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Probing neutrino physics with a self-consistent treatment of the weak decoupling, nucleosynthesis, and photon decoupling epochs

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/ournal of Cosmology and Astroparticle Physics

An IOP and SISSA journal

Probing neutrino physics with a self-consistent treatment of the weak decoupling, nucleosynthesis, and j

photon decoupling epochs A

E. Grohs,®'1 George M. Fuller,® Chad T. Kishimotoa'b O

and Mark W. Parisc

"Department of Physics, University of California, ^—^

San Diego, La Jolla, California 92093-0319, U.S.A. ^Department of Physics, University of San Diego, San Diego, California 92110, U.S.A.

cT-2 Nuclear & Particle Physics, Astrophysics & Cosmology, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, U.S.A.

E-mail: egrohs@physics.ucsd.edu, gfuller@ucsd.edu, ckishimo@physics.ucsd.edu, " ^

mparis@lanl.gov

Received February 12, 2015 Accepted April 16, 2015 Published May 11, 2015

Abstract. We show that a self-consistent and coupled treatment of the weak decoupling, big bang nucleosynthesis, and photon decoupling epochs can be used to provide new insights and constraints on neutrino sector physics from high-precision measurements of light element abundances and Cosmic Microwave Background observables. Implications of beyond-standard-model physics in cosmology, especially within the neutrino sector, are assessed by comparing predictions against five observables: the baryon energy density, helium abundance, deuterium abundance, effective number of neutrinos, and sum of the light neutrino mass eigenstates. We give examples for constraints on dark radiation, neutrino rest mass, lepton numbers, and scenarios for light and heavy sterile neutrinos.

Keywords: cosmological parameters from CMBR, big bang nucleosynthesis, cosmological neutrinos, neutrino masses from cosmology

ArXiv ePrint: 1502.02718

1 Corresponding author.

^ I Content from this work may be used under the terms of the

Creative Commons Attribution 3.0 License. Any further «-l,-,!.

distribution of this work must maintain attribution to the author(s) and aOLi0.!088/!475-75!6/20!5/05/0!7

the title of the work, journal citation and DOI.

Contents

1 Introduction

2 Background

2.1 Big-bang nucleosynthesis

2.2 Cosmic Microwave Background

2.2.1 Sound horizon

2.2.2 Free-electron fraction

2.2.3 Photon diffusion damping length

3 rs/rd as a proxy for Neff

4 Verification of cosmological parameters with dark radiation

5 Examples for neutrino sector BSM physics

5.1 Neutrino rest mass

5.2 Sterile neutrinos

5.2.1 Light sterile neutrinos

5.2.2 Heavy sterile neutrinos

5.3 Lepton numbers

5.3.1 Effect on nucleosynthesis

5.3.2 Effect on Neff

6 Conclusions and outlook

20 22 22

1 Introduction

In this paper we demonstrate how a self-consistent treatment of the physics of the early universe (from temperature T ~ 10MeV down to T ~ 0.1 eV) enables Cosmic Microwave Background (CMB) observations, in concert with light-element abundances, to be used as new probes of beyond-standard-model (BSM) physics in the neutrino sector. In a sense, these probes are tantamount to probes of the CvB (relic Cosmic Neutrino Background). Recent observations [1-4] of the CMB and observationally-inferred primordial abundances of deuterium and helium [5-8] formed in big bang nucleosynthesis (BBN) already place tight constraints on both the cosmological standard model (CSM) and BSM physics. However, future observations will usher in a higher level of precision with even better prospects for BSM probes, see for example ref. [9].

We anticipate that observations will bring about an overdetermined situation where BSM physics may manifest itself if it is present. The existing or anticipated observations and measurements of greatest utility for the present purposes are: (1) high-precision measurements of the baryon-to-photon ratio, or the equivalent baryon density (wb = Qbh2);

(2) high-precision measurements of the "effective number of relativistic degrees of freedom" (Neff); (3) high-precision measurements of the primordial deuterium abundance (D/H) from quasar absorption lines; (4) measurements of the primordial helium abundance (YP) directly from CMB polarization data; and (5) measurements of the sum of the light neutrino masses (^ mv), i.e. the collisionless damping scale associated with neutrinos.

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The physics that determines the relic neutrino energy spectra in weak decoupling and that of primordial nucleosynthesis affects observables of the CMB. These distinct, disparate epochs, however, depend not only on the values of parameters describing the cosmology (such as Neff, relevant cosmic constituents, etc.) but also on the parameters derived from these base, input parameters (such as primordial abundances, recombination history, etc.). This is the basis for what we will term "self-consistency." The requirement for self-consistency between the BBN and CMB epochs, for example, in current data analyses depends on the parametrization of the energy density in terms of Neff and upon the baryon-to-photon ratio. These two parameters, as measured at photon decoupling, are the sole determinants of the primordial helium abundance at the epoch of alpha particle formation (T ~ 0.1 MeV) in "standard" (i.e. zero lepton numbers, no BSM physics) BBN calculations. Though this | procedure is adequate for the CSM and standard model physics, here we argue that it can ^^ be insufficient to probe varieties of BSM physics when confronting model cosmologies with next-generation, high-precision CMB and light element abundance data. In the case of the helium and Neff self-consistency mentioned above, the relationship between the helium yield in BBN and Neff is a non-trivial function of the interplay of expansion rate and neutron-to-proton ratio, as is well known [10]. The latter ratio is, in turn, a sensitive function of the Ve and Ve energy distribution functions, and these can be affected by BSM issues like lepton numbers, flavor mixing, sterile neutrino states, heavy particle decay, etc. ___

Handling and analyzing the observed data while imposing self-consistency over multiple epochs in the early universe can require new procedures. The approach developed in ref. [10] and employed in Planck XVI [11] has been highly successful. There, self-consistency is

imposed approximately by the addition of a term in the log-likelihood function. A very small |—^ theoretical uncertainty ensures the posterior distributions are close to the corresponding theoretical values. However, when we impose self-consistency among the different epochs the need may arise to solve for the various derived parameters iteratively. Returning to the helium/Neff example discussed above, YP and the recombination history of the universe both C )

depend on the radiation energy density, usually parametrized by Neff. Subsequently, the |_

recombination history is affected by the the primordial abundance of helium. The values of these two derived parameters, in turn, affect CMB observables, which recommends an iterative and therefore self-consistent approach.

Ideally, we would want a procedure to self-consistently treat neutrino and BSM physics from the post-QCD epoch to the onset of non-linearities in large scale structure (LSS). Recently, the quantum-kinetic equations (QKEs) governing neutrino flavor evolution in dense environments have been derived from first principles [12-14]. Such a program to treat neutrino physics in the early universe would incorporate a QKE treatment through weak decoupling at the very least. Weak freeze-out and BBN necessitate a neutrino-energy Boltzmann-equation or QKE treatment fully coupled to a nuclear reaction network. The recombination, photon decoupling, and advent of LSS epochs require a Boltzmann-equation treatment of neutrino clustering. Verification of any models related to neutrinos and BSM physics would then rely on agreement with direct cosmological observables, including but not limited to: primordial abundances; CMB power spectra; and the total matter power spectrum.

