Astronomy 1

Cosmology
John Niclasen

Contents:

1. Introduction
1.1 Distances
1.2 Mass
1.3 Luminosity
1.4 Time
1.5 Energy
1.6 Planck scales
2. Fundamental Observations
2.1 The night sky is dark
2.2 The Universe on large scales
2.3 Redshift
2.4 Particles
2.5 Cosmic Microwave Background (CMB)
3. Newton Versus Einstein
3.1 Describing Curvature
3.2 The Robertson-Walker Metric
3.3 Robertson-Walker Metric
3.4 Proper Distance
4. Cosmic Dynamics
4.1 The Friedmann Equation
4.2 The Fluid and Acceleration Equations
4.3 Equation of State
4.4 Learning to Love Lambda
5. Single-Component Universes
5.1 Evolution of Energy Density
5.2 Curvature Only
5.3 Spatially Flat Universes
5.4 Matter only
5.5 Radiation Only
5.6 Lambda Only
6. Multi-Component Universes
6.1 Friedmann Equation
6.2 Matter + Curvature
6.3 Matter + Lambda
6.4 Matter + Curvature + Lambda
6.5 Radiation + Matter
6.6 Benchmark Model

1. Introduction

1.1 Distances

\displaylines{1 AU=1.5\times 10^{11}m\cr 1 pc=3.1\times 10^{16}m\cr 1 Mpc=10^6pc=3.1\times 10^{22}m\cr }

1.2 Mass

\displaylines{1 M_{\odot }=2.0\times 10^{30}kg\cr M_{gal}\approx 10^{12}M_{\odot }\cr }




1.3 Luminosity

\displaylines{1 L_{\odot }=3.8\times 10^{26}watts\cr L_{gal}=3.6\times 10^{10}L_{\odot }\cr }

1.4 Time

\displaylines{1 Gyr=3.2\times 10^{16}s}

1.5 Energy

\displaylines{1 eV=1.6\times 10^{-19}J\cr m_ec^2=511000eV=0.511MeV\cr m_pc^2=938.3MeV\cr }

1.6 Planck scales

\displaylines{G=6.7\times 10^{-11}m^3kg^{-1}s^{-2}\cr c=3.0\times 10^8ms^{-1}\cr \hbar ={h\over 2\pi }=1.1\times 10^{-34}J s=6.6\times 10^{-16}eV s\cr k_B=8.6\times 10^{-5}eV K^{-1}\cr }

1.6.1 Planck length

\displaylines{l_P\equiv \sqrt {{G\hbar \over c^3}}=1.6\times 10^{-35}m}

1.6.2 Planck mass

\displaylines{M_P\equiv \sqrt {{\hbar c\over G}}=2.2\times 10^{-8}kg}

1.6.3 Planck time

\displaylines{t_P\equiv \sqrt {{G\hbar \over c^5}}=5.4\times 10^{-44}s}

1.6.4 Planck energy

\displaylines{E_P=M_Pc^2=2.0\times 10^9J=1.2\times 10^{28}eV}

1.6.5 Planck temperature

\displaylines{T_P={E_P\over k_B}=1.4\times 10^{32}K}

2. Fundamental Observations

2.1 The night sky is dark

2.1.1 Flux

\displaylines{f(r)={L\over 4\pi r^2}}

2.1.2 Intensity

\displaylines{dJ(r)={L\over 4\pi r^2}\cdot n\cdot r^2dr={nL\over 4\pi }dr\cr J=\int _{r=0}^{\infty }dJ={nL\over 4\pi }\int _0^{\infty }dr=\infty \cr }

This is known as Olbers' Paradox.

2.2 The Universe on large scales

On large scales, the Universe is isotropic and homogeneous.

2.3 Redshift

The redskift z:

\displaylines{z\equiv {\lambda _{ob}-\lambda _{em}\over \lambda _{em}}}

Because:

\displaylines{\lambda \cdot f=c\cr E=h\cdot f\cr }

The redshift can be expressed:

\displaylines{z={f_e-f_o\over f_o}\cr z={E_e-E_o\over E_o}\cr z={dt_o-dt_e\over dt_e}\cr }

Relation to scale factor

\displaylines{a={1\over 1+z}}

2.3.1 Hubble's Law

\displaylines{z={H_0\over c}r}

Interpreting the redshifts as Doppler shifts, Hubble's Law takes the form

\displaylines{z={v\over c}\cr \Rightarrow \quad v=H_0r\cr }

2.3.2 Hubble constant

\displaylines{H_0=70\pm 7 km s^{-1} Mpc^{-1}}

2.3.3 Hubble time

\displaylines{t_0={r\over v}={r\over H_0r}=H_0^{-1}\sim 14 Gyr}

2.3.4 Steady State Universe

\displaylines{{dr\over dt}=H_0r\cr \Rightarrow \quad r(t)\propto e^{H_0t}\cr }

Volume:

\displaylines{V={4\pi \over 3}r^3\propto e^{3H_0t}}

Creation of mass:

\displaylines{\dot M_{ss}=\rho _0\dot V=\rho _03H_0V}

2.4 Particles

The building blocks of the Universe known as particles are in reality quantum fluctuations in the fields related to each particle. They are better seen as waves.

