A Mathematical Appendix

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 mathematics. 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 general 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, basic 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 expansions: 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 distribution 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 distributions 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 farther 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 spherical 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 functions. 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 distribution. The terminology is usually borrowed from this example 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 information 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 chemical 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 being created and destroyed, e.g., through reactions such as γγ ↔ e+e−. This is in contrast to the physics of low-temperature 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 energy and temperature may only be es- tablished by probability considerations. Two systems are in statistical equilibrium 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 number of particles will not be fixed since the temperatures considered 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 microstate 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 distribution, 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 determine the number of possible microstates consistent with a distribution and then find the one for which this is a maximum. This is treated in standard books on statistical mechanics, e.g., [47]. Here only the main steps will be reviewed. 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 coordinates 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 product 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 approximation 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 functions 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 momenta 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 momentum 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

(ν + γ − 1)! (ν + γ )! t [f(p)]= i i ≈ i i , BE ν !(γ − 1)! ν !γ ! i i i i i i (B.9) where the product extends over all cells in momentum space. The subscript BE in (B.9) stands for Bose–Einstein since this relation holds for a collection of identical bosons. For fermions, the antisymmetric nature of the total wave system of fermions function implies that it can contain a given one-particle state at most only once. Therefore, each of the γi states in the ith cell in momentum space can be occupied either once or not at all. This implies γi ≥ νi . The number of possible arrangements of νi particles in the γi states where each state is occupied zero or one time is another standard problem of combinatorics, for which one ﬁnds

γi! . (B.10) νi!(γi − νi)!

The total number of combinations for all cells is thus total number of microstates γi! t [f(p)]= , (B.11) FD ν !(γ − ν )! i i i i where FD stands for Fermi–Dirac. As the number of microstates t[f(p)] is astronomically large, it is more convenient to work with its logarithm, and logarithm of the number furthermore one can use Stirling’s approximation, of microstates

ln N!≈N ln N − N, (B.12) valid for large N.Thisgives t [f(p)]≈ ln BE (B.13) ≈ [(νi + γi) ln(νi + γi) − νi ln νi − γi ln γi] i and t [f(p)]≈ ln FD (B.14) ≈ [γi ln γi − νi ln νi − (γi − νi ) ln(γi − νi)] i for bosons and fermions, respectively. The next step is to ﬁnd the distribution f(p) which max- imizes ln t[f(p)]. Before doing this, however, the problem should be generalized to allow for more than one type of more than one type particle. As long as a particle’s mass is small compared to of particle 394 B Results from Statistical Physics: Thermodynamics of the Early Universe

the temperature, it will be continually created and destroyed, and for sufﬁciently early times this will be true for all particle types. So one will have a set of distribution functions fa(p), where the index a = e,µ,τ,u,d,s,... is a label for the particle type. One can write this set of functions as a vector, f ≡ (fe,fµ,fτ ,fu,fd ,...). In order to ﬁnd the set of distributions f (p), one needs maximization to maximize the total number of microstates t[f (p)] subject to two types of constraints. First, it is required that the sum energy conservation of the energies of all of the particles be equal to the total energy U, i.e., 3 U = Eafa(p) d p a = 2 + 2 p 3 p ma fa( ) d p. (B.15) a Second, although particles may be created and destroyed, certain quantities are conserved. Suppose that the system conserved quantities has a total conserved charge Q (e.g., zero), baryon number B, and lepton number L. Suppose further that in thermal equilibrium the system has Na particles of type a.Thesere- quirements can be written as Q = QaNa , (B.16) a B = BaNa , (B.17) a L = LaNa . (B.18) a

Note that the values Na are not explicitly constrained, but rather only the total Q, B,andL. Thus, the quantity that one wants to maximize can be expressed as φ(f (p), α ,α ,α ,β)= Q B L 3 = ln ta[fa(p)]+β U − Eafa (p) d p a a + αQ Q − QaNa + αB B − BaNa a a + αL L − LaNa , (B.19) a

Lagrange multipliers where β, αQ, αB ,andαL are Lagrange multipliers. Setting the derivatives of φ with respect to the Lagrange multipliers B.1 Statistical Mechanics Review 395 to zero ensures that the corresponding constraints are ful- ﬁlled. To ﬁnd the set of distributions f which maximize 3 (B.19), one substitutes fa(pi ) = νai/δ p. Then' the inte- discretizing momentum = grals are converted to sums, and further one has i νai integrals Na. In addition, the number of microstates can be obtained from (B.13) for bosons and (B.14) for fermions. The derivative of φ with respect to νai is ∂φ = ln(γai ± νai) − ln νai − βEai ∂νai − αQQa − αB Ba − αLLa , (B.20) 3 3 where γai = gaδ p/(2π/L) is the number of states available to a particle of type a in the cell i, and for the derivation in this section the upper sign refers to bosons and the lower sign to fermions. Setting (B.20) equal to zero and solving solution for νai gives for number of particles

γai νai = . exp αQQa + αB Ba + αLLa + βEai ∓ 1 (B.21)

Re-expressing this in terms of the functions fa(pi) = 3 νai/δ p gives

3 ga(L/2π) fa(p) = . exp αQQa + αB Ba + αLLa + βEa ∓ 1 (B.22)

The temperature can be deﬁned as deﬁnition of temperature

T = 1/β , (B.23) and it can be shown that this has all of the desired properties of the usual thermodynamic temperature. Furthermore, the chemical potential chemical potential for particle type a can be deﬁned as

µa =−T(αQQa + αB Ba + αLLa), (B.24) which can be modiﬁed in the obvious way to include a different set of conserved quantities. Note that, although the Lagrange multipliers are speciﬁc to the system, i.e., are the same for all particle types, the chemical potential depends on the charge, baryon number, and lepton number of the particle. In a reaction where, say, a + b ↔ c + d, (B.24) implies µa + µb = µc + µd . 396 B Results from Statistical Physics: Thermodynamics of the Early Universe

Using these modiﬁed names for the Lagrange multipliers resulting momentum gives the desired result for the momentum distribution, distribution 3 ga(L/2π) fa(p) = , (B.25) e(Ea−µa )/T ∓ 1 where one uses the minus sign if particle type a is a boson internal degrees of freedom and plus if it is a fermion. The number of internal degrees of freedom, ga, is usually 2J + 1 for a particle of spin J ,butit could include other degrees of freedom besides spin such as colour.

B.2 Number and Energy Densities

“There are 1011 stars in the galaxy. That used to be a huge number. But it’s only a hundred billion. It’s less than the national deﬁcit! We used to call them astronomical numbers. Now we should call them economical numbers.” Richard P. Feynman

From the Planck distribution given by (6.81), (B.1) one can proceed to determine the number and energy per unit volume for all of the particle types. The function (B.25) gives the number of particles of type a in a momentum-space volume number density n d3p. The number density n is obtained by integrating this over all of momentum space and dividing by the volume V = L3, i.e., 1 g d3p n = f(p) d3p = , V (2π)3 e(E−µ)/T ± 1 (B.26) where for clarity the index indicating the particle type has been dropped. Since the integrand only depends on the magnitude of the momentum through E = p2 + m2, one can take the element d3p to be a spherical shell with radius p and thickness dp,sothatd3p = 4πp2 dp.FromE2 = p2 + m2 n: energy integral one gets 2E dE = 2p dp and therefore √ g ∞ E2 − m2E dE n = . (B.27) 2 (E−µ)/T 2π m e ± 1 The integral (B.27) can be carried out in closed form only n: relativistic limit for certain limiting cases. In the limit where the particles are relativistic, i.e., T m,andalsoifT µ, one finds B.2 Number and Energy Densities 397 ⎧ ⎪ζ(3) ⎨⎪ gT 3 for bosons, π2 n = (B.28) ⎪3 ζ(3) ⎩ gT 3 for fermions. 4 π2 Here ζ is the Riemann zeta function and ζ(3) ≈ 1.20 206 .... Notice that in particle physics units the number density has dimension of energy cubed. To convert this to a normal number per unit volume, one has to divide by (hc)¯ 3 ≈ (0.2GeVfm)3. In the non-relativistic limit (T m), the integral (B.27) n: non-relativistic limit becomes 3/2 mT − − n = g e (m µ)/T , (B.29) 2π where the same result is obtained for both the Fermi–Dirac and Bose–Einstein distributions. One sees that for a non- relativistic particle species, the number density is exponen- tially suppressed by the factor e−m/T , the so-called Boltz- mann factor. This may seem counter intuitive, since the density of air molecules in a room is certainly not suppressed by this factor, although they are non-relativistic. One must take into account the fact that the chemical potentials depend in general on the temperature, and this dependence is exactly such that all relevant quantities are conserved. In the case of the air molecules, µ varies with temperature so as to exactly compensate the factor T 3/2 e−m/T . For very high temperatures, to good approximation all very high temperatures of the chemical potentials can be set to zero. The total number of particles will be large compared to the net values of any of the conserved quantum numbers, and the constraints effectively play no rôle. To find the energy density one multiplies the number energy density of particles in d3p by the energy and integrate over all momenta, g E d3p = (2π)3 e(E−µ)/T ± 1 √ g ∞ E2 − m2 E2 dE = . (B.30) 2 (E−µ)/T 2π m e ± 1 As with n, the integral can only be carried out in closed form for certain limiting cases. In the relativistic limit, T m, : relativistic limit one finds 398 B Results from Statistical Physics: Thermodynamics of the Early Universe ⎧ ⎪π2 ⎨⎪ gT 4 for bosons, = 30 ⎪ (B.31) ⎩⎪7 π2 gT 4 for fermions. 8 30 : non-relativistic limit In the non-relativistic limit one has

= mn , (B.32)

with the number density n given by (B.29). From the number and energy densities one can obtain average energy per particle the average energy per particle, E=/n.ForT m one ﬁnds ⎧ ⎪ π4 ⎨⎪ T ≈ 2.701 T for bosons, 30 ζ(3) E= (B.33) ⎪ 7π4 ⎩⎪ T ≈ 3.151 T for fermions. 180 ζ(3) In the non-relativistic limit, the average energy, written as the sum of mass and kinetic terms, reads 3 E=m + T, (B.34) 2 which is dominated by the mass m for low T .

B.3 Equations of State

“If your theory is found to be against the second law of thermodynamics, I give you no hope; there is nothing for it but to collapse in deepest humiliation.” Arthur Eddington

Finally in this appendix an equation of state will be derived, that is, a relation between energy density and pressure P . This will be needed in conjunction with the acceleration and fluid equations in order to solve the Friedmann equation for R(t). There are several routes to the desired relation. The ap- first law of thermodynamics proach that starts from the first law of thermodynamics is the most obvious one,

dU = T dS − P dV, (B.35) B.3 Equations of State 399 which relates the total energy U, temperature T ,entropyS, pressure P ,andvolumeV of the system. The differential dU can also be written as ∂U ∂U dU = dS + dV, (B.36) ∂S V ∂V S where the subscripts indicate what is kept constant when computing the partial derivatives. Equating the coefﬁcients of dV in (B.35) and (B.36) gives the pressure, pressure ∂U P =− . (B.37) ∂V S Recall that the entropy is simply the logarithm of the to- entropy and tal number of microstates Ω, and that to good approximation number of microstates this is given by the number of microstates of the equilibrium distribution t[f(p)].Thatis,

S = ln Ω ≈ ln t[f(p)] . (B.38)

The important thing to notice here is that the entropy is entirely determined by the distribution f(p). Therefore, to keep the entropy constant when computing (∂U/∂V )S, one simply needs to regard the distribution f(p) as remaining constant when V is changed. The total energy U is total energy U = Ef(p) d3p, (B.39) and the pressure is therefore ∂U ∂E P =− =− f(p) d3p. (B.40) ∂V S ∂V The derivative of E with respect to the volume V = L3 is ∂E ∂E ∂p ∂E ∂p ∂V = = . (B.41) ∂V ∂p ∂V ∂p ∂L ∂L One has ∂V /∂L = 3L2 and furthermore E = p2 + m2, so ∂E 1 −1/2 p = p2 + m2 2p = . (B.42) ∂p 2 E

From (B.4) one gets that p ∼ L−1, and therefore ∂p/∂L = −p/L. Substituting these into (B.41) gives 400 B Results from Statistical Physics: Thermodynamics of the Early Universe ∂E p −p 1 −p2 = = . (B.43) ∂V E L 3L2 3EV

general expression Putting this into (B.40) provides the general expression for for the pressure the pressure, 1 p2 P = f(p) d3p. (B.44) 3V E

In the relativistic limit the particle’s rest mass can be neglected, so, E = p2 + m2 ≈ p. Equation (B.44) then becomes 1 P = Ef(p) d3p. (B.45) 3V But the total energy U is (B.39) U = Ef(p) d3p (B.46)

pressure and = U/V, so the ﬁnal result for the pressure for a gas in the relativistic limit of relativistic particles is simply P = . (B.47) 3 This is the well-known result from blackbody radiation, but one realizes here that it applies for any particle type in the relativistic limit T m. In the non-relativistic limit, the pressure is given by the non-relativistic limit: ideal gas law, ideal gas law P = nT . (B.48)

In this case, however, the energy density is simply = mn, so for T m one has P and in the acceleration and ﬂuid equations one can approximate P ≈ 0. Finally, the case of vacuum energy density from a cos- vacuum energy density mological constant can be treated, from a cosmological constant Λ = . (B.49) v 8πG

If one takes U/V = v as constant, then the pressure is ∂U U P =− =− =−v . (B.50) ∂V S V negative pressure Thus, a vacuum energy density leads to a negative pressure. 401 C Deﬁnition of Equatorial and Galactic Coordinates

“In the technical language of astronomy, the richness of the star ﬁeld depends mainly on the galactic latitude, just as the Earth’s cli- mate depends mainly on the geographic latitude, and not to any great extent on the longitude.” Sir James Jeans

Optical astronomers mostly prefer equatorial coordinates, like right ascension and declination, while astrophysicists often use galactic coordinates. For equatorial coordinates, which are centered on the equatorial coordinates Earth, the plane of the Earth’s equator is chosen as the plane of reference. This plane is called the celestial equator. There is another plane which is deﬁned by the motion of the Earth around the Sun. This plane has an inclination with respect to the celestial equator of 23.5 degrees. The plane of the Earth’s orbit is called the ecliptic. At the periphery these two planes ecliptic intersect in two points. The one where the Sun crosses the celestial equator from the south is the vernal equinox. The coordinate measured along the celestial equator east- ward from the vernal equinox is called right ascension,usu- right ascension ally named α. The distance perpendicular to the celestial equator is named declination, named δ. Right ascension declination varies from 0 to 360 degrees, or sometimes – for conve- nience – from 0 to 24 hours. Declination varies from −90 to +90 degrees (see Fig. C.1).

