Non-Equilibrium Aspects of Relic Neutrinos: From Freeze-out to the Present Day

Non-Equilibrium Aspects of Relic Neutrinos: From Freeze-out to the Present Day Program in Applied Mathematics The Dissertation Defense Presentation Advisor J. Rafelski, Department of Physics, March 20, 2014

Outline Part 1: Eras of Cosmology Relic Neutrinos Overview of Cosmology Part 2: Study of Neutrino Distribution using Conservation Laws Chemical vs Kinetic Equilibrium Neutrino Freeze-out Neutrinos Today Part 3: General Relativistic Boltzmann Equation Boltzmann Equation and Scattering Operators Neutrino Freeze-out Revisited

Part 1: Relic Neutrino Background At a temperature of 5 MeV the Universe consisted of e ±, photons, and neutrinos. At around 1 MeV neutrinos stop interacting or freeze-out and free stream through the universe. Today they comprise the relic neutrino background. Photons freeze-out around 0.25 ev and today they make up the Cosmic Microwave Background (CMB), currently at T γ = 0.235 mev. Direct measurement: Relic neutrinos have not been directly measured. Our results give some input into such efforts. Indirect measurement: Impact on speed of Universe expansion constrains neutrino mass and number of massless particles. Theory: This motivates interest in modeling energy flow from e ± annihilation to neutrinos during reheating.

Overview of Standard Cosmology Planck satellite measures CMB which froze out at 0.25 ev. We need to model the Universe evolution between these two eras. The Friedmann Lemaitre Robertson Walker (FRW) Universe assumes the properties Homogeneous Isotropic [ ] dr ds 2 = dt 2 a 2 2 (t) 1 kr 2 + r2 (dθ 2 + sin 2 (θ)dφ 2 ). (1) Standard ΛCDM model is spatially flat (k = 0) [1]. a(t) determines the distance between objects at rest in the Universe frame (comoving). [1] Planck Collaboration, arxiv:1303.5076

Einstein s Equations Einstein s equations for a perfect fluid source ( ) R G µν = R µν 2 + Λ g µν = 8πG N T µν, T ν µ = diag(ρ, P, P, P), ρ ρ + Λ/8πG N, P P Λ/8πG N (2) In a spatially flat FRW universe these reduce to Hubble equation H 2 = ρ 3M 2 p, H ȧ a, M p 1/ 8πG N = 2.435 10 18 GeV Conservation of stress energy tensor (3) ρ = 3(ρ + P)H. (4) Together with an equation of state, P(ρ), one can solve for the large scale dynamics of the Universe.

Reheating In the standard model of neutrino freeze-out, the relic neutrino background and CMB temperatures differ by a reheating factor R ν T ν /T γ = ( ) 4 1/3. (5) 11 This is the result of energy and entropy from e ± annihilation going into photons. In general, the reheating ratio connects the temperature redshift to the geometric redshift R 1 + z T/T now, 1 + z a now a(t). (6)

Reheating and Particle Disappearance History Conservation of entropy relates R at different times ( R1 R 2 ) 3 = i g i(t 1,i /T 1,γ ) 3 j g j(t 2,j /T 2,γ ) 3 (7) Reheating periods occurred regularly throughout the history of the Universe, whenever a particle species disappeared as the temperature dropped below the mass.

Reheating and Particle Disappearance History 3 (1 + z)tnow/t 2.5 2 QGP phase transition, disappearance of resonances and muons Disappearance of bottom, tau, and charm Disappearance of top, Higgs, Z, and W 1.5 e ± annihilation, photon reheating, and neutrino freeze-out 1 10 3 Time 10 2 10 1 10 0 10 1 10 2 10 3 10 4 10 5 T [MeV] Figure: Disappearance of degrees of freedom. The Universe volume inflated by a factor 27 above the thermal red shift scale as massive particles disappeared successively from the inventory. [1] J. Rafelski and J. Birrell, Submitted to: proceedings of Strangeness in Quark Matter Conference July 2013, Birmingham, UK

Deceleration Parameter Definition: q äa ȧ 2. (8) Using Einsteins equations we can derive an expression in terms of matter content q = 1 ( 1 + 3 P ). (9) 2 ρ

Deceleration Parameter In particular: Radiation dominated universe: P = ρ/3 = q = 1. Matter dominated universe: P ρ = q = 1/2. Dark energy (Λ) dominated universe: P = ρ = q = 1.

T [ev] q 10 3 Radiation dominated 1 10 2 0.8 10 1 Matter dominated 0.6 0.4 10 0 0.2 10 1 10 2 T γ Dark Energy dominated 0 0.2 0.4 10 3 T ν q 0.6 0.8 10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 t recomb t reion t [yr] Figure: From the present day until near BBN.

