In the following, we present the fundamental equations, the solution process and results in form of schematic diagrams for the background calculation and for the CMB calculation.
Lambda-CDM background calculation:
Lambda-CDM CMB calculation:
This article gives a state-of-the-art description of the cosmological Lambda-CDM model and in addition, presents extensions of the model with new calculations of background and CMB functions. Chapters 1-4 describe the background part of the model, i.e. the evolution of scale factor and density according to the Friedmann equations, and its extension, which results in a correction of the Hubble parameter, in agreement with new measurements (Cepheids-SNIa and Red-Giants). Based on this improved background calculation presented in chapters 5-9 the perturbation part of the model, i.e. the evolution of perturbation and structure according to the perturbed Einstein equations and continuity-Euler equations, and the power spectrum of the cosmic microwave background (CMB) is calculated with a new own code.
The Lambda-CDM model is widely accepted as the valid description of universe on large scales and its evolution history. It is based on General Relativity and consists of two parts:
- Background part with the ansatz Robertson-Walker (RW) metric, based on Friedmann equations and equations-of-state for the different component particles. It describes the evolution of scale factor and density without perturbations, i.e. without local structure (like galaxies and galaxy groups);
- Perturbation part with the ansatz perturbed RW-metric and locally perturbed density, velocity, and pressure of the component particles. It describes the time-evolution and (quasi-random perturbed spatial distribution) of density, velocity, and pressure, i.e. the actual structure of the universe on inter-galactic scale.
The parameters of the perturbed model are fitted in chap. 10 with the CMB spatial spectrum measured by Planck.
We present here in chap. 2-5 the background part with Friedmann equations and equations-of-state for the components with two notable extensions: explicit temperature dependence and classical gas as baryon eos. From this follows a new solution and own calculation in chap. 5, which offers an explanation for the apparent experimental discrepancy concerning the Hubble parameter.
Based on the improved background calculation, we present the perturbation part in chap. 6-10, with the derivation of the CMB spectrum, and new calculation of it.
In this chapter, we present in concise form the basic equations (Friedmann equations) and equations of state (eos) for density and pressure with their different components radiation γ, neutrinos ν , electrons e, protons p, neutrons n (respectively baryons b), cold-dark-matter cdm d. The presentation relies basically on the four monographies [
-Temperature
The eos depend explicitly on temperature T, resp. thermal energy E t h = k B T , and thermal energy is introduced as a function of time E t h ( t ) , as all other variables, and has to be calculated.
-Baryon eos
The baryons are modeled as classical gas, and not as dust with zero pressure. We shall see in the background calculation in chap. 5, that this model increases the value of the Hubble parameter, which basically solves the Hubble-discrepancy problem.
The metric which fulfills the conditions of space homogeneity and isotropy is the Robertson-Walker (RW) metric [
d s 2 = − c 2 d t 2 + a 2 ( t ) ( d r 2 1 − k r 2 / R H 2 + r 2 d Ω 2 ) (1)
with Hubble radius R H = c H 0 = 1.37 × 10 26 m (Planck value), and scale factor a ( t ) .
The Einstein equations [
( a ˙ a c ) 2 + k a 2 − Λ 3 = κ 3 ρ c 2 , (2a)
2 a ¨ a c 2 + ( a ˙ a c ) 2 + k a 2 − Λ = − κ P , (2b)
a ¨ a c 2 − 1 3 Λ = − κ 2 ( P + ρ c 2 3 ) derived from a, b (2c)
ρ ˙ a 3 + a ˙ ( P c 2 + ρ ) = 0 derived: density equation (2d)
with dimensionless variables using Planck-values: Hubble constant H 0 = 67.74 km ⋅ s − 1 ⋅ Mpc − 1 , normalized Hubble constant h = 0.6774 ,
Einstein constant κ = 8 π G c 4 , κ c 2 ρ c r i t , 0 = ρ c r H R H 2 , relative pressure P r = P c 2 ρ c r i t , 0 = P ρ E c r i t , 0 = P κ R H 2 , relative cosmological constant Λ 1 = Λ R H 2 , relative density Ω = ρ ρ c r i t , 0 with critical density today
ρ E c r i t , 0 = c 2 ρ c r i t , 0 = 3 κ R H 2 ,
ρ c r i t , 0 = 3 κ R H 2 c 2 = 3 H 0 2 8 π G = 0.862 × 10 − 26 kg ⋅ m − 3 = 5.0 m p R H 3 ( 1.37 × 10 26 ) 3 = 13.0 × 10 78 m p R H 3 = 5.0 nucleon / m 3
ρ c r H = κ c 2 R H 2 ρ c r i t , 0 = 3
ρ E c r i t , 0 = 5.0 × 0.963 GeV m 3 = 4.81 GeV m 3 ,
Hubble radius R H = c H 0 = 1.37 × 10 26 m
The Friedmann equations can be reformulated dimensionless with x 0 = t c , a ' = d a d x 0 , ρ c r H = 3
( a ' a ) 2 + k a 2 − Λ 1 3 R H 2 − 1 3 ρ c r H R H 2 Ω = 0 , i.e. ( a ' a ) 2 + k a 2 − Λ 1 3 R H 2 − Ω R H 2 = 0
2 a ' ' a + ( a ' a ) 2 + k a 2 − Λ 1 R H 2 + P r R H 2 = 0
ρ r ' a 3 + a ' ( P r + ρ r ) = 0
rescaled with a R H → a
( a ' ) 2 + k − Λ 1 3 a 2 − ρ r a 2 = 0 sF1 (3a)
a ' ' a − 1 3 Λ 1 a 2 = − 3 a 2 2 ( P r + ρ r 3 ) sF2 (3b)
a ' ' a + 2 ( a ' ) 2 + 2 k − Λ 1 a 2 + 3 2 ( P r − ρ r ) a 2 = 0 sF3 (3c)
ρ r ' a 3 + a ' ( P r + ρ r ) = 0 sF4 (3d)
density eq
with
Ω m r = ρ m a t + ρ r a d ρ E c r i t , 0 , Ω m r = Ω m r , 0 ( H H 0 ) 2 , Ω E c r i t = 3 κ ( H c ) 2 ,
Ω Λ = Λ 3 ( c H ) 2 , Ω k = − k ( c H ) 2 R 0 2 .
Conformal Friedmann equations
In conformal time η, d η = d t a , with comoving distance in η: χ ( η ) = c ∫ t 1 t 0 d t a ( t ) = c ∫ η 1 η d η , or with redshift z = 1 a − 1 : χ ( z ) = c ∫ 0 z d z H ( z ) , follow the Friedmann conformal dimensionless equations [
( a ' ) 2 + k c 2 a 2 R H 2 = Λ c 2 a 4 3 + 8 π G 3 ρ a 4
a ' ' + k c 2 a R H 2 = 4 π G 3 c 2 ( ρ c 2 − 3 P ) a 3 + Λ c 2 a 3 3
and rescaled conformal:
( a ' ) 2 + k a 2 = Λ 1 a 4 3 + ρ c r H 3 ρ a 4 scF1 ( a ' ) 2 a 2 = − k + Λ 1 a 2 3 + ρ c r H 3 ρ a 2 (4a)
a ' ' + k a = ρ c r H 6 ( ρ − 3 P ) a 3 + Λ 1 a 3 3 scF2 (4b)
Friedmann radial equation
It is convenient to reformulate the first Friedmann equation in the form of velocity-potential equation, which we call here Friedmann radial equation [
We get the Friedmann radial equation
( a ˙ ) 2 − K s a 2 − K m a − Λ 3 a 2 + k = 0 (5)
it follows the potential form a ˙ 2 c 2 + V ( a ) = − k with c = 1
V ( a ) = − K s a 2 − K m a − Λ 3 a 2
with Planck data we have
K m = 0.423 × 10 26 m , K s = 1.01 × 10 48 m 2 , Λ = 1.1 × 10 − 52 m − 2
dimensionless
K m 1 = K m / R H = Ω m , 0 = 0.309
K s 1 = K s / R H 2 = Ω r a d , 0 = Ω γ , 0 + Ω ν , 0 = 0.54 × 10 − 4 + 0.0012 = 0.00125
Λ 1 = Λ R H 2 = 1.1 × 1.37 2 = 2.06
from this we get the dimensionless Friedmann radial equation
( a ˙ ) 2 − K s 1 a 2 − K m 1 a − Λ 1 3 a 2 + k = 0 (5a)
In the following, we present the eos for the components radiation γ, neutrinos ν , electrons e, protons p, neutrons n, cdm d [
Relative density & pressure baryons b, CDM c, matter density ρm,r dependent (Eth independent variable)
With thermal energy E t h = k B T matter density ρ m , r = K m 1 a 3 , b = baryon, c = cdm (cold dark matter)
ρ m , r ( a ) = ρ b + ρ c , ρ b ( ρ m , r ) = ρ m , r Ω b , 0 Ω b , 0 + Ω c , 0 , ρ c ( ρ m , r ) = ρ m , r Ω c , 0 Ω b , 0 + Ω c , 0 ,
we have for the pressure before (1) and after (2) nucleosynthesis
P b , 2 ( ρ b , E t h ) = ρ b E t h m p c 2 , E t h > E c , n s ideal gas, E m p = m p c 2 = 0.938 GeV ,
using today’s He-H-ratio Y H , H e = ρ H e ρ H = 4 n H e n H = 0.25 , ρ H e ρ H = 4 n H e n H = 0.25
P b , 1 = 1 + Y H , H e / 4 1 + Y H , H e ρ b E t h m p c 2 = 0.85 ρ b E t h m p c 2 , E t h < E c , n s , E c , n s = 100 keV ,
with the soft-1-0-step function for state-transition at ns = nucleosynthesis with transition energy E c , n s = 100 keV (see chap. 9) we get the pressure
P b ( ρ b , E t h ) = P b , 2 ( ρ b , E t h ) + ( P b , 1 ( ρ b , E t h ) − P b , 2 ( ρ b , E t h ) ) Θ 1 − 0 ( E t h , E c , n s , δ 0 E c , n s ) ,
δ 0 = 0.1 ,
P c ( ρ c , E t h ) = 0 .
Relative density & pressure neutrinos
We have for neutrino density and pressure before (1) and after (2) neutrino decoupling [
ρ ν , 1 ( ρ b , E t h ) = Ω ν b n b E t h c 2 ρ c r i t , 0 = Ω ν b ρ b E t h m p c 2 , n ν = Ω ν b n b
ρ ν , 2 ( ρ b , E t h ) = Ω ν b ρ b E t h m p c 2 , E t h > E c , ν , in thermal equilibrium,
ρ ν , 1 ( ρ b , E t h ) = Ω ν b ρ b E c . v m p c 2 ( E c . v E t h ) − 3 , E t h < E c , ν decrease with ~ a ˜ − 3
P ν ( ρ ν ) = 1 3 ρ ν , parameters today Ω ν , 0 ≈ 10 − 9 , T ν , 0 = 1.95 K ,
E t h , ν 0 = k B T ν , 0 = 1.95 K 300 K × 0.026 eV = 1.69 × 10 − 4 eV , it follows
Ω ν , b = n ν , 0 n b , 0 = Ω ν , 0 Ω b , 0 m p c 2 k B T ν , 0 = 10 − 9 0.049 0.938 GeV 1.69 × 10 − 4 eV = 1.13 × 10 5 .