Particle

Symbol

Rest energy (MeV)

Charge

proton

p

938.3

+1

neutron

n

939.6

0

electron

e^-

0.511

-1

neutrino

\nu _e,\nu _{\mu },\nu _{\tau }

?

0

photon

\gamma

0

0

dark matter

?

?

0

2.4.1 Light

\displaylines{\lambda ={c\over f}\cr E_{\gamma }=hf\quad ,\quad h=2\pi \hbar \cr }

2.4.2 Blackbody radiation

\displaylines{\epsilon (f)df={8\pi h\over c^3} {f^3df\over \exp (hf/k_BT)-1}\cr hf_{peak}\approx 2.82 k_BT\cr \epsilon _{\gamma }=\alpha T^4\cr \alpha ={\pi ^2\over 15} {k^4\over \hbar ^3c^3}=7.56\times 10^{-16} J m^{-3} K^{-4}\cr n_{\gamma }=\beta T^3\cr \beta ={2.404\over \pi ^2} {k^3\over \hbar ^3c^3}=2.03\times 10^7 m^{-3} K^{-3}\cr E_{mean}={\epsilon _{\gamma }\over n_{\gamma }}=hf_{mean}\approx 2.70k_BT\cr }



2.5 Cosmic Microwave Background (CMB)

\displaylines{T_0=2.725\pm 0.001 K\cr \epsilon _{\gamma }=4.17\times 10^{-14} J m^{-3}\cr n_{\gamma }=4.11\times 10^8 m^{-3}\cr E_{mean}=6.34\times 10^{-4}eV\cr }

2.5.1 Photon gas

\displaylines{\epsilon _{\gamma }=\alpha T^4\cr P_{\gamma }={\epsilon _{\gamma }\over 3}\cr dQ=dE+P dV=0\cr {dE\over dt}=-P(t){dV\over dt}\cr \Rightarrow \quad {1\over T} {dT\over dt}=-{1\over 3V} {dV\over dt}\cr V\propto a(t)^3\cr \Rightarrow \quad {d\over dt}(\ln  T)=-{d\over dt}(\ln  a)\cr T(t)\propto a(t)^{-1}\cr }

3. Newton Versus Einstein

The Way of Newton

Mass tells gravity how to exert a force (F=-GMm/r^2),
Force tells mass how to accelerate (F=ma).

The Way of Einstein

Mass-energy tells space-time how to curve,
Curved space-time tells mass-energy how to move.

Einstein was right!

3.1 Describing Curvature

Pythagorean theorem in Cartesian coordinates:

\displaylines{ds^2=dx^2+dy^2}

In planar polar coordinates:

\displaylines{ds^2=dr^2+r^2d\theta ^2}

On a sphere:

\displaylines{ds^2=dr^2+R^2\sin ^2(r/R)d\theta ^2}

Negative curvature (saddle):

\displaylines{ds^2=dr^2+R^2\sinh ^2(r/R)d\theta ^2}

3.1.1 3D Flat Space Metric

\displaylines{\kappa =0\cr ds^2=dx^2+dy^2+dz^2\cr }

In spherical coordinates:

\displaylines{ds^2=dr^2+r^2\lbrack d\theta ^2+\sin ^2\theta d\phi ^2\rbrack }

3.1.2 3D Positive Curvature

\displaylines{\kappa =+1\cr ds^2=dr^2+R^2\sin ^2(r/R)\lbrack d\theta ^2+\sin ^2\theta d\phi ^2\rbrack \cr }

3.1.3 3D Negative Curvature

\displaylines{\kappa =-1\cr ds^2=dr^2+R^2\sinh ^2(r/R)\lbrack d\theta ^2+\sin ^2\theta d\phi ^2\rbrack \cr }

3.1.4 3D Compact Form

\displaylines{ds^2=dr^2+S_{\kappa }(r)^2d\Omega ^2\cr d\Omega ^2\equiv d\theta ^2+\sin ^2\theta d\phi ^2\cr S_{\kappa }(r)=\cases{R\sin (r/R)&(\kappa =+1)\cr r&(\kappa =0)\cr R\sinh (r/R)&(\kappa =-1)\cr }\cr }

Alternative:

\displaylines{x\equiv S_{\kappa }(r)\cr ds^2={dx^2\over 1-\kappa x^2/R^2}+x^2d\Omega ^2\cr }

3.2 The Robertson-Walker Metric

3.2.1 Minkowski Metric

\displaylines{ds^2=-c^2dt^2+dr^2+r^2d\Omega ^2}

Photons follow a null geodesic. So

\displaylines{ds=0\cr ds^2=0=-c^2dt^2+dr^2+r^2d\Omega ^2\cr }

In direction of sight:

\displaylines{d\theta =0\quad ,\quad d\phi =0\cr \Rightarrow \quad c^2dt^2=dr^2\cr {dr\over dt}=\pm c\cr }

3.3 Robertson-Walker Metric

\displaylines{ds^2=-c^2dt^2+a(t)^2\left [{dx^2\over 1-\kappa x^2/R_0^2}+x^2d\Omega ^2\right ]\cr ds^2=-c^2dt^2+a(t)^2\left [dr^2+S_{\kappa }(r)^2d\Omega ^2\right ]\cr }

3.4 Proper Distance

The proper distance d_p(t) between two points is equal to the length of the spatial geodesic between them when the scale factor is fixed at the value a(t).

