# Cosmic Dark Radiation and Neutrinos

Authors: Maria Archidiacono; Elena Giusarma; Steen Hannestad; Olga Mena
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

New measurements of the cosmic microwave background (CMB) by the Planck mission have greatly increased our knowledge about the universe. Dark radiation, a weakly interacting component of radiation, is one of the important ingredients in our cosmological model which is testable by Planck and other observational probes. At the moment, the possible existence of dark radiation is an unsolved question. For instance, the discrepancy between the value of the Hubble constant,H0, inferred from the Planck data and local measurements of H0 can to some extent be alleviated by enlarging the minimal ΛCDM model to include additional relativistic degrees of freedom. From a fundamental physics point of view, dark radiation is no less interesting. Indeed, it could well be one of the most accessible windows to physics beyond the standard model, for example, sterile neutrinos. Here, we review the most recent cosmological results including a complete investigation of the dark radiation sector in order to provide an overview of models that are still compatible with new cosmological observations. Furthermore, we update the cosmological constraints on neutrino physics and dark radiation properties focusing on tensions between data sets and degeneracies among parameters that can degrade our information or mimic the existence of extra species.


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## Body

## 1. Introduction

The connection between cosmological observations and neutrino physics is one of the most interesting and hot topics in astroparticle physics.Earth-based experiments have demonstrated that neutrinos oscillate and therefore have mass (see, e.g., [1] for a recent treatment). However, oscillation experiments are not sensitive to the absolute neutrino mass scale, only the squared mass differences, Δm2. Furthermore, the sign is known for only one of the two mass differences, namely, Δm122, because of matter effects in the Sun. Δm232 is currently only measured via vacuum oscillations which depends only on |Δm232|. Even for standard model neutrinos, there are therefore important unresolved questions which have a significant impact on cosmology. Not only is the absolute mass scale not known, but the hierarchy between masses is also unknown. In any case, the two measured mass squared differences imply that at least two neutrinos are very nonrelativistic today (see, e.g., [2] for a recent overview).Unlike neutrino oscillation experiments, cosmology probes the sum of the neutrino masses (see, e.g., [3, 4]) because it is sensitive primarily to the current neutrino contribution to the matter density. At the moment, cosmology provides a stronger bound on the neutrino mass than laboratory bounds from, for example, beta decay, although the KATRIN experiment is set to improve the sensitivity to ∑‍mν to about 0.6 eV [5].The tightest 95% c.l. upper limits to date are∑‍mν<0.15eV [6] and ∑‍mν<0.23eV [7] from different combinations of data sets and different analyses. This astounding accuracy is possible because neutrinos leave key signatures through their free-streaming nature in several cosmological data sets: the temperature-anisotropy power spectrum of the Cosmic Microwave Background (see Section 1.1) and the power spectrum of matter fluctuations, which is one of the basic products of galaxy redshift surveys (see [8]). However, it should be stressed that cosmological constraints are highly model-dependent and, following the Bayesian method, theoretical assumptions have a strong impact on the results and can lead to erroneous conclusions. For instance in [9, 10], the assumption about spatial flatness is relaxed, testing therefore the impact of a nonzero curvature in the neutrino mass bound. It is also well known that the bound on the neutrino mass is sensitive to assumptions about the dark energy equation of state [11].In the standard model, there are exactly three neutrino mass eigenstates,(ν1ν2,ν3), corresponding to the three flavor eigenstates (νe, νμ, ντ) of the weak interaction.This has been confirmed by precision electroweak measurements at theZ0-resonance by the LEP experiment. The invisible decay width of Z0 corresponds to Nν=2.9840±0.0082 [12], consistent within ~2σ with the known three families of the SM.In cosmology, the energy density contribution of one (Neff=1) fully thermalised neutrino plus antineutrino below the e+e- annihilation scale of T~0.2MeV is at the lowest order given by ρν=(7/8)(4/11)4/3ργ. However, a more precise calculation which takes into account finite temperature effects on the photon propagator and incomplete neutrino decoupling during e+e- annihilation leads to a standard model prediction of Neff=3.046 (see, e.g., [13]). This is not because there is a noninteger number of neutrino species but simply comes from the definition of Neff.In the last few years, the WMAP satellite as well as the high multipole CMB experiments Atacama Cosmology Telescope (ACT) and South Pole Telescope (SPT) provided some hints for a nonstandard value of the effective number of relativistic degrees of freedomNeff, pointing towards the existence of an extra dark component of the radiation content of the Universe, coined dark radiation.A variation inNeff affects both the amplitude and the shape of the Cosmic Microwave Background temperature anisotropy power spectrum (see Section 1.1). Nevertheless, the new data releases of these two experiments (see [14] for ACT and [15] for SPT) seem to disagree in their conclusions on this topic [16, 17]: in combination with data from the last data release of the Wilkinson Microwave Anisotropy Probe satellite (WMAP 9 year), SPT data lead to an evidence of an extra dark radiation component (Neff=3.93±0.68 at 68% c.l.), while ACT data prefer a standard value of Neff (Neff=2.74±0.47 at 68% c.l.). The inclusion of external data sets (Baryonic Acoustic Oscillation [18–22] and Hubble Space Telescope measurements [23]) partially reconciles the two experiments in the framework of a ΛCDM model with additional relativistic species.The recently released Planck data have strongly confirmed the standardΛCDM model. The results have provided the most precise constraints ever on the six “vanilla” cosmological parameters [24] by measuring the Cosmic Microwave Background temperature power spectrum up to the seventh acoustic peak [24] with nine frequency channels (100, 143, and 217 GHz are the three frequency channels involved in the cosmological analysis). Concerning dark radiation, Planck results point towards a standard value of Neff (Neff=3.36-0.64+0.68 at 95% c.l. using Planck data combined with WMAP 9 year polarization measurements and high multipole CMB experiments, both ACT and SPT). However, the ~2.5σ tension among Planck and HST measurements of the Hubble constant value can be solved, for instance, by extending the ΛCDM model to account for a nonvanishing ΔNeff (Neff=3.62-0.48+0.50 at 95% c.l. using Planck + WP + highL plus a prior on the Hubble constant from the Hubble Space Telescope measurements [23]).In this review, after explaining the effects ofNeff on CMB power spectrum (Section 1.1), in Section 1.2, we list the different dark radiation models with their state of art constraints on the effective number of relativistic degrees of freedom. Section 2 illustrates the method and the data sets we use here in order to constrain the neutrino parameters we are interested in (number of species and masses). The results of our analyses are reported in Section 3. In Section 4, we present a forecast of the Euclid results on the neutrino number and mass. Finally in Section 5, we discuss our conclusions in light of the former considerations.


