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Dark Radiation and Inflationary Freedom after Planck 2015 Eleonora Di Valentino,1 Stefano Gariazzo,2,3 Martina Gerbino,4,5,6 Elena Giusarma,6,7 and Olga Mena8 1Institut d’Astrophysique de Paris (UMR7095: CNRS & UPMC- Sorbonne Universities), F-75014, Paris, France 2Department of Physics, University of Torino, Via P. Giuria 1, I–10125 Torino, Italy 3INFN, Sezione di Torino, Via P. Giuria 1, I–10125 Torino, Italy 4The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden 5Nordita (Nordic Institute for Theoretical Physics), Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden 6Physics Department and INFN, Universit`a di Roma “La Sapienza”, P.le Aldo Moro 2, 00185, Rome, Italy 7McWilliams Center for Cosmology, Department of Physics, 6 Carnegie Mellon University, Pittsburgh, PA 15213, USA 1 8IFIC, Universidad de Valencia-CSIC, 46071, Valencia, Spain 0 2 The simplest inflationary models predict a primordial power spectrum (PPS) of the curvature fluctuationsthatcanbedescribedbyapower-lawfunctionthatisnearlyscale-invariant. Ithasbeen y shown, however, that the low-multipole spectrum of the CMB anisotropies may hint the presence a of some features in the shape of the scalar PPS, which could deviate from its canonical power- M law form. We study the possible degeneracies of this non-standard PPS with the active neutrino masses, the effective number of relativistic species and a sterile neutrino or a thermal axion mass. 4 The limits on these additional parameters are less constraining in a model with a non-standard 2 PPS when including only the temperature auto-correlation spectrum measurements in the data analyses. The inclusion of the polarization spectra noticeably helps in reducing the degeneracies, ] O leadingtoresultsthattypicallyshownodeviationfromtheΛCDMmodelwithastandardpower-law PPS. These findings are robust against changes in the function describing the non-canonical PPS. C Albeit current cosmological measurements seem to prefer the simple power-law PPS description, . h the statistical significance to rule out other possible parameterizations is still very poor. Future p cosmological measurements are crucial to improve the present PPS uncertainties. - o r I. INTRODUCTION as a power-law. While the significance of the deviations t s is small for some cases, it is interesting to note that the a CMB temperature power spectra as measured by both [ Inflationisoneofthemostsuccessfultheoriesthatex- WMAP[30]andPlanck[31,32]showsimilarresults: the 2 plains the so-called horizon and flatness problems, pro- differencesfromthepower-lawarelocatedinthelowmul- v viding an origin for the primordial density perturbations tipole region. These deviations could arise from some 7 that evolved to form the structures we observe today [1– statistical fluctuations, or, instead, result from a non- 5 11]. The standard inflationary paradigm predicts a sim- standard inflationary mechanism. 5 ple shape for the primordial power spectrum (PPS) of 7 Ifthefeaturesweobservearetheresultofanon-standard scalarperturbations: inthiscontext,thePPScanbede- 0 inflationary mechanism, we may be using an incomplete . scribed by a power-law expression. However, there also parameterization for the PPS in our cosmological anal- 1 exist more complicated inflationary scenarios which can 0 yses. It has been shown that this could lead to biased give rise to non-standard PPS forms, with possible fea- 6 results in the cosmological constraints of different quan- 1 tures at different scales, see e.g. Refs. [12, 13] and the tities. Namely, the constraints on the dark radiation : reviews [14, 15]. properties [33–35] or on non-Gaussianities [36] can be v i The usual procedure to reconstruct the underlying distorted, leading to spurious conclusions. In this work X PPS is to assume a model for the evolution of the Uni- we aim to study