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Consequences of self-interactions for weak and nucleosynthesis

To cite this article: E. Grohs et al JCAP07(2020)001

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This content was downloaded from IP address 137.110.32.181 on 05/07/2020 at 03:34 JCAP07(2020)001 iclehysics P t a,c ar , a measure of relativistic energy eff strop N [email protected] https://doi.org/10.1088/1475-7516/2020/07/001 and Manibrata Sen b [email protected], osmology and A and Cosmology George M. Fuller 2002.08557 , cosmological parameters from CMBR, cosmological a,1

We calculate and discuss the implications of neutrino self-interactions for the , [email protected] rnal of rnal

ou An IOP and SISSA journal An IOP and Corresponding author. 1 2020 IOP Publishing Ltd and Sissa Medialab Department of Physics, University of366 California, LeConte Hall, Berkeley, Berkeley, CaliforniaDepartment 94720, of U.S.A. Physics, University of9500 California, Gilman San Drive, La Diego, Jolla,Department California of 92093-0319, Physics and U.S.A. 2145 Astronomy, Northwestern Sheridan University, Road, Evanston, Illinois 60208-3112,E-mail: U.S.A. b c a c

E. Grohs, Consequences of neutrino self-interactions for weak decoupling and big bang nucleosynthesis Received February 27, 2020 Revised April 18, 2020 Accepted May 5, 2020 Published July 1, 2020 Abstract. physics of weak decouplingsuch and neutrino-sector big extensions of bangcan stay the nucleosynthesis thermally (BBN) standard coupled in model, toentropy with one the the another. early may photon, Nevertheless, -, not .through the and free-stream, the neutrinos baryon ordinary In exchange yet component weak energy of interaction. and the the We early primordial examine universe the only effects and ofdensity neutrino deuterium at self-interaction abundances for and photonby, decoupling. the physics These into the quantities address weak are decoupling a determined epoch.in number in, the Self-interacting of or Hubble neutrinos anomalies, may have parameter.accompany some including been be of invoked Our as these influenced neutrino calculations aprecision self-interaction show schemes. measurements, possible that Such making means minute surprisingly deuterium signalsmodels of require subtle the explored high- ameliorating best changes here. abundance tension in for BBN BBN constraintsKeywords: in the neutrinos ArXiv ePrint: J JCAP07(2020)001 7 9 4 5 7 7 9 4 7 1 12 15 12 12 13 15 16 17 18 12 14 12 14 17 – 1 – 4.2.1 Neutrino4.2.2 spectra Primordial abundances 5.2.1 Neutrino5.2.2 spectra Primordial abundances 2.2.1 Computational2.2.2 code Results in the standard scenario 3.2.1 Neutrino3.2.2 spectra Primordial abundances 2.1 Physics 4.2 Results 5.2 Results 2.2 Numerics 3.1 Neutrino transport with self-interactions 4.1 Dark radiation parameter 5.1 Differences in implementation with symmetric scenario 6.1 Summary 6.2 Discussion 3.2 Results 1 Introduction In this paper we reportinteracting neutrino detailed standard calculations model of extensionsneutrinos the are evolution have adopted. of been the Though measured, earlyparameters many including in properties universe the the when of unitary mass-squared self- the transformation differences betweenand the and the neutrino three weak mass interaction of (flavor) basisphysics. basis the [1], four Indeed, much many remainsproposed, extensions unexplored and of and for the mysterious aamong standard the in variety neutrinos of model this only — purposes. in sector sometimesnotoriously termed the of Standard difficult self-interactions or neutrino to model particle “secret” constrain interactions extensions sector — inoffer that have are experiments. a posit been However, interactions two window astrophysicalcompact into venues objects perhaps this [3] physics. atneutrino some scattering. Both point The have the their former early energy venue universe and is dynamics [2] a influenced and promising by test collapsing neutrino- bed or for merging this physics because of Contents 1 Introduction 2 Overview of Big Bang Nucleosynthesis 4 Dark radiation addition 5 Lepton asymmetric initial conditions 6 Conclusions 3 Lepton symmetric initial conditions JCAP07(2020)001 ) F G (1.1) (1.2) and equal g . has effective ϕ 21] study how int m L 9]; suppress active and ν s-channel processes g ν , . Spin-flips are possible can also put constraints 2 ϕ 2 2 ij g . g ν m and   = νν . References [20, ij 2 ν 1 1 1 1 1 1 1 1 1 11]. SI E   2 ij ,G is the rest mass of the scalar exchange g ∗ G ϕ is analogous to the Fermi constant ϕ ∼ m ⇒ c jL ij ν ij to be much larger than the range of temper- G σ   iL ϕ ν j eτ ττ µτ m ij g g g – 2 – g . The range of our calculations span and eµ τµ µµ + g g g T i ϕ ee τe µe jL  g g g ν ϕ   c iL m ν on the Dirac spinor indicates charge conjugation. With the ij g ] = c ij 13] and influence flavor evolution of dense neutrino streams in = [g int L between neutrino species ij , but such a mechanism requires other features of an ultraviolet-complete theory, G 3 is a left-handed-neutrino Dirac spinor and that our weak decoupling neutrino transport and nuclear reaction calculations cover. g ϕ. The superscript L T ν ν t-channel processes, where the later manifests in lepton-asymmetric systems. We will There are many extensions in the neutrino sector which incorporate In this work we will consider only processes which couple neutrinos to other neutrinos, We will further restrict the range of neutrino self-interaction parameters we actually ν not consider processes whichto convert represent a the neutrino couplings into in an our anti-neutrino. generic It model is in most flavor-basis convenient matrix form: neutrino self-interactions. Someels of [17–19] envision these, a spontaneously for brokenneutrino symmetry example, self-interactions. with a singlet In Goldstone these Majoron bosoninteractions models, which and mediates exchange among Familon of the mod- thismagnitude boson neutrinos engenders stronger flavor-changing with than interaction theexchange model standard strengths based model that off weak of can interaction. eq. be (2) Here in many we ref. orders consider [9]. a of The scalar- interaction Lagrangian albeit with flavor changing currents. Specifically, we include on models ofcompact inflation objects, [12 –16]. [14 to embed the operator in eq. (1.1) into UV-complete models. and the advent ofin high multi-messenger precision astronomy. cosmology; Cosmologicalmodels the invocations include latter of attempts to: Self-Interacting becausewith Neutrino reconcile cosmological of (SIν bounds sterile the neutrino [4–7];neutrino burgeoning interpretations produce free-streaming capabilities of sterile-neutrino [2]; neutrino and dark act anomalies discordant matter as measurements [8, a of principal participant the in Hubble schemes crafted parameter