### 1.1.Neff Effects on Cosmological Observables

The total radiation content of the Universe below thee+e- annihilation temperature can be parameterized as follows:
(1)ϱr=[1+78(411)4/3Neff]ϱγ,
where ργ is the energy density of photons, 7/8 is the multiplying factor for each fermionic degree of freedom, and (4/11)1/3 is the photon neutrino temperature ratio. Finally, the parameter Neff can account for neutrinos and for any extra relativistic degrees of freedom; namely, are particles are still relativistic at decoupling as follows:
(2)Neff=3.046+ΔNeff.VaryingNeff changes the time of the matter radiation equivalence: a higher radiation content due to the presence of additional relativistic species leads to a delay in zeq as follows:
(3)1+zeq=ΩmΩr=Ωmh2Ωγh21(1+0.2271Neff),
where Ωm is the matter density, Ωr is the radiation density, Ωγ is the photon density, h is defined as H0=100h km/s/Mpc, and in the last equality we have used equation (1). As a consequence at the time of decoupling, radiation is still a subdominant component and the gravitational potential is still slowly decreasing. This shows up as an enhancement of the early Integrated Sachs Wolfe (ISW) effect that increases the CMB perturbation peaks at ℓ~200, that is, around the first acoustic peak. This effect is demonstrated in Figure 1.Figure 1
ISW contribution to the CMB temperature power spectrum. The raise atℓ<30 is due to the late Integrated Sachs Wolfe, while the peak around ℓ~200 is the early Integrated Sachs Wolfe effect. The cosmological model is the ΛCDM with Neff being equal to 3 (black solid line), 5 (red dashed line), and 7 (green dot-dashed line).In [25], the authors stress that the most important effect of changing Neff is located at high ℓ>600 and is not related to the early ISW effect. Indeed, the main effect related to a variation of the number of relativistic species at decoupling is that it alters the expansion rate, H, around the epoch of last scattering. The extra dark radiation component, arising from a value of Neff greater than the standard 3.046, contributes to the expansion rate via its energy density ΩDR as follows:
(4)H2H02=Ωma3+ΩΛ+Ωγa4+Ωνa4+ΩDRa4.
If Neff increases, H increases as well. Furthermore, the delay in matter radiation equality, which causes the early ISW, also modifies the baryon to photon density ratio as follows:
(5)Req=3ρb4ργ∣aeq,
and therefore the sound speed
(6)cs=13(1+Req).
The size of the comoving sound horizon rs is given by
(7)rs=∫0τ′‍dτcs(τ)=∫0a‍daa2Hcs(a)
and is proportional to the inverse of the expansion rate rs∝1/H, when Neff increases, rs decreases. The consequence is a reduction in the angular scale of the acoustic peaks θs=rs/DA, where DA is the angular diameter distance. The overall effect on the CMB power spectrum is a horizontal shift of the peak positions towards higher multipoles. In Figure 2(b), the total temperature power spectrum is corrected for this effect: the ℓ axis is rescaled by a constant factor θs(Neff)/θs(Neff=3) in order to account for the peak shift due to the increase in Neff. Effectively, it amounts to having the same sound horizon for all the models. Considering that θs is the most well-constrained quantity by CMB measures, this is the dominant effect of a varying Neff on the CMB power spectrum.CMB temperature power spectrum. The model and the legend are the same as in Figure1, the grey error bars correspond to Planck data. (a) The total CMB temperature power spectrum. (b) The ℓ axis has been rescaled by a factor θs(Neff)/θs(Neff=3). (c) The total ISW×ISW contribution has been subtracted. (d) The total ISW (both the autocorrelation and the cross correlation) contribution have been subtracted.
(a)
(b)
(c)
(d)Besides the horizontal shift, there is also a vertical shift that affects the amplitude of the peaks at high multipoles where the ISW effect is negligible. Comparing Figure2 with Figure 1, one can also notice that for a larger value of Neff the early ISW causes an increase of power on the first and the second peaks, while the same variation in Neff turns out in a reduction of power in the peaks at higher multipoles. This vertical shift is related to the Silk damping effect. The decoupling of baryon-photon interactions is not instantaneous but rather an extended process. This leads to diffusion damping of oscillations in the plasma, an effect known as Silk damping. If decoupling starts at τd and ends at τls, during Δτ the radiation free streams on scale λd=(λΔτ)1/2 where λ is the photon mean free path and λd is shorter than the thickness of the last scattering surface. As a consequence, temperature fluctuations on scales smaller than λD are damped, because on such scales photons can spread freely both from overdensities and from underdensities. The damping factor is exp[-(2rd/λd)] where rd is the mean square diffusion distance at recombination. An approximated expression of rd is given by [25]
(8)rd2=(2π)2∫0als‍daa3σTneH[R2+(6/15)(1+R)6(1+R2)],
where σT is the Thompson cross section, ne is the number density of free electrons, als is the scale factor at recombination, and the factor in square brackets is related to polarization [26]. This diffusion process becomes more and more effective approaching the last scattering, so we can consider a constant and thus obtain rd∝1/H. Recalling the dependence rs∝1/H and the fact that θs=rs/DA is fixed by CMB observations, we can infer DA∝1/H. The result is that the damping angular scale θd=rd/DA is proportional to the square root of the expansion rate θd∝H and consequently it increases with the number of relativistic species. The effect on the CMB power spectrum can be seen in Figures 2(c) and 2(d), where, in addition to the ℓ rescaling, we have subtracted the ISW power spectrum of Figure 1 in the Figure 2(c), while in the Figure 2(d) we have taken into account the total ISW contribution (both the autocorrelation and the cross correlation). This damping effect shows up as a suppression of the peaks and a smearing of the oscillations that intensifies at higher multipoles.It is important to stress that all these effects (on the redshift of equivalence, on the size of the sound horizon at recombination, and on the damping tail) can be compensated by varying other cosmological parameters [27]. For instance the damping scale is affected by the helium fraction as well as by the effective number of relativistic degrees of freedom: rd∝(1-Yhe)-0.5 [25]. Therefore, at the level of the damping in the power spectrum, a larger value of Neff can be mimicked by a lower value of Yhe (see Figure 5, Section 3.1). The redshift of the equivalence zeq can be kept fixed by increasing the cold dark matter density while increasing Neff. Finally an open Universe with a nonzero curvature can reproduce the same peak shifting of a larger number of relativistic degrees of freedom. All these degeneracies increase the uncertainty on the results and degrade the constraint on Neff.The only effect that cannot be mimicked by other cosmological parameters is the neutrino anisotropic stress. The anisotropic stress arises from the quadrupole moment of the cosmic neutrino background temperature distribution and it alters the gravitational potentials [28, 29]. The effect on the CMB power spectrum is located at scales that cross the horizon before the matter-radiation equivalence and it consists of an increase in power by a factor 5/(1+(4/15)fν) [30], where fν is the fraction of radiation density contributed by free-streaming particles.


### 1.2. Dark Radiation Models

A number of theoretical physics models could explain a contribution to the extra dark radiation component of the universe, that is, toΔNeff.A particularly simple model, based on neutrino oscillation Short BaseLine (SBL) physics results, contains sterile neutrinos. Sterile neutrinos are right-handed fermions which do not interact via any of the fundamental standard model interactions and therefore their number is not determined by any fundamental symmetry in nature. Originally, models with one additional massive mainly sterile neutrinoν4, with a mass splitting Δm142, that is, the so called (3+1) models, were introduced to explain LSND (Large Scintillator Neutrino Detector) [31] SBL antineutrino data by means of neutrino oscillations [32, 33]. A much better fit to both appearance and disappearance data was in principle provided by the (3+2) models [34] in which there are two mostly sterile neutrino mass states ν4 and ν5 with mass splittings in the range 0.1eV2<|Δm142|,|Δm152|<10eV2. In the two sterile neutrino scenarios we can distinguish two possibilities, one in which both mass splittings are positive, named as 3+2, and one in which one of them is negative, named as 1+3+1 [35]. Recent MiniBooNE antineutrino data are consistent with oscillations in the 0.1eV2<|Δm142|,|Δm152|<10eV2, showing some overlapping with LSND results [36]. The running in the neutrino mode also shows an excess at low energy. However, the former excess seems to be not compatible with a simple two-neutrino oscillation formalism [36]. A recent global fit to long baseline, short baseline, solar, and atmospheric neutrino oscillation data [37] has shown that in the 3+1 and 3+2 sterile neutrino schemes there is some tension in the combined fit to appearance and disappearance data. This tension is alleviated in the 1+3+1 sterile neutrino model case with a P value of 0.2%. These results are in good agreement with those presented in [38], which also considered the 3+3 sterile neutrino models with three active and three sterile neutrinos. They conclude that 3+3 neutrino models yield a compatibility of 90% among all short baseline data sets highly superior to those obtained in models with either one or two sterile neutrino species. The existence of this extra sterile neutrinos states can be in tension with Big Bang Nucleosynthesis (see Section 2.2). However, the extra neutrino species may not necessarily be fully thermalised in the early universe. Even though the masses and mixing angles necessary to explain oscillation data would seem to indicate full thermalisation, the presence of, for example, a lepton asymmetry can block sterile neutrino production and lead to a significantly lower final abundance, making the model compatible with BBN bounds, see [39–45].However, an extra radiation component may arise from many other physical mechanisms, as, for instance, QCD thermal axions or extended dark sectors with additional relativistic degrees of freedom. Both possibilities are closely related to minimal extensions to the standard model of elementary particles. Cosmological data provide a unique opportunity to place limits on any model containing new light species, see [46].We first briefly review the hadronic axion model [47, 48] since these hypothetical particles provide the most elegant and promising solution to the strong CP problem. Quantum Chromodynamics (QCD) respects CP symmetry, despite the existence of a natural, four-dimensional Lorentz and gauge invariant operator which violates CP. The presence of this CP violating-term will induce a nonvanishing neutron dipole moment, dn. However, the experimental bound on the dipole moment |dn|<3×10-26e cm [49] would require a negligible CP violation contribution. Peccei and Quinn [50, 51] introduced a new global U(1)PQ symmetry, which is spontaneously broken at a scale fa, generating a new spinless particle, the axion. The axion mass is inversely proportional to the axion decay constant fa which is the parameter controlling the interaction strength with the standard model plasma and therefore the degree of thermalisation in the early universe. The interaction Lagrangian is proportional to 1/fa and high mass axions therefore have a stronger coupling to the standard model and thermalise more easily. Axions produced via thermal processes in the early Universe provide a possible (sub)dominant hot dark matter candidate, similar, but not exactly equivalent to, neutrino hot dark matter. High mass axions are disfavored by cosmological data, with the specific numbers depending on the model and data sets used (see, e.g., [52–55]). Even though moderate mass axions can still provide a contribution to the energy density we also stress that just as for neutrino hot dark matter it cannot be mapped exactly to a change in Neff.Generally, any model with a dark sector with relativistic degrees of freedom that eventually decouple from the standard model sector will also contribute toNeff. Examples are the asymmetric dark matter scenarios (see, e.g., [56, 57] and references therein) or extended weakly interacting massive particle models (see the recent work presented in [58–60]). We will review here the expressions from [57], in which the authors include both light (gℓ) and heavy (gh) relativistic degrees of freedom at the temperature of decoupling TD from the standard model. For high decoupling temperature, TD> MeV, the dark sector contribution to Neff is read [57] as
(9)ΔNeff=13.56g⋆S(TD)4/3(gℓ+gh)4/3gℓ1/3,
where g⋆S(TD) refers to the effective number of entropy degrees of freedom at the dark sector decoupling temperature. If the dark sector decouples at lower temperatures (TD< MeV), there are two possibilities for the couplings of the dark sector with the standard model: either the dark sector couples to the electromagnetic plasma or it couples to neutrinos. In this former case,
(10)Neff=(3+47(gh+gℓ)4/3gℓ1/3)×(3×7/4+gH+gh+gℓ3×7/4+gh+gℓ)4/3,
having gH the number of degrees of freedom that become nonrelativistic between typical BBN temperatures and TD. The authors of [57] have shown that the cosmological constraints on Neff can be translated into the required heavy degrees of freedom heating the light dark sector plasma gh as a function of the dark sector decoupling temperature TD for a fixed value of gℓ. Recent Planck data [24], combined with measurements of the Hubble constant H0 from the Hubble Space Telescope (HST), low multipole polarization measurements from the Wilkinson Microwave Anisotropy Probe (WMAP) 9 year data release [61], and high multipole CMB data from both the Atacama Cosmology Telescope (ACT) [14] and the South Pole Telescope (SPT) [15, 62], provide that the constraint Neff is 3.62-0.48+0.50 at 95% c.l. Using this constraint, the authors of [63] have found that having extra heavy degrees of freedom in the dark sector for low decoupling temperatures is highly disfavored.Another aspect of dark radiation is that it could interact with the dark matter sector. In asymmetric dark matter models (see [56]), the dark matter production mechanism resembles to the one in the baryonic sector, with a particle-antiparticle asymmetry at high temperatures. The thermally symmetric dark matter component eventually annihilates and decays into dark radiation species. Due to the presence of such an interaction among the dark matter and dark radiation sectors, they behave as a tightly coupled fluid with pressure which will imprint oscillations in the matter power spectrum (as the acoustic oscillations in the photon-baryon fluid before the recombination era). The clustering properties of the dark radiation component may be modified within interacting schemes, and therefore the clustering parameters ceff2 and cvis2 may differ from their standard values for the neutrino case ceff2=cvis2=1/3 (see Section 2.2). In the presence of a dark radiation-dark matter interaction, the complete Euler equation for dark radiation, including the interaction term with dark matter, is read as follows:
(11)θ˙dr=3k2ceff2(14δdr-a˙aθdrk2)-a˙aθdr-12k2πdr+andmσdm-dr(θdm-θdr),
where the term andmσdm-dr(θdm-θdr) represents the moment transferred to the dark radiation component and the quantity andmσdm-dr gives the scattering rate of dark radiation by dark matter. The authors of [64] have parameterized the coupling between dark radiation and dark matter through a cross section given by
(12)〈σdm-dr|v|〉~Q0mdm,
if it is constant, or
(13)〈σdm-dr|v|〉~Q2a2mdm,
if it is proportional to T2, where the parameters Q0 and Q2 are constants in cm2 MeV-1 units. It has been shown in [57] that the cosmological implications of both constant and T-dependent interacting cross sections are very similar. Recent cosmological constraints on generalized interacting dark radiation models have been presented in [65]; here the authors have shown that if the dark radiation and the dark matter sectors interact in nature, the errors on the dark radiation clustering properties largely increase.


