Nuclear reactions in the early universe I
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Nuclear reactions in the early universe I Mark Paris – Los Alamos Nat’l Lab Theoretical Division ISSAC 2014 UCSD Acknowledgments ¨ IGPPS University Collaborative subcontract #257842 ¤ ¤ ¤ ¨ Supporting grant: LANL Institutional Computing ¤ ¤ ¤ ¨ “Towards a Unitary and Self-Consistent Treatment of Big Bang Nucleosynthesis” Started FY2014 $45k of LDRD-DR for first year (before -9%) Project Name: w14_bigbangnucleosynthesis Project duration: 2 years (commenced April ‘14) Year1: 1M, Year2: 1M CPU-hours LANL Collaborators ¤ T-2: Gerry Hale, Anna Hayes & Gerry Jungman Paris BBN 2014 May 15 Supporting activities 2013—2014 ¨ Paris – T-2 staff member [Jan. 2012 hire] ¤ ¤ ¤ ¤ ¨ Fuller – Director CASS, UCSD ¤ ¤ ¤ ¨ International conferences (2 invited, 1 contributed), seminars, workshops 4 peer-review publications on light nuclear reactions LANL Institutional Computing 2 year grant LDRD-ER (FY15): BBN proposal oral review 14 May ’14 (yesterday) Conferences, colloquia, workshops (many) Publications (many) NSF Grant No. PHY- 09-70064 at UCSD Grohs – Graduate Program UCSD – ABD ¤ ¤ 15 Feb 2013-Sterile Neutrinos: Dark Matter, Neff, and BBN Implications-CASS Journal Club-UCSD; 10 Sep 2013-Nucleosynthesis, Neff, and Neutrino Mass Implications from Dark Radiation-NUPAC Seminar-UNM; 13 Jan 2014Nucleosynthesis, Neff, and Neutrino Mass Implications from Dark Radiation-HEP Seminar-Caltech; 14 Feb 2014-Evidence (to the trained eye) for Sterile Neutrino Dark Matter-CASS Journal Club-UCSD; 28 Mar 2014-Photon Diffusion in the Early Universe-PCGM30 (Pacific Coast Gravity Meeting)-UCSD; 18 Apr 2014-Neutrinos in Cosmology I-CASS Journal Club-UCSD Dissertation targeted Spring 2015 Paris BBN 2014 May 15 Organization Nuclear reactions in the early universe ¨ Lectures (Paris/E. Grohs) I. II. ¨ Workshop sessions (E. Grohs/Paris) I. II. ¨ Overview of cosmology/Kinetic theory/Big bang nucleosynthesis (BBN) Scattering & reaction formalism/Neutrino energy transport BBN exercises: compute Nuclear Statistical Equilibrium/electron fraction Compute primordial abundances vs Ωb h2: code parallelization Lecture notes ¨ Will be available online (URL TBA) Paris BBN 2014 May 15 Possibly useful references [1] Wikipedia, Recombination (cosmology), http://en.wikipedia.org/wiki/Recombination_ (cosmology)#cite_note-1 (2010), [Online; accessed 01 July 2014]. [2] S. Weinberg, Gravitation and Cosmology (John Wiley & Sons, 1972). [3] S. Detweiler, Classical and Quantum Gravity 22, S681 (2005), URL http://stacks.iop. org/0264-9381/22/i=15/a=006. [4] L. Landau and E. Lifshitz, Mechanics, v. 1 (Elsevier Science, 1982), ISBN 9780080503479, URL http://books.google.com/books?id=bE-9tUH2J2wC. [5] Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. J. Banday, et al., ArXiv e-prints (2013), 1303.5076. [6] D. Baumann, ArXiv e-prints (2009), 0907.5424. [7] S. Dodelson, Modern Cosmology, Academic Press (Academic Press, 2003), ISBN 9780122191411, URL http://books.google.com/books?id=3oPRxdXJexcC. [8] P. Ade et al., Phys. Rev. Lett. 112, 241101 (2014). [9] K. Huang, Statistical mechanics (Wiley, 1987), ISBN 9780471815181, URL http://books. google.com/books?id=M8PvAAAAMAAJ. Cosmology notes 34 [10] L. Brown, Quantum Field Theory (Cambridge University Press, 1994), URL