north celestial pole

declination δ

equator Fig. C.1 ecliptic vernal Deﬁnition of the equatorial coordinates right ascension and equinox α right ascention declination 402 C Deﬁnition of Equatorial and Galactic Coordinates

galactic coordinates In contrast, galactic coordinates are ﬁxed on our galaxy. The spherical coordinates (r,l,b) are centered on the Sun, longitude with l being the galactic longitude and b the latitude. r is the latitude distance of the celestial object from the Sun. The galactic distance from the Sun longitude l is the angle between the direction to the galactic center and the projection of the direction to the star onto the galactic plane. It counts from the direction to the galactic center (l = 0 degrees) via the galactic anticenter (180 degrees, away from the Sun) back to the galactic center. The latitude b varies from the +90 degrees (perpendicular above the galactic plane) to −90 degrees (Fig. C.2).

star

r Sun b l

galactic plane Fig. C.2 galactic center Deﬁnition of the galactic coordinates latitude and longitude 403

D Important Constants for Astroparticle Physics1

“Men who wish to know about the world must learn about it in its particular details.” Heraclitus

relative uncertainty velocity of light c 299 792 458 m/s exact gravitational constant G 6.6742 × 10−11 m3 kg−1 s−2 1.5 × 10−4 Planck’s constant h 6.626 0693 × 10−34 Js 1.7 × 10−7 = 4.135 6675 × 10−15 eV s h¯ = h/2π 1.054 571 68 × 10−34 Js 1.7 × 10−7 = × −16 × −8 √ 6.582 119 15 10 eV s 8.5 10 Planck mass hc/G¯ 2.176 45 × 10−8 kg 7.4 × 10−5 = × 19 2 1.220 90 10 GeV/c 3 × −35 × −5 Planck length Gh/c¯ 1.616 24 10 m7.4 10 Planck time Gh/c¯ 5 5.391 19 × 10−44 s7.4 × 10−5 elementary charge e 1.602 176 53 × 10−19 C8.5 × 10−8 2 ﬁne-structure constant α = e 1/137.035 999 11 3.3 × 10−9 4πε0hc¯ Rydberg energy Ry 13.605 6923 eV 8.5 × 10−8 −31 −7 electron mass me 9.109 3826 × 10 kg 1.7 × 10 0.510 998 918 MeV/c2 8.6 × 10−8 −27 −7 proton mass mp 1.672 621 71 × 10 kg 1.7 × 10 = 938.272 029 MeV/c2 8.6 × 10−8 −27 −7 neutron mass mn 1.674 9287 × 10 kg 6 × 10 = 939.565 36 MeV/c2 3 × 10−7 2 −6 mn − mp = 1.293 3317 MeV/c 6 × 10 23 −1 −7 Avogadro constant NA 6.022 1415 × 10 mol 1.7 × 10 Boltzmann constant k 1.380 6505 × 10−23 JK−1 1.8 × 10−6 = 8.617 343 × 10−5 eV K−1 Stefan–Boltzmann constant σ 5.670 400 × 10−8 Wm−2 K−4 7 × 10−6 electron volt, eV 1.602 176 53 × 10−19 J8.5 × 10−8

1 see also Eidelman et al., Phys. Letters B592, 1 (2004) 404 D Important Constants for Astroparticle Physics

relative uncertainty standard atmosphere, atm 101 325 Pa exact acceleration due to gravity g 9.806 65 m s−2 exact −1 −1 Hubble constant H0 71 km s Mpc 5% 9 age of the universe t0 13.7 × 10 a1.5% Hubble distance c/H0 4 225 Mpc 5% = 2 × −30 3 ≈ critical density c 3H0 /8πG 9.469 10 g/cm 10% density of galaxies ≈ 0.02 Mpc−3 temperature of the blackbody radiation 2.725 K 4 × 10−4 number density of blackbody photons 410.4cm−3 1.2 × 10−3 astronomical unit, AU 1.495 978 706 60 × 1011 m1.3 × 10−10 parsec, pc (1 AE/1arcsec) 3.085 677 5807 × 1016 m1.3 × 10−10 light-year, LY 0.3066 pc = 0.9461 × 1016 m Schwarzschild radius of the Sun: 2GM/c2 2.953 250 08 km 3.6 × 10−4 mass of the Sun M 1.988 44 × 1030 kg 1.5 × 10−4 solar constant 1 360 W/m2 0.2% solar luminosity L 3.846 × 1026 W0.2% − mass of the Earth M♁ 5.9723 × 1024 kg 1.5 × 10 4 radius of the Earth R♁ 6.378 140 × 106 m Schwarzschild radius − of the Earth: 2GM♁/c2 0.887 056 22 cm 1.5 × 10 4 velocity of the solar system about the galactic center 220 km/s9% distance of the Sun from the galactic center 8.0 kpc 6% matter density of the universe Ωm 0.27 18% baryon density of the universe Ωb 0.044 9% dark-matter density of the universe Ωdm 0.22 18% −5 radiation density of the universe Ωγ 4.9 × 10 10% neutrino density of the universe Ων ≤ 0.015 95% C.L. dark-energy density of the universe Λ 0.73 5% total energy density of the universe Ωtot 1.02 2% (Ωtot ≡ Ω) −7 3 number density of baryons nb 2.5 × 10 /cm 4% baryon-to-photon ratio η 6.1 × 10−10 3% 0◦C 273.15 K 405 References

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M. A. Bucher, D. N. Spergel, Was vor dem Urknall geschah, Spektrum der Wissenschaft, 3/1999, S. 55. C. J. Hogan, R. P. Kirshner, N. B. Suntzeff, Die Vermessung der Raumzeit mit Supernovae, Spektrum der Wissenschaft, 3/1999, S. 40. L. M. Krauss, Neuer Auftrieb für ein beschleunigtes Universum, Spektrum der Wissen- schaft, 3/1999, S. 47. J. Trümper, ROSAT und seine Nachfolger, Phys. Bl. 55/9, S. 45, 1999. Craig J. Hogan, Rev. Mod. Phys. 72 (2000) 1149–1161. H. Blümer, Die höchsten Energien im Universum, Physik in unserer Zeit, S. 234–239, Nov. 1999. Brian Greene, “The Elegant Universe”, Vintage 2000, London, 1999. H. Völk, H. Blümer & K.-H. Kampert, H. Krawczynski et. al., Ch. Spiering, M. Simon, J. Jochum & F. von Feilitzsch: Phys. Bl., Schwerpunkt Astroteilchenphysik, März 2000, S. 35–68. Paul J. Steinhardt, “Quintessential Cosmology and Cosmic Acceleration”, Selection of review papers under the heading “Dark Energy and the Cosmological Constant” http://pancake.uchicago.edu/~carroll/reviewarticles.html. Peter K. F. Grieder, Cosmic Rays at Earth, Elsevier Science, 2001. S. Hayakawa, Cosmic Ray Physics, Wiley Interscience, N.Y., 1969. K. C. Cole, TheHoleintheUniverse, The Harvest Book/Harcourt Inc., San Diego, 2001. Brian Greene, The Fabric of the Cosmos: Space, Time and the Texture of Reality, Alfred A. Knopf, N.Y., 2004. John A. Bahcall, Neutrino Astronomy, Cambridge University Press, 1990. Rodney Hillier, Gamma Ray Astronomy, Oxford Studies in Physics, 1984. John A. Peacock, Cosmological Physics, Cambridge University Press, 2003. Martin Rees, Just Six Numbers: The Deep Forces That Shape the Universe, Basic Books, New York, 2001. Martin Rees, New Perspectives in Astrophysical Cosmology, Cambridge University Press, 2000. S. M. Bilenky, The History of Neutrino Oscillations, hep-ph/0410090, 2004. A. V. Filippenko and A. G. Riess, Phys. Rep. 307 (1999) 31. 413 Photo Credits

“Equipped with his ﬁve senses, man explores the universe around him and calls the adventure science.” Edwin Powell Hubble

{1} T. Credner and S. Kohle, University of Bonn, Calar Alto Observatory {2} Prof. Dr. D. Kuhn, University of Innsbruck, Austria {3} Courtesy of The Archives, California Institute of Technology, Photo ID 1.22-5 {4} C. Butler and G. Rochester, Manchester University {5} Prof. Dr. A. A. Penzias, Bell Labs, USA {6} D. Malin, Anglo-Australian Observatory {7} Particle Data Group, European Physical Journal C3, (1998), 144; http://pdg.lbl.gov {8} Prof. Dr. Y. Totsuka, Institute for Cosmic Ray Research, Tokyo, Japan {9} T. Kajita, Y. Totsuka, Rev. Mod. Phys. 73 (2001) p. 85 {10} Y. Fukuda et al., Phys. Rev. Lett. 81 (1998) 1562–1567 {11} Prof. Dr. R. Davis, Brookhaven National Laboratory, USA {12} R. Svoboda and K. Gordan (Louisiana State University), http://antwrp.gsfc.nasa.gov/apod/ap980605.html. Equivalent ﬁgure in: A. B. McDonald et al., Rev. Sci. Instrum. 75 (2004) 293–316, astro-ph/0311343 {13} AMANDA Collaboration, Christian Spiering, private communication 2004 {14} Dr. D. Heck, Dr. J. Knapp, Forschungszentrum Karlsruhe, Germany {15} Cangaroo Collaboration, Woomera, Australia {16} CGRO Project Scientist Dr. Neil Gehrels, Goddard Space Flight Center, USA (1999) {17} BATSE-Team, NASA, M. S. Briggs, (http://www.batse.msfc.nasa.gov/batse/grb/skymap/) {18} Prof. Dr. J. Trümper, Max-Planck Institute for Extraterrestrial Physics, München, Garching {19} ESA/XMM-Newton, http://xmm.vilspa.esa.es/external/xmm_science/ gallery/public courtesy by L. Strüder 414 Photo Credits

{20} ESA/XMM-Newton/Patrick Henry et al., courtesy by L. Strüder {21} Prof. Dr. M. Simon, University of Siegen, Germany {22} R. Tcaciuc, University of Siegen, 2004 {23} ALEPH Collaboration, CERN, courtesy by P. Dornan, Imperial College {24} Frejus-Experiment, France/Italy, courtesy by Chr. Berger, RWTH Aachen {25} Prof. Dr. P. Sokolsky, University of Utah, Salt Lake City, USA {26} LBNL-53543, R. A. Knop et al., Astrophys. J. 598 (2003) 102–137; http://supernova.lbl.gov/, courtesy by Saul Perlmutter {27} Courtesy of the COBE Science Working Group {28} Courtesy of the WMAP Science Team {29} Dr. L. J. King, University of Manchester, UK; NASA; King et al. 1998, MNRAS 295, L41 {30} Prof. Dr. G. G. Raffelt, Max-Planck Institute for Physics (Werner-Heisenberg Insti- tute), München, Germany 415 Index∗

“A document without an index is like a country without a map.” Mike Unwalla

Abell 754, 132 – parameter, 186, 281, 295, – extensive, 9, 10, 79, 154, absolute brightness, 299 299 156, 311 absolute zero, 299 – shock, 66, 67, 68, 74, 167 – – correlations, 162, 163 absorption accelerator, 299 – – longitudinal development, – coefficient, 46 – cosmic, 294 158 – line, neutrino, 275 – data, number of neutrino – – measurement technique, abundance, 299 families, 227 158, 160, 161 – cosmic, 307 – experiments, 293 – measurement, 159 – deuterium, 221, 224 – physics, 293 Air Watch, 160, 161 – helium-4 (4He), 22, 213, accretion disk, 72, 130, 155, ALEPH, 155 221, 222, 224, 227 299 α – light nuclei, 214, 222, 224 acoustic peak, 241, 242, 299 – decay, 300 – lithium-7 (7Li), 223–225 acoustic-horizon distance, 241 – particle, 10, 78 active galactic nuclei, 17, 73, – measurement, 225 Alpha Magnetic Spectrometer, 110, 118, 131, 132, 168, – predicted, 224 206 294, 300 – primordial, 15 Alvarez, L. W., 7 active galaxy, 299 – – deuterium, 223 AMANDA, 88, 107, 300 activity, potassium-40 (40K), 3 Anderson, C. D., 8, 9 – – helium-3 ( He), 224 106 4 Andromeda Nebula, 271, 300 – – helium-4 ( He), 22 Adams, D., 265 angle – – light nuclei, 222 additional Higgs field, 210 accelerated expansion, 252, additional neutrino family, 228 – azimuthal, 235 254, 285 adiabatic expansion, 182 – Cabibbo, 31 acceleration Advanced X-ray Astrophysics – Cherenkov, 55 – by sunspot pairs, 65 Facility, see AXAF – horizon distance, 241 – due to gravity, 404 age – mixing, 31, 91 – equation, 183, 252, 299, 400 – of life, 288 – polar, 235 – in gravitational potentials, – of the universe, 299, 404 Ångström, 300 72 aggregation, gravitational, 266 angular – mechanism, 63,68, 312 AGN, see active galactic nuclei – (diameter) distance, 240 – of particles in binaries, 71 air scintillation, 53, 116, – power spectrum, cosmic – of particles in pulsars, 69, 159–161 microwave background, 236, 294 air shower 238, 239, 242, 243, 385, 388 – of particles in supernovae, – Cherenkov technique, 55, – resolution, COBE and 67, 167, 294 79, 114, 116, 119, 300 WMAP, 238 – of protons, 105 – development, 145 – separation, 236 ∗ Underlined page numbers refer to main entries. Page numbers in italics apply to the glossary. 416 Index anisotropy, 85, 86 ASTRO-E, 132 attenuation coefficient, mass, – cosmic microwave astrobiology, 222, 287, 301 113 background, 235, 238, 241 astronomical unit (AU), 267, attractor, large, 318 – dipole, 235, 236, 310 301, 404 AU, see astronomical unit – – vanishing, 237 astronomy Auger experiment, 83, 158, – small-angle, 237 – bio-, 288, 303 169 ankle, see cosmic rays – γ , 55, 108, 314 Auger, P.-V., 9 annihilation, 112, 300 – – horizon, 116 aurora, 2, 301 – of positrons, 120, 207, 218, – infrared, 109, 317 average 219, 326 – neutrino, 8, 86, 151, 294 – density, 260 – of WIMPs, 277 – proton, 86 – energy per particle, 398 – phase, 171 – radar (radio), 109, 328 – temperature, 236 ¯ – pp, 112, 206, 295 – ultraviolet, 109 AXAF, 131, 301 301 – radiation, 296 – with gravitational waves, axion, 278, 279, 294, anomalies, quantum, 210, 327 133 – coupling to photons, 279 – decay, 279 ANTARES, 277 – X-ray, 56, 109, 123, 336 300 – mass, theoretical, 279 anthropic principle, 290, astroparticle antibaryon, 295, see also azimuthal angle, 235 – interactions, 50,58 baryon – physics, outlook, 293 anticarbon, 84 0 – propagation, 58 B meson anticoincidence counter, 114 astrophysics, 1 – CP violation, 210 anticolour, 26, see also colour – imprints on data, 197 – decay, 29, 278 antigalaxies, 296 background asymmetry antigravitation, 300, see also – microwave, cosmic, – baryon–antibaryon, gravitation see cosmic microwave 205, 206, 208, 302 antihelium, 84 background – matter–antimatter, 211, 296, antilepton, 295, see also lepton – radiation, see cosmic 321 antimatter, 300, see also microwave background asymptotic freedom, 16, 21, antiparticle – radioactivity, 277 301 – primary (cosmic), 84, 295 – X-ray radiation, 132 atmosphere – primordial, 296 Baksan scintillator experiment, – column density, 142 antineutron, 215 2 antinuclei, 214 – particle composition, 145 balloon experiment, 3, 79, 124, – primary (cosmic), 84, 206 – transformation in, 143, 144 143, 284 antiparticle, 295, 300 atmospheric baryogenesis, 206, 211, 215, – in cosmic rays, 84 – Cherenkov technique, see 301 antiproton, 10, 215, 295 air shower Cherenkov baryon, 25, 268, 295, 302 – primary, 84, 143 technique – anti-, 295 – production, 84 – cutoff, 6 – asymmetry of the universe, antiquark, 301, see also quark – depth, relation to zenith 205, 206, 208, 302 antisymmetrization, 391 angle, 143 – density, 214, 215, 224, 225 apastron, 301 – neutrinos, 88, 294 – energy density, 225, 243 aphelion, 301 – – deficit, 272 – fraction of the universe, 225 apogee, 301 – particle flux, 144 – Λ,9 apparent atom, 233 – – decay, 30 – light curve, 270, 271 atomic – number, 30, 295, 302, 394 – magnitude, 301 – mass, 301 – – conservation, 215, 216 Armenteros, R., 9 – number, 301 – – non-zero, 207 Index 417