## 1.1.Neff Effects on Cosmological Observables

The total radiation content of the Universe below thee+e- annihilation temperature can be parameterized as follows:
(1)ϱr=[1+78(411)4/3Neff]ϱγ,
where ργ is the energy density of photons, 7/8 is the multiplying factor for each fermionic degree of freedom, and (4/11)1/3 is the photon neutrino temperature ratio. Finally, the parameter Neff can account for neutrinos and for any extra relativistic degrees of freedom; namely, are particles are still relativistic at decoupling as follows:
(2)Neff=3.046+ΔNeff.VaryingNeff changes the time of the matter radiation equivalence: a higher radiation content due to the presence of additional relativistic species leads to a delay in zeq as follows:
(3)1+zeq=ΩmΩr=Ωmh2Ωγh21(1+0.2271Neff),
where Ωm is the matter density, Ωr is the radiation density, Ωγ is the photon density, h is defined as H0=100h km/s/Mpc, and in the last equality we have used equation (1). As a consequence at the time of decoupling, radiation is still a subdominant component and the gravitational potential is still slowly decreasing. This shows up as an enhancement of the early Integrated Sachs Wolfe (ISW) effect that increases the CMB perturbation peaks at ℓ~200, that is, around the first acoustic peak. This effect is demonstrated in Figure 1.Figure 1
ISW contribution to the CMB temperature power spectrum. The raise atℓ<30 is due to the late Integrated Sachs Wolfe, while the peak around ℓ~200 is the early Integrated Sachs Wolfe effect. The cosmological model is the ΛCDM with Neff being equal to 3 (black solid line), 5 (red dashed line), and 7 (green dot-dashed line).In [25], the authors stress that the most important effect of changing Neff is located at high ℓ>600 and is not related to the early ISW effect. Indeed, the main effect related to a variation of the number of relativistic species at decoupling is that it alters the expansion rate, H, around the epoch of last scattering. The extra dark radiation component, arising from a value of Neff greater than the standard 3.046, contributes to the expansion rate via its energy density ΩDR as follows:
(4)H2H02=Ωma3+ΩΛ+Ωγa4+Ωνa4+ΩDRa4.
If Neff increases, H increases as well. Furthermore, the delay in matter radiation equality, which causes the early ISW, also modifies the baryon to photon density ratio as follows:
(5)Req=3ρb4ργ∣aeq,
and therefore the sound speed
(6)cs=13(1+Req).
The size of the comoving sound horizon rs is given by
(7)rs=∫0τ′‍dτcs(τ)=∫0a‍daa2Hcs(a)
and is proportional to the inverse of the expansion rate rs∝1/H, when Neff increases, rs decreases. The consequence is a reduction in the angular scale of the acoustic peaks θs=rs/DA, where DA is the angular diameter distance. The overall effect on the CMB power spectrum is a horizontal shift of the peak positions towards higher multipoles. In Figure 2(b), the total temperature power spectrum is corrected for this effect: the ℓ axis is rescaled by a constant factor θs(Neff)/θs(Neff=3) in order to account for the peak shift due to the increase in Neff. Effectively, it amounts to having the same sound horizon for all the models. Considering that θs is the most well-constrained quantity by CMB measures, this is the dominant effect of a varying Neff on the CMB power spectrum.CMB temperature power spectrum. The model and the legend are the same as in Figure1, the grey error bars correspond to Planck data. (a) The total CMB temperature power spectrum. (b) The ℓ axis has been rescaled by a factor θs(Neff)/θs(Neff=3). (c) The total ISW×ISW contribution has been subtracted. (d) The total ISW (both the autocorrelation and the cross correlation) contribution have been subtracted.
(a)
(b)
(c)
(d)Besides the horizontal shift, there is also a vertical shift that affects the amplitude of the peaks at high multipoles where the ISW effect is negligible. Comparing Figure2 with Figure 1, one can also notice that for a larger value of Neff the early ISW causes an increase of power on the first and the second peaks, while the same variation in Neff turns out in a reduction of power in the peaks at higher multipoles. This vertical shift is related to the Silk damping effect. The decoupling of baryon-photon interactions is not instantaneous but rather an extended process. This leads to diffusion damping of oscillations in the plasma, an effect known as Silk damping. If decoupling starts at τd and ends at τls, during Δτ the radiation free streams on scale λd=(λΔτ)1/2 where λ is the photon mean free path and λd is shorter than the thickness of the last scattering surface. As a consequence, temperature fluctuations on scales smaller than λD are damped, because on such scales photons can spread freely both from overdensities and from underdensities. The damping factor is exp[-(2rd/λd)] where rd is the mean square diffusion distance at recombination. An approximated expression of rd is given by [25]
(8)rd2=(2π)2∫0als‍daa3σTneH[R2+(6/15)(1+R)6(1+R2)],
where σT is the Thompson cross section, ne is the number density of free electrons, als is the scale factor at recombination, and the factor in square brackets is related to polarization [26]. This diffusion process becomes more and more effective approaching the last scattering, so we can consider a constant and thus obtain rd∝1/H. Recalling the dependence rs∝1/H and the fact that θs=rs/DA is fixed by CMB observations, we can infer DA∝1/H. The result is that the damping angular scale θd=rd/DA is proportional to the square root of the expansion rate θd∝H and consequently it increases with the number of relativistic species. The effect on the CMB power spectrum can be seen in Figures 2(c) and 2(d), where, in addition to the ℓ rescaling, we have subtracted the ISW power spectrum of Figure 1 in the Figure 2(c), while in the Figure 2(d) we have taken into account the total ISW contribution (both the autocorrelation and the cross correlation). This damping effect shows up as a suppression of the peaks and a smearing of the oscillations that intensifies at higher multipoles.It is important to stress that all these effects (on the redshift of equivalence, on the size of the sound horizon at recombination, and on the damping tail) can be compensated by varying other cosmological parameters [27]. For instance the damping scale is affected by the helium fraction as well as by the effective number of relativistic degrees of freedom: rd∝(1-Yhe)-0.5 [25]. Therefore, at the level of the damping in the power spectrum, a larger value of Neff can be mimicked by a lower value of Yhe (see Figure 5, Section 3.1). The redshift of the equivalence zeq can be kept fixed by increasing the cold dark matter density while increasing Neff. Finally an open Universe with a nonzero curvature can reproduce the same peak shifting of a larger number of relativistic degrees of freedom. All these degeneracies increase the uncertainty on the results and degrade the constraint on Neff.The only effect that cannot be mimicked by other cosmological parameters is the neutrino anisotropic stress. The anisotropic stress arises from the quadrupole moment of the cosmic neutrino background temperature distribution and it alters the gravitational potentials [28, 29]. The effect on the CMB power spectrum is located at scales that cross the horizon before the matter-radiation equivalence and it consists of an increase in power by a factor 5/(1+(4/15)fν) [30], where fν is the fraction of radiation density contributed by free-streaming particles.


