

(12.391)

where $\alpha = a_{0}^{2} H_{0}^{2}\Omega _{m0} /2$ and $\beta = (a_{0}^{2} H_{0}^{2}\Omega _{\gamma 0} )^{1/2}$, and $\Omega _{\gamma 0} \,{\text{and}}\,\Omega _{m0}$ are the present density parameters of radiation and matter, and H₀ is the present value of the Hubble parameter.

• 12.12. Event horizons in de Sitter universe models

Find the coordinate distances to the event horizons of the de Sitter universe models with $k > 0,\,\,\,\,k = 0\,\,\,\,\,{\text{and}}\,\,\,\,\,k\,{ < }\, 0$ as function of time.

• 12.13. Flat universe model with radiation and LIVE

  • (a) Find the scale factor as a function of time for a flat universe with radiation and Lorentz invariant vacuum energy represented by a cosmological constant Λ, and with present density parameter of radiation $\Omega _{{{\text{rad}}0}}$.

  • (b) Calculate the Hubble parameter, H, as a function of time, and show that the model approaches a de Sitter model in the far future. Find also the deceleration parameter, q(t).

  • (c) When is the inflection point, t₁, for which the universe went from deceleration to acceleration? What is the corresponding redshift observed at the time t₀?

• 12.14 De Sitter spacetime

Consider a De Sitter spacetime with coordinates $\left( {t,\,\,r} \right)$ and line element

$${\text{d}}s^{2} = - c^{2} {\text{d}}t^{2} + e^{2Ht} \left( {{\text{d}}r^{2} + r^{2} {\text{d}}\Omega ^{2} } \right),$$

(12.393)

where the Hubble parameter H is constant.

• (a) Find the redshift of light emitter from a coordinate r as measured at a point of time $t_{0}$ by an observer at the origin. The Hubble parameter H is assumed to be known.

• (b) What is the 4-acceleration of a reference particle at rest in the coordinate system? What does your result tell about the reference frame in which these coordinates are co-moving?

Will an observer with constant radial coordinate r experience an acceleration of gravity?

Introducing coordinates $\left( {T,\,\,\,R} \right)$ by the transformation

$$R = re^{Ht} , \quad T = t - \ln \left( {1 - H^{2} r^{2} e^{2Ht} } \right)$$

(12.394)

or

$$r = \frac{R}{{e^{HT} \sqrt {1 - H^{2} R^{2} /c^{2} } }}, \quad e^{Ht} = e^{HT} \sqrt {1 - H^{2} R^{2} /c^{2} } ,$$

(12.395)

the line element takes the form (you need not show this)

$${\text{d}}s^{2} = - \left( {c^{2} - H^{2} R^{2} } \right){\text{d}}T^{2} + \frac{{{\text{d}}R^{2} }}{{1 - H^{2} R^{2} /c^{2} }} + R^{2} {\text{d}}\,\Omega ^{2} .$$

(12.396)

• (c) Find the redshift of light emitted from a coordinate R as measured by an observer at the origin. Why is your result different to the one in a)?

• (d) What is the 4-acceleration of a reference particle at rest in the coordinate system? What does your result tell about the reference frame in which these coordinates are co-moving? Will an observer with constant radial coordinate r experience an acceleration of gravity?

• (e) How does a reference particle with $r = r_{0}$ = constant move in the $\left( {T,\,\,\,R} \right)$-coordinate system.

• (f) How is the redshift of light explained in the $\left( {T,\,\,\,R} \right)$-coordinate system? How is it explained in the $\left( {t,\,\,r} \right)$-system?

• 12.15. The Milne Universe

  • (a) The Milne Universe has a line element

$${\text{d}}s^{2} = - c{\text{d}}t^{2} + \left( {\frac{t}{{t_{0} }}} \right)^{2} \left( {\frac{{{\text{d}}r^{2} }}{{1 + r^{2} /c^{2} t_{0}^{2} }} + r^{2} {\text{d}}\,\Omega ^{2} } \right).$$

(12.397)

The $t\,,\,r$ coordinates are co-moving in a reference frame E. Give a physical interpretation of the line-element.

• (b) The Hubble parameter is $H = \dot{a}/a$ where $\dot{a}$ is the derivative of the scale factor with respect to time. Calculate the Hubble parameter of this universe model at an arbitrary point of time. Find the age of the universe in terms of the present value of the Hubble parameter.

• (c) The so-called deceleration parameter is given by $q \equiv - \,a\ddot{a}/\dot{a}^{2}$.

What is the value of q for this universe model? What does this tell about the expansion?