– – processes violating, 204, – deuterium, 220 Boltzmann, see also Stefan– 208, 210, 215 – neutral hydrogen, 233 Boltzmann − – Ω ,25 – nuclear, 323 – constant, 304 – -to-photon ratio, 209, 214, binding, gravitational, see – factor, 213 215, 220, 224–227, 233, 243 gravitational binding Boomerang experiment, 243, – – measured, 243 binomial distribution, 383 284 baryonic matter, 265, 268, 269, binomial expansion and boron, 78 275, 281, 283, 295 coefﬁcients, 381 –-8(8B), 95 BATSE, 121, 302 bioastronomy, 288, 303 Bose–Einstein distribution, beam-dump experiment, 302 biological evolution, 288, 289 195, 304, 389, 393 Becquerel, H., 3 bioluminescence, 106 boson, 24, 304, 389, 392, 393, Bell, J., 8 Birkhoff’s theorem, 178 395, 396 bending of light, see gravita- black hole, 5, 110, 118, 131, – fermionic partner, 276 tional lensing, microlensing 168, 269, 303 – GUT, 203 beryllium, 78 – as WIMP candidate, 278 – Higgs, 276, 316 –-7(7Be), 94, 213 – evaporation, 14 – supersymmetry, 276 –-8(8Be), 95 – γ burst, 122 – symmetrization, 391 ± β decay, 27, 29, 302 – mini, 122, 278, 294, 321 – W , 335 Bethe – particle jet, 119 – X, 249, 336 – –Bloch formula, 51, 302 – primordial, see black hole, – Y , 249, 336 – –Weizsäcker cycle, 302 mini bosonic partner, 276 – –Weizsäcker formula, 302 – supermassive, 332 Bothe, W., 6 Bethe, H. A., 9 – X-ray source, 130 bottom quark, 15, 304 Biermann, L., 10 blackbody, 303 bottom–up scenario, 283 Big Bang, 302 – neutrino, 273 b quark, see bottom quark – echo of the, 12 – photon, 59, 82, 273 Bradt, H. L., 10 – hot, 171 – radiation, 82, 126, 229, Brahe, T., 130, 267, 304 – hypothesis, 12 264, 303, 307, 389, see brane, 33, 304 – model, 213, 225, 295 also cosmic microwave bremsstrahlung, 53, 111, 114, – neutrino echo, 104 background 125, 304 – neutron, 104 – – 2.7 Kelvin, 12, 104, 116, – muon, 54 – nucleosynthesis, 22, 172, 266, 283 – photons, 111 209, 213, 214, 217, 222, 226 – – inhomogeneities, 283 – probability for, 158 – – era, 215 – – measurement, 284 – spectrum, 125 – – measurements, 227 – – spectrum, 59 – thermal, 125 – – start, 215 – – temperature variations, brightness – – timeline, 218 284 – absolute, 299 – proton, 104 – – X rays, 125 – excursion, 270 – quark production, 295 blazar, 118, 168, 294, 303 – – duration, 271 – theory, 302 BL-Lacertae object, 168, 303 – increase, 270 – weak interactions, 104 Bloch, Bethe–, formula, 51, – relation to redshift, 186 Big Crunch, 172, 285, 302 302 broken supersymmetry, 276 Big Rip, 285, 303 blue supergiant, 100 brown star (dwarf), 270, 271, binary, 71, 72, 105, 303 blueshift, 303 304 – γ rays, 105 –cosmicmicrowave – mass spectrum, 270, 271 – X rays, 130, 155 background, 237 burning binary star, see binary B meson, see B0 meson – of carbon, 100 binding energy, 303 bolometer, 277, 304 – of helium, 66 418 Index

– of hydrogen, 94, 222, 223, – longitudinal development, charged higgsino, 276 317 158 chargino, 276 – – star, 223 – muon shower in the ALEPH charm quark, 14, 305 – of oxygen, neon, silicon, experiment, 155 chemical composition sulphur, 100 – positron in, 157 – of cosmic rays, see cosmic Burst and Transient Source Casimir effect, 181, 189, 304 rays Experiment, see BATSE cataclysmic variables, 130, 305 – ofthesolarsystem,see solar burst(er), see γ , X-ray, Z causal system burst(er) – contact, 239, 246, 259 chemical potential, 193, 305, Butler, C. C., 9 – isolation, 250 389, 395, 397 CBI data, 243 – high temperatures, 397 Cabibbo CBR (cosmic background Cherenkov – angle, 31 radiation), see cosmic – angle, 55 – –Kobayashi–Maskawa microwave background and – cone, 115 matrix, 31 blackbody radiation – counter, see Cherenkov Cabibbo, N., 31 CCD camera, 127, 128, 271 detector Calabi–Yau space, 33, 304 – X-ray, 56 – detector, 106, 116 calorimeter CDF collaboration, 15 – – ice, 106 CDMS experiment, 278 – crystal scintillation, 113, – – water, 56, 96, 106, 158 celestial equator, 401 114 – effect, 89, 306 Centaurus A, 305 – electromagnetic, 57, 113 – light, see Cherenkov center, galactic, 108, 117, 118 – sandwich, 58 radiation center-of-mass calorimetric measurement, 277 – radiation, 49, 55, 114, 158, – energy, 37, 38 CANGAROO, 116 159 – system, 44 carbon, 78, 130 – ring, 56 centrifugal force, 267 – anti-, 84 – technique, atmospheric, 55, Cepheid variables, 305 – burning, 100 79, 114, 116, 300 304 CERN, 305 –cycle, – telescope, 16, 119 – Greisen–Zatsepin–Kuzmin CGRO, 16, 117, 121, 168, 305 CKM matrix, see Cabibbo– cutoff, 165 Chadwick, J., 8 Kobayashi–Maskawa Chamberlain, O., 11 – primary energy spectrum, 79 matrix – production, 287 Chandra, 305, see also AXAF Clay, J., 6 Carlson, J. F., 9 Chandrasekhar closed universe, 184, 306 cascade, 330 – limit, 174 cloud – electromagnetic, 10, 51, 55, – mass, 305 – chamber,4,8 114, 115, 143, 145, 150, 157 Chandrasekhar, S., 16, 131 –gas – – longitudinal development, channel-plate multiplier, 129 157 channeltron, 127, 128 – – cold, 269 – – width, 144 characteristic X rays, 95 – – hot, 269 – electron in, 157 charge, 305, 394 – – ultracold, 272 – extensive air shower, see air – colour, 306 – magnetic, 63, 68, 69 shower, extensive – conjugation, 29, 278, 295, cluster, see also galaxy cluster – hadronic, 55, 143, 145, 147, 305 – size, 284 154, 166 – conservation, 305 – stellar, 131, 331 – – width, 144 – magnetic, 249 – super-, 131, 265, 282, 283 – high-energy, 165 – ratio, of muons, 148 – – local, 162, 319 – lateral distribution, 158 charge-coupled device, see – temperature, 284 – lateral spreading, 144 CCD camera – Virgo, 169, 335 Index 419

CMB, see cosmic microwave – galactic, 401, 402 – chemical composition, 77, background – spherical, 235 78, 84, 163 COBE satellite, 12, 232, Copernicus, N., 266 – elemental abundance, 78 235–238, 266, 283, 306 corona, 307 – energy density, 71 cold dark matter, 278, 279, COS-B, 307 – energy spectrum, 73, 74, 79, 282, 283, 306 cosmic abundance, 307 80 cold gas clouds, 269 cosmic accelerator, 294 – extragalactic, 82 cold particle, 282 cosmic antimatter, 295 – high-energy, 169 collapse, gravitational, 66, 69, Cosmic Background Explorer, – highest-energy, 163, 294 70, 314 see COBE – intensity, 146 collider, 306 cosmic background radiation, – knee of primary spectrum, collision, see scattering 307, see cosmic microwave 80, 81, 167 collisions, Hoffmann, 6 background and blackbody – neutrino flux, 104 colour, 25, 26, 306 radiation – origin, 85, 104, 110, 167, –charge,306 cosmic mass density, 269 294 – -neutral hadrons, 205 cosmic microwave back- – – extragalactic, 118, 132 column density, atmosphere, ground, 172, 205, 229, 245, –– γ , point-like, 117 142 295 – primary, 10, 77, 141, 142, comet tail, 10 – angular power spectrum, 326 compactification, 306 236, 238, 239, 242, 243, – – antimatter, 84 COMPTEL, 306 385, 388 – – antinuclei, 84 Compton – anisotropy, 235, 241 – – antiproton, 84, 143 – effect, 56, 306 – blueshift, 237 – – charged component, 78 – Gamma-Ray Observatory, – discovery, 231 – – momentum spectrum, 143 see CGRO – formation, 233 –– powerlaw,75 – scattering, 113 – measurement, 232 – – transformation in the – – inverse, 111, 125, 317 – – angular resolution, 238 atmosphere, 144 – telescope, 57 – prediction and observation, – primordial, 77 Compton, A. H., 7, 111 232 – propagation in the confidence interval, 385 – properties, 231 atmosphere, 142 confinement, 30, 306 – redshift, 237 – secondary, 141, 329 conservation – spectrum, 232 – soft component, 144 – law, particle physics, 30 – temperature, 225, 233–235 – source, see cosmic rays, – of baryon number, 215, 216 – – anisotropy, 238 origin –ofcharge,305 – – isotropy, 259 – toe of primary spectrum, 82 – of energy, 178 – – measurement, 237, 238 – underground, 151 – of entropy, 219 – – variations, 225, 261, 284, cosmic strings, 169, 307 – of helicity, 28, 42 388 cosmic textures, 307 – of mass, 280 cosmic radiation, see cosmic cosmoarcheology, 197, 307 conserved quantity, 307, 394 rays cosmogony, 307 constant, dynamical, 281 cosmic rays, 277, 287, 293, cosmographic map, 238, 307 constellation, 307 307 cosmological constraint, 394 – acceleration model, 63 – constant, 180, 187, 198, 251, contraction, Lorentz, 319 – ankle of primary spectrum, 279, 280, 283–285, 289, contrast, density, 261 80, 82 295, 308, 400 conversion electron, 307 – antinuclei, 206 – limits on neutrino masses, coordinates – antiparticle, 84 273, 274 – equatorial, 401 – at sea level, 147 – neutrinos, 104 420 Index