to [10, reconcile couplings where particle model in eq. (1.1),at order neutrinos cannot undergoe.g., spin-flips eq. at (1.3) order inin ref. the [20]. standardself-interaction weak The model interaction, overall will coupling so scale that with energy typical as cross sections in our adopted schematic strength in allOur neutrino-neutrino generic scattering model channels, then including has two those independently which specifiable change quantities: flavor. the entire weak decoupling and BBN epoch. Generously, and therefore conservatively, we simulate by restricting the mass of the scalar Here the flavor off-diagonalrepresents the entries coupling scheme produce we adopt flavor-changingNucleosynthesis for couplings. (BBN) our early calculations. universe The weak decoupling second We and matrix employ Big Bang an overall coupling strength atures In other words, we will assume JCAP07(2020)001 ), ' 2 and 2 ϕ (1.5) (1.4) (1.3) 2 at the /m T /(4π 30 MeV 2 3 g rad  is the rela- i ≈ ρ ≡ 2 ? ϕ models [22]. (3)T describes the g eff m ν hE 3ζ G 2.2 × 100 MeV. With our = 3 and . , we will consider only , scenario, it will prove ∼ ν ϕ 2 pl n , defined through ν 1/8 2.1 T  m eff ? 11] to alleviate the Hubble ? N g . g 2 4 T > m 3.36 30 π T  26] rules out a universal coupling 2 3 30 π 8π 3/4 # r  eff ∼ N T ν is large enough to produce efficient energy 2. Subsections 1 keV g 4/3 in 2    4 hE 11 is the number density of neutrinos, , to estimate when the neutrinos decouple from 2 – 3 –  ν ϕ  H 4 7 n 2 ϕ 2 m is the Planck mass. If we approximate g m 30 MeV pl 1 keV. Restricting the scalar mass to be 2 +   1 keV, we estimate an effective coupling " m −2 = ' ⇒ = 30 MeV, kaon decay [25, = 0.2 eV, and is parameterized by 10 T ϕ > T > rad × ≈ ρ H m T , this “self-equilibrium” in the neutrino component is effected via ∼ 5.2 T SI ∼ Γ  g sufficiently large to keep these particles in equilibrium with the neutrino ϕ g , in line with the models proposed in ref. [ m g −2 is the average square energy, MeV i 2 , to the Hubble expansion rate, −6 SI hE 10 The neutrino component will be essentially instantaneously self-coupled and energy- The outline of this paper is as follows. To describe the SI Finally, we assume that the overall coupling × and flavor-equilibrated inentropy this with the photon-electron-positron-baryon model. plasma onlymodel through However, the weak ordinary the standard interactions.follow neutrinos self-consistently This the will flow sets exchange ofsimultaneously up energy energy calculating and the a and entropy strong, between formidable electromagneticthe the quantities and transport various that weak components can problem interactions while be that whereindensity. probed, determine i.e., we The the light must latter element abundances quantity and derives relic neutrino from energy the inferred radiation energy density assumption that two-to-two neutrino scattering processes onand timescales neutral-current short weak compared interactionin to times, our standard and charged- calculations. for the entire temperature range employed helpful to givephysics a and brief review numerics of of BBN BBN in with section particular emphasis on the evolution of the neutrino 3.0 in this range [22], implying the mediator mass must be larger than photon decoupling epoch, and flavor exchange among the neutrinos.rate, We compare Γ the self-interacting neutrinothemselves scattering For a decoupling temperature parameter tension [24]. For component. This guaranteesning that time of the our scalar simulations.scalar particles particle Such decays relatively will and large concomitant have couplingsWe entropy disappeared also note, generation preclude during however, by out-of-equilibrium weak that the decoupling cosmology and begin- does BBN. not provide the only constraint on SI take this range tomeans be that 30 we MeV will notparticles have to during be weak concerned decoupling. withsignificant on-shell Moreover, production abundance though and the early population scalar on, ofcoupling particles for these strengths may example be at produced in where tivistic degrees of freedom [23],use and Fermi-Dirac (FD) statistics forwe can 3 flavors solve of for neutrinos to deduce JCAP07(2020)001 , µ . (2.1) (2.8) (2.7) (2.2) (2.4) (2.5) (2.6) (2.3) ν 6. Throughout this 5. We summarize our results and . 4 − . . , , e j , j + j ν e ν ν  + , , e e e − + + + + ν ν e + i j i − ν ν ν ν e + + + results with neutrino-anti-neutrino symmetric p p p ↔ ↔ ↔ ↔ ↔ – 4 – i j j ↔ ↔ ↔ ν ν  ν ν e n n n + + + + + i i i ν + + ν ν ν ν e . Three other neutral-current interactions redistribute + specifies the physics and computational model of SIν ν τ e 27–29] for a treatment on the basic physics of BBN, and or 3.1 µ flavors. = 1. µ/τ B k gives results. To further investigate self-interactions during BBN, we = ~ 3, we present our firstν SI 3.2 = c -flavor only experience neutral-current. As a result, more heat flows into the electron- Weak decoupling concludes at a temperature of a few 10’s of keV. The neutrinos will τ 2.1 Physics Big Bang Nucleosynthesis isout a epoch succession is of out-of-equilibriumthemselves. weak freeze-outs. decoupling, Weak when The decoupling firstinteraction the commences freeze- mediates neutrinos at an decouple temperature exchangetrino from of scales seas the energy of charged from a leptons the few and MeV. and The weak into the neu- ref. [30] forrelevant an topics overview of of standard BBN the here current to status contrast the of standard the scenario field. with that We of give SI a brief review on the 2 Overview of BigWe Bang direct Nucleosynthesis the reader to refs. [23, and discuss further implicationspaper of we set self-interacting neutrinos in section also decouple from thedecoupling baryons have during a the significant Weak overlap Freeze-Outcompared in (WFO) time. to epoch. However, baryons, WFO due andchange WFO to weak in the and large the weak numberrates, -to- of decoupling schematically neutrinos ratio are given of as not the simultaneous. baryons by We monitor calculating the six neutron-to-proton and subsection provide two extensions toanti-neutrino standard asymmetric BBN initial with conditions an in addition sections of dark radiation and neutrino- Both of the abovetron-flavor processes neutrinos include will charged-current experience andand both neutral-current charged interactions. and Elec- neutral-currentflavor interactions, sector while than eitherenergy among the the neutrino seas The above three interactionsamong lead to the a electron “flavor-equilibration” and (see figures 7 and 8 in ref. [31]) initial conditions. Subsection seas. In section JCAP07(2020)001 , i ) ) 4 (2.9) is the P (2.10) (2.11) (4) k f − P 3 − P − )(1 (see table I in 2 (3) P j f + − 1 (1 (P is the pressure exerted (2)  4, to denote the fermion (4) scenarios. Although the f T pl