http://books. google.com/books?id=_qt_nQEACAAJ. [11] E. Lifshitz and L. Pitaevskiı̆, Physical Kinetics, no. v. 10 in Course of theoretical physics (Butterworth-Heinemann, 1981), ISBN 9780750626354, URL http://books.google.com/ books?id=h7LgAAAAMAAJ. [12] M. Peskin and D. Schroeder, An Introduction to Quantum Field Theory, Advanced book classics (Addison-Wesley Publishing Company, 1995). [13] J. Bernstein, Kinetic Theory in the Expanding Universe (Cambridge University Press, 1988). Paris BBN 2014 May 15 Outline Lecture I ¨ Overview ¨ Cosmological dynamics in GR ¨ Big bang nucleosynthesis (BBN) ¨ Boltzmann equation ¤ Flat & curved spacetime Lecture II ¨ Unitary reaction network (URN) of light nuclei ¨ Neutrino energy transport ¨ Evan Grohs: observations of primordial abundances Paris BBN 2014 May 15 Big Bang nucleosynthesis (BBN) in CSM context ¨ ¨ ¨ Cosmological Standard Model – ΛCDM Formation of 4He, deuterium (D), 3H, 3He, 7Be/Li, … in the primordial ‘fireball’ Epochs (Hot/dense > cool/rarified) ¤ ¨ ¨ Time of BBN: ~1sec à ~102 sec; TBBN: ~1 MeV à ~10 keV Relevant physics: cooling thermonuclear reactor ¤ ¤ ¤ ¨ Planck > GUT/Inflation > EWPT > QHT > BBN > RC > GF/LSS work of expansion cools radiation & matter weak (neutrino) & strong nuclear interactions (& ???) Boltzmann transport, non-equilibrium phenomena Comparison to observations ¤ ¤ stunning successes: CMB, helium, deuterium perplexing anomalies: dark matter/energy, lithium problem Paris BBN 2014 May 15 Big Bang nucleosynthesis (BBN) in CSM context Cosmology notes Scale a(t) reheating BBN recombination dark ages reionization galaxy formation 15 dark energy Ia 21 cm LSS BAO QSO Inflation Lyα Cosmic Microwave Background density fluctuations B-mode Polarization gravity waves ? 10 --34 s neutrinos 3 min 10 4 10 15 GeV Lensing 1 MeV 380,000 1,100 13.7 billion 25 6 1 eV 2 0 1 meV Time [years] Redshift Energy FIG. 1: History of the universe in terms of the major epochs defined variously by properties of radiation and matter.[6] Paris BBN 2014 May 15 Big Bang nucleosynthesis (BBN) in CSM context Paris BBN 2014 May 15 Big Bang nucleosynthesis (BBN) in CSM context ¨ ¨ ¨ ¨ ¨ Cosmological Standard Model – ΛCDM Formation of 4He, deuterium (D), 3H, 3He, 7Be/Li, … in the primordial ‘fireball’ Epochs (Hot/dense > cool/rarified) ¤ Planck > GUT/Inflation > EWPT > QHT > BBN > RC > GF/LSS Time of BBN: ~1sec à ~102 sec; TBBN: ~1 MeV à ~10 keV Relevant physics: cooling thermonuclear reactor ¤ ¤ ¤ ¨ work of expansion cools radiation & matter weak (neutrino) & strong nuclear interactions (& ???) Boltzmann transport, non-equilibrium phenomena Comparison to observations ¤ ¤ stunning successes: CMB, helium, deuterium perplexing anomalies: dark matter/energy, lithium problem Paris BBN 2014 May 15 Observations [more from Evan G. tomorrow] !! Observational astronomy !! !! !! !! !! Cosmic microwave background !! !! Planck, WMAP, PolarBear, APT, SPT, CMBPol, … Implications !! !! !! existing 10m-class telescopes: Keck, … #! Gold-plated: 2% D meas. Pettini & Cooke ‘13 adaptive optics space- & ground-based observatories planned 30+m-class telescopes: ELT, TMT, … test physics beyond SM; lab tests difficult/impossible precision constraints expected to test nuclear physics Unprecedented precision for primordial nuclear abundances Paris BBN 2014 May 15 Standard FLRW Cosmology ¨ Robertson, Walker