T [MeV] q 10 1 1 10 0 0.99 0.98 10 1 0.97 T γ T ν 0.96 q t k t i BBN t f BBN 10 3 10 2 10 1 10 0 10 1 10 2 10 3 0.95 10 4 10 2 10 3 Figure: From the end of baryon antimatter annihilation through BBN. t [s]

Outline Part 1: Background Relic Neutrinos Overview of Cosmology Part 2: Study of Neutrino Distribution using Conservation Laws Chemical vs Kinetic Equilibrium Neutrino Freeze-out Neutrinos today Part 3: General Relativistic Boltzmann Equation Boltzmann Equation and Scattering Operators Neutrino Freeze-out Revisited

Part 2: Why Study Neutrino Freeze-out in Detail? Energy in neutrinos impacts speed of Universe expansion. N ν is defined by comparing the total neutrino energy density to the energy density of a massless fermion with two degrees of freedom and standard reheating ratio R ν N ν = ρ ν 7 120 π2 (R ν T γ ) 4, R ν = ( ) 4 1/3. (10) 11 Planck satellite data gives N ν = 3.36 ± 0.34 (CMB no priors) and N ν = 3.62 ± 0.25 (CMB + H 0 ) [1]. The currently used theoretical value is N ν = 3.046 [2]. [1] Planck Collaboration, arxiv:1303.5076 [2] G. Mangano et. al., Nucl. Phys. B 729, 221 (2005)

Why Study Neutrino Freeze-out in Detail? The tension between the Planck result and reheating study motivates our work. This tension has inspired exotic theories introducing new particles [1]. Most conservative explanation is that some extra energy went into neutrinos during e ± annihilation i.e. neutrino reheating. This is the direction we explore. [1] Steven Weinberg Phys. Rev. Lett. 110, 241301 (2013)

Boltzmann equation Particle dynamics given by Boltzmann equation p α x αf Γ j µνp µ p ν p jf = C[f ]. (11) Here Γ α µν is the Levi-Civita connection (Christoffel symbols), the distribution function f is a function on the mass shell g αβ p α p β = m 2, (12) and C[f ] is the collision operator and encodes all information about point interactions between particles. We will return to solution of the Boltzmann equation later, but for now we use a simpler matter model that is independent of the details of C[f ].

Kinetic Equilibrium Model First we perform a model independent study of neutrino freeze-out, assuming instantaneous chemical/kinetic equilibrium and sharp freeze-out transitions. Dynamics will be derived from conservation laws involving the moments T µν = g ν 8π 3 n ν = g ν 8π 3 s µ = g ν 8π 3 f pµ p ν p 0 gd 3 p, (13) f pν gd 3 p, (14) p 0 h pµ gd 3 p, (15) p 0 h = f ln(f ) + (1 f ) ln(1 f ).

Combined Chemical and Kinetic Equilibrium Both number changing processes (chemical) and momentum exchange (kinetic) allowed. Distribution function obtained by maximizing entropy subject to fixed energy. f ch = g ν 8π 3 1 e E/T + 1, T ch < T. (16) T ch is the chemical freeze-out temperature, below which number changing processes cease.

Kinetic (but not chemical) Equilibrium Particles exchange momentum but particle (and antiparticle) numbers fixed. Distribution function obtained by maximizing entropy, subject to fixed particle (and antiparticle) number. f k = g ν 8π 3 1 Υ 1 e E/T + 1 T k < T < T ch. (17) The fugacity, Υ, controls the phase space occupancy. This was first introduced in [1] in the context of QGP. T k is the kinetic freeze-out temperature. Below this, there are no (significant) interactions and the particles free stream. [1]J. Rafelski and B. Muller, Phys. Rev. Lett. 48,1066 (1982)

Relic Neutrino Fugacity is NOT a Chemical Potential Υ is the same for particles and antiparticles (we ignore the small matter-antimatter asymmetry which would yield Υ Υe ±µ/t ) Υ(t) is time dependent, not just an initial condition (initially Υ = 1). We find that Υ < 1 emerges dynamically from chemical equilibrium. It is not put in by hand.

Dynamics with Kinetic Equilibrium Ansatz Divergence freedom of the total stress-energy tensor implies d dt (a3 s) + i σ i d dt (a3 n i ) = 0, Υ i e σ i. (18) where s is entropy density and n i the particle number densities. Assuming instantaneous chemical/kinetic equilibrium and sharp freeze-out transitions, this, along with the Einstein equations, fixes the dynamics of a(t), T(t), Υ i (t). H 2 = ρ 3M 2 p, d dt (a3 s) = 0, d dt (a3 n i ) = 0 ( when T < T ch ). (19)

The Dynamical Emergence of Fugacity Distinct chemical and thermal freeze-out temperatures lead to Υ ν < 1 for T < T ch. This fact has been overlooked in the literature. The mechanism is as follows: If all particles are (effectively) massless there is no effect T 1/a, Υ = constant. (20) When a mass scale, such as m e, becomes important and e + e annihilation begins, conservation of entropy gives d/dt(at) > 0 (reheating). For T < T ch, preserving the comoving number of neutrinos when at is increasing implies Υ < 0. a 3 n ν a 3 T 3 1 Υ 1 ν e u + 1 u2 du (21)