iδ 3, P 2 (1) | 2, j f dt dQ , − , 1 h|M = 1 3H (4) 4 + 1 f − . The time derivative of the plasma  4 2E e (3) ), k pl p 3 T 3 k φ ) P d )f ( =  . We can calculate the heat flow from the + 1 f (2) (2π j ν pl f 3 cm T T ≡ ρ T using a weak-interaction collision operator − )  3  is the amplitude for reaction 2E p 3 (k – 5 – i 3 ) f −1 2 )(1 d |  j exp e, µ, τ (1) (2π pl f 2 = dT is the plasma heat gained/lost from microphysics, and dρ h|M − 2 2E T  p () = 3 (1 3 )  h d e (), i and i f × f (2π k −3H dQ/dt| Z = 1 1 dt dT 2E = cm /T is based off of eq. (D28) in [32] 1 T =E 1 is the energy density of the plasma (less baryons), 

is the temperature derivative of the plasma components (including baryons) [33]. indicates the collision type corresponding to a reaction in eqs. (2.1)–(2.5). )] pl in eq. (2.10) to denote either the flavor (neutrino or antineutrino) or the charged lepton is the electron degeneracy parameter defined below. There exists an implicit subscript 1 ρ j ) e ( As the temperature continues to decrease, the electron-positron pairs will annihilate We begin the weak decoupling process at 10 MeV by dissociating the electromagnetic i (k φ /dT f [f -annihilation and NFO are identical in the standard and SIν pl j  C ref. [31]). We useoccupation the numbers shorthand for either notation distribution neutrinos or charged leptons, where the latter is always a FD four momentum for particle where temperature and on (electron or positron) depending on the collision into photons during theout eponymous epoch electron-positron isstatistical annihilation Nuclear equilibrium epoch. Freeze-Out and (NFO), obtain Thee when their last the primordial freeze- values. nuclear abundances For our go purposes, out the of physics in nuclear components from the neutrinoevolve seas. according The to electromagneticneutrino eq. occupation plasma (2.9) fractions temperature without continues neutrinos. to In the standard scenario, we evolve the We do not consider the energyseas and as entropy this flow from is the a baryon component small into change the compared neutrino to the entropies of the neutrino seas and plasma. physics is distinct, theoverlap with preceding one four another freeze-out and epochs require computational are tools not2.2 to temporally model distinct. the physics. Numerics They At high temperatures, weaklibrium interactions with keep the the electromagnetic neutrinos plasma, in i.e., thermal and chemical equi- by all plasma components, dρ When calculating theQED electromagnetic effects to energy the densities,a electromagnetic equation detailed we of analysis state implement oftemperatures, as finite the explained the in temperature plasma finite-temperature [34–36] consists QED (see of problem ref. photons, [37] in electrons, for the positrons, early neutrinos, and universe). baryons. At high where JCAP07(2020)001 1 . We b (2.13) (2.14) (2.15) (2.16) (2.17) (2.12) /n i n ≡ i is the reaction Y ij term in eq. (2.9). ] T ,[kl )l , ()]}. ) dQ/dt| i (j, k kl [f ] i + e [ij − acts as an energy scale for the j e ! , N j C j a ,

i cm Y !N i ν Y T i i N i ∂t N Z ∂ρ . ()]. Y ()] + i T i i − , , and nuclear abundances [f − X i [f j − b /dt| C Z ij e T νe ] ν n Nuc µ

j dρ {C is the reverse reaction rate coefficient [32]. The [kl = X 3 pl ≡ l – 6 – = ! ∂t kl + l e N ] = l a e ∂ρ φ d  Y n ij !N k = k /dt| () Z N − i ν k dt T N

Y . Although the distinction is important in a thermodynamic sense, − df 6 dρ e . The positron chemical potential is equal in magnitude a i=1 X ( n − , and [ dt dQ e i 2 4 µ cm j,k,l X 2π T , atomic numbers b = implying n = i a a

dt dY ν ∂t ∂ρ or the scale factor T through the nuclear reaction network with an example equation are the number of each nucleus in the reaction with time by ensuring electrons and positrons maintain FD distributions and i Y e φ i,j,k,l N The only terms in the evolution for the neutrino occupation numbers are the collision Equation (2.9) is derived from conservation of energy [32], but does not conserve par- We have switched which variable we hold constant in our derivatives with respect to time, either the 1 can be parameterized using do not explicitly evolveabundances the degeneracy parameters for the nuclei, but instead evolve their plasma temperature T operators, i.e., plasma to theleptons neutrino seas by integrating over the collision terms involving the charged where the first termthe plasma is loses the energy heat to gained the from neutrino nucleosynthesis, seas and during weak the decoupling. minus sign indicates We only evolve thetemperature occupation of numbers 10 MeV. ofnot the When follow neutrinos the an afterdegeneracy equilibrium neutrinos the parameters distribution decouple universe of and in has the so the reached neutrino there a standard components. is scenario, no they need do to evolve temperatures or for electron chemical potential and opposite in signWe if evolve we assume chemicalthe equilibrium between difference the of electrons thefree and two and positrons. number those densities bound is in nuclei) equal to the number density of (both for baryon number density where ticle number. However,degeneracy we parameter do implement conservation of charge by evolving the electron Therefore, the final expression for the heat gain/lost due to microphysics is neutrinos so that we may calculate energy density quantities, e.g., eq. (2.12). rate coefficient to create nucleus Equation (2.12) communicates the change to the plasma using the JCAP07(2020)001 = n (3.1) τ (2.18) (2.20) (2.19) 42]. We is the total with respect tot )} = 10 MeV. We ρ j ( i cm T {f . We caution that the }, i −10 {Y based off of refs. [31, = 30 MeV and terminate at 10 a, and degeneracy quantity , × e cm ν φ T T , , burst T , tot ρ −5 2 pl 10 is the Planck mass, and 8π scenario, and not absolute predictions. 3m × ν ν pl T µ s Li/H = 4.375 m 7 = . .624 = – 7 – ν H η dt da = 0.2478, 1 a P = 3.044, and the following abundances Y ≡ eff D/H = 2 N H scenario. In our model, we assume that the self-interactions ν = 0.02226, consistent with ref. [43], and a neutron lifetime t. Our calculations commence at b ω The last quantity to evolve is the scale factor with the Friedmann equation = 1 keV. We begin calculating the neutrino collision integrals at cm comoving number of protons changesinto as protons, the and weak-charged-current reactions vice-versa.are change We input use into eqs. eq. (19)–(24)We (2.17). can in use Our ref. either rate [38]the FD expressions to neutrino expressions are calculate distribution. or single the out-of-equilibrium integrals rates, occupation over which the fractions neutrino to energy. characterize which defines the Hubble expansion rate for a baryon density energy density of the universe:radiation, photons, dark electrons, matter, positrons, and neutrinos,matter dark baryons/nuclei, and dark energy. dark energy In as practice, they we are ignore2.2.1 negligible the during contribution the from Computational BBN dark epoch. code Public BBN codes exist, see refs.integrate–41]. [39 the We coupled use our ordinary code to differential equations the for coordinateT time integrate the nuclear reaction network at all2.2.2 times. Results inIn the the standard standard scenario, scenario we find 885.1 s. Our value for primordial lithium is abundances are dependent onbe the taken nuclear as reaction a rates, baseline and when so compared the to above quantities the should SIν 3 Lepton symmetric initial3.1 conditions Neutrino transportWith with self-interactions, we self-interactions need towill differentiate between term two the different process epochs/phenomena.pling”, of We neutrinos in decoupling accordance from with thedecouple the electromagnetic from plasma terminology each “weak used other, decou- trino in we decoupling the will are standard call contemporaneous scenario.time/energy in this scales When the process in neutrinos standard “neutrino the model, decoupling”. SI but Weak occur and on neu- vastly different keep the neutrinos in ation FD differs distribution throughout from weak theFor decoupling. electron/positron lepton-symmetric This FD neutrino initial distribution distribu- conditions,anti-neutrinos in and the both we FD number characterize and distribution it energy using is density. a identical temperature for neutrinos and JCAP07(2020)001 a ) in τ (3.5) (3.3) (3.4) (3.6) (3.7) (3.2) using ν ν , ∂ρ/∂t| ) is η τ ν ν η , µ and (; ν ν , T , µ to be (eq) ν f ν , are parameters for T e ν ν (n, ρ) in eq. (3.2). The elec- (n, ρ) η , ν ν e ρ T η ν with the thermodynamic = = and , ν a ν ν and η ν

η T T n ∂t ∂n 2 ( n ). n and , ν ν ν , . ⇒ η ν ν a = T 3T

dt 2 dη (; −9T + 1 n ∂t ∂ρ − ν η ν          ν 1 a ) ) ∂χ (eq)