As outlined, this is a challenging undertaking. We therefore employ a limited approach to show why self-consistency is required and how such a treatment can be efficacious in probing some issues in neutrino sector physics. To that end, we calculate the ratio of sound horizon to photon diffusion length, rs/r¿, as a simple parametrization of the CMB, as described in detail in section 3. It would, of course, be preferable to compute the full CMB

power spectra in the self-consistent manner described above. There are two limiting factors that temper this ambitious proposal, however. The first is that the observable rs/rd is largely insensitive to the poorly constrained equation of state of the dark, vacuum energy component while the CMB power spectra are not. Additionally, current computations of the CMB power spectra [15] would need to be generalized to the BSM scenarios we contemplate here.

Our procedure for calculating derived cosmological quantities utilizes the neutrino occupation probabilities. Since we are developing the capacity to compute the effects of BSM physics that couples to the active neutrino sector, we do not restrict the form of the neutrino distribution functions to that of equilibrium Fermi-Dirac distributions. General forms are permitted and may be handled analytically or numerically. Neutrino occupation probabilities are taken in the present work in the flavor eigenbasis during weak decoupling, weak freeze-out |

and BBN.1 During recombination and last scattering, we transform the occupation probabilities to the mass eigenbasis. When we consider the case of massive neutrinos, the statistic for the sum of the light neutrino masses, mv, is simply the sum of the three lightest mass eigenstates.

We might further clarify the fact that this limited approach does not rely on the assump- .. tion that the radiation energy density can be described in terms of the single parameter Neff. Indeed, an original motivation for using rs/rd as a proxy for Neff (see section 3) is the fact

that the usual definition for Neff does not apply to non-equilibrium neutrino distributions. ___

A corollary of this observation is the possibility that Neff may depend on the scale factor a(t); that is, we do not assume that Neff is independent of scale factor. When extended, our approach is able to handle BSM scenarios such as the decay of massive sterile neutrinos and

other weakly interacting massive particles, which result in neutrino distributions that may differ substantially from a Fermi-Dirac distribution [16].

Our approach aims to be general; it does not rely on particular cosmological models but is appropriate to the class of A cold-dark matter (ACDM) models. It extends and explicates recent work [9, 10, 17, 18] on the consistent incorporation of precision observations of the f

CMB [11] and observations of the light element abundances of helium and deuterium (mass |_^

fraction Yp and relative abundance D/H). The deuterium abundance [6, 8] is now more precisely measured by a significant factor (4 or 5) in relative precision than YP. Our results

indicate that deuterium is sufficiently well determined to be incorporated into CSM analyses as a fixed prior.

The present work is a first result in an ongoing campaign to incorporate neutrino energy transport into a self-consistent treatment of BBN and recombination. Our approach is being implemented in Fortran90/95 as a suite of codes under the working title of "BBN Unitary Recombination Self-consistent Transport" (burst), which will be made publicly available for use on parallel computing platforms using openmpi. The current work is a prerequisite in an ongoing collaboration to develop codes that consistently handle BBN and neutrino energy transport from weak decoupling to the advent of LSS.

The paper is outlined as follows. In the following section, section 2, we discuss our treatment of BBN and relevant CMB physics. Section 3 explains the self-consistent, iterative approach to determining Neff using rs/rd. Section 4 employs a dark-radiation model to test the accuracy of predictions made by burst against observations. Section 5 investigates a number of 'test problems' from BSM physics suited for burst. We conclude in section 6 with future applications of the results given in this work. Throughout this paper, we use natural units where h = c = kB = 1.

1This is a matter that is properly resolved by utilizing a QKE approach [12], the subject of a future study.

2 Background

2.1 Big-bang nucleosynthesis

Recently there has been a dramatic increase in precision in the determination of D/H [8]. This improvement in observational precision drives the need to improve the standard tools used to calculate observables during the BBN and CMB epochs. This development allows us to consider improved descriptions of nuclear physics and non-standard particle and cosmo-logical models.

We have developed a generalized BBN subroutine, part of the larger burst code. The current BBN routine handles general distribution functions for particles out of equilibrium. It numerically integrates the binned neutrino occupation probabilities to obtain thermodynamic variables and number and energy densities for ve, Ve, vt, VT. It is worthwhile, therefore,

to summarize our approach to BBN even though it is substantially similar in some respects to that given in refs. [19-21].

We assume that the cosmic fluid is homogeneous and isotropic and that the comoving number density of baryons is covariantly conserved. Neutrinos can experience, in addition to gravity, essentially arbitrary interactions within the Boltzmann transport approximation.2 We allow for the possibility that neutrinos may decouple from the plasma with non-equilibrium distributions. This assumption implies that there may be deviations from the ^—^ Fermi-Dirac momentum distribution [22-24]. In fact, it was this observation that provided initial motivation for developing the current approach.

BBN codes evolve light nuclide abundances Y¿(t), defined as

yw=m• (21) 5

as a function of comoving time t in the background of the Friedmann-Lemaitre-Robertson-Walker (FLRW) geometry. Here n(t) is the proper number density of nuclide i = n, p, 2H,

H (a) = a = ^ p(a)' (2-2)

where H(a) is the Hubble parameter, mpl is the Planck mass, and the 'dot' indicates differentiation with respect to the FLRW-coordinate time t. The energy density is computed as the integral of the single-particle energy over the momentum distribution:

p(a) = f ^ f (p; a)\Jp2 + m2 (2.3)

where the sum over j reflects all species contributing to the energy density, including but not limited to: photons, baryons, dark matter, e±, and neutrinos. Consistent with the

3He, 4He,... and nb(t) is the proper baryon number density. We will focus on two specific |—

nuclides in this paper: the 4He mass fraction (Yp = 4Y4He), and the 2H relative abundance _____

(D/H = Yd/Yh). The evolution starts from a time when the plasma temperature T is near 30 MeV. Weak equilibrium obtains at this temperature. The time dependence of the metric is determined by the energy density p(a) as

2However, the code can handle limited generalizations of the Boltzmann approximation to incorporate

effects associated with neutrino quantum kinetics.

above discussion, we do not need to assume a well defined temperature for any of the cosmic species. The time dependence of the distribution function /¿(p; a) is indicated by the presence of the scale factor a(t) in its argument. Evolution in time of the distribution functions is accomplished by solving transport equations, such as the Boltzmann equation, in FLRW geometry.

Given initial conditions for the temperature T and momentum distributions of the cos-mological constituents (assumed to be in equilibrium through weak and strong interactions), the BBN code determines the evolution, with respect to the scale factor a(t) (related to time as dt = da/(aH(a))), of the temperature of the plasma T(a), the electron chemical potential ^e(a), and the nuclide abundances relative to baryon number Yi(a).