Relative density & pressure photons
The Stefan-Boltzmann law gives
ρ ( T ) = a T 4 , a = 7.56 × 10 − 16 J m 3 ⋅ K 4 = 4.717 MeV m 3 ⋅ K 4 , a = 51.9 4 π k B 4 c 3 h 3 (6)
ρ ( E t h ) = a S B E t h 4
a S B = 207.6 π h 3 c 3 = a k B 4 = 7.56 × 10 − 16 ( 1.38 × 10 − 23 ) 4 1 J 3 ⋅ m 3 = 2.08 × 10 76 ( 6.24 × 10 18 ) 3 1 eV 3 ⋅ m 3 = 0.856 × 10 20 1 eV 3 ⋅ m 3
a S B = 0.856 × 10 20 1 eV 3 ⋅ m 3 = 1 eV 4 0.856 × 10 11 GeV m 3 = 1 eV 4 0.178 × 10 11 ρ E c r i t , 0
a S B 0 = a S B ρ E c r i t , 0 = 1 eV 4 0.178 × 10 11 .
Before photon decoupling the photon energy density is
ρ γ ( E t h ) = a S B 0 E t h 4 , P γ ( ρ γ ) = 1 3 ρ γ
after photon decoupling at E t h = E c , d c , E c , d c = 0.25 eV , Planck z d c = 1090 , it becomes
ρ γ ( a , E t h ) = a S B ( E c , d c a ( t c , d c ) a ) 4 , E t h < E c , d c , a ( t c , d c ) = 1 z d c + 1 = 1 1091
at e-pair production and above photons lose energy and keep a mean energy
E ≥ m e c 2 , E t h ≈ 2 m e c 2
at p-pair production and above photons lose energy and keep a mean energy
E ≥ m p c 2 , E t h ≈ 2 m p c 2 .
Temperature jumps at phase transitions
At recombination E t h = E c , r e , E c , r e = 0.29 eV temperature goes up due to free electrons forming atoms with baryons,
before recombination:
n = n b + n e = 2 n b , n b = n e , E t h = E c , r e a ( t c , r e ) a ( t ) , a ( t c , r e ) = 1 z r e + 1 = 1 1271 ,
z r e = 1270 , t c , r e = 1.16 × 10 13
after recombination: Saha equation:
X e ( E t h ) = n e n e + n H = n e n b = − 1 + 1 + 4 f ( E t h ) 2 f ( E t h ) (7)
n = n b + n e = n b ( 1 + X e ( E t h ) ) , E H , r e = 13.6 eV
f ( E t h ) = 4 ζ ( 3 ) 2 π η ( E t h m e c 2 ) 3 / 2 exp ( E H , r e E t h ) = 2.26 × 10 − 9 ( E t h m e c 2 ) 3 / 2 exp ( E H , r e E t h ) .
The equation for E t h after recombination with E H = E H , r e , E m = m e c 2 is:
d E t h d a = − E t h 0 a 2 − E H , r e d X e d f d f d a d E t h d a , d E t h d a ( 1 + E H , r e d X e d f d f d a ) = − E t h 0 a 2
with solution E t h , a ( a ) [
E t h , a ( 1 ) = E t h , 0 = 0.000663 eV , E t h , a ( a r e = 1 / ( z r e + 1 ) ) = 0.2842 eV ≈ E c , r e .
At nucleo-synthesis E t h = E c , n s , E c , n s = 100 keV temperature goes up due to helium synthesis with energy released E H e , n s = 12 MeV , thermal energy behavior is analogously for E c , r e < E t h < E H e , n s , z r e = 4 × 10 8
E t h ≈ E c , n s a ( t c , n s ) a ( t ) ( 1 + 0.021 ( ( E c , n s a ( t c , n s ) m p c 2 a ( t ) ) − 3 / 4 exp ( − E H e , n s a ( t ) 2 E c , n s a ( t c , n s ) ) − ( E c , n s a ( t c , n s ) m p c 2 a ( t H e , n s ) ) − 3 / 4 exp ( − E H e , n s a ( t H e , n s ) 2 E c , n s a ( t c , n s ) ) ) )
where the baryon temperature depends on the photon temperature
T b ' = − 2 a ' a T b + 8 3 m b m e ρ ¯ γ ρ ¯ b a n e σ T ( T γ − T b ) with a ' = d a d η [
Density electrons
The density of electrons is described by the Peebles equation with the parameters
C r ( T ) ≡ Λ 2 γ + Λ α Λ 2 γ + Λ α + β α , Λ 2 γ = 8.227 s − 1 , Λ α = 27 128 ζ ( 3 ) H ( T ) ( 1 − X e ) ( n b / n γ ) ( k B T / E I ) 3 , β α = β ( T ) exp ( 3 E I 4 k B T ) , E I = 13.6 eV = hydrogen ionization energy, 1s ionization rate, n 1 s ≈ ( 1 − X e ) n b , n b = η n γ , λ α = 8 π ℏ c 3 E I Lyman wavelength,
β ( T ) = 〈 σ v 〉 ( m e c 2 k B T 2 π ℏ 2 c 2 ) 3 / 2 exp ( − E I k B T )
α ( T ) ≈ 9.8 α 2 ( m e c 2 ) 2 ( E I k B T ) 1 / 2 log ( ( E I k B T ) )
we get the Peebles equation ( [
d X e d z = − C r ( T ) H ( z ) ( 1 + z ) ( ( m e c 2 k B T 2 π ) 1 / 2 ( 1 − X e ) exp ( − E I k B T ) − α ( T ) n b n γ 2 ζ ( 3 ) π 2 ( k B T ) 3 X e 2 ) (8)
where
H ( z ) = Ω m H 0 ( 1 + z ) 3 / 2 ( 1 + 1 + z 1 + z e q ) , H 0 ≈ 1.5 × 10 − 33 eV
T = ( 1 + z ) 0.235 eV .
We get for the electron density before (1) and after (2) recombination
ρ e , 1 ( ρ b , E t h ) = ρ b E t h m p c 2 , E < E c , e p , E c , e p = m e c 2 = 511 keV
n e + ≈ n b 2 n γ 0.17 α ( E t h m e c 2 ) 2 = n b 2 n γ ( E t h m e c 2 ) 2 1.2 × 10 − 3 , n b = Ω b , 0 ρ c r i t , 0 m p
n b n γ = n b , 0 n γ , 0 a 0 3 E t h , 0 3 a 3 E t h 3 = n b , 0 n γ , 0 = 0.242 m − 3 0.41 × 10 − 3 m − 3 = 590 scale-independent
follows n e + n b ≈ n b n γ 0.17 α ( E t h m e c 2 ) 2 = ( E t h m e c 2 ) 2 0.708 ,
ρ e , 2 ( ρ b , E t h ) = ρ b ( 1 + 2 n e + n b ) E t h + m e c 2 m p c 2 , E > E c , e p
due to Saha equation
ρ e , 0 ( ρ b , E t h ) ≈ ρ e , 1 ( ρ b ( t c , r e ) , E c , r e ) exp ( E H , r e ( 1 E c , r e − 1 E t h ) ) = ρ b ( a ( t c , r e ) ) m e m p exp ( E H , r e ( 1 E c , r e − 1 E t h ) )
alternatively
n e = n b X e ( E t h ) , ρ e = m e c 2 m p c 2 ρ b X e ( E t h )
E < E c , r e , E c , r e = 0.29 eV , ρ b ( t c , r e , E c , r e ) = Ω b , 0 z r e ,
Ω b , 0 = 0.0486 , z r e = 1270 , a ( t c , r e ) = 1 z r e + 1 = 1 1271 .
Fermi pressure electrons
The pressure of electrons is the Fermi pressure PFe of a (spin_1/2) fermion gas
P e ( ρ e , E t h ) = P F e ( ρ e , E t h )
with low- and high-density limits P 1 = 1 5 n p F c , P 2 = 2 5 n E F .
Fermi energy E F = ( p F c ) 2 + ( m e c 2 ) 2 , p F c = ℏ c ( 3 π 2 n ) 1 / 3
P F e ( ρ , E ) = P 2 ( ρ ) + ( P 1 ( ρ ) − P 2 ( ρ ) ) Θ 1 − 0 ( E , m e c 2 , δ 0 m e c 2 ) (9)
ρ c r = ρ c r i t , 0 c 2 = 0.77 × 10 − 10 J ⋅ m − 3 = 0.484 × 10 3 MeV ⋅ m − 3
n p , 0 = ρ c r i t , 0 c 2 Ω b , 0 m p c 2 = 0.484 × 10 3 MeV ⋅ m − 3 × 0.047 0.938 GeV = 0.0242 m − 3
ℏ c = 1.96 × 10 − 16 GeV ⋅ m = 1.96 × 10 − 5 eV ⋅ m
n e = ρ c r i t , 0 c 2 ρ e m e c 2 = n p , 0 m p Ω b , 0 m e ρ e = 0.0242 m − 3 ρ e 39.0 × 10 3 = 943.8 ρ e
n e n p , 0 = m p Ω b , 0 m e ρ e = 339055.6 ρ e .
For electrons we get the expressions
P 1 = 1 5 n p F c ρ c r i t , 0 = 1 5 ( n e Ω b , 0 n p , 0 ) p F c m p c 2 = 1 5 ( m p m e ρ e ) p F c m p c 2 = 1 5 ( m p m e ρ e ) 201.78 ( ρ e ) 1 / 3 m p c 2
P 2 = 2 5 n E F ρ c r i t , 0 = 1 5 ( n e Ω b , 0 n p , 0 ) E F m p c 2 = 1 5 ( m p m e ρ e ) ( p F c ) 2 + ( m e c 2 ) 2 m p c 2
p F c = ℏ c ( 3 π 2 n e , 0 ) 1 / 3 ( ρ e ) 1 / 3 33.91 = 1.96 × 10 − 5 eV ⋅ m ( 3 π 2 0.947 × 10 3 m − 3 ) 1 / 3 33.91 ( ρ e ) 1 / 3 = 201.78 ( ρ e ) 1 / 3 eV
p F c = 201.78 ( ρ e ) 1 / 3 eV .
State transitions radiation γ, neutrinos ν , electrons e, protons p, neutrons n, cdm d.
Generally, the density state transition from ρ 1 to ρ 2 at transition temperature Tc (transition thermal energy E c = k B T c ) has the form ρ ( E ) = ρ 2 + ( ρ 1 − ρ 2 ) Θ 1 − 0 ( E , E c , δ E c ) ,
with soft-0-1-step function Θ 0 − 1 ( E , E c , δ E c ) = 1 1 + exp ( E c − E δ E c ) ,
with soft-1-0-step function Θ 1 − 0 ( E , E c , δ E c ) = 1 + exp ( − E c δ E c ) 1 + exp ( E − E c δ E c ) ,
where δ E c is the standard deviation of E c .
We can set approximately δ E c E c = δ T c T c ≈ δ T 0 T 0 , where (measured in CMB) δ T 0 T 0 = Δ T γ , 0 T γ , 0 ≈ 30 μ K 2.72 K = 1.1 × 10 − 5 .