– parameters, 243 – spectrometer, 127 – pion, 28, 42, 88, 111, 112, – – determination, 238 – ultrapure, 277 144, 147, 150 – principle, 175, 308 current, charged and neutral, – three-body, 43 – redshift, 308 see interaction, weak – tritium, 102 cosmology, 171, 308 curvature, 308 – two-body, 41 – equations, 213, 214 – early universe, 284 – width, Z, 22, 275 cosmoparticle physics, 204, – negative, 241 deceleration parameter, 186 290, 308 – parameter, 178, 199 declination, 309, 401 cosmos, see universe curve, rotational, see rotational decoupling counter curve – distance, 235 – anticoincidence, 114 cyclotron mechanism, 64, 308 – of neutrinos, 218 – Cherenkov, see Cherenkov Cygnus – photons from matter, 233, detector – X1, 14, 309 234 – Geiger, 124 – X3, 11, 108, 117, 155, 309 – temperature, 222, 233, 234 – proportional, 127, 128 – time, 234 – semiconductor, 56, 113, 127 DAMA collaboration, 278 deflection of light, see – silicon semiconductor, 128 dark energy, 17, 32, 186, 265, gravitational lensing, Cowan, C. L., 8 280, 283, 295, 309 microlensing CP – present-day, 262 deflector, 270 – invariance, 278, 308 dark matter, 17, 172, 265, 266, degeneracy pressure, 309 – problem, strong, 279 272, 280–282, 285, 294, 309 degenerate matter, 309 – symmetry, see CP invariance – circumstantial evidence, 267 degrees of freedom, 309 – transformation, 29 – cold, 278, 279, 282, 283, – effective number, 196, 214, – violation, 278, 295 306 218, 219, 226 CPT – dark stars, 269 – internal, 194, 389, 396 – invariance, 29, 308 – hot, 273, 282, 283, 316 deleptonization, 100, 309 – symmetry, see CPT – LSP, 276 ∆ resonance, 59, 82, 164 invariance – neutrinos, 272, 273 density c quark, see charm quark – non-baryonic, 266 – average, 260 Crab Nebula, 2, 11, 16, 108, – resumé, 283 – baryon, 214, 215, 224, 225 117, 119, 308 – WIMPs, 275 – column, atmosphere, 142 Crab Pulsar, 130 Davis experiment, 13, 96, 309 – contrast, 261 critical Davis, R., 13, 96 – critical, see critical density – density, 172, 185, 225, 246, de Broglie wavelength, 309 – effect, 53 247, 256, 263, 266, 269, decay, 309 – electron flux, 141 273, 279, 308 – α, 300 –energy,see energy density – temperature, 249, 250 – axion, 279 – fluctuations, 237, 240, 256, cross section, 35,46, 49, 308 – B0 meson, 29, 278 260, 309 – differential, 47 – β, 27, 29, 302 – galaxy, 261 – nuclear interaction, 49, 213 – D meson, 149 – mass, see mass density – reaction of protons and – kaon, 29, 144, 150, 278 – matter, see matter density neutrons, 216 – Λ baryon, 30 – monopole, 259 cryogenic detection techniques, – muon, 28, 43 – nucleon, 214 58 – – electron spectrum, 44 – – BBN phase, 231 ‘cryptons’, 169 – neutrino, 103 – number, see number density crystal – neutron, 27, 28, 217, 218, – parameter, see Ω parameter – calorimeter, 113, 114 219, 323 – photon, 225 – detectors, 57 – nucleus, γ rays, 112 – position dependent, 260 Index 421

– proton flux, 141 differential equations, 386 – zenith-angle, muon, 148, departure from thermal differential gravitation, 310 153, 154 equilibrium, 202, 210 dilution D meson, decay and lifetime, depth–intensity relation, 152, – of matter density, 280 149 153 – of monopoles, 260 domain wall, 169, 310 detection dimension DONUT experiment, 13 – of electrons, 114 – extended, 311 Doppler effect, 310, see also – of neutrinos, 49, 51, 56, 106, –extra,32 blueshift and redshift 107 – number, 289 double – of particles, 49,51 dipole – image, 5, 270 – of photons, 50, 56, 128 – anisotropy, 235, 236, 310 – pulsar, 310 – of radiation, 49 – – vanishing, 237 – ratio, 89 – of WIMPS, 277 – moment, neutron, 279 –star,see binary – of X rays, 56, 126, 127, 128 – term, 236 doublet (isospin), 29 detector Dirac, Fermi–, distribution, see down quark, 27, 310 – characteristic features, 60 Fermi–Dirac distribution d quark, see down quark – Cherenkov, see Cherenkov Dirac, P. A. M., 8 dust, galactic, 269 detector directionality, in neutrino dwarf – cryogenic, 58 detection, 96 – brown, 270, 304 –crystal,see crystal detector disk, 310 – star, 269 335 – neutrino, 277 – accretion, 72, 130, 155, 299 – white, 6, 130, dynamical constant, 281 – particle, see particle detector – galactic, 118, 267, 271 dynamics – sampling, 159 distance – of galactic clusters, 269, 283 – semiconductor, see – angular (diameter), 240 – of galaxies, 17, 269, 272, semiconductor detector – from the Sun, 402 281, 283 – semiconductor pixel, see – horizon, 234, 241, 242 – of the universe, 131, 266 semiconductor pixel detector – – acoustic, 241 – silicon semiconductor, – Hubble, 259, 404 Earl, J. A., 11 see silicon semiconductor – ladder, 310 early universe, see universe, detector – luminosity, 173, 187, 320 early – (water) Cherenkov, see – of decoupling, 235 EAS, see air shower, extensive (water) Cherenkov detector – proper, 240 east–west effect, 7, 310 deuterium, 94, 213, 220, 222, distortions of images, 272 ecliptic, 401 309 distribution Eddington, A. S., 5 – abundance, 221, 224 – binomial, 383 effect – – primordial, 223 – Bose–Einstein, 195, 304, – Casimir, 181, 189, 304 – binding energy, 220 389, 393 – Cherenkov, 55, 89, 306 – fraction, 224 – equilibrium, 390, 392, 399 – Compton, 56, 306 – fusion, 231 – Fermi–Dirac, 195, 312, 389, – density, 53 –Lyman-α line, 224 393 – Doppler, 310, see also – production, 219–221, 223, – Gaussian, 384 blueshift and redshift 288 – Landau, 53, 384 – east–west, 7, 310 – stability, 288 – momentum, 390, 391, 396 – latitude, 6, 318 – synthesis, 218 – of microstates, 390 – micro-gravitational-lens, see – -to-hydrogen ratio, 224 – Planck, 59, 325 microlensing development of life, 287 – Poisson, 384 – MSW, 98, 322 deviation, standard, 384 – probability, 383 – photoelectric, 49, 56, 113, differential cross section, 47 – thermal, of velocities, 202 128, 325 422 Index effective number of degrees – reaction, scattering, see – – matter, 243 of freedom, 196, 214, 218, electron interaction – – matter domination, 230 219, 226 – secondary, 144, 150 – – monopole, 250 efficiency of energy generation – solar wind, 141 – – of cosmic rays, 71 in stars, 289 – spectrum from muon decay, – – perturbations, 266 EGRET, 310 44 – – photon, 250 eigenstate – synchrotron radiation, 125 – – photon contribution, 232 – mass, 31, 90 – volt, 311, 403 – – present-day, 230, 256 – weak interaction, 31, 90 electroweak – – scalar field, 254 Einstein – force, 293 – – spatial variations, 256 – Bose–, distribution, see – interaction, see interaction – – total, of the universe, 195, Bose–Einstein distribution – scale, 204, 293 214, 219, 243, 245, 246 – Observatory, 129 – theory, 13, 311 – – vacuum, see vacuum – ring, 270, 271 – unification, 171 energy density Einstein, A., 5, 21, 280, 295 element – equipartition of, 390 electromagnetic – abundance, in cosmic rays, – Fermi, 274, 312 – calorimeter, 57, 113 78 – generation in stars, 10 – cascade, see cascade, – formation, 77, 120 – – efficiency, 289 electromagnetic – light, abundance, 214 –high – interaction, see interaction – – primordial, 222 – – cascade, 165 – radiation, 160, see also γ – production, in supernovae, – – events, 163, 166, 168, 169 and X rays, photon 66, 112, 120 – – muon, 155 electron, 21, 111, 310 – radioactive, primordial, 287 – – neutrino, 82, 95, 105, 166, – annihilation with positrons, – synthesis, see nucleosynthe- 168 see annihilation of positrons sis – – photon, 82, 165 – atmospheric flux, 144 elementary particles, see – highest – β decay, 29 particle – – cosmic rays, 163, 294 – bremsstrahlung, 53, 125 – Standard Model, see – – event, nucleus, 163 – capture, 94, 95, 310 Standard Model –– events,82 – collision with positron, 226 elliptical galaxy, 311 – – observed, 293 – conversion, 307 Elster, J., 3 – – proton, 163 – detection, 114 endotherm fusion, 66 –loss – evidence, 89 energy – – of charged particles, 51, 53 – flux density, 141 – average per particle, 398 – – of muons, 54, 151, 152 – in rock, 154 – binding, 303 – measurement, 159 – in shower, 157 – – deuterium, 220 – –momentum tensor, 180 – interaction, 218, 219 – – neutral hydrogen, 233 – muon – – with neutrino, 27 – – nuclear, 323 – – definition, 54 – knock-on, 150 – center-of-mass, 37, 38 – – determination, 106 – mass, temperature, 218 – conservation, 178 – Planck, 192, 294 – neutrino, 150 – consumption, world, 102 – potential, 280 – – energy spectrum, 150 –dark,see dark energy – –range relation, 152 – number, 311 – density, 194, 214, 262, 311, – recoil, 277 – – density, 215 389, 396, 397, 400 – Rydberg, 403 – –positron pair production, – – baryon, 225, 243 – spectrum see pair production – – dependence on scale – – of cosmic rays, 73, 74, 79, – primary, 10, 84 factor, 230 80 – radiation length in air, 143 – – gravitational wave, 263 – – of electron neutrinos, 150 Index 423

– – of muon neutrinos, 150 – rate, 49 – SAGE, 13, 96, 329 – – of muons, 146, 147, 154 evolution – satellite, 79, 124 – – of primary nuclei, 79 – biological, 288, 289 – SNO, 18, 283 – – of protons, 146 – stellar, 287 – Super-Kamiokande, 17, 88, – threshold, 37 exchange particle, 39 96, 283, 332 – total, 252, 394, 399 excitation, 51 – underground, 161 – – relativistic, 36 – of fields, 253 Explorer I satellite, 7 – vacuum, see vacuum energy expansion, see also scale factor explosion, thermonuclear, 130 – zero-point, 253, 336 – accelerated, 252, 254, 285 exponential expansion, 252, entropy, 399 – adiabatic, 182 254, 266, 281, see also – conservation, 219 – binomial, 381 inflation equality of matter and radia- – discovery, 6 exponential function, 382 tion, see matter–radiation – exponential, 252, 254, 266, extended dimension, 311 equality 281, see also inflation extensive air shower, see air equation – Hubble, 12, 173, 240 shower, extensive – acceleration, 183, 252, 299, – model, 311 extra dimensions, 32 400 – multipole, 236, 387 extragalactic – differential, 386 – non-relativistic, 184 – matter sample, 77 – fluid, 182, 200, 312, 400 – quasi-exponential, 263 – neutrinos, 104 – Friedmann, 177, 178, 179, – rate, 176, 201, 214, – radiation, 82, 311 183, 198, 247, 251, 313 216–219, 226, 257, 258, 281 – sources, 118, 132 – – matter dominance, 230 – – constant, 252 extrasolar – – solving, 199 Expérience pour la Recherche – planets, 18 – of cosmology, 213, 214 d’Objets Sombres, see – X rays, 124 – of state, 182, 198, 252, 398 EROS extraterrestrial intelligence, 18 –– parameter, see w experiment parameter – accelerator, 293 false vacuum, 255, 279, 311 – rate, 222 – ALEPH, 155 families, 311 equator, celestial, 401 – AMANDA, 88, 107, 300 – neutrino, see neutrino, equatorial coordinates, 401 – ANTARES, 277 families and generation equilibrium – Auger, 83, 158, 169 – of particles, 21 – distribution, 390, 392, 399 – Baksan scintillator, 2 Fermi – number density, 217 – balloon, 3, 79, 124, 143, 284 – acceleration mechanism, 10, – thermal, 214–217, 222, 394 – beam-dump, 302 68, 312 – – departure from, 202, 210 – Boomerang, 243, 284 – constant, 216 equipartition of energy, 390 – CDMS, 278 – –Dirac distribution, 195, EROS, 17, 271, 311 – Davis, 13, 96, 309 312, 389, 393 error (propagation), 384 – DONUT, 13 – energy, 274, 312 escape velocity, 274, 311 –fixed-target,312 Fermi, E., 10 evaporation of neutrons, 323 – Fly’s Eye, 160, 166, 313 fermion, 23, 24, 312, 389, 393, even–even nuclei, 79 – Frejus, 156 395, 396 even–odd nuclei, 79 – GALLEX, 13, 96, 313 – antisymmetrization, 391 event – HERA, 166 – bosonic partner, 276 – high-energy, 163, 166, 168, – IMB, 103 – gas, relativistic, 274 169 – Kamiokande, 13, 96, 101, – generation, 15 – highest-energy, 82, 294 103, 318 – relativistic, 215 – – nucleus, 163 – Maxima, 243, 284 – supersymmetry, 276 – horizon, 14, 118, 192, 311 – radiochemical, 95 fermionic partner, 276 424 Index

Feynman diagram, 26, 98, 216, – WIMP, 278 fundamental particle, 313 276, 312 Fly’s Eye, 160, 166, 313 fusion, 10, 66, 79, 94, 130, Feynman graph, see Feynman – technique, 53, 159, 161 313, 323 diagram Forbush, S. E., 10 – deuterium, 231 field force – endotherm, 66 – excitation, quantum, 253 – centrifugal, 267 – of hydrogen, 288 – Higgs, see Higgs field – electroweak, 293 – processes, solar, 94, 95 – inflaton, 255 – gravitational, 267 – successive, 66 – magnetic, see magnetic field –Lorentz,6 – scalar, see scalar field formation galactic – theory – neutron star, 66, 100 – center, 108, 117, 118 – – interaction, 255 –ofcosmicmicrowave – coordinates, 401, 402 – – potential, 254 background, 233 – disk, 118, 267, 271 – – quantum, 253, 327 – of elements, 77, 120 – dust, 269 γ fine-tuned parameters, 222, – of galaxies, 13, 237, 281, – rays, 117 289 282, 284 – gas, 267 – halo, 267, see halo, galactic fission, nuclear, 312, 323 – of structure, universe, 261, – latitude, 402 fixed-target experiment, 312 266 – longitude, 402 flare, 312 four – lower limits on neutrino –solar,330 – -momentum, 40, 313 masses, 274 flash, X-ray, 130 – – vector, 39 – magnetic field, 81, 313 flat geometry, 241 – -vector, 39, 313 – neutrinos, 104 flat rotational curve, 267, 268 Fourier series, coefficients, 261 – nucleus, 267 flat universe, see universe, flat fragmentation, 78, 313 – – active, see active galactic flatness problem, 246, 248, frame, rest, local, 237 nuclei 312 256, free parameters, Standard galaxy, 129, 131, 176, 177, flavour, 23, 25, 312 Model, 287 265, 269 – neutrino, 91, 98, 101 freedom, asymptotic, 16, 21, – 3C134, 162 – quark, 28, 30 301 – 3C273, 108 – – violation, 31 freeze-out temperature, – active, 299 fluctuations neutron-to-proton ratio, – Andromeda, 271, 300 – density, 237, 240, 256, 260, 217–219, 226, 228 – at high redshift, 132 309 Freier, P., 10 – cluster, 129, 131, 265, 282, – potential, 284 Frejus experiment, 156 306, 313 – quantum, 181, 256, 266, Friedmann – – Abell 754, 132 281, 282 – equation, 177, 178, 179, – – dynamics, 269, 283 fluid equation, 182, 200, 312, 183, 198, 247, 251, 313 – – X rays from, 130 400 – – matter dominance, 230 – density, 261 fluorescence technique, 79, – – solving, 199 – dynamics, 17, 269, 272, 281, 159 – –Lemaître universes, 313 283 flux full width at half maximum – elliptical, 311 – atmospheric, of particles, (FWHM), 385 – formation, 13, 237, 281, 144 functions 282, 284 – density, electron and proton, – orthogonal, 387 – groups, 131 141 – spherical harmonics, 385, – M81, 130 – muon, at sea level, 147 386 – M87, 320 – neutrino, cosmic, 104 – trigonometric and exponen- – Markarian, 17, 108, 119, – of gamma rays, 207 tial, 382 165, 321 Index 425