\displaylines{ds^2=a(t)^2\left [dr^2+S_{\kappa }(r)^2d\Omega ^2\right ]}

Along direction of sight:

\displaylines{ds=a(t)dr\cr d_p(t)=a(t)\int _0^rdr=a(t)r\cr }

Or

\displaylines{d_p(t)=a(t)r(x)=\cases{a(t)R_0\sin ^{-1}(x/R_0)&(\kappa =+1)\cr a(t)x&(\kappa =0)\cr a(t)R_0\sinh ^{-1}(x/R_0)&(\kappa =-1)\cr }\cr }

3.4.1 Rate of Change

\displaylines{\dot d_p=\dot ar={\dot a\over a}d_p}

For t=t_0:

\displaylines{v_p(t_0)=H_0d_p(t_0)\cr v_p(t_0)\equiv \dot d_p(t_0)\cr H_0=\left ({\dot a\over a}\right )_{t=t_0}\cr }

3.4.2 Radius of Curvature

\displaylines{R(t)=a(t)R_0}

3.4.3 Hubble Distance

\displaylines{d_H(t_0)\equiv c/H_0\cr v_p=\dot d_p>c\cr }

In our universe

\displaylines{H_0=70\pm 7km s^{-1} Mpc^{-1}\cr d_H(t_0)=4300\pm 400Mpc\cr }

3.4.4 Null Geodesic

\displaylines{ds=0\cr c^2dt^2=a(t)^2dr^2\cr \Leftrightarrow \quad c{dt\over a(t)}=dr\cr }








A wave crest emitted at time t_e and observed at time t_0:

\displaylines{c\int _{t_e}^{t_0}{dt\over a(t)}=\int _0^rdr=r\cr \int _{t_e}^{t_e+\lambda _e/c}{dt\over a(t)}=\int _0^{t_0+\lambda _0/c}{dt\over a(t)}\cr {\lambda _e\over a(t_e)}={\lambda _0\over a(t_0)}\cr 1+z={a(t_0)\over a(t_e)}={1\over a(t_e)}\cr }

3.4.5 Scale Factor

\displaylines{a(t_0)=1}

4. Cosmic Dynamics

4.1 The Friedmann Equation

4.1.1 Poisson's Equation in Newtonian Dynamics

\displaylines{\nabla ^2\Phi =4\pi G\rho }

Gravitational acceleration at the surface of an expanding sphere

\displaylines{{d^2R_S\over dt^2}=-{GM_S\over R_S(t)^2}\cr \Leftrightarrow \quad {1\over 2}\left ({dR_S\over dt}\right )^2={GM_S\over R_S(t)}+U\cr E_{kin}={1\over 2}\left ({dR_S\over dt}\right )^2\cr E_{pot}=-{GM_S\over R_S(t)}\cr }

Because mass is constant

\displaylines{M_S={4\pi \over 3}\rho (t)R_S(t)^3}

Since expansion is isotropic

\displaylines{R_S(t)=a(t)r_S}

4.1.2 Friedmann Equation in Newtoniam Mechanics

\displaylines{\left ({\dot a\over a}\right )^2={8\pi G\over 3}\rho (t)+{2U\over r_S^2} {1\over a(t)^2}\cr }

Max scale factor

\displaylines{a_{\max }=-{G M_S\over U r_S}}


4.1.3 Friedmann Equation incl. General Relativity

\displaylines{\left ({\dot a\over a}\right )^2={8\pi G\over 3c^2}\epsilon (t)-{\kappa c^2\over R_0^2} {1\over a(t)^2}\cr }

4.1.4 Energy of a Particle

\displaylines{E=\sqrt {m^2c^4+p^2c^2}\cr E_{non-rel}\approx mc^2(1+v^2/c^2)\approx mc^2+{1\over 2}mv^2\cr E_{rel}=pc=hf\cr }

4.1.5 Relation between Recession Speed and Proper Distance

\displaylines{v(t)=H(t)d(t)\cr H(t)\equiv \left ({\dot a\over a}\right )\cr }

Thus, the Friedmann equation can be rewritten as

\displaylines{H(t)^2={8\pi G\over 3c^2}\epsilon (t)-{\kappa c^2\over R_0^2a(t)^2}}

At the present moment

\displaylines{H_0=H(t_0)=\left ({\dot a\over a}\right )_{t=t_0}=70\pm 7km s^{-1} Mpc^{-1}\cr H_0^2={8\pi G\over 3c^2}\epsilon _0-{\kappa c^2\over R_0^2}\cr }

4.1.6 In an Empty Universe with Negative Curvature

\displaylines{R_0(\min )=c/H_0}

4.1.7 In a Spatial Flat Universe

\displaylines{H(t)^2={8\pi G\over 3c^2}\epsilon (t)}

4.1.8 Critical Density

\displaylines{\epsilon _c(t)\equiv {3c^2\over 8\pi G}H(t)^2}

4.1.9 Mass Density

\displaylines{\rho _{c,0}\equiv {\epsilon _{c,0}\over c^2}}

4.1.10 Density Parameter

\displaylines{\Omega (t)\equiv {\epsilon (t)\over \epsilon _c(t)}\cr \Rightarrow \quad 1-\Omega (t)=-{\kappa c^2\over R_0^2a(t)^2H(t)^2}\cr }