## 1.2. Dark Radiation Models

A number of theoretical physics models could explain a contribution to the extra dark radiation component of the universe, that is, toΔNeff.A particularly simple model, based on neutrino oscillation Short BaseLine (SBL) physics results, contains sterile neutrinos. Sterile neutrinos are right-handed fermions which do not interact via any of the fundamental standard model interactions and therefore their number is not determined by any fundamental symmetry in nature. Originally, models with one additional massive mainly sterile neutrinoν4, with a mass splitting Δm142, that is, the so called (3+1) models, were introduced to explain LSND (Large Scintillator Neutrino Detector) [31] SBL antineutrino data by means of neutrino oscillations [32, 33]. A much better fit to both appearance and disappearance data was in principle provided by the (3+2) models [34] in which there are two mostly sterile neutrino mass states ν4 and ν5 with mass splittings in the range 0.1eV2<|Δm142|,|Δm152|<10eV2. In the two sterile neutrino scenarios we can distinguish two possibilities, one in which both mass splittings are positive, named as 3+2, and one in which one of them is negative, named as 1+3+1 [35]. Recent MiniBooNE antineutrino data are consistent with oscillations in the 0.1eV2<|Δm142|,|Δm152|<10eV2, showing some overlapping with LSND results [36]. The running in the neutrino mode also shows an excess at low energy. However, the former excess seems to be not compatible with a simple two-neutrino oscillation formalism [36]. A recent global fit to long baseline, short baseline, solar, and atmospheric neutrino oscillation data [37] has shown that in the 3+1 and 3+2 sterile neutrino schemes there is some tension in the combined fit to appearance and disappearance data. This tension is alleviated in the 1+3+1 sterile neutrino model case with a P value of 0.2%. These results are in good agreement with those presented in [38], which also considered the 3+3 sterile neutrino models with three active and three sterile neutrinos. They conclude that 3+3 neutrino models yield a compatibility of 90% among all short baseline data sets highly superior to those obtained in models with either one or two sterile neutrino species. The existence of this extra sterile neutrinos states can be in tension with Big Bang Nucleosynthesis (see Section 2.2). However, the extra neutrino species may not necessarily be fully thermalised in the early universe. Even though the masses and mixing angles necessary to explain oscillation data would seem to indicate full thermalisation, the presence of, for example, a lepton asymmetry can block sterile neutrino production and lead to a significantly lower final abundance, making the model compatible with BBN bounds, see [39–45].However, an extra radiation component may arise from many other physical mechanisms, as, for instance, QCD thermal axions or extended dark sectors with additional relativistic degrees of freedom. Both possibilities are closely related to minimal extensions to the standard model of elementary particles. Cosmological data provide a unique opportunity to place limits on any model containing new light species, see [46].We first briefly review the hadronic axion model [47, 48] since these hypothetical particles provide the most elegant and promising solution to the strong CP problem. Quantum Chromodynamics (QCD) respects CP symmetry, despite the existence of a natural, four-dimensional Lorentz and gauge invariant operator which violates CP. The presence of this CP violating-term will induce a nonvanishing neutron dipole moment, dn. However, the experimental bound on the dipole moment |dn|<3×10-26e cm [49] would require a negligible CP violation contribution. Peccei and Quinn [50, 51] introduced a new global U(1)PQ symmetry, which is spontaneously broken at a scale fa, generating a new spinless particle, the axion. The axion mass is inversely proportional to the axion decay constant fa which is the parameter controlling the interaction strength with the standard model plasma and therefore the degree of thermalisation in the early universe. The interaction Lagrangian is proportional to 1/fa and high mass axions therefore have a stronger coupling to the standard model and thermalise more easily. Axions produced via thermal processes in the early Universe provide a possible (sub)dominant hot dark matter candidate, similar, but not exactly equivalent to, neutrino hot dark matter. High mass axions are disfavored by cosmological data, with the specific numbers depending on the model and data sets used (see, e.g., [52–55]). Even though moderate mass axions can still provide a contribution to the energy density we also stress that just as for neutrino hot dark matter it cannot be mapped exactly to a change in Neff.Generally, any model with a dark sector with relativistic degrees of freedom that eventually decouple from the standard model sector will also contribute toNeff. Examples are the asymmetric dark matter scenarios (see, e.g., [56, 57] and references therein) or extended weakly interacting massive particle models (see the recent work presented in [58–60]). We will review here the expressions from [57], in which the authors include both light (gℓ) and heavy (gh) relativistic degrees of freedom at the temperature of decoupling TD from the standard model. For high decoupling temperature, TD> MeV, the dark sector contribution to Neff is read [57] as
(9)ΔNeff=13.56g⋆S(TD)4/3(gℓ+gh)4/3gℓ1/3,
where g⋆S(TD) refers to the effective number of entropy degrees of freedom at the dark sector decoupling temperature. If the dark sector decouples at lower temperatures (TD< MeV), there are two possibilities for the couplings of the dark sector with the standard model: either the dark sector couples to the electromagnetic plasma or it couples to neutrinos. In this former case,
(10)Neff=(3+47(gh+gℓ)4/3gℓ1/3)×(3×7/4+gH+gh+gℓ3×7/4+gh+gℓ)4/3,
having gH the number of degrees of freedom that become nonrelativistic between typical BBN temperatures and TD. The authors of [57] have shown that the cosmological constraints on Neff can be translated into the required heavy degrees of freedom heating the light dark sector plasma gh as a function of the dark sector decoupling temperature TD for a fixed value of gℓ. Recent Planck data [24], combined with measurements of the Hubble constant H0 from the Hubble Space Telescope (HST), low multipole polarization measurements from the Wilkinson Microwave Anisotropy Probe (WMAP) 9 year data release [61], and high multipole CMB data from both the Atacama Cosmology Telescope (ACT) [14] and the South Pole Telescope (SPT) [15, 62], provide that the constraint Neff is 3.62-0.48+0.50 at 95% c.l. Using this constraint, the authors of [63] have found that having extra heavy degrees of freedom in the dark sector for low decoupling temperatures is highly disfavored.Another aspect of dark radiation is that it could interact with the dark matter sector. In asymmetric dark matter models (see [56]), the dark matter production mechanism resembles to the one in the baryonic sector, with a particle-antiparticle asymmetry at high temperatures. The thermally symmetric dark matter component eventually annihilates and decays into dark radiation species. Due to the presence of such an interaction among the dark matter and dark radiation sectors, they behave as a tightly coupled fluid with pressure which will imprint oscillations in the matter power spectrum (as the acoustic oscillations in the photon-baryon fluid before the recombination era). The clustering properties of the dark radiation component may be modified within interacting schemes, and therefore the clustering parameters ceff2 and cvis2 may differ from their standard values for the neutrino case ceff2=cvis2=1/3 (see Section 2.2). In the presence of a dark radiation-dark matter interaction, the complete Euler equation for dark radiation, including the interaction term with dark matter, is read as follows:
(11)θ˙dr=3k2ceff2(14δdr-a˙aθdrk2)-a˙aθdr-12k2πdr+andmσdm-dr(θdm-θdr),
where the term andmσdm-dr(θdm-θdr) represents the moment transferred to the dark radiation component and the quantity andmσdm-dr gives the scattering rate of dark radiation by dark matter. The authors of [64] have parameterized the coupling between dark radiation and dark matter through a cross section given by
(12)〈σdm-dr|v|〉~Q0mdm,
if it is constant, or
(13)〈σdm-dr|v|〉~Q2a2mdm,
if it is proportional to T2, where the parameters Q0 and Q2 are constants in cm2 MeV-1 units. It has been shown in [57] that the cosmological implications of both constant and T-dependent interacting cross sections are very similar. Recent cosmological constraints on generalized interacting dark radiation models have been presented in [65]; here the authors have shown that if the dark radiation and the dark matter sectors interact in nature, the errors on the dark radiation clustering properties largely increase.


## 2. Analysis Method

The parameter space (see Section2.2) is sampled through a Monte Carlo Markov Chain performed with the publicly available package CosmoMC [66] based on the Metropolis-Hastings sampling algorithm and on the Gelman Rubin convergence diagnostic. The calculation of the theoretical observables is done through CAMB [67] (Code for Anisotropies in the Microwave Background) software. The code is able to fit any kind of cosmological data with a bayesian statistic; in our case, we focus on the data sets reported in the following section.


### 2.1. Data Sets

Our basic data set is the Planck temperature power spectrum (both at lowℓ and at high ℓ) in combination with the WMAP 9 year polarization data (hereafter WP) and the high multipole CMB data of ACT and SPT (hereafter highL). These data sets are implemented in the analysis following the prescription of the Planck likelihood described in [24]. The additional data sets test the robustness at low redshift of the predictions obtained with CMB data. These data sets consist of a prior on the Hubble constant from the Hubble Space Telescope measurements [23] (hereafter H0) and the information on the dark matter clustering from the matter power spectrum extracted from the Data Release 9 (DR9) of the CMASS sample of galaxies [68] from the Baryon Acoustic Spectroscopic Survey (BOSS) [19], part of the program of the Sloan Digital Sky Survey III [69].


### 2.2. Parameters

In Table1, the parameters used in the analyses are listed together with the top-hat priors on them. The six standard parameters of the ΛCDM model are: the physical baryon density, ωb≡Ωbh2; the physical cold dark matter density, ωc≡Ωch2; the angular scale of the sound horizon, θs; the reionization optical depth, τ; the amplitude of the primordial spectrum at a certain pivot scale, As; and the power law spectral index of primordial density (scalar) perturbations, ns.Table 1
Priors on the cosmological parameters considered in this work. All priors are uniform (top hat) in the given intervals.
Parameter
Prior
ω
b
0.005
→
0.1
ω
cdm
0.001
→
0.99
θ
s
0.5
→
10
τ
0.01
→
0.8
ln
(
1
0
10
A
s
)
2.7
→
4
n
s
0.9
→
1.1
A
L
0
→
5
N
eff
0
→
7
∑
‍
m
ν [eV]
0
→
7
Y
he
0.1
→
0.5
c
eff
2
0
→
1
c
vis
2
0
→
1We include the effective number of relativistic degrees of freedomNeff, and, in addition, our runs also contain one or a combination of the following parameters: the sum of neutrino masses ∑‍mν, the primordial helium fraction Yhe, and the neutrino perturbation parameters, namely, the effective sound speed ceff2 and the viscosity parameter cvis2. Finally we also investigated the impact of a varying lensing amplitude AL.We assume that massive neutrinos are degenerate and share the same mass. Indeed given the present accuracy of CMB measurements, cosmology cannot extract the neutrino mass hierarchy but only the total hot dark matter density. Even if the future measurements of the Euclid survey will achieve an extreme accurate measurement of the neutrino mass (σmν≃0.01eV [70]), the neutrino mass hierarchy would not be pin down.


#### 2.2.1. Primordial Helium Fraction

The primordial helium fraction,Yhe, is a probe of the number of relativistic species at the time of Big Bang Nucleosynthesis. As we have seen in Section 1.1, when Neff increases, the expansion rate increases as well. This means that free neutrons have less time to convert to protons through beta decay before the freeze out and so the final neutron-to-proton ratio is larger. The observable consequence is that the helium fraction is higher.Measurements of the primordial light element abundances seem consistent with a standard number of relativistic species at the time of BBN at 95% c.l. (NeffBBN=3.80-0.70+0.80 at 2σ [71]). This result is also consistent with the CMB value NeffCMB=3.68-0.64+0.68 at 95% c.l. obtained with the combination of data sets Planck + WP + highL. Nevertheless a tension among NeffBBN and NeffCMB arises if the H0 prior is taken into account; indeed in this case NeffCMB=3.62-0.48+0.50 at 95% c.l. However the value of Neff at BBN (T~1MeV) and the value measured by CMB at the last scattering epoch (T~1eV) may be different because of the unknown physics in the region 1MeV<T<1eV (see [72] for a recent review). Several efforts have been carried out in order to reconcile NeffBBN with the existence of extra species: decay of massive particles (1MeV<m<1 eV) in additional relativistic species [73, 74], decay of gravitino into axino and axion [75], or neutrino asymmetries [76].The BBN consistency relation implies that the number of relativistic species present at BBN is the same as the number measured by CMB at recombination. In order to impose the BBN consistency, we use the standard option implemented inCosmoMC [77]. This routine calculates Yhe as a function of Neff and Ωbh2 using a fitting formula obtained with the ParthENoPE code [78].