The interactions of the light nuclear species is governed by the nuclear reaction network.3 C_| The reaction network, which is determined by a chosen set of nuclides and the thermally averaged reactivities naina2 {a^ava}, is proportional to the rate of change of the number density of nuclides participating either in the initial state nai or final state where i,j indexes particles (up to 3) in the initial a or final 3 channel for the process a ^ 3; that is, ' ^ a and 3 are two- or three-body reaction channels and ai is the ith nuclide of the channel a. ^ The nuclear reaction network includes nuclides with mass number A < 9. Our code allows for the inclusion of additional nuclides with A > 9, but we maintain the smaller network as the larger network provides no new insights for this paper. The nuclides are taken to be in thermal equilibrium with the photon-electron plasma. We are currently employing an updated [25] version of the reaction network from ref. [21]. We also couple in all relevant e± and neutrino-induced weak interactions (charged and neutral current) [26].

2.2 Cosmic Microwave Background

3The nuclear reaction network should be derived from reaction cross sections that are governed by the

principles of quantum mechanics, such as unitarity. We are currently developing a unitary reaction network for application in future work.

The physics of BBN and photon decoupling are related through the epoch of recombination by several mechanisms including, importantly, He i and He ii recombination. The recombination rates depend sensitively on the amount of 4He created in BBN [10, 27, 28]. The 4He |—

dependence also determines the photon diffusion damping wave number kd or, equivalently, _____

the photon diffusion length, defined as rd = n/kd. This quantity depends strongly on the recombination history through the free-electron fraction, Xe(a), which, in turn, depends on the helium abundance (see section 2.2.3 below).

We relate the diffusion length directly to an observed angular size dd through the angular diameter distance Da at the epoch of last scattering (as described in detail below). Since Da depends on the vacuum (dark) energy equation of state, which is poorly understood, we employ the ratio ds/0d, equivalent to rs/rd (see eqs. (2.6) and (2.13)), where ds is the angle subtended by the sound horizon, which eliminates explicit dependence on the dark energy equation of state.

Modern computations of the recombination history of the early universe are available in accurate and well tested codes such as HyReo [29], CosmoRec [30], and the fast, approximate computations given in ref. [31] by the code RecFast [32]. Our recombination history compares well with RecFast, agreeing at the level of < 2% over the recombination epoch. The present accuracy suffices for the purposes of the current exploratory work.

2.2.1 Sound horizon

The sound horizon is defined as

/ra-fd 1

0 a2H (aV3(l + R(a))

where aYd is the scale factor at photon decoupling.4 The ratio R(a) is given as

faYd l

rs{ald)-Jo da a2h(aK/3(1 + R(a)) (2.4)

R(a) = ^ (2-5)

where pb and pY are the baryon rest mass and photon field energy densities, respectively.

The angular size of the sound horizon 0s,5 determined from the spacing of the acoustic peaks in the CMB temperature power spectrum, is related to the sound horizon by the angular diameter distance DA(ajd) at photon decoupling (scale factor ajd) as

SsKd) = «M-T^H • (2.6)

DA(ajd)

since the angle ds is small. The angular diameter distance is

f ao -1

Da (a) = a / da' [a/2H (a') 1. (2.7)

The quantity DA(ald) depends on the vacuum (dark) energy equation of state, which is not very well understood. Our approach will eliminate dependence on this poorly constrained component of the energy density.

2.2.2 Free-electron fraction

We have written an independent code to calculate the free-electron fraction, Xe. The results agree well with ref. [32] (RecFast). In fact, the agreement is within 2% for most of the range 10-4 < a/a0 < 10-3 (104 > z > 103). We have not included additional fit parameters, |— as in ref. [32], that modify recombination and ionization terms of the three-level treatment to obtain agreement with the full 300-level computation [31]. The code allows significant deviations from the model parameters of ACDM; it should be used with caution, however, since the effective three-level treatments for helium and hydrogen recombination have been optimized for near-standard model parameters.

The number density of free and total electrons is

n(free) = np + nHe „ + 2nHe m, (2.8)

ntot) = n6( 1 - Yf) , (2.9)

where nb, np, nHe n, nHe ni are the proper number densities for baryons, protons, singly- and

doubly-ionized helium, respectively. We write the free-electron fraction as

n(free)

Xe = -(tot) ^ XP + XHe II + 2XHe ill, (2.10)

so defined to take values in the range 0 < Xe < 1.

4Reference [11] takes a„fd ^ a* where the optical depth is unity.

5Reference [11] takes 0s ^ d*.

104 100

* 10-2

10-4 010

Figure 1. Free-electron fraction Xe as a function of scale factor a/a0 and redshift z (at top.) The different curves correspond to various values of The cold dark matter contribution is held fixed at = 0.12029. The vertical shaded bar is the scale factor at photon decoupling, aYd given by ref. [11].

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We follow refs. [33, 34] and consider the simplification of the multi-level hydrogen and helium atoms to that of an effective three-level system which includes the ground n = 1 state, the first excited n = 2 states, and the continuum. All other excited states are assumed to be in equilibrium with the 2s state. We treat He ii recombination approximately [31] (via the Saha equation) since it is essentially complete at the advent of the epoch of He i recombination. The He iii contribution, therefore, in the Boltzmann equation for He ii is negligible.

Boltzmann equations for H ii and He ii contain a thermally-averaged cross section and relative velocity {av). We use Case B recombination coefficients for the {av) of both neutral hydrogen [35] and helium [36]. Along with a Saha equation for He iii, the Boltzmann equations for H ii and He ii are a coupled set of ordinary differential equations constituting a recombination network to model the ionization history of the universe prior to photon decoupling.

Fig. 1 shows the evolution of the free-electron fraction with scale factor for various values of and concomitant values of YP. The vertical shaded bar is centered on ald, the scale factor at photon decoupling, given by ref. [11] for a best-fit-value ub = 0.022068. The recombination rate increases with increasing which results in a lower freeze-out value of the free-electron fraction at large a/a0. The recombination history is largely insensitive to the helium fraction but we calculate Yp for each value of u to maintain self-consistency between BBN and recombination. The values of YP are YP = 0.220,0.238, 0.243,0.246 for increasing ub.

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2.2.3 Photon diffusion damping length

In the tightly coupled limit, photon diffusion damping is characterized through the damping wave number [37-40]

2 = faYd da 1 R2(a) + H (1 + R(a)) (211)

Kd l0 a2H(a) ane(a)ar 6(1 + R(a))2 , (.)

where jt is the Thomson cross section. Here, we have assumed that moments of the temperature fluctuation higher than the quadrupole make a negligible contribution in the linearized Boltzmann equation for the photon distribution.