the impact of a general PPS form in r verse and calculate the transfer function, and then use the constraints obtained for the properties of dark ra- a different techniques to constrain a completely unknown diation candidates, such as the active neutrino masses PPS,comparingthetheoreticalpredictionwiththemea- and their effective number, sterile neutrino species and sured power spectrum of the Cosmic Microwave Back- thermal axion properties. The outline of the Paper is as ground radiation (CMB). Among the methods devel- follows: wepresentthebaselinestandardΛCDM cosmo- oped in the past, we can list regularization methods as logical model, the PPS parameterization and the cosmo- the Richardson-Lucy iteration [16–19], truncated singu- logical data in Sec. II. The results obtained within the larvaluedecomposition[20]andTikhonovregularization ΛCDM framework are presented in Sec. III. Concerning [21, 22], or methods as the maximum entropy deconvo- possible extensions of the ΛCDM scenario, we study the lution [23] or the cosmic inversion methods [24–28]. Re- constraints on the effective number of relativistic species cently, the Planck collaboration presented a wide discus- in Sec. IV, on the neutrino masses in Sec. V, on massive sion about constraints on inflation [29]. All these meth- neutrinos with a varying effective number of relativistic ods provide hint for a PPS which may not be as simple speciesinSec.VI,onmassivesterileneutrinosinSec.VII, 2 andonthethermalaxionpropertiesinSec.VIII.Finally, the assumed PPS. Several cosmological parameters are in Sec. IX we show the reconstructed PPS shape, com- known to present degeneracies with the standard PPS paring different possible approaches, and we draw our parameters, as, for example, the existing one between conclusions in Sec. X. effective number of relativistic species N and the tilt eff of the power-law PPS n . These degeneracies could be s even stronger when more freedom is allowed for the PPS II. BASELINE MODEL AND COSMOLOGICAL shape. We adopt here a non-parametric description for DATA thePPSofscalarperturbations: wedescribethefunction P (k)astheinterpolationamongaseriesofnodesatfixed s In this Section we outline the baseline theoretical wavemodes k. Unless otherwise stated, we shall consider model that will be extended to study the dark radiation twelve nodes kj (j ∈ [1,12]) that cover a wide range of properties. For our analyses we use the numerical Boltz- values of k: the most interesting range is explored be- mann solver CAMB [37] for the theoretical spectra calcu- tween k2 = 0.001Mpc−1 and k11 = 0.35Mpc−1, that is lation, and the Markov Chain Monte Carlo (MCMC) al- approximately the range of wavemodes probed by CMB gorithm CosmoMC [38] to sample the parameter space. experiments. In this range we use equally spaced nodes in logk. Additionally, we consider k =5×10−6Mpc−1 1 and k =10Mpc−1 in order to ensure that all the PPS 12 evaluationsareinsidethecoveredrange. Weexpectthat A. Standard Cosmological Model thenodesattheseextremewavemodesareunconstrained by the data. The baseline model that we will extend to study vari- Having fixed the position of all the nodes, the free pa- ous dark radiation properties is the ΛCDM model, de- rametersthatareinvolvedinourMCMCanalysesarethe scribed by the six usual parameters: the current en- values of the PPS at each node, P =P (k )/P , where ergydensityofbaryonsandofColdDarkMatter(CDM) s,j s j 0 P istheoverallnormalization, P =2.2×10−9 [39]. We (Ω h2, Ω h2), the ratio between the sound horizon and 0 0 b c use a flat prior in the interval [0.01,10] for each P , for the angular diameter distance at decoupling (θ), the op- s,j which the expected value will be close to 1. tical depth to reionization (τ), plus two parameters that ThecompleteP (k)isthendescribedastheinterpola- describe the PPS of scalar perturbations, P (k). The s s tion among the points