A new version of the Lambda-CDM cosmological model, with extensions and new calculations

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


Introduction
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.9 with the CMB spatial spectrum measured by Planck.
We present here in chap.1-4 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.4,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.5-9, with the derivation of the CMB spectrum, and new own calculation of it.1 Friedmann equations 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  , electrons e, protons p, neutrons n (respectively baryons b),cold-dark-matter cdm d .The presentation relies basically on the four monographies [1] [2] [10] [11] , with two notable extensions. -Temperature The eos depend explicitly on temperature T, resp.thermal energy th B E k T  , and thermal energy is introduced as a function of time   th E 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.4, that this model increases the value of the Hubble parameter, which basically solves the Hubble-discrepancy problem.

Friedmann equations and metric
The metric which fulfills the conditions of space homogeneity and isotropy is the Robertson-Walker (RW) metric [1] [2] [10] [11]: with Hubble radius ) and two derived equations c (acceleration eq.) and d (density equation) The Friedmann equations can be reformulated dimensionless with 0   ' ' 0 3

Relative density & pressure neutrinos
We have for neutrino density and pressure before (1) and after (2) neutrino decoupling [26] with threshold energy , 1.95 0.026 1.69 10 300 .938 1.13 10 0.049 1.69 10 The equation for th E after recombination with  where the baryon temperature depends on the photon temperature

Density electrons
The density of electrons is described by the Peebles equation with the parameters . s we get the Peebles equation ( [11] 3.153) for the hydrogen ionization percentage where We get for the electron density before (1)   .
We can set approximately , where (measured in CMB) Transition thermal energies and eos -neutrino decoupling c ,  -e-p-annihilation

Some specifications
The amplitude As , is determined by the CMB power spectrum The fluctuation amplitude is defined by Hot Big-bang problems -the observed homogeneity of the present universe (distances >200Mly) 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. a H const a    , and we have an inflating universe [15].This inflating universe can be described by the de-Sitter metric For the case of exponential expansion, the equation of state is P    , with world radius The expansion generates an almost-flat and large-scale-homogeneous universe, as it is observed today.
Furthermore, horizon reaches a minimum at the end of inflation, and then rises again, this explains superluminal correlations in the present universe.[32] Inflation takes place between

equation-of-state of dark energy   generating temporary inflation
We get the Friedmann equations (radiation-matter density rm and the Klein-Gordon equation We get dimensionless 2 equations in Planck-units and for the scale factor

Square potential
We use the square potential , slow-roll condition: , so condition for convergence is: The analytical solution avoids the convergence problem, and this solution scheme is used in the calculation of results presented below.

Numerical solution
We solve for dimensionless function variables , r a  , in dimensionless relative time variable , where the upper limit is the relative cosmic time today 0 0 00 0 96 The two independent (3c and 3d is derived) equations (3a, 3d) are non-linear second-order differential equations quadratic in the variables , r a  .
Alternatively, one can solve for function variables , th B a E k T  , the latter with thermal energy th The additional equation for pressure is the equation-of-state (eos) for the pressure r P :  

Solution 3
One solves numerically [28] (3a) with boundary conditions   0 1 a x  as differential equation for function variable a , with ansatz for 4 3 . This is the usual solution method for background functions, used in CAMB [29] and in CMBquick ( [23], [27]) .
The solution exists until 1 0.0055 c x  , where numerical integration stops converging, and the solution becomes complex (i.e.

Analytic solution
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 , using the solution   a x in the next iteration: , as shown in the schematic in chap.10.
The zero iteration is the "naive" thermal energy The variables are scale factor and density , r a  The boundary conditions are The two equations (3ad) are non-linear first-order differential equations quadratic in the variables , r a  The third equation is the equation-of-state (eos) for the pressure r P :   i.e.