∂η 3n ν ν −η η η  ∂t ∂ρ ρ, and the FD expression 4ρn, − −9T + e η η a (; (; ν d f

n, dt and ∂t ν ) = dT ∂n Z – 8 – (eq) (eq) ν 4ρn, T ν 3 f f n 2 ν η 4ρ 2 3 π T + ∂χ ∂T ν (; ν = d  d  is the following expression N = ν (eq) η Z Z dt = as functions of energy and number density in eq. (3.2). . As the universe evolves through weak decoupling, there f dη dt −HT n, dχ ν η 2 2 scenario, unlike how we conserve energy and charge for the 4 3 ν ν ν µ η n, T T = 2π 2π ν ν ν n. Therefore, we find the time derivative of ν N N dt η and as functions of dT or ν ν scenario. If we add energy or number to the neutrino seas, we change ρ ) = ) = η T ν ν ν , η , η ν ν and have familiar meanings in an out-of-equilibrium system, temperature and ν n is the number of particle species and is equal to 6 ( via the definitions of energy and number density T ρ(T n(T is the number density added from out-of-equilibrium weak decoupling, ν ν a η N and ρ denotes either and ν χ ∂n/∂t| T We have written and the time derivative of In eq. (3.2), the non-degenerate SI where where is the energy density added, and Although tromagnetic plasma acts asbetween a the heat two source systems for the precludes neutrinos, the but equivalence the of lack of a tight-coupling both chemical potential (i.e., degeneracy parameter)ing do as not they have the would same in thermodynamic an mean- equilibrium system. Simply stated, for neutrino chemical potential will be aweak transfer interactions. The of self bothergy interactions among only energy the thermalize constituent and the particles.number neutrino particle through These seas number communication interactions by neither exchanging with tonumber en- create the are the nor conserved electromagnetic destroy in neutrino plasma. neutrino theelectromagnetic seas plasma SIν As components through a in eqs. result, the (2.9) energy and and (2.16). We must evolve energy and number distributions.the Equations two (3.5) parameters and and (3.6) are give consistent the with equations the of definitions motion of for the definitions of energy and number density where we write JCAP07(2020)001  (3.9) (3.8) (3.15) (3.10) (3.12) (3.13) (3.11) (3.14) (), of the neutrino )]}. i ν ν for the energy scale. η changes both number f scenario + cm ν (; e T − e (eq) C using the following definition [f + are the same collision operators e cm − + ), T -flavor (red) which are degenerate e e ν , C τ − instead of e C or ν , and )], ) for both terms and integrate over , T ν ν + )] + (; 0) ∆(aT ν µ , η η e ν  T ν and η e ≡ curve corresponds to a higher temperature + (eq) (; 0) (; (; as temperature and degeneracy parameter, 1 + νe f (; − C ν e ν − (eq) − (eq) (eq) −HT η f (eq) ν f [f ↔ ↔ = – 9 – = 0, () [f + aT ν e  ν f ν and . In either scenario, there exists an overabundance νe e − dt dt e + ν dη dT ≡ cm C {C + T 1 = ν 2 3 T ν − () scenario. The dotted black line at zero represents a non- d  d  ν δf , respectively. cm ν T + Z Z T e − 6 6 e i=1 i=1 X X C 2 2 4 3 ν ν T T 2π 2π and = = νe . We compare the energy scales C a a . The positive slope of theν SI

cm e-flavor neutrinos (green) and either T ∂t ∂t ∂ρ ∂n gives the relative change of the neutrino occupation fractions from non-degenerate 1 There are two collision terms we use in the SI There is neither number nor energy density added to the neutrino seas after weak preserves the neutrino number and changes the energy, while compared to ν νe which are the free-streaming conditions for relativistic3.2 particles. Results 3.2.1 Neutrino spectra Figure FD equilibrium, i.e., Equation (3.11) is a copy of eq.2.12) ( except we use of particles at high T variables of temperature andertheless, degeneracy we parameter will of colloquially the refer grand to canonical ensemble. Nev- with collision terms versus the dimensionless comovingtrino energy. spectra The for dashed linesin are our Boltzmann the scenario. out-of-equilibrium The neu- any solid species blue of line is neutrinodegenerate the in degenerate spectrum the FD at equilibrium SI temperature spectrum for respectively, throughout this manuscript. decoupling concludes. At this point, eqs. (3.5) and (3.6) reduce to for the standard scenario ofFor our out-of-equilibrium purposes, neutrino we spectra do (seespectra. not appendix evolve the C Instead, individual in we occupation ref.to use fractions, [31]). find the the FD total expression C added neutrino number and energyand density energy. for Therefore, a the givenweak number time decoupling step. and is energy Note density that added during the out-of-equilibrium JCAP07(2020)001 e (3.18) (3.19) (3.16) (3.17) scenario (solid) ν 10 8 there exists a deficit of particles, ,  0, 0, . −3 cm 6 > > −3 10 ν = 10 MeV. At freeze-out (f.o.), we find ν 10 /T × dt ν dt ν dη dT × T E ) ) . . ⇒ ⇒ = = = 4 in eq. (3.5). Assuming non-degenerate statistics, ) = 2.463  – 10 – −4.282 n ν η . Vertical axis is the relative change to a non-degenerate 4ρ ν 3n .o. /dt τ f (Stand = (Stand n, (SI ν 3T )| e µ < . Conversely, at low or ν .o. ν ν ν f dT > | cm µ ∂ρ ν ∂n = = = T 2 η i i i ∂ρ ∂n ∆(aT , as both elastic scattering and electron-positron annihilation over ν ν T 0 by examining eqs. (3.5) and (3.6). The denominator in both equations 5 0 5 0 5 0 1 ......

0 0 0 2 1 1

× −

i

δf 10 = 0. If we solve the numerators for the energy added per particle we find 2 . In the standard scenario of Boltzmann transport, the green dashed curve is the ν η . Freeze-out neutrino spectra in the standard scenario (dashed) and the SI In the standard out-of-equilibrium scenario, the deficit arises from the up-scattering of where we ignore the redshift term for plotted against Figure 1 flavor and the red dashed curve is either showing a slight increase in FD spectrum in eq. (3.14). where we ascribeproduct a is dimensionless unit and conversion equal to between unity scale when factor and temperature such that the which corresponds to a negative degeneracy parameter [eq. (3.9)] add energy andthe a behavior in modest figure number ofis particles positive to if the neutrino seas. We can explain low-energy neutrinos on higher-temperature chargedThe leptons, situation given schematically is in similar eq. for (3.8). SI JCAP07(2020)001 cm T (3.20) (3.21) over

can only

ν

× ν

T η

ν are non-zero. η 10 3 a spectrum lies in ν 0 5 0 5 0 5 5 0 5 ...... shows the evolution 2 2 3 3 4 4 0 1 1 0 ∂ρ/∂t| . 2 2 − − − − − − 0 − − − . When calculating the − ν (dashed red curve) plotted η and ν 10 η a . Figure and the quantity ν = 3.045, a small overall increase ν 1, implying a slight enhancement /T ∂n/∂t| e(µ)-flavor neutrinos. There is also 1 eff aT cm − N T , . ν ν 10 1. and ν η case would imply an increase in 3.288T 4.202T ν ' ' (MeV) 2.2. Although we evolve a non-zero degeneracy – 11 – 0 n µ-flavor in figure η cm 4ρ ν 3n factors given in eq. (2.10). The SI 10 . The product n, ) T 3T cm (k (dashed red line) from the initial values of zero to the T f ν η 1 e-flavor and ) from eq. (3.15) (solid blue curve) and shows that the occupation numbers at freeze-out differ between ν 10 1 for any flavor for a substantially negative aT 

, which is borne out by figure 0 5 0 5 0 5

ν ......

