the neutrino puzzle: anomalies, interactions, and cosmological … · 03-04-2019 · interactions...

The Neutrino Puzzle: Anomalies, Interactions, and Cosmological Tensions

Christina D. Kreisch,1, ∗ Francis-Yan Cyr-Racine,2, 3, † and Olivier Dore4

1Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544 USA2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

3Department of Physics and Astronomy, University of New Mexico,1919 Lomas Blvd NE, Albuquerque, New Mexico 87131, USA

4Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA(Dated: April 3, 2019)

New physics in the neutrino sector might be necessary to address anomalies between differentneutrino oscillation experiments. Intriguingly, it also offers a possible solution to the discrepantcosmological measurements of H0 and σ8. We show here that delaying the onset of neutrino free-streaming until close to the epoch of matter-radiation equality can naturally accommodate a largervalue for the Hubble constant H0 = 72.3± 1.4 km s−1Mpc−1 and a lower value of the matter fluctu-ations σ8 = 0.786± 0.020, while not degrading the fit to the cosmic microwave background (CMB)damping tail. We achieve this by introducing neutrino self-interactions in the presence of a non-vanishing sum of neutrino masses. Without explicitly incorporating additional neutrino species, thisstrongly interacting neutrino cosmology prefers Neff = 4.02 ± 0.29, which has interesting implica-tions for particle model-building and neutrino oscillation anomalies. We show that the absence ofthe neutrino free-streaming phase shift on the CMB can be compensated by shifting the value ofseveral cosmological parameters, hence providing an important caveat to the detections made inthe literature. Due to their impact on the evolution of the gravitational potential at early times,self-interacting neutrinos and their subsequent decoupling leave a rich structure on the matter powerspectrum. In particular, we point out the existence of a novel localized feature appearing on scalesentering the horizon at the onset of neutrino free-streaming. While the interacting neutrino cos-mology provides a better global fit to current cosmological data, we find that traditional Bayesiananalyses penalize the model as compared to the standard cosmological scenario due to the relativelynarrow range of neutrino interaction strengths that is favored by the data. Our analysis shows thatit is possible to find radically different cosmological models that nonetheless provide excellent fitsto the data, hence providing an impetus to thoroughly explore alternate cosmological scenarios.

PACS numbers: 98.80.-k,14.60.St,98.70.Vc

I. INTRODUCTION

The neutrino sector of the Standard Model (SM) ofparticle physics is a promising area to search for new phe-nomena that could help pinpoint the Ultraviolet comple-tion of the SM. Indeed, terrestrial neutrino experimentshave identified several anomalies that could potentiallyindicate the presence of new physics in the neutrino sec-tor (see, e.g., Ref. [1] for a recent review). Of particularsignificance are the νµ → νe appearance results from theMiniBooNE [2] and LSND [3] collaborations which, if in-terpreted within a neutrino oscillation framework that in-cludes an extra sterile neutrino, would indicate the pres-ence of such a sterile neutrino at very high statistical sig-nificance. Within this “3+1” neutrino oscillation frame-work, these results are, however, very difficult to reconcilewith the absence of anomalies in the νµ → νµ disappear-ance as probed by recent atmospheric [4, 5] and short-baseline [6, 7] experiments. If these results are confirmedby future analyses, it is likely that new physics beyondthe sterile+active oscillation models would be necessary

∗ [email protected]† [email protected]

to resolve the tension between neutrino appearance anddisappearance data.

Astrophysical and cosmological observations providecomplementary means of probing the properties of neu-trinos. This is perhaps best illustrated by the cosmolog-ical constraints on the sum of neutrino masses

∑mν <

0.12 eV [8] obtained by combining cosmic microwavebackground (CMB) data from the Planck satellite withbaryon acoustic oscillation (BAO) measurements. Cos-mological observables such as the CMB and large-scalestructure (LSS) are also sensitive to the presence of newinteractions (see e.g. Refs. [9–42]) in the neutrino sectorthat would modify their standard free-streaming behav-ior during the radiation-dominated epoch following theirweak decoupling. In the literature, a phenomenologicaldescription based on the ceff and cvis parametrization[43] has often been used to test the free-streaming natureof neutrinos in the early Universe [44–54]. While theseanalyses generally find results consistent with the stan-dard neutrino cosmology, they are difficult to interpretin terms of possible new interactions among neutrinos, asemphasized in Refs. [55, 56]. Other works [55–67] haveused more physical parameterizations that make the con-nection to the underlying particle nature of the neutrinointeraction more transparent.

In particular, Ref. [56] has developed a rigorous treat-

arX

iv:1

902.

0053

4v2

[as

tro-

ph.C

O]

2 A

pr 2

019

mailto:[email protected]

mailto:[email protected]

2

ment of the evolution of cosmological neutrino fluctua-tions in the presence of neutrino self-interactions medi-ated by either a massive or massless new scalar particle.Using this framework, Ref. [66] used CMB data to putconstraints on the strength of neutrino self-interactionsin the early Universe for the case of a massive media-tor. These results largely confirmed earlier constraintsfrom Refs. [55, 62, 65] obtained using an approximate(but nonetheless accurate) form of the neutrino Boltz-mann hierarchy. Interestingly, these studies, which fo-cused on four-neutrino interactions parametrized by aFermi-like coupling constant Geff , found a bimodal pos-terior distribution for this latter parameter. While thefirst (and statistically dominant) posterior mode is con-sistent with the onset of neutrino free-streaming beingin the very early Universe, the second posterior modecorresponds to a much delayed onset of free-streamingto zν,dec ∼ 8300. In Ref. [65], a previously unknownmulti-parameter degeneracy involving the amplitude ofscalar fluctuations, the scalar spectral index, the Hubbleconstant, and the neutrino self-interacting strength wasidentified as being responsible for the existence of thissecond posterior mode. While intriguing, the neutrinointeraction strength favored by this mode is nearly tenorders of magnitude above the standard weak interaction.Taken at face value, this likely constitutes a very seriouschallenge from a model-building perspective.

Nevertheless, given the current tensions among ter-restrial and atmospheric neutrino experiments describedabove, is the “interacting” neutrino mode hinting at thepresence of new physics beyond the SM? The simplifiedinteraction models used in Refs. [55, 65–67] are likely cap-turing parts of a more realistic neutrino interaction sce-nario, hence leading to a somewhat suboptimal fit to thecosmological data. One aspect that has been neglected instudies of self-interacting neutrinos so far is the presenceof neutrino mass. The impact of massive neutrinos on theCMB and matter clustering has been studied extensivelyin the literature (see e.g. Refs. [68–72]). One of the aimsof this paper is to understand how the effects of massiveneutrinos on cosmological observables are modified whenself-interactions are present in the early Universe.

Tensions are also growing between different late-timemeasurements of the Hubble constant H0 [73–76] andthose based on CMB data [8]. Measurements of theamplitude of matter fluctuations at low redshifts (oftenparametrized using σ8) from weak gravitational lensingand cluster counts are all consistently lower than thatinferred from the CMB [77–79]. While the statistical sig-nificance of the deviation of each individual measurementis less than 3σ, all recent measurements of the amplitudefluctuations in the local universe are below the Planckvalue. Physics beyond ΛCDM has been proposed to rec-oncile these tensions, such as early dark energy [80–83],dark matter interactions [84, 85], decaying dark matter[86–89], modified gravity [90, 91], and new relativisticspecies [92], among others. However, these propositionsoften struggle to remedy both tensions simultaneously.

In this paper, we study how the presence of self-interacting massive neutrinos in the early Universe af-fect cosmological observables such as the CMB, with aneye on how these new effects could help relieve the cur-rent tensions among different datasets. In Sec. II, wedescribe the simplified neutrino interaction model usedin this work. In Sec. III, we present the cosmologicalperturbation equations for massive self-interacting neu-trinos. In Sec. IV, we describe the physical impacts thatmassive self-interacting neutrinos have on the CMB andthe matter power spectrum. In Sec. V, we outline thedata and method used in our cosmological analyses ofself-interacting. The results from these analyses are pre-sented in Sec. VI and discussed in Sec. VIII. We concludeand highlight future directions in Sec. IX.

II. NEUTRINO INTERACTION MODEL

In this work, we focus on a simple framework that cap-tures the most important cosmological aspects of realis-tic neutrino interaction models. We note however thatbuilding a successful model of neutrino self-interactionthat respects the gauge and flavor structure of the SMlikely requires the introduction of a light sterile specieswhich mass-mixes with the active neutrinos and is itselfcoupled to a massive scalar or vector mediator (see e.g.Refs. [32, 93–100]). The presence of these new interac-tions in the sterile sector suppresses the effective mix-ing angle between the active and sterile species at earlytimes, ensuring that Big Bang nucleosynthesis (BBN)constraints are respected . At later times once the active-sterile oscillation rate becomes comparable to the finitetemperature effective potential resulting from the new in-teraction, the mixing angle is no longer suppressed henceallowing the active and sterile sectors to partially ther-malize with each other [30, 33, 35].

Diagonalizing the mass matrix of such a model leads toan effective interaction Lagrangian between the differentneutrino mass eigenstates of the generic form

Lint = gij νiνjϕ, (1)

where gij is a (generally complex) coupling matrix, νiis a left-handed neutrino Majorana spinor, and the in-dices i, j labeled the neutrino mass eigenstates. Here wehave assumed a Yukawa-type interaction with a massivescalar ϕ, but note that the results presented in this workalso apply if a massive vector is assumed instead. TheLagrangian given in Eq. (1) could also arise in modelswhere neutrinos couple to a Majoron [11, 101, 102].

In models where the new interaction arises throughactive-sterile mixing, the structure of the coupling matrixgij would generally depend on the flavor content of eachmass eigenstate. For instance, a mass eigenstate madeof mostly active flavors will couple very weakly to themassive scalar ϕ, while an eigenstate being largely com-posed of the sterile species would couple more stronglyto the mediator. In other models of neutrino interaction,

3

the structure of the coupling matrix could be more ar-bitrary. In all cases though, gij is subject to importantflavor-dependent bounds [103–105] arising from meson,tritium, and gauge boson decay kinematics.

In this work, we consider the simple case of a universalcoupling gν between every neutrino mass eigenstate

gij ≡ gνδij , (2)

where δij is the Kronecker delta. While the universalcoupling case is likely unrealistic for the reason outlinedabove, it does provide a simple benchmark to test thesensitivity of cosmological data to new neutrino physics.

We work in the contact-interaction limit in which themass of the ϕ mediator is much larger than the typicalenergy of the scattering event. In this case, one can inte-grate out this massive mediator and write the interactionas a four-fermion contact interaction. This is an excellentapproximation at the energy scale probed by the CMBfor mϕ & 1 keV. In this limit, the squared scatteringamplitude for a neutrino νi to interact with any otherneutrino in the thermal bath is

|M|2νi =∑

spins

|M|2νi+νj→νk+νl

= 2G2eff

(s2 + t2 + u2

), (3)

where we have defined the dimensionfull coupling con-stant Geff ≡ |gν |2/m2

ϕ. Here, s, t, and u are the stan-dard Mandelstam variables. While our phenomenologicalmodel described by Geff is unlikely to accurately captureall the complexity of novel neutrino interactions, it isnonetheless a useful framework to identify the interest-ing parameter space, as described in Ref. [31].

Introducing new neutrino interactions has an impactbeyond cosmology. For a low mass mediator (< 10MeV), SN 1987A [15], Big Bang nucleosynthesis (BBN)[106, 107], and the detection of ultra-high energy neutri-nos at IceCube [31, 34, 108] provide some of the strongestconstraints, with the latter bound having the potentialof being the most stringent in the near future. Otherlimits [10, 109, 110] coming from Z-boson decay do notdirectly apply at the energy scale probed by the CMB.Also, elastic collisions caused by the new interaction donot affect the time it takes for neutrinos to escape super-novae [111, 112], although they could lead to interestingphenomena (see e.g. Refs. [113–118]). Finally, supernovacooling puts bounds on the coupling of majorons to SMneutrinos [119–122], but the applicability of these likelydepends on the details of the exact coupling matrix used.

III. COSMOLOGICAL PERTURBATIONS

In this section we summarize the key ingredients andsimplifications entering our derivation of the Boltzmannequation governing the evolution of massive and self-interacting neutrino fluctuations, at first order in per-turbation theory. Our computation uses two main ap-proximations:

• Based on previous studies [55, 65], we assume thatneutrinos decouples while still in the relativisticregime. We thus neglect the presence of the smallneutrino mass in the computation of collision inte-grals. As we shall see, our final results are consis-tent with this approximation.

• We assume that the neutrino distribution functionremains exactly thermal throughout the epoch atwhich neutrinos decouple and start free-streaming.This thermal approximation (also called, relaxationtime approximation) implies that the only possibleneutrino perturbations are local temperature fluc-tuations. This approximation was shown to be veryaccurate in Ref. [66] for the type of interaction weconsider here.

Conformal Newtonian gauge is used throughout thissection.

A. Neutrino distribution function andperturbation variables

We present a detailed derivation of the left-hand sideof the Boltzmann equation for massive neutrino in Ap-pendix B (see also Ref. [123]). Our starting point is toexpand the neutrino distribution function as

fν(x,p, τ) = f (0)ν (p, τ)[1 + Θν(x,p, τ)], (4)

where x denotes the spatial coordinates, τ is confor-mal time, and p is the proper momentum. The back-ground (spatially uniform) neutrino distribution functionis taken to be of a Fermi-Dirac shape

f (0)ν (p, τ) =

1

ep/Tν + 1, (5)

where p = |p|. In the ultra-relativistic regime, for whichthe thermal approximation implies that the only possi-ble neutrino perturbations are local temperature fluctu-ations, the perturbation variable Θν admits the form

Θν(x,p, τ) = −d ln f(0)ν

d ln p

δTν(x, τ)

Tν(τ), (6)

where Tν is the background neutrino temperature, andδTν is its perturbation. It is therefore convenient to in-troduce the temperature fluctuation variables Ξν

Ξν(x,p, τ) ≡ −4Θν(x,p, τ)

d ln f(0)ν

d ln p

(7)

which is independent of p in the thermal approximationfor massless neutrinos. However, the presence of a non-vanishing neutrino mass and the non-negligible momen-tum transfered in a typical neutrino-neutrino collisionwould in general introduce some extra p-dependence to

4

Ξν [123]. This turns the Boltzmann equation of self-interacting neutrinos into a differentio-integral equationthat is particularly difficult to solve exactly [56]. In prac-tice though, the absence of energy sources or sinks cou-pled to the neutrino sector implies that the momentumdependence of the right-hand side of Eq. (7) should bevanishingly small at early times when neutrinos form ahighly-relativistic tightly-coupled fluid. This allows usto neglect the momentum-dependence of Ξν in the com-putation of the collision integrals, an approximation thatwas found to be accurate in Ref. [66]. We do retain, how-ever, the momentum dependence of Ξν in the left-handside of the Boltzmann equation.

In this work, we only consider scalar perturbations andthus expand the angular dependence of the Ξν variable(the Fourier transform on Ξν) in Legendre polynomialsPl(µ)

Ξν(k,p, τ) =

∞∑

l=0

(−i)l(2l + 1)νl(k, p, τ)Pl(µ), (8)

where µ is the cosine of the angle between k and p. Be-fore presenting the equation of motion for the neutrinomultipole moments νl, we discuss the structure of thecollision integrals.

B. Collision term

The details of the collision term calculation for theνν → νν process is given in Appendix C. As explainedabove, the main simplification entering this calculationis the use of the thermal approximation in which we ne-glect the momentum dependence of the νl variables. Un-der this assumption, the collision term at first order in

perturbation theory C(1)ν can be written as

C(1)ν [p] =

G2effT

6ν

4

∂ ln f(0)ν

∂ ln p1(9)

×∞∑

l=0

(−i)l(2l + 1)νlPl(µ)

(A

(p

Tν

)

+Bl

(p

Tν

)− 2Dl

(p

Tν

)),

where the functions A(x), Bl(x), and Dl(x) are givenin Eqs. (C52), (C53), and (C54), respectively. Here, wehave adopted the notation Tν ≡ Tν to avoid clutter.

C. Boltzmann equation for self-interactingneutrinos

Substituting the collision term from Eq. (9) intoEq. (B10) and performing the µ integral yields the equa-tion of motion for the different neutrino multipoles νl.

They can be summarized in the following compact form

∂νl∂τ

+ kq

ε

(l + 1

2l + 1νl+1 −

l

2l + 1νl−1

)(10)

− 4

[∂φ

∂τδl0 +

k

3

ε

qψδl1

]

= −aG2effT

5ν νl

f(0)ν (q)

(Tν,0q

)(A

(q

Tν,0

)

+Bl

(q

Tν,0

)− 2Dl

(q

Tν,0

)),

where we have introduced the comoving momentum q ≡ap, q = |q|, ε =

√q2 + a2m2

ν , a is the scale factor nor-malized to a = 1 today, δmn is the Kronecker delta func-tion, and Tν,0 is the current (a = 1) temperature of theneutrinos. The fact that the collision term is directlyproportional to νl is a consequence of our use of thethermal approximation. We note that energy and mo-mentum conservation ensures that A+B0−2D0 = 0 andA+B1 − 2D1 = 0, respectively.

As is standard in analyses of massive neutrino cosmolo-gies, we shall consider our neutrino sector to be composedof a mix of massive and massless neutrinos. In the mass-less case (q = ε), one can integrate Eq. (10) over thecomoving momentum to yield a simpler neutrino multi-pole hierarchy [55, 65]

∂Fl∂τ

+ k

(l + 1

2l + 1Fl+1 −

l

2l + 1Fl−1

)(11)

− 4

[∂φ

∂τδl0 +

k

3ψδl1

]= −aG2

effT5ναlFl,

where

αl =120

7π4

∫ ∞

0

dxx2

[A (x) +Bl (x)− 2Dl (x)

], (12)

and where we denoted the massless perturbations as Flto distinguish them from the massive neutrino variablesνl.

We implement these modified Boltzmann equations inthe cosmological code CAMB [124]. For computationalspeed, we precompute the functions A, Bl and Dl on agrid of q/Tν,0 values and use an interpolation routine toaccess them when solving the cosmological perturbationequations. As in standard CAMB, we use a sparse 3-pointgrid of q/Tν,0 values to evaluate the integrals requiredto compute the energy density and momentum flux ofmassive neutrinos. We have checked convergence of ourscheme against a 5-point momentum grid and found neg-ligible difference in the CMB and matter power spec-trum in the parameter space of interest. We also precom-pute the coefficient αl and tabulate them. We emphasizethat energy and momentum conservation ensures thatα0 = α1 = 0, which we have checked with high accuracy.