-neutrino decoupling E c , ν = 1 MeV , t c , ν = 1 s , ρ 1 c , ν = ρ 1 , ν ( t c , ν ) , ρ 1 , ν ( E t h ) = E t h , ρ 2 , ν ( E t h , a ) = ρ 1 c , ν ( a a ( t c , ν ) ) 4 ;
-e-p-annihilation
E c , e p = 0.5 MeV , t c , e p = 6 s , n γ = a S B E t h 4 for all t a = 7.56 × 10 − 16 J / m 2 ⋅ K 4 , a = 51.9 4 π k B 4 c 3 h 3
ρ 1 , e = ( n b + n e + ( t c , e p ) ) m e , ρ 2 , e = n b m e with n e + ≈ n b 2 n γ 0.17 α ( E t h m e c 2 ) 2 = n b 2 n γ ( E t h m e c 2 ) 2 1.2 × 10 − 3 ;
-photon recombination
E c , r e = 0.29 eV , t c , r e = 29 0 ky , ρ 2 c , r e = ρ 1 c , r e + n b ( t c , r e ) E c , r e
ρ 1 , e = n b m e , ρ 2 , e = 1 2 ρ 1 , e exp ( E t h − E c , r e E t h ) ;
-photon decoupling
E c , γ = 0. 25 eV , t c , γ = 37 0 ky , ρ 1 c , γ = ρ 1 , γ ( t c , γ ) ,
ρ 1 , γ ( E t h ) = E t h , ρ 2 , γ ( E t h , a ) = ρ 1 c , γ ( a a ( t c , γ ) ) 4 ;
-nucleo-synthesis helium
E c , n s = 100 keV , t c , n s = 3 min , 4 p + + 2 e − → He 2 + , ratio ρ H e ρ p = 0.25 , eos transition 1 → 2 with ideal gas P 1 = n b E t h = ρ b E t h m p , t < t c , n s , with ideal gas P 2 = n b , 1 ( 0.75 + 0.25 / 4 ) E t h = n b , 1 0.81 E t h = 0.81 ρ b E t h m p , t < t c , n s .
The simple ΛCDM model is based on seven parameters: physical baryon density parameter Ωbh2; physical matter density parameter Ωmh2; the age of the universe t0; scalar spectral index ns; curvature fluctuation amplitude As; and reionization optical depth τ, dark energy density ΩΛ.
The parameters of the ΛCDM are given in the following table (
11 independent parameters: Ωbh2, Ωch2, t0, ns, Δ R 2 , τ, Ωt, w, ∑mν, Neff(ν), As;
7 fixed parameters r, dns/d lnk, H0, Ωb, Ωc, Ωm, ΩΛ;
5 calculated parameters ρcrit, σ8, zdec, tdec, zre;
13 total parameters Ωb, Ωc, t0, ns, As, τ, ΩΛ, w, ∑mν, Neff(ν), r, dns/dk, H0;
derived parameters ρcrit, σ8, zdec, tdec, zre, ωb = Ωbh2, ωm = Ωmh2.
| Description | Symbol | Value | |
|---|---|---|---|
| Independent parameters 11 | Physical baryon density parameter | Ωbh2 | 0.02230 ± 0.00014 |
| Physical dark matter density parameter | Ωch2 | 0.1188 ± 0.0010 | |
| Age of the universe | t0 | 13.799 ± 0.021 × 109 years | |
| Scalar spectral index | ns | 0.9667 ± 0.0040 | |
| Curvature fluctuation amplitude, k0 = 0.002 Mpc−1 | Δ R 2 | 2.441 + 0.088 − 0.092 × 10−9 | |
| Reionization optical depth | τ | 0.066 ± 0.012 | |
| Fixed parameters 7 | Total density parameter | Ωtot | 1 |
| Equation of state of dark energy | w | −1 | |
| Sum of three neutrino masses | ∑mν | 0.06 eV/c2 | |
| Effective number of relativistic degrees of freedom | Neff | 3.046 | |
| Scalar amplitude | As | (2.215 ± 0.13) | |
| Tensor/scalar ratio | r | 0 | |
| Running of spectral index | dns/dlnk | 0 | |
| Calculated values 5 | Hubble constant | H0 | 67.74 ± 0.46 km·s−1·Mpc−1 |
| Baryon density parameter | Ωb | 0.0486 ± 0.0010 | |
| Dark matter density parameter | Ωc | 0.2589 ± 0.0057 | |
| Matter density parameter | Ωm | 0.3089 ± 0.0062 | |
| Dark energy density parameter | ΩΛ | 0.6911 ± 0.0062 | |
| Critical density | ρcrit | (8.62 ± 0.12) × 10−27 kg/m3 | |
| Fluctuation amplitude at 8 h−1 Mpc | σ8 | 0.8159 ± 0.0086 | |
| Redshift at decoupling | z* | 1089.90 ± 0.23 | |
| Age at decoupling | t* | 377,700 ± 3200 y | |
| Redshift of reionization (with uniform prior) | zre | 8.5 + 1.0 − 1.1 |
The additional parameters of the extended ΛCDM are given in the second table (
Some specifications
The amplitude As, is determined by the CMB power spectrum
Δ R 2 ( k 2 ) = A s ( k k 0 ) n s − 1 , k 0 ≈ 0.05 Mpc − 1 .
The relative current Hubble parameter is h = H 0 100 .
The fluctuation amplitude is defined by σ 8 = σ ( ρ m a t , R ) R = 8 h − 1 Mpc , where σ ( ρ m a t , R ) = stdev ( ρ m a t ) smoothed by distance R ( [
Key cosmological events
Key cosmological events calculated from the ΛCDM model with temperature, energy scale and cosmic time are given below [
| Description | Symbol | Value |
|---|---|---|
| Total density parameter | Ωtot | 1.0023 + 0.0056 − 0.0054 |
| Equation of state of dark energy | w | −0.980 ± 0.053 |
| Tensor-to-scalar ratio | r | <0.11, k0 = 0.002 Mpc−1 (2σ) |
| Running of the spectral index | dns/dlnk | −0.022 ± 0.020, k0 = 0.002 Mpc−1 |
| Physical neutrino density parameter | Ωνh2 | <0.0062 |
| Sum of three neutrino masses | ∑mν | <0.58 eV/c2 (2σ) |
| Event | Temperature | Energy | Time |
|---|---|---|---|
| Inflation ends | 1029 K | 1016 GeV | 10−35 s |
| CDM decouples, GUT scale | 1029 K | 1015 GeV | 10−36 s |
| Baryons form | 1016 K | 1 TeV? | 10−12 s |
| El-weak force | 1015 K | 100 GeV | 10−11 s |
| Hadrons form | 1012 K | 150 MeV | 10−5 s |
| Neutrinos decouple | 1010 K | 1 MeV | 1 s |
| Nuclei form | 109 K | 100 keV | 200s |
| Atoms form | 3460 K | 0.29 eV | 290 ky |
| Photons decouple | 2970 K | 0.25 eV | 370 ky |
| First stars | 50 K | 4 meV | 100 My |
| First galaxies | 12 K | 1 meV | 400 My |
| Dark energy domination | 3.8 K | 0.33 meV | 9 Gy |
| Now | 2.7 K | 0.24 meV | 13.8 Gy |
The “naive” so called Hot-Big-Bang model has several aspects, which are in disagreement with cosmological observations.
Hot Big-bang problems
- the observed homogeneity of the present universe (distances > 200 Mly) should arise from arbitrary initial conditions: horizon problem;
- the observed curvature is small: flatness problem;
- the observed correlation regions in the CMB have supraluminal distance: superhorizon correlations.
Cosmological inflation
In the approximation that the expansion is exactly exponential, the horizon is static, i.e. H = a ˙ a ≈ c o n s t , and we have an inflating universe [
d s 2 = − ( 1 − Λ r 2 ) c 2 d t 2 + 1 1 − Λ r 2 d r 2 + r 2 d Ω 2 (10a)
For the case of exponential expansion, the equation of state is P = − ρ , with world radius
R ( t ) = R 0 exp ( c t Λ 3 ) (10b)
The expansion generates an almost-flat and large-scale-homogeneous universe, as it is observed today.
Furthermore, horizon R H = a ˙ − 1 = ( H a ) − 1 reaches a minimum at the end of inflation, and then rises again, this explains superluminal correlations in the present universe.
Inflation in Ashtekar-Kodama quantum gravity [
Inflation takes place between r i = l p = 1.61 × 10 − 35 m and R inf = r g r = 3.1 × 10 − 5 m with expansion factor f inf = exp ( r inf Λ 3 ) = 1.9 × 10 30 , r inf = 2 × 10 − 26 m , E inf = ℏ c r inf = 1.96 × 10 − 16 GeV 2 × 10 − 26 m = 0.98 × 10 10 GeV , t inf = r inf c = 0.66 × 10 − 34 s , R inf = 10 − 2 m .
Inflation with standard assumptions ( [
r i = 3 × 10 − 28 m , t inf = 10 − 36 s , f inf = 10 30 , a inf = 10 − 28 , R inf = 3 × 10 2 m ,
f inf = exp ( r inf Λ 3 ) , Λ = 3 ( log ( f inf ) r inf ) 2 = 1.4 × 10 60 m − 2 ,
H = Λ 3 = log ( f inf ) r inf = 6.9 × 10 29 m − 1 .
Assessment of the inflation factor ( [
f = end inflation, i = start inflation, eq = matter-radiation-equality, 0 = today, ER = f = expansion rate
a ( t f ) a ( t i n ) = exp N , N ≫ log ( T f T e q ) + 1 2 log ( T e q T 0 ) ,
T f ≃ 10 16 GeV , T e q ≃ 1 eV , T 0 ≃ 10 − 4 eV
N ≥ 60 , Δ t ≥ 60 H ( t f ) ≃ 60 3 8 π G ρ E R ( T 0 T f ) 2 ≈ 10 − 37 s .
Inflaton model ϕ ( t , x ) with GR-action
The action is ( [
S = ∫ d 4 x − g ( L E H + L ϕ )
with the Einstein-Hilbert action of GR
S E H = ∫ ( R − 2 Λ 2 κ ) − g d 4 x
L E H = R − 2 Λ 2 κ
and the inflaton action
S ϕ = ∫ d 4 x − g ( ℏ c 2 g μ ν ∂ μ ϕ ∂ ν ϕ − V ( ϕ ) )
L ϕ = ℏ c 2 g μ ν ∂ μ ϕ ∂ ν ϕ − V ( ϕ )
with energy-momentum T μ ν = ℏ c ∂ μ ϕ ∂ ν ϕ − g μ ν ( ℏ c 2 g μ ν ∂ μ ϕ ∂ ν ϕ − V ( ϕ ) )
T 0 0 = ℏ c ϕ ˙ 2 2 + V ( ϕ ) , T i j = − δ i j ( ℏ c ϕ ˙ 2 2 − V ( ϕ ) ) .
For RW-metric the action is S = ∫ d 4 x − g ( ℏ c ( − ϕ ˙ 2 2 + 1 2 a 2 ( ∇ ϕ ) 2 ) − V ( ϕ ) )
with eom = Klein-Gordon equation ϕ ¨ + 3 H ϕ ˙ + 1 ℏ c d V ( ϕ ) d ϕ = 0
which represents an oscillator with Hubble-friction 3 H ϕ ˙
and energy density ρ ϕ = ℏ c ϕ ˙ 2 2 + V ( ϕ ) ,
and pressure P ϕ = ℏ c ϕ ˙ 2 2 − V ( ϕ ) (4.50).
If E k i n ≡ 1 2 ϕ ˙ 2 ≪ E p o t ≡ V ( ϕ ) , E k i n = ℏ c ϕ ˙ 2 2 ≪ E p o t = V ( ϕ ) , we have P ϕ ≈ − ρ ϕ i.e. equation-of-state of dark energy Ω Λ generating temporary inflation.
We get the Friedmann equations (radiation-matter density ρ r m added)
H 2 = κ 3 ρ E = κ 3 ( ℏ c ϕ ˙ 2 2 + V ( ϕ ) + ρ r m ) (11a)
H ˙ = − κ 2 ( ρ ϕ + P ϕ − ρ r m − P r m ) = − κ 2 ( ℏ c ϕ ˙ 2 − 4 3 ρ r m ) (11b)
and the Klein-Gordon equation
ϕ ¨ + 3 H ϕ ˙ + 1 ℏ c d V ( ϕ ) d ϕ = 0 (11c)
We get dimensionless 2 equations in Planck-units l P l = 1.62 × 10 − 35 m ,
ρ r m = 3 8 π H 2 − ϕ ˙ 2 2 − V ( ϕ )
Friedmann H ˙ = − 4 π ( ϕ ˙ 2 − 4 3 ( 3 8 π H 2 − ϕ ˙ 2 2 − V ( ϕ ) ) ) = − 4 π ( 3 ϕ ˙ 2 2 − H 2 2 π + 4 3 V ( ϕ ) ) .