At the present moment

\displaylines{1-\Omega _0=-{\kappa c^2\over R_0^2H_0^2}\cr \Leftrightarrow \quad {\kappa \over R_0^2}={H_0^2\over c^2}(\Omega _0-1)\cr }

4.2 The Fluid and Acceleration Equations

4.2.1 1. Law of Thermodynamics

\displaylines{dQ=dE+PdV}

For the expanding universe

\displaylines{dQ=0\cr \Rightarrow \quad \dot E+P\dot V=0\cr }

For a sphere

\displaylines{R_S(t)=a(t)r_S\cr V(t)={4\pi \over 3}r_S^3a(t)^3\cr \dot V={4\pi \over 3}r_S^3(3a^2\dot a)=V\left (3{\dot a\over a}\right )\cr }

The internal energy

\displaylines{E(t)=V(t)\epsilon (t)\cr \dot E=V\dot \epsilon +\dot V\epsilon =V\left (\dot \epsilon +3{\dot a\over a}\epsilon \right )\cr \Rightarrow \quad V\left (\dot \epsilon +3{\dot a\over a}\epsilon +3{\dot a\over a}P\right )=0\cr }

4.2.2 The Fluid Equation

\displaylines{\dot \epsilon +3{\dot a\over a}(\epsilon +P)=0}

4.2.3 The Acceleration Equation

\displaylines{{\ddot a\over a}=-{4\pi G\over 3c^2}(\epsilon +3P)}

4.3 Equation of State

4.3.1 Nonrelativistic Gas

\displaylines{P_{nonrel}=\omega \epsilon _{nonrel}\cr \omega \approx {\langle v^2\rangle \over 3c^2}\ll 1\cr }

4.3.2 Relativistic Gas (Photons)

\displaylines{P_{rel}={1\over 3}\epsilon _{rel}}

4.3.3 Sound Speed

\displaylines{c_S^2=c^2\left ({dP\over d\epsilon }\right )\cr \omega >0\cr \Rightarrow \quad c_S=\sqrt {\omega }c\cr \omega \le 1\cr }

4.4 Learning to Love Lambda

4.4.1 Poisson's Equations

\displaylines{\nabla ^2\Phi =4\pi G\rho }

4.4.2 Gravitational Acceleration

\displaylines{\bar a=-\bar \nabla \Phi }

For constant \Phi

\displaylines{\rho ={1\over 4\pi G}\nabla ^2\Phi =0}

4.4.3 Lambda

What Einstein did in Newtonian terms

\displaylines{\nabla ^2\Phi +\Lambda =4\pi G\rho }

In general relativistic terms

\displaylines{\left ({\dot a\over a}\right )^2={8\pi G\over 3c^2}\epsilon -{\kappa c^2\over R_0^2a^2}+{\Lambda \over 3}\cr }

The fluid equation is the same

\displaylines{\dot \epsilon +3{\dot a\over a}(\epsilon +P)=0}

The acceleration equation becomes

\displaylines{{\ddot a\over a}=-{4\pi G\over 3c^2}(\epsilon +3P)+{\Lambda \over 3}}

4.4.4 Extra energy density

\displaylines{\epsilon _{\Lambda }\equiv {c^2\over 8\pi G}\Lambda }

4.4.5 Pressure

\displaylines{P_{\Lambda }=-\epsilon _{\Lambda }=-{c^2\over 8\pi G}\Lambda }

In the static model, the acceleration equation reduces to

\displaylines{0=-{4\pi G\over 3}\rho +{\Lambda \over 3}\cr \Leftrightarrow \quad \Lambda =4\pi G\rho \cr }

4.4.6 The Friedmann Equation

\displaylines{\dot a=0\cr \Rightarrow \quad 0={8\pi G\over 3}\rho -{\kappa c^2\over R_0^2}+{\Lambda \over 3}=4\pi G\rho -{\kappa c^2\over R_0^2}\cr }

The static model had to be positively curved (\kappa =+1), with a radius of curvature

\displaylines{R_0={c\over 2\sqrt {\pi G\rho }}={c\over \sqrt {\Lambda }}}

4.4.7 Vacuum Energy

For pairs of particle-antiparticles

\displaylines{\Delta E\Delta t\le h}

5. Single-Component Universes

The relation among the energy density \epsilon (t), the pressure P(t), and the scale factor a(t) is given by three equations:

  • The Friedmann Equation
\displaylines{\left ({\dot a\over a}\right )^2={8\pi G\over 3c^2}\epsilon -{\kappa c^2\over R_0^2a^2}}
  • The Fluid Equation
\displaylines{\dot \epsilon +3{\dot a\over a}(\epsilon +P)=0}
  • The Equation of State
\displaylines{P=\omega \epsilon }

5.1 Evolution of Energy Density

\omega is a dimensionless number.