For simplicity, we assume throughout this paper thatthe neutrino sector contains one massive neutrino, with

5

the remaining neutrino species being massless. All neu-trinos are assumed to interact with the same couplingstrength Geff . We find that varying the number of mas-sive neutrinos and number of mass eigenstates, whileholding Neff and

∑mν constant, has a very small impact

on the CMB and matter power spectra for all values ofGeff consistent with the data used here. It is howeverpossible that future data might be sensitive to the way∑mν is spread among different mass eigenstates.At early times, the large self-interaction rate of neu-

trinos renders the equations of motion for multipolesl ≥ 2 extremely stiff. To handle this, we employ a tight-coupling scheme [126] in which multipole moments withl ≥ 2 are set to zero at early times. Once the neutrinoself-interaction rate falls to about a 1000 times the Hub-ble expansion rate, we turn off this tight-coupling ap-proximation and allows power to flow to the higher mul-tipoles. We have checked that this switch happens earlyenough as to not affect the accuracy of our results. Afterneutrino decoupling, once they become non-relativistic,we revert to the standard velocity-integrated truncatedBoltzmann hierarchy as described in Ref. [127]. We alsomodified the adiabatic initial conditions for the cosmo-logical perturbations to take into account the absence offree-streaming neutrinos at early times. Finally, through-out this work, we use the standard BBN predictions tocompute the helium abundance given the abundance ofrelativistic species and the baryon-to-photon ratio.

IV. EFFECT ON COSMOLOGICALOBSERVABLES

A. Cosmic microwave background

In the standard cosmological paradigm, free-streamingneutrinos travel supersonically through the photon-baryon plasma at early times, hence gravitationallypulling photon-baryon wavefronts slightly ahead of wherethey would be in the absence of neutrinos [42, 128, 129].As a result, the free-streaming neutrinos imprint a netphase shift in the CMB power spectra towards largerscales (smaller `), as well as a slight suppression of itsamplitude. Free-streaming neutrinos thus lead to a phys-ical size of the photon sound horizon at last scattering r∗that is slightly larger than it would otherwise be. Thisphase shift is thought to be a robust signature of thepresence of free-streaming radiation in the early Universe[42, 130, 131].

The neutrino self-interactions mediated by the cou-pling constant Geff delay the time at which neutrinosbegin to free-stream. Fourier modes entering the causalhorizon while neutrinos are still tightly-coupled will notexperience the gravitational tug of supersonic neutrinosand will therefore not receive the associated phase shiftand amplitude reduction. Compared to the standardΛCDM model, neutrino self-interactions thus shift theCMB power spectra peaks towards smaller scales (larger

`) and boost their fluctuation amplitude. This leads toa net reduction of the physical size of the photon soundhorizon at last scattering r∗. As we shall see, this is thekey feature of our model that helps reconcile CMB andlate-time measurements of the Hubble constant H0.

The left panels of Fig. 1 show the temperature CMBpower spectra and their relative difference to a ΛCDMmodel for different values of Geff ,

∑mν , and Neff to

illustrate the effects of neutrino self-scattering in thepresence of a non-vanishing mass term. Here, we keepΩm fixed as

∑mν changes, and use the best-fit Planck

TT+lowP+lensing ΛCDM values as our fiducial cosmol-ogy [132]. The middle left panel of Fig. 1 displays thecombined effect of changing both Geff and

∑mν . For the

minimal sum of neutrinos masses∑mν = 0.06 eV, an in-

teraction strength of Geff = 10−4 MeV−2 (solid blue line)has for only effect a slight increase of power at large mul-tipoles. On the other hand, increasing the neutrino cou-pling strength to Geff = 10−2 MeV−2 (solid red line) sig-nificantly boosts the amplitude of the TT spectrum andintroduces a clear phase shift (identifiable from the oscil-latory pattern of the residuals), which are the two telltalesignatures of self-scattering neutrinos as described above.

Increasing the sum of neutrinos masses to∑mν =

0.23 eV (at fixed Ωm) delays the time of matter-radiationequality. The delay slightly increases the amplitude ofthe TT spectrum near the first few acoustic peaks anddampens the spectrum at smaller scales (see dashed blackline in Fig. 1). The resulting changes to the photon-baryon sound horizon at recombination and to the an-gular diameter distance to the surface of last scatteringcreate a net phase shift towards low ` [69], that is, in theopposite direction to that caused by increasing Geff . Thisopens the door for possible cancellations between the rel-ative phase shift (as compared to ΛCDM) caused by neu-trino self-scattering and that resulting from a large sumof neutrino masses. Such cancellation partially occurs inthe middle left panel of Fig. 1 for Geff = 10−2 MeV−2

as∑mν is increased from 0.06 to 0.23 eV (dashed red

line). Similarly, the boost in amplitude from Geff can alsocompensate for the damping effects of increasing

∑mν

at small scales (see e.g. the dashed blue line). Overall,we see that the effect of massive neutrinos and increasedinteraction strength are nearly additive1, reflecting thefact that the physical processes associated with each ofthese properties take place at different times in the cos-mological evolution.

The lower left panel of Fig. 1 displays the impact ofincreasing the energy density of the neutrino fluid, whichwe parametrized here through the standard parameter

1 Indeed, combining the spectrum for Geff = 10−2 MeV−2,Σmν = 0.06 eV (solid red line) with that of the Σmν = 0.23 eVΛCDM model (dashed black line) yields a spectrum similar tothe model with Geff = 10−2 MeV−2, Σmν = 0.23 eV (dashedred line).

6

0 500 1000 1500 2000 2500 3000

`

−0.14

−0.10

−0.06

−0.02

0.02

0.06

0.10

0.14

( CTT

`,Iν−CTT

`,Λ

CD

M

) /CTT

`,Λ

CD

M

Fixed∑mν = 0.06 eV

Fixed Ωbh2, zeq, θ∗



ΛCDM (Neff = 3.046)


Geff = 10−2 MeV−2, Neff = 3.046

Geff = 10−2 MeV−2, Neff = 4.046

−0.06

−0.02

0.02

0.06

0.10

( CTT

`,Iν−CTT

`,Λ

CD

M

) /CTT

`,Λ

CD

M

FixedNeff = 3.046FixedNeff = 3.046

ΛCDM (Σmν = 0.06 eV)


Geff = 10−2 MeV−2, Σmν = 0.06 eV

Geff = 10−2 MeV−2, Σmν = 0.23 eV

Geff = 10−4 MeV−2, Σmν = 0.06 eV

Geff = 10−4 MeV−2, Σmν = 0.23 eV

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

`3(`

+1)CTT

`/2π[ ×

109µK

2]

Planck 2015Planck 2015

0 500 1000 1500 2000 2500 3000

`

−0.2

−0.1

0.0

0.1

0.2

( CEE

`,Iν−CEE

`,Λ

CD

M

) /CEE

`,Λ

CD

MFixed

∑mν = 0.06 eV






Geff = 10−2 MeV−2, Neff = 3.046

Geff = 10−2 MeV−2, Neff = 4.046

−0.2

−0.1

0.0

0.1

0.2

( CEE

`,Iν−CEE

`,Λ

CD

M

) /CEE

`,Λ

CD

M




Geff = 10−2 MeV−2, Σmν = 0.06 eV

Geff = 10−2 MeV−2, Σmν = 0.23 eV

Geff = 10−4 MeV−2, Σmν = 0.06 eV

Geff = 10−4 MeV−2, Σmν = 0.23 eV

0.0

0.2

0.4

0.6

0.8

1.0

`3(`

+1)CEE

`/2π[ ×

108µK

2]


FIG. 1: Effects of∑mν , Geff , and Neff on the phase and amplitude of the TT and EE power spectra. Colors denote

different values of Geff . Solid spectra correspond to∑mν = 0.06 eV and dashed spectra correspond to∑

mν = 0.23 eV. Measurements from the Planck 2015 data release are included [125].

Neff , defined via the relation

ρR =

[1 +Neff

7

8

(4

11

)4/3]ργ , (13)

where ρR and ργ are the total energy density in radiationand in photons, respectively. The effects on the CMB ofincreasing Neff have been well-studied in the literature(see e.g. Ref. [133]) for the case of free-streaming neutri-nos. For fixed values of the angular scale of the soundhorizon, the epoch of matter-radiation equality, and thephysical baryon abundance, it was found that the mostimportant net impact of increasing Neff was to damp thehigh-` tail of the TT spectrum and to induce a phaseshift towards larger scales (low-`). Interestingly, self-interacting neutrinos can partially compensate for these

effects, hence pointing to a possible degeneracy betweenGeff and Neff . An example of this can be seen in thedotted red line in the lower left panel of Fig. 1, wherethe excess of damping caused by Neff = 4.046 (dottedblack line) is compensated by suppressing neutrino free-streaming with Geff = 10−2 MeV−2.

Geff affects the EE polarization power spectrum in asimilar manner as the temperature spectrum. The rightpanel of Fig. 1 shows that the phase shift between thestandard ΛCDM model and that with self-interactingneutrinos is more visible in this case due to the sharp, welldefined peaks of the polarization spectrum [129]. Thisallows to directly see in which direction the spectrum isshifted compared to ΛCDM since the oscillations in theresiduals lean in the direction of the phase shift, that is,there is a sharper drop off in the residuals in the direction

7

that the spectrum is shifted. Once again, we clearly seethat the absence of phase shift caused by a large valueof Geff can be partially canceled by increasing

∑mν , in

a nearly additive fashion. For the EE polarization spec-trum, suppressing neutrino free-streaming can somewhatcompensate the extra damping caused by a large Neff (atfixed θ∗, zeq, and Ωbh

2; see lower right panel of Fig. 1).

B. Matter power spectrum

The growth of matter fluctuations is sensitive to thepresence of self-interacting neutrinos through the neu-trinos’ impact on the two gravitational potentials φand ψ. Indeed, neutrino self-interactions suppress theanisotropic stress of the universe, leading to φ − ψ = 0before the onset of neutrino free-streaming. This con-trasts with the ΛCDM case for which φ = (1 + 2Rν/5)ψon large scales at early times for the adiabatic mode [123],where Rν is the radiation free-streaming fraction. Thisdifference in the evolution of the potentials modifies thegravitational source term driving the growth of matterfluctuations. The equation describing the evolution ofdark matter fluctuations can be written in Fourier spaceas [128]

dc +a

adc = −k2ψ, (14)

where

dc ≡ δc − 3φ, (15)

and where δc = δρc/ρc is the standard dark matter en-ergy density contrast in Newtonian gauge. Here, an over-head dot denotes a derivative with respect to conformaltime τ . The gauge-invariant variable dc represents thefractional dark matter number density perturbation byunit coordinate volume. At late times, dc is nearly equalto δc and it is thus a useful quantity to understand thestructure of the matter power spectrum at z = 0. In theradiation-dominated epoch where a/a = τ−1, the solu-tion to Eq. (14) can be written [134]

dc(k, τ) = −9

2φp + k2

∫ τ

0

dτ ′τ ′ψ(k, τ ′) ln (τ ′/τ), (16)

where φp is the primordial value of φ on large scales. Theintegral appearing in Eq. (16) obtains most of its contri-bution when kτ ∼ 1. The changes to the growth of darkmatter fluctuations can thus be understood by examiningthe behavior of the ψ potential at horizon entry.

We compare the evolution of ψ in the presence of self-interacting neutrinos with Geff = 10−2 MeV−2 to that ofstandard ΛCDM in the left panel of Fig. 2. There, wetrack the evolution of three different Fourier modes: k =10h/Mpc which enters the horizon during the radiationdominated era while neutrinos are still tightly-coupled toeach other, k = 0.3h/Mpc which roughly corresponds tothe scale entering the horizon when neutrinos begin to

free-stream, and k = 10−3 h/Mpc which does not enterthe horizon until far after neutrino decoupling. We usehere the same cosmological parameters as in Fig. 1. Theresulting evolution of dark matter fluctuations for thesethree modes is shown in the right panel of Fig. 2.

When modes enter the horizon during the radiation-dominated era, the gravitational potential ψ decays inan oscillatory fashion [134]. The absence of anisotropicstress implies that ψ starts its oscillatory decaying behav-ior from a larger amplitude. This boosts the amplitudeof the envelope of the decaying oscillations as comparedto ΛCDM, leading to an overall slower decay. While thisat first increases the amplitude of dark matter fluctua-tions at horizon entry as compared to ΛCDM (see bottomright panel of Fig. 2), the subsequent oscillations of theintegrand appearing in Eq. (16) lead to a net damping ofthe dark matter perturbation amplitude. Another way tothink about this is that the slower decay of the potentialψ in the presence of self-interacting neutrinos reduces thehorizon-entry boost that dark matter fluctuations expe-rience as compared to ΛCDM.

For modes entering the horizon at the time of neu-trino decoupling, the potential ψ begins decaying from itslarger value with Rν = 0 but rapidly locks into its stan-dard ΛCDM evolution due to the onset of neutrino free-streaming. This case thus displays the quickest dampingof the ψ potential after horizon entry, which leads to anet boost of dark matter fluctuations as compared toΛCDM. Indeed, these modes receive a positive contribu-tion near horizon entry from the integral in Eq. (16), butwithout the subsequent extra damping due to the ψ po-tential quickly converging to its ΛCDM behavior. Theevolution of the k = 0.3h/Mpc mode in Fig. 2 displaysthis behavior.

Finally, modes entering the horizon well-after theonset of neutrino free-streaming behave exactly liketheir ΛCDM counterparts, as illustrated by the k =10−3 h−1Mpc mode in Fig. 2. Taking together the evolu-tion of the different Fourier modes entering before, dur-ing, and after neutrino decoupling, we expect the matterpower spectrum to have the following properties (at fixedneutrino mass). For large wavenumbers entering the hori-zon while neutrinos are tightly coupled, we expect thematter power spectrum to be suppressed compared toΛCDM. As we go to larger scales and approach modesentering the horizon at the onset of free-streaming, we ex-pect a “bump”-like feature displaying an excess of poweras compared to ΛCDM. As we go to even larger scales,the matter power spectrum is expected to asymptote toits standard ΛCDM value.

These expectations are indeed realized as shown inFig. 3. The middle panel shows the power spectrum ra-tios between the interacting neutrino models and ΛCDM.Focusing for the moment on the cases with

∑mν = 0.06

eV, we see that the matter power spectrum is damped atlarge wavenumbers and then displays a broad peak-likefeature with an excess of power as compared to ΛCDM.The shape of this power excess is determined by the neu-

8

100101102103104105106107108

z

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

0.08

0.10

−3

(ψIν−ψ

ΛC

DM

)/2ζ

k = 1× 10−3 [hMpc−1]

k = 3× 10−1 [hMpc−1]

k = 1× 101 [hMpc−1]

k = 1× 10−3 [hMpc−1]

k = 3× 10−1 [hMpc−1]

k = 1× 101 [hMpc−1]

0.0

0.2

0.4

0.6

0.8

1.0

−3ψ/2ζ

Iν ΛCDMIν ΛCDM

100101102103104105106107108

z

0.94

0.96

0.98

1.00

1.02

1.04

1.06

dc,

Iν/d

c,Λ

CD

M

100

101

102

103

104

105

dc

Iν ΛCDM

k = 1× 10−3 [hMpc−1]

k = 3× 10−1 [hMpc−1]

k = 1× 101 [hMpc−1]

Iν ΛCDM

FIG. 2: The evolution of the ψ gravitational potential (left) and of the gauge invariant dark matter density contrastdc (right) for different k-modes as a function of redshift. Solid lines correspond to the interacting neutrino case withGeff = 10−2 MeV−2, Neff = 3.046, and

∑mν = 0.06 eV, whereas dashed lines correspond to the ΛCDM case. On

the left, we plot −3ψ/(2ζ), where ζ is the gauge-invariant curvature perturbation. The lower left panel shows thenormalized difference between the interacting neutrino and ΛCDM ψ potential, while the lower right panel shows

the ratio of the dark matter fluctuations in the two models. The onset of neutrino free-streaming for the interactingneutrino model shown here occurs at zdec,ν ' 104. Dark matter fluctuations entering the horizon while neutrinos arestill tightly coupled decay and appear damped at present relative to ΛCDM, while those entering the horizon during

neutrino decoupling receive a net boost that persists until the present epoch.

trino visibility function [55] encoding the details of neu-trino decoupling. Increasing the sum of neutrino masses(at fixed Ωm) leads to a damping of the matter powerspectrum on small scales [69, 72]. This standard reduc-tion of power is shown for ΛCDM as the thick blackdashed line in Fig. 3. Interestingly, this small-scale sup-pression is also present for self-interacting neutrinos andoccurs in addition to that caused by the slower decayof the gravitational potential ψ discussed above. Thus,the matter power spectrum for massive self-interactingneutrinos is even more suppressed at large k than in thestandard ΛCDM case with massive neutrinos.

This fact might seem counterintuitive at first since thereduction of small-scale power from massive neutrinos isoften refereed to as “free-streaming” damping. We seethis moniker is somewhat of a misnomer since the damp-ing is present whether or not neutrinos are actually free-streaming. Instead, the small-scale reduction of power issimply caused by the large pressure term that prohibitsneutrino clustering on these scales. This pressure termis always there as long as neutrinos are relativistic, even

when neutrinos are self-scattering. As was the case forthe CMB, the effects of a non-vanishing sum of neutrinomasses and large Geff are largely additive. Comparingthe

∑mν = 0.23 eV cases to that with

∑mν = 0.06

eV in Fig. 3 for both interacting neutrino models shownillustrates this well. Again, this near additivity reflectsthe fact that part of the effect comes from the behaviorof dark matter fluctuations at horizon entry, while therest is caused by the large pressure term of relativisticneutrinos on small scales.

The lowest panel of Fig. 3 shows the effect of increas-ing Neff (at fixed θ∗, zeq, and Ωbh

2) on the matter powerspectrum. For ΛCDM, the main impact is to increase theamplitude of Fourier modes that enter the causal hori-zon during radiation domination. This results from thelarger radiation density and free-streaming fraction Rν[128] at early times. Suppressing neutrino free-streamingfor Neff = 4.046 (dotted red line) nullifies this increaseof power on small-scales, even leading to a net dampingcompared to ΛCDM for k > 10h/Mpc. However, as theneutrinos start to decouple from one another, the larger

9

10−4 10−3 10−2 10−1 100 101 102

k [h/Mpc]

0.85

0.95

1.05

1.15

1.25

Pm,Iν(k

)/P

m,Λ

CD

M(k

)







Geff = 10−2 MeV−2, Neff = 3.046

Geff = 10−2 MeV−2, Neff = 4.046

0.75

0.85

0.95

1.05

1.15

Pm,Iν(k

)/P

m,Λ

CD

M(k

)




Geff = 10−2 MeV−2, Σmν = 0.06 eV

Geff = 10−2 MeV−2, Σmν = 0.23 eV

Geff = 10−4 MeV−2, Σmν = 0.06 eV

Geff = 10−4 MeV−2, Σmν = 0.23 eV

100

102

104

Pm

(k)

FIG. 3: Effects of Geff ,∑mν , and Neff on the matter

power spectrum. Colors denote different values of Geff .Solid spectra correspond to

∑mν = 0.06 eV and dashed

spectra correspond to∑mν = 0.23 eV. Dotted lines in

the bottom panel have Neff = 4.046. Note the localizedincrease in amplitude at the scales entering the horizon

at the onset of neutrino free-streaming.

radiation density leads to a higher amplitude feature onscales entering the horizon at that time.

We thus see that taken together, the joint effect ofGeff ,

∑mν , and Neff can lead to matter power spectra

having a significantly different structure and shape thanthe standard ΛCDM paradigm.

V. DATA & METHODOLOGY

We use our modified versions of CAMB [124] andCosmoMC + Multinest [135, 136] to place constraints onGeff , Neff , and

∑mν , as well as the standard cosmolog-

ical parameters. We use nested sampling [137] to ensurethat we properly sample our posterior, which we expectto be multi-modal as in previous cosmological studies ofself-interacting neutrinos [55, 65, 66].