Equation (2.11) requires the free-electron fraction, ne(a), determined in the recombination history, as discussed in the previous section. The free-electron fraction, over the course of the recombination history, depends strongly on the primordial helium mass fraction Yp. The fraction YP, in turn, depends on the relativistic energy density, which is often parametrized

in terms of the parameter Neff:

pr = 2

7/4\ 4/3

1 + 7 (H)

—T4. (2.12)

30 ( )

Equation (2.12) is only valid after the epoch of e± annihilation. The point that BBN and recombination are related is well known [10] but has not been implemented self-consistently ISJ as a constraint for general, BSM physics model cosmologies. We return to it after describing f )

the relation between rs and kd to directly observable quantities given by the CMB power |_^

spectrum.

The observed diffusion angle 9d(aYd) [11] is related to the diffusion damping length, rd = n/kd as

dd(aYd) = aYd rd(a 7dl , (2.13)

DA(ald)

with the same stipulation regarding the smallness of the angle in eq.(2.6). 3 Vs/Vd as a proxy for Nff

In this section we describe our method for determining Neff in detail. We introduce two variants of Neff. When referring to the radiation energy density equation (2.12), which takes as an input the quantity Neff, we designate Neff as Neffh). When considering general cosmologies, perhaps with BSM physics, we deduce the value of Neff from the observable quantity rs/rd = ds/dd, described in this section, and designate it as Neff. The simplest cosmologies for which eq. (2.12) obtains, having negligible neutrino mass, standard model constituents and no energy transfer between species have Ne(ffh) =Neff.

We consider a test input cosmology that is non-standard yet substantively similar to ACDM. We proceed by determining Yp at temperature T ~ 0.1 MeV using the BBN network of burst [25]. The principal observational cosmological input at this time is Also input and incorporated into the BBN subroutine is any neutrino and BSM physics that constitute the test cosmology. Subsequently, we compute the recombination history of the universe from early times (a/a0 ~ 10-7) to the current epoch. Specific observational inputs include wc (where for cold dark matter wc = Qch2), YP, and H0 (the Hubble constant, H0 = H(a0)).

For the purposes of the present discussion, other inputs of particular importance include neutrino occupation probabilities (as output from a neutrino transport calculation of weak decoupling that is fully coupled to BBN) and neutrino rest masses. The neutrino energy density of the neutrino seas is calculated by writing the occupation probabilities in the mass eigenbasis [41]. We emphasize the fact that is not input as a base parameter; this is

of paramount import in the present approach. The recombination history, ne(a) determines the optical depth as a function of scale factor:

fao da!

t(a) = / —¡2 a'ne(a')aT (3.1)

We define the scale factor at photon decoupling ald such that t(ald) = 1. In this definition, we C_| do not include the effects of cosmic reionization when calculating ne(a) for use in eq. (3.1) [11]. We apply ald and the input cosmology to equations (2.4) and (2.11) to compute the sound horizon and photon diffusion length, respectively. We arrive in this way at a ratio (rs/rd)(inp) for our input cosmology.

Our immediate objective is to determine a value of Neff (here termed Nff corresponding Q to this value for (rs/rd)(inp). We map out a range of values of rs/rd that correspond to the same input cosmology as that used in calculating (rs/rd)(inp), with one significant difference. We parametrize all of the neutrino and BSM physics into the single Ne(ffffh) parameter. We then use to calculate the radiation energy density in eq. (2.12) to determine the Hubble ISJ rate. We vary Ne(ffffh) to compute the function C )

rs/rd = rs/rd Ub, Uc, Yp ,...; Ne(ffffh) , (3.2)

shown in figure 2. Since rs/rd is a one-to-one function of N^, we may invert eq. (3.2) to " ^ obtain Ne(ffffh) = NffWrd]. The final step is to evaluate the previous function with our input cosmology ratio, i.e. Neff = Ne(ffffh)[rs/rd = (rs/rd)(inp)], to obtain a value of Neff. As an example, we take the best-fit values from ref. [11] combined with WMAP Polarization data (1000s = 1.04136 & 1006>d = 0.161375) to obtain (rs/rd)(lnp) = 1006>s/1006>d = 6.45304. This corresponds to a value NVeff = 3.31 on figure 2, in line with the best-fit value Neff = 3.25 (3.51+°'f4 at 95% limits) [11]. We again note that for the simplest cosmologies the two generally distinct functions Ne(ffffh) [rs/rd] and Neff[rs/rd] reduce to the same function and have

Ne(ffffh) =

Figure 2, which shows the function rs/rd[Neffffh)], demonstrates an important constraint between phenomena occurring during the epochs of BBN and recombination/photon decoupling. For a given input cosmology (ub,uc,...), the graph of rs/rd as a function of requires a computation of ne(a) to obtain rd for each value of Ne(ffffh).

We can understand figure 2 qualitatively by a simple scaling argument. We expect that, as the radiation energy density increases with increasing N^), the sound horizon and the diffusion length will decrease with the increasing Hubble rate H(a). The sound horizon decreases due to the increased energy density driving a more rapid expansion and a decrease in the sound speed, cs = [3(1 + R(a))]-1/2. The diffusion length increases, naively, due to a decrease in the scattering rate driven by the reduced Hubble time. Caution should be taken when using such naive scaling arguments. For example, the non-trivial dependence of the recombination history leads to counterintuitive effects in the Neff dependence on ^ mv [42].

Figure 2. Ratio of the comoving coordinate of the sound horizon radius rs to that of the photon diffusion length rd as a function of N^ff!1'' for cosmological parameter values YP = 0.2425, = 0.22068, and = 0.12029.

Ref. [42] demonstrates that where a naive scaling argument would suggest an increase in with increasing ^ mv, the non-trivial dependence of the recombination history on ^ mv implies Neff decreases monotonically and rapidly with increasing ^mv (See section 5.1).

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4 Verification of cosmological parameters with dark radiation

We adopt the point of view advocated in refs. [10, 27, 28] that the constraint provided by predictions of BBN should be incorporated simultaneously with constraints due to recombination effects in the extraction of cosmological parameters. We also require, as previously discussed, that BBN and recombination be solved iteratively. In this section, we demonstrate the self-consistent extraction of these parameters by employing a simple model for the radiation energy density that avoids solving, for example, the Boltzmann equation, for the set of distribution functions of the cosmic constituents, in particular the neutrinos.

We follow the standard paradigm for the BBN epoch as originally discussed in ref. [19] and subsequently in refs. [20, 21]. To that extent, we use burst to verify our results with those of other theoretical groups [43-45]. For example, at = 0.022068 and Neff = 3.046, ref. [11] finds:

YP = 0.24725 ± 0.00032 D/H = (2.656 ± 0.067) x 10-5

using the code PArthENoPE [46]. This compares favorably to our values:

Yp = 0.24306 D/H = 2.631 x 10

ref. [11] includes effects of non-thermal neutrino spectral distortions [46], whereas our model only includes dark radiation. The non-thermal spectra alters the neutron to proton ratio which is the source of the disagreement for YP. We give a detailed description of the dark-radiation model to distinguish between it and the models of section 5.

The model we explore in this section is identical to ACDM except for one additional constituent: dark radiation. The dark radiation energy density is radiation at all epochs and does not interact with the other energy-density constituents through any force except gravitation.