Conponent

\omega

Density

nonrelativistic matter

0

\epsilon _m

radiation

{1\over 3}

\epsilon _r

cosmological constant

-1

\epsilon _{\Lambda }

5.1.1 Total Energy Density

\displaylines{\epsilon =\sum _{\omega }\epsilon _{\omega }}

5.1.2 Total Pressure

\displaylines{P=\sum _{\omega }P_{\omega }=\sum _{\omega }\omega \epsilon _{\omega }}

5.1.3 The Fluid Equation

For each component:

\displaylines{\dot \epsilon _{\omega }+3{\dot a\over a}(\epsilon _{\omega }+P_{\omega })=0\cr sim\quad \dot \epsilon _{\omega }+3{\dot a\over a}(1+\omega )\epsilon _{\omega }=0\cr \Leftrightarrow \quad {d\epsilon _{\omega }\over \epsilon _{\omega }}=-3(1+\omega ){da\over a}\cr }

For constant \omega :

\displaylines{\epsilon _{\omega }(a)=\epsilon _{\omega ,0}a^{-3(1+\omega )}}

5.1.4 Expanding Universe

\displaylines{\epsilon _m(a)={\epsilon _{m,0}\over a^3}\cr \epsilon _r(a)={\epsilon _{r,0}\over a^4}\cr }

Density in general:

\displaylines{\epsilon =nE}

, where n is the number of particles.

Type of energy

Number density

Matter

n\propto a^{-3}

Radiation

n\propto a^{-4}

5.1.5 Cosmic Microwave Background (CMB)

\displaylines{T_0=2.725K\cr \epsilon _{CMB,0}=\alpha T_0^4=4.17\times 10^{-14} J m^{-3}=0.260 MeV m^{-3}\cr \Omega _{CMB,0}\equiv {\epsilon _{CMB,0}\over \epsilon _{c,0}}={0.260 MeV m^{-3}\over 5200 MeV m^{-3}}=5.0\times 10^{-5}\cr }

5.1.6 Neutrinos

There are three flavors of neutrinos. For each neutrino flavor:

\displaylines{\epsilon ={7\over 8}\left ({4\over 11}\right )^{4/3}\epsilon _{CMB}\approx 0.227 \epsilon _{CMB}}

For the sum of all three flavors

\displaylines{\Omega _{\nu }=0.681 \Omega _{CMB}}

as long as all neutrino flavors are relativistic.

Mean energy per neutrino

\displaylines{E_{\nu }\approx {5\times 10^{-4} eV\over a}}

as long as E_{\nu }>m_{\nu }c^2.

5.1.7 Total radiation

\displaylines{\Omega _{r,0}=\Omega _{CMB,0}+\Omega _{\nu ,0}=5.0\times 10^{-5}+3.4\times 10^{-5}=8.4\times 10^{-5}}

5.1.8 Benchmark Model

\displaylines{\Omega _{r,0}=8.4\times 10^{-5}\cr \Omega _{m,0}=0.3\cr \Omega _{\Lambda ,0}=1-\Omega _{r,0}-\Omega _{m,0}\approx 0.7\cr {\epsilon _{\Lambda ,0}\over \epsilon _{m,0}}={\Omega _{\Lambda ,0}\over \Omega _{m,0}}\approx {0.7\over 0.3}\approx 2.3\cr {\epsilon _{\Lambda }(a)\over \epsilon _m(a)}={\epsilon _{\Lambda ,0}\over \epsilon _{m,0}/a^3}={\epsilon _{\Lambda ,0}\over \epsilon _{m,0}} a^3\cr }

Matter-\Lambda  equality happened at

\displaylines{a_{m \Lambda }=\left (\Omega _{m,0}\over \Omega _{\Lambda ,0}\right )^{1/3}\approx 0.75}

Radiation-matter equality happened at

\displaylines{a_{r m}={\epsilon _{m,0}\over \epsilon _{r,0}}\approx 2.8\times 10^{-4}}

5.1.9 Many Components

In a universe with many components, the Friedmann equation can be written in the form

\displaylines{\dot a^2={8\pi G\over 3c^2}\sum _{\omega }\epsilon _{\omega ,0}a^{-(1+3\omega )}-{\kappa c^2\over R_0^2}}

5.2 Curvature Only

For an empty universe, the Friedmann equation is

\displaylines{\dot a^2=-{\kappa c^2\over R_0^2}}

One solution has \dot a=0 \land  \kappa =0. The geometry is described by the Minkowski metric. All the transformations of special relativity hold true.

5.2.1 Negatively Curved Empty Universe

It's also possible to have \kappa =-1, which is negatively curved.

\displaylines{\dot a=\pm {c\over R_0}}

If it's expanding

\displaylines{a(t)={t\over t_0}\quad ,\quad t_0={R_0\over c}\cr t_0={1\over H_0}\cr 1+z={1\over a(t_e)}={t_0\over t_e}\cr \Leftrightarrow \quad t_e={t_0\over 1+z}={1\over H_0(1+z)}\cr }



Proper distance:

\displaylines{d_p(t_0)=ct_0\int _{t_e}^{t_0}{dt\over t}=ct_0 \ln \left ({t_0\over t_e}\right )\cr d_p(t_0)={c\over H_0} \ln (1+z)\cr d_p\propto \ln  z\cr d_p(t_e)={c\over H_0} {\ln (1+z)\over 1+z}\cr }

5.3 Spatially Flat Universes

Flat means \kappa =0.