   
1/4 ,0 and all the pressure becomes a function of a , then we can integrate (3d) in a : and then can integrate (3a) in a :  The Lambda-CDM model is locally homogeneous, but during inflation the quantum fluctuations are "blownup", 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 a 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 dt d a   ), and space location vector i x , and are not random variables.The randomness is introduced by initial conditions for perturbations (see chap. 7).
We introduce metric perturbations and split-up in scalar, vector, tensor parts scalar where Furthermore, we form the gauge-invariant Bardeen variables with 8=1scalar(A)+3vector(Bi)+4tensor(Eij) degrees-of-freedom (dof's) Since we have 6 Einstein equations.we can remove the 8-6=2 dof's by gauge-fixing ▪ Newtonian gauge The relativistic Euler equation is , the Euler equation in the RW metric becomes   where i j  is the anisotropic stress with the decomposition Finally, we get 10 fundamental equations: 6 Einstein equations [11] 4 conservation equations : continuity +Euler [11] (6.91,92)

Fundamental equations in k-space ([22] Ma)
In the following, we transform the fundamental equations via Fourier-transform into k-space.
We use Newtonian gauge , conformal time  , da a' d  , the metric in Newtonian gauge reduces to and 2 continuity-Euler equs in k-space   with the definitions where is the k-unit-vector, i j  anisotropic stress and the relations We have here 6 variables , , , , , P , which are functions of  

k , 6 Evolution of distribution momenta
We introduce here density distribution momenta for density components radiation γ , neutrinos  , 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.9.

Evolution of distribution function momenta (Ma [22])
We have for Newtonian gauge , conformal time

Phase space distribution
With phase space element , temperature T, today temperature T0 , We change variables: i j x P to i j x q , and get the expressions: scaled momentum j j j q ap qn   , unit momentum vector n with 1 Finally we get for the neutrino distribution perturbation function   i j x ,q,n ,   (not equal to the metric perturbation  ) for the distribution of energy tensor x ,q,n , , with collision term and Boltzmann equation becomes with fluid equations cdm

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  distribution perturbation function are developed in Legendre polynomials of the angle Boltzmann equation yields for evolution of perturbation momenta 0 0 1

Evolution equations photons
We assume e   Thomson scattering with the Thomson cross-section 0 665 10 and momenta evolution becomes

Evolution equations baryons
We have the fluid equations ' k k Initial conditions in k-space for density components (radiation γ , neutrinos  , electrons e, baryons b, colddark-matter c ) and metric perturbations ,   generate the random (Gaussian distributed) inhomogeneities required for structure formation.

Initial conditions k-space
For Newtonian gauge in conformal time  , initial conditions are chosen in such a way, that only the largest order in k is present (Ma [22]) Structure formation In the following, we present in concise form cross sections, reaction rates and densities for important cosmological particle processes[12] [14] [2] [10] [11].They are used in the background eos equations in chap. 1, and in the evolution equations of density distribution momenta in chap.6.

Cosmic neutrino background
and corresponding Hubbble rate the number density

Gamma pair production
The gamma-pair production reaction is with the cross-section   where and Mandelstamm variables , where

General photon eos
For T>Tan in pair-production regime, we have in equilibrium (relativistic))

Thomson scattering ([25] Hu)
We get density of free electrons  

Photons and neutrinos
After photon decoupling we have the relation for neutrino and photon temperature , where From this follows cosmic Boltzmann equation with collision term with particle number , where 1

Dark matter cdm decoupling
The reaction for cdm particle X , light particle l : X X l l    with Boltzmann equation particles in co-moving volume, and reduced mass , we get the Riccati equation x reduced mass at freeze-out.
The cdm density is   (≈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

  
with density ratio , and with For n X we get the equation where  

Deuterium
The density ratio is

H He
Li    

Hydrogen recombination
Fig. 13 Hydrogen recombination state diagram [11] We have the Peebles equation for free electron density Xe with an improved calculation in redshift z [33]  .s 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.10.Finally, we present the self-calculated results and a comparison with data.

CMB spectrum today [18]
CMB as measured today has the parameters [18]

Temperature anisotropy
The temperature anisotropy of the CMB has the following contributions: The temperature anisotropy has the form are the initial curvature anisotropies.
We get for the anisotropy the series in Legendre polynomials with the transfer function including ISW The two-point temperature correlation (scalar TT-correlation) spectrum measured in CMB is , with directions n , n' , angle ˆĉos n n'    , and the series in Legendre polynomials is the power amplitude , and where sound horizon is Weinberg proposed a semi-analytic solution for photon density perturbations with Weinberg semi-analytic transfer functions for SW and Doppler with