– mass density, 267, 268 Gaussian distribution, 384 – constant, 403 – metal-poor, 222 Geiger counter, 124 – force, 267 – NGC 6503, 267 Geitel, H., 3 – instabilities, 241, 266, 282, – proto-, 282 Gell-Mann, M., 13, 14 314 –radio,328 Geminga, 117 –lens,315 – rotational curve, 267 general relativity, 178, 314 – lensing, 5, 270–272, see also – Seyfert, 329 – principle, 314 microlensing –starburst,331 – theory, 5, 280, 314 – – weak, 272 – super-, 86, 168 generation, 15, 22, 23, 28, 314 – redshift, 315 Galilei transformation, 313 – fourth, lepton, 275 – wave, 14, 263, 315 GALLEX, 13, 96, 313 – neutrino, see neutrino, – – astronomy, 133 gallium families and generation – – background, 264 – arsenide, 128 geomagnetic cutoff, 6 – – energy density, 263 – experiment, 96 geometry gravitino, 276, 315 γ astronomy, 108, 314 – flat, 241 graviton, 24, 276, 315 – horizon, 116 – of the universe, 241 gravity, 315 –inTeV,55 germanium, 128 – acceleration due to, 404 γ burster, 15, 120, 162, 314 Giacconi, R., 11 – Newtonian, 178 – black hole as, 122 giant, red, 6, 174, 328 – particle acceleration, 72 – discovery, 120 Glashow, S., 13 – quantum, 171, 191, 293, 327 – neutron stars as, 122 GLAST, 314 Great Wall, 315 – quasi-periodic, 123 gluino, 276, 314 Greisen–Zatsepin–Kuzmin γ rays, 109, 314 gluon, 26, 276, 314 cutoff, 59, 87, 161, 164, 315 – flux, 207 – residual interaction, 26 – carbon, 165 – from pion decay, 111 Golden, R. L., 11 Gross, D. J., 16 – galactic, 117 graceful exit problem, 255 group, local, 319 – line emission, 112, 120 Gran Sasso laboratory, 278 GUT, see Grand Unified – measurement, 113 grand unification, 314, see also Theory – nucleus decay, 112 Grand Unified Theory gyroradius, 315 – of binaries, 105 Grand Unified Theory, 32, 197, – point source, 117 204, 254, 293, 296, 315 hadron, 25, 315 – production mechanisms, 110 – bosons, 203 – colour-neutral, 205 – supernova, 110, 118 – creation, 293 – in rock, 154 γ satellite, 11 – epoch, 315 – interaction length in air, 143 γ -ray burster, see γ burster – particles, 169 – interaction, inelasticity, 157 γγ interaction, 82, 86, 106, – scale, 204, 250, 293, 315 hadronic 116 gravitation, 289, see also – cascade, see cascade, Gamow, G., 12 interaction, gravitative hadronic gas – differential, 310 – matter, 21 – cloud – repulsive, 284, 289 Halley, E., 2 – – cold and hot, 269 – singularity, 14 halo, galactic, 267, 269, 271, – – ultracold, 272 – super-, 32, 33 313 – galactic, 267 gravitational – mass content, 271 – ideal, law, 400 – aggregation, 266 – metal-poor star, 223 – proportional chamber, 128 – binding, 282 – neutrino, 275 – relativistic fermions, 274 – – of neutrinos, 273 – object, non-luminous, 270 Gassendi, P., 2 – – of WIMP, 277, 278 – WIMP, 278 gaugino, 276 – collapse, 66, 69, 70, 314 Harari, H., 86 426 Index

Harrison–Zel’dovich spectrum, – field, 204, 249, 254, 259, – parameter, 176, 186, 187 261 316 – telescope, 16, 281, 282, 317 Hawking – – additional, 210 Hubble, E. P., 6 – radiation, 14, 316 – mechanism, 31 Hulse, R. A., 14 – temperature, 316 – potential, 254 hydrogen, 125, 270 Hawking, S., 14, 293 higgsino, charged and neutral, – binding energy, 233 HEAO, 129, 316 276 – burning, 94, 222, 223, 317 heat, specific, 277 High Energy Astronomy – – star, 223 heavy neutrino, 275 Observatory, see HEAO – fusion, 288 Heisenberg’s uncertainty High Resolution Imager, see –Lyman-α line, 224 relation, see uncertainty HRI – neutral, 233 relation high-energy – primary energy spectrum, 79 Heisenberg, W., 8, 22 – events, 163, 166, 168, 169 hypernova, 122 Heitler, W., 9 – muon, 155 hyperon, 9 helicity, 28, 29, 97 – neutrino, 82, 95, 105, 166, – conservation, 28, 42 168 ice Cherenkov detector, 106 – suppression, 28, 150 – photon, 82, 165 IceCube, 277 heliocentric, 316 highest observed energy, 293 ideal gas law, 400 identification of particles, 147 heliosphere, 316 highest-energy image helium, 79, 125, 130, 220, 222, – cosmic rays, 163 – distortions, 272 270 – events, 82, 294 – double, 5, 270 – anti-, 84 – proton, 163 – sensor, silicon, 128 – burning, 66 Hoffmann’s collisions, 6 IMB, 2, 101, 103, 317 – from fusion reaction, 94 Hoffmann, G., 6 impact parameter, 270 – primary energy spectrum, 79 holographic universe, 32 implosion, 317 – production, 218 homogeneity, universe, 175 imprints on astrophysical data, helium-3 (3He), 94, 213 – early, 266 197 – primordial abundance, 224 horizon, 316 4 index of refraction helium-4 ( He), 94, 213 – distance, 234, 241, 242 – in the keV range, 126 – abundance, 213, 221, 222, – event, 14, 118, 192, 311 – of air, 114 224, 225, 227 – γ astronomy, 116 index, scalar spectral, 261 – – primordial, 22 – particle, 239, 240, 250, 258, inelasticity, 157 – mass fraction, 221, 226, 227 324 inertia, 317 – – predicted, 226 – problem, 246, 258, 316 inflation, 171, 205, 245, 251, – – primordial, 223 Hot Big Bang, 171 253, 254, 256, 259, 260, – reaction chain, 220 hot dark matter, 273, 282, 283, 262, 263, 266, 281, 317 – synthesis, 220 316 – causal contact, 259 HERA, 166 hot gas clouds, 269 – end of, 255–257 Hertzsprung–Russell diagram, hot particle, 282 – graceful exit problem, 255 5, 316 h parameter, 175 – gravitational waves, 263 HESS telescope, 119 HRI, 129, 130, 317 – models, 261 Hess, V. F., 3 Hubble – new, 255, 256 Hewish, A., 8 – constant, 175, 186, 187, 225, – number of e foldings, 258 hidden quantum number, 25 243, 263, 317, 404 – Ω parameter, 257 hierarchy of masses, 98, 273, – diagram, 187 – period of, 258 274 – distance, 259, 404 – start of, 257, 258 Higgs – expansion, 12, 173, 240 – time, 263 – boson, 276, 316 – law, 175, 317 inflationary epoch, 172 Index 427 inflaton (field), 255 – rate, 46 – group, 66 infrared – residual, 26, 328 – line, 132 – astronomy, 109, 317 – strong, 24, 332 – primary energy spectrum, 79 – photons, 116 – – asymptotic freedom, 16, – production, 287 – radiation, 106 21, 301 – solid-, momentum – slavery, 21, 317 – – carrier, 15 spectrometer, 148 inhomogeneities –– CP violation, 278 Irvine–Michigan–Brookhaven, – blackbody radiation, 283 – – GUT, 293 see IMB – in the universe, 265, 266 – – of nuclei, 50 isomer, 318 initial conditions of the – – parity, 29 isospin, 29, 30, 318 universe, 260 – – strangeness, 29, 332 – doublet, triplet, multiplet, 29 instabilities, gravitational, 241, – superweak, 167 isotope, 318 266, 282, 314 – unified, 32 isotopic shift of wavelengths, integrals – united, 31 224 – indefinite, 382 – weak, 24, 27, 275, 293, 295, isotropic radiation, 235 – specific, 383 335 isotropy intensity attenuation, photon, – – Big Bang, 104 – temperature, cosmic 113 – – carrier, 16 microwave background, 259 intensity of cosmic-ray – – charge conjugation, 29, – universe, 175 particles, 146, see also 278 Jeans mass, 318 depth–intensity relation – – charged current, 27, 305 Jesse, W. P., 10 interaction, 24, 255, 317 –– CP violation, 278 jet, 119, 318 – astroparticle, 50,58 – – eigenstate, 31, 90 Johnson, T. H., 7 – electromagnetic, 24, 293, – – in supernovae, 101 310 – – neutral current, 27, 322 ± K ,K0, see kaon – – of nuclei, 50 – – of nuclei, 50 Kamiokande, 13, 96, 101, – – parity, 29 – – parity, 29, 278 103, 318, see also – electroweak, 13, 24, 216, – – strangeness, 29 Super-Kamiokande 293 – – strength, 216 Kant, I., 3, 281 γγ – , 82, 86, 106, 116 – WIMP, 277 kaon, 318 – gravitational, 24, 275, 315, interchange particles, 27 – as secondary particle, 143 see also gravitation internal degrees of freedom, – at sea level, 150 – – gravitino, 276 194, 389, 396 – CP violation, 210 – – graviton, 24, 276, 315 International Space Station, – decay, 29, 144, 150, 278 – – weakness, 32 207 – discovery, 9 – hadron, inelasticity, 157 interstellar medium, 317 – lifetime, 144 – kinematics, 44 interval, confidence, 385 – mass, 42 – length, 46, 143, 317 invariance – strangeness, 29 – mechanism, 51 – CP, 278, 308 Kepler, J., 100, 267 – neutrino–air, 51 – CPT, 29, 308 Keplerian motion, 267 – neutrino–neutron, 27 – time reversal, 334 kinematics – neutrino–nucleon, 51, 89 inverse Compton scattering, – interaction, 44 – nuclear, 54 111, 125, 317 – relativistic, 35 – – cross section, 49, 213 inverse reaction of protons and K meson, see kaon – probability, 46, 49 neutrons, 217 knee, see cosmic rays – processes, 26,51 ionization, 51, 114, 277, 318 knock-on electron, 150 – properties, 25 – minimum, 53 Kobayashi, M., 31 – proton–air, 50, 88 iron, 78, 79, 100, 132, 151, 163 Kohlhörster, W., 4, 6 428 Index