Neutrino Freeze-out 1. Conservation of comoving neutrino number: T1 3V 1 Tk 3V k = 2 3ζ(3) 0 u 2 du Υ 1 ν (T k )e u + 1. (22) 2. Conservation of e ±, γ, neutrino entropy before neutrino freeze-out: ( 7 8 g ν + 7 ) 2π 2 8 g e ± + g γ 45 T3 1 V 1 = (23) ( ) 2π 2 45 g γtk 3 + S e ±(T k) + S ν (T k ) V k. 3. Conservation of e ±, γ entropy after neutrino freeze-out: 2π 2 ( ) 2π 45 g γtγ,2 3 2 V 2 = 45 g γtk 3 + S e ±(T k) V k. (24)

Deceleration Parameter and Neutrino Fugacity Υ 1 0.9 0.8 0.7 0.6 T k = 0.3m e T k = m e 1 0.99 0.98 q 0.97 0.5 Υ q 0.96 0.4 0.95 10 1 10 0 10 1 T/m e [1]J. Birrell, C.T. Yang, P. Chen, and J. Rafelski, Mod. Phys. Lett. A 28, 1350188 (2013)

Effective Number of Neutrinos and Reheating Ratio 5 1.4 4.75 4.5 1.3 Nν 4.25 4 3.75 3.5 3.25 Nν = 3.62 1 3 10 1 10 0 10 1 Time T k /m e [1]J. Birrell, C.T. Yang, P. Chen, and J. Rafelski, Phys. Rev. D 89, 023008 (2014) Nν = 3.36 Nν = 3.046 N ν T γ /T ν 1.2 1.1 Tγ/Tν

Reheating Ratio T γ = aυ b ( 1 + cσ 2 + O(σ 3 ) ), Υ = e σ T ν ( a = 1 + 7 ) g 1/3 ( ) e 11 1/3 = 1.4010, 8 g γ 4 b = π2 1 + 7 g ν+g e 8 g γ 3645 ζ(3) 2 g ν 8π 6 g γ 27ζ(3) 1 + 7 g e 8 0.367, c 0.0209. g γ (25a) (25b) (25c) (25d) (25e)

Reheating Ratio T γ T ν = aυ b ( 1 + cσ 2 + O(σ 3 ) ) (26) The first order approximation has a maximum relative error of 2% relative error) in the region of interest.4 Υ ν 1. The second order approximation brings the relative error down to less than 5 10 4. Power law behavior first observed numerically in [1] [1]J. Birrell, C.T. Yang, P. Chen, and J. Rafelski, Mod. Phys. Lett. A 28, 1350188 (2013)

Effective Number of Neutrinos N ν = 360 7π 4 e 4bσ (1 + cσ 2 ) 4 0 u 3 e u σ + 1 du ( 1 + O(σ 3 ) ). The relative error of this approximation is less than 0.002 over the range 0.9 σ 0. (27)

Equation of State First physically consistent derivation of equation of state of free-streaming neutrinos, including dependence on N ν and neutrino mass (β = m ν /T γ ). Fugacity is a crucial aspect. ρ EV /ρ 0 = N ν + 0.1016 βi 2 + 0.0015δN ν i i 0.0001δNν 2 βi 2 0.0022 βi 4, i i P EV /P 0 = N ν 0.0616 βi 2 0.0049δN ν i i + 0.0005δNν 2 βi 2 + 0.0022 βi 4. i i β 2 i β 2 i

Neutrinos Today 420 410 number density [1/cm 3 ] 400 390 380 370 360 350 340 neutrinos (exact) neutrinos (linearized) photons 3 3.2 3.4 3.6 3.8 4 N ν Figure: Conservation of comoving neutrinos gives present day neutrino number density.

10 2 10 3 m ν = 0.05 ev m ν = 0.25 ev m ν = 0.5 ev m ν = 1 ev m ν = 2 ev fv 10 4 10 2 10 3 10 4 v [km/s] Figure: Normalized neutrino velocity distribution in the Earth frame for N ν = 3.046 (solid lines) and N ν = 3.62 (dashed lines). [1]J. Birrell and J. Rafelski arxiv:1402.3409 [hep-ph]

Part 2 Summary The effective number of neutrinos is probed by BBN element abundances and the CMB. N ν larger than predicted by two body scattering? We performed a model independent parametric study of the dependence of Υ, T γ /T ν, and N ν on T k < T ch. Measurement of N ν constitutes a measurement of T k. N ν > 3.046 indicates that there are significant factor(s) other than two body scattering in neutrino freeze-out. Characterization of the effects of reheating on neutrino spectrum today is a small step towards detector design. Drag force on a millimeter sized coherent detector is encouraging, considering precision force measurements on the order of yocto-newtons (10 24 N).