P : simplestmodelsofinflationpredictapower-lawformfor s,j the PPS: P (k)=P ×PCHIP(k;P ,...,P ), (2) s 0 s,1 s,12 P (k)=A (k/k )ns−1 , (1) s s ∗ wherePCHIPisthepiecewise cubic Hermite interpolating where k =0.05Mpc−1 is the pivot scale, while the am- polynomial [41, 42] (see also Ref. [34] for a detailed de- ∗ scription∗). Inthefollowing,whenpresentingourresults, plitude (A ) and the scalar spectral index (n ) are free s s we will compare the constraints obtained in the context parameters in the ΛCDM model. From these fundamen- of the standard ΛCDM model with a standard power- tal cosmological parameters we will compute other de- rivedquantities,suchastheHubbleparametertodayH law PPS to those obtained with the free PCHIP PPS, de- 0 and the clustering parameter σ , defined as the mean of scribedby(atleast)sixteenfreeparameters(Ωbh2,Ωch2, 8 matter fluctuations inside a sphere of 8h Mpc radius. θ, τ, Ps,1,...,Ps,12). This minimal model will be ex- tended to include the dark radiation properties we shall From what concerns the remaining cosmological pa- study in the various analyses. rameters, we follow the values of Ref. [39]. In particular, unlesstheyarefreelyvarying,weconsiderthesumofthe The impact of the assumptions on the PPS param- active neutrino masses to be Σm =0.06 eV, and the ef- eterization will also be tested. We shall compare the ν fective number of relativistic species to be N = 3.046 results obtained with twelve nodes to the ones derived eff [40]. using a PCHIP PPS described by eight nodes. The posi- tion of these eight nodes k(8) is selected with the same j rules as above: equally spaced nodes in logk between B. Primordial Power Spectrum of Scalar k(8) = k = 0.001Mpc−1 and k(8) = k = 0.35Mpc−1, 2 2 7 11 Perturbations plus the external nodes k(8) =k =5×10−6Mpc−1 and 1 1 k(8) =k =10Mpc−1. Asstatedbefore,possiblehintsofanon-standardPPS 8 12 of scalar perturbations were found in several analyses, including both the WMAP and the Planck CMB spec- tra [16–29, 34, 35]. From the theoretical point of view, ∗ The PCHIP method is similar to the natural cubic spline, but there are plenty of well-motivated inflationary models it has the advantage of avoiding the introduction of spurious that can give rise to non-standard PPS forms. Our ma- oscillationsintheinterpolation: thisisobtainedwithacondition jorgoalhereistostudytherobustnessoftheconstraints onthefirstderivativeinthenodes,thatisnullifthereisachange on different cosmological quantities versus a change in inthemonotonicityofthepointseries. 3 To ease comparison bewteen the power-law and the PCHIP PPS approaches, we list in all the tables the re- 6000 ΛCDM (PL): Planck TT+lowP (PCHIP): Planck TT+lowP sults obtained for these two schemes. When considering 5000 (PL): Planck TT,TE,EE+lowP a power-law PPS model, we show the constraints on n (PCHIP): Planck TT,TE,EE+lowP s and A , together with the values of the nodes Pbf to 4000 Planck2015 Pansb,df12Atsha(tnwbfoualnddcAorbrfe)s.poInndottohetrhewboersdts-,fitinvaelaucehssot,1afbnles TT2D[µK]‘3000 s s s presenting the marginalized constraints for the different 2000 cosmological parameters, in the columns corresponding 1000 to the analysis involving a power-law PPS, we shall list the values 0.004 5 1015202530 TTD‘ 0.02 Abf (cid:18)k (cid:19)nbsf−1 TTD/‘ 00..0002 Pbf ≡ s j with j ∈[1,...,12], (3) ∆ 0.04 s,j P0 k∗ 5 1015202530 500 1000 1500 2000 ‘ thatcanbeexploitedforcomparisonpurposesamongthe two PPS approaches. ΛCDM (PL): Planck TT+lowP 150 (PCHIP): Planck TT+lowP (PL): Planck TT,TE,EE+lowP 100 (PCHIP): Planck TT,TE,EE+lowP Planck2015 C. Cosmological data 2K] 50 TED[µ‘ 0 We base our analyses on the recent release from the 50 Planck Collaboration [32], that obtained the most pre- 100 cise CMB determinations in a very wide range of multi- poles. We consider the full temperature power spectrum 150 2 5 1015202530 