× η

0 2 2 1 1 0

5 in the standard scenario. As this is larger than the two values above, the ν ) aT ∆( 10 3 '  . ) (solid blue line) and illustrates the weak-decoupling epoch, starting at a few MeV, and finishing slightly . Evolution of ∆( cm compared to the result in section ν T 2 Neutrino self-interactions only indirectly influence the heat flow from the plasma into We have given the values at freeze-out for −3 between the two spectra(suppression) for for scattering between chargedan leptons enhancement and at low radiation energy density atof low 10 temperatures, we find the two scenarios. This isoccupation also numbers the and case during associated thelepton Pauli entirety scattering blocking of rates factors weak decoupling. through will the change Modifying the the neutrino-charged- the neutrino seas. Figure Figure 4 in ref.is [31] located shows at that the weak-decoupling induced distortionand in a the decrease neutrino in seas Figure after 100 keV. same energy per particle injection in the SI against we calculate of ∆(aT freeze-out values aschange a during weak function decoupling, of i.e., when the derivatives Figure 2 JCAP07(2020)001 ν )p /T − (4.1) (4.2) cm , e (3.22) (3.23) T e by adding n(ν eff scenario. We N . dr 46]. There exists a slight increase model in the last row of table 1 δ . The increase is less than 1%, so dr .o. 4/3 f µ,τ δ , introduced in [42] , . ν compared to out-of-equilibrium occu-  . T dr ν −4 −4 δ dr η cm T = δ . The increase in all abundances is due to T 10 10 e 4 cm −4  ν × × and T T 4 2 10 2 ν 4/3 T to calculate the changes to the neutrino spectra × ' '  120 7π – 12 – P dr 4 11 = δ δY  scenario. dr to be larger than ρ (D/H) = ν δ case [47]. (dr) eff (dr) eff agrees with the ∆N eff for the standard scenario. The difference results in small relative N in the SIν ()} e e ν , causingN ∆ ν , e ν 1/3 {f and D/H /11) to good approximation. P Li/H with a relative change of 2 Y 7 dr in eq. (1.5) δ scenario uses FD distributions with ≈ rad ρ We do a parameter-space scan over (dr) eff to dr After electron-positron annihilation terminates, weρ can calculate the change to scenarios. The onlyThe differenceν SI between the twopation scenarios numbers is in the neutron-to-proton rates. Both changes are well within observational uncertaintiesin [45, the ratio as there exists fewer 4 Dark radiation addition 4.1 Dark radiationThis parameter section and the succeedingbegin one by explore considering cosmological an extensionsdark addition to radiation of the dark only SIν radiationcomputation, interacts [48] with only to the the the universal otherdark Hubble energy radiation components expansion density. energy through rate The density gravitation. in by using eq. Within the (2.18) quantity the changes. We parameterize the Weak decoupling and QEDof corrections entropy to from the the plasma chargedlarger equation than leptons of (4 to state the modify photons. the The transfer result is a freeze-out ratio changes for an increase in the neutron-to-proton ratio which results from a decrease in the rate parameter, our result for of ref.]. [44 3.2.2 Primordial abundances The calculation for the primordial abundances is nearly identical in the standard and SI ∆N 4.2 Results 4.2.1 Neutrino spectra By construction, the dark radiationweak component has decoupling. no direct effect Whenan on adding neutrino earlier scattering extra and epoch radiation of energy weak density decoupling to precipitated the by universe, the we more expect rapid expansion rate. This and primordial abundances in the SI JCAP07(2020)001 3 P Y (4.3) . The solid dr δ uncertainty in 5 . in eq. (4.2). We are qualitatively 0 3, suggesting that but is not a factor ν .o. η f | dr comes from a change δ scenario but is directly = 1.0) in the following /T , and eff . dr cm −5 ν and close to the change in 4 T dr . T (δ 10 δ dr 0 eff δ × N 9 ' against the parameter 3 scenarios. The final freeze-out spectra . eff = 0) = 0) 0 N = 0.5) to dr dr plotted against (δ (δ dr dr eff eff eff (δ δ N N N eff – 13 – N 2 − − . 1, and the evolution of S4 0 - = 0.5) = 1.0) 3 and includes neutrino transport, QED corrections, and dr dr ) Planck ) CMB − eff eff (δ (δ eff 1 . eff eff eff (N (N 0 dr N N σ σ ∆N δ N scenario. The dashed green curve is Figure 3. Change in . We emphasize this analysis is for the SIν − ≡ ν eff 2 N eff 0 . we plot four quantities related to from neutrino transport/QED corrections and dark radiation add linearly. 0 2 1 0 6 5 4 3 precipitates an earlier epoch of weak freeze-out between the neutrinos and ...... 3 eff 0 0 0 0 0 0 0 dr N δ The blue and green lines appear to run parallel to one another in figure In figure from the addition of the dark radiation. The horizontal lines give the 1σ from Planck (dashed red line) [43] and a projection from Cosmic Microwave Background eff eff applicable to the standard scenario. 4.2.2 Primordial abundances Increasing baryons, and results in a higher neutron-to-proton ratio during nuclear freeze-out. Both conclusion is identical forlook either similar the to standard the or blue SIν curve in figure indicating that the degreeare of parallel non-linearity to is high small precision. and the The non-linearity blue grows and with green increasing curves in figure N (CMB) Stage IV (dashed black line) [49]. changes to in the expansion ratelowers which the leads amount to ofboth an heat the earlier flow from energy epoch the density ofhave plasma in verified weak into the this decoupling the and neutrino behaviormanner neutrino subsequently seas by seas. comparing and This also effect the changes ratio for modest values of This conclusion is indeed theIn case models to good with precision dark and radiation, is also the true non-linear in the contribution standard to scenario. ∆N the same for the scenarios with and withoutblue dark curve radiation. gives ∆N N dark radiation in the SI JCAP07(2020)001 with (5.3) (5.1) (5.2) (4.4) . Transport and self- = 0.4009), = 0.4009). eff N dr dr scenario by introducing an . Equation (5.3), however, ν t-channel scattering between T 52] shows the hyper-sensitivity scenario with asymmetric initial = ν T e, µ, τ, , i = ν, ν as it is a comoving invariant throughout = 2.161% (δ n i + γ P ? ν − n ν , i L i δY i ν ν ↔ 3 n to describe the asymmetry for any flavor. cm n (D/H) = 5.385% (δ – 14 – T ν δ ? ν = − L and D/H with relative changes from the standard 2 + (3) i i π ν ν ν 2ζ P n L ⇒ ⇒ Y . We use = 0 for all flavors at the commencement of BBN. We will = ≡ T i ? ν i ? ν = L L −5 cm 10 T × taken in eq. (4.4). Figure 10 of ref. [ = 0.4009 dr δ dr .769 δ . The Lagrangian in eq. (1.1) allows for ν = 0.2532 = scenario of flavor-violation and lepton-number conservation in this section. η P .202 is the Riemann Zeta function of argument 3. The above definition is ν Y D/H = 2 and ν 51] for more information on extra radiation energy density in BBN, and in particular T (3) = 1 ζ We highlight that eq. (4.4) shows that deuterium, not helium, has a larger relative /H to extra energy density. The reason for the result in eq. (4.4) is the dark radiation = 0. We give specific values of dr holds the twodoes seas not in hold thermal the equilibrium, two implying seas in chemical equilibrium. As a result, our model of SIν consider the SI conditions. In general, we neednamely, temperature and degeneracy quantities for the anti-neutrinos, Therefore, we only use a single quantity electron-positron annihilation and label iteq. the (5.2). comoving lepton If number self-interactionslepton to violate distinguish number flavor it conservation, to from than belepton we number, the would we expect would same expect the in comoving all three flavors. Additionally, if self-interactions violate 5.1 Differences inNeutrinos implementation and with anti-neutrinos will symmetric evolve scenario uniquely in the SIν identical with the standard definition of lepton number baseline for the case where at high temperature where neutrinos and anti-neutrinos, so the interaction change with the additionspecific of to dark the radiation. value of In fact, this is a general trend in BBN and not and D/H increase in therefs. [50, presence of an over-abundanceref. of52] [ neutrons. for We directinteractions quantitative the scaling do reader little forms to to ofδ modify the the abundances scaling with forms ∆ and instead renormalize the values when of D precipitates an earlier freeze-outD/H, of any nuclear BBN reaction scenario which rates. includeson a Given deuterium dark the production. radiation precision component must in consider measuring the impact 5 Lepton asymmetric initialWe give conditions a second example ofasymmetry a between one-parameter numbers of extension neutrinos to the and SIν anti-neutrinos with the following quantity [53] JCAP07(2020)001 t- (5.6) (5.7) (5.4) (5.5) follow 4. The ν . ρ . The set of  , . ν and clearly shows w ) are identical η E E

