381 A Mathematical Appendix
A.1 Selected Formulae “Don’t worry about your difficulties in mathematics; I can assure you that mine are still greater.” Albert Einstein
The solution of physics problems often involves mathemat- ics. In most cases nature is not so kind as to allow a precise mathematical treatment. Many times approximations are not only rather convenient but also necessary, because the gen- eral solution of specific problems can be very demanding and sometimes even impossible. In addition to these approximations, which often involve power series, where only the leading terms are relevant, ba- sic knowledge of calculus and statistics is required. In the following the most frequently used mathematical aids shall be briefly presented.
1. Power Series
Binomial expansion: binomial expansion (1 ± x)m = m(m − 1) m(m − 1)(m − 2) 1 ± mx + x2 ± x3 +··· 2! 3! m(m − 1) ···(m − n + 1) + (±1)n xn +··· . n! For integer positive m this series is finite. The coefficients are binomial coefficients m(m − 1) ···(m − n + 1) m = . n! n If m is not a positive integer, the series is infinite and con- vergent for |x| < 1. This expansion often provides a simpli- fication of otherwise complicated expressions. 382 A Mathematical Appendix
A few examples for most commonly used binomial ex- pansions: examples 1 1 1 5 (1 ± x)1/2 = 1 ± x − x2 ± x3 − x4 ±··· , for binomial expansions 2 8 16 128 − 1 3 5 35 (1 ± x) 1/2 = 1 ∓ x + x2 ∓ x3 + x4 ∓··· , 2 8 16 128 − (1 ± x) 1 = 1 ∓ x + x2 ∓ x3 + x4 ∓··· , (1 ± x)4 = 1 ± 4x + 6x2 ± 4x3 + x4 finite . trigonometric functions Trigonometric functions:
x3 x5 x2n+1 sin x = x − + −···+(−1)n ±··· , 3! 5! (2n + 1)! x2 x4 x2n cos x = 1 − + −···+(−1)n ±··· , 2! 4! (2n)! 1 2 17 π tan x = x + x3 + x5 + x7 +··· , |x| < , 3 15 315 2 1 x x3 2x5 cot x = − − − −··· , 0 < |x| <π. x 3 45 945 exponential function Exponential function:
x x2 x3 xn ex = 1 + + + +···+ +··· . 1! 2! 3! n! natural logarithm Logarithmic function:
2 3 4 n x x x + x ln(1 + x) = x − + − +···+(−1)n 1 . 2 3 4 n
2. Indefinite Integrals xn+1 powers xn x = ,(n=− ), d + 1 n 1 dx = ln x, x 1 + powers of linear functions (ax + b)n x = (ax + b)n 1 ,(n=− ), d + 1 a(n 1) dx 1 = ln(ax + b) , ax + b a A.1 Selected Formulae 383 ex dx = ex , exponential function eax x ax x = (ax − ), e d 2 1 a dx 1 eax = ln , 1 + eax a 1 + eax sin x dx =−cos x, trigonometric functions cos x dx = sin x, tan x dx =−ln cos x, ln x dx = x ln x − x. natural logarithm
3. Specific Integrals π/2 1 cos x sin x dx = , triginometric 0 2 ∞ − 2 1 π e ax dx = , Gaussian 0 2 a ∞ x dx π2 = , exponentials ex − 1 6 0 ∞ x x π2 d = x , 0 e + 1 12 ∞ ) sin ax π a> = 2 for 0 ax/x dx − π , sin 0 x 2 for a<0 1 ln x π2 dx = . logarithm 0 x − 1 6
4. Probability Distributions
Binomial: binomial distribution n! f(r,n,p) = pr qn−r , r!(n − r)! r = 0, 1, 2,...,n, 0 ≤ p ≤ 1 ,q= 1 − p ; mean: r=np , variance: σ 2 = npq . 384 A Mathematical Appendix
Poisson distribution Poisson: µr e−µ f(r,µ)= ,r= 0, 1, 2,... , µ>0 ; r! mean: r=µ, variance: σ 2 = µ.
Gaussian distribution Gaussian: 1 (x − µ)2 f(x,µ,σ2) = √ exp − ,σ>0 ; σ 2π 2σ 2
mean: x=µ, variance: σ 2 .
Landau distribution Approximation for the Landau distribution: ) * 1 1 − L(λ) = √ exp − (λ + e λ) , 2π 2 where λ is the deviation from the most probable value.