athtempuoltlaipriozlaetsio2n≤p(cid:96)ow≤er25s0p0ec(t“rPalainnctkheTTra”ngheer2ea≤fte(cid:96)r)≤an2d9 TE∆D‘ 011 (“lowP”). We shall also include the polarization data at 2 5 1015202530 500 1000 1500 2000 30 ≤ (cid:96) ≤ 2500 (“TE, EE”) [43]. Since the polarization ‘ spectra at high multipoles are still under discussion and some residual systematics were detected by the Planck 70 ΛCDM (PL): Planck TT+lowP Collaboration [39, 43], we shall use as baseline dataset (PCHIP): Planck TT+lowP 60 (PL): Planck TT,TE,EE+lowP thecombination“PlanckTT+lowP”.Theimpactofpo- (PCHIP): Planck TT,TE,EE+lowP larizationmeasurementswillbeseparatelystudiedinthe 50 Planck2015 dataset “Planck TT,TE,EE+lowP”. 2K]40 Additionally, we will consider the two CMB datasets EED[µ‘30 above in combination with the following cosmological 20 measurements: 10 • BAO: Baryon Acoustic Oscillations data as ob- 0 tained by 6dFGS [44] at redshift z = 0.1, by the EED‘ 00..0150 5 1015202530 SDSS Main Galaxy Sample (MGS) [45] at redshift EED/‘ 00..0005 zeff = 0.15 and by the BOSS experiment in the ∆ 0.10 DR11 release, both from the LOWZ and CMASS 5 1015202530 500 1000 1500 2000 ‘ samples [46] at redshift z =0.32 and z =0.57, eff eff respectively; FIG. 1. Comparison of the Planck 2015 data [32] with the TT, TE and EE spectra obtained using the marginal- • MPkW: the matter power spectrum as measured ized best-fit values from the analyses of Planck TT+lowP by the WiggleZ Dark Energy Survey [47], from (black) and Planck TT,TE,EE+lowP (blue) in the ΛCDM measurements at four different redshifts (z =0.22, modelwiththepower-law(PL)PPS,andfromtheanalysesof z = 0.41, z = 0.60 and z = 0.78) for the scales PlanckTT+lowP(red)andPlanckTT,TE,EE+lowP(green) 0.02hMpc−1 <k <0.2hMpc−1; in the ΛCDM model with the PCHIP PPS. The adopted val- ues for each spectrum are reported in Tab. 1. We plot the D = (cid:96)((cid:96)+1)C /(2π) spectra and the relative (absolute for (cid:96) (cid:96) • lensing: the reconstruction of the lensing poten- thecaseoftheTEspectra)differencebetweeneachspectrum tial obtained by the Planck collaboration with the and the one obtained in the ΛCDM (power-law PPS) model CMB trispectrum analysis [48]. from the Planck TT+lowP data (black line). 4 III. THE ΛCDM MODEL ΛCDM+Neff (PL): Planck TT+lowP +MPkW In this section we shall consider a limited number +BAO of data combinations, including exclusively the datasets +lensing that can improve the constraints on the PCHIP PPS at (PCHIP): Planck TT+lowP +MPkW small scales, namely, the Planck polarization measure- +BAO mentsathigh-(cid:96)andtheMPkWconstraintsonthematter +lensing power spectrum. (PL): Planck TT,TE,EE+lowP +MPkW The results we obtain for the ΛCDM model are re- +BAO ported in Tab. 1 in the Appendix. In general, in the ab- +lensing sence of high multipole polarization or large scale struc- (PCHIP): Planck TT,TE,EE+lowP +MPkW ture data, the parameter errors are increased. Those +BAO associated to Ω h2, Ω h2, H and σ show a larger +lensing b c 0 8 (PCHIP 8 nodes): Planck TT,TE,EE+lowP+MPkW difference, with deviations of the order of 1σ in the PCHIPPPScasewithrespecttothepower-lawPPScase. 2.0 2.5 3.0 3.5 4.0 4.5 5.0 The differences between the PCHIP and the power-law Neff PPSparameterizationsaremuchsmallerforthe“Planck TT,TE,EE+lowP+MPkW” dataset, and the two de- FIG. 2. 68% and 95% CL constraints on N , obtained eff scriptionsofthePPSgiveboundsfortheΛCDMparam- in the ΛCDM + N model. Different colors indicate eff eters tha fully agree. Therefore, the addition of the high Planck TT+lowP with PL PPS (black), Planck TT+lowP multipole polarization spectra has a profound impact in with PCHIP PPS (red), Planck TT,TE,EE+lowP with our analyses, as we carefully explain in what follows. PL PPS (blue) and Planck TT,TE,EE+lowP with PCHIP Figure 1 depicts the CMB spectra measured by Planck PPS (green). For each color we plot 4 different datasets: [32], together with the