ν 4 ν dt dt , we find in figure d d ν ρ w , η ∂t + ∂ρ ν −3 (T /dt| 10 ν,η )]} )]} − ν ν ν ν n × n η η 2 ν dρ n a . These two curves evolve

(; (; − ν cm and T w ) and ∂t (eq) (eq)

∂n ν = 2.897 w ν [f [f )] ? ν , η + + ν ∂t ν e e ∂ρ L n /dt| − − ) ν (T e e ν,η 3 as the weak-collision terms conserve − ν C C ξ n dρ n ν a ν 2 + −1 n (n )] + )] + ν )] ξ  /∂t| ν ν 2 n ν ν η η ν,η ν (π n n ∂n (; (; ν 2 ν n (3) + 9T 1 = – 15 – (eq) (eq) + calculation with  a 12ζ [f [f w ν

) + 9T ν,η νe νe ν = n /∂t| ν,η . 2 ν ∂t /dt ν ? ν {C {C ∂ρ n 3 3 E L ν ν (n −ξ d ∂n ρ ρ ν ν d  d  = n − 9T ν Z Z w . In addition, the weak interaction ensures neutrinos and anti- η − −

= 3 for three flavors of neutrinos/anti-neutrinos. T ) ν 3 3 ν ν i=1 i=1 ν,η X X ∂t = ρ and ∂ρ n N 2 2 4 4 ν ν ν ν ν show the evolution of the degeneracy parameters for neutrinos (solid ξ + ρ ρ T T T evolve in accordance with the definitions of energy and number density 2π 2π ν ν 4  = } (ρ = = ν ν η ν,η a a [4(n , η ν,η

n ν ν follow from eqs. (3.5) and3.6). ( n ν ν ν,η 3T ν , η ∂t ∂t interaction will be much more efficient and will thermally equilibrate the two ν,η ∂ρ ∂ρ ν η  n ν {T +4n × = [4 , and Initially at high temperature, neutrinos and anti-neutrinos are in thermal equilibrium The asymmetry in the system implies there will be heat flow between the neutrinos ν E η dt d , ν lepton number. ThoseT number-flow terms are analogous to eq. (3.10). The derivatives for 5.2 Results 5.2.1 Neutrino spectra We give an example of antwo asymmetric curves SI of figure blue curve) and anti-neutrinos (solidsimilarly red albeit curve) as with a a function noticeable of offset due to the initial asymmetry. Figure with the plasma and variables to eq. (3.2), except now In writing eq. (5.6), we have set neutrinos are in chemicaland equilibrium opposite and in sign. have degeneracy The parameters equation equal in magnitude relates the comoving leptonlation number by to setting the degeneracy parameter [23]. We begin the calcu- and anti-neutrinos. Although the weak interaction can precipitate such a heat flow, the lepton degeneracy only requires the introduction of onefor new neutrinos variable, and namely anti-neutrinos. The expressions for from analogy. channel SI seas instantaneously. Toterms account with for an equilibration this term fact, we need to supplement the weak collision The equilibration term is ainteraction. weighted average If of we the group energy the and weak-collision number flows terms due into to the weak JCAP07(2020)001 ν 5 = 0, T ? ν < L . The ? ν ? ν . Both ? ν L L would be L once weak to examine 4 ν ? ν 2 0. If L −η − > 6= scenario. Figure 10 ν ν ? ν 0 and the red curve ν L η = = > i i ? ν L . ? ν 2. L 1 for − P 3 Y − 10 10 in figure as its behavior is similar to that of . The comoving lepton number is ν ν × T T cm T 5, but the other abundances have similar (MeV) 897 . 0 cm 10 T – 16 – = 2 in figure ? ν only has sensitivity to the magnitude of P L is immune to the sign of 2. The paucity of anti-neutrinos leads to a suppression eff Y N eff as a function of does change in an asymmetric scenario as this system has N system and the results corresponding to figure ν eff 1 0. Overall, the increase in the anti-neutrino scattering rates ν, N |, where the blue curve gives 10 ? ν CP = |L , i /dt > i E η d 0 2 4 6 8 6 4 2 . Therefore,

? ν 10

, the primordial abundances are indeed sensitive to the sign of 0. We only show − − − −

× − eff

i L < η

10 N 3 ? ν L . as a function of −3 for P 10 P Y Y × We have shown results for the neutrino parameters in the case conjugates. The quantity in the lepton-asymmetric case differs only slightly from the lepton-symmetric case. The ν decoupling begins. We do not show the evolution of 2.897 the composite neutrino/anti-neutrino system is not inin the equilibrium lepton-symmetric as case in figure of the Pauli blockingscattering factors rates in ofneutrinos the charged enhances weak leptons the collision with Paulieffects terms, anti-neutrinos. blocking together factors and imply Conversely, and therefore the decreases an those overabundance increase scattering of rates. in The the two Figure 4. Evolution of and the decrease inT the neutrino rateslepton-asymmetric nearly case cancel does and inducethe the a difference larger end is heat below result flow the is than resolution the of the evolution the lepton-symmetric of plot case, for but the larger energy densitysymmetrical from for degenerate FD statistics and neutrino energy transport are shapes as seen in figure 14 in ref. [53]. In particular, D/H follows qualitatively the same then we are in the opposite 5.2.2 Primordial abundances As opposed to neutrino spectra are inputstion into of the neutron-to-proton the rates neutron-to-protonhow and ratio. the therefore We change lepton execute the asymmetry ashows evolu- affects parameter the space primordial scangives over abundances in the SIν a larger energy density. However, CP JCAP07(2020)001 at all shows P 2 Y . Although the scenarios do not 3 has the potential 1 P − Y 10 scenario and is directly scenario. ν ν 1/2 as compared to [49]. Self-interacting neutrinos ∼ eff N | ? ν L 2 [eq. (5.1)] in the SI | − | ? ν 10 |L – 17 – 0 0 . Our conclusion is for the SI shows that these deviations from equilibrium are small, would be difficult to reconcile with current observational eff > < P 1 N ? ν ? ν Y L L and has a relative change of plotted against 5 than P Y ? ν 3 L − 10 22 20 18 34 32 30 28 26 24 ......