5. Errors and Error Propagation
mean value Mean value of n independent measurements:
n 1 x= x ; n i i=1 variance variance of n independent measurements:
n n 1 1 s2 = (x −x )2 = x2 −x2 , n i i n i i=1 i=1 standard deviation where s is called the standard deviation. A best estimate for the standard estimation of the mean is s σ = √ . n − 1
If f(x,y,z) and σx , σy, σz are the function and standard independent, uncorrelated deviations of the independent, uncorrelated variables, then variables ∂f 2 ∂f 2 ∂f 2 σ 2 = σ 2 + σ 2 + σ 2 . f ∂x x ∂y y ∂z z If D(z) is the distribution function of the variable z around thetruevaluez0 with expectation value z and standard deviation σz, the quantity A.2 Mathematics for Angular Variations of the CMB 385 z+δ 1 − α = D(z) dz z−δ represents the probability that the true value z0 lies in the in- confidence interval terval ±δ around the measured value z; i.e., 100(1 − α)% measured values are within ±δ.IfD(z) is a Gaussian distri- bution one has δ 1 − α 1σ 68.27% . 2σ 95.45% 3σ 99.73%
In experimental distributions frequently the full width at half full width at half maximum maximum, z, is easily measured. For Gaussian distribu- tions z is related to the standard deviation by √ z(fwhm) = 2 2ln2σz ≈ 2.355 σz .
A.2 Mathematics for Angular Variations of the CMB
“As physics advances farther and far- ther every day and develops new ax- ioms, it will require fresh assistance from mathematics.” Francis Bacon
In this appendix the mathematics needed to describe the variations in the CMB temperature as a function of direction is reviewed. In particular, some of the important properties of the spherical harmonic functions Ylm(θ, φ) will be col- spherical harmonics lected. More information can be found in standard texts on mathematical methods of physics such as [46]. First it will be recollected what these functions are needed for in astroparticles physics. Suppose a quantity (here the temperature) as a function of direction has been dependence on the direction measured, which one can take as being specified by the standard polar coordinate angles θ and φ. This applies, e.g., for the directional measurements of the blackbody radiation. But one is not able – or at least it is highly impractical – to try to understand individually every measurement for every direction. Rather, it is preferable to parameterize the data with some function and see if one can understand the most important characteristics of this function. 386 A Mathematical Appendix
When, however, a function to describe the measured temperature as a function of direction is chosen, one can- ‘periodic functions’ not take a simple polynomial in θ and φ, because this would not satisfy the obvious continuity requirements, e.g., that the function at φ = 0 matches that at φ = 2π. By using spheri- cal harmonics as the basis functions for the expansion, these requirements are automatically taken into account. Now one has to remember how the spherical harmon- important differential ics are defined. Several important differential equations of equations mathematical physics (Schrödinger, Helmholtz, Laplace) can be written in the form ∇2 + v(r) ψ = 0 , (A.1)
nabla operator where ∇ is the usual nabla operator, as defined by ∂ ∂ ∂ ∇ = e + e + e . (A.2) x ∂x y ∂y z ∂z Here v(r) is an arbitrary function depending only on the ra- dial coordinate r. In separation of variables in spherical co- separation of variables ordinates, a solution of the form
ψ(r,θ,φ) = R(r) Θ(θ) Φ(φ) (A.3)
is tried. Substituting this back into (A.1) gives for the angu- angular parts lar parts
d2Φ =−m2Φ, 2 dφ d2Θ cos θ dΘ m2 + + l(l + 1) − Θ = 0 , dθ 2 sin θ dθ sin2 θ (A.4)
where l = 0, 1,...and m =−l,...,l are separation con- azimuthal solution stants. The solution for Φ is 1 Φ(φ) = √ eimφ . (A.5) 2π polar solution The solution for Θ is proportional to the associated Leg- m endre function Pl (cos θ). The product of the two angular spherical harmonic function parts is called the spherical harmonic function Ylm(θ, φ), Y (θ, φ) = Θ(θ)Φ(φ) lm 2l + 1 (l − m)! = P m(cos θ)eimφ . (A.6) 4π (l + m)! l A.2 Mathematics for Angular Variations of the CMB 387
Some of the spherical harmonics are given below: 1 Y00(θ, φ) = √ , (A.7) 4π 3 iφ Y11(θ, φ) =− sin θ e , (A.8) 8π 3 Y10(θ, φ) = cos θ, (A.9) 4π 15 2 2iφ Y22(θ, φ) = sin θ e , (A.10) 32π 15 iφ Y21(θ, φ) =− sin θ cos θ e , (A.11) 8π 5 Y (θ, φ) = (3cos2 θ − 1). (A.12) 20 16π The importance of spherical harmonics for this investi- gation is that they form a complete orthogonal set of func- tions. That is, any arbitrary function f(θ,φ) can be ex- complete orthogonal set panded in a Laplace series as of functions Laplace series ∞ l f(θ,φ)= almYlm(θ, φ) . (A.13) l=0 m=−l
To determine the coefficients alm, one uses the orthogonality orthogonality relation relation ∗ = sin θ dθ dφYlm(θ, φ)Ylm (θ, φ) δl lδm m . (A.14)
If both sides of (A.13) are multiplied by Ylm , integration over θ and φ leads to calculation of coefficients = ∗ alm sin θ dθ dφYlm(θ, φ)f (θ, φ) . (A.15)