theoretical spectra obtained from from top to bottom, we have CMB only, CMB+MPkW, CMB+BAO and CMB+lensing. We also illustrate the re- the best-fit values arising from our analyses. More con- sults, in the context of the 8-nodes parameterization, for cretely, we use the marginalized best-fit values reported the Planck TT,TE,EE+lowP+MPkW dataset (last point in in Tab. 1 for the ΛCDM model with a power-law PPS black). obtained from the analyses of the Planck TT+lowP (in black) and Planck TT,TE,EE+lowP (in blue) datasets, plus the best-fit values in the ΛCDM model with whereN =3.046[40]forthethreeactiveneutrinostan- eff a PCHIP PPS, from the Planck TT+lowP (red) and dardscenario. DeviationsofN fromitsstandardvalue eff Planck TT,TE,EE+lowP (green) datasets. We plot the may indicate that the thermal history of the active neu- D = (cid:96)((cid:96)+1)C /(2π) spectra of the TT, TE and EE (cid:96) (cid:96) trinosisdifferentfromwhatweexpect,orthatadditional anisotropies, as well as the relative (absolute for the TE relativistic particles are present in the Universe, as addi- spectra) difference between each spectrum and the one tional sterile neutrinos or thermal axions. obtained from the Planck TT+lowP data in the ΛCDM A non-standard value of N may affect the Big Bang eff model with the power-law PPS. Notice that, in the case Nucleosynthesisera,andalsothematter-radiationequal- oftheTTandEEspectra,thebest-fitspectraareingood ity. A shift in the matter-radiation equality would cause agreement with the observational data, even if there are a change in the expansion rate at decoupling, affecting variations among the ΛCDM parameters, as they can be the sound horizon and the angular scale of the peaks of compensatedbythefreedominthePPS.However,inthe the CMB spectrum, as well as in the contribution of the TEcross-correlationspectrumcase,suchacompensation early Integrated Sachs Wolfe (ISW) effect to the CMB isnolongerpossible: theinclusionoftheTEspectrumin spectrum. To avoid such a shift and its consequences, theanalysesisthereforeexpectedtohaveastrongimpact it is possible to change simultaneously the energy den- on the derived bounds. sities of matter and dark energy, in order to keep fixed all the relevant scales in the Universe. In this case, the CMB spectrum will only be altered by an increased Silk IV. EFFECTIVE NUMBER OF RELATIVISTIC damping at small scales (see e.g. Refs. [49–52]). SPECIES The constraints on N are summarized in Fig. 2, eff where we plot the 68% and 95% CL constraints on N eff The amount of energy density of relativistic species in obtained with different datasets and PPS combinations the Universe is usually defined as the sum of the photon for the ΛCDM + N model. eff contributionργ plusthecontributionofalltheotherrela- The introduction of N as a free parameter does not eff tivisticspecies. Thisisdescribedbytheeffectivenumber change significantly the results for the ΛCDM parame- of relativistic degrees of freedom Neff: ters if a power-law PPS is considered. However, once (cid:34) 7(cid:18) 4 (cid:19)4/3 (cid:35) thefreedominthePPSisintroduced, somedegeneracies ρrad = 1+ 8 11 Neff ργ, (4) btheetwleesse,netvheenPiCfHtIhPe ncoondsetsraPins,tjsaonndNNeffaarpepleoaors.enNedevfeorr- eff 5 Planck TT+lowP Planck TT+lowP+MPkW Planck TT,TE,EE+lowP Planck TT,TE,EE+lowP+MPkW 5 4 Neff3 2 1 2 4 6 8 0.8 1.2 1.6 2.0 0.3 0.6 0.9 1.2 1.5 0.75 1.00 1.25 1.50 1.75 0.9 1.0 1.1 1.2 1.3 0.96 1.04 1.12 1.20 P P P P P P s,1 s,2 s,3 s,4 s,5 s,6 5 4 Neff3 2 1 0.88 0.96 1.04 1.12 1.20 0.88 0.96 1.04 1.12 1.20 0.8 0.9 1.0 1.1 1.2 1.3 0.75 0.90 1.05 1.20 1.35 1.5 3.0 4.5 6.0 2 4 6 8 P P P P P P s,7 s,8 s,9 s,10 s,11 s,12 FIG. 3. 