#### 2.2.2. Lensing Amplitude

Massive neutrinos suppress the growth of dark matter perturbations both through free streaming and through the equivalence delay. As a consequence, the matter power spectrum is damped on scales smaller than the scale of the horizon when neutrinos become nonrelativistic. The accuracy level of Planck allows for a detection of this clustering suppression in the CMB lensing potential, so it is timely to investigate the correlation amongAL and neutrino parameters. Planck analysis [24] provides an anomalous value of the lensing amplitude AL=1.23±0.11 (68% c.l., Planck + WP + highL). This anomaly was already revealed by ACT data (AL=1.70±0.38 at 68% c.l. [14]) even if with a lower precision. On the contrary, the SPT value (AL=0.86-0.13+0.15 at 68% c.l. [62]) is consistent with the standard prediction AL=1 within 1σ. Subsequent analyses [79] have confirmed this anomaly and studied the impact on massless Neff.Even if a modification of General Relativity cannot be ruled out, this anomaly is most likely a spurious signal related to the bias induced by the combination of data sets belonging to different experiments with different experimental techniques and different analysis methods. However, it is important to account for its effect in order to get unbiased constraints on the sum of neutrino masses, that is correlated withAL, as we will see in Section 3.2.


#### 2.2.3. Neutrino Perturbation Parameters

As we have seen in Section1.2, there is a wide variety of models that can explain an excess in the number of relativistic degrees of freedom at decoupling. In order to distinguish between these models, we introduce the neutrino perturbation parameters, the effective sound speed, and the viscosity parameter, ceff2 and cvis2, respectively [80, 81]. The reason is that these parameters can characterize the properties of the component that accounts for extra relativistic species.Following [82, 83], we encode ceff2 and cvis2 in the massless neutrino perturbation equations as follows:
(14)δ˙ν=a˙a(1-3ceff2)(δν+3a˙aqνk)-k(qν+23kh˙),q˙ν=kceff2(δν+3a˙aqνk)-a˙aqν-23kπν,π˙ν=3cvis2(25qν+815σ)-35kFν,3,2l+1kF˙ν,l-lFν,l-1=-(l+1)Fν,l+1,l≥3.
Here, the equations are written in the synchronous gauge (the one used in CAMB package [67]), the dot indicates the derivative respect to conformal time τ, a is the scale factor, k is the wavenumber, δν is the neutrino density contrast, qν is the neutrino velocity perturbation, πν is the neutrino anisotropic stress, and Fν,ℓ are higher-order moments of the neutrino distribution function and σ is the shear.The viscosity parameter is related to the clustering properties of particles, because it parameterizes the relationship between velocity/metric shear and anisotropic stress:cvis2=0 indicates a perfect fluid with undamped perturbations, while an increased value of cvis2 causes an overdamping of the oscillations. Free streaming particles, such as neutrinos, lead to anisotropies in the Cosmic Neutrino Background that are characterized by cvis2=1/3.Whenceff2 decreases, the internal pressure of the dark radiation fluid decreases and its perturbations can grow and start clustering; on the contrary, if ceff2 increases the oscillations are damped. Furthermore, an increase (decrease) in ceff2 leads to an increase (decrease) in the neutrino sound horizon and, as a consequence, also in the scale at which neutrino perturbations affect the dark radiation fluid.If the additional relativistic species we are dealing with consist of free streaming particles, such as neutrinos, the perturbation parameters would beceff2=cvis2=1/3.


## 2.1. Data Sets

Our basic data set is the Planck temperature power spectrum (both at lowℓ and at high ℓ) in combination with the WMAP 9 year polarization data (hereafter WP) and the high multipole CMB data of ACT and SPT (hereafter highL). These data sets are implemented in the analysis following the prescription of the Planck likelihood described in [24]. The additional data sets test the robustness at low redshift of the predictions obtained with CMB data. These data sets consist of a prior on the Hubble constant from the Hubble Space Telescope measurements [23] (hereafter H0) and the information on the dark matter clustering from the matter power spectrum extracted from the Data Release 9 (DR9) of the CMASS sample of galaxies [68] from the Baryon Acoustic Spectroscopic Survey (BOSS) [19], part of the program of the Sloan Digital Sky Survey III [69].


## 2.2. Parameters

In Table1, the parameters used in the analyses are listed together with the top-hat priors on them. The six standard parameters of the ΛCDM model are: the physical baryon density, ωb≡Ωbh2; the physical cold dark matter density, ωc≡Ωch2; the angular scale of the sound horizon, θs; the reionization optical depth, τ; the amplitude of the primordial spectrum at a certain pivot scale, As; and the power law spectral index of primordial density (scalar) perturbations, ns.Table 1
Priors on the cosmological parameters considered in this work. All priors are uniform (top hat) in the given intervals.
Parameter
Prior
ω
b
0.005
→
0.1
ω
cdm
0.001
→
0.99
θ
s
0.5
→
10
τ
0.01
→
0.8
ln
(
1
0
10
A
s
)
2.7
→
4
n
s
0.9
→
1.1
A
L
0
→
5
N
eff
0
→
7
∑
‍
m
ν [eV]
0
→
7
Y
he
0.1
→
0.5
c
eff
2
0
→
1
c
vis
2
0
→
1We include the effective number of relativistic degrees of freedomNeff, and, in addition, our runs also contain one or a combination of the following parameters: the sum of neutrino masses ∑‍mν, the primordial helium fraction Yhe, and the neutrino perturbation parameters, namely, the effective sound speed ceff2 and the viscosity parameter cvis2. Finally we also investigated the impact of a varying lensing amplitude AL.We assume that massive neutrinos are degenerate and share the same mass. Indeed given the present accuracy of CMB measurements, cosmology cannot extract the neutrino mass hierarchy but only the total hot dark matter density. Even if the future measurements of the Euclid survey will achieve an extreme accurate measurement of the neutrino mass (σmν≃0.01eV [70]), the neutrino mass hierarchy would not be pin down.


### 2.2.1. Primordial Helium Fraction

The primordial helium fraction,Yhe, is a probe of the number of relativistic species at the time of Big Bang Nucleosynthesis. As we have seen in Section 1.1, when Neff increases, the expansion rate increases as well. This means that free neutrons have less time to convert to protons through beta decay before the freeze out and so the final neutron-to-proton ratio is larger. The observable consequence is that the helium fraction is higher.Measurements of the primordial light element abundances seem consistent with a standard number of relativistic species at the time of BBN at 95% c.l. (NeffBBN=3.80-0.70+0.80 at 2σ [71]). This result is also consistent with the CMB value NeffCMB=3.68-0.64+0.68 at 95% c.l. obtained with the combination of data sets Planck + WP + highL. Nevertheless a tension among NeffBBN and NeffCMB arises if the H0 prior is taken into account; indeed in this case NeffCMB=3.62-0.48+0.50 at 95% c.l. However the value of Neff at BBN (T~1MeV) and the value measured by CMB at the last scattering epoch (T~1eV) may be different because of the unknown physics in the region 1MeV<T<1eV (see [72] for a recent review). Several efforts have been carried out in order to reconcile NeffBBN with the existence of extra species: decay of massive particles (1MeV<m<1 eV) in additional relativistic species [73, 74], decay of gravitino into axino and axion [75], or neutrino asymmetries [76].The BBN consistency relation implies that the number of relativistic species present at BBN is the same as the number measured by CMB at recombination. In order to impose the BBN consistency, we use the standard option implemented inCosmoMC [77]. This routine calculates Yhe as a function of Neff and Ωbh2 using a fitting formula obtained with the ParthENoPE code [78].


### 2.2.2. Lensing Amplitude

Massive neutrinos suppress the growth of dark matter perturbations both through free streaming and through the equivalence delay. As a consequence, the matter power spectrum is damped on scales smaller than the scale of the horizon when neutrinos become nonrelativistic. The accuracy level of Planck allows for a detection of this clustering suppression in the CMB lensing potential, so it is timely to investigate the correlation amongAL and neutrino parameters. Planck analysis [24] provides an anomalous value of the lensing amplitude AL=1.23±0.11 (68% c.l., Planck + WP + highL). This anomaly was already revealed by ACT data (AL=1.70±0.38 at 68% c.l. [14]) even if with a lower precision. On the contrary, the SPT value (AL=0.86-0.13+0.15 at 68% c.l. [62]) is consistent with the standard prediction AL=1 within 1σ. Subsequent analyses [79] have confirmed this anomaly and studied the impact on massless Neff.Even if a modification of General Relativity cannot be ruled out, this anomaly is most likely a spurious signal related to the bias induced by the combination of data sets belonging to different experiments with different experimental techniques and different analysis methods. However, it is important to account for its effect in order to get unbiased constraints on the sum of neutrino masses, that is correlated withAL, as we will see in Section 3.2.