show homogen., isotropic > Friedmann, Lemaître solution to GR unique: ◆2 ✓ G00 = 8⇡T00 ; ¨ gij = a2 (t), Kspace ⌘ 0; ȧ(t) a(t) = 8⇡G ⇢(t) 3 The ‘Old’, Big Three observations ¤ ¤ ¤ ¨ g00 = 1, expansion: Hubble “constant,” H0= 67.1 km/s/Mpc (Planck) CMB: T = 2.73 K BBN: concordance at baryon/photon ratio HIF universe ¤ ¤ ¤ may only tune RHS of Einstein-Friedmann Eqn radiation: photons, neutrinos, dark radiation matter: baryonic, dark ΛCDM model: set of assumptions to confront data n Wayne Hu (Uchicago): “alive and well” but issues with growth of density fluctuations Paris BBN 2014 May 15 Einstein-Friedmann equations (0) !! An enduring legacy… Paris BBN 2014 May 15 Einstein-Friedmann equations (0) !! An enduring legacy… Paris BBN 2014 May 15 Einstein-Friedmann equations (I) ¨ Universe dynamics from GR energy-momentum density Gµ⌫ = 8⇡Tµ⌫ Tµ⌫ = ¨ Einstein/Ricci/Curv Cosmology Scalar notes Gµ⌫ = Rµ⌫ 1 2 gµ⌫ R Rµ⌫ = g ↵ R↵µ R=g Rµ⌫⇢ = + µ⌫ @ ⌫ @ ⌫ @x⇢ ⌧ =R ⌫ µ µ ⌫ = R↵µ↵⌫ ⌫⇢ @x µ ⇢⌧ Metric/connection g00 = gij This gives for the components of 1, the connection: Rµ⌫ µ ¨ pgµ⌫ + (p + ⇢)uµ u⌫ 0 00 0 i0 0 ij i 00 i ⌧ ⌫⇢ µ ⌧ i j0 jk = 12 g 0↵ (2g↵0,0 = a2 (t)g̃ij g00,↵ ) = 0 = 12 g 0↵ (g↵i,0 + g↵0,i gi0,↵ ) = 0 = 12 g 0↵ (g↵i,j + g↵j,i gij,↵ ) = = 1 g 2 ij,0 = ȧag̃ij 1 i↵ g (2g↵0,0 2 g00,↵ ) = 0 2 ȧ 1 i↵ 1 1 ik @[a g̃kj ] = 2 g g↵j,0 = 2 2 g̃ = 0 a @x a i 1 il ˜ = 2 g̃ (g̃lj,k + g̃lk,j g̃jk,l ) ⌘ jk . i j = i 0j Equipped with the components of the connection, ijk of the FLRW me consider the equation of motion of objects in free-fall – the geodesic equation [ Paris BBN 2014 May 15 ⇢ ẍ + ⇢ ẋµ ẋ⌫ = 0. Einstein-Friedmann equations (II) logy notes ¨ Knowing energy density (ρ) and pressure (p) ting this relation with Eq.(87) gives Gµ⌫ = 8⇡Tµ⌫ ✓ ◆2 k 8⇡ ȧ ⇢, + 2 = a a 3 4⇡ ä = (⇢ + 3p), a 3 ual form quoted for the Friedmann equations. Due to the fact that the first equ ¨ Covariantly conserved energy-momentum (not indep. eqn.) is an (non-linear) ordinary di↵erential equation of first-order ȧ in time and tha µ⌫ T ;⌫ =there 0 = 3(⇢ +that p) arises if we di↵ere equation is second-order, is a consistency⇢˙ condition a st equation once: ¨ Two equations for three unknowns: ¤ Equation of state: a(t), ⇢(t), p(t) ȧ x3(⇢ + p) . ⇢ ˙ = p = w⇢ a hat we have two equations here for three unknowns, a(t), ⇢(t), p(t) since this equ nsequence of the Friedmann equations. In fact, ParisEq.(92) BBN 2014arises May 15 from the fact th Cosmology notes Einstein-Friedmann equations (III) ¨ a(t) Solution classes Collecting this relation with Eq.(87) gives ✓ ◆2 k 8⇡ ȧ ⇢, + 2 = a a 3 4⇡ ä = (⇢ + 3p), a 3 the usual form quoted for the Friedmann equations. Due to the fa above is an (non-linear) ordinary di↵erential equation of first-or ¨ Acceleration parameter second equation is second-order, there is a consistency condition th ä/ȧ the first equation once: q(t) = ȧ/a ¨ Hubble constant ȧ ⇢˙ = 3(⇢ + p) . ȧ(t 0) a >0 H0 = a(t ) 0 Note that we have two equations here for three unknowns, a(t), ⇢( ¨ Redshift is a consequence of the Friedmann equations.a(t In0 fact, Eq.