Part 3: Boltzmann Equation in an FRW Spacetime Particle dynamics given by Boltzmann equation p α x αf Γ j µνp µ p ν p jf = C[f ]. (29) No longer assuming chemical or kinetic equilibrium. For a general distribution in an FRW spacetime the Boltzmann equation is t f ph p f = 1 ȧ C[f ], H = E a For 2 2 reactions the collision operator is C[f 1 ] = 1 2 F(p 1, p 2, p 3, p 4 )S M 2 (2π) 4 δ( p) 4 i=2 (30) d 3 p i 2(2π) 3 E i, (31) F =f 3 (p 3 )f 4 (p 4 )f 1 (p 1 )f 2 (p 2 ) f 1 (p 1 )f 2 (p 2 )f 3 (p 3 )f 4 (p 4 ), (32) f i =1 f i (Fermi blocking factors).

Neutrino Scattering Matrix Elements Process S M 2 ν e + ν e ν e + ν e 128G 2 F (p 1 p 4 )(p 2 p 3 ) ν e + ν e ν e + ν e 64G 2 F (p 1 p 2 )(p 3 p 4 ) ν e + ν e ν j + ν j 32G 2 F (p 1 p 4 )(p 2 p 3 ) ν e + ν j ν e + ν j 32G 2 F (p 1 p 4 )(p 2 p 3 ) ν e + ν j ν e + ν j 32G 2 F (p 1 p 2 )(p 3 p 4 ) ν e + ν e e + + e 128G 2 F [g2 L (p 1 p 4 )(p 2 p 3 ) + g 2 R (p 1 p 3 )(p 2 p 4 ) + g L g R m 2 e(p 1 p 2 )] ν e + e ν e + e 128G 2 F [g2 L (p 1 p 2 )(p 3 p 4 ) + g 2 R (p 1 p 4 )(p 2 p 3 ) g L g R m 2 e(p 1 p 3 )] ν e + e + ν e + e + 128G 2 F [g2 R (p 1 p 2 )(p 3 p 4 ) + g 2 L (p 1 p 4 )(p 2 p 3 ) g L g R m 2 e (p 1 p 3 )] Table: Matrix elements for electronic neutrino processes where j = µ, τ, g L = 1 2 + sin2 θ W, g R = sin 2 θ W, sin 2 (θ W ) = 0.23 is the Weinberg angle, and G 2 F = 1.16637 10 5 GeV 2 is Fermi s constant. [1] A. Dolgov, S.H. Hansen, and D.V. Semikoz, Nucl.Phys. B543 (1999) 269.

Neutrino Scattering Matrix Elements Process S M 2 ν i + ν i ν i + ν i 128G 2 F (p 1 p 4 )(p 2 p 3 ) ν i + ν i ν i + ν i 64G 2 F (p 1 p 2 )(p 3 p 4 ) ν i + ν i ν j + ν j 32G 2 F (p 1 p 4 )(p 2 p 3 ) ν i + ν j ν i + ν j 32G 2 F (p 1 p 4 )(p 2 p 3 ) ν i + ν j ν i + ν j 32G 2 F (p 1 p 2 )(p 3 p 4 ) ν i + ν i e + + e 128G 2 F [ g2 L (p 1 p 4 )(p 2 p 3 ) + g 2 R (p 1 p 3 )(p 2 p 4 ) + g L g R m 2 e (p 1 p 2 )] ν i + e ν i + e 128G 2 F [ g2 L (p 1 p 2 )(p 3 p 4 ) + g 2 R (p 1 p 4 )(p 2 p 3 ) g L g R m 2 e (p 1 p 3 )] ν i + e + ν i + e + 128G 2 F [g2 R (p 1 p 2 )(p 3 p 4 ) + g 2 L (p 1 p 4 )(p 2 p 3 ) g L g R m 2 e(p 1 p 3 )] Table: Matrix elements for µ and τ neutrino processes where i = µ, τ, j = e, µ, τ, j i, g L = g L 1 = 1 2 + sin2 θ W, g R = sin 2 θ W, sin 2 (θ W ) = 0.23 is the Weinberg angle, and G 2 F = 1.16637 10 5 GeV 2 is Fermi s constant. [1] A. Dolgov, S.H. Hansen, and D.V. Semikoz, Nucl.Phys. B543 (1999) 269.

Chemical Equilibrium Spectral Method Has been solved via a spectral method based on the orthonormal basis of polynomials generated by the weight [1], [2] f ch = 1 e y, y = a(t)p. (33) + 1 Similar to method of moments of Grad in hydrodynamics, except with ultra-relativistic fermions. Implicit assumptions Near chemical equilibrium (Υ 1). Reheating is insignificant (T scales as 1/a). [1] S. Esposito et. al., Nucl. Phys. B590 (2000) 539 [2] G. Mangano et. al., Phys. Lett. B534 (2002) 8

Chemical Non-Equilibrium Spectral Method Change variables z = p/t(t), T is effective temperature. ( t f z H + Ṫ ) z f = 1 C[f ]. (34) T E To model a distribution near kinetic equilibrium decompose f (t, z) = f Υ (z, Υ(t))ψ(t, z), f Υ (z, Υ) = 1 Υ 1 e z + 1. (35) We will solve the Boltzmann equation using a dynamical basis of orthogonal polynomials that is adapted to systems where ψ 1.