We use a combination of CMB and low-redshift datasets in our analysis:

• TT: low-` and high-` CMB temperature powerspectrum from the Planck 2015 release2 [125].

• EE, TE: low-` and high-` CMB E-mode polariza-tion and their temperature cross-correlation fromthe Planck 2015 data release3 [125]. The 2015 po-larization data is known to have residual systemat-ics and results drawn using this dataset should beinterpreted with caution. While our main conclu-sions will not make use of this dataset, we nonethe-less present results including this dataset for com-pleteness.

• lens: CMB lensing data from the Planck 2015 datarelease [138].

• BAO: Baryon Acoustic Oscillation (BAO) mea-surements from the 6dF Galaxy Survey constrain-ing DV at z = 0.106 [139], Sloan Digital Sky Survey(SDSS-III) Baryon Oscillation Spectroscopic Sur-vey (BOSS) data release 11 low-z data measuringDV at z = 0.32 and CMASS data measuring DV

at z = 0.57 [140], and data from the SDSS MainGalaxy Sample measuring DV at z = 0.15 [141]

• H0: Local measurement4 of the Hubble parameterH0 = 73.0 ± 1.75 km s−1Mpc−1 at z = 0.04 fromRef. [73].

We use the lite high-` likelihood, which marginal-izes over nuisance parameters, to reduce the number offree parameters in our analysis. We use the followingdata set combinations for our nested sampling analysis:‘TT+lens+BAO’, ‘TT+lens+BAO+H0’, ‘TT,TE,EE’,and ‘TT,TE,EE+lens+H0’.

In Table I we list our adopted prior ranges. We placeuniform priors on all these parameters, except for the

2 Explicitly, we use the likelihood plik lite v18 TT for high-` andcommander rc2 v1.1 l2 29 B at low-`.

3 Explicitly, we use the likelihood plik lite v18 TTTEEE for high-`and lowl SMW 70 dx11d 2014 10 03 v5c Ap at low-`.

4 We note that the mean value of H0 used in our analysis is slightlylower (∼ 0.14σ) than the value quoted in the published versionof Ref. [73] (ours corresponds to the value found in an earlierversion of their manuscript). We do not expect this very smalldifference to impact our results in any way.

10

TABLE I: Adopted prior ranges

Parameter Prior

log10(GeffMeV2) [−5.5,−0.000001]∑mν [eV] [0.0001, 1.5]Neff [2.0, 5.0]

Ωbh2 [0.01, 0.04]

Ωch2 [0.08, 0.16]

100θMC [1.03, 1.05]τ [0.01, 0.25]

ln(1010As) [2, 4]ns [0.85, 1.1]

ycal [0.9, 1.1]

Planck calibration parameter ycal, for which we use aGaussian prior ycal = 1.0000 ± 0.0025. For the analy-ses using the Planck polarization data, we also include aGaussian prior on the optical depth to reionization givenby τ = 0.058± 0.012 from Ref. [142].

We use 2000 live points in our nested sampling runs,setting the target sampling efficiency to 0.3. We im-pose an accuracy threshold on the log Bayesian evidenceof 20%, which ensures that our confidence intervals arehighly accurate. We use the mode-separation feature ofMultinest to isolate each posterior mode and computetheir respective summary statistics.

VI. RESULTS

In this section, we first present the main highlights ofour analysis, before discussing the physical properties ofthe two categories of interacting neutrino models that arefavored by the data. We end this section with a brief dis-cussion about which properties of interacting neutrinomodels help alleviate current tensions in cosmologicaldata. Throughout this section, we quote and analyzeresults for the TT + lens + BAO+H0 data set combina-tion unless otherwise specified. We discuss the impactof other data (including CMB polarization) in Sec. VIIIand list parameter constraints for these other data setcombinations in Table VI and Table VII, in Appendix A.

A. Highlights

Similarly to previous works [55, 65, 66], we find twounique neutrino cosmologies preferred by the data: astrongly interacting neutrino cosmology (hereafter de-noted SIν model) characterized by log

(Geff MeV2

)=

−1.35+0.12−0.07 for the TT+lens+BAO+H0 combination,

and a moderately interacting neutrino cosmology (here-after, MIν model) characterized by log

(Geff MeV2

)=

−3.90+1.00−0.93 for the same data set. Values of Geff between

these two modes are strongly disfavored by the data since

they either prefer to have a phase shift that is largely con-sistent with free-streaming neutrinos, or no phase shift atall. We present constraints on cosmological parametersfor the SIν and MIν modes in Table II. While the MIνcosmology was nearly indistinguishable from the ΛCDMscenario with massless neutrinos in previous work [65],the addition of neutrino mass and Neff in combinationwith the H0 measurement from Ref. [73] leads to a slightpreference for a delayed onset of neutrino free-streaming.We expand more on this new development in Sec. VI Cbelow.

Cosmological parameters in the SIν cosmology admitvalues that are significantly different from ΛCDM:

1. The angular scale of the baryon-photon sound hori-zon at last scattering 100θ∗ = 1.04604 ± 0.00056(68% C.L.) takes a value that is radically different(> 5σ away) than in the ΛCDM scenario, reflectingthe absence of the free-streaming neutrino phaseshift.

2. The large Neff value 4.02 ± 0.29 (68% C.L.) sug-gests the presence of an additional neutrino species,which might help reduce tensions between differentneutrino oscillation experiments.

3. A smaller value of the baryon drag scale rdrag =138.8 ± 2.5 Mpc (68% C.L.) helps reconcile BAOwith local Hubble constant measurements, leadingto H0 = 72.3± 1.4 km s−1 Mpc−1 (68% C.L.).

4. The impact of self-interacting neutrinos on thegrowth of dark matter perturbations and a pre-ferred suppressed spectrum of primordial scalarfluctuations lead to σ8 = 0.786± 0.020 (68% C.L.).

To illustrate the ability of neutrino self-interactions tohelp resolve current cosmological tensions, we comparethe S8 ≡ σ8Ω0.5

m and H0 2D posteriors for the SIν modeland MIν model with the base ΛCDM model in Fig. 4. Weoverlay bands for HSC constraints on S8 [79] and localmeasurements of H0 [73]. In order for our analysis to beindependent from these measurements, we show posteri-ors for the TT+lens+BAO constraints for both the neu-trino self-interaction models and ΛCDM. Intriguingly,the strong neutrino self-interactions in the SIν model areable to independently produce the preferred values for S8

and H0, even without using these measurements in ouranalysis. The base ΛCDM model is unable to achievethese values, and the weak neutrino interactions of theMIν model can only achieve such values with weak sig-nificance.

In Table III we compute the ∆χ2 values between thetwo neutrino self-interaction models and ΛCDM. Thedata favor the strongly interacting neutrino model overΛCDM with ∆χ2

Total = −7.91. This is a significant differ-ence, even after accounting for the three extra parametersin the SIν model (see Sec. VII B for further discussionabout this point). The preference for the self-interacting

11

64 68 72 76H0 [km/s/Mpc]

0.40

0.42

0.44

0.46

8

0.5

m

CDM SI

64 68 72 76H0 [km/s/Mpc]

0.40

0.42

0.44

0.46

8

0.5

m

CDM SI

63 66 69 72 75H0 [km/s/Mpc]

0.42

0.44

0.46

0.48

8

0.5

m

CDM MI

63 66 69 72 75H0 [km/s/Mpc]

0.42

0.44

0.46

0.48

8

0.5

m

CDM MI

FIG. 4: 2D posteriors for S8 and H0 illustrating how neutrino self-interactions can remedy cosmological tensions.We compare the Planck TT + lens + BAO ΛCDM posterior to the SIν and MIν posteriors for TT + lens + BAO.

We overlay 2σ bands for the measurements S8 = 0.427± 0.016 [79] and H0 = 73± 1.75 km/s/Mpc [73].

TABLE II: TT + lens + BAO + H0 Constraints: Parameter 68% Confidence Limits

Parameter Strongly Interacting Neutrino Mode Moderately Interacting Neutrino Mode

Ωbh2 0.02245+0.00029

−0.00033 0.02282± 0.00030Ωch

2 0.1348+0.0056−0.0049 0.1256+0.0035

−0.0039

100θMC 1.04637± 0.00056 1.04062+0.00049−0.00056

τ 0.080± 0.031 0.127+0.034−0.029∑

mν [eV] 0.42+0.17−0.20 0.40+0.17

−0.23

Neff 4.02± 0.29 3.79± 0.28log10(GeffMeV2) −1.35+0.12

−0.066 −3.90+1.0−0.93

ln(1010As) 3.035± 0.060 3.194+0.068−0.056

ns 0.9499± 0.0098 0.993+0.013−0.012

H0 [km/s/Mpc] 72.3± 1.4 71.2± 1.3Ωm 0.3094± 0.0083 0.3010± 0.0080σ8 0.786± 0.020 0.813+0.023

−0.020

109As 2.08+0.11−0.13 2.44± 0.15

109Ase−2τ 1.771± 0.016 1.892+0.019

−0.017

r∗ [Mpc] 136.3± 2.4 139.1± 2.3100θ∗ 1.04604± 0.00056 1.04041+0.00058

−0.00064

DA [Gpc] 13.03± 0.23 13.37± 0.21rdrag [Mpc] 138.8± 2.5 141.6± 2.3

TABLE III: Comparison to ΛCDM for TT + lens + BAO + H0


∆χ2low ` 0.66 −0.75

∆χ2high ` −1.15 1.08

∆χ2lens 0.06 −0.24

∆χ2H0

−6.68 −6.12∆χ2

BAO −0.81 −0.36

∆χ2Total −7.91 −6.39

∆AIC −1.91 −0.39

12

neutrinos comes from the local measurements of H0, thehigh-` TT data, and the BAO data.

In Fig. 5, we separate the posterior modes and plottheir separate statistical distribution for the most salientparameters. For comparison, we also show the marginal-ized posteriors for the standard ΛCDM paradigm, as wellas for its Neff +

∑mν two-parameter extension. In Fig. 6,

we show the different covariances between the most rele-vant model parameters for three of the dataset combina-tions used in this work5.

B. Strongly interacting neutrino mode

The existence of the SIν mode was first pointed outin Ref. [55], and further studied in Refs. [65, 66]. Asdiscussed there, the SIν cosmology arises due to a multi-parameter degeneracy that opens up in CMB data whenthe onset of neutrino free-streaming is delayed until red-shift z ∼ 8000. This approximately coincides with theepoch when Fourier modes corresponding to multipole` ≈ 400 enters the causal horizon [65], which lies some-where between the first and second peak of the CMBtemperature spectrum. We review below the propertiesof this alternate cosmology, emphasizing its differenceswith the standard ΛCDM model.Sound horizon One of the most striking features of theSIν model is the significantly larger value of the angularsize of the sound horizon θ∗. This is probably the mostconfusing aspect of our results since the angular size ofthe CMB sound horizon at last scattering is thought tobe the best measured quantity in all of cosmology. Tounderstand this apparent discrepancy, it is important torealize that the angular sound horizon is defined as θ∗ ≡r∗/DA, where

r∗ =

∫ a∗

0

cs(a)

a2H(a)da, DA =

∫ 1

a∗

da

a2H(a), (17)

where cs is the baryon-photon sound speed, H is the Hub-ble rate, and a∗ is the scale factor at last scattering. Wethus see that θ∗ is purely defined in terms of backgroundquantities, independent of the behavior of cosmologicalperturbations. In particular, it is independent of thegravitational tug that neutrinos exert on the photons.

Of course, when fitting CMB data we use the fulltemperature and polarization spectra computed from theBoltzmann equation which includes the effect of neutri-nos. For the SIν model, the absence of free-streaming

5 Due to the presence of two posterior modes with different width,it is difficult to choose a smoothing scale that faithfully capturesthe intrinsic shape of the whole posterior while removing sam-pling noise. This particularly affects the SIν mode and results ina significantly reduced height which appears to visually suppressits statistical significance. See Fig. 11 in Appendix A for a figurewith a smoothing scale more appropriate for the SIν mode.

neutrinos means that the CMB spectra do not receive thestandard phase shift, and thus appear slightly displacedtoward larger ` as compared to the corresponding ΛCDMspectra. In order to fit the data, we must compensate forthis shift by increasing the value of θ∗. Thus, the dif-ference between the values of θ∗ in the SIν and ΛCDMmodels directly reflects the absence of the free-streamingneutrino phase shift in the former.

We note that it was a priori far from obvious thatsuch a dramatic change in the angular size of the soundhorizon was possible without introducing other artifactsthat would significantly worsen the fit to CMB and BAOdata. Our analysis shows that the larger value of θ∗ isachieved by increasing H0 and Ωch

2 above their ΛCDMvalues.

Primordial spectrum In addition to removing theCMB phase shift, suppressing neutrino free-streamingalso increases the amplitude of the temperature and po-larization spectra, as discussed in Sec. IV A. In the SIνmodel, these changes are reabsorbed by modifying theprimordial spectrum of scalar fluctuations parametrizedby the amplitude As and spectral index ns. As was foundin Refs. [55, 65], lower values of both As and ns are re-quired to fit the temperature data in the SIν mode. Thedifference between this alternative cosmology and ΛCDMis even more apparent if we compare the values of the pa-rameter Ase

−2τ which directly determines the amplitudeof the CMB temperature spectrum. As shown in Fig. 5,this amplitude parameter admits values that are radically(> 5σ) different than in ΛCDM, again reflecting the largeimpact that suppressing neutrino free-streaming has onthe CMB.

Neutrino properties The SIν model is consistent withhaving an entire additional neutrino species (Neff =4.02 ± 0.29, see Fig. 5), which has interesting implica-tions for neutrino oscillation experiments. By comparingthe SIν cosmology with a more standard ΛCDM + Neff +∑mν model, we can understand how much of this prefer-

ence is driven by the neutrino self-interaction. As shownin Fig. 5, the two-parameter extension of the ΛCDM cos-mology already favors a larger Neff , but the introductionof strong neutrino self-interactions shifts the posterior toeven larger values. To a certain extent, this shift is drivenby the need to fit the large value of the local Hubble ratefrom Ref. [73] by reducing the size of the sound horizonat the baryon drag epoch (see e.g. Ref. [143]). However,in the presence of free-streaming neutrinos, increasingNeff also leads to a larger phase shift toward low ` whichputs a limit on how much extra free-streaming radiationcan be added before severely degrading the fit to CMBdata. For the SIν model, the absence of this phase shiftallows for larger Neff , which leads to a smaller value ofrdrag and, in turn, a larger Hubble constant. This is thekey feature of the SIν model that allows it to severelyreduce the Hubble rate tension between CMB and late-time measurements, as we shall discuss in Sec. VI D.

The SIν model also statistically prefers a nonzero valuefor the sum of neutrino masses. This preference was how-

13

135 140 145 150rdrag

135 140 145 150rdrag

135 140 145 150rdrag

135 140 145 150rdrag

67.5 70.0 72.5 75.0 77.5H0 [km/s/Mpc]

0.72 0.76 0.80 0.84 0.888

1.0400 1.0425 1.0450 1.0475100

1.74 1.80 1.86 1.92109Ase

20.925 0.950 0.975 1.000 1.025

ns

0.0216 0.0224 0.0232 0.0240bh

20.12 0.13 0.14 0.15

ch2

0.285 0.300 0.315 0.330m

CDM CDM + Ne↵ +P

m SI MI

3.0 3.5 4.0 4.5 5.0Ne↵

3.0 3.5 4.0 4.5 5.0Ne↵

0.0 0.3 0.6 0.9 1.2m [eV]

135 140 145 150rdrag

67.5 70.0 72.5 75.0 77.5H0 [km/s/Mpc]

0.72 0.76 0.80 0.84 0.888

1.0400 1.0425 1.0450 1.0475100

1.74 1.80 1.86 1.92109Ase

20.925 0.950 0.975 1.000 1.025

ns

0.0216 0.0224 0.0232 0.0240bh

20.12 0.13 0.14 0.15

ch2

0.660 0.675 0.690 0.705 0.720

CDM + Ne↵ +P

m SI MI

5 4 3 2 1 0log10(Ge↵MeV2)

3.0 3.5 4.0 4.5 5.0Neff

0.0 0.3 0.6 0.9 1.2m

135 140 145 150rdrag

67.5 70.0 72.5 75.0 77.5H0

0.72 0.76 0.80 0.84 0.888

1.0400 1.0425 1.0450 1.0475100

1.74 1.80 1.86 1.92109Ase

20.925 0.950 0.975 1.000 1.025

ns

0.0216 0.0224 0.0232 0.0240bh

20.12 0.13 0.14 0.15

ch2

0.660 0.675 0.690 0.705 0.720

SI MI

FIG. 5: 1D posteriors for the TT+lens+BAO+H0 data combination after separating the SIν and MIν modes andplotting them independently. For this reason, the peak locations and posterior shapes are of physical interest rather

than the relative heights of the peaks.

14

0.3 0.6 0.9 1.2Σmν [eV]

1.0400

1.0425

1.0450

1.0475

100θ∗

1.74

1.80

1.86

1.92

109A

se−

2τ

0.93

0.96

0.99

1.02

ns

2.4

3.0

3.6

4.2

4.8

Neff

−4.5 −3.0 −1.5

log10(Geff MeV2)

0.3

0.6

0.9

1.2

Σmν

[eV

]

1.0400 1.0425 1.0450 1.0475

100θ∗

1.74 1.80 1.86 1.92

109Ase−2τ

0.93 0.96 0.99 1.02

ns

2.4 3.0 3.6 4.2 4.8

Neff

TT,TE,EE

TT + lens + BAOTT + lens + BAO + H0

FIG. 6: Marginalized posterior distributions for select parameters for three of the data set combinations used in thiswork. Here, we focus on parameters illustrating the difference between the two modes in relation to sound horizon,

the amplitude of the spectrum, and the neutrino properties. Posterior for H0 and σ8 are shown in Fig. 7.

15

ever already present at a less significant level (< 2σ) inthe Neff +

∑mν extension of the ΛCDM scenario. In

this latter case, nonzero neutrino masses arise from theneed to suppress the amplitude of matter of fluctuationsat late times (as measured here through CMB lensing) inthe presence of a larger Neff and Ωm. For the SIν model,the preference for a nonzero sum of neutrino masses is in-creased (> 2σ) due to the even larger Neff and Ωm valuesfavored by this scenario.

We note that, in our analysis, the primordial heliumabundance YP is highly correlated with Neff due to ouruse of the BBN consistency condition. Allowing YP totake a different set of values in the SIν scenario couldlead to an even better fit to cosmological data.Matter clustering Several competing effects act to setthe amplitude of late-time matter fluctuations (as cap-tured by the parameter σ8) in the SIν model. First,the large values of Neff and Ωm (the latter necessary tokeep the epoch of matter-radiation equality fixed) tend toboost the amplitude of matter fluctuation as discussed inSec. IV B. On scales entering the horizon before the onsetof neutrino free-streaming, this increase is counteractedby both a nonzero sum of neutrino masses and the reduc-tion of the horizon entry boost for dark matter fluctua-tions in the presence of self-interacting neutrinos. Darkmatter fluctuations entering the horizon during neutrinodecoupling, which for the SIν model are coincidentallythose primarily contributing to σ8, are however enhancedby the rapid decay of the gravitational potential on thesescales. Finally, the lower amplitude and spectral indexof the primordial scalar spectrum in the SIν model tendto suppress power on scales probed by σ8. Putting allof these effects together leads to a net lower value ofσ8, which, as discussed in the previous section, might befavored by some probes of late-time matter clustering.The overall shape of the matter power spectrum in theSIν model will be further discussed in Sec. VIII.