The total energy density as given in terms of radiation, matter, and vacuum energy components is

p(a) = pr (a) + pm(a) + pv (a), (4.5)

Pr = Py + Pv + Pdr (4.7)

where pY is the photon energy density, pv is the neutrino energy density, and pdr is the dark-radiation energy density. We parametrize pdr as

7 ( 4 \4/3 n2 , Pdr -(H) 30T4 (4-8>

where ¿dr is the dark-radiation parameter and is always assumed to be non-negative. In principle, we could entertain negative values of ¿dr since it is an adjustable parameter of Pr. Such a change requires a fundamental reworking of the ACDM model so that i fi = 1. These non-standard cosmologies obtain when considering, for example, neutrino oscillations. These models, however, are not continuously connected with our model at ¿dr = 0, for any values of the parameters, thus motivating the maintenance of ¿dr > 0.

We write Ne(ffffh) = 3 + ANff and assume that the contribution to Ne(ffffh) from pv is 3. Then the contribution from Pdr is given as ANe(ffffh). We see then that ANeffffh) = ¿dr. This is simply a restatement of the fact from the previous section, section 3, that Neff = 3+ANeffffh) = Ne(ffffh) for 'standard' cosmologies. It is, therefore, unnecessary for this simple dark-radiation model, to deduce NVeff from rs/rd since =NVeff by construction. This model of dark

which depend on the scale factor a as a-4, a-3, a0, respectively, as long as there is no energy transfer between the species. We assume for the purposes of this section that the neutrinos are massless, always acting as radiation energy density. The matter energy density consists of contributions from baryons and cold dark matter. Modeling the matter as a pressureless gas and observing that the comoving matter energy density is conserved, we write the proper energy density as

Pm =3-Hm- (fib + fic)( a0)3 . (4.6)

8n V a )

The vacuum energy is the least understood of the energy densities. Assuming the |_.

universe to be critically closed, the sum of the three energy densities must be equal to the critical energy density, specified only by the Hubble rate at the current epoch. Hence, Pv = Pc — Pr,0 — Pm,0. The vacuum energy density is negligible at all epochs of interest in this paper but is included for completeness.

The radiation component is given as:

radiation is the usual model applied, for example in ref. [11] and we explore, in this section, the predictions of the present burst code to verify our results against those of prior results within the community. We will use Neg, for the remainder of this section, to denote the "effective number of relativistic degrees of freedom".6

Figures 3-6 show the results of computations in which the four parameters wb, YP, D/H, and are varied. We begin by varying the two model inputs: wb and Neff. The upper panel in figure 3 shows the dependence of Neff on wb for curves of constant YP; the vertical band is the Planck value of wb = 0.02207 ± 0.00033. The figure is generated by first choosing a value for the baryon density in the range 0.004 < wb < 0.029. Each selected value of the dark radiation parameter in the range 3 < Neff < 4.5 allows for the prediction of Yp and D/H by parametrizing the radiation energy density as in eq. (2.12). The values so obtained are plotted C | as contours in the Neff-wb plane in the upper and lower panels of figure 3. The solid curve ^^ is the preferred value YP = 0.2465 ± 0.0097 of ref. [7], which is a selection of observations of metal poor extragalactic H ii regions. The contours are spaced by roughly 0.0097/3 showing that Nff is not strongly constrained by values of YP alone; this is a manifestation of the h^ degeneracy of Neff and YP. For example, at YP = 0.2465 ± 0.0035, corresponding to the contours closest to YP = 0.2465, the range allowed is nearly consistent with both the standard, calculated value Neff = 3.046 and the ref. [11] derived value of Neff = 3.30 ± 0.27.

Predictions of the primordial deuterium abundance are a much more sensitive constraint upon allowed values of Neff. This can be seen in the lower panel of figure 3. The solid line contour with 105 x D/H = 2.530 ± 0.04 corresponds to the recent measurement of ref. [8]. Contours in this figure are separated by the one standard deviation of ref. [8]. There are two

points of interest regarding the deuterium figure. First, as noted in ref. [8], observation of the |— primordial component of deuterium is precise enough to begin to constrain the microscopic # ^ physics of the thermally averaged nuclear reaction rates and their cross sections. Additionally, given the precision of the current and forthcoming deuterium measurements and the strong dependence of Neff on its value (at constant wb), we advocate using D/H as a prior, over YP, C ) for future base model parameter searches as recommended by refs. [6, 8, 47]. i_^

The degeneracy between Neff and YP is again evident in figure 4. Each plot explores the D/H vs. Wb contour space, where the upper plot contains contours of constant Yp and the

lower plot contains contours of constant Neff. The shaded bands in each figure indicate the one-sigma observations of wb and D/H from refs. [8, 11], respectively. Deuterium is not an input parameter into our model. We compute it by choosing a baryon number and iteratively change the dark-radiation parameter, until matching the chosen deuterium target. The outputs from the process are Nff and YP. Values of wb, D/H and Nff are in satisfactory agreement with the standard cosmology at the precision of current observations.

The quantities D/H and wb are the tightest observationally-constrained parameters we are currently investigating. Figure 5 shows two plots in the YP-Neff plane with contours of constant wb (upper plot), and contours of constant 105 x D/H (lower plot). The horizontal band in each figure indicates the one-sigma observation of Yp from ref. [7]. Like D/H, Yp is not an input into our model. Consequently, we adopt the identical iterative method for YP in figure 5 as we do for D/H in figure 4. For the wb (upper) plot of figure 5, the solid contour line is the best-fit value of ref. [11] with nine-sigma spacing of the contours. The contours exist in a subspace of the YP-Neff plane which is well within current observations, but nevertheless could span a range of radically different physics. Similarly, the bottom plot

6 We refer to Nff as the effective number of relativistic degrees of freedom although there are factors that complicate this interpretation, among them the temperature parameter, and fermionic nature of neutrinos.

0.005 0.010 0.015 0.020 0.025

4.4-4.2-4.0-J 3.8-

3.0 — 0.021

0.023 0.024 ujb

Figure 3. (Top) Neff plotted against ujb for contours of constant values of Yp (labeled by mass fraction). The solid curve is the preferred value of ref. [7]. The contours are spaced by A Yp « 0.003. (Bottom) Neff versus ujb for contours of constant values of 105 x D/H. The solid curve is the preferred value of ref. [8]. The contours are spaced by A(105 x D/H) = 0.04. In each case, abundances are determined in a self-consistent BBN calculation.

0.015 0.020 0.025

Figure 4. (Top) 105 x D/H plotted against ujb for contours of constant YP. The solid curve is the preferred value of ref. [7]. The contours are spaced by AYP « 0.003. (Bottom) 105 x D/H versus ujb for contours of constant Neff. The contours are spaced by ANeff = 0.3

£ 0.250

0.230 3.0

Figure 5. (Top) Yp plotted against iV"eff for contours of constant The solid curve is the best-fit value of ref. [11]. The contours are spaced by Aw6 = 0.003. (Bottom) Yp versus Neff for contours of constant 105 x D/H. The solid curve is the preferred value of ref. [8]. The contours are spaced by A(105 x D/H) = 0.6.