(92) ari ) , t0 > t1 1 + zconserved: = energy-momentum tensor is covariantly a(t1 ) 1 µ⌫ t [MeV ] ;⌫ = 0, ¨ CurrentTcritical density x p = w⇢ 3 2So 2the conservation protons e = 0. a consequence of the identity⇢c,0 Gµ⌫= H m ⇡ 5 ;⌫ 0 Pl ¨ w > 0 =) ä < 0 a(t) negative curvature 3 system of equations for (a, ⇢, p). It’s a8⇡general feature ofmGR that ¨ w < 0 =) ä > 0 a(t) positive curvature; momentum is inflation a consequence of the geometric structure of the t Einstein tensor. If we assume that we know the equation of state Paris BBN matter 2014 May 15 for radiation, then we non-interacting, non-relativistic and Einstein-Friedmann equations (IV) ¨ Maximally symmetric subspace ¤ ¤ ¤ ¤ ¨ Consequence of homogeneity & isotropy ‘Maximal’ number L.I. Killing vector fields N(N+1)/2 (dim N) Flows of Killing vector fields generate isometries of manifold Friedman universe has MS spacelike hypersurfaces Tensors in MS spaces ¤ scalar: @µ S(x) = 0 ¤ vector: Ai (x) ⌘ 0 ¤ rank-2 tensor: (A0 (x) 6= 0) Bij = Bji = Cgij C 6= C(x) Paris BBN 2014 May 15 Standard BBN – 7Li anomaly n/p ratio ¤ ¨ Helium ¤ ¤ ¨ 0.27 ¤ mass A=7 n 3—5σ discrepancy > Li anomaly 0.03 0.25 0.23 10 −3 D/H p 10 − 4 3He/H p 10 − 5 10 − 9 5 7Li/H Lithium ¤ 0.02 Yp D 0.24 ___ H exquisite sensitivity to neutrons mass fraction Yp~1/4 (p:primordial) ~1:105 Pettini & Cooke obs. better by fact 5 0.01 4He 0.26 Deuterium ¤ ¨ exquisite sensitivity to neutrino distribution ~1:5 Baryon density Ωbh2 0.005 CMB ¤ h=H0/(100 km/s/Mpc) BBN ¨ p 2 10 − 10 1 2 3 4 5 6 7 8 9 10 Baryon-to-photon ratio η × 1010 Figure 20.1: The abundances of 4 He, D, 3 He, and 7 Li as Review Particle Properties Workshop II: generate ‘Schramm plot’ Paris BBN 2014 May 15 The New, ‘Big Five’ observations GF: “VERY EXCITING situa&on developing . . . because of the advent of . . .” ¨ comprehensive cosmic microwave background (CMB) observations (WMAP, Planck, ACT, SPT, PolarBear, CMBPol,...) ¤ ¨ 10/30-meter class telescopes, adaptive optics, and orbiting observatories ¤ ¨ Neff : “effective number” of relativistic species;Yp : 4He mass fraction (relative to proton);η(Ωb): baryon-to-photon number fraction; Primordial deuterium abundance (D/H)p;Σmν e.g., precision determinations of deuterium abundance dark energy/ matter content, structure history etc. Laboratory neutrino mass/mixing measurements ¤ mini/micro-BooNE, EXO, LBNE GF: “is se6ng up a nearly over-­‐determined situa>on where new Beyond Standard Model neutrino physics likely must show itself!” Paris BBN 2014 May 15 ΛCDM: Possible discrepancies (I) ¨ Hubble expansion Planck Collaboration: Cosmological parameters ¨ Planck XVI (2013) There have been remarkable improvements in ¤ sion “tension the CMB-based of the between cosmic distance scale in the last dec The final results the Hubble Space Telescope ( estimates andof the astrophysical Project (Freedman et al. 2001), which used Cepheid c H0 is intriguing ofmeasurements secondary distanceof indicators, resulted in a Hubb 1 1 (72 ± 8) km s Mpc (where the error inc ofand H0 =merits further discussion” mates of both 1 random and systematic errors). Th been used widely in combination with CMB ob ¤ has “highly model dependent” and other cosmological data sets to constrain cosmo ¤ rameters ΛCDM extraction (e.g., Spergel et al. 2003, 2007). It has also b nized that an accurate measurement of H0 with aroun n requires assumptions cision, when combined with CMBabout and other cosmolo has the potential toenergy reveal exotic new physics, relativistic density (RED) for e time-varying dark energy equation of state, additional particles, or neutrino massesexplain (see e.g., Suyu et al. 201 n extra