C. Moderately interacting neutrino mode

Within the MIν mode, the onset of neutrino free-streaming occurs before most Fourier modes probed bythe Planck high-` data enter the causal horizon. As such,the cosmological parameter values preferred by this modeare very similar to those from the Neff +

∑mν exten-

sion of the ΛCDM scenario (see Fig. 5). The main differ-ence here is that high-` CMB modes do not receive thefull amplitude suppression associated with free-streamingneutrinos due to the finite width of the neutrino visibil-ity function. In other words, even though these high-`modes enter the horizon after most neutrinos have startedto free-stream, residual scattering in the neutrino sec-tor still influences the amplitude of the CMB dampingtail (see, e.g., the model with Geff = 10−4 MeV−2 and∑mν = 0.06 eV in Fig. 1). This increased small-scale

power allows for a larger Neff , which, by reducing thebaryon drag scale, leads to slightly larger Hubble con-

stant. This shift is however quite small.A surprising fact about the MIν mode (also pointed out

in Ref. [66]) is that it shows a slight statistical preferencefor a nonzero value of Geff . As we can see in Fig. 6, thispreference is nearly entirely driven by the local Hubbleconstant measurement of Ref. [73]. Indeed, removing thisdataset from our analysis (blue contours) eliminates mostof the preference for a nonzero value of Geff .

D. Mediating Controversy: Effects on H0 and σ8

We show in Fig. 7a the impact of Geff , Neff and∑mν

on the inferred value of the Hubble parameter. As de-scribed above, the large values of Neff allowed in thepresence of neutrino self-interactions reduce the size ofthe baryon drag scale, which allows a larger value of H0

without damaging the fit to the BAO scale [143] andwithout introducing extra damping at large multipoles(see Fig. 1 for an illustration of this latter effect). Inthe SIν model, this effect is compounded by the largervalue of θ∗ necessary to compensate for the absence of thefree-streaming neutrino phase shift. This further slightlyincreases the value of H0 necessary to fit the data, ascan be seen by comparing the two modes in the left-mostpanel of Fig. 7a.

It is worth noting that when Neff is fixed at 3.046,H0 and

∑mν are usually negatively correlated (see

e.g. Ref. [132]). If both Neff and∑mν are allowed to

vary, there is not a strong correlation between Neff and∑mν for CMB data alone. However, when H0 or BAO

data are added, Neff and∑mν become positively cor-

related [144] as shown in Fig. 6. The tight correlationbetween Neff and H0 then permits a positive correlationbetween H0 and

∑mν , seen in the third panel from the

left of Fig. 7a. Thus, instead of larger∑mν being corre-

lated with a smaller Hubble constant, here a larger sumof neutrino masses corresponds to a slightly larger H0.Allowing the neutrinos to self-interact does not dramat-ically change the direction of this degeneracy, but doesallow it to stretch to larger H0 values.Geff ’s direct effects on matter clustering are scale de-

pendent. As discussed in Sec. IV B, dark matter fluctu-ations that enter the horizon during neutrino decouplingreceive a boost, while fluctuations that enter the hori-zon before neutrino decoupling are damped. For the SIνmode, neutrino decoupling is coincident with the modesentering the horizon that contribute most to σ8, givingthem a gravitational boost. However, low values of As

and ns must accompany a large Geff for the SIν, as dis-cussed in Sec. VI B, which consequently damp these samescales. The combination of Geff ’s effects thus leads to anoverall decrease in matter clustering at scales probed byσ8, seen in the SIν island in the left panel of Fig. 7b.

The sum of neutrino masses is negatively correlatedwith σ8 since massive neutrinos do not contribute tomatter clustering for small scales where their pressureterm is large (see third panel in Fig. 7b). Typically large

16

4.5 3.0 1.5log10(Ge↵ MeV2)

60

65

70

75

H0[k

m/s

/Mpc]

2.4 3.0 3.6 4.2 4.8Ne↵

0.3 0.6 0.9 1.2m [eV]

0.66 0.72 0.78 0.848

0.3 0.6 0.9 1.2m [eV]

TT, TE, EE TT + lens + BAO TT + lens + BAO+H0

4.5 3.0 1.5log10(Ge↵ MeV2)

H0[k

m/s

/Mpc]

2.4 3.0 3.6 4.2 4.8Ne↵

0.3 0.6 0.9 1.2m [eV]

0.66 0.72 0.78 0.848

60 65 70 75H0 [km/s/Mpc]


(a) H0 correlations.

4.5 3.0 1.5log10(Ge↵ MeV2)

0.66

0.72

0.78

0.84

8

2.4 3.0 3.6 4.2 4.8Ne↵

0.3 0.6 0.9 1.2m [eV]

0.3 0.6 0.9 1.2m [eV]


4.5 3.0 1.5log10(Ge↵ MeV2)

8

2.4 3.0 3.6 4.2 4.8Ne↵

0.3 0.6 0.9 1.2m [eV]

0.66 0.72 0.78 0.848


(b) σ8 correlations.

FIG. 7: Correlations between H0 and σ8 with neutrino properties.

Neff boosts the dark matter fluctuations upon horizonentry, leading to a positive correlation between Neff andσ8. However, the positive correlation between Neff and∑mν when including BAO data causes Neff and σ8 to be

negatively correlated (see second panel in Fig. 7b). Thisallows the interacting neutrino model to both be compat-ible with a large value of the Hubble constant and notoverpredict the amplitude of matter fluctuations at latetimes.

E. Impact of CMB polarization data

As can be seen in Fig. 6 and Fig. 7, the addition of EEpolarization data tends to significantly reduce the statis-tical significance of the SIν cosmology. This is in con-trast with Ref. [65] which found that polarization dataslightly increased the significance of the SIν mode (seealso Ref. [66]). The degradation of the fit for the SIνmodel in our case is the result of (i) our use of the reion-ization optical depth prior from Ref. [142] whenever weuse polarization data, and (ii) our use of BBN calcula-tions to predict the helium abundance for a given value ofNeff . As we discuss in Sec. IX, it is likely that the fit couldimprove significantly by replacing this strong prior withthe actual low-` polarization data used to obtain it, andby letting the helium fraction float freely in the fit. Asmight be expected, the addition of the local Hubble con-stant measurement increases the statistical significance ofthe SIν mode, as can be seen from TT,TE,EE+lens+H0

data set in Fig. 11 in Appendix A.

VII. STATISTICAL SIGNIFICANCE

In this section, we quantify the relative statistical sig-nificance of the two modes of the posterior, and comparethe maximum likelihood values between our interactingneutrino models and standard extensions of the ΛCDMparadigm.

A. Mode Comparison

To determine the statistical significance of the SIνmode relative to the MIν mode, we can compare their rel-ative Bayesian evidence. It is defined as the parameter-averaged likelihood of the data

Z ≡ Pr (d|M) =

∫

Ωθ

Pr (d|θ,M) Pr (θ|M) dθ, (18)

where d is the data, M is the cosmological model, θ arethe parameters in model M, and Ωθ is the domain ofthe model parameters. We use Multinest’s [145] modeseparation algorithm to compute the Bayesian evidencefor each mode. In practice, this mode separation occursnear a neutrino coupling value of log10

(Geff MeV2

)≈

−2.2. This separation in parameter space defines Ωθ foreach mode.

To compare the SIν to the MIν mode, we compute thefollowing Bayes factor:

BSIν ≡Pr (MSIν |d)

Pr (MMIν |d)=ZSIν

ZMIν

Pr (MSIν)

Pr (MMIν). (19)

We place a uniform prior on log10

(Geff MeV2

)rather

than a uniform prior on Geff to avoid introducing a pre-ferred energy scale. With our choice of prior, small valuesof Geff can be thoroughly explored, which is particularly

17

important since the actual Fermi constant governing neu-trino interaction in the Standard Model takes the valueGF ∼ O

(10−11 MeV−2

). Taking a uniform prior on Geff

would greatly increase the statistical significance of theinteracting mode (see Ref. [55]). We thus consider it con-servative to adopt a uniform prior on log10

(Geff MeV2

),

but note that the statistical significance of the SIν modecould be greatly enhanced by a different choice of prior.

The probability of the prior is equivalent for each mode(or cosmological model), so Pr (MSIν) /Pr (MMIν) = 1.In Table IV we show the Bayes factor for each data setcombination we consider in this work. A Bayes factorless than unity indicates the data prefer the MIν modefor the specified parameter space. All values are belowunity, indicating the data, on average, do not prefer theSIν mode. As expected though, incorporating the localHubble rate measurement does increase the significanceof the SIν mode.

A useful method to understand if the SIν mode isever preferred and to further investigate the significance’sdependence on LSS data is to compare the maximum-likelihood value of each model:

RSIν =max [L (θSIν |d)]

max [L (θMIν |d)]. (20)

In Table IV we show the maximum-likelihood ratios forthe data set combinations in our analysis. Again, addingH0 and CMB lensing data increases the likelihood of thestrongly interacting mode. Intriguingly, the SIν modehas a larger maximum-likelihood (by a factor larger than2) than the MIν mode for TT+lens+BAO+H0 (see theunsmoothed posteriors in Fig. 11). It is reasonable thatthe Bayes factor for TT+lens+BAO+H0 is below unitywhile the maximum-likelihood ratio is above unity sincethe former is a global, parameter-averaged statistic whilethe latter is based on a single set of best-case scenarioparameters. This indicates that the parameter space forwhich strong neutrino interactions are preferred has asmall volume.

It is also informative to look at the individual χ2 valuesfor the different data sets. To compare the two modes, welist the ∆χ2 = χ2

SIν−χ2MIν values in Table IV. A positive

∆χ2 value thus means that the MIν mode is preferred,and vice versa. The H0 and high-` TT,TE,EE data showpreference for the SIν mode for TT,TE,EE+lens+H0,but this is compensated by a poorer fit to low-` and CMBlensing data. For the TT+lens+BAO data combinations,the BAO and high-` TT data display a slight preferencefor the SIν mode, which is again overshadowed by thelow-` data. We see that the slight preference for the SIνmode with the TT+lens+BAO+H0 data combination islargely due to improvement of the BAO and high-` like-lihoods.

B. Comparison to ΛCDM and its extensions

Comparing how well each mode fits the data rel-ative to ΛCDM and its common extensions tells usif these neutrino self-interaction models offer a viableimprovement to current cosmological theory. For theTT+lens+BAO+H0 data set, we list the ∆χ2 = χ2

SIν −χ2

ΛCDM+ext values and the ∆χ2 = χ2MIν−χ2

ΛCDM+ext val-ues for each observable in Table V. Here, ΛCDM + extrefers to the Neff +

∑mν two-parameter extension of

the ΛCDM cosmology. Comparison to plain ΛCDM wasgiven in Table III above. For all data sets except the low-` TT data, both modes offer a better fit to the data thanΛCDM + ext. In fact, the SIν mode has a total ∆χ2 of−3.33, a significant difference. The improvement of thehigh-` CMB data is notable since jointly fitting CMBand local H0 data usually results in a worse fit to theCMB damping tail. For the SIν model, this is somewhatcompensated by a degradation of the low-` fit.

What if the strong improvement in fit over ΛCDMis due to overfitting from the extra parameter we haveadded? To take this into account we compute the Akaikeinformation criterion (AIC) [146]. The AIC takes intoaccount how well the model fits the data and penalizesextra parameters, thereby discouraging overfitting. TheAIC is defined as

AIC = −2 ln (L) + 2k = χ2Total + 2k, (21)

where χ2Total = χ2

low ` + χ2high ` + χ2

lens + χ2H0

+ χ2BAO, L

is the maximum-likelihood, and k is the number of fitparameters. Then we can write

∆AIC = AICIν −AICΛCDM = ∆χ2 + 2∆k, (22)

where ∆k is the difference in the number of parametersbetween the two models. The lower AIC between twomodels corresponds to the preferred model. Thus, forus, a negative ∆AIC value indicates the data prefer thespecified Iν model over ΛCDM, while a positive ∆AICvalue indicates the data prefer ΛCDM over the Iν model.

We list the ∆AIC values relative to ΛCDM + Neff +∑mν in Table V. Here ∆k = 1, and the SIν mode has a

negative ∆AIC = −1.33, indicating a genuine statisticalpreference for the suppression of neutrino free-streamingin the early Universe for the TT+lens+BAO+H0 dataset. On the other hand, ∆AIC = 0.19 for the MIν mode,indicating that the neutrino self-interactions do not addvalue to the fit beyond what is already provided by theNeff +

∑mν extension. Values of ∆AIC between the Iν

models and standard ΛCDM (∆k = 3) are also given inTable III. The fact that ∆AIC values for the SIν cosmol-ogy are similar (−1.91 versus −1.33) when comparing itto plain ΛCDM and ΛCDM +Neff +

∑mν means that

suppressing neutrino free-streaming is the true drivingfactor behind the improvement of the fit. Thus, even afterpenalizing the self-interacting neutrino models for incor-porating additional parameters, the TT+lens+BAO+H0

data still significantly prefer the strongly interacting neu-trino cosmology over ΛCDM.

18

TABLE IV: Mode Comparison. Here, BSIν is the Bayes factor between the SIν and the MIν mode, RSIν is theiemaximum likelihood ratio, and ∆χ2 = χ2

SIν − χ2MIν . The low-` dataset refers to low-` TEB if polarization was

included and low-` TT if only temperature was used. Similarly, the high-` dataset refers to high-` TT,TE,EE ifpolarization was included and high-` TT if only temperature was used.

Parameter TT,TE,EE TT,TE,EE + lens + H0 TT + lens + BAO TT + lens + BAO + H0

BSIν 0.03± 0.01 0.10± 0.04 0.13± 0.04 0.37± 0.10RSIν 0.26 0.63 0.81 2.14

∆χ2low ` 2.47 2.18 2.00 1.41

∆χ2high ` 0.22 −0.16 −1.53 −2.23

∆χ2lens – 1.34 0.16 0.30

∆χ2H0

– −2.12 – −0.56∆χ2

BAO – – −0.20 −0.44∆χ2

Total 2.69 0.92 0.43 −1.52

TABLE V: Comparison of the interacting neutrino cosmology to ΛCDM +Neff +∑mν for TT + lens + BAO + H0


∆χ2low ` 2.40 0.99

∆χ2high ` −3.40 −1.17

∆χ2lens −0.20 −0.50

∆χ2H0

−1.32 −0.76∆χ2

BAO −0.81 −0.36

∆χ2Total −3.33 −1.81

∆AIC −1.33 0.19

VIII. DISCUSSION

A. Cosmic microwave background

In Fig. 8, we plot the high-` TT and EE power spec-tra residuals between the maximum likelihood parame-ters for each data set combination used and the best-fitPlanck ΛCDM model. For the SIν mode (upper panels),the most striking feature of the residuals is the deficitof power at high multipoles (` > 1500) as compared toΛCDM for the TT+lens+BAO and TT+lens+BAO+H0

data combinations. This is caused by the large value ofNeff and the resulting high helium abundance6 YP forthis category of models. This implies that the multi-parameter degeneracy that allows the SIν cosmology toprovide a decent fit to CMB temperature data at ` <1500 could be broken by the addition of high-resolutionCMB data (see e.g. Refs. [147, 148]). However, it is rea-sonable to assume that the BBN helium abundance ismodified in the presence of the new neutrino physics weexplore here, and that the deficit of power at large mul-tipoles could be compensated by a smaller value of YP[133]. We leave the study of the impact of a free he-

6 We remind the reader that we use the standard BBN predic-tions to compute the helium abundance for given Neff and Ωbh

2

values.

lium fraction on interacting neutrino cosmologies to fu-ture works.

The EE polarization residuals shown in the right panelof Fig. 8 for the TT+lens+BAO and TT+lens+BAO+H0

data combinations also display strong oscillations for theSIν mode (upper panel). This implies that the shift in θ∗(and other parameters, see Sec. VI B) that was requiredto compensate for the absence of the free-streaming neu-trino phase shift in the temperature spectrum does notfully realign the peaks of the polarization spectrum withthe data. This is a consequence of the polarization databeing more sensitive to the phase of the acoustic peaks[129]. With the current size of the Planck error bars, thisdoes not constitute an overwhelmingly strong constrainton the absence of a neutrino-induced phase shift, but itis possible that future CMB polarization data could en-tirely rule out this possibility.

The TT+TE+EE CMB-only data combination in theupper panels of Fig. 8 display an excess of power as com-pared to ΛCDM at nearly all scales, resulting in an over-all poorer fit to the CMB data. At large multipoles,this is of course in contrast with the deficit of power thatthe TT+lens+BAO and TT+lens+BAO+H0 fits display.Our use of the polarization-driven prior on the reioniza-tion optical depth from Ref. [142] is largely responsiblefor this excess of power as compared to ΛCDM for theSIν mode with the TT+TE+EE data set. Again, thisshows that polarization data could in principle break themulti-parameter degeneracy that allows the SIν cosmol-

19

0 500 1000 1500 2000 2500 3000

`

−0.03

−0.01

0.00

0.01

0.03

0.05

( CTT

`,M

Iν−CTT

`,Λ

CD

M

) /CTT

`,Λ

CD

M


TT, TE, EE


−0.04

−0.02

0.02

0.04

( CTT

`,S

Iν−CTT

`,Λ

CD

M

) /CTT

`,Λ

CD

M


0 500 1000 1500 2000 2500 3000

`

−0.04

−0.02

0.00

0.02

0.04

0.06

0.08

( CEE

`,M

Iν−CEE

`,Λ

CD

M

) /CEE

`,Λ

CD

M


TT, TE, EE


−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

0.08

( CEE

`,S

Iν−CEE

`,Λ

CD

M

) /CEE

`,Λ

CD

M

Planck 2015

FIG. 8: Relative difference between the SIν mode (upper panels) or MIν mode (lower panels) and ΛCDM for thehigh-` TT (left) and EE (right) power spectra. The SIν mode and MIν mode spectra are produced using themaximum likelihood parameter values for each respective mode. Colors denote the data set combination used.

Measurements from the Planck 2015 data release are included [125].

ogy to exist.

All dataset combinations we consider display an excessof power at ` < 50 for the SIν mode. This is largelycaused by the lower value of the scalar spectral index ns

which adds power on large scales. While error bars arelarge in this regime due to cosmic variance, the dip inpower around ` ∼ 20-30 in the CMB temperature datatends to penalize any model displaying more low-` powerthan ΛCDM. If this dip were to be explained by someother physics (from the inflationary epoch, say), then itis possible that the fit to the data using the SIν cosmologycould significantly improve.

It is important to emphasize how the suppression ofneutrino free-streaming plays a very important role inthe existence of the SIν cosmology. To illustrate this,we plot in Fig. 9 the spectra corresponding to the best-fit SIν parameters for the TT+lens+BAO+H0 data setbut allow neutrino to free-stream at all times by settingGeff = 0 (red dashed-dot line), along with the originalTT+lens+BAO+H0 best-fit SIν model (solid red) and aΛCDM model with

∑mν = 0.23 eV (black dashed), for

reference. Here, the difference between the dashed-dotand solid red lines is entirely driven by the streamingproperty of neutrinos. Figure 9 reinforces our discussionfrom Sec. VII B that the Geff parameter plays a statis-tically significant role in improving the fit to the CMBdata, beyond what is already provided by theNeff+

∑mν

two-parameter extension of ΛCDM.