Calculation of CMB spectrum coefficients Cl ([24] Hu)
The temperature and photon polarization Stokes parameters anisotropy are expanded in a series in angular momentum (l,m) , and with temperature basis functions In this representation, the spectrum coefficients Cl are where the power spectrum on the angular momentum l is We use the variables averaged pressure   The temperature (l,m)-moments are calculated from the evolution equations with sources 00 00

CMB calculations results
The metric perturbations ,   in k-space for k=5 are shown in Fig. 14, as a function of relative scale factor eq a / a , where   In Fig. 18 is shown the scalar TT-correlation power spectrum from Fig. 16, together with measurement data and its error bars.We carry out an iterated calculation with two steps i=1 and i=2 , the results are shown graphically in chap10.2.Note the deviation the temperature from the conventional linear behavior (brown) to the calculated firstiteration-value (blue) for later times.This produces also a slight "bump" for the Hubble parameter   H a , and there is a slight "kink" in   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   with parameters, which are calculated from the fundamental equations.The actual numerical calculation is performed in program [30], based on a function library from [27].
Then a fit is carried out between the calculated parameterized coefficients   are calculated by the Planck collaboration [31], and are not recalculated here.
The fitted [31] and measured coefficients Cl are shown in a plot .

(
Planck value), , and scale factor   a t .The Einstein equations [1] [5] [6] [7] [9] for this metric are the two original Friedmann equations a and b (with da a dt  

)
In the following, we present the eos for the components radiation γ , neutrinos  , electrons e, protons p,neutrons n, cdm d [2] [10] [11] [12] [3].Relative density & pressure baryons b, CDM c , matter density ρm,r dependent ( at e-pair production and above photons lose energy and keep a mean energy pair production and above photons lose energy and keep a mean energy

1
Fig.4The scale factor a(x) in dependence of relative time

Fig. 5
Fig. 5 The scale factor a[x] in dependence of relative time is dust-like, i.e. pressure is almost zero.The densities have the form r mat rad

( 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 Fig. 12 e-p annihilation e n  results from equality of both n  from pair-production-annihilation and Stefan-
Hydrogen recombination ([11] ch.2)For hydrogen recombination we have the reaction e p Photon decouplingThe photon decoupling reaction is e e        ,

▪
SW The first term is the so-called Sachs-Wolfe term .It represents the intrinsic temperature fluctuations associated to the photon density fluctuations 4 /   and the metric perturbation  at last scattering.▪ Doppler The second term is the Doppler term b n v   caused by local velocity, this contribution is small on large scales.▪ ISW The last term describes the additional gravitational redshift

Fig. 18
Fig. 18 Temperature scalar TT-correlation power spectrum with measured data [27] [30], for measurements Planck, WMAP, ACBAR, CBI, and BOOMERANG t ,H , A ,n , ,w, m ,N ,r, dk results for the background part are presented in schematic form in chap.10.1 Lambda-CDM background calculation.We start with the Friedmann equations in dependence of the scale factor a (inverting the scalefactor-time relation for components radiation γ , neutrinos  , electrons e, protons p, neutrons n, cdm d , where the pressure   the conventional ansatz, -the temperature resp.thermal energy is introduced as explicit function of time   th E t , -we use the ideal gas eos for baryons, instead of the usual setting 0 b P (dust eos).As we show in chap.4, this leads to a correction of 4.3% for the present value of Hubble parameter H0c= 1.043 H0 , which brings it into agreement with the measured Red-Giant-result, and within error margin with the Cepheids-SNIa-measurement.
the perturbation part are presented in schematic form in chap.10.3 Lambda-CDM CMB calculation.We start with the perturbed metric calculated already in the background part.and  =reionization optical depth is a parameter used for the CMB calculation.

synthesis He
The simple ΛCDM model is based on seven parameters: physical baryon density parameter Ωbh 2 ; physical matter density parameter Ωmh 2 ; the age of the universe t0 ; scalar spectral index ns; curvature fluctuation amplitude As ; and reionization optical depth τ , dark energy density ΩΛ .The additional parameters of the extended ΛCDM are given in the second table .

Relativistic perturbations and the perturbed Lambda-CDM model
Cepheids-SNIa becomes 73.04/1.015=72. 22 The CMB power coefficients Cl depend on the angular of temperature correlation lm