68% and 95% CL constraints in the (N , P ) planes, obtained in the ΛCDM + N model. We eff s,j eff show the results for Planck TT+lowP (gray), Planck TT+lowP+MPkW (red), Planck TT,TE,EE+lowP (blue) and Planck TT,TE,EE+lowP+MPkW (green). the PCHIP PPS case, all the dataset combinations give ΛCDM+Σmν constraints on Neff that are compatible with the stan- (PL): Planck TT+lowP dard value 3.046 at 95% CL, as we notice from Fig. 2 +MPkW +BAO and Tab. 2 in the Appendix. The mild preference for +lensing Neff > 3.046 arises mainly as a volume effect in the (PCHIP): Planck TT+lowP Bayesian analysis, since the PCHIP PPS parameters can +MPkW +BAO be tuned to reproduce the observed CMB temperature +lensing spectrumforawiderangeofvaluesofN . Asexpected, (PL): Planck TT,TE,EE+lowP eff the degeneracy between the nodes P and N shows +MPkW s,j eff +BAO up at high wavemodes, where the Silk damping effect is +lensing dominant, see Fig. 3. As a consequence of this correla- (PCHIP): Planck TT,TE,EE+lowP tion, the values preferred for the nodes P to P are +MPkW s,6 s,10 +BAO slightly larger than the best-fit values in the power-law +lensing PPS at the same wavemodes. The cosmological limits (PCHIP 8 nodes): Planck TT,TE,EE+lowP+MPkW for a number of parameters change as a consequence of 0.5 1.0 1.5 2.0 thevariousdegeneracieswithN . Forexample,tocom- Σm [eV] eff ν pensatetheshiftofthematter-radiationequalityredshift due to the increased radiation energy density, the CDM (cid:80) FIG. 4. As Fig. 2 but for the ΛCDM plus m case. energy density Ω h2 mean value is slightly shifted and ν c its constraints are weakened. At the same time, the un- certaintyontheHubbleparameterH isconsiderablyre- 0 with 3.046, the ΛCDM + N model gives results that laxed,becauseH mustbealsochangedaccordingly. The eff 0 are very close to those obtained in the simple ΛCDM introduction of the polarization data helps in improving model, but with slightly larger parameter uncertainties, theconstraintsinthemodelswithaPCHIPPPS,sincethe in particular for H and Ω h2. effectsofincreasingN andchangingthePPSarediffer- 0 c eff ent for the temperature-temperature, the temperature- polarizationandthepolarization-polarizationcorrelation V. MASSIVE NEUTRINOS spectra, as previously discussed in the context of the ΛCDM model (see Tab. 3 in the Appendix): the pre- Neutrinososcillationshaverobustlyestablishedtheex- ferred value of N is very close to the standard value eff istence of neutrino masses. However, neutrino mixing 3.046. Apparently, the Planck polarization data seem to data only provide information on the squared mass dif- prefer a value of N slightly smaller than 3.046 for all eff ferencesandnotontheabsolutescaleofneutrinomasses. the datasets except those including the BAO data, but Cosmology provides an independent tool to test it, as the effect is not statistically significant (see the blue and massiveneutrinosleaveanonnegligibleimprintindiffer- green points in Fig. 2). ent cosmological observables [53–64]. The primary effect In conclusion, as the bounds for N are compatible of neutrino masses in the CMB temperature spectrum eff 6 is due to the early ISW effect. The neutrino transition shift in the Hubble constant toward lower values. This from the relativistic to the non-relativistic regime affects occurs because there exists a strong, well-known degen- thedecayofthegravitationalpotentialsatthedecoupling eracy between the neutrino mass and the Hubble con- period,producinganenhancementofthesmall-scaleper- stant, see Fig. 6. In particular, considering CMB data turbations,especiallynearthefirstacousticpeak. Anon- only, a higher value of Σm will shift the location of the ν zero value of the neutrino mass also induces a higher ex- angular diameter distance to the last scattering surface, pansion rate, which suppresses the lensing potential and change that can be compensated with a smaller value the clustering on scales smaller than the horizon when of the Hubble constant H . The mean values of the 0 neutrinos become non-relativistic. However, the largest clustering parameter σ are also displaced by ∼ 2σ (ex- 8 effect of