Klein-Gordon ϕ ¨ + 3 H ϕ ˙ + d V ( ϕ ) d ϕ = 0 .
Slow-roll approximation
If E k i n ≡ 1 2 ϕ ˙ 2 ≪ E p o t ≡ V ( ϕ ) or ε H ≪ 1 , ε H ≡ − H ˙ H 2 (slow-roll parameter 1), and almost constant velocity, η H = − ϕ ¨ H ϕ ˙ ≪ 1 (slow-roll parameter 2), we have persisting slow-roll condition ε H ≪ 1 , η H ≪ 1 (slow-roll approximation), which yields approximate fundamental equations with approximations 3 H ϕ ˙ ≈ − V ′ and 3 H 2 ≈ 8 π G V and ε H = − H ˙ H 2 = − V ′ 2 V ϕ ˙ H = 1 16 π G ( V ′ V ) 2 and η H = − ϕ ¨ H ϕ ˙ = V ″ 3 H 2 = 1 8 π G ( V ″ V ) and for the scale factor
a ( t ) = a ( t i n ) exp ( ∫ t i n t H ( t ) d t ) = a ( t i n ) exp ( − 8 π G ∫ t i n t V V ′ d ϕ ) .
Square potential
We use the square potential V ( ϕ ) = c 1 + c 2 ( ϕ − ϕ 0 ) 2 , c 1 = 1.16 × 10 − 124 , slow-roll condition: c 1 ≪ c 2 with the minimum value V ( ϕ 0 ) = c 1 = Λ κ = 1.16 × 10 − 124 and r inf = 2 × 10 − 26 m , we get the following relations:
a ( t ) = a ( t i n ) exp ( ∫ t i n t H ( t ) d t ) = a ( t i n ) exp ( − 8 π ∫ t i n t V V ′ d ϕ )
a ( t ) = a ( t i n ) exp ( 4 π ∫ 0 ϕ 0 ( ϕ − ϕ 0 ) d ϕ ) = a ( t i n ) exp ( 2 π ϕ 0 2 )
ϕ 0 = 1 2 π log ( a ( t ) a ( t i n ) ) = 1 2 π log ( f inf ) = 3.31
ε H = 1 16 π ( V ′ V ) 2 = 1 16 π ( 2 c 1 c 2 ( ϕ − ϕ 0 ) + ( ϕ − ϕ 0 ) ) 2 ≈ 1 4 π 1 ( ϕ − ϕ 0 ) 2
η H = 1 8 π ( V ″ V ) = 1 8 π ( 2 c 2 c 1 + c 2 ( ϕ − ϕ 0 ) 2 ) ≈ 1 4 π 1 ( ϕ − ϕ 0 ) 2
ρ r m = 3 8 π H 2 − ϕ ˙ 2 2 − V ( ϕ )
for t → ∞ , ϕ ˙ = δ c 1 ≪ 1 , H = H 0 , ϕ → ϕ 0 , ρ r m = ( 3 8 π H 0 2 − c 1 ) = 0 ,
so condition for convergence is: c 1 = 3 8 π H 0 2 .
The fundamental equations become
Friedmann H ˙ = − 4 π ( ϕ ˙ 2 − 4 3 | 3 8 π H 2 − ϕ ˙ 2 2 − V ( ϕ ) | ) ;
Klein-Gordon ϕ ¨ + 3 H ϕ ˙ + d V ( ϕ ) d ϕ = 0 ;
slow-roll H ˙ ≈ − 6 π ϕ ˙ 2 ;
3 boundary conditions for t = l P l = 1 : H ( 1 ) = H 1 , ϕ ( 1 ) = ϕ 1 , ϕ ˙ ( 1 ) = ϕ d 1 ;
with 3 potential parameters c 1 , c 2 , ϕ .
Example: δ c 1 = 0.05 , H 0 = 5 , ϕ 0 = 2.3 , c 1 = 3 , c 2 = 1 [
Below in
There are basically two possible ways for background calculation:
-numerical solution of two Friedmann equations in two variables, calculating backward from boundary conditions at present time x0;
-analytical solution, where the second equation is solved analytically, and inserted into the first, which gives an integral, which is calculated numerically.
The numerical solution encounters the problem of limited convergence: it stops at some time xc.
The analytical solution avoids the convergence problem, and this solution scheme is used in the calculation of results presented below.
We solve for dimensionless function variables a , ρ r , in dimensionless relative time variable x = t c R H , limits 0 ≤ x ≤ x 00 = 0.96 , where the upper limit is the relative cosmic time today x 00 = c t 0 R H = R 0 R H = 0.96 , from Planck data t 0 = 13.9 × 10 9 y , with boundary conditions: ρ r ( x 0 ) = Ω m , 0 + Ω r a d , 0 , a ( x 0 ) = 1 , a ' ( x 0 ) = 1 (because H ( x 0 ) = R H ) from a ' ( x 0 ) = 1 follows k 0 = − 0.0042 which is compatible with Planck data
( a ' ) 2 + k 0 − Λ 1 3 a 2 − ρ r a 2 = 0 sF1 (3a)
a ' ' a − 1 3 Λ 1 a 2 = − a 2 ρ c r H 2 ( P r + ρ r 3 ) sF2 (3b)
a ' ' a + 2 ( a ' ) 2 + 2 k − Λ 1 a 2 + ρ c r H 2 ( P r − ρ r ) a 2 = 0 sF3 (3c)
ρ r ' a 3 + a ' ( P r + ρ r ) = 0 sF4 (3d)
The two independent (3c and 3d is derived) Equations (3a, 3d) are non-linear second-order differential equations quadratic in the variables a , ρ r .
Alternatively, one can solve for function variables a, E t h = k B T , the latter with thermal energy E t h = k B T , photon density ρ γ = a S B 0 E t h 4 , P γ ( ρ γ ) = 1 3 ρ γ , mattter density ρ m a t = ρ b + ρ c = K m a K s + K m a ρ r , baryon density ρ b = ρ m a t Ω b , 0 Ω b , 0 + Ω c , 0 , cold-dark-matter (cdm) density ρ c = ρ m a t Ω c , 0 Ω b , 0 + Ω c , 0
P b ( ρ b , E t h ) = ρ b E t h m p c 2 .
The additional equation for pressure is the equation-of-state (eos) for the pressure P r : P r = P ( a , ρ r ) .
Solution 1
One solves numerically [
Solution 2
One solves numerically [
Plot a(x) is shown below [
The solution limit x 1 c = 0.0196 indicates the transition from matter-dominated to the radiation-dominated regime, which happens approximately at photon decoupling time t r e = 370 ky , x r e = 0.000026 . For x ≤ x 1 c solution is continued by pure radiation density ( [
Solution 3
One solves numerically [
The solution exists until x 1 c = 0.0055 , where numerical integration stops converging, and the solution becomes complex (i.e. Im ( a ) ≠ 0 ).
Plot a(x) is shown below [
The solution limit x 1 c = 0.0055 indicates the transition from matter-dominated to the radiation-dominated regime, which happens approximately at photon decoupling time t r e = 370 ky , x r e = 0.000026 . For x ≤ x 1 c solution is continued by pure radiation density ( [
The analytic solution scheme transforms the two basic equations into a parameterized integral x ( a ) , which is the inverted scale factor a ( x ) .
In order to calculate the thermal energy, we apply an iteration, we calculate the temperature E t h ( a ) from ρ r a d ≡ ρ γ + ρ ν = K s K s + K m a ρ r , using the solution a ( x ) in the next iteration: E t h ( n + 1 ) = E t h ( n ) ( a ( n ) ( x ) ) , as shown in the schematic in chap. 11.
The zero iteration is the “naive” thermal energy E t h ( 0 ) = E t h , 0 / a .
The variables are scale factor and density a , ρ r .
The boundary conditions are ρ r ( x 0 ) = Ω m , 0 + Ω r a d , 0 , a ( x 0 ) = 1 , a ' ( x 0 ) = 1 , from a ' ( x 0 ) = 1 follows k = − 0.0042 which is compatible with Planck data
( a ' ) 2 + k 0 − Λ 1 3 a 2 − ρ r a 2 = 0 sF1 (3a)
ρ r ' a 3 + a ' ( P r + ρ r ) = 0 sF4 (3d)
The two Equations (3ad) are non-linear first-order differential equations quadratic in the variables a , ρ r .
The third equation is the equation-of-state (eos) for the pressure P r : P r = P ( a , ρ r ) .
The density and pressure have the form: relative energy density ρ r = ρ b + ρ γ + ρ c + ρ e + ρ ν for baryons, photons, dark matter, free electrons, neutrinos, relative pressure P r = P b + P γ + P c + P e + P ν , where radiation pressure P r a d = P γ + P ν = ρ γ + ρ ν 3 , and matter pressure (neglecting electrons) is the baryon ideal gas pressure P m a t = P b = ρ b k B T m b c 2 , for under-nuclear temperature k B T ≪ m b c 2 = 0.94 GeV the baryon matter is dust-like, i.e. pressure is almost zero.
The densities have the form
ρ r = ρ m a t + ρ r a d
ρ m a t = ρ b + ρ c = K m a K s + K m a ρ r , ρ r a d = ρ γ + ρ ν = K s K s + K m a ρ r
ρ c = ρ m a t Ω c , 0 Ω b , 0 + Ω c , 0 , ρ b = ρ m a t Ω b , 0 Ω b , 0 + Ω c , 0
ρ γ = a S B 0 E t h 4 , ρ ν = Ω ν , 0 a 3
We calculate the temperature E t h ( a ) from ρ r a d ≡ ρ γ + ρ ν = K s K s + K m a ρ r (12a)
i.e. E t h ( a ) = 1 a S B 0 1 / 4 ( K s K s + K m a ρ r ( a ) − Ω ν , 0 a 3 ) 1 / 4 (12a1)
and all the pressure becomes a function of a,
P r ( a , ρ r ) = P r a d + P m a t = ( K s K s + K m a + K m a K s + K m a Ω b , 0 Ω b , 0 + Ω c , 0 E t h ( a ) m b c 2 ) ρ r (12b)
i.e. P r ρ r = P ρ ( a ) = ( K s K s + K m a + K m a K s + K m a Ω b , 0 Ω b , 0 + Ω c , 0 E t h ( a ) m b c 2 )
then we can integrate (3d) in a :
log ( ρ r ( a ) ) = ρ r ' a 3 + a ' ( P r + ρ r ) = − ∫ 0 a d a ( 3 + P ρ ( a ) a ) + c 1 (12c)
and then can integrate (3a) in a :
x ( a ) = ∫ 0 a d a a Λ 1 3 + ρ r ( a ) − k 0 a 2 + c 2 , (12d)
where c 1 and c 2 are set to fulfill the boundary conditions
ρ r ( x 0 ) = Ω m , 0 + Ω r a d , 0 , a ( x 0 ) = 1 , Ω = ρ ρ c r i t , 0
Results for density and relative time in dependence of scale factor ρ r ( a ) , x ( a ) , are shown below [
Relative density in ρ c r i t , 0 units is shown over scale factor a, in double-logarithmic plot
There is a critical point a T ≈ 0.5 × 10 − 4 , where the density changes its behavior, it coincides roughly with the critical point in temperature. The corresponding time is x T ≈ 10 − 8 , thermal energy E t h ≈ 1 eV .