5.3.1 Single Component

The Friedmann equation takes the form

\displaylines{\dot a^2={8\pi G\epsilon _0\over 3c^2}a^{-(1+3\omega )}\cr LHS\propto t^{2q-2}\cr RHS\propto t^{-(1+3\omega )q}\cr \Rightarrow \quad q={2\over 3+3\omega }\quad ,\quad \omega \not =-1\cr }

Other properties

\displaylines{a(t)=\left ({t\over t_0}\right )^{2/(3+3\omega )}\cr t_0={1\over 1+\omega }\sqrt {{c^2\over 6\pi G\epsilon _0}}\cr H_0\equiv \left ({\dot a\over a}\right )_{t=t_0}={2\over 3(1+\omega )}t_0^{-1}\cr \Rightarrow \quad t_0={2\over 3(1+\omega )}H_0^{-1}\cr }
  • In a spatially flat universe,, if \omega >-{1\over 3}, the universe is younger than the Hubble time.
  • In a spatially flat universe,, if \omega <-{1\over 3}, the universe is older than the Hubble time.

Energy density:

\displaylines{\epsilon (a)=\epsilon _0a^{-3(1+\omega )}\cr \Rightarrow \quad \epsilon (t)=\epsilon _0\left ({t_0\over t}\right )^2\cr \Leftrightarrow \quad \epsilon (t)={1\over 6\pi (1+\omega )^2} {c^2\over G} t^{-2}\cr }

In terms of Planck units

\displaylines{\epsilon (t)={1\over 6\pi (1+\omega )^2} {E_P\over l_P^3}\left ({t_P\over t}\right )^2}


Redshift:

\displaylines{1+z={a(t_0)\over a(t_e)}=\left ({t_0\over t_e}\right )^{2/(3+3\omega )}\cr \Rightarrow \quad t_e={t_0\over (1+z)^3(1+\omega )/2}={2\over 3(1+\omega )H_0} {1\over (1+z)^{3(1+\omega )/2}}\cr }

Proper distance:

\displaylines{d_p(t_0)=c\int _{t_e}^{t_0}{dt\over a(t)}=ct_0{3(1+\omega )\over 1+3\omega }\lbrack 1-(t_e/t_0)^{(1+3\omega )/(3+3\omega )}\rbrack \quad ,\quad \omega \not =-{1\over 3}\cr d_p(t_0)={c\over H_0} {2\over 1+3\omega }\lbrack 1-(1+z)^{-(1+3\omega )/2}\rbrack \cr }

The horizon, if \omega >-{1\over 3}:

\displaylines{d_{hor}(t_0)=ct_0{3(1+\omega )\over 1+3\omega }={c\over H_0} {2\over 1+3\omega }}

In a flat universe dominated by matter (\omega =0)[m](omega = 0)[m] or by radiation (\omega ={1\over 3})[m](omega = {1 over 3})[m], the horizon distance is finite.

With \omega \le -{1\over 3}, the horizon distance is infinite.

5.4 Matter only

\displaylines{\omega =0}

The age of such a universe

\displaylines{t_0={2\over 3H_0}}

The horizon distance:

\displaylines{d_{hor}(t_0)=3ct_0={2c\over H_0}}

The scale factor:

\displaylines{a_m(t)=\left ({t\over t_0}\right )^{2/3}}

Proper distance

\displaylines{d_p(t_0)=c\int _{t_e}^{t_0}{dt\over (t/t_0)^{2/3}}=3ct_0\left [1-\left ({t_e\over t_0}\right )^{1/2}\right ]={2c\over H_0}\left [1-{1\over \sqrt {1+z}}\right ]\cr d_p(t_e)={2c\over H_0(1+z)}\left [1-{1\over \sqrt {1+z}}\right ]\cr }







5.5 Radiation Only

\displaylines{t_0={1\over 2H_0}\cr d_{hor}(t_0)=2ct_0={c\over H_0}\cr a(t)=\left ({t\over t_0}\right )^{1/2}\cr d_p(t_0)=c\int _{t_e}^{t_0}{dt\over (t/t_0)^{1/2}}=2ct_0\left [1-\left ({t_e\over t_0}\right )^{1/2}\right ]={c\over H_0} {z\over 1+z}\cr d_p(t_e)={c\over H_0(1+z)}\left [1-{1\over 1+z}\right ]={c\over H_0} {z\over (1+z)^2}\cr \epsilon _r(t)=\epsilon _0\left ({t_0\over t}\right )^2={3\over 32} {E_P\over l_P^3}\left ({t_P\over t}\right )^2\approx 0.094 {E_P\over l_P^3}\left ({t_P\over t}\right )^2\cr }

Temperature of a universe dominated by blackbody radiation:

\displaylines{T(t)=\left ({45\over 32\pi ^2}\right )^{1/4} T_P\sqrt {{t_P\over t}}\approx 0.61 T_P\sqrt {{t_P\over t}}\quad ,\quad T_P=1.4\times 10^{32} K}

5.5.1 Mean Energy per Photon

\displaylines{E_{mean}(t)\approx 2.70 kT(t)\approx 1.66 E_P\sqrt {{t_P\over t}}}

5.5.2 Number Density of Photons

\displaylines{n(t)={\epsilon _r(t)\over E_{mean}(t)}\approx {0.057\over l_P^3}\left ({t_P\over t}\right )^{3/2}}

