Big Bang Nucleosynthesis :The First Twenty Minutes (A Computational Modelling Approach)
Sayantan Bhattacharya 16PHMP17
A Report Presented For The Completion Of IIIrd Sem Project Master Of Physics
School Of Physics UniversityOf Hyderabad Telengana,India 5th November,2017
Big Bang Nucleosynthesis :The First Twenty Minutes Sayantan Bhattacharyaa a
∗
School of Physics,University Of Hyderabad,Telengana,India 16PHMP17 (Dated: November 5, 2017)
Big Bang Nucleosynthesis is the method of formation of light elements after Big Bang.The universe was very hot in the beginning and not at all suitable for matter formation,When the universe cooled down,after 3 minutes of big bang the formation of baryons (proton and neutrons) started.This marks the beginning of Nucleosynthesis.The abundances of the light elements predicted by standard big bang cosmological model have matched to a great extent with observational data (WMAP,COBE etc.).Also we can model the Big Bang Nucleosynthesis using various computer programs i.e Alterbbn,Parthenope,kawano and Wagnnor code,bbn new123 etc.In this project I have studied the Big Bang Neucleosynthesis and tried to decipher various codes(already existing) and parameters used(η,T,Xn etc.)
I.
INTRODUCTION
When the universe was much hotter and denser, when the temperature was of order an MeV/kB , there were no neutral atoms or even bound nuclei. The vast amounts of radiation in such a hot environment ensured that any atom or nucleus produced would be immediately destroyed by a high energy photon. As the universe cooled well below the binding energies of typical nuclei,light elements began to form.This Process is known as the Big Bang Nucleosynthesis;Knowing the conditions of the early universe and the relevant nuclear cross-sections, we can calculate the expected primordial abundances of all the elements.
A.
Salient Features Of Cosmology
The study of cosmos i.e cosmology is governed by some laws and axioms,those are prerequisites for conducting any study of cosmology. Those have been discussed here. a. Cosmological Principle The universe is homogeneous and isotropic.Homogeneity means The universe looks the same from all locations and Isotropy means it looks the same from all directions. b. Hubble’s Law The galaxies are all moving away from us with a speed which is proportional to the distance of their separation.The constant of proportionality H0 is Hubble’s constant.The value of H0 is 100h,where h = 0.70kms−1 M pc−1 Combining these two we are led to a model of universe which has the following properties: • The universe has no boundaries. • It is filled with galaxies(0.1 per Mpc3 ) • All of them are moving away from each other. • If we extrapolate in the past,we will have a singularity. c. Co-moving co-ordinate system Co-moving coordinates or galaxy fixed coordinates are used for all the cosmological calculations.The co-moving coordinates continues to be the same though physical separation increases.These are related to physical co-ordinates as ~r = a(t)~x.Where a(t) is scale factor.Calculating the expansion velocity from here we can get the Hubble constant as ~a˙ H0 = ~a t=t0
∗
M.sc student at University Of Hyderabad,Telengana,India Corresponding author E-mail address:
[email protected](Sayantan Bhattacharya)