B. Matter clustering

We show in Fig. 10a the linear matter power spec-trum residuals between the best-fit SIν (and MIν) mod-els and the corresponding ΛCDM models. The moststriking feature for the SIν mode is the overall red tiltof the matter power spectrum residuals for all datacombinations shown. This tilt is due to the low pre-ferred value of ns for this mode. Despite this globalshape difference with ΛCDM, the enhancement of mat-ter fluctuations on scales entering the horizon at theonset of neutrino free-streaming discussed in Sec. IV Bcauses the matter power spectrum to only slightly de-viate from the CDM prediction on scales contributingthe most to σ8 (0.02hMpc−1 . k . 0.2hMpc−1). Forthe TT+lens+BAO and TT+lens+BAO+H0 data com-binations, this difference is less than 5% on these scalesand results in a σ8 value that is slightly lower than inΛCDM, potentially bringing low-redshift measurementsof the amplitude of matter fluctuations in agreement withCMB data as discussed in Sec. VI D.

Nevertheless, it is important to note that the localizedfeature in the matter power spectrum caused by the lateonset of neutrino free-streaming nearly coincides with the

20

0 500 1000 1500 2000 2500 3000

`

−0.26

−0.22

−0.18

−0.14

−0.10

−0.06

−0.02

0.02

0.06

( CEE

`,S

Iν−CEE

`,Λ

CD

M

) /CEE

`,Λ

CD

M



TT + lens + BAO + H0

SI νMode withGeff = 0 MeV−2

−0.22

−0.18

−0.14

−0.10

−0.06

−0.02

0.02

( CTT

`,S

Iν−CTT

`,Λ

CD

M

) /CTT

`,Λ

CD

M


FIG. 9: Illustration of the importance of the neutrinoself-interaction to the fit to CMB data for the SIν

cosmology. The red solid spectra corresponds to thebest-fit SIν model, while the red dashed-dot spectra use

the same best-fit cosmological parameters but allowsneutrino free-streaming by setting Geff = 0.

BAO scale, that is, it is on scales where we have a largeamount of data from, for example, spectroscopic galaxysurveys (see e.g. Ref. [140]). While an analysis that takesinto account the full shape of the measured galaxy powerspectrum at these scales is beyond the scope of this work,we note that both the SIν and MIν cosmologies onlymildly deviate from the ΛCDM model near the BAOscale. On smaller scales, the SIν mode displays a netsuppression of power which has implications for probesof small-scale structure such as the Lyman-α forest [149]and the satellite galaxy count surrounding the Milky Way[150]. It is an interesting possibility that the SIν cosmol-ogy could help alleviate the small-scale structure prob-lems [151] without introducing a nongravitational cou-pling between neutrinos and dark matter.

The MIν residuals (lower panel) in Fig. 10a displayan even richer structure than those shown in Fig. 3. In-deed, even in the case of relatively weak neutrino inter-actions, their impact on the matter power spectrum issignificant, and potentially provide a different channelto constrain new physics in the neutrino sector. Sincethe dominant constraining power of the data used herecomes from k ∼ 0.1hMpc−1, we observe that the MIνpower spectra have values similar to ΛCDM near thisscale. Outside the scales probed by σ8, the linear matter

power spectra deviate more significantly (up to ∼ 20%)from ΛCDM.

The lensing potential power spectrum in Fig. 10bshows a similar pattern to the matter power spectrumfor the different best-fit models, as expected. The cur-rent large error bars of the Planck lensing measurementsallow substantial freedom to the SIν and MIν cosmolo-gies. As shown in Table IV, the lensing data prefer theMIν mode for all data-set combinations, but we note thatthe SIν modes are typically within the error bars of thelensing data.

IX. CONCLUSIONS

The presence of yet-unknown neutrino interactionstaking place in the early Universe could delay the onset ofneutrino free-streaming, imprinting the CMB and probesof matter clustering with distinct features. We have per-formed a detailed study of the impact of neutrino selfinteractions with a rate scaling as Γν ∼ G2

effT5ν on the

CMB and the matter power spectrum, taking into ac-count the presence of nonvanishing neutrino masses andof a nonstandard neutrino thermal history. Using recentmeasurements of the BAO scale, the local Hubble rate,and of the CMB, we find that a cosmological scenario(originally pointed out in Ref. [55]) in which the onsetof neutrino free-streaming is delayed until close to theepoch of matter-radiation equality can provide a goodfit to CMB temperature data while also being consistentwith the Hubble constant inferred from the local distanceladder [73].

This strongly interacting neutrino cosmology has thefollowing properties:

• Using the data combination TT+lens+BAO+H0,it displays a strong preference (> 3σ) for an ad-dition neutrino species (Neff = 4.02 ± 0.29, 68%C.L.). This can have important implications giventhe current anomalies in neutrino oscillation exper-iments. It also prefers a nonvanishing value of thesum of neutrino masses

∑mν = 0.42+0.17

−0.20 eV (68%C.L.).

• It can easily accommodate a larger value of H0 andsmaller σ8, hence possibly alleviating tensions be-tween current measurements. Quantitatively, thedata combination TT+lens+BAO+H0 favors H0 =72.3± 1.4 km s−1 Mpc−1 and σ8 = 0.786± 0.020 at68% C.L.

It is remarkable that a cosmological model admittingparameter values that are so different (see Fig. 5) thanin the standard ΛCDM paradigm can provide a betterfit to the data at a statistically-significant level (∆AIC=−1.91). We believe that this is the most important lessonto be drawn from our work: While most analyses havefocused on mild deformation from the standard ΛCDMscenario in trying to reconcile the current cosmological

21

10−4 10−3 10−2 10−1 100 101 102

k [h/Mpc]

0.85

0.95

1.05

1.15

1.25

Pm,M

Iν(k

)/P

m,Λ

CD

M(k

)


TT, TE, EE


0.75

0.85

0.95

1.05

1.15

Pm,S

Iν(k

)/P

m,Λ

CD

M(k

)

(a) Linear matter power spectrum

10 100 1000

L

−0.2

−0.1

0.0

0.1

0.2

( Cφφ

L,M

Iν−Cφφ

L,Λ

CD

M

) /Cφφ

L,Λ

CD

M


TT, TE, EE


−0.2

−0.1

0.0

0.1

0.2

( Cφφ

L,S

Iν−Cφφ

L,Λ

CD

M

) /Cφφ

L,Λ

CD

M

Planck 2015 MV

(b) CMB Lensing power spectrum

FIG. 10: Relative difference between the SIν mode (upper panels) or MIν mode (lower panels) and ΛCDM for thelinear matter power spectrum (left) and the CMB lensing power spectrum (right). The SIν and MIν spectra are

produced using the maximum likelihood parameter values for each respective mode. Measurements from the Planck2015 data release [125] are included in the right panel.

datasets, it is important to entertain the possibility thata radically different scenario (i.e. statistically disjointin cosmological parameter space) could provide a betterglobal fit to the data.

Despite the success of the strongly interacting neutrinocosmology in addressing tensions between certain cosmo-logical data sets, there are several important obstaclesthat still tilt the balance towards the standard ΛCDMcosmology. First, the addition of polarization data seemsto degrade the quality of the fit for the strongly interact-ing neutrino cosmology. We have traced this deterio-ration of the fit to our use of a Gaussian prior on thereionization optical depth from Ref. [142]. This priorwas utilized as a way to capture the constraint on theoptical depth from low-` HFI Planck polarization databefore the full likelihood is made available. It it likelythat the Gaussian form of the prior leads to constraintsthat are too strong as compared to what the full like-lihood will provide. Only a complete analysis with thelegacy Planck data, once available, will allow us to deter-mine whether this is the case. An important fact to keepin mind is that Ref. [65] found that the addition of CMBpolarization data (without an additional τ prior) tendsto increase the statistical significance of the strongly in-teracting neutrino cosmology.

Second, the low values of the Bayes factor (see Ta-

ble IV) consistently favor either very weakly interactingneutrinos or no interaction at all. This reflects the factthat strongly interacting neutrinos can only fit the databetter for a narrow window of interaction strengths, whileΛCDM provides a decent (but overall less good) fit overa broader part of the parameter space. This is a funda-mental feature of Bayesian statistics and it is unlikely tochange in future analyses. This highlights the need toconsider a portfolio of statistical measures to assess thequality of a given cosmological model.

Third, it might be difficult from a particle model-building perspective to generate neutrino self interactionswith the strength required by the strongly interactingneutrino cosmology while not running afoul of other con-straints on neutrino physics. A viable model might looksimilar to that presented in Ref. [32], but it remains to beseen whether the necessary large interaction strength canbe generated while evading current constraints [103] onnew scalar particles coupling to Standard Model neutri-nos. It is also possible that a successful self-interactingneutrino model could have a different temperature de-pendence than that considered in this work (Γν ∝ T 5

ν ).This would change the shape of the neutrino visibilityfunction (see Refs. [55, 65]) and potentially improve theglobal fit to the data. We leave the study of differenttemperature scalings of the neutrino interacting rate to

22

future works.

Our analysis could be improved in a few different ways.Given the computational resources we had at our disposaland the need to obtain accurate values of the Bayesianevidence, we used the “lite” version of the Planck high-`likelihoods in our analysis. Since some of the assump-tions that went into generating these likelihoods [152]might not apply to the interacting neutrino cosmologies,it would be interesting, given sufficient computing power,to reanalyze these models with the complete version ofthe likelihoods that include all the nuisance parameters.In particular, it is possible that some of the foregroundnuisance parameters might be degenerate with the effectof self-interacting neutrinos. For simplicity, we have alsoassumed that the helium fraction is determined by thestandard big-bang nucleosynthesis calculation through-out our analysis. Given the new physics and the result-ing modified thermal history of the neutrino sector forthe type of models we explore here, it reasonable to as-sume that the helium fraction would in general be differ-ent than in ΛCDM. While the details of the helium pro-duction within any interacting neutrino model are likelymodel-dependent, a sensible way to take these effects intoaccount would be to let the helium fraction float freelyin the fit to CMB data. We leave such analysis to futureworks.

Given the structure of the residuals between thebest-fit interacting neutrino cosmologies and the ΛCDMmodel presented in Sec. VIII, it is clear that future high-` CMB polarization and matter clustering measurementswill play an important role in constraining or ruling outthese models [see e.g. 153]. In particular, the overall redtilt of the matter power spectrum in the strongly inter-acting neutrino cosmology could have important conse-quences on both large and small scales. Since currentanomalies in terrestrial neutrino experiments [2, 3] mayindicate the presence of new physics in the neutrino sec-tor, it is especially timely to use the complementary na-ture of cosmological probes to look for possible clues

about physics beyond the Standard Model.

ACKNOWLEDGMENTS

We thank Kris Sigurdson and Roland de Putter for col-laboration at early stages of this work. We also thank JoDunkley, David Spergel, and Lyman Page for commentson an early version of this manuscript, and PrateekAgrawal and David Pinner for useful conversations.C. D. K. acknowledges the support of the NationalScience Foundation award number DGE1656466 atPrinceton University and of the Minority UndergraduateResearch Fellowship at the Jet Propulsion Laboratory.F.-Y. C.-R. acknowledges the support of the NationalAeronautical and Space Administration (NASA) ATPgrant NNX16AI12G at Harvard University. This workwas performed in part at the California Institute ofTechnology for the Keck Institute for Space Studies,which is funded by the W. M. Keck Foundation. Partof the research described in this paper was carried outat the Jet Propulsion Laboratory, California Institute ofTechnology, under a contract with NASA. The compu-tations in this paper were run on the Odyssey clustersupported by the FAS Division of Science, ResearchComputing Group at Harvard University.

Appendix A: Results for all data sets

We display in Table VI and Table VII the 68% confi-dence limits for the strongly interacting and moderatelyinteracting neutrino modes, respectively. In Fig. 11, weshow the marginalized posteriors for key cosmological pa-rameters for a choice of smoothing kernel that representsmore accurately the shape of the SIν mode. In Fig. 12,we compare the marginalized posterior distribution of theSIν mode for the four data set combinations consideredin this work.

Appendix B: Perturbation equations for interacting massive neutrinos

In this appendix, we derive the Boltzmann equation governing the evolution of the distribution function of massiveself-interacting neutrinos which we denote by fν(x,P, τ), where P is the canonical conjugate variable to the positionx, and τ is the conformal time. In the scenario considered here, neutrinos can exchange energy and momentum via2-to-2 scattering of the type νi + νj → νk + νl. The Boltzmann equation of neutrino species i can be written as

dfνidλ

=

3∑

j,k,l=1

Cνi+νj→νk+νl [fνi , fνjfνk , fνl ] (B1)

where λ is an affine parameter that described the trajectory of the observer (see below) and Cνi+νj→νk+νl is thecollision term for the process νi + νj → νk + νl. In the conformal Newtonian gauge, the space-time metric takes theform

ds2 = a2(τ)[−(1 + 2ψ)dτ2 + (1− 2φ)d~x2], (B2)

23

TABLE VI: Strongly interacting neutrino cosmology: Parameter 68% Confidence Limits


Ωbh2 0.02219+0.00024

−0.00022 0.02257± 0.00018 0.02239+0.00029−0.00036 0.02245+0.00029

−0.00033

Ωch2 0.1189+0.0032

−0.0038 0.1222± 0.0032 0.1311+0.0090−0.0065 0.1348+0.0056

−0.0049

100θMC 1.04590+0.00061−0.00045 1.04622+0.00054

−0.00042 1.04623+0.00067−0.00044 1.04637± 0.00056

τ 0.0633+0.0089−0.0066 0.0624+0.0089

−0.0074 0.082+0.028−0.036 0.080± 0.031∑

mν [eV] 0.166+0.064−0.18 0.069+0.027

−0.066 0.39+0.16−0.20 0.42+0.17

−0.20

Neff 2.88+0.19−0.22 3.20± 0.18 3.80± 0.45 4.02± 0.29

log10(GeffMeV2) −1.60+0.14−0.089 −1.55+0.12

−0.080 −1.41+0.20−0.066 −1.35+0.12

−0.066

ln(1010As) 2.995+0.019−0.015 2.994± 0.017 3.036+0.054

−0.071 3.035± 0.060ns 0.9273± 0.0080 0.9412± 0.0061 0.947± 0.011 0.9499± 0.0098

H0 [km/s/Mpc] 66.2+2.3−1.9 70.1± 1.3 71.1± 2.2 72.3± 1.4

Ωm 0.327+0.013−0.026 0.2961+0.0075

−0.011 0.3115± 0.0090 0.3094± 0.0083σ8 0.799+0.041

−0.017 0.824+0.015−0.010 0.786± 0.020 0.786± 0.020

109As 1.998+0.039−0.030 1.998± 0.034 2.09+0.10

−0.15 2.08+0.11−0.13

109Ase−2τ 1.760± 0.014 1.763± 0.013 1.766± 0.016 1.771± 0.016

YP 0.2430± 0.0029 0.2476± 0.0024 0.2549+0.0060−0.0048 0.2577± 0.0034

r∗ [Mpc] 145.8± 2.0 143.0± 1.7 138.2+3.2−4.3 136.3± 2.4

100θ∗ 1.04626+0.00060−0.00046 1.04629+0.00053

−0.00044 1.04604+0.00060−0.00046 1.04604± 0.00056

DA [Gpc] 13.93± 0.19 13.67± 0.16 13.21+0.30−0.41 13.03± 0.23

rdrag [Mpc] 148.5± 2.1 145.6± 1.8 140.8+3.3−4.3 138.8± 2.5

TABLE VII: Moderately Interacting Neutrino Mode: Parameter 68% Confidence Limits


Ωbh2 0.02203± 0.00023 0.02246± 0.00018 0.02254+0.00030

−0.00035 0.02282± 0.00030Ωch

2 0.1191± 0.0031 0.1220± 0.0027 0.1220+0.0039−0.0046 0.1256+0.0035

−0.0039

100θMC 1.04085± 0.00044 1.04063± 0.00040 1.04086± 0.00058 1.04062+0.00049−0.00056

τ 0.0642+0.0095−0.0082 0.0645+0.0090

−0.0073 0.108± 0.033 0.127+0.034−0.029∑

mν [eV] 0.150+0.054−0.16 0.052+0.020

−0.052 0.28+0.12−0.23 0.40+0.17

−0.23

Neff 2.95± 0.19 3.29± 0.16 3.44+0.30−0.38 3.79± 0.28

log10(GeffMeV2) −4.44+0.58−0.77 −4.26± 0.69 −4.12± 0.77 −3.90+1.0

−0.93

ln(1010As) 3.059+0.022−0.019 3.067+0.019

−0.016 3.150± 0.067 3.194+0.068−0.056

ns 0.9548± 0.0089 0.9718± 0.0073 0.980+0.014−0.015 0.993+0.013

−0.012

H0 [km/s/Mpc] 65.3+2.2−1.7 69.3± 1.2 69.3+1.7

−1.9 71.2± 1.3Ωm 0.335+0.012

−0.025 0.3021+0.0077−0.010 0.3075± 0.0092 0.3010± 0.0080

σ8 0.798+0.038−0.016 0.826+0.014

−0.011 0.809+0.021−0.018 0.813+0.023

−0.020

109As 2.132± 0.043 2.148+0.039−0.035 2.34+0.14

−0.18 2.44± 0.15109Ase

−2τ 1.875± 0.018 1.888± 0.016 1.880± 0.021 1.892+0.019−0.017

YP 0.2439± 0.0027 0.2486± 0.0022 0.2506+0.0041−0.0048 0.2550± 0.0035

r∗ [Mpc] 145.5± 1.9 142.8± 1.5 141.9+3.0−2.7 139.1± 2.3

100θ∗ 1.04117± 0.00054 1.04066± 0.00047 1.04086± 0.00070 1.04041+0.00058−0.00064

DA [Gpc] 13.97± 0.17 13.72± 0.14 13.63+0.28−0.25 13.37± 0.21

rdrag [Mpc] 148.3± 1.9 145.4± 1.6 144.5+3.1−2.8 141.6± 2.3

where a is the cosmological scale factor and φ and ψ are the two gravitational potentials. We can define the affineparameter in terms of the four-momentum P of an observer

Pµ ≡ dxµ

dλ, (B3)

where x = (τ, ~x) is a four-vector parametrizing the trajectory of the observer. Using Eq. (B2), we can then write

d

dλ=dτ

dλ

d

dτ= P 0 d

dτ=E(1− ψ)

a

d

dτ, (B4)

where we have used the dispersion relation gµνPµP ν = −m2

ν . Here, we have defined E =√p2 +m2

ν , where p = |p|is the proper momentum, which is related to the conjugate momentum P via the relation p2 = gijP

iP j . We note

24

−4.5 −3.0 −1.5 0.0log10(Geff MeV2)

1.0400 1.0425 1.0450 1.0475100θ∗

1.74 1.80 1.86 1.92109Ase

−2τ0.93 0.96 0.99 1.02

ns

TT,TE,EE TT,TE,EE + lens + H0 TT + lens + BAO TT + lens + BAO + H0

FIG. 11: 1D posteriors for bimodal parameters with low smoothing.

that Eq. (B4) is valid to first-order in perturbation theory. As in other work in the literature, we choose to writethe distribution function in terms of the proper momentum p. This choice is valid as long as we also modify thephase-space volume element as d3P → a3(1 − 3φ)d3p. The left-hand side of the Boltzmann equation takes the form[123, 134]

dfνidτ

=∂fνi∂τ

+p

E· ∇fνi + p

∂fνi∂p

[−H+

∂φ

∂τ− E

p2p · ∇ψ

], (B5)

where H ≡ a/a is the conformal Hubble parameter, a overhead dot denoting a derivative with respect to conformaltime. We expand the neutrino distribution function as

fνi(x,p, τ) = f (0)νi (p, τ)[1 + Θνi(x,p, τ)]. (B6)

At early times, neutrinos form a relativistic tightly-coupled fluid with an equilibrium background distribution function

f(0)ν (p, τ) given by the Fermi-Dirac distribution. If the interactions mediated by the Lagrangian in Eq. (1) go out

of equilibrium while neutrinos are relativistic, the background distribution function would maintain this shape, witha temperature red shifting as Tν ∝ a−1. As mentioned in Sec. III, we work under this approximation here andassume that the background distribution function maintains its equilibrium shape throughout the epoch of neutrinodecoupling. In the absence of energy source or sink, and for the type of interaction we consider in this work, this isan excellent approximation [66].