• (d) Introduce new coordinates T and R by the transformation

$$T = t\sqrt {1 + \left( {\frac{r}{{ct_{0} }}} \right)^{2} }, \quad R = \frac{r\,t}{{t_{0} }}.$$

(12.398)

Show that the inverse transformation is

$$c\,t = \sqrt {c^{2} T^{2} - R^{2} } \,\,\,\,\,\,,\,\,\,\,\,\,r = \frac{{ct_{0} R}}{{\sqrt {c^{2} T^{2} - R^{2} } }},$$

(12.399)

and that

$$R = \frac{rcT}{{\sqrt {c^{2} t_{0}^{2} + r^{2} } }}.$$

(12.400)

• (e) Make a Minkowski diagram with reference to the $cT,\,R$-system, and draw the world lines of the reference particles of E, i.e. those with $r =$ constant and the simultaneity curves of E, i.e. those with $t =$ constant.

• (f) Use the transformation Eqs. (3) to show that the differentials of the coordinates co-moving in the reference frame E are

$${\text{d}}t = \frac{{c^{2} T\,{\text{d}}T - R\,{\text{d}}R}}{{c\sqrt {c^{2} T^{2} - R^{2} } }},\;{\text{d}}r = \frac{{c^{3} t_{0} T\left( {T\,{\text{d}}R - R\,{\text{d}}T} \right)}}{{\left( {c^{2} T^{2} - R^{2} } \right)^{3/2} }}.$$

(12.401)

Use these differentials and the expressions for t and r in Eq. (3) to calculate the line element in the $T,\,\,R$ coordinate system. What does your result tell about the spacetime described by the line element you just have found and the line element (1)?

• (g) Calculate the cosmic redshift, z, of a star with $r = r_{1} =$ constant in terms of $r_{1}$ and the point of time $t_{0}$ of the observation. Then calculate the redshift of a star with $R =$ constant. Explain the results you found.

• (h) Einstein’s field equations as applied to an isotropic and homogeneous universe model lead to the Friedmann equations

$$\frac{{\dot{a}^{2} + kc^{2} }}{{a^{2} }} = \frac{8\pi G}{3}\rho \,\,\,\,\,\,,\,\,\,\,\,\,\frac{{\ddot{a}}}{a} = - \,\frac{4\pi G}{3}\left( {\rho + 3\frac{p}{{c^{2} }}} \right)$$

(12.402)

where a is the scale factor, $\rho$ and p are the density and pressure of the cosmic fluid, respectively, and k is the spatial curvature index.

Apply these equations to the line-element (1). What does your result tell about this universe model?

• 12.16. Natural Inflation

The natural inflation model has potential

$$V\left( \phi \right) = V_{0} \left( {1 + \cos \tilde{\phi }} \right)\,,$$

(12.403)

where $\tilde{\phi } = \phi /M$, and M is the spontaneous symmetry breaking scale. We shall here write Einstein’s gravitational constant as $\kappa = 1/M_{P}^{2}$, where $M_{P}$ is the Planck mass.

• (a) Show that for this model the spectral parameters are

$$\delta_{ns} = b\frac{{3 - \cos \tilde{\phi }}}{{1 + \cos \tilde{\phi }}},\;n_{T} = - \,b\frac{{1 - \cos \tilde{\phi }}}{{1 + \cos \tilde{\phi }}},\;r = 8\,b\frac{{1 - \cos \tilde{\phi }}}{{1 + \cos \tilde{\phi }}},\;b = \left( {\frac{{M_{P} }}{M}} \right)^{2}$$

(12.404)

Observations have given the results $\delta_{ns} = 0.032$ and $r < 0.04$.

• (b) Show that

$$b = \delta_{ns} - \frac{r}{4},$$

(12.405)

and use this to calculate the requirement from the observations upon the symmetry breaking scale. Is there any problem with the result?

• (c) Show that for this model the number of e-folds is

$$N = \frac{1}{b}\ln \frac{{1 - \cos \left( {\tilde{\phi }_{f} } \right)}}{{1 - \cos \left( {\tilde{\phi }} \right)}}.$$

(12.406)

Use this to express the spectral parameters as

$$\delta_{ns} = b\frac{{\left( {2 + b} \right)e^{b\,N} + 2}}{{\left( {2 + b} \right)e^{b\,N} - 2}},\;n_{T} = - \frac{2b}{{\left( {2 + b} \right)e^{b\,N} - 2}},\;r = \frac{16b}{{\left( {2 + b} \right)e^{b\,N} - 2}},$$

(12.407)

In order to solve the horizon- and flatness problems the number of e-folds must be larger than 50. Insert $N = 50$, $r = 0.04$ and make a judgement of this model.