### 2.2.3. Neutrino Perturbation Parameters

As we have seen in Section1.2, there is a wide variety of models that can explain an excess in the number of relativistic degrees of freedom at decoupling. In order to distinguish between these models, we introduce the neutrino perturbation parameters, the effective sound speed, and the viscosity parameter, ceff2 and cvis2, respectively [80, 81]. The reason is that these parameters can characterize the properties of the component that accounts for extra relativistic species.Following [82, 83], we encode ceff2 and cvis2 in the massless neutrino perturbation equations as follows:
(14)δ˙ν=a˙a(1-3ceff2)(δν+3a˙aqνk)-k(qν+23kh˙),q˙ν=kceff2(δν+3a˙aqνk)-a˙aqν-23kπν,π˙ν=3cvis2(25qν+815σ)-35kFν,3,2l+1kF˙ν,l-lFν,l-1=-(l+1)Fν,l+1,l≥3.
Here, the equations are written in the synchronous gauge (the one used in CAMB package [67]), the dot indicates the derivative respect to conformal time τ, a is the scale factor, k is the wavenumber, δν is the neutrino density contrast, qν is the neutrino velocity perturbation, πν is the neutrino anisotropic stress, and Fν,ℓ are higher-order moments of the neutrino distribution function and σ is the shear.The viscosity parameter is related to the clustering properties of particles, because it parameterizes the relationship between velocity/metric shear and anisotropic stress:cvis2=0 indicates a perfect fluid with undamped perturbations, while an increased value of cvis2 causes an overdamping of the oscillations. Free streaming particles, such as neutrinos, lead to anisotropies in the Cosmic Neutrino Background that are characterized by cvis2=1/3.Whenceff2 decreases, the internal pressure of the dark radiation fluid decreases and its perturbations can grow and start clustering; on the contrary, if ceff2 increases the oscillations are damped. Furthermore, an increase (decrease) in ceff2 leads to an increase (decrease) in the neutrino sound horizon and, as a consequence, also in the scale at which neutrino perturbations affect the dark radiation fluid.If the additional relativistic species we are dealing with consist of free streaming particles, such as neutrinos, the perturbation parameters would beceff2=cvis2=1/3.


## 2.2.1. Primordial Helium Fraction

The primordial helium fraction,Yhe, is a probe of the number of relativistic species at the time of Big Bang Nucleosynthesis. As we have seen in Section 1.1, when Neff increases, the expansion rate increases as well. This means that free neutrons have less time to convert to protons through beta decay before the freeze out and so the final neutron-to-proton ratio is larger. The observable consequence is that the helium fraction is higher.Measurements of the primordial light element abundances seem consistent with a standard number of relativistic species at the time of BBN at 95% c.l. (NeffBBN=3.80-0.70+0.80 at 2σ [71]). This result is also consistent with the CMB value NeffCMB=3.68-0.64+0.68 at 95% c.l. obtained with the combination of data sets Planck + WP + highL. Nevertheless a tension among NeffBBN and NeffCMB arises if the H0 prior is taken into account; indeed in this case NeffCMB=3.62-0.48+0.50 at 95% c.l. However the value of Neff at BBN (T~1MeV) and the value measured by CMB at the last scattering epoch (T~1eV) may be different because of the unknown physics in the region 1MeV<T<1eV (see [72] for a recent review). Several efforts have been carried out in order to reconcile NeffBBN with the existence of extra species: decay of massive particles (1MeV<m<1 eV) in additional relativistic species [73, 74], decay of gravitino into axino and axion [75], or neutrino asymmetries [76].The BBN consistency relation implies that the number of relativistic species present at BBN is the same as the number measured by CMB at recombination. In order to impose the BBN consistency, we use the standard option implemented inCosmoMC [77]. This routine calculates Yhe as a function of Neff and Ωbh2 using a fitting formula obtained with the ParthENoPE code [78].


## 2.2.2. Lensing Amplitude

Massive neutrinos suppress the growth of dark matter perturbations both through free streaming and through the equivalence delay. As a consequence, the matter power spectrum is damped on scales smaller than the scale of the horizon when neutrinos become nonrelativistic. The accuracy level of Planck allows for a detection of this clustering suppression in the CMB lensing potential, so it is timely to investigate the correlation amongAL and neutrino parameters. Planck analysis [24] provides an anomalous value of the lensing amplitude AL=1.23±0.11 (68% c.l., Planck + WP + highL). This anomaly was already revealed by ACT data (AL=1.70±0.38 at 68% c.l. [14]) even if with a lower precision. On the contrary, the SPT value (AL=0.86-0.13+0.15 at 68% c.l. [62]) is consistent with the standard prediction AL=1 within 1σ. Subsequent analyses [79] have confirmed this anomaly and studied the impact on massless Neff.Even if a modification of General Relativity cannot be ruled out, this anomaly is most likely a spurious signal related to the bias induced by the combination of data sets belonging to different experiments with different experimental techniques and different analysis methods. However, it is important to account for its effect in order to get unbiased constraints on the sum of neutrino masses, that is correlated withAL, as we will see in Section 3.2.


## 2.2.3. Neutrino Perturbation Parameters

As we have seen in Section1.2, there is a wide variety of models that can explain an excess in the number of relativistic degrees of freedom at decoupling. In order to distinguish between these models, we introduce the neutrino perturbation parameters, the effective sound speed, and the viscosity parameter, ceff2 and cvis2, respectively [80, 81]. The reason is that these parameters can characterize the properties of the component that accounts for extra relativistic species.Following [82, 83], we encode ceff2 and cvis2 in the massless neutrino perturbation equations as follows:
(14)δ˙ν=a˙a(1-3ceff2)(δν+3a˙aqνk)-k(qν+23kh˙),q˙ν=kceff2(δν+3a˙aqνk)-a˙aqν-23kπν,π˙ν=3cvis2(25qν+815σ)-35kFν,3,2l+1kF˙ν,l-lFν,l-1=-(l+1)Fν,l+1,l≥3.
Here, the equations are written in the synchronous gauge (the one used in CAMB package [67]), the dot indicates the derivative respect to conformal time τ, a is the scale factor, k is the wavenumber, δν is the neutrino density contrast, qν is the neutrino velocity perturbation, πν is the neutrino anisotropic stress, and Fν,ℓ are higher-order moments of the neutrino distribution function and σ is the shear.The viscosity parameter is related to the clustering properties of particles, because it parameterizes the relationship between velocity/metric shear and anisotropic stress:cvis2=0 indicates a perfect fluid with undamped perturbations, while an increased value of cvis2 causes an overdamping of the oscillations. Free streaming particles, such as neutrinos, lead to anisotropies in the Cosmic Neutrino Background that are characterized by cvis2=1/3.Whenceff2 decreases, the internal pressure of the dark radiation fluid decreases and its perturbations can grow and start clustering; on the contrary, if ceff2 increases the oscillations are damped. Furthermore, an increase (decrease) in ceff2 leads to an increase (decrease) in the neutrino sound horizon and, as a consequence, also in the scale at which neutrino perturbations affect the dark radiation fluid.If the additional relativistic species we are dealing with consist of free streaming particles, such as neutrinos, the perturbation parameters would beceff2=cvis2=1/3.


## 3. Results

In what follows, the results of our analyses are presented. These results cover a wide range of different parameter spaces and they are obtained using different combinations of data sets. In Section3.1, we study the impact of a varying helium fraction and of the BBN consistency relation on the effective number of relativistic degrees of freedom. Section 3.2 analyzes the dependence of the neutrino abundances and masses on the varying lensing amplitude and on the matter power spectrum information. Finally in Section 3.3, we provide constraints on the neutrino perturbation parameters.


### 3.1. Constraints onNeff: Number of Relativistic Species

In Table2, the constraints on the number of effective relativistic degrees of freedom are shown with different priors.Table 2
Marginalized 68% c.l. constraints onNeff in a standard cosmology with Neff massless neutrinos. In the second and in the fourth columns, we also apply a prior on the Hubble constant from the Hubble Space Telescope measurements. In the third and in the fourth columns, we apply the BBN consistency relation. In the last column, we also vary the helium fraction Yhe and we show its marginalized 68% c.l.
Planck + WP + highL
Planck + WP + highL +H0
Planck + WP + highL + BBNc
Planck + WP + highL +H0 + BBNc
Planck + WP + highL
N
eff
3.63
±
0.41
3.81
±
0.29
3.44
±
0.35
3.65
±
0.26
3.32
±
0.70
Y
he
0.24
0.24
BBN
BBN
0.260
±
0.036First of all, in order to recall the effects of the number of effective relativistic degrees of freedom on CMB, we show in Figure3 the degeneracies among Neff and the parameters that are directly measured by the CMB temperature power spectrum: the redshift of the equivalence zeq, the angular scale of the sound horizon θs, and the damping scale θd. We can notice that zeq is proportional to the increase of Neff as expected from (3), while θd is correlated to Neff through the expansion rate at recombination H, because it scales as H.Figure 3
68
% and 95% c.l. 2D marginalized posterior in the plane Neff-H0.The inclusion of theH0 prior moves the mean value of Neff towards a higher value and reduces the error on Neff (Neff=3.81±0.29 with respect to Neff=3.63±0.41, 68% c.l.). The effect can be noticed in Figure 4. The final result is a ~2.6σ evidence for an extra dark radiation component. Instead, applying the BBN consistency relation leads to a constraint on Neff much closer to the standard value than in the case of Yhe fixed to 0.24; that is, Neff=3.44±0.35 (68% c.l.).Figure 4
68
% and 95% c.l. 2D marginalized posterior in the plane Neff-H0.Figure 5
68
% and 95% c.l. 2D marginalized posterior in the plane Neff-Yhe. Dotted line shows the BBN consistency relation.Finally, if we consider the helium fraction as a free parameter (last column of Table2), the evidence for an extra number of relativistic degrees of freedom disappears and we obtain a milder constraint on Neff (Neff=3.32±0.70, 68% c.l.) that makes it perfectly consistent with the prediction of the Standard Model. Figure 5 shows the anticorrelation between Neff and Yhe from CMB data (blue contours) and the BBN consistency relation among these two parameters (dotted line). We can notice that an increase in Neff requires a lower value of Yhe to reproduce the same CMB power spectrum, as we have explained in Section 1.1. Concerning the comparison between the models with and without varying the primordial helium fraction, the Δχ2 at the best fit point is negligible, meaning that a higher value of Yhe is preferred by the data but a lower value can be accommodated by tuning the other parameters.All the cases described above are illustrated in Figure6 where the one-dimensional posterior of Neff is shown for the different cases of Table 2. We can notice that both the inclusion of H0 and BBN consistency narrow the posterior and reduce the error on Neff. However, H0 moves the best fit of Neff toward a higher value of the number of effective relativistic degrees of freedom, while BBN consistency prefers a lower value and brings back Neff closer to the standard value. In subsequent analyses, we will follow a conservative approach, applying the BBN consistency relation in all our MCMC analyses, accordingly also with Planck team strategy.Figure 6
1D posterior ofNeff. The different cases reported in Table 2 are shown: black line corresponds to Planck + WP + highL, blue line to Planck + WP + highL + H0, red line to Planck + WP + highL + BBNc, and magenta line to Planck + WP + highL + BBNc + H0. Finally, the green line refers to the analysis that includes also a varying Yhe.