Substituting Eq. (B6) in Eq. (B5) and keeping terms that are first order in the perturbation variables, we obtain

f (0)νi (p, τ)

[∂Θνi

∂τ+

p

E· ∇Θνi

]+ p

∂f(0)νi (p, τ)

∂p

[−HΘνi +

∂φ

∂τ− E

p2p · ∇ψ

]=

a

EC(1)νi [p], (B7)

where the superscript C(1)νi denotes the part of the collision term that is first order in the perturbation variables Θνi .

It is useful at this point to introduce the comoving momentum q ≡ ap and comoving energy ε ≡ aE. Going to Fourierspace, Eq. (B7) becomes

f (0)νi (q, τ)

[∂Θνi

∂τ+ i

q

εkµΘνi

]+ q

∂f(0)νi (q, τ)

∂q

[∂φ

∂τ− i ε

qkµψ

]=a2

εC(1)νi [q], (B8)

25

−2.1 −1.8 −1.5 −1.2log10(GeffMeV2)

2.4 3.2 4.0 4.8Neff

0.0 0.3 0.6 0.9 1.2Σmν[eV]

132 138 144 150 156rdrag

60 65 70 75H0[km/s/Mpc]

0.65 0.70 0.75 0.80 0.85σ8

1.0435 1.0450 1.0465 1.0480100θ∗

1.71 1.74 1.77 1.80 1.83109Ase

−2τ0.90 0.92 0.94 0.96 0.98

ns

0.0216 0.0224 0.0232Ωbh

20.105 0.120 0.135 0.150

Ωch2

0.28 0.32 0.36 0.40 0.44Ωm

TT,TE,EE TT,TE,EE + lens + H0 TT + lens + BAO TT + lens + BAO + H0

FIG. 12: SIν mode posteriors for all data set combinations.

where Θνi is the Fourier transform of the perturbation variable Θνi , k is the Fourier conjugate of x, k = |k|, µ ≡ q · k,

and k = k/k. In this work, we focus on (helicity) scalar perturbations and expand the angular dependence of the Θνi

variable in Legendre polynomials Pl(µ)

Θνi(q,k, τ) =

∞∑

l=0

(−i)l(2l + 1)θl(k, q, τ)Pl(µ). (B9)

We note that this decomposition is always valid for scalar perturbations since they must be azimuthally symmetric withrespect to k, independently of the structure of the collision term. Substituting the above expansion in the first-order

Boltzmann equation and and integrating both sides with 12(−i)l

∫ 1

−1dµPl(µ) yields the hierarchy of equations

f (0)νi

[∂θl∂τ

+ kq

ε

(l + 1

2l + 1θl+1 −

l

2l + 1θl−1

)]+ q

∂f(0)νi

∂q

[∂φ

∂τδl0 +

k

3

ε

qψδl1

]=a2

ε

1

2(−i)l∫ 1

−1

dµPl(µ)C(1)νi [q], (B10)

where δij is the Kroenecker delta and where we have suppressed the arguments of f(0)νi and θl for succinctness. We

now turn our attention to the collision integral.

26

Appendix C: Collision Integrals

We now compute the first-order collision term for neutrino scattering, νi(p1) + νj(p2)↔ νk(p3) + νl(p4). We startfrom the general expression [154]

Cν [p1] =1

2

∫dΠ2dΠ3dΠ4|M|2ν(2π)4δ4(P1 + P2 − P3 − P4)F (p1,p2,p3,p4), (C1)

where |M|2ν here is the spin-summed (not averaged) matrix element for the scattering as defined in Eq. (3), pi denotesthe ith three-momentum, pi = |pi|, and where

dΠi =d3pi

(2π)32Ei, (C2)

and

F (p1,p2,p3,p4) = fν(p4)fν(p3)(1− fν(p2))(1− fν(p1))− fν(p2)fν(p1)(1− fν(p4))(1− fν(p3)). (C3)

Using Eq. (B6) and keeping only the first order term in the perturbation variable Θν , we can rewrite the collisionterm as:

C(1)ν =

1

2

∫dΠ2dΠ3dΠ4|M|2ν

(2π)4δ4(P1 + P2 − P3 − P4)e(p1+p2)/T

(ep1/T + 1)(ep2/T + 1)(ep3/T + 1)(ep4/T + 1)

×(

2(1 + e−p3/T )Θν(p3)− (1 + e−p2/T )Θν(p2)− (1 + e−p1/T )Θν(p1)), (C4)

where we have suppressed the x and τ dependence of the Θν variables to avoid clutter, and where we have used thesymmetry p3 ↔ p4 to simplify the integrand. Here, we have taken the background neutrino distribution functionto have a relativistic Fermi-Dirac shape. As mentioned in Sec. III, we assume neutrinos decouple in the relativisticregime and thus neglect the small neutrino mass in the computation of the collision integrals, gµνP

µP ν ≈ 0 andE ≈ p. We use the technique developed in Refs. [56, 155, 156] to perform the majority of the integrals. We firstperform the p4 integration using the identity

d3pi2Ei

≡ d4Piδ(P2i )H(P 0

i ), (C5)

where H(x) is the Heaviside step function. The collision term then reduces to:

C(1)ν = π

∫dΠ2dΠ3|M|2νδ(2(P1 · P2 − P1 · P3 − P2 · P3))H(p1 + p2 − p3)F (p1, p2, p3, p1 + p2 − p3), (C6)

where we have gathered all the terms dependent on the distribution functions inside F . To make progress, we haveto choose a coordinate system. We take p1 to point in the z-direction, and p3 to lie in the x-z plane. and define thefollowing angles:

p1 · k = µ p1 · p2 = cosα, p1 · p3 = cos θ, p2 · p3 = cosα cos θ + sinα sin θ cosβ, (C7)

where k is the Fourier wavenumber of the perturbations and β is the azimuthal angle for p2 to wrap around p1. Theintegration measure than takes the form

d3p3 = p23dp3d cos θdφ d3p2 = p2

2dp2d cosαdβ. (C8)

The φ angle is the azimuthal angle for p3 to wrap around p1. Since we are only dealing with scalar perturbations here,we are free to redefine this angle at will since no physical quantity depends on it. Within this coordinate system, we canwrite P1 ·P2 = −p1p2+p1p2 cosα, P1 ·P3 = −p1p3+p1p3 cos θ, and P2 ·P3 = −p2p3+p2p3(cosα cos θ+sinα sin θ cosβ).We can now use the delta function to do the β integration. Setting the argument of the delta function to zero andsolving for cosβ yields

cosβ = −p1p2 − p1p3 − p2p3 − p1p2 cosα+ p1p3 cos θ + p2p3 cosα cos θ

p2p3 sinα sin θ. (C9)

27

Performing the integration introduces a Jacobian

C(1)ν =

1

8(2π)5

∫p2dp2p3dp3d(cosα)d(cos θ)dφ√

aα cos2 θ + bα cos θ + cα|M|2ν

×H(p1 + p2 − p3)F (p1, p2, p3, p1 + p2 − p3)H(aα cos2 θ + bα cos θ + cα). (C10)

In the above, aα, bα, and cα are

aα = −p23(p2

1 + p22 + 2p1p2 cosα), (C11)

bα = 2p3(p1 + p2 cosα)(p2p3 + p1(p3 − p2) + p1p2 cosα), (C12)

cα = −(p2p3 + p1(p3 − p2) + p1p2 cosα)2 + p22p

23(1− cos2 α). (C13)

The above form of the collision term is useful when the θ integration needs to be performed first. In some instances,it will be easier to first perform the α integral first. In this latter case, the collision term can equivalently be writtenas:

C(1)ν =

1

8(2π)5


aθ cos2 α+ bθ cosα+ cθ|M|2ν

×H(p1 + p2 − p3)F (p1, p2, p3, p1 + p2 − p3)H(aθ cos2 α+ bθ cosα+ cθ). (C14)

In the above, aθ, bθ, and cθ are

aθ = −p22(p2

1 + p23 − 2p1p3 cos θ), (C15)

bθ = 2p2(p1 − p3 cos θ)(p1p2 − p3(p1 + p2) + p1p3 cos θ), (C16)

cθ = −(p1p2 − p3(p1 + p2) + p1p3 cos θ)2 + p22p

23(1− cos2 θ). (C17)

We remark that using Eq. (C9) simplifies P2 · P3

P2 · P3 → p1p3 − p1p2 + p1p2 cosα− p1p3 cos θ = P1 · P2 − P1 · P3. (C18)

We note that since the matrix element for the type of interaction of interest (see Eq. (3)) only depends on a sum ofMandelstam variables or squares of Mandelstam variables, we can write

|M|2ν = 16G2eff

(∆2(θ) cos2 α+ ∆1(θ) cosα+ ∆0(θ)

)or |M|2ν = 16G2

eff

(∆2(α) cos2 θ + ∆1(α) cos θ + ∆0(α)

),

(C19)depending on which of the θ or α integral we want to perform first. The coefficients are as follow:

∆2(θ) = p21p

22

∆1(θ) = p21p2 (p3 − 2p2 − p3 cos θ)

∆0(θ) =(p2

1(p22 − p2p3 + p2

3) + p1p3 cos θ(p1(p2 − 2p3) + p1p3 cos θ)), (C20)

∆2(α) = p21p

23

∆1(α) = p1p3 (p1(p2 − 2p3)− p1p2 cosα)

∆0(α) =(p2

1(p22 − p2p3 + p2

3) + p1p2 cosα(p1(−2p2 + p3) + p1p2 cosα)). (C21)

We note that we can perform the cos θ or the cosα integration using the following results:

∫ ∞

−∞

dx√ax2 + bx+ c

H(ax2 + bx+ c) =π√−aH(b2 − 4ac), (C22)

∫ ∞

−∞

xdx√ax2 + bx+ c

H(ax2 + bx+ c) = − bπ

2a√−aH(b2 − 4ac), (C23)

28

∫ ∞

−∞

x2dx√ax2 + bx+ c

H(ax2 + bx+ c) =π(3b2 − 4ac)

8a2√−a H(b2 − 4ac). (C24)

To make further progress in performing the angular integration, we need to specify the angular dependence of theΘν(pi) variables. As in Eq. (B9), we expand their angular dependence in Legendre polynomials with respect to theangle between the vector k and the pi vectors. Within our coordinate system, these angles are

p1 · k = cos γ ≡ µ, p2 · k = cosα cos γ + sinα sin γ cos (φ− β), p3 · k = cos θ cos γ + sin θ sin γ cosφ. (C25)

The following identity will be useful later in order to perform the remaining azymuthal integral (the “φ” integral)

∫ 2π

0

dφPl(cos θ cos γ + sin θ sin γ cosφ) = 2πPl(cos θ)Pl(cos γ). (C26)

We now consider separately the different terms in the perturbative expansion in Θν(pi).

1. Terms involving Θν(p1)

This is the simplest case since Θν(p1) can be carried outside the integrals. The azymuthal φ integral is trivial andyield an extra factor of 2π.

−Θν(p1)

8(2π)4

∫p2dp2p3dp3d(cosα)d(cos θ)√aθ cos2 α+ bθ cosα+ cθ

〈|M|2ν〉H(p1 + p2 − p3)H(aθ cos2 α+ bθ cosα+ cθ)

× ep2/Tν

(ep2/Tν + 1)(ep3/Tν + 1)(e(p1+p2−p3)/Tν + 1)(C27)

Performing the α integration first using Eqs. (C22)-(C24), we obtain

−16G2effΘν(p1)

128π3

∫p2dp2p3dp3d(cos θ)√−aθ

(∆2(θ)

3b2θ − 4aθcθ8a2θ

−∆1(θ)bθ

2aθ+ ∆0(θ)

)H(p1 + p2 − p3)H(b2θ − 4aθcθ)

× ep2/Tν

(ep2/Tν + 1)(ep3/Tν + 1)(e(p1+p2−p3)/Tν + 1).(C28)

Writing η ≡ cos θ, we have

−G2effΘν(p1)

8π3

∫dp2p3dp3dη√

(p21 + p2

3 − 2p1p3η)|Mη(p1, p2, p3, η)|2H(p1 + p2 − p3)H(b2θ − 4aθcθ)

× ep2/Tν

(ep2/Tν + 1)(ep3/Tν + 1)(e(p1+p2−p3)/Tν + 1), (C29)

where we have use the definition(

∆2(θ)3b2θ − 4aθcθ

8a2θ

−∆1(θ)bθ

2aθ+ ∆0(θ)

)≡ |Mη(p1, p2, p3, η)|2. (C30)

The Heaviside step function H(b2θ − 4aθcθ) determines the range of integration of both η and p3. It yields

Max[η−,−1] ≤ η ≤ 1 for 0 ≤ p3 ≤ p1 + p2, (C31)

where

η− =(p1 + 2p2)p3 − 2p2(p1 + p2)

p1p3. (C32)

We can then write

−G2effΘν(p1)

8π3

∫ ∞

0

dp21

(e−p2/Tν + 1)

∫ p1+p2

0

p3dp3

(ep3/Tν + 1)(e(p1+p2−p3)/Tν + 1)

×∫ 1

Max[η−,−1]

dη|Mη(p1, p2, p3, η)|2√

(p21 + p2

3 − 2p1p3η). (C33)

29

Defining xi ≡ pi/Tν , we obtain

−G2effT

6νΘν(p1)

8π3

∫ ∞

0

dx21

(e−x2 + 1)

∫ x1+x2

0

x3dx3

(ex3 + 1)(e(x1+x2−x3) + 1)

×∫ 1

Max[η−,−1]

dη|Mη(x1, x2, x3, η)|2√(x2

1 + x23 − 2x1x3η)

. (C34)


For the term involving Θν(p2), we start from Eq. (C10) and substitute the expansion from Eq. (B9).

−∞∑

l=0

(−i)l(2l + 1)

8(2π)5


aα cos2 θ + bα cos θ + cα〈|M|2ν〉H(p1 + p2 − p3)H(aα cos2 θ + bα cos θ + cα)

×θl(p2)Pl(cosα cos γ + sinα sin γ cos (φ))ep1/Tν

(ep1/Tν + 1)(ep3/Tν + 1)(e(p1+p2−p3)/Tν + 1), (C35)

where we have used the available freedom to redefine the azymuthal angle φ. We can now perform the φ integralusing the identity given in Eq. (C26)

−∞∑

l=0

(−i)l(2l + 1)Pl(µ)

8(2π)4

∫p2dp2p3dp3d(cosα)d(cos θ)√aα cos2 θ + bα cos θ + cα

〈|M|2ν〉H(p1 + p2 − p3)H(aα cos2 θ + bα cos θ + cα)

× θl(p2)Pl(cosα)ep1/Tν

(ep1/Tν + 1)(ep3/Tν + 1)(e(p1+p2−p3−µν)/Tν + 1). (C36)

Performing the cos θ integral yields

−16G2eff

∞∑

l=0

(−i)l(2l + 1)Pl(µ)

128π3(e−p1/Tν + 1)

∫p2dp2p3dp3d(cosα)√−aα

(∆2(α)

3b2α − 4aαcα8a2α

−∆1(α)bα

2aα+ ∆0(α)

)

×H(p1 + p2 − p3)H(b2α − 4aαcα)θl(p2)Pl(cosα)

(ep3/Tν + 1)(e(p1+p2−p3)/Tν + 1). (C37)

Similarly to the previous section, we define(

∆2(α)3b2α − 4aαcα

8a2α

−∆1(α)bα

2aα+ ∆0(α)

)≡ |Mρ(p1, p2, p3, ρ)|2. (C38)

We use the Heaviside step function H(b2α − 4aαcα) to determine the range of integration for ρ ≡ cosα and p3

Max[ρ−,−1] ≤ ρ ≤ 1 for 0 ≤ p3 ≤ p1 + p2, (C39)

where

ρ− =p1p2 − 2(p1 + p2)p3 + 2p2

3

p1p2. (C40)

We thus obtain

−∞∑

l=0

G2eff(−i)l(2l + 1)Pl(µ)

8π3(e−p1/Tν + 1)

∫ ∞

0

dp2θl(p2)p2

∫ p1+p2

0

dp3

(ep3/Tν + 1)(e(p1+p2−p3)/Tν + 1)

×∫ 1

Max[ρ−,−1]

dρ|Mρ(p1, p2, p3, ρ)|2Pl(ρ)√

(p21 + p2

2 + 2p1p2ρ). (C41)

The remaining difficulty is the p2 dependence of θl(p2). Since we are working in the thermal approximation in whichthe only possible neutrino perturbations are local temperature fluctuations, we note that the perturbation variableΘ(p2) admits the form

Θ(x,p2, τ) = −d ln f(0)ν

d ln p2

δTν(x, τ)

T(0)ν (τ)

. (C42)

30

It is therefore convenient to introduce the temperature fluctuation variables (see Eq. (7) in main text) νl

νl ≡−4θl(p2)

d ln f(0)ν

d ln p2

, (C43)

which are independent of p2 in the ultra-relativistic limit. We note that as the neutrinos transition to the non-relativistic regime, the νl variables will develop a momentum dependence due to the presence of the mass term in theleft-hand side of the Boltzmann equations. However, since we expect the neutrinos to self-decoupled in the relativisticregime, we can safely assume that νl is independent of the neutrino momentum. Substituting

θl(p2) =1

4

ep2/Tν

1 + ep2/Tνp2

Tννl (C44)

in Eq. (C41) and writing down the answer in terms of xi, we obtain

−∞∑

l=0

G2effT

6ν (−i)l(2l + 1)νlPl(µ)

32π3(e−x1 + 1)

∫ ∞

0

dx2x22

ex2

1 + ex2

∫ x1+x2

0

dx3

(ex3 + 1)(e(x1+x2−x3) + 1)

×∫ 1

Max[ρ−,−1]

dρ|Mρ(x1, x2, x3, ρ)|2Pl(ρ)√

(x21 + x2

2 + 2x1x2ρ). (C45)


For the term involving Θν(p3), we begin from Eq. (C14) and substitute the expansion from Eq. (B9)