RED could erences therein). Establishing a more accurate cosm discrepancy scale is, of course, an important problem in its own possibility of uncovering new fundamental physics p additional incentive. Two recent analyses have greatly improved the p the cosmic distance scale. Riess et al. (2011) use HS tions of Cepheid variables in the host galaxies of eigh calibrate the supernova magnitude-redshift relation. T Fig. 16. Comparison of H0 measurements, with estimates of estimate” of the Hubble constant, from fitting the calib ±1 errors, from a number of techniques (see text for details). magnitude-redshift Paris BBN 2014 May 15 is relation, These are compared with the spatially-flat ⇤CDM model con- ΛCDM: Possible discrepancies (II) ¨ Clustering ¤ ¤ Abundance of rare massive DM halos exponentially sensitive to growth of structure rms fluct. total mass 8 h-1 Mpc spheres with variance 2 R ¤ ¤ ¨ = Z dk 3j1 (kR) Pm (k) k kR 2 Discrepancy b/w CMB & lensing extra RED can reconcile CMBinferred σ8 with direct observational determinations Dark matter & structure formation 10 1 σlnM ms M* dnh ρ 0 d ln Mh As 0.1 Mth N(Mh) occupation 0.01 0.001 1011 1012 Mh 1013 1014 (h-1M ) 1015 Wayne Hu/2013 October Paris BBN 2014 May 15 CDM: Possible discrepancies (III) !! Big Bang nucleosynthesis !! !! !! !! Lithium anomaly YP,YDP exquisite sensitivity to active neutrino spectrum: #! Most neutrons " 4He #! Yp $ n/p $ f(p,T) Thermal effects #! Hotter later: less neutrons #! Non-equilibrium : less neutrons Probe neutrino sector by studying constraints on various scenarios imposed by precision BBN Planck XVI (2013) Paris BBN 2014 May 15 Possible solutions to lithium anomaly in BBN J. Meléndez et al.: Li depletion in Spite plateau stars ¨ Astronomical explanation ¤ ¤ ¨ Spite-Spite plateau Robust? Melendez et.al.(2010) ‘broken’ Nuclear physics Fig. 2. Li abundances vs. T eff for our sample stars in different metallicity ranges. A typical error bar is shown. ¤ resonant destruction of mass 7 nuclides work is that now we select stars with errors in EW below 5% (typically ∼2−3%), instead of the 10% limit adopted in 9 MR04. only exceptions are the cool dwarfs HD 64090 n B compound The system example (below) and BD+38 4955, which are severely depleted in Li and have Fig. 3. Li abundances for stars with T > 5700 K (open circles) EW errors of 8% and 10%, respectively, and the very metal-poor >6100 K (filled squares), >6350 K (filled triangles) and ≥5850−180 × Unitarity: fundamental, neglected property QM stars (B07) BPS CS29518-0020 (5.2%) and BPS CS29518-0043of[Fe/H] (stars). In the bottom panel stars above the cutoff in T fall into eff ¤ ¨ (6.4%), which were kept due to their low metallicity. The main sources of error are the uncertainties in equivalent widths and T eff , which in our work have typical values of only 2.3% and 50 K, implying abundance errors of 0.010 dex and 0.034 dex, respectively, and a total error in ALi of ∼0.035 dex. This low error in ALi is confirmed by the star-tostar scatter of the Li plateau stars, which have similar low values (e.g. σ = 0.036 dex for [Fe/H] < −2.5, see below). Our Li abundances and stellar parameters are given in Table 1 (available in the online edition). eff two flat plateaus with σ = 0.04 and 0.05 dex for [Fe/H] < −2.5 (dotted line) and [Fe/H] ≥ −2.5 (solid line), respectively. Physics beyond the standard model (BSM) ¤ new particles’ effect on thermal history, etc. keeps the most metal-rich stars in the Spite plateau. We propose such a metallicity-dependent cutoff below. 