5.5.3 Horizon

\displaylines{d_{hor}(t)=2ct=2l_P\left ({t\over t_P}\right )}

5.5.4 Visible Volume

\displaylines{V_{hor}(t)={4\pi \over 3}d_{hor}^3\approx 34l_P^3\left ({t\over t_P}\right )^3}

5.5.5 Number of Photons inside Horizon

\displaylines{N(t)=V_{hor}(t)n(t)\approx 1.9\left ({t\over t_P}\right )^{3/2}}

5.5.6 At Planck Time

\displaylines{t\sim t_P\sim 5\times 10^{-44} s\cr n\sim l_P^{-3}\sim 2\times 10^{104} m^{-3}\cr E_{mean}\sim E_P\sim 1\times 10^{28} eV\cr }

5.6 Lambda Only

\displaylines{\omega =-1}

5.6.1 Friedmann Equation

\displaylines{\dot a^2={8\pi G\epsilon _{\Lambda }\over 3c^2}a^2}

where \epsilon _{\Lambda } is constant with time.

\displaylines{\dot a=H_0a\cr H_0=\sqrt {{8\pi G\epsilon _{\Lambda }\over 3c^3}}\cr }

In an expanding universe

\displaylines{a(t)=e^{H_0(t-t_0)}}

5.6.2 Proper Distance

\displaylines{d_p(t_0)=c\int _{t_e}^{t_0}e^{H_0(t_0-t)}dt={c\over H_0}\lbrack e^{H_0(t_0-t_e)}-1\rbrack ={c\over H_0}z\cr d_p(t_e)={c\over H_0} {z\over 1+z}\cr }

6. Multi-Component Universes

6.1 Friedmann Equation

\displaylines{H(t)^2={8\pi G\over 3c^2}\epsilon (t)-{\kappa c^2\over R_0^2a(t)^2}\quad ,\quad H\equiv {\dot a\over a}\cr {\kappa \over R_0^2}={H_0^2\over c^2}(\Omega _0-1)\cr \Rightarrow \quad H(t)^2={8\pi G\over 3c^2}\epsilon (t)-{H_0^2\over a(t)^2}(\Omega _0-1)\cr \Leftrightarrow \quad {H(t)^2\over H_0^2}={\epsilon (t)\over \epsilon _{c,0}}+ {1+\Omega _0\over a(t)^2}\cr \epsilon _{c,0}\equiv {3c^2H_0^2\over 8\pi G}\cr }

Component

\omega

Energy density

\Omega

Matter

0

\epsilon _m=\epsilon _{m,0}/a^3

\Omega _{m,0}=\epsilon _{m,0}/\epsilon _{c,0}

Radiation

{1\over 3}

\epsilon _r=\epsilon _{r,0}/a^4

\Omega _{r,0}=\epsilon _{r,0}/\epsilon _{c,0}

Cosmological constant

-1

\epsilon _{\Lambda }=\epsilon _{\Lambda ,0}=constant

\Omega _{\Lambda ,0}=\epsilon _{\Lambda ,0}/\epsilon _{c,0}

In our universe, we expect

\displaylines{{H^2\over H_0^2}={\Omega _{r,0}\over a^4}+{\Omega _{m,0}\over a^3}+\Omega _{\Lambda ,0}+{1-\Omega _0\over a^2}\cr \Omega _0=\Omega _{r,0}+\Omega _{m,0}+\Omega _{\Lambda ,0}\cr }

In the Benchmark Model:

\displaylines{\Omega _0=1}

6.1.1 Radiation Dominans

During the early stages of expansion, radiation dominate, so

\displaylines{H_0t\approx \int _0^a{ada\over \sqrt {\Omega _{r,0}}}\approx {1\over 2\sqrt {\Omega _{r,0}}}a^2\cr \Leftrightarrow \quad a(t)\approx \sqrt {2\sqrt {\Omega _{r,0}}H_0t}\cr }

6.2 Matter + Curvature

\displaylines{\omega =0\cr a(t)=\left ({t\over t_0}\right )^{2/3}\cr }

6.2.1 Friedmann Equation

\displaylines{{H(t)^2\over H_0^2}={\Omega _0\over a^3}+{1-\Omega _0\over a^2}\cr \Leftrightarrow \quad {\dot a^2\over H_0^2}={\Omega _0\over a}+(1-\Omega _0)\cr }

6.2.2 Maximum Expansion

\displaylines{0={\Omega _0\over a_{\max }^3}+{1-\Omega _0\over a_{\max }^2}\cr a_{\max }={\Omega _0\over \Omega _0-1}\cr }

Density

Curvature

Ultimate fate

\Omega _0<1

\kappa =-1

Big Chill (a\propto t)

\Omega _0=1

\kappa =0

Big Chill (a\propto t^{2/3})

\Omega _0>1

\kappa =+1

Big Crunch

6.2.3 The Age t

\displaylines{H_0t=\int _0^a{da\over \sqrt {\Omega _0/a+(1-\Omega _0)}}}

Solution for \Omega _0>1:

\displaylines{a(\theta )={1\over 2} {\Omega _0\over \Omega _0-1}(1-\cos  \theta )\cr t(\theta )={1\over 2H_0} {\Omega _0\over (\Omega _0-1)^{3/2}}(\theta -\sin  \theta )\cr \theta \in \lbrack 0;2\pi \rbrack \cr }