Chemical Non-Equilibrium Spectral Method Expand ψ(t, z) in the orthonormal basis of polynomials, ˆψ(z, Υ), generated by the Υ-dependent weight via the recurrence relation [1] w Υ = z 2 f Υ (z, Υ). (36) ψ 0 = 1, ψ 1 = ψ 0 (x x ˆψ 0, ˆψ 0 ) ˆψ 0, (37) [( ψ n+1 = ψ n x x ˆψ n, ˆψ ) n ˆψn x ˆψ n, ˆψ n 1 ˆψ ] n 1. (38) [1] See e.g. F. Olver, Asymptotics and Special Functions. New York: Academic Press, 1974.

Chemical Non-Equilibrium Spectral Method Expand ψ(t, z) in the orthonormal basis of polynomials, ˆψ(z, Υ) ψ(t, z) = i b i (t) ˆψ i (z, Υ(t)) (39) ḃ k = i i ( b i H + Ṫ ) ( z ) ˆψi z f Υ, T f ˆψ k + z z ˆψi, ˆψ k Υ b i Υ ( 1 f Υ f Υ Υ ˆψ i, ˆψ k + ˆψ i Υ, ˆψ k ) + 1 f Υ E C[f ], ˆψ k [1] J. Birrell and J. Rafelski, arxiv:1403.2019.

Chemical Non-Equilibrium Spectral Method Up to this point, T and Υ were arbitrary functions. Fix the dynamics of T(t) and Υ(t) by imposing In other words b 0 (t) ˆψ 0 (z, Υ(t)) = 1, b 1 (t) = 0. (40) f (t, z) = f Υ (t, z)(1 + φ(t, z)), φ = N b i ˆψi. (41) i=2 The non-thermal distortions are contained in φ.

Chemical Non-Equilibrium Spectral Method These condition lead to evolution equations for T(t), Υ(t) (Ab) Υ/Υ 1 1 f = C[f ], ˆψ ΥE 0 (Ab) 0 1 f C[f ], ˆψ ΥE 1 [Υ Υ 1, 1 /(2 ψ 0 ) + (Bb) 0 ](Ab) 1 (Ab) 0 (Bb) 1, (42) Ṫ/T = (Bb)1 1 f C[f ], ˆψ ΥE 0 1 f C[f ], ˆψ ΥE 1 [Υ Υ 1, 1 /(2 ψ 0 ) + (Bb) 0 ] [Υ Υ 1, 1 /(2 ψ 0 ) + (Bb) 0 ](Ab) 1 (Ab) 0 (Bb) 1 H = 1 ( ) (Ab) 1 (Bb) 1 1 Υ/Υ f Υ E C[f ], ˆψ 1 H. (43) where (Ab) n = N j=0 An j bj and similarly for B.

Chemical Non-Equilibrium Spectral Method Physical motivation for the weight function: n = g νt 3 2π 2 1, 1 = g νt 3 2π 2 ρ = g ν 2π 2 T4 1, z = g νt 4 2π 2 0 0 f Υ z 2 dz, f Υ z 3 dz. (44a) (44b) The particle number and energy densities are captured by f Υ and the non-thermal distortions from kinetic equilibrium are entirely contained in φ. Only two degrees of freedom are necessary. [1] J. Birrell and J. Rafelski, arxiv:1403.2019.

Chemical Non-Equilibrium Spectral Method n = g νt 3 2π 2 1, 1 = g νt 3 2π 2 ρ = g ν 2π 2 T4 1, z = g νt 4 2π 2 0 0 f Υ z 2 dz, f Υ z 3 dz. (45a) (45b) Only the zeroth order term f Υ is used here, not the full distribution i.e. computed as if in kinetic equilibrium. This is only possible with the correct evolution of T(t), Υ(t) shown earlier. It is in this sense that T, Υ are the averaged temperature and fugacity of the non-equilibrium distribution f.

Key Differences and Improvements Adapted to kinetic equilibrium (i.e. non-thermal distortions φ small) but allows for arbitrary Υ. Generalized averaged temperature (i.e. T not required to scale as 1/a). Introduction of z 2 factor means only 2 modes are required to capture particle number and energy flow, compared to 4 for the chemical equilibrium method. Numerical evaluation of scattering integrals dominates the numerical cost, so this last point results in a large reduction in computer time.

Generalizations In general, one can introduce additional parameters in the weight to capture additional properties/impose conditions. For example pressure for massive particles local flow direction for non-isotropic distributions (e.g. hydrodynamics for fermions) We have begun work on the generalization to massive fermions/bosons.