Koshiba, M., 13 – Planck, 192, 325, 403 – emission, γ rays, 112, 120 kpc, 318, see also parsec lens, gravitational, 315, see – hydrogen, Lyman-α, 224 krypton, 56 also gravitational lensing, – iron, X rays, 132 microlensing – of sight, 270 laboratory system, 37, 44 LEP, 15, 22, 227, 275, 319 lithium, 78 Lagrange multipliers, 394 leptogenesis, 206, 319 –-6(6Li), 213 Lambda baryon, see Λ baryon lepton, 21, 30, 214, 295, 319 –-7(7Li), 94, 213 Λ baryon, 9 – anti-, 295 – – abundance, 223–225 – decay, 30 – fourth generation, 275 – -to-hydrogen ratio, 223 Landau distribution, 53, 384 – mass hierarchy, 98, 273, 274 LMC, see Large Magellanic Landau, L. D., 8 – number, 28, 30, 319, 394 Cloud Laplace series, 235, 387, 388 –– tau,333 local large attractor, 318 – supersymmetry, 276 – group, 319 Large Electron–Positron LHC, 276, 293, 319 – supercluster, 162, 319 Collider, see LEP life logarithm, natural, 382 Large Hadron Collider, see – age of, 288 longitude, galactic, 402 LHC – development of, 287 Lorentz Large Magellanic Cloud, 86, lifetime – contraction, 319 100, 120, 270, 319 – D meson, 149 – factor, 35 large-scale structure of the – kaon, 144 – force, 6 universe, 265, 272 – transformation, 44, 45, 319 – muon, 88 – development, 266 low temperatures, 277 – neutrino, 16, 103 Las Campanas observatory, lower mass limit, galactic, for – neutron, 217, 220, 221, 288 100 neutrinos, 274 – pion, 144 last scattering LSP, see supersymmetry, LSP light – surface of, 235, 240 luminosity, 320 – bending and deflection, – time of, 239 – distance, 173, 187, 320 see gravitational lensing, latitude luminous matter and stars, 269 microlensing – effect, 6, 318 lunar X rays, 124, 132 – Cherenkov, see radiation, – galactic, 402 Lyman-α line, hydrogen, 224 Lattes, C. M. G., 9 Cherenkov law – curve, apparent, 270, 271 MACHO, 17, 270–272, 280, – conservation, particle – northern, 2, 301 320 physics, 30 – scintillation, 160, 161 Magellanic Cloud, 123, 320, – first, of thermodynamics, – velocity of, 35, 335, 403 see also Large Magellanic 398 – -year, 319, 404 Cloud – Hubble, 175, 317 light elements, see light nuclei magic numbers, 79, 320 – ideal gas, 400 light nuclei, 94 MAGIC telescope, 119 – radiation, Planck, 126, 325 – abundances, 214 magnetar, 123, 320 – Stefan–Boltzmann, 126, – – predicted, 224 magnetic 194, 331 – – primordial, 222 – charge, 249 Lederman, L. M., 13, 15 – mass fractions, 223 – cloud, 63, 68, 69 Lemaître, Friedmann–, – nucleosynthesis, 220 –field universes, 313 lightest particle, supersym- – – galactic, 81, 313 length metric, see supersymmetry, – – planetary, 287 – of interaction, see interaction LSP – – solar, 141 length lighthouse model, 319 – moment, neutrino, 97 – of radiation, see radiation line – monopole, see monopole, length – absorption, neutrino, 275 magnetic Index 429 magnitude, 187, 320 matter – power spectrum, 262 – apparent, 301 – anti-, see antimatter – proton, 50 magnitudo, see magnitude – asymmetry with antimatter, – scintillation light, 160 main-sequence star, 320 211, 296, 321 – technique of extensive air Mairan, J.-J. d’Ortous de, 3 – baryonic, 265, 268, 269, showers, 158, 160, 161 map, cosmographic, 238, 307 275, 281, 283, 295 medium, interstellar, 317 Markarian galaxies, see galaxy, –dark,see dark matter meson, 25, 321, see also kaon, Markarian – degenerate, 309 pion, B0 meson Maskawa, T., 31 – density, 243, 280 – charmed, 14, 149, 150 mass, 321 – – dilution, 280 metal-poor star and galaxy, –atomic,301 – – total, universe, 275 222, 223 – attenuation coefficient, 113 – dominance, 229, 230, 239, metric tensor, 179 – axion, theoretical, 279 240, 247, 259, 260, 296 metric, Robertson–Walker, – Chandrasekhar, 305 –– Ω parameter, 257 179, 329 – conserved, 280 – energy density, 243 Meyer, P., 11 – density – extragalactic, 77 micro-gravitational-lens effect, – – cosmic, 269 – hadronic, 21 see microlensing – – galaxy, 267, 268 – luminous, 269 microlensing, 17, 270, 271, – – neutrino, 274 – non-baryonic, 281 321 – – universe, 269 – non-luminous, 269, 283 microstates – difference, of neutron and – non-relativistic, 200 – distribution, 390 proton, 215, 217 – oscillation, 321 – number, 390, 392, 393 – eigenstate, 31, 90 – photons decoupling from, – – logarithm, 399 – electron, temperature, 218 233, 234 – – total, 392 – fraction – –radiation equality, 229, microwave background, cos- – – helium-4, see helium-4 230, 247, 259, 260, 321 mic, see cosmic microwave mass fraction – – before, 239 background – – light nuclei, 223 – visible, 265, 269 Mikheyev, S. P., 98 – galactic halo, content, 271 Maxima experiment, 243, 284 Milky Way, see galaxy – hierarchy, 98, 273, 274 Mayor, M., 18 mini black hole, see black hole, – Jeans, 318 McNaught, R., 100 mini – kaon, 42 mean free path, 233, 234 minimum-ionizing particles, – missing, 321 mean value, 384 53 – muon, 42 measurement missing mass, 321 – neutrino, see neutrino mass – abundances, 225 mixing angle and matrix, 31, – nucleon, 214, 221 – air shower, 159 91 – number, 321 – axion decay, 279 Mk 421, Mk 501, see galaxy, – pion, 39, 42 – baryon-to-photon ratio, 243 Markarian – Planck, 192, 325, 403 – Big Bang nucleosynthesis, model – quark, 24, 288 227 – cosmology, see Standard – rest, 35, 39, 328 – blackbody radiation, 284 Cosmological Model – shell, 39 – calorimetric, 277 – for cosmic-ray acceleration, – spectrum, of brown stars, –cosmicmicrowave 63 270, 271 background, 232, 237, 238 – lighthouse, 319 – WIMP, 278 – – angular resolution, 238 – nucleus, shell, 79 Massive Compact Halo Object, – energy, 159 – of expansion, 311 see MACHO – nucleus, 50 – of inflation, 261 massive quark stars, 271 –ofγ rays, 113 – of the universe, 321 430 Index

– standard, see Standard – bremsstrahlung, 54 neon burning, 100 Model – charge ratio, 148 neutral modulation, solar, 141 – decay, 28, 43 – higgsino, 276 momentum – – electron spectrum, 44 – hydrogen, 233 – distribution, 390, 391, 396 – depth–intensity relation, neutralino, 172, 276, 322 – four-, 40, 313 152, 153 neutrino, 8, 21, 282, 322 – space, 389–391, 393, 396 – directions, 156 – absorption line, 275 – – division, 392 – discovery, 9 – as (hot) dark matter, 272, – spectrometer, solid-iron, 148 –energy 273, 282 – spectrum – – definition, 54 – as fermion gas, 274 – – muon, 146 – – determination, 106 – astronomy, 8, 86, 151, 294 – – muon, at sea level, 148 – – loss, 54, 151, 152 – atmospheric, 88, 272, 294 – – primary cosmic rays, 143 – – spectrum, 146, 147, 154 – blackbody, 273 – – proton, 146 – evidence, 89 – blazar as source, 168 – tensor, energy–, 180 – flux at sea level, 147 – burst of SN 1987A, 100–102 – transverse, 144 – high-energy, 155 – cosmic flux, 104 – vector, 391 – in an extensive air shower, – cosmological, 104 – – four-, 39 158 – decay, 103 monopole – in rock, 152, 154 – decoupling, 218 – inclined horizontal direction, – magnetic, 169, 245, 294, – deficit, 96 147 320 – – atmospheric, 272 – lifetime, 88 – – as topological defect, 249 – – muon, 91, 272, 294 – mass, 42 – – density, 259 –– solar,13 – momentum spectrum, 146 – – dilution, 260 – detection, 49, 51, 56, 106, – – at sea level, 148 – – energy density, 250 107 – neutrino, 88–90, 150, see – – number density, 250, 260 – detectors, 277 also neutrino – – search, 250 – distinguishing of ν /ν ,89 – – deficit, 91, 272, 294 e µ – – stability, 249 – dominance, 282 – – energy spectrum, 150 – problem, 248, 251, 259, 321 – – mass, 274 – echo of the Big Bang, 104 – term, 236 – number, 322 – electron, 150 Moon in X-ray light, 132 – ratio, proton-to-, 149 – energy spectrum, 150 Mpc, 322, see also parsec – shower in the ALEPH – evidence, 88 MSW effect, 98, 322 experiment, 155 – extragalactic, 104 M theory, 33, 322 – spectrum at sea level, 147 –– burst,16 Muirhead, H., 9 – zenith-angle distribution, – families, 214, see also multiplate spark chamber, 113 148, 153, 154 neutrino, generation multiple scattering, 144 – – additional, 228 multiplet nabla operator, 386 – – number of, 226, 227 – isospin, 29 NACHO, 272 – – number of, accelerator – super-, 276 natural logarithm, 382 data, 227 multiplicity, 322 natural radioactivity, 277 – – number of, light, multipole expansion, 236, 387 nebula, see galaxy equivalent, 227 multiverse, 290, 322 – planetary, 325 – flavour, see neutrino, multiwire proportional Neddermeyer, S., 9 generation chamber, 56, 129 negative – galactic, 104 muon, 9, 21, 322 – curvature, 241 – – halo, 275 – at sea level, 147 – pressure, 186, 198, 252, 284, – generation, 13, 15, 22, 91, – atmospheric flux, 144 322, 400 98, 101 Index 431

– gravitational binding, 273 – telescope, 88, 106, 107 non-relativistic matter, 200 – heavy, 275 – temperature, 196, 219 non-zero baryon number, 207 – high-energy, 82, 95, 105, neutron, 21, 213, 323 northern light, 2, 301 166, 168 – anti-, 215 nova, 323 – in rock, 154 – attachment, 66 nuclear – interaction, 218 – Big Bang, 104 – binding energy, 323 – – with air, 51 – composition, see neutron, – fission, 312, 323 – – with electron, 27 quark content – fusion, 323, see also fusion – – with neutron, 27 – decay, 27, 28, 217, 218, 219, – interaction, 54 – – with nucleon, 51, 89 323 – – cross section, 49, 213 – lifetime, 16, 103 – dipole moment, 279 – reaction rate, 213, 222, 231 – light, relativistic, 283 – discovery, 8 nucleon, 21, 214, see also – magnetic moment, 97 – evaporation, 323 proton and neutron – mass, 16, 23, 97, 102, 272, – freeze-out temperature, see – at sea level, 149 275, 294 freeze-out temperature – density, 214 – – density, 274 – interaction, 213, 216 – – BBN phase, 231 – – muon, 274 – – with neutrino, 27 – interaction with neutrino, – – non-zero, 17 – isospin, 29 51, 89 – – tau, 274 – lifetime, 217, 220, 221, 288 – mass, 214, 221 – mass limit – number, 215 – number, 221, 225 – – cosmological, 273, 274 – – density, 216, see also – primary, 145 – – direct, 272 freeze-out temperature – -to-photon ratio, 220 – – fourth generation, 275 – primary, 145 nucleosynthesis, 120, 222, 288 – – lower, 273 – quark content, 25, 288 – Big Bang, see Big Bang – – lower, galactic, 274 – reaction, see neutron nucleosynthesis – – tau, 273 interaction – deuterium, 218 – massive, 17 – star, 8, 69, 110, 269, 323 – helium-4 (4He), 220 – mixing, see neutrino, – – formation, 66, 100 oscillation –– γ burst, 122 – of light nuclei, 220 – muon, 88–90, 150 – – in binary, 71 – primordial, see Big Bang nucleosynthesis – number, 272 – – rotating, see pulsar – – density, 215, 216, 218, 274 – – X-ray source, 130 – stellar, 222 – – density, primordial, 272 – -to-proton ratio, 213, 216, nucleus, 50, 165 – oscillation, 13, 17, 24, 90, 218, 220, 222, 226, see also – active galactic, see active 272, 274, 293, 294, 323 freeze-out temperature galactic nuclei – – in matter, 97 – – temperature dependence, – anti-, see antinuclei – – mixing matrix, 91 220 – β decay, 27 – – vacuum, 323 new inflation, 255, 256 – electromagnetic interaction, – postulate, 8 Newton observatory, see XMM 50 – range, 58 Newtonian gravity, 178 – even–even and even–odd, 79 – reaction, scattering, see nitrogen fluorescence, 159 – events of highest energy, 163 neutrino interaction Niu, K., 14 – formation, 77, 120 –solar,1,94, 95, 96, 104 ‘no hair’ theorem, 323 – galactic, 267 – – puzzle, 96 non-baryonic (dark) matter, – γ decay, 112 – spectrum, 107 266, 281 – light, see light nuclei – sterile, 99 non-luminous –magic,79 – supernova, 100, 104 – matter, 269, 283 – measurement, 50 –tau,92 – object, 270 – odd–even and odd–odd, 79 432 Index

– primary, energy spectrum, – of neutrons, 215 oxygen, 78, 79 79 – of nucleons, 221, 225 – burning, 100 – recoil, 277 – of one-particle states, 392 – shell model, 79 – of photons, 221 pair annihilation, see – strong interaction, 50 – of protons, 215 annihilation – synthesis, see (Big Bang) – of relativistic particles, 222 pair creation, see pair nucleosynthesis – of states, 395 production – weak interaction, 50 – of X-ray sources, 129 pair production, 38, 40, 50, 57, number – particle, variable, 389 59, 113, 114, 165, 324 –atomic,301 – quantum, see quantum – direct, 54 – baryon, see baryon number number parallax, 324 – density, 194, 214, 216, 217, parameter – acceleration, 186, 281, 295, 396 observatory 299 – – electron, 215 – CGRO, see CGRO – curvature, 178, 199 – – equilibrium, 217 – Einstein, 129 – deceleration, 186 – – monopole, 250, 260 –HEAO,see HEAO – density, see Ω parameter – – neutrino, 215, 216, 218, – Las Campanas, 100 – h, 175 274 – Newton, see XMM – Hubble, 176, 186, 187 – – neutrino, primordial, 272 Occhialini, G. P. S., 9 – impact, 270 – – neutron, 216, see also odd–even nuclei, 79 – w, 252, 254, 335 freeze-out temperature odd–odd nuclei, 79 parameters – – of particles, 389 OGLE, 271, 324 – cosmological, 243 – – proton, 216 Olbert’s paradox, 324 – – determination, 238 – – WIMP, 277 Ω parameter, 241, 247, 266, – (fine-)tuning, 221, 222, 289 – effective, degrees of 281, 283, 285, 289, 295 – free, Standard Model, 287 freedom, 196, 214, 218, 219, – at Planck time, 248 parity, 29, 324 226 – during inflation, 257 – R, 276, 277, 329 – electron, 311 – matter dominance, 257 − – violation, 29, 278, 295 – equivalent, light neutrino Ω baryon, 25 parsec (pc), 81, 324, 404 families, 227 one-particle state, 392 particle, 295 – fraction, light nuclei, 223 open universe, 184, 242, 324 – acceleration – internal degrees of freedom, Oppenheimer, J. R., 9 – – in binaries, 71 194, 389, 396 Optical Gravitational Lens – – in gravitational potentials, – lepton, 28, 30, 319, 394 Experiment, see OGLE 72 –– tau,333 orbit, 267, 324 – – in pulsars, 294 – magic, 79, 320 orbital velocity, 267 – – in supernovae, 67, 167, – mass, 321 origin of cosmic rays, 85, 104, 294 – muon, 322 110, 167, 294 – α, 10, 78 – of e foldings during – extragalactic, 118, 132 – anti-, see antiparticle inflation, 258 – point-like γ sources, 117 – at sea level, 150 – of dimensions, 289 orthogonal functions, 387 – average energy, 398 – of microstates, 390, 392, 393 orthogonality relation, 387 – charged, energy loss, 51, 53 – – logarithm, 399 oscillation hypothesis, length, – cold, 282 – – maximization, 394 and model, see neutrino – composition in the – of neutrino families, 226, oscillation atmosphere, 145 227 oscillation, matter, 321 – content of the universe, 226 – – accelerator data, 227 OSSE, 324 – dark matter, 282 – of neutrinos, 272 outlook, 293 – detection, 49,51 Index 433