The analytic solution yields directly the inverse scale factor function x ( a ) , it shown in
There is a critical point at photon decoupling, a d e c = 0.9 × 10 − 3 , x d e c = 0.3 × 10 − 4 ≙ 370 ky , redshift z d e c = 1090 , thermal energy E t h = 0.25 eV .
The scale factor changes its power-law dependence on time:
a ( x ) ≅ { x , x > x d e c x 1 / 2 , x < x d e c
It is useful to compare the result for x ( a ) from the analytical solution and the standard CAMB solution ( [
The plots of density ρ r ( a ) (blue) and radiation density ρ r a d ( a ) are shown in comparison below ( [
The Hubble parameter is approximately linear in x, as it should be. However, there is a small deviation at critical point x c H ≈ 10 − 8 , scale factor a c H ≈ 0.5 × 10 − 4 , redshift z c H ≈ 1 / a ≈ 20000 .
This is apparently responsible for the small correction of the present Hubble constant H0, compared to CAMB solution.
The plot of the Hubble parameter is shown in
The “naive” temperature E t h ( 0 ) ( a ) from (12a) is compared to the iterated temperature E t h ( 1 ) ( a ) calculated from the first analytic solution in (12a1) is shown in
Hubble parameter
Baryon pressure correction
Baryon pressure correction yields t 0 c = t 0 / 1.043 t 0 , so H 0 c = 1.043 H 0 , the corrected Planck-value is H 0 P c = H 0 P × 1.043 = 70.6 ± 0.4 ;
H 0 R = 69.8 ± 1.7 Red-Giants Freedmann 09/21;
H 0 S = 73.04 ± 1.04 Cepheids-SNIa SHOES 12/21;
H 0 P = 67.66 ± 0.42 Planck 07/18.
H0R Red-Giants is in agreement with corrected Planck within error margin.
Assessed correction of the Cepheids-SNIa-measurement
Cepheids-SNIa-measurement based on time-brightness calibration for small redshift z, peak power P max ~ T ( t c r ) ~ m ¯ b , with average nucleus mass m ¯ b percentage of higher-mass nuclei at present: r ( O ) = 1.04 % , r ( C ) = 0.46 % , so P max ( z ≫ 1 ) P max ( z ≪ 1 ) ≈ ( 1 + r ( O ) + r ( C ) ) = 1.015 so z-corrected Cepheids-SNIa becomes 73.04/1.015 = 72. H 0 S c = H 0 S / 1.015 = 72. ± 1. , which is at error margin.
The Lambda-CDM model is locally homogeneous, but during inflation the quantum fluctuations are “blown-up”, and the universe becomes inhomogeneous on small (galactic) scales and remains homogeneous on large scales. These local inhomogeneities generate structure, which we observe today.
In order to reproduce these local inhomogeneities in the perturbed Lambda-CDM model, we introduce small perturbations in the metric and in the density distribution. These perturbations are functions of conformal time η (defined by d η = d t a ), and space location vector x i , and are not random variables. The randomness is introduced by initial conditions for perturbations (see chap. 8).
We introduce metric perturbations A , B i , E i j in the RW-metric [
d s 2 = a 2 ( η ) ( − ( 1 + 2 A ) d η 2 + 2 B i d x i d η + ( δ i j + 2 E i j ) d x i d x j ) (13)
and split-up in scalar, vector, tensor parts:
scalar A
B i = ∂ i B + B ^ i , scalar B, vector B ^ i
E i j = C δ i j + ∂ i ∂ j E + ( ∂ i E ^ j − ∂ j E ^ i ) + E ^ i j , scalar C E, vector E ^ i , tensor E ^ i j , where ∑ i E i i = 3 C
Furthermore, we form the gauge-invariant Bardeen variables with 8 = 1scalar (A) + 3vector (Bi) + 4tensor (Eij) degrees-of-freedom (dof’s)
Ψ = A + H ( B − E ' ) + ( B − E ' ) ' , Φ = − C + 1 3 ∇ 2 E − H ( B − E ' ) ,
Φ ^ i = B ^ i − E ^ i ' , E ^ i j
Since we have 6 Einstein equations, we can remove the 8 − 6 = 2 dof’s by gauge-fixing.
▪ Newtonian gauge B = E = 0
d s 2 = a 2 ( η ) ( − ( 1 + 2 Ψ ) d η 2 + ( 1 − 2 Φ ) δ i j d x i d x j )
A = Ψ , C = − Φ (6.30)
▪ Spatially flat gauge C = E = 0
▪ Synchronous gauge A = B = 0
From now on, we use the Newtonian gauge.
We get for the energy-density tensor
T 0 0 = − ( ρ ¯ + δ ρ )
T 0 i = − ( ρ ¯ + P ¯ ) v i
T j i = − ( P ¯ + δ P ) δ j i + Π j i , Π i i = 0 ∀ i (14)
The relativistic Euler equation is
( ρ c 2 1 − ( v / c ) 2 + p ) 1 c 2 1 − ( v / c ) 2 d d t ( v i 1 − ( v / c ) 2 ) + ∂ i p + 1 c 2 1 − ( v / c ) 2 d p d t v i = 0 ,
The Euler equation in the RW metric becomes
v i ' = − ( H + P ¯ ' P ¯ + ρ ¯ ) v i − 1 P ¯ + ρ ¯ ( ∂ i δ P + ∂ j Π i j ) − ∂ i Ψ (6.76)
where Π i j is the anisotropic stress with the decomposition
Π i j = ∂ i ∂ j Π + ( ∂ i Π ^ j − ∂ j Π ^ i ) + Π ^ i j (6.39)
Finally, we get 10 fundamental equations:
6 Einstein equations
[
∇ 2 Φ − 3 H ( Φ ′ + H Ψ ) = π G a 2 δ ρ
Φ ′ + H Ψ = π G a 2 a ″ a ′ H
∂ i ∂ j ( Φ − Ψ ) = 8 π G a 2 Π i j , i < j
Φ ″ + H Ψ ′ + 2 H Φ ′ + 1 3 ∇ 2 ( Φ − Ψ ) + ( 2 H ′ + H 2 ) Ψ = π G a 2 δ P (15a-d)
4 conservation equations: continuity +Euler
[
δ ′ = − ( 1 + P ¯ ρ ¯ ) ( ∂ i v i − 3 Φ ′ ) − 3 H ( δ P δ ρ − P ¯ ρ ¯ ) δ
v i ' = − ( H + P ¯ ' P ¯ + ρ ¯ ) v i − 1 P ¯ + ρ ¯ ( ∂ i δ P + ∂ j Π i j ) − ∂ i Ψ (15ef)
q i = ( ρ ¯ + P ¯ ) v i , δ = δ ρ ρ ¯ decelaration conformal q = − a ″ a ′ H , T i 0 = ∂ i q ,
for 10 variables 4 scalar Φ , Ψ , δ , δ P , 3 vector v i , 3 tensor Π j i ;
initial conditions 6
Φ 2c, Ψ 1c, v i 3c, ( δ , δ P ) 0c;
background parameters
H = a ′ a , q = − a ″ a ′ H , a , ρ ¯ , P ¯ .
Fundamental equations in k-space ( [
In the following, we transform the fundamental equations via Fourier-transform into k-space.
We use Newtonian gauge, conformal time η , a ′ = d a d η , the metric in Newtonian gauge reduces to
d s 2 = a ( η ) ( − ( 1 + 2 Ψ ) d η 2 + ( 1 − 2 Φ ) d x i d x i )
We get 4 Einstein equations in k-space
k 2 Φ − 3 H ( Φ ′ + H Ψ ) = π G a 2 δ ρ
k 2 ( Φ ′ + H Ψ ) = π G a 2 ( P ¯ + ρ ¯ ) θ
k 2 ( Φ − Ψ ) = 12 π G a 2 ( P ¯ + ρ ¯ ) σ
Φ ″ + H ( Ψ ′ + 2 Φ ′ ) + 1 3 k 2 ( Φ − Ψ ) + ( 2 H ′ + H 2 ) Ψ = 4 π G a 2 δ P (16a-d)
and 2 continuity-Euler equs in k-space
δ ′ = − ( 1 + P ¯ ρ ¯ ) ( θ − 3 Φ ′ ) − 3 H ( δ P ρ ¯ δ − P ¯ ρ ¯ ) δ density equ
θ ′ = − ( H + P ¯ ′ P ¯ + ρ ¯ ) θ − δ P P ¯ + ρ ¯ k 2 − k 2 σ + k 2 Ψ velocity equ (16ef)
with the definitions
δ = δ ρ ρ ¯ , θ = i k j v j , σ = − ( k ^ i k ^ j − δ i j 3 ) Π i j P ¯ + ρ ¯ ,
where k ^ = k → k is the k-unit-vector, Π j i anisotropic stress
and the relations
T 0 0 = − ( ρ ¯ + δ ρ ) , T i 0 = ( ρ ¯ + P ¯ ) v i , T j i = ( P ¯ + δ P ) δ j i + Π j i , δ = δ ρ ρ ¯ = − T 0 0 ρ ¯
Π i i = 0 , i = 1 , 2 , 3 , Π j i ≡ T j i − T k k δ j i
θ = i k j v j , ( ρ ¯ + P ¯ ) θ = i k j δ T j 0 , ( ρ ¯ + P ¯ ) σ = − ( k ^ i k ^ j − 1 3 δ i j ) Π j i .
We have here 6 variables Φ , Ψ , θ , σ , δ , δ P , δ P = δ T i i , δ ρ = δ T 0 0 , which are functions of ( k , η ) .
We introduce here density distribution momenta for density components radiation γ, neutrinos ν , electrons e, baryons b, cold-dark-matter d. The densities acquire their random nature from random initial conditions, and have therefore a (Gaussian) probability distribution. These distribution momenta are used in the calculation of CMB spectrum in chap. 10.
Evolution of distribution function momenta (Ma [
We have for Newtonian gauge, conformal time η , a ′ = d a d η
d s 2 = a ( η ) ( − ( 1 + 2 Ψ ) d η 2 + ( 1 − 2 Φ ) d x i d x i ) .
Phase space distribution
With phase space element d x 1 d x 2 d x 3 d P 1 d P 2 d P 3
d N = f ( x i , P j , η ) d x 1 d x 2 d x 3 d P 1 d P 2 d P 3 particle number in element (32)
P i = a ( 1 − Φ ) p i co-moving disturbed momentum
density distribution for matter fermions (Fermi-Dirac distribution +), density distribution for radiation bosons (Bose-Einstein distribution -)
f 0 ( ε , T ) = g s h 3 1 exp ( ε k B T ) ± 1 (17)
energy ε = a p 2 + m 2 = P 2 + a 2 m 2 , temperature T, today temperature T0.
We change variables: x i P j to x i q j , and get the expressions:
scaled momentum q j = a p j = q n j , unit momentum vector n ^ with n i n i = 1
energy ε = q 2 + a 2 m 2 ;
change distribution f ( x i , P j , η ) to f ( x i , q , n j , η ) .