Neutrino Scattering Integrals The collision operator inner products must be analytically reduced to fewer dimensions for numerical computation M T3 2π 2 1 f Υ E C[f ], ˆψ k = ˆψ k (p 1 )C[f 1 ](p 1 ) = G k (p 1, p 2, p 3, p 4 )S M 2 (2π) 4 δ( p) d 3 p 1 (2π) 3 E 1 4 i=1 δ 0 (p 2 i m 2 i ) d4 p i (2π) 3, (46) G k = ˆψ k (z 1 ) [ f 3 (p 3 )f 4 (p 4 )f 1 (p 1 )f 2 (p 2 ) f 1 (p 1 )f 2 (p 2 )f 3 (p 3 )f 4 (p 4 ) ], (47) f i = 1 f i. (48)

Neutrino Scattering Integrals Integral is initially 16 dimensional. Mass shell delta functions reduce it to 12 dimensions. 4-momentum conservation reduces it to 8 dimensions. Spend symmetries to reduce the dimensionality further. Using isotropy, we reduced this to three iterated integrals and improved on previous methods by obtaining a smooth integrand over a simple integration region to numerically integrate. Numerically, the matrix element inner products computed by our method agree with the method used in [1]. [1] A. Dolgov, S.H. Hansen, and D.V. Semikoz, Nucl.Phys. B543 (1999) 269.

K(s, p) = 8πrr s 1 M = 1 [ 1 1 256(2π) 7 1 ( 2π 0 s 0 0 K(s, p) p2 dpds, (49) p0 S M 2 (s, t(cos(ψ) ) 1 y 2 1 z2 + yz))dψ ] (50) G 34((q ) 0 α r δy)dy G 12(q 0 α rδz)dz. (51) For neutrino reactions, K splits into a linear combination of products of 1-d integrals. p 0 = p 2 + s, α = p0, δ = p, q 0 = m2 1 m2 2, (q ) 0 = m2 3 m2 4, (52) s s s s r = 1 s (s (m 1 + m 2 ) 2 )(s (m 1 m 2 ) 2 ), r = 1 s (s (m 3 + m 4 ) 2 )(s (m 3 m 4 ) 2 ), (53) t(x) = 1 4 ((q0 (q ) 0 ) 2 r 2 (r ) 2 + 2rr x), s 0 = max{(m 1 + m 2 ) 2, (m 3 + m 4 ) 2 }. (54) F(p, q, q ) =F 12 ((p + q) U/2, (p q) U/2)F 34 ((p + q ) U/2, (p q ) U/2) (55) G 12 (p U, q U)G 34 (p U, q U).

Solving the Boltzmann Equation We tested the chemical non-equilibrium method against the chemical equilibrium method to show their equivalence. Define the following z fin = T γ /a, ρ ν0 = 7 120 π2 a 4, δ ρ ν = ρ ν ρ ν0 1. (56) Method Modes z fin δ ρ νe δ ρ νµ,τ N ν Chemical Eq 4 1.39785 0.009230 0.003792 3.044269 Chemical Non-Eq 2 1.39784 0.009269 0.003799 3.044383 Chemical Non-Eq 3 1.39785 0.009230 0.003791 3.044264 Chemical non-equilibrium method with minimum number of modes 20 faster.

Applications: Dependence on Weinberg Angle Parameter that impacts strength of weak force processes. Measured value in vacuum θ W 30. cos(θ W ) = m W m Z. (57) Value not determined by Standard Model. Neutrino freeze-out is a probe of possible time or temperature dependence.

Dependence on Weinberg Angle 3.12 3.11 3.1 3.09 Nν 3.08 3.07 3.06 3.05 3.04 3.03 0 0.2 0.4 0.6 0.8 1 sin 2 (θ W ) Figure: Effective number of neutrinos as a function of Weinberg angle. Vertical line is sin 2 (θ W ) =.23.

Applications: Dependence on Interaction Strength The Einstein Boltzmann system depends on a single dimensionless parameter,which we call the interaction strength, η M p m 3 eg 2 F. (58) Neutrino freeze-out is a probe of possible time or temperature dependence of this combination of natural constants. We let η 0 0.0442 be the presently measured value.

Dependence on Interaction Strength 10 0 Nν 3 10 1 10 0 10 1 10 2 η/η 0 Figure: Effective number of neutrinos as a function of interaction strength ratio.

Part 3 Summary We developed an improved method for analytically simplifying the scattering integrals that substantially reduced the numerical integration cost. We developed a spectral method for solving the Boltzmann equation that is adapted to chemical non-equilibrium. The method incorporates three key improvements: a modified weight function together with a dynamical effective temperature and fugacity. We investigated impact of varying sin 2 (θ W ) on neutrino freeze-out. Maximum value of N ν = 3.13 for sin 2 (θ W ) = 1. We investigated impact of varying the dimensionless parameter η = M p m 3 eg 2 F on neutrino freeze-out. N ν = 3.30 for η/η 0 = 36.3 and N ν = 3.62 for η/η 0 = 147.3.