– exchange, 39 Perl, M. L., 9 – virtual, 40 – family, 21 perturbations in the energy photoproduction – flux, atmospheric, 144 density, 266 – of electron–positron pair, 40 – fundamental, 313 Peters, B., 10 – of pions, 39, 82, 161, 164 – GUT, 169 PETRA, 325 physics – horizon, 239, 240, 250, 258, Pfotzer maximum, 9, 325 – accelerator, 293 324 Pfotzer, G., 9 – astro-, 1 – hot, 282 phase transition, 204, 250 – astroparticle, outlook, 293 – identification, 143, 147 – early universe, 248 – beyond the Standard Model, – in rock, 154 phonon, 253 293 – interchange, 27 photino, 276, 325 – cosmoparticle, 204, 290, 308 – jet, 119, 168 photoelectric effect, 49, 56, – of particle and radiation – minimum ionizing, 53 113, 128, 325 detection, 49 – numbers, variable, 389 photomultiplier, 49, 53, 89, – particle, conservation law, – periodic table, 23, 24 106, 127, 128, 159, 160 30 – physics, conservation law, photon, 26, 27, 214, 325, see – statistical, 389 30 also Xandγ rays – thermal, 213, 214 – primary, 50, 145 – blackbody, 59, 82, 273 pion, 325 – primordial, 277, 326 – bremsstrahlung, 111 – as secondary particle, 143 – pseudoscalar, 279 – contribution to energy – at sea level, 150 – relativistic, 214 density, 232 – decay, 28, 42, 88, 111, 112, – – number of, 222 – decoupling from matter, 144, 147, 150 – sampling, 116 233, 234 – discovery, 9 – secondary, 143, 144, 150 – density, 225 – isospin, 30 – supersymmetric, 17, 203, – detection, 50, 56, 128 – lifetime, 144 276 – energy density, 250 – mass, 39, 42 – – creation, 276 – from electromagnetic – photoproduction, 39, 59, 82, – – lightest, 276 cascades, 150 161, 164 – – primordial, 277 – high-energy, 82, 165 – production, 88, 106, 111, – type, 393, 396 – in rock, 154 112, 157 – virtual, 39, 335 – infrared, 116 – quark content, 25 – Yukawa, 336 – intensity attenuation, 113 – tertiary, 147 partner, bosonic and fermionic, – interaction, 219 pixel detector, semiconductor, 276 – – with photon, see γγ 57 Pauli principle, 69, 325, 391 interaction pixel-lensing technique, 271 Pauli, W., 8 – mean free path, 233, 234 Planck pc, see parsec – number, 221 – constant, 22, 109, 403 peak, acoustic, 241, 242, 299 – pair annihilation, 112 – distribution, 59, 325 Penzias, A., 12, 232, 264 – pair production, 38, 59 – energy, 192, 294 perchlorethylene, 95 – radiation length in air, 143 – length, 192, 325, 403 periastron, 325 – ratio, baryon-to-, see – mass, 192, 325, 403 – rotation, 15 baryon-to-photon ratio – radiation law, 126, 325 perigee, 325 – ratio, nucleon-to-, 220 – scale, 191, 293 perihelion, 325 – real, 40 – tension, 325 – rotation of the planet – space-like, 40 – time, 191, 192, 248, 325, Mercury, 15 – starlight, 82, 106, 111, 116 403 periodic table of elementary – temperature, 199, 219, 231 –– Ω parameter at, 248 particles, 23, 24 – time-like, 41 PLANCK project, 243 434 Index planet, extrasolar, 18 – relativistic limit, 400 –ofγ rays, 110 planetary nebula, 325 – scalar field, 254 – of helium, 218 plasma, primordial, 240, 241 – wave, 284 – of iron, 287 plethora of universes, 290 primordial – of neutron stars, 66, 100 Poisson distribution, 384 – abundance, 15 – of pions, 88, 106, 111, 112, polar angle, 235 – – deuterium, 223 157 Politzer, H. D., 16 – – helium-3 (3He), 224 – – by photons, 39, 59, 82, Pontecorvo, B., 13 – – helium-4 (4He), 22 161, 164 positron, 326 – – light elements, 222 – of positrons, 84 – annihilation, 120, 207, 218, – antimatter, 296 – of protons, 38 219, 326 – black hole, see black hole, – of quarks in the Big Bang, – collision with electron, 226 mini 295 – discovery, 8 – cosmic rays, 77 –ofstars,66 – generation, 84 – elements, mass and number – of supersymmetric particles, – in rock, 154 fraction, 223 276 – in shower, 157 – helium-4 (4He) mass – of X rays, 109, 123, 124 – primary, 84 fraction, 223 – pair, 50, 114, 165, 324 – secondary, 144, 150 – neutrinos, number density, – – direct, 54 potassium-40 (40K) activity, 272 propagation of astroparticles, 106 – nucleosynthesis, 326, 58 potential see also Big Bang propagation of errors, 384 – chemical, 193, 305, 389, nucleosynthesis proper distance, 240 395, 397 – particle, 326 proportional chamber – – high temperatures, 397 – plasma, 240, 241 – gas, 128 – energy, 280 – radioactive elements, 287 – multiwire, 56, 129 – fluctuations, 284 – supersymmetric particles, proportional counter, 127, 128, – Higgs, 254 277 see also gas proportional – in field theory, 254 principle chamber and multiwire Powell, C. F., 9 – anthropic, 290, 300 proportional chamber power law, 261 – cosmological, 175, 308 protogalaxy, 282 – primary cosmic rays, 75 – of general relativity, 314 proton, 21, 161, 162, 213, 295, power series, 381 – of relativity, 328 326 power source, 63 – Pauli, 69, 325, 391 – acceleration, 105 power spectrum, 261, 284, 326 – symmetry, 181 – annihilation with antiproton, – angular, cosmic microwave – uncertainty, 22, 181, 316, 112, 206, 295 background, 236, 238, 239, 334 – anti-, see antiproton 242, 243, 385, 388 probability – astronomy, 86 – measurement, 262 – distributions, 383 – at sea level, 147 predicted – interaction, 46, 49 – atmospheric flux, 144 – 4He mass fraction, 226 production – Big Bang, 104 – abundances, 224 – of antiprotons, 84 – composition, see proton, pressure, 214, 252, 389, 399 – of carbon, 287 quark content – degeneracy, 309 – of deuterium, 219–221, 223, – energy spectrum, 146 – general expression, 400 288 – flux density, 141 – negative, 186, 198, 252, 284, – of electron and positron, 38, – highest energy, 163 322, 400 40, 57, 59, 113 – in solar wind, 141 – non-relativistic, 400 – of elements in supernovae, – interaction, 213, 216 – radiation, 66, 241, 327 66, 112 – – with air, 50, 88 Index 435

– – with proton, 50 –foam,327 Rabi, I. I., 9 – isospin, 29 – gravitation, 171, 191, 293, radiation – measurement, 50 327 – annihilation, 296 – momentum spectrum, 146 – mechanics, 327 – at sea level, 147 – number, 215 – – tunneling, 255, 256 – belts, 7, 141, 142, 327, 334 – – density, 216 – number, 28 – blackbody, see also cosmic – pion production, 59, 88, 106, – – hidden, 25 microwave background, see 111, 112 – theory of gravity, see blackbody radiation – primary, 10, 50, 59, 78, 82, quantum gravitation – bremsstrahlung, see 143, 145, 149, 168 quark, 13, 14, 21, 25, 30, 327 bremsstrahlung – production, 38 – anti-, 301 – Cherenkov, see Cherenkov 326 – –proton chain, – asymptotic freedom, 16, 21, radiation – quark content, 25, 288 301 – cosmic microwave back- – ratio, neutron-to-, see – baryon number, 30 ground, see also blackbody neutron-to-proton ratio – bottom, 15, 304 radiation, see cosmic – reaction, scattering, see – charm, 14, 305 microwave background proton interaction – colour, 25 – cosmic rays underground, – stability, 288 151 – structure function, 166 – confinement, 30, 306 – content – detection, 49 – -to-muon ratio, 149 – dominance, 196, 214, 248, – – neutron, 25, 288 protostar, 326 258, 260 – – pion, 25 Proxima Centauri, 326 –– Ω parameter, 257 – – proton, 25, 288 pseudoscalar, 326 – electromagnetic, 160, see – down, 27, 310 – particle, 279 also γ and X rays, photon – flavour, 23, 28, 30 PSPC, 326 – equality with matter, see – – violation, 31 pulsar, 69, 110, 118, 326 matter–radiation equality – Crab, 130 see – generation, quark flavour –era,327, see also radiation – creation rate, 71 – mass, 24, 288 dominance – Cygnus X3, 155, see also – mixing, 31 – extragalactic, 82, 311 Cygnus X3 – nugget, 294 – γ rays, see γ rays – discovery, 8 – production in Big Bang, 295 – Hawking, 14, 316 – double, 310 – sea, 25 – infrared, 106 – in a binary, 71, 105 – spectator, 27, 30 – isotropic, 235 – particle acceleration, 69, 294 – stars, massive, 271 327 – Vela, 2, 117, 129, 334 – length, 54, 143, – strange, 29, 30, 332 – Planck law, 126, 325 – supersymmetry, 276 QCD, see quantum chromody- – pressure, 66, 241, 327 – top, 15, 334 namics – synchrotron, see synchrotron – up, 27, 334 quantity, conserved, 307, 394 radiation quantum, 326 – valence, 25 – underground, 151 – anomalies, 210, 327 quasar, 118, 132, 168, 327 – X-ray, see Xray(s) – chromodynamics, 13, 21, – 3C134, 162 radio 278, 279, 327 – 3C273, 108 – (radar) astronomy, 109, 328 – – scale, 204 – discovery, 11 – galaxy, 328 – field excitation, 253 – redshift, 12 radioactive elements, – field theory, 253, 327 quasi-exponential expansion, primordial, 287 – – potential, 254 263 radioactivity, 3 – fluctuations, 181, 256, 266, Queloz, D., 18 – natural, 277 281, 282 quintessence, 284, 327 radiochemical experiment, 95 436 Index range redshift, 6, 11, 173, 224, 234, Röntgen, W. C., 3, 123 – relation, energy–, 152 281, 328 ROSAT, 16, 129–131, 329 – straggling, 152 –cosmicmicrowave Rossi curve, 7 rate background, 237 Rossi, B., 7 – equations, 222 – cosmological, 308 rotation, periastron and –event,49 – gravitational, 315 perihelion, 15 – interaction, 46 – high, galaxies at, 132 rotational curve, 267–269 – of expansion, 176, 201, 214, – quasar, 12 – flat, 267, 268 216–219, 226, 257, 258, 281 – relation to brightness, 186 – galaxy NGC 6503, 267 – – constant, 252 reflection, total of X rays, 127 – of planets, 267 – of pulsar creation, 71 refraction, index of R parity, 276, 277, 329 – reaction, 202, 213, 216, 217, – keV range, 126 Rubbia, C., 16 226 – of air, 114 Russell, Hertzsprung–, – – nuclear, 213, 222, 231 Reines, F., 8 diagram, 5, 316 ratio relation Rutherford scattering, 26 – baryon to photon, see – depth–intensity, 152, 153 Rutherford, E., 3 baryon-to-photon ratio – energy–range, 152 Rydberg energy, 403 – charge, of muons, 148 – expansion rate and SAGE, 13, 96, 329 – deuterium to hydrogen, 224 temperature, 219 Sakharov conditions, 209, 210, – double, 89 – temperature and time, 219 329 – lithium to hydrogen, 223 – uncertainty, 22, 181, 316, Salam, A., 13 – nucleon to photon, 220 334 sampling – proton to muon, 149 relativistic – detectors, 159 rays, cosmic, γ ,X,see cosmic, – fermion gas, 274 – of particles, 116 γ ,Xray(s) – fermions, 215 Sanduleak, 100, 102 reaction, see also scattering – kinematics, 35 sandwich calorimeter, 58 – chain to helium-4 (4He), 220 – particle, 214 sapphire, 277 – neutron, 213, 216 – – number, 222 SAS-2, SAS-3, 329 – of electrons Relativistic Heavy Ion Collider satellite – – and neutrinos, 218 (RHIC), 205 –COBE,see COBE satellite – – and photons, 219 relativity – experiment, 79, 124 – of protons and neutrons, – general, 178, 314 – Explorer I, 7 213, 216 – – principle, 314 – γ ,11 – – cross section, 216 – – theory, 5, 280, 314 – ROSAT, 16, 129–131, 329 – – inverse, 217 – principle, 328 –WMAP,see WMAP satellite – proton, 213, 216 – special, theory, 5, 35, 331 – X-ray, 11, 127, 128, 129 – rate, 202, 213, 216, 217, 226 repulsive gravitation, 284, 289 scalar field, 253 – – nuclear, 213, 222, 231 residual interaction, 26, 328 – energy density, 254 – thermonuclear, 333 rest frame, local, 237 – pressure, 254 real photon, 40 rest mass, 35, 39, 328 scalar spectral index, 261 recoil, energy and nucleus, 277 Richter, B., 14 scale recollapse of the universe, 251 Riemann zeta function, 194, – electroweak, 204, 293 recombination, 284, 328 397 – factor, 177, 185, 230, 234, – atoms and electrons, 233 right ascension, 328, 401 257, 329 – temperature, 233, 328 Robertson–Walker metric, 179, – – exponential increase, 252 – time, 233, 234 329 – – relation to temperature, red giant, 6, 174, 328 Rochester, G. D., 9 199 red supergiant, 100 rocket flight, 124 – – time dependence, 230, 231 Index 437