∞∑

l=0

(−i)l(2l + 1)

8(2π)5


aθ cos2 α+ bθ cosα+ cθ〈|M|2ν〉H(p1 + p2 − p3)H(aθ cos2 α+ bθ cosα+ cθ)

×e(p1+p2−p3)/Tνθl(p3)Pl(cos θ cos γ + sin θ sin γ cosφ)

(ep1/Tν + 1)(ep2/Tν + 1)(e(p1+p2−p3)/Tν + 1). (C46)

We can now perform the φ integral using the identity given in Eq. (C26)

∞∑

l=0

(−i)l(2l + 1)Pl(µ)

8(2π)4

∫p2dp2p3dp3d(cosα)d(cos θ)√aθ cos2 α+ bθ cosα+ cθ

〈|M|2ν〉H(p1 + p2 − p3)H(aθ cos2 α+ bθ cosα+ cθ)

× e(p1+p2−p3)/Tνθl(p3)Pl(cos θ)


Performing the cosα integral yields

16G2eff

∞∑

l=0

(−i)l(2l + 1)Pl(µ)

128π3

∫p2dp2p3dp3d(cos θ)√−aθ

(∆2(θ)

3b2θ − 4aθcθ8a2θ

−∆1(θ)bθ

2aθ+ ∆0(θ)

)

×H(p1 + p2 − p3)H(b2θ − 4aθcθ)e(p1+p2−p3)/Tνθl(p3)Pl(cos θ)


Writing η ≡ cos θ and using the same integration limits as in Eq. (C31), we obtain

∞∑

l=0

G2eff(−i)l(2l + 1)Pl(µ)

8π3(ep1/Tν + 1)

∫ ∞

0

dp2

ep2/T + 1

∫ p1+p2

0

dp3p3θl(p3)

e−(p1+p2−p3)/Tν + 1

×∫ 1

Max[η−,−1]

dη|Mη(p1, p2, p3, η)|2Pl(η)√

p21 + p2

3 − 2p1p3η. (C49)

Again, writing down the p3 dependence of θl(p3) in terms of the νl variables and writing the integrals in terms of thedimensionless variables xi, we get

∞∑

l=0

G2effT

6ν (−i)l(2l + 1)νlPl(µ)

32π3(ex1 + 1)

∫ ∞

0

dx2

ex2 + 1

∫ x1+x2

0

dx3x2

3ex3

(1 + ex3)(e−(x1+x2−x3) + 1)

×∫ 1

Max[η−,−1]

dη|Mη(x1, x2, x3, η)|2Pl(η)√

x21 + x2

3 − 2x1x3η. (C50)

31

4. Total Collision Term

The complete collision term can then be written as

C(1)ν [p1] =

G2effT

6ν

4

∂ ln f (0)(p1)

∂ ln p1

∞∑

l=0

(−i)l(2l + 1)νlPl(µ)

(A

(p1

Tν

)+Bl

(p1

Tν

)− 2Dl

(p1

Tν

)), (C51)

where

A(x1) =1

8π3

∫ ∞

0

ex2dx2

ex2 + 1

∫ x1+x2

0

x3 dx3

(ex3 + 1)(e(x1+x2−x3) + 1)

∫ 1

Max[η−,−1]

dη|Mη(x1, x2, x3, η)|2√(x2

1 + x23 − 2x1x3η)

, (C52)

Bl(x1) =1

8π3x1

∫ ∞

0

ex2x22dx2

ex2 + 1

∫ x1+x2

0

dx3

(ex3 + 1)(e(x1+x2−x3) + 1)

∫ 1

Max[ρ−,−1]

dρ|Mρ(x1, x2, x3, ρ)|2Pl(ρ)√

(x21 + x2

2 + 2x1x2ρ),(C53)

Dl(x1) =e−x1

8π3x1

∫ ∞

0

dx2

ex2 + 1

∫ x1+x2

0

ex3x23dx3

(ex3 + 1)(e−(x1+x2−x3) + 1)

∫ 1

Max[η−,−1]

dη|Mη(x1, x2, x3, η)|2Pl(η)√

x21 + x2

3 − 2x1x3η,(C54)

where xi = pi/Tν .

[1] M. Dentler, . Hernndez-Cabezudo, J. Kopp, P. A. N.Machado, M. Maltoni, I. Martinez-Soler, andT. Schwetz, JHEP 08, 010 (2018), arXiv:1803.10661[hep-ph].

[2] A. A. Aguilar-Arevalo et al. (MiniBooNE), Phys. Rev.Lett. 121, 221801 (2018), arXiv:1805.12028 [hep-ex].

[3] A. Aguilar-Arevalo et al. (LSND), Phys. Rev. D64,112007 (2001), arXiv:hep-ex/0104049 [hep-ex].

[4] M. G. Aartsen et al. (IceCube), Phys. Rev. D95, 112002(2017), arXiv:1702.05160 [hep-ex].

[5] M. G. Aartsen et al. (IceCube), Phys. Rev. Lett. 117,071801 (2016), arXiv:1605.01990 [hep-ex].

[6] P. Adamson et al. (NOvA), Phys. Rev. D96, 072006(2017), arXiv:1706.04592 [hep-ex].

[7] P. Adamson et al. (MINOS), Submitted to: Phys. Rev.Lett. (2017), arXiv:1710.06488 [hep-ex].

[8] N. Aghanim et al. (Planck), (2018), arXiv:1807.06209[astro-ph.CO].

[9] Z. Bialynicka-Birula, Nuovo Cim. 33, 1484 (1964).[10] D. Y. Bardin, S. M. Bilenky, and B. Pontecorvo,

Phys.Lett. B32, 121 (1970).[11] G. Gelmini and M. Roncadelli, Phys.Lett. B99, 411

(1981).[12] Y. Chikashige, R. N. Mohapatra, and R. Peccei,

Phys.Rev.Lett. 45, 1926 (1980).[13] V. D. Barger, W.-Y. Keung, and S. Pakvasa, Phys.Rev.

D25, 907 (1982).[14] G. Raffelt and J. Silk, Phys.Lett. B192, 65 (1987).[15] E. W. Kolb and M. S. Turner, Phys.Rev. D36, 2895

(1987).[16] R. V. Konoplich and M. Yu. Khlopov, Sov. J. Nucl.

Phys. 47, 565 (1988), [Yad. Fiz.47,891(1988)].[17] A. V. Berkov, Yu. P. Nikitin, A. L. Sudarikov, and

M. Yu. Khlopov, Sov. J. Nucl. Phys. 48, 497 (1988),[Yad. Fiz.48,779(1988)].

[18] K. M. Belotsky, A. L. Sudarikov, and M. Yu.Khlopov, Phys. Atom. Nucl. 64, 1637 (2001), [Yad.Fiz.64,1718(2001)].

[19] S. Hannestad, Journal of Cosmology and AstroparticlePhysics 0502, 011 (2005).

[20] Z. Chacko, L. J. Hall, S. J. Oliver, and M. Perel-stein, Phys. Rev. Lett. 94, 111801 (2005), arXiv:hep-ph/0405067 [hep-ph].

[21] S. Hannestad and G. Raffelt, Phys.Rev. D72, 103514(2005), arXiv:hep-ph/0509278 [hep-ph].

[22] R. F. Sawyer, Phys.Rev. D74, 043527 (2006).[23] G. Mangano, A. Melchiorri, P. Serra, A. Cooray, and

M. Kamionkowski, Phys. Rev. D 74, 043517 (2006).[24] A. Friedland, K. M. Zurek, and S. Bashinsky, (2007),

arXiv:0704.3271 [astro-ph].[25] D. Hooper, Phys. Rev. D 75, 123001 (2007), arXiv:hep-

ph/0701194 [hep-ph].[26] P. Serra, F. Zalamea, A. Cooray, G. Mangano, and

A. Melchiorri, Phys.Rev. D81, 043507 (2010).[27] L. G. van den Aarssen, T. Bringmann, and

C. Pfrommer, Phys. Rev. Lett. 109, 231301 (2012),arXiv:1205.5809 [astro-ph.CO].

[28] K. S. Jeong and F. Takahashi, Phys. Lett.B 725, 134 (2013), 10.1016/j.physletb.2013.07.001,arXiv:1305.6521 [hep-ph].

[29] R. Laha, B. Dasgupta, and J. F. Beacom, Phys.Rev.D89, 093025 (2014), arXiv:1304.3460 [hep-ph].

[30] M. Archidiacono, S. Hannestad, R. S. Hansen,and T. Tram, Phys. Rev. D 91, 065021 (2015),arXiv:1404.5915 [astro-ph.CO].

[31] K. C. Y. Ng and J. F. Beacom, Phys.Rev. D90, 065035(2014), arXiv:1404.2288 [astro-ph.HE].

[32] J. F. Cherry, A. Friedland, and I. M. Shoemaker,(2014), arXiv:1411.1071 [hep-ph].

[33] M. Archidiacono, S. Hannestad, R. S. Hansen,

http://dx.doi.org/10.1007/JHEP08(2018)010

http://arxiv.org/abs/1803.10661


http://dx.doi.org/10.1103/PhysRevLett.121.221801



http://dx.doi.org/10.1103/PhysRevD.64.112007


http://arxiv.org/abs/hep-ex/0104049













http://dx.doi.org/10.1007/BF02749481

http://dx.doi.org/10.1016/0370-2693(70)90602-7

http://dx.doi.org/10.1016/0370-2693(81)90559-1

http://dx.doi.org/10.1016/0370-2693(81)90559-1




http://dx.doi.org/10.1016/0370-2693(87)91143-9



http://dx.doi.org/10.1134/1.1409505

http://dx.doi.org/10.1088/1475-7516/2005/02/011

http://dx.doi.org/10.1088/1475-7516/2005/02/011


http://arxiv.org/abs/hep-ph/0405067






http://dx.doi.org/ 10.1103/PhysRevD.74.043517








http://dx.doi.org/10.1016/j.physletb.2013.07.001












32

and T. Tram, Phys. Rev. D93, 045004 (2016),arXiv:1508.02504 [astro-ph.CO].

[34] J. F. Cherry, A. Friedland, and I. M. Shoemaker,(2016), arXiv:1605.06506 [hep-ph].

[35] M. Archidiacono, S. Gariazzo, C. Giunti, S. Hannestad,R. Hansen, M. Laveder, and T. Tram, JCAP 1608, 067(2016), arXiv:1606.07673 [astro-ph.CO].

[36] G. Dvali and L. Funcke, Phys. Rev. D93, 113002 (2016),arXiv:1602.03191 [hep-ph].

[37] F. Capozzi, I. M. Shoemaker, and L. Vecchi, JCAP1707, 021 (2017), arXiv:1702.08464 [hep-ph].

[38] C. Brust, Y. Cui, and K. Sigurdson, JCAP 1708, 020(2017), arXiv:1703.10732 [astro-ph.CO].

[39] F. Forastieri, Proceedings, Neutrino Oscillation Work-shop (NOW 2016): Otranto (Lecce), Italy, September4-11, 2016, PoS NOW2016, 084 (2017).

[40] O. Balducci, S. Hofmann, and A. Kassiteridis, Phys.Rev. D 98, 023003 (2018), arXiv:1710.09846 [hep-ph].

[41] C. S. Lorenz, L. Funcke, E. Calabrese, and S. Hannes-tad, Phys. Rev. D99, 023501 (2019), arXiv:1811.01991[astro-ph.CO].

[42] G. Choi, C.-T. Chiang, and M. LoVerde, JCAP 1806,044 (2018), arXiv:1804.10180 [astro-ph.CO].

[43] W. Hu, Astrophys.J. 506, 485 (1998).[44] R. Trotta and A. Melchiorri, Phys. Rev. Lett. 95,

011305 (2005).[45] A. Melchiorri and P. Serra, Phys. Rev. D 74, 127301

(2006).[46] F. De Bernardis, L. Pagano, P. Serra, A. Melchiorri,

et al., Journal of Cosmology and Astroparticle Physics0806, 013 (2008).

[47] T. L. Smith, S. Das, and O. Zahn, Phys.Rev. D85,023001 (2012).

[48] M. Archidiacono, E. Calabrese, and A. Melchiorri,Phys. Rev. D 84, 123008 (2011).

[49] M. Archidiacono, E. Giusarma, A. Melchiorri, andO. Mena, Phys.Rev. D86, 043509 (2012).

[50] M. Gerbino, E. Di Valentino, and N. Said, Phys.Rev.D88, 063538 (2013), arXiv:1304.7400 [astro-ph.CO].

[51] M. Archidiacono, E. Giusarma, S. Hannestad, andO. Mena, Adv. High Energy Phys. 2013, 191047 (2013),arXiv:1307.0637 [astro-ph.CO].

[52] A. Melchiorri, M. Archidiacono, E. Calabrese, andE. Menegoni, Journal of Physics: Conference Series 485,012014 (2014).

[53] E. Sellentin and R. Durrer, Phys. Rev. D92, 063012(2015), arXiv:1412.6427 [astro-ph.CO].

[54] P. A. R. Ade et al. (Planck), Astron. Astrophys. 571,A16 (2014), arXiv:1303.5076 [astro-ph.CO].

[55] F.-Y. Cyr-Racine and K. Sigurdson, Phys. Rev. D90,123533 (2014), arXiv:1306.1536 [astro-ph.CO].

[56] I. M. Oldengott, C. Rampf, and Y. Y. Y. Wong, Journalof Cosmology and Astroparticle Physics 4, 016 (2015),arXiv:1409.1577.

[57] Z. Chacko, L. J. Hall, T. Okui, and S. J. Oliver, Phys.Rev. D 70, 085008 (2004).

[58] J. F. Beacom, N. F. Bell, and S. Dodelson, Phys. Rev.Lett. 93, 121302 (2004).

[59] N. F. Bell, E. Pierpaoli, and K. Sigurdson, Phys.Rev.D73, 063523 (2006).

[60] M. Cirelli and A. Strumia, JCAP 0612, 013 (2006),arXiv:astro-ph/0607086 [astro-ph].

[61] A. Basboll, O. E. Bjaelde, S. Hannestad, and G. G.Raffelt, Phys. Rev. D 79, 043512 (2009).

[62] M. Archidiacono and S. Hannestad, JCAP 1407, 046(2014), arXiv:1311.3873 [astro-ph.CO].

[63] F. Forastieri, M. Lattanzi, and P. Natoli, JCAP 1507,014 (2015), arXiv:1504.04999 [astro-ph.CO].

[64] F. Forastieri, M. Lattanzi, G. Mangano, A. Mirizzi,P. Natoli, and N. Saviano, JCAP 1707, 038 (2017),arXiv:1704.00626 [astro-ph.CO].

[65] L. Lancaster, F.-Y. Cyr-Racine, L. Knox, and Z. Pan,JCAP 7, 033 (2017), arXiv:1704.06657.

[66] I. M. Oldengott, T. Tram, C. Rampf, and Y. Y. Y.Wong, JCAP 11, 027 (2017), arXiv:1706.02123.

[67] G. A. Barenboim, P. B. Denton, and I. M. Oldengott,(2019), arXiv:1903.02036 [astro-ph.CO].

[68] M. Kaplinghat, L. Knox, and Y.-S. Song, Phys. Rev.Lett. 91, 241301 (2003), arXiv:astro-ph/0303344 [astro-ph].

[69] J. Lesgourgues and S. Pastor, Phys. Rep. 429, 307(2006), astro-ph/0603494.

[70] J. Lesgourgues and S. Pastor, New J. Phys. 16, 065002(2014), arXiv:1404.1740 [hep-ph].

[71] S. Vagnozzi, E. Giusarma, O. Mena, K. Freese,M. Gerbino, S. Ho, and M. Lattanzi, Phys. Rev. D96,123503 (2017), arXiv:1701.08172 [astro-ph.CO].

[72] M. Lattanzi and M. Gerbino, Front.in Phys. 5, 70(2018), arXiv:1712.07109 [astro-ph.CO].

[73] A. G. Riess, L. M. Macri, S. L. Hoffmann, D. Scolnic,S. Casertano, A. V. Filippenko, B. E. Tucker, M. J.Reid, D. O. Jones, J. M. Silverman, R. Chornock,P. Challis, W. Yuan, P. J. Brown, and R. J. Foley,Astrophys. J. 826, 56 (2016), arXiv:1604.01424.

[74] J. L. Bernal, L. Verde, and A. G. Riess, JCAP 1610,019 (2016), arXiv:1607.05617 [astro-ph.CO].

[75] S. Birrer, T. Treu, C. E. Rusu, V. Bonvin, C. D. Fass-nacht, J. H. H. Chan, A. Agnello, A. J. Shajib, G. C. F.Chen, M. Auger, F. Courbin, S. Hilbert, D. Sluse, S. H.Suyu, K. C. Wong, P. Marshall, B. C. Lemaux, andG. Meylan, Mon. Not. R. Astron. Soc. 484, 4726 (2019),arXiv:1809.01274 [astro-ph.CO].

[76] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, andD. Scolnic, arXiv e-prints , arXiv:1903.07603 (2019),arXiv:1903.07603 [astro-ph.CO].

[77] C. Heymans, L. Van Waerbeke, L. Miller, T. Erben,H. Hildebrandt, H. Hoekstra, T. D. Kitching, Y. Mel-lier, P. Simon, C. Bonnett, J. Coupon, L. Fu, J. HarnoisDeraps, M. J. Hudson, M. Kilbinger, K. Kuijken,B. Rowe, T. Schrabback, E. Semboloni, E. van Uitert,S. Vafaei, and M. Velander, Mon. Not. R. Astron. Soc.427, 146 (2012), arXiv:1210.0032.

[78] S. Joudaki et al., Mon. Not. Roy. Astron. Soc. 474, 4894(2018), arXiv:1707.06627 [astro-ph.CO].

[79] C. Hikage et al. (HSC), Submitted to: Publ. Astron.Soc. Jap. (2018), arXiv:1809.09148 [astro-ph.CO].

[80] E. Mortsell and S. Dhawan, Journal of Cosmol-ogy and Astro-Particle Physics 2018, 025 (2018),arXiv:1801.07260 [astro-ph.CO].

[81] V. Poulin, T. L. Smith, T. Karwal, andM. Kamionkowski, arXiv e-prints , arXiv:1811.04083(2018), arXiv:1811.04083 [astro-ph.CO].

[82] V. Poulin, K. K. Boddy, S. Bird, andM. Kamionkowski, Phys. Rev. D 97, 123504 (2018),arXiv:1803.02474 [astro-ph.CO].