The time between Big Bang at \theta =0 and the Big Crunch at \theta =2\pi is

\displaylines{t_{crunch}={\pi \over H_0} {\Omega _0\over (\Omega _0-1)^{3/2}}}

Solution for \Omega _0<1:

\displaylines{a(\eta )={1\over 2} {\Omega _0\over 1-\Omega _0}(\cosh  \eta -1)\cr t(\eta )={1\over 2} {\Omega _0\over (1-\Omega _0)^{3/2}}(\sinh  \eta -\eta )\cr \eta \in \lbrack 0;\infty \rbrack \cr }

6.3 Matter + Lambda

6.3.1 Flat Space

\displaylines{\Omega _{\Lambda ,0}=1-\Omega _{m,0}\cr {H^2\over H_0^2}={\Omega _{m,0}\over a^3}+(1-\Omega _{m,0})\cr }

Maximum scale factor

\displaylines{a_{\max }=\left ({\Omega _{m,0}\over \Omega _{m,0}-1}\right )^{1/3}}

Collapse to a=0 at a cosmic time

\displaylines{t_{crunch}={2\pi \over 3H_0} {1\over \sqrt {\Omega _{m,0}-1}}}

For \Omega _{\Lambda ,0}<0:

\displaylines{H_0t={2\over 3\sqrt {\Omega _{m,0}-1}}\sin ^{-1}\left [\left ({a\over a_{\max }}\right )^{3/2}\right ]}

For \Omega _{m,0}<1 \land  \Omega _{\Lambda ,0}>0, the density contribution of matter and the cosmological constant are equal at the scale factor

\displaylines{a_{m \Lambda }=\left ({\Omega _{m,0}\over \Omega _{\Lambda ,0}}\right )^{1/3}=\left ({\Omega _{m,0}\over 1-\Omega _{m,0}}\right )^{1/3}}

For \Omega _{\Lambda ,0}>0:

\displaylines{H_0t={2\over 3\sqrt {1-\Omega _{m,0}}}\ln \left [\left ({a\over a_{m \Lambda }}\right )^{3/2}+\sqrt {1+\left ({a\over a_{m \Lambda }}\right )^3}\right ]}

6.3.2 The Age

\displaylines{t_0={2H_0^{-1}\over 3\sqrt {1-\Omega _{m,0}}} \ln \left [{\sqrt {1-\Omega _{m,0}}+1\over \sqrt {\Omega _{m,0}}}\right ]}

The age at which matter and the cosmological constant had equal energy density was

\displaylines{t_{m \Lambda }={2H_0^{-1}\over 3\sqrt {1-\Omega _{m,0}}} \ln \lbrack 1+\sqrt {2}\rbrack =0.702 H_0^{-1}=9.8\pm 1.0 Gyr}

6.4 Matter + Curvature + Lambda

6.4.1 Friedmann Equation

\displaylines{{H^2\over H_0^2}={\Omega _{m,0}\over a^3}+{1-\Omega _{m,0}-\Omega _{\Lambda ,0}\over a^2}+\Omega _{\Lambda ,0}}

See p. 92!

6.5 Radiation + Matter

6.5.1 Radiation-Matter Equality

\displaylines{a_{r m}\equiv {\Omega _{r,0}\over \Omega _{m,0}}\approx 2.8\times 10^{-4}\cr {H^2\over H_0^2}={\Omega _{r,0}\over a^4}+{\Omega _{m,0}\over a^3}\cr \Leftrightarrow \quad H_0dt={ada\over \sqrt {\Omega _{r,0}}}\left [1+{a\over a_{r m}}\right ]^{-1/2}\cr \Leftrightarrow \quad H_0t={4a_{r m}^2\over 3\sqrt {\Omega _{r,0}}}\left [1-\left (1-{a\over 2a_{r m}}\right )\sqrt {1+{a\over a_{r m}}}\right ]\cr t_{r m}={4\over 3}\left (1-{1\over \sqrt {2}}\right ){a_{r m}^2\over \sqrt {\Omega _{r,0}}}H_0^{-1}\approx 0.391 {\Omega _{r,0}^{3/2}\over \Omega _{m,0}^2}H_0^{-1}\cr }

6.6 Benchmark Model

6.6.1 Properties

List of Ingredients:

photons

\Omega _{\gamma ,0}=5.0\times 10^{-5}

neutrinos

\Omega _{\nu ,0}=3.4\times 10^{-5}

total radiation

\Omega _{r,0}=8.4\times 10^{-5}

baryonic matter

\Omega _{\bar y,0}=0.04

nonbaryonic dark matter

\Omega _{dm,0}=0.26

total matter

\Omega _{m,0}=0.30

cosmological constant

\Omega _{\Lambda ,0}\approx 0.70

Important Epochs:

radiation-matter equality

a_{r m}=2.8\times 10^{-4}

t_{r m}=4.7\times 10^4 yr

matter-lambda equality

a_{m \Lambda }=0.75

t_{m \Lambda }=9.8 Gyr

Now

a_0=1

t_0=13.5 Gyr

6.6.2 Horizon

\displaylines{d_{hor}(t_0)={3.24c\over H_0}=3.12ct_0=14000 Mpc}

NicomDoc - 26-Jun-2007 - niclasen@fys.ku.dk