### 3.2. Constraints onNeff and ∑‍mν: Massive Neutrinos

The constraints on massive neutrinos are summarized in Table3.Table 3
Marginalized 68% c.l. constraints onNeff and AL and 95% cl upper bounds on ∑‍mν in extended models with Neff massive neutrinos. We also include the lensing amplitude as a free parameter.
Planck + WP + highL
Planck + WP + highL
Planck + WP + highL +H0
Planck + WP + highL + DR9
Planck + WP + highL + DR9 +H0
N
eff
3.38
±
0.36
3.65
±
0.38
3.81
±
0.28
3.33
±
0.31
3.65
±
0.26
∑
‍
m
ν [eV]
<0.64
<1.03
<0.66
<0.66
<0.51
A
L
1
1.36
±
0.14
1.36
±
0.14
1.10
±
0.08
1.10
±
0.07We also marginalize over the lensing amplitude and we study this effect in Figure7 and in Figure 8 for our basic data set (Planck + WP + highL). As we already discussed in Section 2.2, Planck analysis points towards a value of the lensing amplitude higher than the standard one. This anomaly is confirmed by our results AL=1.36±0.14 (68% c.l.) related to the model with a varying number of massive neutrinos. Nevertheless, including BOSS DR9, data shift the AL parameter towards a value consistent with the standard AL=1 value within 2σ (AL=1.10±0.08, 68% c.l.). It is clear from Figure 7(a) that the neutrino mass has a strong degeneracy with the lensing amplitude: allowing for a higher value of AL leading to a larger value of the neutrino mass; the 95% upper bound moves from 0.64eV to 1.03eV. Figure 7(b) shows that there is no preferred direction for a correlation between Neff and AL, but the side effect of the degeneracy among ∑‍mν and AL is also an increasing value of Neff (3.65±0.38 against 3.38±0.36, 68% c.l.) related to the correlation among Neff and ∑‍mν. This conclusion arises from Figure 8(a) where the increasing value of AL is located along the bisecting line in the plane Neff-∑‍mν. We summarize the effect of the lensing amplitude on the neutrino parameters in Figure 8(b): a varying AL parameter will lead to a larger neutrino mass and, consequently, to a larger Neff. Finally, we shall comment that a larger value of AL will provide a better fit to the data, lowering the χ2 by 4.2 units.68
% and 95% c.l. 2D marginalized posterior in the plane ∑‍mν-AL (a) and in the plane Neff-AL (b).
(a)
(b)(a) Scatter plot in the∑‍mν-Neff plane with points colored by the value of the AL parameter (second column of Table 3). (b) 68% and 95% c.l. 2D marginalized posterior in the plane ∑‍mν-Neff; blue contours refer to the case with a varying lensing amplitude (second column of Table 3) and red contours illustrate the AL=1 case (first column of Table 3).
(a)
(b)Concerning the effects of external non-CMB data sets, we include in the analyses of aΛCDM model with massive neutrinos and a varying lensing amplitude the H0 prior and the BOSS DR9 data. On one hand with the inclusion of the H0 prior, we obtain a better constraint on Neff, driving Neff from Neff=3.65±0.38 to Neff=3.81±0.28 (68% c.l.). So the combination of the data sets Planck + WP + highL + H0 provides a stronger evidence (2.7σ) for an extra dark radiation component. On the other hand, H0 leads to tighter constraints on the 95% c.l. upper bound of the sum of neutrino masses, moving it from ∑‍mν<1.03eV to ∑‍mν<0.66eV at 95% c.l. (see Figure 9(a)). The same effect on ∑‍mν can be achieved by including BOSS DR9, but in this case Neff remains close to the standard value Neff=3.26±0.30 (68% c.l.) (see Figure 9(b)). The joint effect of adding both an H0 prior and the galaxy clustering information from BOSS DR9 is shown in Figure 9(c): the 95% upper bound on the sum of neutrino masses is tightened both by the prior on H0 and the BOSS DR9 galaxy clustering information, and an extra dark radiation component is favored at 2.3σ level (Neff=3.65±0.26, 68% c.l.).68
% and 95% c.l. 2D marginalized posterior in the ∑‍mν-Neff plane. Blue contours refer to the constraints from the combination of Planck + WP + highL and red contours include also H0 (a), BOSS DR9 (b), and H0 and BOSS DR9 (c).
(a)
(b)
(c)


### 3.3. Constraints onceff2 and cvis2: Perturbation Parameters

Table4 reports the constraints on the perturbation parameters of a varying number of relativistic species. The neutrino perturbation parameters are not strongly affected by the inclusion of the H0 prior: the constraints on ceff2 and cvis2 remain almost the same. Interestingly both the effective sound speed and the viscosity parameter show a deviation from the standard value 0.33 having ceff2=0.309±0.012 and cvis2=0.56±0.17 at 68% c.l. for the basic data set Planck + WP + highL, consistent with the results of [84]. Furthermore, we can notice that varying the neutrino perturbation parameters does not change our conclusions on the effective number of relativistic species; the bounds on Neff turn out to be almost the same as those reported in Table 2: varying ceff2 and cvis2, we get Neff=3.40±0.34 (68% c.l.), while we obtained Neff=3.44±0.35 (68% c.l.) with standard ceff2 and cvis2.Table 4
Marginalized 68% constraints onNeff, ceff2, and cvis2.
Planck + WP + highL
Planck + WP + highL +H0
N
eff
3.40
±
0.34
3.56
±
0.26
c
eff
2
0.309
±
0.012
0.310
±
0.012
c
vis
2
0.56
±
0.17
0.57
±
0.17


## 3.1. Constraints onNeff: Number of Relativistic Species

In Table2, the constraints on the number of effective relativistic degrees of freedom are shown with different priors.Table 2
Marginalized 68% c.l. constraints onNeff in a standard cosmology with Neff massless neutrinos. In the second and in the fourth columns, we also apply a prior on the Hubble constant from the Hubble Space Telescope measurements. In the third and in the fourth columns, we apply the BBN consistency relation. In the last column, we also vary the helium fraction Yhe and we show its marginalized 68% c.l.
Planck + WP + highL
Planck + WP + highL +H0
Planck + WP + highL + BBNc
Planck + WP + highL +H0 + BBNc
Planck + WP + highL
N
eff
3.63
±
0.41
3.81
±
0.29
3.44
±
0.35
3.65
±
0.26
3.32
±
0.70
Y
he
0.24
0.24
BBN
BBN
0.260
±
0.036First of all, in order to recall the effects of the number of effective relativistic degrees of freedom on CMB, we show in Figure3 the degeneracies among Neff and the parameters that are directly measured by the CMB temperature power spectrum: the redshift of the equivalence zeq, the angular scale of the sound horizon θs, and the damping scale θd. We can notice that zeq is proportional to the increase of Neff as expected from (3), while θd is correlated to Neff through the expansion rate at recombination H, because it scales as H.Figure 3
68
% and 95% c.l. 2D marginalized posterior in the plane Neff-H0.The inclusion of theH0 prior moves the mean value of Neff towards a higher value and reduces the error on Neff (Neff=3.81±0.29 with respect to Neff=3.63±0.41, 68% c.l.). The effect can be noticed in Figure 4. The final result is a ~2.6σ evidence for an extra dark radiation component. Instead, applying the BBN consistency relation leads to a constraint on Neff much closer to the standard value than in the case of Yhe fixed to 0.24; that is, Neff=3.44±0.35 (68% c.l.).Figure 4
68
% and 95% c.l. 2D marginalized posterior in the plane Neff-H0.Figure 5
68
% and 95% c.l. 2D marginalized posterior in the plane Neff-Yhe. Dotted line shows the BBN consistency relation.Finally, if we consider the helium fraction as a free parameter (last column of Table2), the evidence for an extra number of relativistic degrees of freedom disappears and we obtain a milder constraint on Neff (Neff=3.32±0.70, 68% c.l.) that makes it perfectly consistent with the prediction of the Standard Model. Figure 5 shows the anticorrelation between Neff and Yhe from CMB data (blue contours) and the BBN consistency relation among these two parameters (dotted line). We can notice that an increase in Neff requires a lower value of Yhe to reproduce the same CMB power spectrum, as we have explained in Section 1.1. Concerning the comparison between the models with and without varying the primordial helium fraction, the Δχ2 at the best fit point is negligible, meaning that a higher value of Yhe is preferred by the data but a lower value can be accommodated by tuning the other parameters.All the cases described above are illustrated in Figure6 where the one-dimensional posterior of Neff is shown for the different cases of Table 2. We can notice that both the inclusion of H0 and BBN consistency narrow the posterior and reduce the error on Neff. However, H0 moves the best fit of Neff toward a higher value of the number of effective relativistic degrees of freedom, while BBN consistency prefers a lower value and brings back Neff closer to the standard value. In subsequent analyses, we will follow a conservative approach, applying the BBN consistency relation in all our MCMC analyses, accordingly also with Planck team strategy.Figure 6
1D posterior ofNeff. The different cases reported in Table 2 are shown: black line corresponds to Planck + WP + highL, blue line to Planck + WP + highL + H0, red line to Planck + WP + highL + BBNc, and magenta line to Planck + WP + highL + BBNc + H0. Finally, the green line refers to the analysis that includes also a varying Yhe.