Figure 6. Yp plotted against for contours of constant The contours are spaced by

A Neii = 0.33.

shows the 105 x D/H value of ref. [8] as the solid contour with the other contours spaced fifteen-sigma apart. Clearly Yp and Nefj do not constrain the cosmological model as tightly as uij) and D/H. This observation indicates the import of using the next generation of 30-meter class telescopes and CMB observation to better determine the light, element abundances, particularly, Yp with high precision.

Figure 6 shows how Neg changes in the Yp-u)b plane. The shaded bands in each figure indicate the one-sigma observations of and Yp from refs. [7, 11], respectively. We may conclude from this plot that there could exist many different values of Neg consistent with the observations of Yp and However, we caution against such a conclusion without considering the effect on D/H. In fact, we choose not to include a figure of contours of D/H in the Yp-u)b plane because the strong sensitivity of D/H to produces too large a range of values for Yp to be useful as a constraint of the cosmological model.

All calculations show a consistency between ujb, AW, D/H, and Yp to a conservative limit, of t.wo-sigma. error range in each observation. We expect, the uncertainties in each observation to improve in the coming years with large ground-based CMB experiments [48, 49] and 30-meter class telescopes [50-52]. Future high-precision measurements may result, in tensions for the best-fit. values of ujb, AW, and D/H. These tensions could be indications of the need for more precise theoretical and numerical approaches or could signal the presence of physics beyond the standard model. As it. stands here, the bottom panels of figures 3 and 4 shows that the tension between and D/H cannot, be resolved with the addition of extra, radiation energy density. Uncertainties in nuclear reactions may produce disagreement, between and D/H, allowing D/H to become a. probe of nuclear physics. It. is also possible that the spectroscopic determination of D/H may be subject, to small systematic errors only recognizable at such precision. An exciting prospect, is the need to revise the CSM to resolve

W0.4 0.2

■ - -0.010

■ -0.001

0.005 0.010 0.015 0.020 0.025

Figure 7. Determination of J2 mv plotted against at constant ANeff. The contours are spaced by « 0.01 in values of ANeff. All contours correspond to ANeff < 0.

tensions with the observations of D/H and ujb, possibly leading to the conclusion of BSM physics active during BBN.

5 Examples for neutrino sector BSM physics

We describe three examples of unresolved issues in BSM/CSM physics, which require the fully self-consistent, parameter determination described in previous sections. We consider, in turn, models incorporating neutrino rest mass, sterile neutrinos, and non-zero lepton numbers. We use, throughout this section, the observationally-inferred definition of Neg, N^.

5.1 Neutrino rest mass

Section 4 details a self-consistent, treatment, of the BBN observables Yp, D/H, Ne¡j, and The sum of the light, neutrino masses, denoted has no bearing on the determination

of primordial abundances in BBN calculations due to the high temperatures relevant, there. Here, however, we explore the epochs and energy scales in the history of the universe associated with the ^ mv energy scale in order to investigate the relationship between ^ mv and the other four observables of interest, (uib, Neñ, Yp and D/H). Specific examples of such epochs that we might, consider include the surface of last, scattering (z ~ 1100) and the advent, of LSS (z < 10). We focus on the surface of last, scattering and implications for the CMB in this paper.

Reference [42] (hereafter GFKPI) investigates the effect, of neutrino rest, mass on Neg using the burst suite of codes. Conventional estimates based on the energy density added by non-zero neutrino rest, masses suggest, an increase in However, using the method

outlined in section 3, GFKPI shows a decrease in NVeff. The decrease is due to an effect on the recombination history stemming from an increase in the Hubble rate, which results in a larger free-electron fraction. This counterintuitive result is termed the "neutrino-mass/recombination (vMR) effect". The vMR effect manifests itself only in a self-consistent treatment, such as that employed by GFKPI.

In addition, GFKPI investigates the dependence of the vMR effect on We revisit this physics here in preparation for a discussion on the effect of non-zero lepton number Lv later, in section 5.3.2. Increasing leads to an enhancement of the vMR effect when ^ mv is held constant, as is evident by the curvature of the contours in figure 7. The enhancement is a consequence of the effect that changing has on the the recombination history. We might naively expect a larger change in the Hubble rate relative to the massless neutrino |

case resulting in an enhanced vMR effect for the smaller case. This is opposite to that ^^ observed in figure 7. This result also is counterintuitive based on expectations from a simple scaling of the energy density and the resulting change in the recombination history [42].

The origin of the enhancement of the vMR effect can be understood by considering a simplification of the Boltzmann equation that determines the recombination history [eq. (2.10)]. We take Yp = 0 for the purposes of this argument since the vMR enhancement is insensitive to Yp, as we have verified numerically for the ranges of parameters we are considering. In this simple scenario, Xe = Xp and we obtain an expression for the change in the free-electron fraction: 2 »

^ = (1 - Xe)^ - XM-V2), (5.1)

where ft = a(2)(meT/2n)3/2e-AQ/T is the ionization coefficient and a(2) is the recombination coefficient with AQ = 13.6 eV.

Equation (5.1) for the recombination history shows that the free-electron disappearance rate is proportional to the total electron number density, which in turn is proportional to wb through eq. (2.9). Equation (2.9) also shows how

ntot) relates to Yp. Note that and YP affect ntot) differently. However, due to the relative insensitivity of Yp to the dependence dominates in eq. (2.9). The increase in energy density from ^ mv = 0 and the increase in the free-electron disappearance rate combine to alter the recombination history so as to enhance the vMR effect for increasing ub.

5.2 Sterile neutrinos

We next consider the possibility that there exists either single or multiple sterile-neutrino species, which could have profound implications in cosmology. We entertain two possibilities of either light or heavy sterile neutrinos.

5.2.1 Light sterile neutrinos

Observations of neutrino events in large scintillating detectors may have revealed anomalies that could be interpreted as sterile neutrinos with rest masses mVs ~ 1eV [53-55]. We investigate the presence of a single sterile neutrino in the early universe by employing a model where the sterile state populates a thermal Fermi-Dirac shaped distribution with temperature parameter Ts, possibly through flavor mixing. The sterile neutrino temperature, Ts is taken to be less than or equal to the active neutrino temperature Tv. The ratio Ts/Tv is assumed to be the same throughout weak decoupling, BBN, and recombination. For this analysis, we do not investigate smaller active-sterile neutrino mixing angles with resultant non Fermi-Dirac-shaped energy spectra [41, 56]. Future work will consider such physics [57].

1.0 0.8 , 0.6 0.4 0.2

..3.9UU

. 3.600 ■ . 3.350 ■ • 3.200 ■

. 3.100

3.010 —

■ 3.000

mVs (eV)

Figure 8. Ratio of the sterile to active neutrino temperatures, Ts/T„, plotted against mVs for contours of constant iVeff for J2 n>" = 0.06 eV. Horizontal dotted lines show the prediction if the sterile neutrino was massless, i.e. m„a = 0.