Finally we get for the neutrino distribution perturbation function ψ ( x i , q , n j , η ) (not equal to the metric perturbation Ψ )
f ( x i , P j , η ) = f 0 ( ε , T ) ( 1 + ψ ( x i , q , n j , η ) ) (35)
for the distribution of energy tensor
T 0 0 = a − 4 ∫ d q d Ω q 2 ε f 0 ( ε , T ) ( 1 + ψ )
T i 0 = a − 4 ∫ d q d Ω q n i f 0 ( ε , T ) ( 1 + ψ )
T j i = a − 4 ∫ d q d Ω n i n j q 2 ε f 0 ( ε , T ) ( 1 + ψ )
Boltzmann equation in ( x i , q , n j , η ) , with collision term ∂ f C ∂ η becomes
D f d η = ∂ f ∂ η + ∂ x i ∂ η ∂ f ∂ x i + ∂ q ∂ η ∂ f ∂ q + ∂ n i ∂ η ∂ f ∂ n i = ∂ f C ∂ η
GR geodesic equation P 0 d P μ d η + Γ α β μ P α P β = 0 gives
d q d η = q Φ ˙ − ε ( q , η ) n i ∂ i Ψ (39)
and Boltzmann equation becomes
∂ ψ ∂ η + i q ε ( k → ⋅ n ^ ) ψ + d ln f 0 d ln q ( Φ ˙ − i ε q ( k → ⋅ n ^ ) Ψ ) = 1 f 0 ∂ f C ∂ η (18)
with fluid equations cdm
δ c ' = − θ c + 3 Φ ' , θ c ' = − a ' a θ c + k 2 Ψ (19a)
Component evolution equations
In the following we present the evolution equations for l-momenta in k-space for important components.
Evolution equations massive neutrinos
We have for (average) background density, pressure
ρ ¯ h = a − 4 ∫ d q d Ω q 2 ε f 0 ( ε , T ) , P ¯ h = 1 3 a − 4 ∫ d q d Ω q 2 q 2 ε f 0 ( ε , T )
the perturbations
δ ρ h = a − 4 ∫ d q d Ω q 2 ε f 0 ( ε , T ) ψ , δ P h = 1 3 a − 4 ∫ d q d Ω q 2 q 2 ε f 0 ( ε , T ) ψ
δ T h 0 i = a − 4 ∫ d q d Ω q n i f 0 ( ε , T ) ψ ,
δ Π h 0 i = 1 3 a − 4 ∫ d q d Ω q 2 q 2 ε ( n i n j − 1 3 δ i j ) f 0 ( ε , T ) ψ
distribution perturbation function are developed in Legendre polynomials of the angle ( k ^ ⋅ n ^ )
ψ ( k → , n ^ , q , η ) = ∑ l = 0 ∞ ( − i ) l ( 2 l + 1 ) ψ l ( k → , q , η ) P l ( k ^ ⋅ n ^ ) (54)
δ ρ h = 4 π a − 4 ∫ d q q 2 ε f 0 ( ε , T ) ψ 0 , δ P h = 4 π 3 a − 4 ∫ d q q 2 q 2 ε f 0 ( ε , T ) ψ 0
( ρ ¯ h + P ¯ h ) θ h = 4 π k a − 4 ∫ d q q 3 f 0 ( ε , T ) ψ 1 ,
( ρ ¯ h + P ¯ h ) σ h = 4 π 3 a − 4 ∫ d q q 2 q 2 ε f 0 ( ε , T ) ψ 0 .
Boltzmann equation yields for evolution of perturbation momenta
ψ 0 ' = − q k ε ψ 1 − Φ ' d ln f 0 d ln q , ψ 1 ' = q k 3 ε ( ψ 0 − 2 ψ 2 ) − ε k 3 q Ψ d ln f 0 d ln q
ψ l ' = q k ( 2 l + 1 ) ε ( l ψ l − 1 − ( l + 1 ) ψ l + 1 ) , l ≥ 2 (19b)
truncating order l max
ψ l max + 1 = ( 2 l max + 1 ) ε q k η ψ l max − ψ l max − 1 .
Evolution equations photons
We assume γ − e Thomson scattering with the Thomson cross-section
d σ d Ω = 3 σ T 1 + cos 2 θ 16 π , σ T = 0.665 × 10 − 24 cm 2
with F γ ( k , n ^ , η ) distribution total intensity
with G γ ( k , n → , η ) distribution difference polarization components
with collision terms
( ∂ F γ ∂ η ) C = a n e σ T ( − F γ + F γ 0 + 4 ( n ^ ⋅ v → e ) − ( F γ 2 + G γ 0 + G γ 2 ) P 2 )
( ∂ G γ ∂ η ) C = a n e σ T ( − G γ + 1 2 ( F γ 2 + G γ 0 + G γ 2 ) ( 1 − P 2 ) )
with expansion
( ∂ F γ ∂ η ) C = a n e σ T ( 4 i k ( θ γ − θ b ) P 1 + ( 9 σ γ − 1 2 G γ 0 − 1 2 G γ 2 ) P 2 − ∑ l = 3 ∞ ( − i ) l ( 2 l + 1 ) F γ l P l )
( ∂ G γ ∂ η ) C = a n e σ T ( 1 2 ( F γ 2 + G γ 0 + G γ 2 ) ( 1 − P ) 2 − ∑ l = 0 ∞ ( − i ) l ( 2 l + 1 ) G γ l P l ) .
Resulting fluid equations are then
δ γ ' = − 4 3 θ γ + 4 Φ ' , θ γ ' = k 2 ( 1 4 δ γ − σ γ ) + k 2 Ψ + a n e σ T ( θ b − θ γ ) (19c1)
and momenta evolution becomes
F γ 2 ' = 2 σ γ ' = 8 15 θ γ − 3 5 k F γ 3 − 9 5 a n e σ T σ γ ( θ γ − θ b ) + 1 10 a n e σ T ( θ γ − θ b ) ( G γ 0 + G γ 2 )
F γ l ' = k 2 l + 1 ( l F γ ( l − 1 ) − ( l + 1 ) F γ ( l + 1 ) ) − a n e σ T F γ l , l ≥ 3 (19c2)
G γ l ' = k 2 l + 1 ( l G γ ( l − 1 ) − ( l + 1 ) G γ ( l + 1 ) ) + a n e σ T ( − G γ l + 1 2 ( F γ 2 + G γ 0 + G γ 2 ) ( δ l 0 + δ l 2 5 ) ) (19c3)
Evolution equations baryons
We have the fluid equations
δ b ' = − θ b + 3 Φ ' , θ b ' = − a ' a θ b + c s 2 k 2 δ b − 4 ρ ¯ γ 3 ρ ¯ b a n e σ T ( θ b − θ γ ) + k 2 Ψ (19d1)
with sound speed c s 2 = k B T b μ ( 1 − 1 3 d ln T b d ln a ) , μ mean baryon mass.
The temperature equation becomes
T b ' = − 2 a ' a T b + 8 3 μ m e ρ ¯ γ ρ ¯ b a n e σ T ( T γ − T b )
Before recombination tight-coupling γ − b , we have
θ b − θ γ = τ c ( θ γ ' − k 2 ( 1 4 δ γ − σ γ ) − k 2 Ψ ) (19d2)
σ γ = τ c 9 ( 8 3 θ γ − 10 σ γ ' − 3 k F γ 3 ) (19d3)
θ γ ' = − 3 4 ρ ¯ b ρ ¯ γ ( θ b ' + a ' a θ b − c s 2 k 2 δ b ) + k 2 ( 1 4 δ γ − σ γ ) + ( 1 + 3 4 ρ ¯ b ρ ¯ γ ) k 2 Ψ (19d4)
Initial conditions in k-space for density components (radiation γ, neutrinos ν , electrons e, baryons b, cold-dark-matter c) and metric perturbations Ψ , Φ generate the random (Gaussian distributed) inhomogeneities required for structure formation.
Initial conditions k-space
For Newtonian gauge in conformal time η , initial conditions are chosen in such a way, that only the largest order in k η is present (Ma [
δ γ = − 40 C 3 ( P ¯ + ρ ¯ ) = − 2 Ψ
δ c = δ b = 3 4 δ ν = 3 4 δ γ
θ γ = θ ν = θ b = θ c = 10 C 15 + 4 R ν ( k 2 η ) = k 2 η 2 Ψ
σ ν = 4 C 3 ( 15 + 4 R ν ) ( k η ) 2 = ( k η ) 2 15 Ψ
Ψ = 20 C 15 + 4 R ν , Φ = ( 1 + 2 5 R ν ) Ψ
with neutrino density ratio R ν = ρ ¯ ν ρ ¯ γ + ρ ¯ ν
In the following, we present in concise form cross sections, reaction rates and densities for important cosmological particle processes [
Cosmic neutrino background
The reaction is ν e + ν ¯ e ↔ e + + e − , e − + ν ¯ e ↔ e − + ν ¯ e
with reaction rate Γ = n σ v ≈ G F 2 T 5 , G F ≈ 1.2 × 10 − 5 GeV − 2 (3.58)
and corresponding Hubbble rate H ≈ T 2 M P l , Γ H ≈ ( T 1 MeV ) 3 ,
neutrinos decouple at T ν , d = 1 MeV , t ν , d = 1 s ,
the number density n ν ∝ a − 3 ∫ d 3 q 1 exp ( q a T ν + 1 ) ,
with T ν ∝ a − 1 for T ν > T ν , d .
Gamma pair production
The gamma-pair production reaction is γ + A → e + + e − + A [
with the cross-section σ = α r e 2 Z 2 P ( E , Z ) , where Z = atomic number of material A, k = E γ E e , α fine-structure-constant, and
P ( E , Z ) ≈ 2 π 3 ( | k − 2 | k ) 3 , 2 < k < 4 ,
P ( E , Z ) ≈ 28 9 ln ( 2 k ) − 218 27 = 3.11 ln ( 2 E γ E e ) − 8.07 , k > 4 ,
wih reaction rate Γ = n σ c .
Electron-positron annihilation
The ep-annihilation reaction is e + + e − → γ + γ shown in
wih the cross-section
σ e + e − ( ω 0 ) = ( 1 + π α v ) σ 0 ( β ) − 2 α π ( − 1 + β 2 2 β log ( 1 + β 1 − β ) − 1 ) log ( s 2 ω 0 ) σ 0 ( β ) [
where σ 0 ( β ) = π α 2 s β ( − 3 − β 4 β log ( 1 + β 1 − β ) − 2 ( 2 − β 2 ) ) Born cross-section, and Mandelstamm variables s = ( p 1 + p 2 ) 2 , t = ( p 1 − p 3 ) 2 , u = ( p 1 − p 4 ) 2 , where
β = 1 − 4 ( m c 2 ) 2 / s , z = 1 + β 1 − β
ω 0 soft cut-off, v = 2 β 1 + β 2 relative velocity, dof number g S = { 2 + 7 8 × 4 = 11 2 T ≥ m e 2 T < m e with photons decoupling at T e , d = 0.5 MeV , t e , d = 6 s , duration Δ t e , d = α 2 m e = 10 − 18 s
T ν = ( 4 11 ) 1 / 3 T γ , t > t e , d after ep-annihilation, so T γ , 0 = 2.73 K , T ν , 0 = 1.95 K .
Planck data yield ∑ i m ν i < 0.13 eV , Ω ν < 0.003 .
General photon eos
For T > Tan in pair-production regime, we have in equilibrium (relativistic)
σ 0 ( β ) = 2 π α 2 s β , β = v e c
Γ e e γ = 2 n e + v σ ≈ 2 n e + β c 2 π α 2 ℏ 2 c 2 s β ( 1 + π α β )
Γ γ e e = 2 n γ c σ ≈ 2 n γ c α r e 2 Z e f 2 ( 3.1 ln ( E γ E e ) − 8.1 ) with Z e f = 1 n b n γ , s = 4 E t h 2
Γ e e γ = Γ γ e e results n γ = n b 2 n e + E t h 2 r e 2 3.1 ln ( E γ E e ) 4 π α ℏ 2 c 2 ( 1 + π α c v e ) , i.e. n γ ~ n b n e + n b E t h 2 ∼ E t h 4 , with thermal energy E t h = k B T .
In the black-body regime we have the Stefan-Boltzmann relation n γ = a S B E t h 4 .