Looking to the Future Investigate systems of massive particles. Extend to spatially varying systems (generalized hydrodynamics for bosons/fermions). We focused on algorithmic optimization. Can also look at code optimization (will be more critical for spatially dependent systems).

Acknowledgments Thanks to my advisor Dr. Rafelski for his invaluable physical insight, providing the physical compass for our work. Thanks to Cheng-Tao Yang, a student from National Taiwan University who visited U of A and worked on evaluating matrix elements for neutrino processes, supervisor Dr. Pisin Chen. Special thanks to my parents for their unwavering emotional support. This work has been supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. Supported in part by a grant from the U.S. Department of Energy, DE-FG02-04ER41318 (PI Dr. Johann Rafelski).

10 0 Energy Density Fraction 10 1 10 2 10 3 10 4 Dark Energy Dark Matter Baryons e ± γ ν µ ± π ±,0 10 3 10 2 10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 T [ev] Figure: Current era: 69% dark energy, 26% dark matter, 5% baryons, < 1% photons and neutrinos, 1 massless and 2.1 ev neutrinos.

Effective Number of Neutrinos 4 4 3.9 3.9 3.8 3.8 3.7 3.6 Nν = 3.62 3.7 3.6 Nν = 3.62 Nν 3.5 Nν 3.5 3.4 Nν = 3.36 3.4 Nν = 3.36 3.3 3.3 3.2 3.1 Nν = 3.046 3 1.15 1.2 1.25 1.3 1.35 1.4 T γ /T ν 3.2 3.1 Nν = 3.046 3 0.5 0.4 0.3 0.2 0.1 0 σ [1]J. Birrell, C.T. Yang, P. Chen, and J. Rafelski, Phys. Rev. D 89, 023008 (2014)

Neutrinos Today Neutrino distribution in Earth rest frame: f (p µ 1 ) =. (59) Υ e 1 (p µ U µ) 2 m 2 ν /Tν + 1 The 4-vector characterizing the rest frame of the neutrino distribution: U µ = (γ, 0, 0, V γ), γ = 1/ 1 V 2, V = 300 ± 30 km/s c (60) Note that spatial inhomogeneity of gravity and of the neutrino distribution are not included in our considerations.

1.4 1.2 1 0.8 m ν = 0.05 ev m ν = 0.1 ev m ν = 0.25 ev m ν = 0.5 ev m ν = 1 ev m ν = 2 ev fe 0.6 0.4 0.2 0 1 0.5 0 0.5 1 1.5 T 1 2 m νv 2 [µev] Figure: Neutrino energy distribution in the Earth frame for N ν = 3.046 (solid lines) and N ν = 3.62 (dashed lines). [1]J. Birrell and J. Rafelski arxiv:1402.3409 [hep-ph]

10 0 m ν = 0.05 ev m ν = 0.25 ev m ν = 0.5 ev m ν = 1 ev m ν = 2 ev fλ 10 1 10 2 10 3 0.5 1 1.5 2 2.5 3 3.5 4 λ [mm] Figure: Neutrino de Broglie wavelength distribution in the Earth frame for N ν = 3.046 (solid lines) and N ν = 3.62 (dashed lines). [1]J. Birrell and J. Rafelski arxiv:1402.3409 [hep-ph]

10 2 m ν = 0.05 ev m ν = 0.25 ev m ν = 0.5 ev m ν = 1 ev m ν = 2 ev f λ 3.62 /f 3.046 λ 10 1 10 0 10 1 10 0 10 1 λ [mm] Figure: Ratio of neutrino de Broglie wavelength distribution with N ν = 3.62 to the distribution with N ν = 3.046. [1]J. Birrell and J. Rafelski arxiv:1402.3409 [hep-ph]

Neutrino Drag Drag force on spherical detector due to hard sphere scattering 1 dp A dt = 2 π f (p)p 2 r d 3 p cos φ sin φdφẑ, m ν 0 p r<0 1 f (p) =, (61) Υ e 1 (E V p ẑ) 2 γ 2 m 2 ν/t ν + 1 p ẑ = p r cos φ p φ sin φ.

Least squares fit to drag force in hard sphere limit, valid to approximately 1% relative error in the region 270 km/s V 330 km/s; 3 N ν 3.7; 0.05 ev m ν 2 ev. Define δn ν = N ν 3, δv = V /300 km/s 1 and recalling the CMB temperature T CMB = 0.235 mev, x = m ν /10 4 T CMB. F drag /A zn/mm 2 = K 1 + K 2 δv + K 3 δn ν + K 4 δv δn ν, K 1 (x) = 0.789 + 4.09x 2 6.01x 4 + 7.37x 6 3.72x 8, K 2 (x) = 0.821 + 10.1x 2 12.5x 4 + 7.72x 6, K 3 (x) = 0.260 0.0631x 2 + 0.513x 4 0.392x 6, K 4 (x) = 0.248 + 0.308x 2 + 0.126x 4. [1]J. Birrell and J. Rafelski arxiv:1402.3409 [hep-ph]