Figure 8 displays contours of constant Nefj. The vertical axis is the ratio Ts/Tv and the horizontal axis the sterile neutrino rest mass m,Vs. We maintain the ratio Tu/T = (4/11)1/3, assuming covariant conservation of entropy, starting at the end of the epoch of e±-annihilation and continuing throughout the remainder of the history of the universe. The dotted lines show the expectation from the dark radiation analysis of section 4 without employing the self-consistent, iterative approach developed in section 3. The deviation of the contours from the dotted lines is again due to an effect similar to the ¡/MR effect but, in this instance, due to the sterile state. Figure 8 takes the sum of the active neutrino masses to be 0.06 eV. This is inconsequential for large Ts/Tv < 1. For Ts/Tv < 0.1, A^ < 0 due to the vMR effect in the active neutrino sector. As a consequence, the contour for AW = 3 is not coincident with the rnVs axis. Since mVa is too small to be of any significant kinematic effect during BBN, we need only compute BBN once for a given value of Ts/Tv. During recombination, mVs is kinematically important and affects the Hubble rate. Hence, for every point in the Ts/Tv-mVa plane of figure 8 we calculate recombination. This figure clearly emphasizes the need for a self-consistent, treatment between BBN and recombination when considering this BSM physics.

5.2.2 Heavy sterile neutrinos

Heavy sterile neutrinos that decay out of equilibrium in the early universe can affect weak decoupling and, as a consequence, primordial nucleosynthesis [16, 58, 59]. Sterile neutrinos in the rest, mass range 0.1 GeV < mUs < 1.0 GeV, with lifetimes > 1 s decaying during the weak decoupling, weak freeze-out., and/or BBN epochs can have constrainable, sometimes dramatic, cosmological effects.

Such sterile neutrinos have mass and vacuum mixings with ve, vM,vr constrained by accelerator and other laboratory oscillation experiments/observations [60-67], beta-decay experiments [68], and cosmological considerations, including constraints on ^ mv and Neff [69-76]. In fact, stringent constraints can be obtained from Neff limits alone [16], as sterile neutrinos decaying out of equilibrium can lead to dilution (entropy production) which, in the weak decoupling epoch, can lead to distortions in the relic neutrino energy spectrum, affecting Y^, mv, and have significant impact on the relativistic energy content and, hence, Neff. A sophisticated theoretical and computational treatment of dilution physics is a challenging endeavor. We have developed burst to address this specific problem. burst employs individual neutrino spectra for each species, binned according to the co-moving quantity e = E/Tv. The binned-spectra evolve with the universe as heavy particles decay, injecting entropy into | the neutrino seas, and subsequently equilibrate by scattering on background neutrinos and ^^ electrons and positrons. We track multiple decades of e values over many Hubble times. Our Boltzmann solver calculates the rates for each individual scattering process so we can decipher the contributions of each process to the shape of the neutrino spectra. Even though these heavy sterile neutrinos may decay away before an epoch where T ~ 10 keV, they can .. nevertheless alter the relationship between YP, D/H, Neff, ^ mv, and wb, necessitating the need for a self-consistent treatment between the weak decoupling, weak freeze-out, BBN, recombination, photon decoupling, and advent of LSS epochs [77]. ___

5.3 Lepton numbers

We examine how lepton numbers affect the primordial abundances and Neff. We define the |_

lepton number Lv for a neutrino species v in a flavor eigenstate as

Lv ^ Uv-n,, (52) s^

where nv is the number density of neutrino species v, n, is the number density of anti- |—

neutrino species v, and nY is the number density of photons. Here, for illustrative purposes, _____

we take Lve = LvM = LvT = Lv.

The efficiency of neutrino oscillations in equating lepton numbers is approximate, and indeed dependent on neutrino physics [78-80]. In fact, ref. [79] shows that oscillations with solar mass-splitting scales cause disparate lepton numbers in e, t neutrinos to equilibrate to within an order of magnitude of one another. The limitations of existing calculations revolve around how quantum damping and neutrino energy dependence are handled.

We use the comoving-invariant neutrino degeneracy parameter £v = ^v/Tv, where ^v is the chemical potential of neutrino v, to compute Lv. The present model assumes that the lepton number evolves through the epoch of e± annihilation only in response to the relative increase in nY. We do not consider any BSM physics which could alter the difference nv — n,; that is, we fix £v throughout weak decoupling, BBN, and photon decoupling. We relate the degeneracy parameter to the lepton number using the following expression [56, 81, 82]:

Lv = ——^r (n2£v + £3) , (5.3)

v 1112Z(3) v s,v; v !

where Z(3) ~ 1.202. The factor 4/11 in eq.(5.3) implies that our lepton numbers refer to the post e± annihilation epoch, where Tv/T = (4/11)1/3.

0.04 -

.............. _ ......0.210

0.02 _____0.225 --------

4 o.oo ✓ ✓ _____0.240

-0.04 ___ 0.285 ----

0.005 0.010 ~ 0.015 0.020 0.025

Figure 9. Lepton asymmetry Lv [eq.(5.2)] plotted against ujb for contours of constant Yp. The contours are spaced by A Yp = 0.015.

Figure 10. Lepton asymmetry Lv plotted against ujb for contours of constant 105 x D/H. The solid curve is the preferred value of ref. [8]. The contours are spaced by A(105 x D/H) = 0.04.

5.3.1 Effect on nucleosynthesis

The helium mass fraction is sensitive to the neutron-to-proton ratio, n/p. We determine n/p by calculating the weak rates associated with neutrino-nucleon reactions, namely:

ve + n ^ e- + p+ (5.4)

ve + p+ ^ e+ + n (5.5)

and in addition, neutron and inverse neutron decay:

n ^ e- + ve + p+ (5.6)

The net rates in reactions (5.4) and (5.5) fall below the Hubble rate at the epoch of weak freeze-out. Weak freeze-out largely precedes the alpha-particle formation process in BBN, though unlike the brief time/temperature range of a-formation, weak freeze-out occurs over several Hubble times at this epoch. The rates in reactions (5.4) through (5.6) are sensitive to h^ the neutrino and e± distributions. We follow ref. [21] to evolve T and the electron chemical ——^ potential in order to maintain equilibrium between the electrons, positrons and photons. For the electron-flavor neutrinos, we use the comoving invariants aTv and to compute the neutrino distributions. We set ^ mv = 0 as neutrinos of sub-eV rest mass remain ultra-relativistic throughout weak freeze-out.

Figures 9 and 10 show the helium mass fraction and the relative deuterium abundance, respectively. Each plot is in the Lvplane for contours of constant primordial abundance.