The positron density n e + results from equality of both n γ from pair-production-annihilation and Stefan-Boltzmann
n e + ≈ n b 2 n γ 0.17 α ( E t h m e c 2 ) 2 = n b 2 n γ ( E t h m e c 2 ) 2 1.2 × 10 − 3 .
Thomson scattering ( [
We get density of free electrons
n e = ( 1 − Y p 2 ) X e n b ≈ Ω b h 2 ( 1 + z ) 3 × 10 − 5 cm − 3 , ionization fraction X e ≈ 1 , where Y p ≈ 0.24 Helium mass fraction.
The optical depth τ results from the Thomson equation d τ d η = n e σ T a , where σ T = 8 π α 2 3 m e 2 = 6.65 × 10 − 25 cm 2 is the Thomson cross-section in photon-electron scattering.
Photons and neutrinos
After photon decoupling we have the relation for neutrino and photon temperature
T ν = ( 4 11 ) 1 / 3 T γ (3.62)
Hydrogen recombination ( [
For hydrogen recombination we have the reaction e − + p + → H + γ ,
and number density ( n H n e 2 ) = ( 2 π m e T ) 3 / 2 exp ( E i o n T ) ,
with ionization energy E i o n = m p + m e − m H = 13.6 eV , E H , r e = 13.6 eV
and free electron fraction X e ≡ n e n p + n H = n e n b .
The free electron fraction obeys Saha equation
1 − X e X e 2 = 2 ζ ( 3 ) π 2 ( 2 π m e T ) 3 / 2 η exp ( E i o n T ) (3.78) ζ ( 3 ) = 1.202
where n b n γ = n b , 0 n γ , 0 = 0.242 m − 3 0.41 × 10 9 m − 3 = 0.59 × 10 − 9 , and baryon-photon ratio η ≈ 6 × 10 − 10 .
The solution is X e = − 1 + 1 + 4 f ( E t h ) 2 f ( E t h ) ,
f ( E t h ) = 4 ζ ( 3 ) 2 π η ( E t h m e c 2 ) 3 / 2 exp ( E H , r e E t h ) = 2.26 × 10 − 9 ( E t h m e c 2 ) 3 / 2 exp ( E H , r e E t h ) ,
with limits
f ≫ 1 , X e ≈ 1 f ( E t h ) , n e = n b , n b n H ≪ 1
f ≪ 1 , X e ≈ 1 , n e = n b , n H = 0 ,
and recombination temperature T r e c ≈ 0.32 eV = 3760 K , t r e c ≈ 290 ky .
Photon decoupling
The photon decoupling reaction is e − + γ ↔ e − + γ , with reaction rate Γ γ ≈ n e σ T , σ T ≈ 2 × 10 − 3 MeV − 2 , and decoupling temperature Γ γ ( T d e c ) ≈ H ( T d e c ) , X e ( T d e c ) T d e c 3 / 2 ≈ π 2 2 ζ ( 3 ) H 0 Ω m η σ T T 0 3 / 2 , T d e c ≈ 0.25 eV = 2970 K for t d e c ≈ 370 ky .
The Boltzmann equation is ∂ f ∂ t + p → m ∇ f + F → ⋅ ∂ f ∂ p → = C ( f ) , for reaction 1 + 2 ↔ 3 + 4 collision term is C i [ { n j } ] = − α c n 1 n 2 + α c β c n 3 n 4 , where α c = 〈 σ v 〉 thermally averaged cross-section, β c = ( n 1 n 2 n 3 n 4 ) e q detailed balanced coefficient.
From this follows cosmic Boltzmann equation with collision term
1 a 3 d ( n i a 3 ) d t = − 〈 σ v 〉 ( n 1 n 2 − β c n 3 n 4 ) (3.96)
where the particle number is N i ≡ n i s ∝ n i a 3 , d ( log N 1 ) d ( log a ) = − Γ 1 H ( 1 − ( N 1 N 2 N 3 N 4 ) e q N 3 N 4 N 1 N 2 ) , where Γ 1 ≡ n 2 〈 σ v 〉 (1,2) interaction rate.
Dark matter cdm decoupling
The reaction for cdm particle X, light particle l: X + X ¯ ↔ l + l ¯ with Boltzmann equation 1 a 3 d ( n X a 3 ) d t = − 〈 σ v 〉 ( n X 2 − ( n X ) e q 2 ) , with Y X ≡ n X T 3 particles in co-moving volume, and reduced mass x ≡ M X T , d x d t = H x .
Using λ ≡ Γ ( M X ) H ( M X ) = M X 3 〈 σ v 〉 H ( M X ) , we get the Riccati equation d Y X d x = − λ x 2 ( Y X 2 − ( Y X ) e q 2 ) .
The asympotic value is Y X , ∞ ≈ x f λ with x f reduced mass at freeze-out.
The cdm density is Ω X ~ 0.1 x f g s ( M X ) 10 − 8 GeV − 2 〈 σ v 〉 with reaction rate 〈 σ v 〉 ~ 10 − 8 GeV − 2 ~ 0.1 G F (≈weak interaction).
Baryo-genesis
In the following we present important cosmological processes of nuclei, with density evolution equation, cross-section, and charasteristic (freeze-out) time.
Neutron-proton decay
The reaction here is n + ν e ↔ p + + e − , n + e + ↔ p + + ν ¯ e with density ratio ( n n n p ) e q = exp ( − E n p k B T ) , E n p = ( m n − m p ) c 2 = 1.30 MeV , and with X n ≡ n n n n + n p relative n-abundance.
For X n we get the equation
d X n d t = − Γ n ( x ) ( X n − ( 1 − X n ) exp ( − E n p k B T ) )
where
Γ n ( x ) = 255 τ n 12 + 6 x + x 2 x 5 , x = E n p k B T , τ n = 886.7 ± 0.8 s neutron lifetime.
With freeze-out abundance X n , ∞ = 0.15 it becomes X n ( t ) = X n , ∞ exp ( − t τ n ) .
Deuterium
The density ratio is ( n D n p ) e q = 3 4 n n , e q ( 4 π ℏ 2 c 2 m p c 2 k B T ) 3 / 2 exp ( E n p D k B T ) , with E n p D = ( m n + m p − m D ) c 2 = 2.22 MeV and temperature T n u c = 0.06 MeV at ( n D n p ) e q ( T = T n u c ) = 1 , the corresponding time is t n u c = ( 0.1 MeV T n u c ) 2 120 s ≈ 330 s .
Helium
The reactions are
D + p + ↔ H e 3 + γ , H 3 + p + ↔ H e 3 + n
D + D ↔ H 3 + p + , H 3 + D ↔ H e 4 + p +
D + D ↔ H e 3 + n , H e 3 + D ↔ H e 4 + p +
helium-hydrogen ratio is then
Y P = 4 n H e n H = 4 n H e n p ≈ 2 X n ( t n u c ) 1 − X n ( t n u c ) ~ 0.25 , which is observed.
Lithium beryllium
The reactions are
B e 7 + n ↔ L i 7 + p + , L i 7 + p + ↔ H e 4 + H e 4 , B e 7 + e − ↔ L i 7 + ν e
H e 3 + H e 4 ↔ B e 7 + γ , H 3 + H e 4 ↔ L i 7 + γ .
Hydrogen recombination
The process of hydrogen recombination is shown in
We have the Peebles equation for free electron density Xe with an improved calculation in redshift z [
d X e d z = − C r ( T ) H ( z ) ( 1 + z ) ( ( m e c 2 k B T 2 π ) 1 / 2 ( 1 − X e ) exp ( − E I k B T ) − α ( T ) n b n γ 2 ζ ( 3 ) π 2 ( k B T ) 3 X e 2 ) (20)
with
C r ( T ) ≡ Λ 2 γ + Λ α Λ 2 γ + Λ α + β α ,
Λ α = 27 128 ζ ( 3 ) H ( T ) ( 1 − X e ) ( n b / n γ ) ( k B T / E I ) 3 ,
Λ 2 γ = 8.227 s − 1 ,
λ α = 8 π ℏ c 3 E I Lyman wavelength, β α = β ( T ) exp ( 3 E I 4 k B T ) ,
α ( T ) ≈ 9.8 α 2 ( m e c 2 ) 2 ( E I k B T ) 1 / 2 log ( E I k B T ) ,
H ( z ) = Ω m H 0 ( 1 + z ) 3 / 2 ( 1 + 1 + z 1 + z e q ) ,
H 0 ≈ 1.5 × 10 − 33 eV , T = ( 1 + z ) 0.235 eV .
In this chapter, we present first in concise way the contributions to the temperature anisotropy of the cosmic microwave background CMB.
Then we describe the scheme for the calculation of the CMB spectrum coefficients Cl.
The schematic of the calculation is shown in chap. 11.
Finally, we present the self-calculated results and a comparison with data.
CMB spectrum today
CMB as measured today has the parameters [
temperature T γ , 0 = 2.7255 ± 0.0006 K .
CMB dipole is around 3.3621 ± 0.0010 mK
relative density Ω γ = 6 × 10 − 5
temperature anisotropy Δ T γ , 0 ≈ 30 μ K , so Δ T γ , 0 T γ , 0 ≈ 30 μ K 2.72 K = 1.1 × 10 − 5 .
Temperature anisotropy
The temperature anisotropy of the CMB has the following contributions:
δ T T ¯ ( n ^ ) = ( SW = ( 1 4 δ γ + Ψ ) * ) + ( Dop = − ( n ^ ⋅ v → b ) * ) + ( ISW = ∫ η * η 0 d η ( Φ ′ + Ψ ′ ) ) (7.29)
at conformal time η = η * = η d e c .
▪ SW The first term is the so-called Sachs–Wolfe term. It represents the intrinsic temperature fluctuations associated to the photon density fluctuations δ γ / 4 and the metric perturbation Ψ at last scattering.
▪ Doppler The second term is the Doppler term n ^ ⋅ v → b caused by local velocity, this contribution is small on large scales.
▪ ISW The last term describes the additional gravitational redshift ∫ η * η 0 d η ( Φ ′ + Ψ ′ ) due to the evolution of the metric.
The temperature anisotropy has the form
Θ ( n ^ ) ≡ δ T T ¯ ( n ^ ) = ∫ d 3 k ( 2 π ) 3 exp ( i k → ⋅ n ^ c t ( η * ) ) ( F ( η * , k → ) + i ( k → ⋅ n ^ ) G ( η * , k → ) ) ,
where F ( η * , k → ) = ( 1 4 δ γ + Ψ ) , G ( η * , k → ) = v b , F * ( k ) = F ( η * , k → ) R ( η = 0 , k → ) , G * ( k ) = G ( η * , k → ) R ( η = 0 , k → ) and R ( η = 0 , k → ) are the initial curvature anisotropies.
We get for the anisotropy the series in Legendre polynomials
Θ ( n ^ ) = ∑ l i l ( 2 l + 1 ) ∫ d 3 k ( 2 π ) 3 Θ l ( k ) R ( 0 , k → ) P l ( k → ⋅ n ^ )
with the transfer function including ISW
Θ l ( k ) = Θ l ( k ) = ( F * ( k ) j j ( χ * k ) − G * ( k ) j j ' ( χ * k ) ) + ∫ η * η 0 d η ( Φ ′ + Ψ ′ ) j j ( c t ( η ) k ) ,
with χ * = c t ( η * ) .