neutrino masses on the different cosmological cept for the BAO case) toward lower values in the PCHIP observables comes from the suppression of galaxy clus- PPS approach with respect to the mean values obtained tering at small scales. After becoming non-relativistic, when using the power-law PPS, as can be noticed from the neutrino hot dark matter relics possess large veloc- Fig. 7. Concerning the P parameters, the bounds on s,i itydispersions, suppressingthegrowthofmatterdensity P with i ≥ 5 are weaker with respect to the ΛCDM s,i fluctuationsatsmallscales. Thebaselinescenariowean- case(seeTab.1intheAppendix),andonlythecombina- alyzeherehasthreeactivemassiveneutrinospecieswith tionof PlanckTT+lowP datawiththe MPkWmeasure- degeneratemasses. Inaddition,weconsiderthePPSap- ments provides an upper limit for the P (concretely, s,12 proach outlined in Sec. II. For the numerical analyses, P <3.89 at 95% CL). s,12 when considering the power-law PPS, we use the follow- Alsowhenconsideringthehigh-(cid:96)polarizationmeasure- ing set of parameters: ments,theboundsonthesumoftheneutrinomassesare larger when using the PCHIP parameterization with re- {Ωbh2,Ωch2,θ,τ,ns,log[1010As],Σmν} . (5) spect to the ones obtained with the power-law approach. However,theseboundsaremorestringentthanthoseob- We then replace the n and A parameters with the s s tained using the Planck TT+lowP data only (see Tab. 5 othertwelveextraparameters(P withi=1,...,12)re- s,i in the Appendix). The reason for this improvement is lated to the PCHIP PPS parameterization. The 68% and duetothefactthattheinclusionofthepolarizationmea- 95% CL bounds on Σm obtained with different dataset ν surements removes many of the degeneracies among the and PPS combinations are summarized in Fig. 4. parameters. Concerning the CMB measurements only, Notice that, when considering Planck TT+lowP CMB we find an upper limit Σm < 0.880 eV at 95% CL in measurements plus other external datasets, for all the ν the PCHIP approach. The addition of the matter power data combinations, the bounds on neutrino masses are spectrummeasurementsleadstoavalueofΣm <0.458 weaker when considering the PCHIP PPS with respect ν eV at 95% CL in the PCHIP parameterization, improv- to the power-law PPS case (see also Tab. 4 in Ap- ing the Planck TT,TE,EE+lowP constraint by a factor pendix). Concerning CMB data only, the bound we find of two. Notice that, as in the Planck TT+lowP results, in the PCHIP approach is Σm < 2.16 eV at 95% CL, ν the data combination that gives the most stringent con- muchlessconstrainingthantheboundΣm <0.75eVat ν straintsistheoneinvolvingthePlanckTT,TE,EE+lowP 95%CLobtainedinthepower-lawapproach. Thislarger and BAO datasets, since it provides a 95% CL upper value is due to the degeneracy between Σm and the ν bound on Σm of 0.218 eV in the PCHIP PPS case. Fi- nodesP andP ,asillustratedinFig.5. Inparticular, ν s,5 s,6 nally,whenthelensingmeasurementsareadded,thecon- these two nodes correspond to the wavenumbers where straintontheneutrinomassesisshiftedtoahighervalue thecontributionoftheearlyISWeffectislocated. There- (agreeing with previous findings from the Planck collab- fore, the change induced on these angular scales by a oration), being Σm <1.17 eV at 95% CL for the PCHIP largerneutrinomasscouldbecompensatedbyincreasing ν case. The degeneracies between Σm and H , σ , even P andP . Theadditionofthematterpowerspectrum ν 0 8 s,5 s,6 if milder than those without high multipole polarization measurements,MPkW,leadstoanupperboundonΣm ν data,arestillpresent(seeFigs.6and7). Theconstraints of 1.15 eV at 95% CL in the PCHIP parameterization, ontheP parametersdonotdiffermuchfromthoseob- which is twice the value obtained when considering the s,i tained with the Planck TT+lowP data. power-law PPS with the same dataset. The most strin- gent constraints on the sum of the three active neutrino masses are obtained