Fdrag/A [zn/mm 2 ] 3 2.5 2 1.5 m ν = 0.05 ev m ν = 0.25 ev m ν = 0.5 ev m ν = 1 ev m ν = 2 ev 1 270 280 290 300 310 320 330 V [km/s] Figure: Drag per unit area on spherical detector due to hard sphere neutrino scattering for N ν = 3.046 (solid) and N ν = 3.62 (dashed). [1]J. Birrell and J. Rafelski arxiv:1402.3409 [hep-ph]

Scattering Length We want to define a characteristic length between scatterings. We have 1 d a 3 dt (a3 n) + = g ν 2π 2 T2 C[f ]zdz, (63) where the plus indicates that we want the one way flow and hence the one way scattering operator (denoted by arrow) in order to count the number of reactions and not just the net flow. Define the scattering length a 3 n L v d dt (a3 n) = 1 0 Υ 1 e z +1 z2 dz 0 C[f ]z 2 /Edz. (64)

Scattering Length νe Scattering Length [m] 1e5 1 1e 05 1e 10 1e 15 Annihilation Length ν e ± Scattering Length ν ν Scattering Length c/h νµ,τ Scattering Length [m] 1e5 1 1e 05 1e 10 1e 15 Annihilation Length ν e ± Scattering Length ν ν Scattering Length c/h 10 1 10 0 T γ [MeV] 10 1 10 0 T γ [MeV] Figure: Comparison of Hubble parameter to neutrino scattering length for various types of processes.

Dependence on Weinberg Angle δρν 0.03 0.025 0.02 0.015 Υ 1 0.998 0.996 0.994 0.992 0.99 ν e ν µ,τ 0.01 0.005 ν e ν µ,τ 0 0 0.2 0.4 0.6 0.8 1 sin 2 (θ W ) 0.988 0.986 0.984 0.982 0 0.2 0.4 0.6 0.8 1 sin 2 (θ W ) Figure: Fractional increase in neutrino energy and neutrino fugacities, as functions of Weinberg angle. Vertical line is sin 2 (θ W ) =.23.

Dependence on Weinberg Angle atγ 1.4 1.399 1.398 1.397 1.396 1.395 1.394 Tγ/Tν 1.4 1.395 1.39 1.385 1.38 ν e ν µ,τ 1.393 0 0.2 0.4 0.6 0.8 1 sin 2 (θ W ) 1.375 0 0.2 0.4 0.6 0.8 1 sin 2 (θ W ) Figure: Photon reheating and neutrino to photon temperature ratios, as functions of Weinberg angle. Vertical line is sin 2 (θ W ) =.23.

Dependence on Weinberg Angle νe Freeze-out Temperatures [MeV] 3 2.5 2 1.5 1 ν e ± Chemical Freeze-out ν e ± Kinetic Freeze-out ν ν Kinetic Freeze-out 0.5 0 0.5 1 sin 2 (θ W ) νµ,τ Freeze-out Temperatures [MeV] 4 3.5 3 2.5 2 1.5 1 ν e ± Chemical Freeze-out ν e ± Kinetic Freeze-out ν ν Kinetic Freeze-out 0.5 0 0.5 1 sin 2 (θ W ) Figure: Freeze-out temperatures for various types of processes, as functions of Weinberg angle. Vertical line is sin 2 (θ W ) =.23.

Dependence on Interaction Strength 0.2 0.18 0.16 0.14 0.12 1 0.98 0.96 ν e ν µ,τ δρν 0.1 Υ 0.94 0.08 0.06 0.92 0.04 0.02 ν e ν µ,τ 0.9 0 50 100 150 200 250 300 η/η 0 0.88 50 100 150 200 250 300 η/η 0 Figure: Fractional increase in neutrino energy and neutrino fugacities, as functions of interaction strength ratio.

Dependence on Interaction Strength 1.4 1.395 1.4 ν e ν µ,τ 1.39 atγ 1.385 1.38 1.375 1.37 1.365 Tγ/Tν 1.35 1.3 1.36 1.355 1.35 50 100 150 200 250 300 1.25 50 100 150 200 250 300 η/η 0 η/η 0 Figure: Photon reheating and neutrino to photon temperature ratios, as functions of interaction strength ratio.

Dependence on Interaction Strength νe Freeze-out Temperatures [MeV] 2 1.5 1 0.5 ν e ± Chemical Freeze-out ν e ± Kinetic Freeze-out ν ν Kinetic Freeze-out 0 10 0 10 1 10 2 η/η 0 νµ,τ Freeze-out Temperatures [MeV] 3.5 3 2.5 2 1.5 1 0.5 ν e ± Chemical Freeze-out ν e ± Kinetic Freeze-out ν ν Kinetic Freeze-out 0 10 0 10 1 10 2 η/η 0 Figure: Freeze-out temperatures for various types of processes, as functions of interaction strength ratio.