The relationships between lepton number and nucleosynthesis are well known [47, 83]. In- |—1 creasing LVe leads to an overabundance of neutrinos compared to anti-neutrinos. The forward rate of reaction (5.4) freezes-out after the forward rate of reaction (5.5). The imbalance lowers n/p which lowers YP as seen in figure 9. The decrease in n/p also leads to a decrease in D/H, although deuterium is not as sensitive to Lv as helium. However, D/H is known to C ) much higher precision than is Yp. |_

Comparing with recent observations [7, 8], the two light element abundances achieve consistency at 2ct. Yp prefers a value of Lv < 0 whereas D/H prefers a positive value of Lv. If future observations of the light-element abundances were to show a larger disagreement than 2ct, lepton numbers of identical value could not solely rectify the tension. Future analyses will consider scenarios with multiple facets of BSM physics including non-zero lepton numbers [84]. This analysis will use NVeff as a discriminating factor.

5.3.2 Effect on Neff

We consider how two aspects of non-CSM/BSM physics (Lv = 0 and/or ^ mv = 0) modify NVeff. If we set the neutrino rest mass to ^ mv = 0, we can investigate whether the vMR effect still applies with non-zero Lv. Figure 11 shows the changes to NVeff in the ^ mv-plane for three values of Lv = -0.05,0, 0.05 corresponding to blue, magenta, and red contours, respectively. The non-zero lepton number increases NVeff for small values of ^ mv in accordance with ref. [9]. However, for values of ^ mv ~ 1.0 eV, the vMR effect overwhelms the extra energy density from more particles to lower NVeff below three.

Figure 11 also shows that, despite the total energy density being insensitive to the sign of Lv, the contours for non-zero values of Lv with opposite sign do not overlap because YP depends sensitively on its value. YP is largest for the blue contours, so more helium suppresses the vMR effect.

Figure 11. plotted against ujf, for contours of constant AÑef¡. The blue contours are for

Lv = —0.05. The magenta contours are for Lv = 0. The red contours are for Lv = 0.05. Solid contours are for positive values; dashed contours are for negative values.

Note that taking Lv / 0 conflates the interpretation that the effect of ^ mv is identical to perturbations in the matter power spectrum, which are used in calculating the suppression of power on small scales. This is borne out by the present model where the ^ mv statistic cannot be equated to the cosmological measurement. In our model, ^ mv is simply the sum of the active vacuum neutrino mass eigenvalues. The observationally determined value of Y '"»i/ depends on quantities other than the sum of the active neutrino masses such as their energy distributions.

6 Conclusions and outlook

Cosmological considerations are a key route to exploring BSM physics. This is especially true for the neutrino sector, where there are many outstanding questions and where laboratory experiments are limited in what aspects of this physics can be addressed. In this paper we have argued that a self consistent treatment of BSM issues, across all epochs from weak decoupling to photon decoupling, is the best way to take advantage of the expected coming increase in precision of CMB measurements and observationally-inferred primordial abundances of the light elements. We employ a limited prescription to link the salient, features of self consistency between early-time neutrino dynamics and the surface of last, scattering. We couple the weak decoupling and nucleosynthesis of early times to CMB observables, including baryon-to-photon ratio (equiva.lent.ly uib), sound horizon, and photon diffusion length.

We have shown that such a. self consistent, treatment, is necessary, in part, because new neutrino physics can alter the relationships between different, cosmological epochs. For example, üüb and other CMB observables affect, the calculated yields of deuterium and helium.

In addition, the calculated relic neutrino energy spectra after weak decoupling affects the predicted value of Aeff at photon decoupling. The principal tool in our analysis is the suite of burst codes for nucleosynthesis and neutrino interactions and energy transport.

A case in point is our investigation of the relationship between neutrino rest masses, i.e. J2mv, and the four potential observables Aeff, YP, and D/H. This analysis reveals the "neutrino-mass/recombination" (vMR) effect first described in ref. [42]. The vMR effect is below the threshold of current CMB capabilities, but may not be in future observations [85].

There have been spectacular advances in the measurements/observations of neutrino properties. We know the neutrino mass-squared differences and three of the four parameters in the unitary transformation between the energy eigenstates (mass states) and the weak interaction eigenstates (flavor states) of the three active neutrinos (only the CP -violating phase remains unmeasured). As active neutrinos mix in vacuum and have non-zero rest masses, the question indubitably arises of whether there exist "sterile" neutrino states. If indeed sterile neutrinos do exist, we acknowledge that the parameter space of mass, vacuum mixing angle, and number is enormous. However, sterile neutrinos could have profound effects in all of the epochs under study in this paper. This possibility makes a self consistent treatment of these effects a powerful basis for constraining sterile neutrino states. U1

We have also studied the effects of non-zero lepton numbers on the relationship between CMB observables, nucleosynthesis, and neutrino physics. Our conclusion is that the primordial deuterium abundance is a potentially powerful probe of lepton number. However, an eventual CMB-only measurement of the primordial helium abundance Yp will be the most powerful probe of lepton number and many other issues. Determining Yp from CMB observables will require a sophisticated self-consistent approach to BBN, neutrino physics, and photon decoupling transport physics.

Finally, our study has revealed a potential tension between Aeff, and the primordial deuterium abundance, D/H, inferred from high redshift QSO absorption systems. In fact, if the advent of 30-m class telescopes in the near future allows for a decrease in errors in observationally-inferred D/H to the ~ 1% level, while observed and Aeff maintain their respective current central values, then tension is unavoidable. This may signal BSM or non-CSM physics, likely in the neutrino sector, or it could point to not understanding systematics in the damped Lyman-a cloud measurements of the isotope-shifted hydrogen absorption lines. We advocate using future instruments to explore the rich physics of weak decoupling, nucleosynthesis, and photon decoupling to discover what role BSM neutrino physics has in these epochs.

In this paper we have considered scenarios for both "light" (mass ~ 1 eV) and "heavy" ^ ^ (mass ~ 0.1-1 GeV) sterile neutrinos. In the former case we consider cases where the sterile neutrino relic energy spectra are Fermi-Dirac black-body shaped, though with a temperature parameter Ts differing from that characterizing the relic energy spectra of active neutrino species. We show here that the vMR effect has interesting consequences and that this case demands a self consistent treatment of recombination and BBN. Additionally, heavy sterile neutrino decay out of equilibrium can lead to dilution and high energy relic active neutrinos, and both of these features potentially can have dramatic and constrainable effects on CMB-epoch observables. This implies that CMB observations can indirectly probe the

CvB and explore active-sterile mass/vacuum-mixing parameter space unavailable to current |—^ accelerator-based experiments.

Acknowledgments

We would like to acknowledge the Institutional Computing Program at Los Alamos National Laboratory for use of their HPC cluster resources. EG acknowledges the San Diego Supercomputer Center for their use of HPC resources and helpful technical support. This work was supported in part by NSF grant PHY-1307372 at UC San Diego, by the Los Alamos National Laboratory Institute for Geophysics, Space Sciences and Signatures subcontracts, and the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. We thank Lauren Gilbert, Jeremy Ariche, and J.J. Cherry for helpful discussions.

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