So, in principle, once a function f(θ,φ) is specified, the coefficients of its Laplace series can be found. The same spherical harmonics are found in the multipole multipole expansion expansion of the potential from an electric charge distribu- tion. The terminology is usually borrowed from this exam- ple and the terms in the series are referred to as multipole moments. The l = 0 term is the monopole, l = 1 the dipole, etc. 388 A Mathematical Appendix
To quantify the temperature variations of the CMB, the Laplace series for the Laplace series can be used to describe temperature variations = − of the CMB T (θ, φ) T(θ,φ) T l = almYlm(θ, φ) , (A.16) l≥1 m=−l where T is the temperature averaged over all directions. Here the sum starts at l = 1, not l = 0, since by construc- tion the l = 0 term gives the average temperature, which has been subtracted off. In some references one expands T /T rather than T . This gives the equivalent informa- tion but with the coefficients simply differing from those in (A.16) by a factor of T . In practice one determines the coefficients alm up to finite series: practical limit some lmax by means of a statistical parameter estimation technique such as the method of maximum likelihood. This procedure will use as input the measured temperatures and information about their accuracy to determine estimates for the coefficients alm and their uncertainties. Once one has estimates for the coefficients alm, one can summarize the amplitude of regular variation with angle by defining
l 1 2 Cl = |alm| . (A.17) 2l + 1 m=−l
angular power spectrum The set of numbers Cl is called the angular power spectrum. The value of Cl represents the level of structure found at an angular separation
180◦ θ = . (A.18) l The measuring device will in general only be able to resolve angles down to some minimum value; this determines the maximum measurable l. 389 B Results from Statistical Physics: Thermodynamics of the Early Universe
“Scientists speak of the Law of Inertia or the Second Law of Thermodynamics as if some great legislative in the sky once met and set down the rules to govern the universe.” Victor J. Stenger
In this appendix some results from statistical and thermal physics will be recalled that will be needed to describe the early universe. To start with, the Fermi–Dirac and Bose– Fermi–Dirac, Bose–Einstein Einstein distributions for the number of particles per unit distribution volume of momentum space will be derived: V 1 f(p) = g , (B.1) (2π)3 e(E−µ)/T ± 1 where E = p2 + m2 is the energy and g is the number of internal degrees of freedom for the particle, V is the volume of the system, T is the temperature, and µ is the chemi- cal potential. (The Boltzmann constant k is set to unity as chemical potential usual.) The minus sign in (B.1) is used for bosons and the plus for fermions. These distributions will be required to de- rive the number density n of particles, the energy density , and the pressure P . Some of the relations may differ from those covered in a typical course in statistical mechanics. This is for two main reasons. First, the particles in the very hot early universe relativistic treatment typically have velocities comparable to the speed of light, therefore the relativistic equation E2 = p2 + m2 will be needed to relate energy and momentum. Second, the tem- variable particle numbers peratures will be so high that particles are continually be- ing created and destroyed, e.g., through reactions such as γγ ↔ e+e−. This is in contrast to the physics of low-tem- perature systems, where the number of particles in a system is usually constrained to be constant. The familiar exception is blackbody radiation, since massless photons can be cre- blackbody radiation ated and destroyed at any non-zero temperature. For a gas of relativistic particles it will be found that the expressions for n, ,andP are similar to those for blackbody radiation. 390 B Results from Statistical Physics: Thermodynamics of the Early Universe
B.1 Statistical Mechanics Review
“The general connection between en- ergy and temperature may only be es- tablished by probability considerations. Two systems are in statistical equilib- rium when a transfer of energy does not increase the probability.” Max Planck
Consider a system with volume V = L3 and energy U, which could be a cube of the very early universe. The num- ber of particles will not be fixed since the temperatures con- sidered here will be so high that particles can be continually created and destroyed. For the moment only a single particle type will be considered but eventually the situation will be generalized to include all possible types. The system can be in any one of a very large number of possible microstates. The fundamental postulate of sta- fundamental postulate: tistical mechanics is that all microstates consistent with the equipartition of energy imposed constraints (volume, energy) are equally likely. A given microstate, e.g., an N-particle wave function ψ(x1, ...,xN ) specifies everything about the system, but this is far more than one wants to know. To reduce the information to a more digestible level, one can determine from the mi- crostate the momentum distribution of the particles, i.e., the expected number of particles in each cell d3p of momentum space. There will be many microstates that lead to