– GUT, 204, 250, 293, 315 shower, see cascade – momentum, solid iron, 148 – Planck, 191, 293 –air,see air shower spectrum – QCD, 204 silicon, 128 – blackbody radiation, 59 scattering, see also reaction – burning, 100 – bremsstrahlung, 125 – Compton, 113 – image sensor, 128 – cosmic microwave – – inverse, 111, 125, 317 – semiconductor counters, 128 background, 232 – cross section, differential, 47 singularity, 14, 330, see also – cosmic rays, primary – electron–positron, 226 black hole – – ankle of, 80, 82 – multiple, 144 slepton, 276, 330 – – knee of, 80, 81, 167 – neutrino–electron, 27 Sloan Digital Sky Survey, 261 –– toeof,82 – proton–proton, 50 Small Magellanic Cloud, 330 – electron, from muon decay, – Rutherford, 26 small-angle anisotropy, 237 44 – surface of last, 235, 240 SMC, see Small Magellanic –energy,see energy spectrum – Thomson, 234 Cloud – Harrison–Zel’dovich, 261 – time of last, 239 Smirnov, A. Yu., 98 – mass, of brown stars, 270, Schein, M., 10 SNAP, 330 271 Schwartz, M., 13 SNO experiment, 18, 283 –momentum,see momentum Schwarzschild radius, 192, 329 SNR, see supernova remnant spectrum Schwarzschild, K., 5, 14 Soft-Gamma-Ray Repeater, – muon, at sea level, 147 scintillation, 277, 329 see SGR – neutrino, 107 – crystal calorimeter, 113, 114 solar –power,see power spectrum – in air, 53, 116, 159–161 – flare, 330 – X rays, 125 – mechanism, 53 – modulation, 141 spherical coordinates, 235 scintillator experiment, – neutrinos, 1, 94, 95, 96, 104 spherical harmonics, 385, 386 Baksan, 2 – – deficit, 13 spin, 331, 396 sea quark, 25 – – puzzle, 96 spontaneous symmetry Segrè, E. G., 11 –system breaking, 204, 331 selectron, 329 – – chemical composition, 78 squark, 276, 331 semiconductor – – rotational curves, 267 s quark, see strange quark – counter, 56, 113, 127 – wind, 10, 141, 330 standard – – silicon, 128 solid-iron momentum – candle, 2, 173, 281, 331 – pixel detector, 57 spectrometer, 148 – rock, 152 separation sound waves in primordial Standard Cosmological Model, – angular, 236 plasma, 240, 241, 284 172, 173, 245 – of variables, 386 sources of cosmic rays, see standard deviation, 384 series origin of cosmic rays Standard Model, 13, 21, 226, – Fourier, 261 Soviet–American Gallium 294, 331 – Laplace, 235, 387, 388 Experiment, see SAGE – free parameters, 287 – power, 381 space-like photon, 40 – of electroweak interactions, Seyfert galaxy, 329 spaghettification, 330 216 SGR, 123, 329 spallation, 330 – physics beyond, 293 shell model, 79 spark chamber, multiplate, 113 – supersymmetric extension, Shelton, I., 100 special relativity, 5, 35, 331 276 shock specific heat, 277 star – acceleration, 66, 67, 68, 74, spectator quark, 27, 30 – as X-ray source, 132 167 spectral index, scalar, 261 –binary,see binary – front, 67, 329 spectrometer – brown, 270, 271 –wave,330 – crystal, 127 – cluster, 131, 331 438 Index

– dark, 269 – cosmic, 169, 307 – particle acceleration, – double, see binary – super-, theory, 289 67, 167, 294, see also – dwarf, see dwarf – theory, 31, 332 shock-wave acceleration – energy generation, 10 strip, superconducting, 277 – remnant, 69, 108, 123, 129, – hydrogen burning, 223 strong CP problem, 279 130, 330, 332 – luminous, 269 strong interaction, see – SN 1987A, 1, 16, 100, 103, – main sequence, 320 interaction 120, 330 – mass spectrum, 270, 271 structure function, proton, 166 – – discovery, 100 – metal-poor, 223 structure of the universe, 260, – SNR 1572, 130 – neutron, see also pulsar, see 283 – total number, 270 neutron star – formation, 261, 266 – type-Ia, 174, 186 – Vela, 2, 108 – of antimatter, 84 – growth, 260 – weak interaction processes, – production of new – large-scale, 265, 272 101 generation, 66 – – development, 266 – X rays, 129 – proto-, 326 see SUGRA, supergravity Supernova Cosmology Project, – quake, 123, 331 sulphur burning, 100 – quark, massive, 271 281 Sun superpartner, 276, 333 – spot, 64 – cycle, 141 starburst galaxies, 331 superstring theories, 289 – distance from, 402 starlight photons, 82, 106, 111, supersymmetric particle, 17, – fusion process, 94, 95 116 203, 276 – in X-ray light, 128 state – creation, 276 – magnetic field, 141 – lightest, 276 – equation of, 182, 198, 252, – neutrinos, see solar neutrinos – primordial, 277 398 332 supersymmetry, 276, 294, 333 –– parameter, see w sunspot, 64, – broken, 276 parameter – cycle, 79, 332 – Feynman diagram, 276 – number, 395 –pair,65 – LSP, 276 – one-particle, 392 supercluster, 131, 265, 282, superweak interaction, 167 static universe, 280 283, 332, see also galaxy surface of last scattering, 235, statistical physics (mechanics), cluster 240 389, 390 – local, 162, 319 SUSY, see supersymmetry steady-state universe, 12, 331 superconducting strip, 277 symmetrization, 391 Stefan–Boltzmann supergalactic plane, 162, 168, symmetry, see also invariance – constant, 126, 403 332 supergalaxy, 86, 168 – breaking, 333 – law, 126, 194, 331 – – mechanisms, 249 supergiant, blue and red, 100 Steinberger, J., 13 – – spontaneous, 204, 331 supergravity, 32, 33 stellar – C and CP, violation, 210 Super-Kamiokande, 17, 88, 96, – cluster, 131, 331 – matter and antimatter, 283, 332 – evolution, 287 microscopic, 211 superluminal speed, 332 – nucleosynthesis, 222 – principle, 181 – wind, 331 supermultiplet, 276 synchrotron, 333 sterile neutrinos, 99 supernova, 281, 332 – radiation, 110, 125, 333 Stirling approximation, 393 –asγ source, 110, 118 – – X rays, 125 Störmer, C., 6 – element production, 66, 112, synthesis, see nucleosynthesis storage ring, 37, 331 120 strange quark, 29, 30, 332 – explosion, 100, 287 tachyon, 333 strangeness, 29, 332 – γ burst, 122 tail of comet, 10 string, 332 – neutrino, 100, 104 Tarantula Nebula, 16, 100 Index 439 tau, 9, 21, 333 – map, 285 – dilatation, 333 – lepton number, 333 – neutrino, 196, 219 – -like photon, 41 – neutrino, 92 – photon, 199, 219, 231 – of inflation, 263 – – mass, 274 – recombination, 233, 328 – of last scattering, 239 – – mass limits, 273 – relation to scale factor, 199 – Planck, 191, 192, 248, 325, Tau Boötis, 18 – rise, 277 403 Taylor, J. H., 14 – time dependence, 231 – recombination, 233, 234 technique – uniform, of the universe, 246 – -reversal invariance, 334 – Cherenkov, atmospheric, 55, tension, Planck, 325 Ting, S. C. C., 14 79, 114, 116, 119, 300 tensor TOE, see Theory of Everything – cryogenic, 58 – energy–momentum, 180 toeofcosmicrays,82 – extensive air shower – metric, 179 Tomasini, G., 9 measurement, 158, 160, 161 ter Haar, D., 10 top quark, 15, 334 – fluorescence, 79, 159 TeV γ astronomy, 55 top–down scenario, 282 – Fly’s Eye, 53, 159, 161 textures, cosmic, 307 topological defect, 169, 249, – pixel-lensing, 271 theory 334 telescope – electroweak, 13, 311 total energy, 252, 394, 399 – Cherenkov, 16, 119 –field,see (quantum) field – density of the universe, 195, – Compton, 57 theory 214, 219, 243, 245, 246 – HESS, 119 – M, 33, 322 – relativistic, 36 – Hubble, 16, 281, 282, 317 – of relativity total matter density, universe, – MAGIC, 119 – – general, 5, 280, 314 275 – neutrino, 88, 106, 107 – – special, 5, 35, 331 total reflection, X rays, 127 – Wolter, 127 – string, 31, 332 t quark, see top quark – X-ray, 56, 127 – superstring, 289 tracking chamber, 113 temperature, 219, 226, 389, Theory of Everything, 31, 290, transformation 399 294, 333 – between neutrons and – average, 236 thermal protons, 213 – clusters, 284 – bremsstrahlung, 125 – CP,29 – cosmic microwave – distribution of velocities, – Galilei, 313 background, see cosmic 202 – in the atmosphere, 143, 144 microwave background – equilibrium, 214–217, 222, –Lorentz,44, 45, 319 temperature 394 transverse momentum, 144 – critical, 249, 250 – – departure from, 202, 210 trigonometric functions, 382 – decoupling, 222, 233, 234 – physics, 213, 214 triple-alpha process, 334 – definition, 395 – X rays, 125 triplet (isospin), 30 – dependence, 214, 217, 220, thermodynamics, 193, 389 tritium, 213, 334 226 – first law, 398 – decay, 102 – deuterium production, 220, thermonuclear true vacuum, 255, 280 221 – explosion, 130 tuning (fine-) of parameters, – electron mass, 218 – reaction, 333 221, 222, 289 – freeze-out, see freeze-out Thomson scattering, 234 tunneling, 255, 256 temperature Thomson, J. J., 3 two-body decay, 41 – freeze-out, neutron-to- t’Hooft, G., 13 type-Ia supernova, 174, 186 proton ratio, 217–219, 226, three-body decay, 43 228 threshold energy, 37 ultracold gas clouds, 272 – Hawking, 316 time, 219 ultrapure crystal, 277 – low, 277 – decoupling, 234 ultraviolet astronomy, 109 440 Index uncertainty – matter asymmetry, density, – temperature, cosmic mi- – principle, see uncertainty dominance, see matter crowave background, 225, relation asymmetry, density, 261, 284, 388 – relation, 22, 181, 316, 334 dominance vector underground – models, 321 – four-, 39, 313 – cosmic rays, 151 – neutrino-dominated, 282 – – momentum, 39 – experiments, 161 – open, 184, 242, 324 – momentum, 391 unification – particle content, 226 Vela – electroweak, 171 – plethora, 290 – pulsar, 2, 117, 129, 334 – supernova, 2, 108 – grand, 314, see also Grand – radiation dominance, 196, – X1, 2, 11, 117, 129 Unified Theory 214, 248, 258, 260 velocity Unified Theory, 334, see also – recollapse, 251 – escape, 274, 311 Grand Unified Theory – static, 280 – of light, 35, 335, 403 uniform temperature, 246 – steady-state, 12, 331 – orbital, 267 universe – structure, see structure of the universe – superluminal, 332 –age,299, 404 – uniform temperature, 246 – thermal distribution, 202 – alternatives, 287 up quark, 27, 334 Veltman,M.J.G.,13 – baryon asymmetry, u quark, see up quark vernal equinox, 401 205, 206, 208, 302 violation – baryon fraction, 225 – baryon number, 204, 208, vacuum – closed, 184, 306 210, 215 – energy, 205, 243, 251, 253, – clumpiness, 256 –ofC and CP, 210 257, 262, 263, 285 – dynamics, 131, 266 –ofCP, 278, 295 – – density, 172, 180, 243, – early, 104, 179, 191, 287, – – strong, 278 251, 252, 263, 279, 280, 293, 296 – of parity P, 29, 278, 295 295, 334, 400 – – curvature, 284 – quark flavour, 31 – – experimental evidence, – – homogeneity, 266 Virgo cluster, 169, 335 186 – – phase transition, 248 virtual – expectation value of the – – thermal history, 203 – particle, 39, 335 Higgs field, 204 – photon, 40 – – thermodynamics, 193, 389 – false, 255, 279, 311 – energy density, total, 195, virtuality, 40, 335 – neutrino oscillation, 323 visible matter, 265, 269 214, 219, 243, 245, 246 – true, 255, 280 – eras, 245 Vogt, R., 11 – velocity of light, 35, 403 volume, 399 – expansion, see also inflation, valence quark, 25 see expansion ± Van Allen, J. A., 7 W , 335, see also interaction, – flat, 184, 242, 243, 247, 266, Van Allen belts, 7, 141, 142, weak 279, 281, 285, 289, 295 327, 334 – discovery, 16 – Friedmann–Lemaître, 313 van der Meer, S., 16 water Cherenkov detector, 56, – geometry, 241 variables 96, 106, 158 – holographic, 32 – cataclysmic, 130, 305 wave – homogeneity, 175 – Cepheid, 305 – gravitational, 14, 263, 315 – inflation, see inflation – independent, uncorrelated, – – background, 264 – inhomogeneities, 265, 266 384 – – energy density, 263 – initial conditions, specific, – separation of, 386 – pressure, 284 260 variance, 384 – shock, see shock wave – isotropy, 175 variation – sound, in primordial plasma, – mass density, 269 – spatial, energy density, 256 240, 241, 284 Index 441 wave function, 335 WMAP satellite, 235, 237– – background radiation, 132 – N-particle, 390 239, 243, 266, 283, 284, –burster,336 wavelength, 181, 261 335 – CCD, 56, 127, see also CCD – de Broglie, 309 Wolfenstein, L., 98 – flashes, 130 – shift, isotopic, 224 Wollan, E. O., 10 – satellite, 11, 127, 128, 129 weak gravitational lensing, 272 Wolter telescope, 127 – telescope, 56, 127 Wolter, H., 127 weak interaction, see xenon, 56, 127 world energy consumption, interaction, weak XMM-Newton, 131, 336 102 Weakly Interacting Massive X-ray Multi-Mirror Mission, wormhole, 335 Particle, see WIMP see XMM w parameter, 198, 252, 254, Weber, J., 14 335 Y boson, 249, 336 Weinberg, S., 13 Wright, T., 281 Weizsäcker Wulf,Th.,3 ylem, 336 – Bethe–, cycle, 302 Yukawa – Bethe–, formula, 302 X boson, 249, 336 – particle, 336 Weizsäcker, C. F., 10 Xrays – postulate, 8 – blackbody radiation, 125 Yukawa, H., 8 white dwarf, 6, 130, 335 – by synchrotron radiation, Wilczek, F., 16 125 Z, 336, see also interaction, Wilson chamber, see cloud – characteristic, 95 weak chamber – detection, 56, 126, 127, 128 – burst, 275 Wilson, C. T. R., 3 – direction of incidence, 126 – decay width, 22, 275 Wilson, R. W., 12, 232, 264 – discovery, 3, 124 – discovery, 16 WIMP, 275, 276, 280, 282, – extrasolar, 124 – exchange, 27 283, 285, 294, 335 – from binaries, 130, 155 – resonance, 22, 226, 227 – annihilation (signal), 277 – from black holes, galactic Zatsepin, see Greisen– – flux, 278 clusters, neutron stars, 130 Zatsepin–Kuzmin cutoff – gravitational binding, 277, – from stars, 132 Zeeman splitting, 64 278 – from supernovae, 129 Zel’dovich, Harrison–, – halo, 278 – lunar, 124, 132 spectrum, 261 – interaction and detection, – number of sources, 129 – penetration power, 123 zenith angle, relation to 277 atmospheric depth, 143 – mass, 278 – production, 109, 123, 124 – solar, 128 zenith-angle distribution of – number density, 277 – spectrum, 125 muons, 148, 153, 154 – primordial, 278 – thermal, 125 zero, absolute, 299 wind – total reflection, 127 zero-point energy, 253, 336 – solar, 10, 141, 330 X-ray zeta function, 194, 397 – stellar, 331 – astronomy, 56, 109, 123, zino, 276 wino, 276 336 Zweig, G., 13, 14