4.2. Two flat Spite plateaus Even if the lithium anomaly is not nuclear or BSM in origin, precision cosmology forces better treatment of nuclear and astroparticle physics Giving the shortcomings of a constant T eff cutoff, we propose an empirical cutoff of T eff = 5850−180 × [Fe/H]. The stars above this cutoff are shown as stars in the bottom panel of Fig. 3. 4. Discussion Our empirical cutoff excludes only the most severely Li-depleted stars, i.e., the stars that remain in the Spite plateau may still 4.1. The Teff cutoff of the Spite plateau be affected by depletion. The less Li-depleted stars in the botDespite the fact that Li depletion depends on mass tom panel of Fig. 3 show two well-defined groups separated at (e.g. Pinsonneault et al. 1992), this variable has been ig- [Fe/H] ∼ −2.5 (as shown below, this break represents a real disnored by most previous studies. Usually a cutoff in T eff is continuity), which have essentially zero slopes (within the error imposed to exclude severely Li-depleted stars in the Spite bars) and very low star-to-star scatter in their Li abundances plateau, with a wide range of adopted cutoffs, such as 5500BBN K The first group has −2.5 Paris 2014 May 15 ≤ [Fe/H] < −1.0 and &ALi '1 = 2.272 (Spite & Spite 1982), 5700 K (B05), 6000 K (MR04; S07) and (σ = 0.051) dex and a slope of 0.018 ± 0.026, i.e., flat ∼6100 K for stars with [Fe/H] < −2.5 (Hosford et al. 2009). Possible refinements to BBN ¨ Physics beyond the standard model ¤ ¤ ¨ Fundamental principle of nuclear physics: Unitarity ¤ ¤ ¤ ¨ Increasing observational precision requires “sharpening the tool” n improve on existing BBN codes from late 60’s n replace equilibrium thermal history > full neutrino transport BBN can be used to test BSM & nuclear physics Existing codes’ nuclear reaction networks don’t observe unitarity LANL-developed unitary reaction network (URN) for thermonuclear boost & burn Two objectives from nuclear physics perspective n Test LANL URN in similar (but different, high-entropy) environment n Address fundamental problem in cosmology NB: without correct URN, req’d. by QM, BSM physics uncertain Paris BBN 2014 May 15 BBN project: introduce a new theoretical tool ¨ Outline for the rest of talk ¤ ¤ 1st refinement: neutrino sector 2nd refinement: nuclear physics Paris BBN 2014 May 15 BBN project: introduce a new theoretical tool !! Outline for the rest of talk !! !! 1st refinement: neutrino sector 2nd refinement: nuclear physics Paris BBN 2014 May 15 Unitary, self-consistent primordial nucleosynthesis ¨ BBN as a tool for precision cosmology ¤ ¤ incorporate unitarity into strong & electroweak interactions couple unitary reaction network (URN) to full Boltzmann transport code n neutrino energy distribution function evolution/transport code n fully coupled to nuclear reaction network n calculate light primordial element abundance for non-standard BBN n n ¤ active-sterile neutrino mixing massive particle out-of-equilibrium decays→energetic active SM particles New tools/codes for nuc-astro-particle community: n test new physics w/BBN n existing codes are based on Wagoner’s (1969) code n we will improve this situation dramatically