the same dis- tribution, but one distribution in particular will have over- whelmingly more possible microstates than the others. To good approximation all the others can be ignored and this equilibrium distribution equilibrium distribution can be regarded as the most likely. Once it has been found, one can determine from it the other quantities needed, such as the energy density and pressure. So, to find the equilibrium distribution one needs to de- termine the number of possible microstates consistent with a distribution and then find the one for which this is a max- imum. This is treated in standard books on statistical me- chanics, e.g., [47]. Here only the main steps will be re- viewed. N-particle wave function It is assumed that the system’s N-particle wave function can be expressed as a sum of N terms, each of which is the product of N one-particle wave functions of the form
ipA·x ψA(x) ∼ e . (B.2) B.1 Statistical Mechanics Review 391
The total wave function is thus
ψ(x1,...,xN ) = 1 = √ P(i,j,...)ψA(xi)ψB (xj ) ··· , (B.3) N! where the sum includes all possible permutations of the co- ordinates xi. For a system of identical bosons, the factor P is equal to one, whereas for identical fermions it is plus or mi- symmetrization for bosons nus one depending on whether the permutation is obtained antisymmetrization from an even or odd number of exchanges of particle coor- for fermions dinates. This results in a wave function that is symmetric for bosons and antisymmetric for fermions upon interchange of any pair of coordinates. As a consequence, the total wave function for a system of fermions is zero if the same one- particle wave function appears more than once in the prod- uct of terms; this is the Pauli exclusion principle. Pauli exclusion principle Although the most general solution to the N-particle Schrödinger equation does not factorize in the way ψ has been written in (B.3), this form will be valid to good ap- proximation for systems of weakly interacting particles. For high-temperature systems such as the early universe, (B.3) is assumed to hold. Further, one assumes that the one-particle wave func- tions should obey periodic boundary conditions in the vol- periodic boundary ume V = L3. The plane-wave form for the one-particle conditions: discretizing wave functions in (B.2) then implies that the momentum momentum vectors p cannot take on arbitrary values but that they must satisfy 2π p = (n ,n ,n ), (B.4) L x y z where nx , ny ,andnz are integers. Thus, the possible mo- menta for the one-particle states are given by a cubic lattice of points in momentum space with separation 2π/L. For a given N-particle wave function, where N will in general be very large, the possible momentum vectors for the one-particle states will follow some distribution in mo- mentum space. That is, one will find a certain number dN of one-particle states for each element d3p in momentum space, and momentum distribution
d3N f(p) = (B.5) d3p will be called the momentum distribution. 392 B Results from Statistical Physics: Thermodynamics of the Early Universe
A given distribution f(p) could result from a number of distinct N-particle wave functions, i.e., from a number of different microstates. All available microstates are equally likely, but the overwhelming majority of them will corre- spond to a single specific f(p), the equilibrium distribution. This is what one needs to find. To find this equilibrium momentum distribution, one number of microstates must determine the number of microstates t for a given f(p). To do this, one considers the momentum space to be divided into cells of size δ3p. The number of particles in the ith cell is
3 νi = f(pi)δ p. (B.6)
The number of possible one-particle momentum states in the cell is δ3p divided by the number of states per unit volume total number in momentum space, (2π/L)3. The total number of one- of one-particle states particle states in δ3p is therefore1
δ3p γi = g , (B.7) (2π/L)3
number of degrees where g represents the number of internal (e.g., spin) de- of freedom grees of freedom for the particle. For an electron with spin 1/2, for example, one has g = 2. It is assumed that the element δ3p is large compared to the volume of momentum space per available state, which is (2π/L)3, but small compared to the typical momenta of the particles. Within this approximation, the set of numbers νi for all i contains the same information as f(p). system of bosons For a system of bosons, there is no restriction on the number of particles that can have the same momentum. Therefore, each of the γi states can have from zero up to νi particles. The number of ways of distributing the νi particles among the γi states is a standard problem of combinatorics (see, e.g., [47]). One obtains
(νi + γi − 1)! (B.8) νi !(γi − 1)!
total number of microstates possible arrangements. The total number of microstates for the distribution is therefore 1 In many references the number of particles is called ni and the number of states gi. Unfortunately, these letters need to be used with different meanings later in this appendix, so here νi and γi will be used instead. B.1 Statistical Mechanics Review 393