## 3.2. Constraints onNeff and ∑‍mν: Massive Neutrinos

The constraints on massive neutrinos are summarized in Table3.Table 3
Marginalized 68% c.l. constraints onNeff and AL and 95% cl upper bounds on ∑‍mν in extended models with Neff massive neutrinos. We also include the lensing amplitude as a free parameter.
Planck + WP + highL
Planck + WP + highL
Planck + WP + highL +H0
Planck + WP + highL + DR9
Planck + WP + highL + DR9 +H0
N
eff
3.38
±
0.36
3.65
±
0.38
3.81
±
0.28
3.33
±
0.31
3.65
±
0.26
∑
‍
m
ν [eV]
<0.64
<1.03
<0.66
<0.66
<0.51
A
L
1
1.36
±
0.14
1.36
±
0.14
1.10
±
0.08
1.10
±
0.07We also marginalize over the lensing amplitude and we study this effect in Figure7 and in Figure 8 for our basic data set (Planck + WP + highL). As we already discussed in Section 2.2, Planck analysis points towards a value of the lensing amplitude higher than the standard one. This anomaly is confirmed by our results AL=1.36±0.14 (68% c.l.) related to the model with a varying number of massive neutrinos. Nevertheless, including BOSS DR9, data shift the AL parameter towards a value consistent with the standard AL=1 value within 2σ (AL=1.10±0.08, 68% c.l.). It is clear from Figure 7(a) that the neutrino mass has a strong degeneracy with the lensing amplitude: allowing for a higher value of AL leading to a larger value of the neutrino mass; the 95% upper bound moves from 0.64eV to 1.03eV. Figure 7(b) shows that there is no preferred direction for a correlation between Neff and AL, but the side effect of the degeneracy among ∑‍mν and AL is also an increasing value of Neff (3.65±0.38 against 3.38±0.36, 68% c.l.) related to the correlation among Neff and ∑‍mν. This conclusion arises from Figure 8(a) where the increasing value of AL is located along the bisecting line in the plane Neff-∑‍mν. We summarize the effect of the lensing amplitude on the neutrino parameters in Figure 8(b): a varying AL parameter will lead to a larger neutrino mass and, consequently, to a larger Neff. Finally, we shall comment that a larger value of AL will provide a better fit to the data, lowering the χ2 by 4.2 units.68
% and 95% c.l. 2D marginalized posterior in the plane ∑‍mν-AL (a) and in the plane Neff-AL (b).
(a)
(b)(a) Scatter plot in the∑‍mν-Neff plane with points colored by the value of the AL parameter (second column of Table 3). (b) 68% and 95% c.l. 2D marginalized posterior in the plane ∑‍mν-Neff; blue contours refer to the case with a varying lensing amplitude (second column of Table 3) and red contours illustrate the AL=1 case (first column of Table 3).
(a)
(b)Concerning the effects of external non-CMB data sets, we include in the analyses of aΛCDM model with massive neutrinos and a varying lensing amplitude the H0 prior and the BOSS DR9 data. On one hand with the inclusion of the H0 prior, we obtain a better constraint on Neff, driving Neff from Neff=3.65±0.38 to Neff=3.81±0.28 (68% c.l.). So the combination of the data sets Planck + WP + highL + H0 provides a stronger evidence (2.7σ) for an extra dark radiation component. On the other hand, H0 leads to tighter constraints on the 95% c.l. upper bound of the sum of neutrino masses, moving it from ∑‍mν<1.03eV to ∑‍mν<0.66eV at 95% c.l. (see Figure 9(a)). The same effect on ∑‍mν can be achieved by including BOSS DR9, but in this case Neff remains close to the standard value Neff=3.26±0.30 (68% c.l.) (see Figure 9(b)). The joint effect of adding both an H0 prior and the galaxy clustering information from BOSS DR9 is shown in Figure 9(c): the 95% upper bound on the sum of neutrino masses is tightened both by the prior on H0 and the BOSS DR9 galaxy clustering information, and an extra dark radiation component is favored at 2.3σ level (Neff=3.65±0.26, 68% c.l.).68
% and 95% c.l. 2D marginalized posterior in the ∑‍mν-Neff plane. Blue contours refer to the constraints from the combination of Planck + WP + highL and red contours include also H0 (a), BOSS DR9 (b), and H0 and BOSS DR9 (c).
(a)
(b)
(c)


## 3.3. Constraints onceff2 and cvis2: Perturbation Parameters

Table4 reports the constraints on the perturbation parameters of a varying number of relativistic species. The neutrino perturbation parameters are not strongly affected by the inclusion of the H0 prior: the constraints on ceff2 and cvis2 remain almost the same. Interestingly both the effective sound speed and the viscosity parameter show a deviation from the standard value 0.33 having ceff2=0.309±0.012 and cvis2=0.56±0.17 at 68% c.l. for the basic data set Planck + WP + highL, consistent with the results of [84]. Furthermore, we can notice that varying the neutrino perturbation parameters does not change our conclusions on the effective number of relativistic species; the bounds on Neff turn out to be almost the same as those reported in Table 2: varying ceff2 and cvis2, we get Neff=3.40±0.34 (68% c.l.), while we obtained Neff=3.44±0.35 (68% c.l.) with standard ceff2 and cvis2.Table 4
Marginalized 68% constraints onNeff, ceff2, and cvis2.
Planck + WP + highL
Planck + WP + highL +H0
N
eff
3.40
±
0.34
3.56
±
0.26
c
eff
2
0.309
±
0.012
0.310
±
0.012
c
vis
2
0.56
±
0.17
0.57
±
0.17


## 4. Future Constraints

We present here a forecast of the impact of the Euclid survey [85] in constraining Neff and ∑‍mν (for a recent and complete analysis see [70]). We perform a Fisher matrix analysis following the prescription of [86]. The fiducial values of the standard cosmological parameters are fixed at the best fit values obtained by Planck [24] (see Table 5). Concerning the neutrino parameters, the fiducial value of Neff is fixed at the standard cosmological value Neff=3.046, while the neutrino mass fiducial value is ∑‍mν=0.2eV (we recall here that the minimum mass sum in the inverted hierarchy is 0.11eV, while in the normal hierarchy it is 0.06 eV). Furthermore, we add priors on the standard cosmological parameters from Planck results [24].Table 5
Fiducial values of the cosmological parameters considered in the forecast analysis.
Parameter
Fiducial value
ω
b
0.02207
ω
cdm
0.1203
h
0.671
ln
(
1
0
10
A
s
)
3.098
n
s
0.962
N
eff
3.046
∑
‍
m
ν [eV]
0.2The1σ marginalized errors on the parameters of the two different fiducial cosmological models (with either massless or massive neutrinos) are reported in Table 6. The relative errors show that Euclid will improve the constraints on both Neff and ∑‍mν. Nevertheless, even Euclid will not be able to reveal the hierarchy, it would provide a detection of the neutrino mass sum only if ∑‍mν>0.1eV [87].Table 6
1
σ marginalized errors and relative errors for all parameters considering the two different fiducial cosmological models described in the text: ΛCDM model with either massless or massive neutrinos.
Euclid + Planck prior
ΛCDM + Neff
ΛCDM + Neff + ∑‍mν
1
σ
rel. err.
1
σ
rel. err.
ω
b
0.00031
1.4
%
0.00031
1.4
%
ω
cdm
0.0021
1.7
%
0.0022
1.8
%
h
0.007
1.0
%
0.007
1.1
%
ln
(
1
0
10
A
s
)
0.064
2.1
%
0.072
2.3
%
n
s
0.009
0.9
%
0.009
0.9
%
N
eff
0.23
7.5
%
0.29
9.7
%
∑
‍
m
ν [eV]
—
—
0.10
51.4
%


## 5. Conclusions

The newly released Planck data have provided us with an extremely precise picture of the cosmic microwave background, confirming the standardΛCDM model. However, the exact properties of the dark sector are still under discussion and, in particular, there is no strong argument against the existence of a dark radiation component. On the contrary, combining CMB data with measurements of galaxy clustering and of the Hubble constant leads to an evidence for a nonstandard number of relativistic species.In this paper we have illustrated the effects of an additional relativistic component on the temperature power spectrum and we have reviewed the most promising models to explain the presence of this component: sterile neutrinos, axions, decay of massive particles, and interactions between dark matter and dark radiation sectors.We have focused on the hypothesis of a link between cosmology and neutrino physics that can explain the cosmologically inferred excess in the number of relativistic species in terms of sterile neutrinos whose existence could explain some short baseline neutrino oscillations results. In this framework, we have updated the cosmological constraints on massive neutrinos including the new Planck CMB data and the matter power spectrum from BOSS DR9. Including also a prior on the Hubble constant from the Hubble Space Telescope measurements, our results show a preference for a nonstandard number of neutrino species at2.3σ with Neff=3.65±0.26 at 68% c.l. and an upper bound on the sum of neutrino masses of 0.51eV at 95% c.l.However, the relevance of these cosmological constraints on dark radiation depends on the model and on the data sets.We have stressed the impact of the lensing amplitude on these results; the inclusion of a varying lensing amplitude drives the results towards a more statistically significant detection of dark radiation.Concerning the data sets, theH0 prior also leads to a better constraint on Neff. The former effect is related to the 2.5σ tension among Planck and HST measurements of the Hubble constant that must be fixed [88].Finally our results confirm a significant deviation from the standard values (ceff2=1/3 and cvis2=1/3) expected for a free streaming dark radiation component. We find ceff2=0.309±0.012 and cvis2=0.56±0.17 at 68% c.l., allowing for further consideration on the nature of dark radiation.In conclusion, there is still ample room for interesting new discoveries of physics beyond the standard model in the form of dark radiation.



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*Source: 191047-2013-11-27.xml*