0 0 0 0 0 0 0 0 0 Figure 5. P Y . ? ν = 0.06, which is consistent with ref. [43]. We conclude that L 0.073. A 10% change in eff ' Before we conclude, we provide an example model with specific values. If we desire to We have extended our BSM model to include self-interactions and dark radiation. With ? ν L to be a betterapplicable probe to for the standard scenario extension with lepton asymmetry. 6 Conclusions 6.1 Summary Self-interacting neutrinos constitute a modelimplications of for Beyond Standard BBN. Modelin Neutrinos (BSM) the physics go with standard out scenario. of thermal Figure and chemical equilibrium during BBN bounds [45]. The extrato energy ∆N density from this value of comoving lepton number translates although potentially detectable in future measurements of values of physics governing neutrino interactionspredict is any different, substantial thefuture differences standard precision and [see for SIν eqs. cosmological (3.22) observables/parameters and3.23)]. ( atthe present inclusion of or extra radiation energyand density, weak less decoupling terminates heat atshows flows an earlier the from epoch change the in electromagnetic neutrino plasma energy to density the caused neutrino by seas. the Equation earlier (4.3) freeze-out is small. As decrease the helium mass fractionof by 10%, we would need a positive comoving lepton number shape as the curves in figure have little influence onenergy the among magnitude the of flavorsthat while the self-interactions preserving deviations, modify the and general the instead shape spectral reshuffle of the parameters the neutrino by spectra. a Figure few parts in 10 JCAP07(2020)001 3 ν will , and ? ν is large L ϕ D/H} behaves in . Although , P eff −3 ,Y shows different N 10 eff 4 × changes with the {N eff N . As a result, predictions scenario, = 2.897 3 ν ? ν L shows how gives the primordial mass fraction 3 5 is for the SI 3 while self-interactions simply translate the scenario where is a complex scalar; (4) the mass of ? ν ϕ in eq. (1.2), is agnostic to any particular flavor; L ij – 18 – g . Our impetus for this decision is two-fold. First, ν scenario. Lepton degeneracy sources an increasing scenario to include lepton asymmetry. In the class of to zero, then we would have needed separate parameters models for this work. Namely, we operate in the following ij scenario which amount to less than a 1% change from the g -symmetric; (3) ν in the SIν CP ? ν L forced all flavors of neutrinos to have the same distribution. Had ij g with decreasing (increasing) P Y for each flavor. Given how little self-interactions changes cosmological parame- i ν η We chose a limited class of SIν Our choice for Our intent for this work was to quantify how self-interactions, and solely self- Finally, we extended the SIν framework: (1) the interaction matrix, He as a function of and 4 i ν we set the off-diagonal terms of presents the results for the SI scenarios occur when we addBBN dark does radiation or not leptonneutrinos. provide asymmetry to any BBN. additional We conclude constraints that on aSIν scenario with only self-interacting compared to BBN energyBBN. scales; We and contemplate finally the repercussions (5), the if we neutrinos relax stay each self-coupled one throughout of theseT limitations in turn. ters/observables, we do not expect this class of models to alter our conclusions significantly. standard BBN scenario. The same level of relative changes between the standard and SI interactions, modify the neutrino spectra and concomitant light-element synthesis. Section either place stronger constraintskinetic or regime reveal and the signatures ability ofoperating to make new during measurements this physics. which epoch provide Coupled diagnostics provide of together, researchers the with the physics a tool to explore new physical theories. (2) the self-interactions are the neutrinos experience kineticlarge evolution enough during to this maintainneutrino epoch, equilibrium physics as manifests nor during scattering small BBN, standard-model enough ratessignals. physics to are will not preclude neither Second, be thermal able cosmologicalthe contact. to parameters/observables erase BBN If these directly epoch new influenced have by already been the measured. physics of Advances in precision on (decreasing) curves to slightly higher values. 6.2 Discussion The motivation for this workdial was nucleosynthesis to in determine the the dynamics presence of of weak SI decoupling and primor- self-interactions produce higher-orderof effects. Figure on cosmological parameters/observables mainly scale with the presence of a non-zero quantitatively the same manner for the one-parameter extensionmodels to we the explore, standard the scenario. self-interactionsthe always neutrino maintain and flavor-identical distributions anti-neutrinoin among seas. the The lepton flavor-independence numbersdifference of necessarily the between being self-interactions this results identical scenarioneutrino to and and one the anti-neutrino another components ones for can with the have symmetric three different initial spectra. flavors.the conditions degeneracy Figure The parameters is undergo that differentthe the evolutions net compared change to from the the initial symmetric starting scenario, point is still a few parts in 10 evolutions for the degeneracy parameters in a SIν a result, self-interactions perturbwithout the the neutrino additionaddition seas of of in dark the much radiation, which thethe dark is effects same nearly radiation. from an manner incoherent weak as sum decoupling. Figure of they the Although did extra figure energy density and JCAP07(2020)001 58]. The SIν models is for a ν while the neutrinos to resolve cosmolog- 1 ϕ. For Majorana character, L ν , then we might expect some . Moreover, had we introduced R µµ ν eff g N scenarios which remedy the Hubble 6= ν ee g – 19 – -symmetric, then models which use SIν CP shows there is little difference between these two kinds of distributions, so 1 interactions but with the addition of dark radiation. Alternatively, if we were to . The spin-flip interactions would have to freeze-out well before BBN [59] for a eff N Returning to the scalar-mediator scenario, we only considered the case when the mass In addition to all flavors being self-interacting, SI In our analyses of neutrino-anti-neutrino asymmetry, we have always made the assump- model with right-handed neutrinos to be viable. As a corollary, BBN would proceed SIν without SIν consider vector mediators, then leptona number symmetric would system. not We be dodecoupling conserved not if anticipate and the any we mediator further would implications is expect for a either vector. BBN or Another photon possibility in the class of SI of the mediator is much larger than BBN energy scales. If this were not the case, then it is left-handed states couldthe undergo processes spin-flips which equilibrate to thepersist the neutrino/anti-neutrino in seas. right-handed such states A aopposite lepton at scenario. asymmetry helicity the cannot states. If same These neutrinoshave states rate are ramifications are as Dirac inactive on under fermions,raise BBN the then weak as the interaction, but they spin-flip would populates contribute still the to the radiation energy density and would pseudoscalar mediator. In thisSome scenario, late neutrinos portion may ofBBN recouple would recombination after occur could weak with decoupling transpire out-of-equilibrium [60]. with distributions equilibrium identical neutrino to the spectra, standard but scenario. again we expect littlethere effect on exists parameters. lepton What asymmetries.then may lepton be numbers If can different the be wouldinant self-interacting different be effect for Lagrangian the during different case is flavors,severe BBN where and indeed repercussions [54–56]. oscillations on flavor would neutrino biased, play Furthermore, free-streaminggas a during the dom- is atomic flavor-asymmetric only recombination, scenario if partiallytrino the could self-coupled. oscillations, neutrino but have Such wefree-streaming can a give neutrinos scenario a cause requires qualitative a a description here. calculation phase with In shift the vacuum in standard neu- the cosmology, CMB acoustic peaks [57, tion that the netchange. lepton For the number Lagrangian (i.e., into eq. include the (1.1), right-handed sum this neutrinos, is with of certainly an the true. operator three such We as could flavor alter lepton the numbers) Lagrangian cannot Table 1 in ref. [44] appears to bear outflavors this of conclusion neutrinos for tofashion. stay Figure in self-equilibrium and others to evolve in an out-of-equilibrium gas eliminates that phasetemperature shift power-spectrum and defect. utilizesthe If free-streaming extra neutrino not dark component all radiationdark would the induce to radiation a neutrinos remedy would small were necessarily thisincrease phase to the be shift, particular be Hubble and smaller. parameter the