The two-point temperature correlation (scalar TT-correlation) spectrum measured in CMB is C ( θ ) = 〈 Θ ( n ^ ) Θ ( n ^ ′ ) 〉 , with directions n ^ , n ^ ′ , angle cos θ = n ^ ⋅ n ^ ′ , and the series in Legendre polynomials
C ( θ ) = ∑ l 2 l + 1 4 π C l P l ( cos θ )
with series coefficients C l
C l = 2 π ∫ − 1 1 d ( cos θ ) C ( θ ) P l ( cos θ ) = 4 π ∫ d k k Θ l 2 ( k ) Δ R 2 ( k ) (7.6)
where Δ R 2 ( k ) = A s ( k k 0 ) n s − 1 is the power amplitude, and where sound horizon is r s = ∫ d η 3 ( 1 + R ( η ) ) , with curvature R ( η ) .
Weinberg semi-analytic solution [
Weinberg proposed a semi-analytic solution for photon density perturbations
δ γ = 4 5 R ( η = 0 , k → ) ( S ( k ) ( 1 + R ( η , k → ) ) 1 / 4 cos ( k r s + θ ( k ) ) − ( 1 + 3 R ( η , k → ) ) T ( k ) )
with Weinberg semi-analytic transfer functions for SW and Doppler with
F * ( k ) = 1 5 ( exp ( − k 2 k D * 2 ) S ( k ) ( 1 + R ( η * , k → ) ) 1 / 4 cos ( k r s * + θ ( k ) ) − 3 R ( η * , k → ) T ( k ) )
G * ( k ) = − 3 5 exp ( − k 2 k D * 2 ) S ( k ) ( 1 + R ( η * , k → ) ) 1 / 4 sin ( k r s * + θ ( k ) ) where k D * − 1 = 8.8 Mpc
and the resulting CMB power spectrum
l ( l + 1 ) 2 π C l = ∫ 1 ∞ d β β 2 β 2 − 1 ( F * 2 ( l β χ * ) + β 2 − 1 β 2 G * 2 ( l β χ * ) ) Δ R 2 ( l β χ * ) with χ * = c t ( η * )
where
S ( κ ) = ( 1 + ( 1.209 κ ) 2 + ( 0.5611 κ ) 4 + 5 ( 0.1567 κ ) 6 1 + ( 0.9459 κ ) 2 + ( 0.4249 κ ) 4 + ( 0.167 κ ) 6 ) 2
T ( κ ) = log ( 1 + ( 0.124 κ ) 2 ) ( 0.124 κ ) 2 ( 1 + ( 1.257 κ ) 2 + ( 0.4452 κ ) 4 + ( 0.2197 κ ) 6 1 + ( 1.606 κ ) 2 + ( 0.8568 κ ) 4 + ( 0.3927 κ ) 6 ) 1 / 2
θ ( κ ) = ( ( 1.1547 κ ) 2 + ( 0.5986 κ ) 4 + 5 ( 0.2578 κ ) 6 1 + ( 1.723 κ ) 2 + ( 0.8707 κ ) 4 + ( 0.4581 κ ) 6 + ( 0.2204 κ ) 8 ) 1 / 2 .
Calculation of CMB spectrum coefficients Cl ( [
The temperature and photon polarization Stokes parameters anisotropy are expanded in a series in angular momentum (l, m),
Θ ( η , x → , n ^ ) = ∫ d 3 k ( 2 π ) 3 ∑ l = 0 ∞ ∑ m = − 2 2 Θ l m G l m (21a)
( Q ± i U ) ( η , x → , n ^ ) = ∫ d 3 k ( 2 π ) 3 ∑ l = 0 ∞ ∑ m = − 2 2 ( E l m ± i B l m ) G l m
with temperature (l, m)-moments
Θ l ( m ) = ∫ d n Y l m * ( n → ) Θ ( n → ) (21b)
and with temperature basis functions
G l m = ( − i ) l 4 π 2 l + 1 Y l m ( n ^ ) exp ( i k → ⋅ x → ) = ∑ l ( − i ) l 4 π ( 2 l + 1 ) j l ( k r ) Y l 0 ( θ , φ ) ,
G l ' m = ∑ l ( − i ) l 4 π ( 2 l + 1 ) j l l ' m ( k r ) Y l m ( θ , φ ) ,
where
exp ( i k → ⋅ x → ) = ∑ l ( − i ) l 4 π ( 2 l + 1 ) j l ( k r ) Y l 0 ( θ , φ ) .
In this representation, the spectrum coefficients Cl are
〈 Θ l ( m ) , Θ l ′ ( m ′ ) 〉 η ≡ ∫ d η Θ * l ( m ) Θ l ′ ( m ′ ) = δ l l ′ δ m m ′ C l (21c)
where the power spectrum on the angular momentum l is
Δ T 2 ( l ) = l ( l + 1 ) 2 π C l T 2 in μK2 (21d)
We use the variables:
averaged pressure V ( η ′ , k ) = − 8 π G k a 2 ∫ 0 η ′ d η a 4 δ P , V ′ ( η , k ) = − 8 π G k a 2 δ P
optical depth τ ( η ′ ) = σ T ∫ 0 η ′ d η n e a , τ ′ ( η ) = n e σ T a .
The temperature (l, m)-moments are calculated from the evolution equations
Θ ' l m = k ( κ 0 l m 2 l − 1 Θ l m − κ 0 l + 1 m 2 l + 3 Θ l + 1 m ) − τ ' Θ l m + S l m (21e)
with sources
S 00 = τ ' Θ 00 − Φ ′ , S 10 = τ ′ v b 0 + k Ψ , S 11 = τ ′ v b 1 + V ′
S 20 = 1 10 τ ′ ( Θ 20 − 6 E 20 ) , S 21 = 1 10 τ ′ ( Θ 21 − 6 E 21 ) ,
S 22 = 1 10 τ ′ ( Θ 22 − 6 E 22 ) − Φ ′
S 20 = 1 10 τ ′ ( Θ 20 − 6 E 20 ) , S 21 = 1 10 τ ′ ( Θ 21 − 6 E 21 ) ,
S 22 = 1 10 τ ′ ( Θ 22 − 6 E 22 ) − Φ ′
Θ l m ( η 0 , k ) 2 l + 1 = ∫ 0 η 0 d η exp ( − τ ) ∑ l ′ S l ′ m ( η ) j l l ′ m ( k ( η 0 − η ) )
and j l l ′ m are spherical Bessel functions
j l 00 ( x ) = j l ( x ) , j l 10 ( x ) = j l ' ( x ) , j l 20 ( x ) = 1 2 ( 3 j l ' ' ( x ) + j l ( x ) )
j l 11 ( x ) = l ( l + 1 ) 2 j l ( x ) x , j l 21 ( x ) = 3 l ( l + 1 ) 2 d d x ( j l ( x ) x ) ,
j l 22 ( x ) = 3 8 ( l + 2 ) ! ( l − 2 ) ! j l ( x ) x 2 .
The metric perturbations Ψ , Φ in k-space for k = 5 are shown in
Density fluctuations for baryons, radiation, cdm δb, δr, δc, for k = 5 are shown in
The calculated normalized scalar TT-correlation power spectrum of CMB, Δ T 2 ( l ) = l ( l + 1 ) 2 π C l T 2 , is shown in
The background Hubble parameter H0 influences the CMB spectrum, but the deviation δ = 1.3% caused by the calculated correction from chap. 5 is within measurement error.
The plot in
In
In the following, we present the fundamental equations, the solution process and results in form of schematic diagrams for the background calculation and for the CMB calculation.
Lambda-CDM background calculation:
Lambda-CDM CMB calculation:
The results for the background part are presented in schematic form in chap. 11 Lambda-CDM background calculation.
We start with the Friedmann equations
( a ' ) 2 + k − Λ 1 3 a 2 − ρ a 2 = 0
ρ ' a 3 + a ' ( P + ρ ) = 0
with the variables in dependence of the scale factor a (inverting the scalefactor-time relation a = a ( x ) ,
x ( a ) time,
ρ i ( a ) density of component i,
E t h ( a ) temperature,
for components radiation γ, neutrinos ν , electrons e, protons p, neutrons n, cdm d, where the pressure P i ( a ) is eliminated using the component eos P i = P i ( ρ i , E t h ) .
In difference to the conventional ansatz,
-the temperature resp. thermal energy is introduced as explicit function of time E t h ( t ) ;
-we use the ideal gas eos for baryons, instead of the usual setting P b = 0 (dust eos).
As we show in chap. 5, this leads to a correction of 4.3% for the present value of Hubble parameter H 0 c = 1.043 H 0 , which brings it into agreement with the measured Red-Giant-result, and within error margin with the Cepheids-SNIa-measurement.
We carry out an iterated calculation with two steps i = 1 and i = 2, the results are shown graphically in chap. 10.2.
Note the deviation of the temperature from the conventional linear behavior (brown) to the calculated first-iteration-value (blue) for later times. This produces also a slight “bump” for the Hubble parameter H ( a ) , and there is a slight “kink” in x ( a ) .
The results for the perturbation part are presented in schematic form in chap. 11 Lambda-CDM CMB calculation.
We start with the perturbed metric
d s 2 = a ( η ) ( − ( 1 + 2 Ψ ) d η 2 + ( 1 − 2 Φ ) d x i d x i )
perturbations Φ , Ψ , θ , σ , δ , δ P , where
δ P pressure
θ = i k j v j velocity
δ = δ ρ / ρ ¯ relative density
σ = − ( k ^ i k ^ j − 1 3 δ i j ) Π j i / ( ρ ¯ + P ¯ ) stress
ρ ¯ , P ¯ , a , E t h are background functions calculated already in the background part.
And τ = reionization optical depth is a parameter used for the CMB calculation.
The perturbations result from (random) initial conditions and represent the random nature of structure formation.
The resulting fundamental equations are transformed to k-space (i.e. Fourier transformed), and consist of two parts.
The Einstein equations in k-space resulting from the perturbed metric ansatz
k 2 Φ − 3 H ( Φ ′ + H Ψ ) = π G a 2 δ ρ
k 2 ( Φ ′ + H Ψ ) = π G a 2 ( P ¯ + ρ ¯ ) θ
k 2 ( Φ − Ψ ) = 12 π G a 2 ( P ¯ + ρ ¯ ) σ
Φ ″ + H ( Ψ ′ + 2 Φ ′ ) + 1 3 k 2 ( Φ − Ψ ) + ( 2 H ′ + H 2 ) Ψ = 4 π G a 2 δ P
and the thermodynamic: density and Euler (relativistic fluid) equation, resulting from the relativistic Boltzmann transport equation
δ ′ = − ( 1 + P ¯ ρ ¯ ) ( θ − 3 Φ ′ ) − 3 H ( δ P ρ ¯ δ − P ¯ ρ ¯ ) δ
θ ′ = − ( H + P ¯ ′ P ¯ + ρ ¯ ) θ − δ P P ¯ + ρ ¯ k 2 − k 2 σ + k 2 Ψ
The CMB power spectrum coefficients Cl depend on the angular moments of temperature correlation Θ l m , which obey the iterative differential equation in k-space
Θ ' l m = k ( κ 0 l m 2 l − 1 Θ l m − κ 0 l + 1 m 2 l + 3 Θ l + 1 m ) − τ ' Θ l m + S l m
with parameters, which are calculated from the fundamental equations.
The actual numerical calculation is performed in program [
Then a fit is carried out between the calculated parameterized coefficients C l ( p i ) and tthe measured values C l , e x p .
The 13 fitted parameters
p i = ( Ω b , Ω c , Ω Λ , t 0 , H 0 , A s , n s , τ , w , Σ m ν , N ν , r t , d n s d k ) are calculated by the Planck collaboration [
The fitted [
The author declares no conflicts of interest regarding the publication of this paper.
Helm, J. (2024) A New Version of the Lambda-CDM Cosmological Model, with Extensions and New Calculations. Journal of Modern Physics, 15, 193-238. https://doi.org/10.4236/jmp.2024.152011