[83] V. Poulin, T. L. Smith, D. Grin, T. Karwal, andM. Kamionkowski, Phys. Rev. D98, 083525 (2018),arXiv:1806.10608 [astro-ph.CO].




http://dx.doi.org/ 10.1088/1475-7516/2016/08/067

http://dx.doi.org/ 10.1088/1475-7516/2016/08/067




http://dx.doi.org/10.1088/1475-7516/2017/07/021

http://dx.doi.org/10.1088/1475-7516/2017/07/021


http://dx.doi.org/10.1088/1475-7516/2017/08/020

http://dx.doi.org/10.1088/1475-7516/2017/08/020


http://dx.doi.org/10.22323/1.283.0084







http://dx.doi.org/10.1088/1475-7516/2018/06/044

http://dx.doi.org/10.1088/1475-7516/2018/06/044


http://dx.doi.org/10.1086/306274





http://dx.doi.org/10.1088/1475-7516/2008/06/013

http://dx.doi.org/10.1088/1475-7516/2008/06/013








http://dx.doi.org/10.1155/2013/191047


http://stacks.iop.org/1742-6596/485/i=1/a=012014

http://stacks.iop.org/1742-6596/485/i=1/a=012014




http://dx.doi.org/10.1051/0004-6361/201321591

http://dx.doi.org/10.1051/0004-6361/201321591





http://dx.doi.org/10.1088/1475-7516/2015/04/016

http://dx.doi.org/10.1088/1475-7516/2015/04/016








http://dx.doi.org/10.1088/1475-7516/2006/12/013

http://arxiv.org/abs/astro-ph/0607086


http://dx.doi.org/10.1088/1475-7516/2014/07/046

http://dx.doi.org/10.1088/1475-7516/2014/07/046


http://dx.doi.org/10.1088/1475-7516/2015/07/014

http://dx.doi.org/10.1088/1475-7516/2015/07/014


http://dx.doi.org/ 10.1088/1475-7516/2017/07/038


http://dx.doi.org/10.1088/1475-7516/2017/07/033


http://dx.doi.org/10.1088/1475-7516/2017/11/027







http://dx.doi.org/10.1016/j.physrep.2006.04.001

http://dx.doi.org/10.1016/j.physrep.2006.04.001


http://dx.doi.org/10.1088/1367-2630/16/6/065002

http://dx.doi.org/10.1088/1367-2630/16/6/065002





http://dx.doi.org/10.3389/fphy.2017.00070

http://dx.doi.org/10.3389/fphy.2017.00070


http://dx.doi.org/ 10.3847/0004-637X/826/1/56


http://dx.doi.org/10.1088/1475-7516/2016/10/019

http://dx.doi.org/10.1088/1475-7516/2016/10/019


http://dx.doi.org/10.1093/mnras/stz200



http://dx.doi.org/ 10.1111/j.1365-2966.2012.21952.x

http://dx.doi.org/ 10.1111/j.1365-2966.2012.21952.x


http://dx.doi.org/10.1093/mnras/stx2820

http://dx.doi.org/10.1093/mnras/stx2820



http://dx.doi.org/10.1088/1475-7516/2018/09/025

http://dx.doi.org/10.1088/1475-7516/2018/09/025







33

[84] E. Di Valentino, A. Melchiorri, and O. Mena, Phys.Rev. D 96, 043503 (2017), arXiv:1704.08342 [astro-ph.CO].

[85] E. Di Valentino, C. Bœhm, E. Hivon, andF. R. Bouchet, Phys. Rev. D 97, 043513 (2018),arXiv:1710.02559 [astro-ph.CO].

[86] K. L. Pandey, T. Karwal, and S. Das, arXiv e-prints, arXiv:1902.10636 (2019), arXiv:1902.10636 [astro-ph.CO].

[87] K. Vattis, S. M. Koushiappas, and A. Loeb, arXive-prints , arXiv:1903.06220 (2019), arXiv:1903.06220[astro-ph.CO].

[88] K. Enqvist, S. Nadathur, T. Sekiguchi, and T. Taka-hashi, Journal of Cosmology and Astro-Particle Physics2015, 067 (2015), arXiv:1505.05511 [astro-ph.CO].

[89] T. Bringmann, F. Kahlhoefer, K. Schmidt-Hoberg,and P. Walia, Phys. Rev. D98, 023543 (2018),arXiv:1803.03644 [astro-ph.CO].

[90] J. Renk, M. Zumalacarregui, F. Montanari, and A. Bar-reira, Journal of Cosmology and Astro-Particle Physics2017, 020 (2017), arXiv:1707.02263 [astro-ph.CO].

[91] N. Khosravi, S. Baghram, N. Afshordi, and N. Al-tamirano, arXiv e-prints , arXiv:1710.09366 (2017),arXiv:1710.09366 [astro-ph.CO].

[92] F. D’Eramo, R. Z. Ferreira, A. Notari, and J. L. Bernal,Journal of Cosmology and Astro-Particle Physics 2018,014 (2018), arXiv:1808.07430 [hep-ph].

[93] T. Bringmann, J. Hasenkamp, and J. Kersten, JCAP1407, 042 (2014), arXiv:1312.4947 [hep-ph].

[94] B. Dasgupta and J. Kopp, Phys.Rev.Lett. 112, 031803(2014), arXiv:1310.6337 [hep-ph].

[95] S. Hannestad, R. S. Hansen, and T. Tram,Phys.Rev.Lett. 112, 031802 (2014), arXiv:1310.5926[astro-ph.CO].

[96] Y. Tang, Phys. Lett. B750, 201 (2015),arXiv:1501.00059 [hep-ph].

[97] C. Kouvaris, I. M. Shoemaker, and K. Tuominen, Phys.Rev. D91, 043519 (2015), arXiv:1411.3730 [hep-ph].

[98] X. Chu, B. Dasgupta, and J. Kopp, JCAP 1510, 011(2015), arXiv:1505.02795 [hep-ph].

[99] A. Ghalsasi, D. McKeen, and A. E. Nelson, Phys. Rev.D95, 115039 (2017), arXiv:1609.06326 [hep-ph].

[100] J. Hasenkamp, Phys. Rev. D93, 055033 (2016),arXiv:1604.04742 [hep-ph].

[101] F. Simpson, R. Jimenez, C. Pena-Garay, and L. Verde,Phys. Dark Univ. 20, 72 (2018), arXiv:1607.02515[astro-ph.CO].

[102] A. Berlin and N. Blinov, (2018), arXiv:1807.04282 [hep-ph].

[103] A. P. Lessa and O. L. G. Peres, Phys.Rev. D75, 094001(2007).

[104] P. Bakhti and Y. Farzan, Phys. Rev. D95, 095008(2017), arXiv:1702.04187 [hep-ph].

[105] G. Arcadi, J. Heeck, F. Heizmann, S. Mertens, F. S.Queiroz, W. Rodejohann, M. Slezk, and K. Valerius,JHEP 01, 206 (2019), arXiv:1811.03530 [hep-ph].

[106] B. Ahlgren, T. Ohlsson, and S. Zhou, Phys.Rev.Lett.111, 199001 (2013), arXiv:1309.0991 [hep-ph].

[107] G.-y. Huang, T. Ohlsson, and S. Zhou, Phys. Rev. D97,075009 (2018), arXiv:1712.04792 [hep-ph].

[108] K. Ioka and K. Murase, PTEP 2014, 061E01 (2014),arXiv:1404.2279 [astro-ph.HE].

[109] M. S. Bilenky, S. M. Bilenky, and A. Santamaria,Phys.Lett. B301, 287 (1993).

[110] M. S. Bilenky and A. Santamaria, (1999), arXiv:hep-ph/9908272 [hep-ph].

[111] A. Manohar, Phys.Lett. B192, 217 (1987).[112] D. A. Dicus, S. Nussinov, P. B. Pal, and V. L. Teplitz,

Phys.Lett. B218, 84 (1989).[113] H. Davoudiasl and P. Huber, Phys. Rev. Lett. 95,

191302 (2005).[114] M. Sher and C. Triola, Phys. Rev. D 83, 117702 (2011).[115] P. Fayet, D. Hooper, and G. Sigl, Phys. Rev. Lett. 96,

211302 (2006).[116] K. Choi and A. Santamaria, Phys. Rev. D 42, 293

(1990).[117] M. Blennow, A. Mirizzi, and P. D. Serpico, Phys.Rev.

D78, 113004 (2008), arXiv:0810.2297 [hep-ph].[118] S. Galais, J. Kneller, and C. Volpe, J. Phys. G39,

035201 (2012), arXiv:1102.1471 [astro-ph.SR].[119] M. Kachelriess, R. Tomas, and J. Valle, Phys.Rev. D62,

023004 (2000), arXiv:hep-ph/0001039 [hep-ph].[120] Y. Farzan, Phys.Rev. D67, 073015 (2003), arXiv:hep-

ph/0211375 [hep-ph].[121] S. Zhou, Phys.Rev. D84, 038701 (2011),

arXiv:1106.3880 [hep-ph].[122] Y. S. Jeong, S. Palomares-Ruiz, M. H. Reno, and

I. Sarcevic, JCAP 1806, 019 (2018), arXiv:1803.04541[hep-ph].

[123] C.-P. Ma and E. Bertschinger, Astrophys.J. 455, 7(1995).

[124] A. Lewis, A. Challinor, and A. Lasenby, The Astro-physical Journal 538, 473 (2000), astro-ph/9911177.

[125] Planck Collaboration, N. Aghanim, M. Arnaud,M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Ban-day, R. B. Barreiro, J. G. Bartlett, N. Bartolo, andet al., A&A 594, A11 (2016), arXiv:1507.02704.

[126] F.-Y. Cyr-Racine and K. Sigurdson, Phys.Rev. D83,103521 (2011).

[127] A. Lewis and A. Challinor, Phys. Rev. D66, 023531(2002), arXiv:astro-ph/0203507 [astro-ph].

[128] S. Bashinsky and U. Seljak, Phys.Rev. D69, 083002(2004).

[129] D. Baumann, D. Green, J. Meyers, and B. Wal-lisch, JCAP 1601, 007 (2016), arXiv:1508.06342 [astro-ph.CO].

[130] B. Follin, L. Knox, M. Millea, and Z. Pan, Phys.Rev. Lett. 115, 091301 (2015), arXiv:1503.07863 [astro-ph.CO].

[131] D. Baumann, D. Green, and M. Zaldarriaga, JCAP1711, 007 (2017), arXiv:1703.00894 [astro-ph.CO].

[132] Planck Collaboration, Ade, P. A. R., et al., A&A 594,A13 (2016).

[133] Z. Hou, R. Keisler, L. Knox, M. Millea, andC. Reichardt, Phys. Rev. D 87, 083008 (2013),arXiv:1104.2333 [astro-ph.CO].

[134] S. Dodelson, Modern Cosmology, Academic Press (Aca-demic Press, 2003) iSBN: 9780122191411.

[135] A. Lewis and S. Bridle, Phys.Rev. D66, 103511 (2002).[136] F. Feroz and M. P. Hobson, Monthly Notices of

the Royal Astronomical Society 384, 449 (2008),arXiv:0704.3704.

[137] J. Skilling, Bayesian Anal. 1, 833 (2006).[138] Planck Collaboration, P. A. R. Ade, N. Aghanim,

M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi,A. J. Banday, R. B. Barreiro, J. G. Bartlett, and et al.,A&A 594, A15 (2016), arXiv:1502.01591.

[139] F. Beutler, C. Blake, M. Colless, D. H. Jones,











http://dx.doi.org/10.1088/1475-7516/2015/09/067

http://dx.doi.org/10.1088/1475-7516/2015/09/067




http://dx.doi.org/10.1088/1475-7516/2017/10/020

http://dx.doi.org/10.1088/1475-7516/2017/10/020



http://dx.doi.org/10.1088/1475-7516/2018/11/014

http://dx.doi.org/10.1088/1475-7516/2018/11/014


http://dx.doi.org/10.1088/1475-7516/2014/07/042

http://dx.doi.org/10.1088/1475-7516/2014/07/042













http://dx.doi.org/10.1088/1475-7516/2015/10/011

http://dx.doi.org/10.1088/1475-7516/2015/10/011







http://dx.doi.org/10.1016/j.dark.2018.04.002










http://dx.doi.org/ 10.1007/JHEP01(2019)206








http://dx.doi.org/10.1093/ptep/ptu090


http://dx.doi.org/10.1016/0370-2693(93)90703-K



http://dx.doi.org/10.1016/0370-2693(87)91171-3

http://dx.doi.org/10.1016/0370-2693(89)90480-2











http://dx.doi.org/10.1088/0954-3899/39/3/035201

http://dx.doi.org/10.1088/0954-3899/39/3/035201










http://dx.doi.org/10.1088/1475-7516/2018/06/019



http://dx.doi.org/10.1086/176550

http://dx.doi.org/10.1086/176550

http://dx.doi.org/10.1086/309179

http://dx.doi.org/10.1086/309179


http://dx.doi.org/10.1051/0004-6361/201526926









http://dx.doi.org/ 10.1088/1475-7516/2016/01/007



http://dx.doi.org/ 10.1103/PhysRevLett.115.091301

http://dx.doi.org/ 10.1103/PhysRevLett.115.091301



http://dx.doi.org/10.1088/1475-7516/2017/11/007

http://dx.doi.org/10.1088/1475-7516/2017/11/007


http://dx.doi.org/10.1051/0004-6361/201525830

http://dx.doi.org/10.1051/0004-6361/201525830




http://dx.doi.org/10.1111/j.1365-2966.2007.12353.x

http://dx.doi.org/10.1111/j.1365-2966.2007.12353.x


http://dx.doi.org/10.1214/06-BA127

http://dx.doi.org/10.1051/0004-6361/201525941


34

L. Staveley-Smith, L. Campbell, Q. Parker, W. Saun-ders, and F. Watson, Mon. Not. R. Astron. Soc. 416,3017 (2011), arXiv:1106.3366.

[140] L. Anderson, E. Aubourg, S. Bailey, F. Beutler,V. Bhardwaj, M. Blanton, A. S. Bolton, J. Brinkmann,J. R. Brownstein, A. Burden, C.-H. Chuang, A. J.Cuesta, K. S. Dawson, D. J. Eisenstein, S. Escoffier,J. E. Gunn, H. Guo, S. Ho, K. Honscheid, C. Howlett,D. Kirkby, R. H. Lupton, M. Manera, C. Maras-ton, C. K. McBride, O. Mena, F. Montesano, R. C.Nichol, S. E. Nuza, M. D. Olmstead, N. Padmanabhan,N. Palanque-Delabrouille, J. Parejko, W. J. Percival,P. Petitjean, F. Prada, A. M. Price-Whelan, B. Reid,N. A. Roe, A. J. Ross, N. P. Ross, C. G. Sabiu, S. Saito,L. Samushia, A. G. Sanchez, D. J. Schlegel, D. P. Schnei-der, C. G. Scoccola, H.-J. Seo, R. A. Skibba, M. A.Strauss, M. E. C. Swanson, D. Thomas, J. L. Tinker,R. Tojeiro, M. V. Magana, L. Verde, D. A. Wake, B. A.Weaver, D. H. Weinberg, M. White, X. Xu, C. Yeche,I. Zehavi, and G.-B. Zhao, Mon. Not. R. Astron. Soc.441, 24 (2014), arXiv:1312.4877.

[141] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival,A. Burden, and M. Manera, Mon. Not. R. Astron. Soc.449, 835 (2015), arXiv:1409.3242.

[142] R. Adam et al. (Planck), Astron. Astrophys. 596, A108(2016), arXiv:1605.03507 [astro-ph.CO].

[143] K. Aylor, M. Joy, L. Knox, M. Millea, S. Raghunathan,and W. L. Kimmy Wu, Astrophys. J. 874, 4 (2019),arXiv:1811.00537 [astro-ph.CO].

[144] Z. Hou, C. L. Reichardt, K. T. Story, B. Follin,R. Keisler, K. A. Aird, B. A. Benson, L. E. Bleem, J. E.Carlstrom, C. L. Chang, H.-M. Cho, T. M. Crawford,A. T. Crites, T. de Haan, R. de Putter, M. A. Dobbs,S. Dodelson, J. Dudley, E. M. George, N. W. Halver-son, G. P. Holder, W. L. Holzapfel, S. Hoover, J. D.Hrubes, M. Joy, L. Knox, A. T. Lee, E. M. Leitch,M. Lueker, D. Luong-Van, J. J. McMahon, J. Mehl,

S. S. Meyer, M. Millea, J. J. Mohr, T. E. Montroy,S. Padin, T. Plagge, C. Pryke, J. E. Ruhl, J. T. Sayre,K. K. Schaffer, L. Shaw, E. Shirokoff, H. G. Spieler,Z. Staniszewski, A. A. Stark, A. van Engelen, K. Van-derlinde, J. D. Vieira, R. Williamson, and O. Zahn,Astrophys. J. 782, 74 (2014), arXiv:1212.6267 [astro-ph.CO].

[145] F. Feroz, M. P. Hobson, E. Cameron, and A. N. Pettitt,ArXiv e-prints (2013), arXiv:1306.2144 [astro-ph.IM].

[146] H. Akaike, IEEE Transactions on Automatic Control19, 716 (1974).

[147] T. Louis et al. (ACTPol), JCAP 1706, 031 (2017),arXiv:1610.02360 [astro-ph.CO].

[148] J. W. Henning et al. (SPT), Astrophys. J. 852, 97(2018), arXiv:1707.09353 [astro-ph.CO].

[149] V. Irsic, M. Viel, M. G. Haehnelt, J. S. Bolton, S. Cris-tiani, G. D. Becker, V. D’Odorico, G. Cupani, T.-S.Kim, T. A. M. Berg, S. Lopez, S. Ellison, L. Chris-tensen, K. D. Denney, and G. Worseck, Phys. Rev. D96, 023522 (2017).

[150] S. Y. Kim, A. H. G. Peter, and J. R. Hargis, Phys.Rev. Lett. 121, 211302 (2018), arXiv:1711.06267 [astro-ph.CO].

[151] J. S. Bullock and M. Boylan-Kolchin, Annual Review ofAstronomy and Astrophysics 55, 343 (2017).

[152] Planck Collaboration, R. Adam, P. A. R. Ade,N. Aghanim, M. I. R. Alves, M. Arnaud, M. Ashdown,J. Aumont, C. Baccigalupi, A. J. Banday, and et al.,ArXiv e-prints (2015), arXiv:1502.01588.

[153] M. Park, C. D. Kreisch, J. Dunkley, and F.-Y. Cyr-Racine, (in prep.).

[154] E. W. Kolb and M. S. Turner, The Early Universe(Addison-Wesley, 1990) pp. 1–547, frontiers in Physics,69.

[155] W. R. Yueh and J. R. Buchler, Astrophysics and SpaceScience 39, 429 (1976).

[156] S. Hannestad and J. Madsen, Phys.Rev. D52, 1764(1995), arXiv:astro-ph/9506015 [astro-ph].

http://dx.doi.org/ 10.1111/j.1365-2966.2011.19250.x

http://dx.doi.org/ 10.1111/j.1365-2966.2011.19250.x


http://dx.doi.org/10.1093/mnras/stu523

http://dx.doi.org/10.1093/mnras/stu523


http://dx.doi.org/ 10.1093/mnras/stv154

http://dx.doi.org/ 10.1093/mnras/stv154


http://dx.doi.org/ 10.1051/0004-6361/201628897

http://dx.doi.org/ 10.1051/0004-6361/201628897


http://dx.doi.org/ 10.3847/1538-4357/ab0898


http://dx.doi.org/10.1088/0004-637X/782/2/74




http://dx.doi.org/10.1088/1475-7516/2017/06/031


http://dx.doi.org/ 10.3847/1538-4357/aa9ff4

http://dx.doi.org/ 10.3847/1538-4357/aa9ff4








http://dx.doi.org/10.1146/annurev-astro-091916-055313

http://dx.doi.org/10.1146/annurev-astro-091916-055313


http://dx.doi.org/10.1007/BF00648341

http://dx.doi.org/10.1007/BF00648341




the neutrino puzzle: anomalies, interactions, and cosmological … · 03-04-2019 · interactions...

Documents