Paris BBN 2014 May 15 Kinetic theory: flat spacetime !! distribution function d3 p dN (r, p, t) = d r dn(r, p, t) = d r f (r, p, t) (2⇡)3 d @ @r @ @p @ · · f (r, p, t) = f (r, p, t) + f (r, p, t) + f (r, p, t) dt @t @t @r @t @p ✓ ◆ d @f f (r, p, t) = !! Boltzmann equation dt @t c ✓ ◆ Cosmology notes 21 3 3 collisionless: @f =0 @t c The collision integral is defined as the di↵erence of these two rates !! collisional: ✓ ◆ !! @f @t ✓ c ◆ = Ri Z = 1 d = 2E1 2E2 |v1 4 (4) ⇥ (2⇡) (p01 + p02 (127) Ro , d3 p01 d3 p02 d 3 p2 2E2 (2⇡)3 2E10 (2⇡)3 2E20 (2⇡)3 ⇥ (2⇡)4 (4) (p01 + p02 (p1 + p2 )) |Mf i |2 (F10 20 d3 p02 d3 p01 v2 | 2E10 (2⇡)3 2E20 (2⇡)3 2 (p1 + p2 )) |Mf i | , dj d" F12 ) . (128) Z ! @f from left p2 right, we first encounter the integral over the the mod3to Considering the factors = d |v1 is vrequired F12 2 |(F10 20 since mentum of the projectile, we) consider that a particle 3 which @t c particle (2⇡)‘2’, with any initial momentum impinging upon the target is capable of scattering it from its PSE. The next two integration measures is the final state of the two-body system; no matter the final state, if a collision has taken place we consider that the target has been removed Fig.7.1:H. Schematic Haberzettl Figure description of the quantum-mechanical scattering from its the PSE. process. Part of the incident particle current j through the cross-sectional area A is scattered under the angle θ with respect to the beam axis. The The (4) function enforces conservation of four-momentum. It ensures,Paris among BBNothers, 2014current May 15 into the solid-angle aperture dΩ of the deteccorresponding scattered that if the incoming energy isn’t sufficient to create the masses of the final state particles, tor is dj. The asymptotic picture of the scattering process (i.e., the picture 0 2 Kinetic theory: curved spacetime ¨ Liouville operator Geodesic equation dxµ µ p ⌘ d µ dp + µ⌫⇢ p⌫ p⇢ = 0 d @f dpµ @f dxµ d µ µ f (x ( ), p ( )) = + µ d @xµ d @p d ȧ @ 1 @ LF (f (p, t)) = p f (p, t) = C(E) @t a @p E p0 = E = (p2 + m2 )1/2 ¨ Relativistic Boltzmann equation Z 3 d p f (p, t) (2⇡)3 Z d d3 p C(E) 3 3 ṅ(t) + 3H(t)n(t) = a (a n(t)) = dt (2⇡)3 E ȧ H(t) = a Hubble expansion n(t) = Paris BBN 2014 May 15 ⇠˙ + 3H⇠ = 0 ṅ + 3Hn = C[n] ✓ ◆ 1 n @ = C[n] @t ⇠ ⇠ Entropy production ¨ Boltzmann H-theorem ¤ Entropy current Z d 3 p pµ S = [f log f ⌥ (1 ± f ) log(1 ± f )] 3 0 (2⇡) p Z d3 p µ S ;µ = log f C(E) 0 (2⇡)3 Equilibrium =) S µ;µ ⌘ 0 µ ¨ Equivalence relations ¤ ¤ Collision integral is zero; proper entropy is constant; equilibrium distributions Collision integral non-zero; proper entropy generation; non-equilibrium Paris BBN 2014 May 15 Equilibrium distributions 19 of a quantum state with energy ✏ is ¨ Fermi-Dirac XX 1 1 hN i = X ⇣ ZF D =Z e N s(N ) N =0 ⌘(✏Ns(N ) µN ) N e , (✏ µ) =1+e (✏ µ) . 1 @ fF D== hN iF D log=Z . (✏ µ) e +1 @( µ) derivative and defining fBE and fF D we obtain: ¨ Bose-Einstein (113) (114) The equilibrium distributions satisfy the condition that the collision integral is zero. But (115) here we derive them from the grand canonical ensemble. (116) 1 1 ⇣ ⌘ X N , = hN i = fBE 1 BE (✏ µ) (✏ µ) e e ZBE = =1 , (✏ µ) 1 e 1 N =0 . fF D = hN iF D = (✏ 1µ) e + 1 fBE = hN iBE = (✏ µ) e 1 perature is high, (✏ µ) ⌧ 1, these distribution functions have the ¨ Maxwell-Boltzmann fBE = fF D ⇡ e (✏ µ) = fM B , (117) assical, Maxwell-Boltzmann limit for distinguishable particles. later to have the number and energy density Paris BBN 2014 May 15 Z 3 Kinetic regimes !! Equilibrium !! !! !! !! !! !! !! !! Hubble exp. negligible for kinetics Forward/Reverse rates detail balance Reaction rate sufficiently fast to explore much phase space Caveat: FLRW no timelike Killing field Kinetic !! H(t) Reaction rate d 34,12 = dn2 h ' H(t) Hubble exp. and reactions compete Non-zero net=F-R rate Boltzmann H-theorem: dS/dt>0 but ~ 0 #! However, assume adiabatic Decoupled !! ⌧ H(t) e.g. Relativistic: T~a-1 !! Free-streaming; distribution frozen Paris BBN 2014 May 15 34,12 v12,rel i —Derive (3.1.45). Notice that this is nothing but the ideal gas law, P V = N kB T . Cosmological transitions (Caveat Emptor) paring the relativistic (T m) and non-relativistic (T ⌧ m) limits, we see that er density, energy density, and pressure of a particle species fall exponentially (are nn suppressed”) as the temperature drops below the mass of the particle. We interpret Z JHTêmL q these annihilations annihilation of particlesX and anti-particles. 8 also d3 pAt higher energies 2 2 ⇢(T ) = by particle-antiparticle pair fproduction. (p) p At+low mtemperatures, they are balanced the i 3 i (2⇡) article energies aren’t for pair production. Numerically, one finds that the i= sufficient ,⌫j ,`± ,... 6 ntiparticle annihilation takes place mainly (about 80%) in the interval T = m ! 16 m. 4 X Tevent, at this isn’t an instantaneous but takes several Hubble times. i gi 2 J⇡i (xi ) = 4 ⇤ 2⇡ i the non-relativistic limit, show that —Restoring finite µ in J⇡i (xi ) = Z n 1 n̄ 0 ✓ p ◆ ⇠ 2e +m/Tx2isinh ⇣ µ ⌘ . d⇠ ⇠ p 2 2 , T xi = mi /T ⇠ +xi e + ⇡i = 2g2 mT 2⇡ 3/2 2 (3.1.46) 0 0.0 0.5 1.0 1.5 Têm 2.0 ⇡2 4 ⇢(T ) = g? (T ) T 30 X ✓ Ti ◆4 J⇡ (xi ) i g? = gi T J (0) i EW QCD NB: Ti =T ˙ ⇣ Ji (xi ) ! ✓ T Paris BBN mi ⌘ J(xi ) ! arbitrary! 6 2014 May 15 Reaction network reduction of Boltzmann eqn ¨ Reaction network reduction of BEq. (classical, non-degenerate) X 1 d 3 (a n↵1 ) = a3 dt ↵2 Z ⇥ |M = p 1p 2 p↵1 p↵2 ↵| X ↵2 2 (2⇡)4 (f 1 f 2 (0) n(0) ↵ 1 n↵ 2 h h (0) ni = gi Z d3 p e (2⇡)3 Ei /T ⇡ gi ✓ (4) (p 1 +p (p↵1 + p↵2 )) 2 f↵1 f↵2 ) " n↵1 n↵2 v i ↵ ↵ (0) (0) n↵ 1 n↵ 2 Z n 1n n (0) (0) 1 n 2 Z 1 d3 p 2 d p 1 |v1 v2 |d ↵ f↵1 f↵2 ↵ v↵ i ⌘ N (2⇡)3 (2⇡)3 Z 3 Z 3 d p 2 d p 1 (0) (0) f f ⌘ n N = ↵ ↵ ↵ 1 n↵ 1 1 2 (2⇡)3 (2⇡)3 mi T 2⇡ ◆3/2 e 3 2 # mi /T Paris BBN 2014 May 15 n-p weak equilibrium [Workshop exercise] ¨ ¨ At high T ~ 10’s MeV Xn ~ Xp~ 1/2 At 10 MeV > T > 1 MeV (ignore nucleons) ne+ $ p¯ ⌫e , n⌫e $ pe , ¨ Equilibrium condition n $ pe ⌫¯e . (0) µp = µn =) Q = mn nn (0) np = eQ/T mp ' 1.293 MeV Electron Fraction vs. Plasma Temperature (⌦b h2 =2.207E-02) 1.1 Equil Actual dXn = (n ! p)Xn + (p ! n)(1 dt (0) (i ! j) = n` h ji vi i 1.0 Ye 0.9 Xn = 0.8 nn nb nb = nn + np X p ⇡ Xe 0.7 0.6 0.5 101 100 10 T (MeV) 1 10 2 10 3 Paris BBN 2014 May 15 Xn ) Big bang nucleosynthesis [Workshop exercise] !! Full reaction network [NB: should be unitary] Xh dY↵1 = dt nb h ↵2 !! ↵ iY↵1 Y↵2 + nb h ↵ iY 1 Y 2 i Nuclear statistical equilibrium Relative abundances wrt YH vs. Plasma Temp. (⌦b h2 =2.207E-02) Yi /YH 100 10 2 10 4 10 6 10 8 10 10 10 12 10 14 10 16 10 18 10 20 10 22 10 24 N D T 3 He 4 He 6 Li 7 Li 7 Be 101 100 10 T (MeV) 1 10 2 Paris BBN 2014 May 15 Yi = ni nb End Lecture I Paris BBN 2014 May 15