3 B.
Dynamics Of Expanding Universe
In order to calculate the dynamics of expanding universe we have to consider a spherical mass distribution(symmetric),around the observer,gravity is going to slow down expansion of the universe.This treatment gives the following relations. ρa3 = constant 4 d2 a = − πaGρ 2 dt 3 a˙ 2 4 1 = πρ0 G + E 2 3 a
(1) (2) (3)
(ρ is the density and terms with zero as subscript represents present epoch) Solving differential equation (3) for a,we can achieve value for Hubble’s constant as, " # ρ0 2 H0 1 − 3H 2 = 2E 0
8πG 2
0 The term 3H 8πG ,has the dimension of density and is termed as critical density ρc .This leads us to the idea of density parameter Ω,which is defined as
Ω=
ρ ρc
Depending on the values of Ω we can propose different models of the Universe,The model for universe where Ω = 1 hence E=0,is called the Critical Universe or the Einstein De-sitter Universe.This type of universe is flat at infinity and if Ω > 1 it leads to Big Crunch and if Ω < 1 we can conclude about the origin of the universe from a big bang. Generalized Equation Of Dynamics : The previous paragraph deals only with the contribution of matter(non-relativistic) towards the expansion of universe,but there are other elements contributing to the rate of expansion also i.e radiation,curvature and dark energy.To account for those contributions we will take a generalized equation of state P = wu;P is the pressure exerted by the contributor and u is the energy associated whereas w is a constant. In co-moving co-ordinate,from this equation of state we can derive ρa3(1+w) = ρ0 (constant) the equation of motion will be actually the superposition of contribution from all the constituents(matter,radiation,curvature,dark energy). following same method as before we get the following relations,i.e X 4 a ¨ = − πG (1 + 3wi )ρi a 3 i X 1 d ˙2 4 d ρi0 a−3(1+3wi ) a = πG 2 dt 3 dt i 2 X a˙ 8 2E = πG ρi0 a−3(1+wi ) + 2 a 3 a i
(4) (5) (6)
Writing this in terms of Hubble’s constant in present epoch we get, 2 a˙ H0 2 X = ρi0 a−3(1+wi ) a ρc0 i X H 2 (t) = H0 2 Ωi0 a−3(1+wi )
(7) (8)
i
If we know the two parameters (i)H0 and (ii)Ωi0 ,we will be able to know the dynamics of the universe from Eqn.8.
4 Components of standard cosmological model: are tabulated as:
The 4 components contributing to the evolution of universe
Constituent w Ωi0 Eqn. Of state Matter(baryonic matter,dark matter etc) 0 Ωm0 P=0 1 Radiation(i.e CMBR) Ωr0 P= 31 u 3 Dark Energy -1 Ωλ0 P=-u 1 Curvature − 3 ΩK0 P= - 13 The contribution of these components to the dynamics of universe can be quantified using the parameter deceleration parameter q(t) = −
a ¨ a a˙ 2 a
In terms of density parameter it can be written as, q0 =
1X (1 + 3wi )Ωi0 2 i
From this deceleration parameter we get that,the dark energy component opposes the contribution by other constituents(having -ve sign). II.
BIG BANG NUCLEOSYNTHESIS
Big Bang Nucleosynthesis gives us the information about formation of matter in the early radiation dominated universe.Let us now consider what happens to the nucleons in this early era (T ≈ 1010 K). The nucleons,i.e the protons and neutrons get’s converted into one another by six processes viz. − • n + e− * ) − p + ν¯ − • n+ν * ) − p + e− * • n− ) − p + e + ν¯ Binding energy Q = (mn − mp )c2 =1.293eV Temperature scale:
T =
Q(inJoules) kB
=
1.293×106 ×1.6×10−19 1.38×10−23
= 1.5 × 1010 K
(At this temperature the conversion of the nucleons started occurring.) The total rates at which an individual neutron is converted to a proton or a proton to a neutron take the form: 1/2 Z m2e (Q + q)2 q 2 dq (9) λ(n → − p) = A 1− 2 q/K (Q + q) (1 + e B T (1 + e−(Q+q)/KB T 1/2 Z m2e (Q + q)2 q 2 dq λ(p → − n) = A 1− (10) (Q + q)2 (1 + e−q/KB T (1 + e(Q+q)/KB T where,A is a constant and the integral over q runs from+∞ to -∞ leaving out the part q = −Q−me to q = Q+me ,where the square root would be imaginary.With the rates known in principle, we can calculate the change in the ratio Xn of neutrons to all nucleons from the differential equation: dXn = −λ(n → − p)Xn + λ(p → − n)(1 − Xn ) dt From these rate equations we can derive the following relations viz.: λ(p → − n) = exp(−Q/KB T ) λ(n → − p) Xn Xn = = exp(−Q/KB T ) Xp 1 − Xn
(11) (12)
5 For kB T