self-coupled, only amount At then of slightly. the currentparameter epoch, tension the also effect stipulateLagrangian would the in be eq. anti-neutrinos (1.1) to to isical be not tensions self-interacting. will also If confrontabove. the the interacting From same the problems as perspectivethen the the of flavor-asymmetric anti-neutrino BBN, models spectra if discussed followed would anti-neutrinos the follow do solid the line. not dashedin remain Again, lines cosmological in we in parameters do a figure related not self-equilibrium, still expect to would the expect BBN. asymmetry equal Withvor to lepton the content cause numbers introduction in significant in of the changes allnumbers neutrino a flavors of sector as lepton neutrinos and self-interactions number, and standard will we anti-neutrinos model change such the interactions that fla- will all then have equilibrate identical the chemical potentials. a flavor asymmetry in the interaction matrix, i.e., JCAP07(2020)001 may ϕ φ, during + φ ϕ-sea will heat → system. Second, ν + –ϕ ν ν scenarios. In this case, ν in eqs. (3.5) and ()3.6 no longer ν η and utilizing Majorana neutrinos can D/H}. The more interesting scenario and , ϕ ν P m T ,Y eff {N – 20 – framework presented in this paper, is if we remove ν . The evolution of ν /H, are sensitive to this entropy flow. Given the precision η and would not be able to alleviate the Hubble tension. As the F . This extra energy density is not dark radiation and will have G eff ∼ N [62] to be constrained. Future tonne-scale double detectors will eff ϕ-sea acts as a heat bath for neutrinos without conserving number. With a G /H, BBN has the potential of putting stricter limits for the “weak-self- −3 and vanishing ν 10 T system which will become only neutrino and therefore increase the radiation g > –ϕ ν -width measurement obviously provides a stringent limit on the triplet versions but 0 There can be other, non-cosmology constraints on some self-interacting sterile neutrino Neutrino physics can play a significant, even dominant role in the dynamics, evolution, Our final retrospection on the SI ϕ-sea persists to times after weak decoupling, then there will be extra energy density Z would be for the caseBBN. of If mediators with the massesstill self-interactions low be freeze-out enough present to prior and haveequilibrium non-zero to decay densities process into or during high-energy and during neutrinos. wouldweak BBN, decoupling, Such require the there a a exists decay population transportseas the is of calculation. into possibility necessarily the that an If electromagnetic energy out-of- plasma,The this and opposite entropy decay abundances, flow of occurs from the especially during theon direction D neutrino in measuring standard weak D decoupling. produce a unique neutrino-lessdecay double nuclear beta mass decay differenceThis signature. is distinctive shared phase This space among is signaturecouplings two where has electrons allowed the Majoron-like and double models an beta [17–19] undetected with scalar effective [61]. interacting” case than the stronger case considered in thismodels. work. Such models with low enough scalar mass have higher sensitivity andthe therefore likely will benot able the to singlet extend versions of theseconstraints, the constraints. for Majoron-like example Finally, models on discussed four-neutrino above. couplings There [63]. can beand other such nucleosynthesis in some compactcollapse object supernovae and environments. binary Theseinteractions include, neutron may have for star dramatic example, effects mergers. core inwhich these Not can venues. complement surprisingly, These those BSM effects derivedwork. neutrino could from lead self- However, the to the constraints early origin universerooted of considerations in some discussed the in of same this self-equilibration the and potentially flavor constrainable changing physics compact that object affects effects weak decoupling is the stipulation that theof neutrinos stayed models in where self-equilibrium throughoutself-interactions BBN. only This is reshuffle aobservables neutrino class to energy be density, we somewhere would between expect the changes standard to and BBN full SI the neutrinos to higherneutrino temperatures. temperature will Ifhelium. cause this a occurs lower prior neutron-to-proton to ratio helium formation, and the a higher smaller amount of we would not expect significant changes in energy density and ramifications on temperature power spectra [22]. Lastly, the energy in the in the apply, and we wouldif have a to construct an analog of eq. (2.9) for the heat bath and strong coupling, thetemperature neutrino seas are in thermal and chemical equilibrium with possible for the mediator toBBN. be produced As on shell, we throughthroughout reactions take BBN, like the light mediators self-interactionsseas until will strong the continue neutrinos enough to decoupleof from remain to this. themselves in at ensure First, late equilibrium the times. neutrino with There self-equilibrium the are three neutrino ramifications JCAP07(2020)001 P Y /H and show P 5 Y and D P and Y 4 are to be used /H observable, ? ν scenario, picking L ν and the abundances and eff dr δ (N ? ν L species, opening phase space for τ 64]. Moreover, after core bounce, the and µ – 21 – Li, as the sign of the relative abundance changes is 7 [72]). However, the results of sections ? ν . Very quickly the electron lepton number residing in L e ν + He and 3 and other neutrino cosmology parameters (i.e., the sum of the , our analysis applies to other BSM scenarios. Cosmological n ν — or any other BSM scenario which must reconcile eff → and D/H. At this point, other physical mechanisms would need to N ν p P Y 0. This can be remedied using a positive lepton number, as both + − > e mediator for neutrino entropy creation during BBN could change dr δ ϕ — will narrowly constrict the available parameter space. eff N In closing, we mention the importance of deuterium in BBN constraints. Although be invoked to mitigate internal BBN inconsistencies. Returning to the SI different compared to that the two abundancesin do not concert. have identical Intensions scaling other in relations words, if abundances. darkchanges radiation Given of even and the a lepton few precision percentwith asymmetry in could precise cause cannot measuring a measurements together strong the of tension. assuage primordial The D situation would be even worse a mass of the D/H such that theSuch a two requirement scaling on relationshipswith SI are identical when dark radiation is included. Acknowledgments We thank Luke Johns,and Amol Patwardhan, Julien Andr´ede Froustey Gouvˆea,Yue Zhang, forwork Walter Tangarife, useful for discussions. Neutrinos,by Nuclear This National Astrophysics Science research Foundation, and was grant2017-228. PHY-1630782, conducted Symmetries and through collaboration We Heising-Simons also the Foundation, and grant acknowledgeof Net- is NSF California supported grants San Diego. PHY-1614864Laboratory and This Institutional PHY-1914242 Computing research Program, used at which resourcesEnergy University is National provided supported Nuclear by by Security the the Administration U.S.We Los under Department thank Alamos of Contract the National No. referee 89233218CNA000001. for their helpful comments. models which positthe extra abundances. radiation In energy particular,radiation, density helium i.e., during and BBN deuterium increase [68–71] if necessarily that change energy density is dark the topic of this work is SI neutrinos can be “cooled”This by could adiabatic lead expansionthe of to IMB the a strongly [65] constraint self-coupled andsupernova if neutrino SN1987a Kamiokande [67]. gas. II neutrino [66] energies experiments are which reduced detected below the the neutrino burst thresholds from for the degenerate seain of dramatic electrons alteration is of redistributed the and physics entropy of is collapse generated. [3, This can result and D/H decrease, while scale differently for large values of light neutrino masses) are little changed by modest values of , and BBN in the early universe.difference between When compact it comes objects to andin neutrino the the self-interactions early and former flavor, universe venue. the is key Forentropy the example, and very highly the low degenerate collapsing entropy electron cores per lepton ofneutrinos. number baryon supernovae carried are by seas When characterized of by the electrons low heavy and neutrinos electron nuclei, become self-interactions trapped canthe by do neutrinos, coherent two very neutral things:and much current (2) like rapidly (1) scattering re-distribute the effect on theflavors. self-equilibration rapid degenerate electron in self-equilibration Electron lepton the among number neutrinos among early can all universe three be neutrino discussed converted above; to JCAP07(2020)001 B Nucl. 07 , INSPIRE]. [ (2020) Phys. Lett. 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