when we use the BAO data, since the geometrical information they provide helps breaking VI. EFFECTIVE NUMBER OF RELATIVISTIC SPECIES AND NEUTRINO MASSES degeneracies among cosmological parameters. In partic- ular,wehaveΣm <0.261eV(Σm <0.220eV)at95% ν ν CLwhenconsideringthePCHIP(power-law)PPSparam- AfterhavinganalyzedtheconstraintsonNeff andΣmν eterization. Finally,thecombinationofPlanckTT+lowP separately,westudyinthissectiontheirjointconstraints datawiththePlanckCMBlensingmeasurementsprovide in the context of the ΛCDM + N + Σm extended eff ν a bound on neutrino masses of Σm < 1.64 eV at 95% cosmological model, focusing mainly on the differences ν CL in the PCHIP case. with the results presented in the two previous sections. It can be noticed that in the PCHIP PPS there is a The68%and95%CLconstraintsonN andΣm are eff ν 7 ΛCDM+Σmν:PlanckTT+lowP ΛCDM+Σmν:PlanckTT+lowP+BAO ΛCDM+Σmν:PlanckTT,TE,EE+lowP ΛCDM+Σmν:PlanckTT,TE,EE+lowP+BAO 2.5 V]2.0 m,[eν1.5 Σ1.0 0.5 2 4 6 8 0.8 1.2 1.6 2.0 0.3 0.6 0.9 1.2 1.5 0.75 1.00 1.25 1.50 1.75 0.90 1.05 1.20 1.35 0.96 1.04 1.12 1.20 1.28 Ps,1 Ps,2 Ps,3 Ps,4 Ps,5 Ps,6 2.5 V]2.0 m,[eν1.5 Σ1.0 0.5 0.96 1.04 1.12 1.20 0.88 0.96 1.04 1.12 1.20 0.88 0.96 1.04 1.12 0.88 0.96 1.04 1.12 1.20 1.5 3.0 4.5 6.0 2 4 6 8 Ps,7 Ps,8 Ps,9 Ps,10 Ps,11 Ps,12 (cid:80) FIG. 5. As Fig. 3 but for the ΛCDM plus m case. ν 70 1.0 ΛCDM+mν: Planck TT+lowP ΛCDM+mν: Planck TT+lowP ΛCDM+mν: Planck TT+lowP+BAO ΛCDM+mν: Planck TT+lowP+BAO ] 65 ΛCDM+mν: Planck TT,TE,EE+lowP 0.9 ΛCDM+mν: Planck TT,TE,EE+lowP pc ΛCDM+mν: Planck TT,TE,EE+lowP+BAO ΛCDM+mν: Planck TT,TE,EE+lowP+BAO M 0.8 / 60 s m/ σ80.7 55 k [ 0.6 0 H 50 0.5 45 0.5 1.0 1.5 2.0 2.5 0.4 0.5 1.0 1.5 2.0 2.5 Σm [eV] Σm [eV] ν ν FIG. 6. 68% and 95% CL allowed regions in the (Σmν, H0) FIG. 7. 68% and 95% CL allowed regions in the (Σmν, σ8) plane, using different combinations of datasets, within the plane, using different combinations of datasets, within the PCHIP PPS parameterization. PCHIP PPS parameterization. reportedinFigs.8and9respectively,fordifferentdataset model, but we have verified that they are qualitatively combinations and PPS choices, see also Tabs. 6 and 7 in similar to those depicted in Fig. 3 (Fig. 5) for the Neff the Appendix. Notice that the qualitative conclusions (Σmν) parameter. When considering CMB data only, drawn in the previous sections do not change here. The theconstraintsareslightlyloosenedwithrespecttothose PCHIP PPS parameterization still allows for a significant obtained when the Neff and Σmν parameters are freely freedom in the values of N and Σm , as these param- variedseparatelyandnotsimultaneously. Whencompar- eff ν eters have an impact on the CMB spectrum that can be ing the power-law PPS and the PCHIP PPS models, we easilymimickedbysomevariationsinthePPSnodes. In cannoticethatthevariationsoftheneutrinoparameters particular,asignificantdegeneracybetweenN andthe lead to several variations in other cosmological parame- eff nodes P to P appears, in analogy to what happens ters. Such is the case of the baryon and CDM densities s,6 s,10 intheΛCDM+N model(seeFig.3andthediscussion andtheangularscaleofthepeaks,whichareshiftedbya eff in Sec. IV). At the same time, the strongest degeneracy significant amount, as a consequence of the degeneracies involving the total neutrino mass appears between Σmν withbothNeff andΣmν. AstheeffectsofNeff andΣmν and Ps,5. This corresponds to a rescaling of the PPS on the Hubble parameter H0 and the clustering param- that compensates the change in the early ISW contri- eter σ8 are opposite, we find an increased uncertainty in bution driven by massive neutrinos (see Fig. 5 and the theseparameters,beingtheirallowedrangessignificantly discussion in Sec. V). We do not show here the degen- enlarged. eracieswiththeP nodesfortheΛCDM+N +Σm While the tightest neutrino mass bound arises from s,i eff ν 8 for the PCHIP PPS). ΛCDM+Σmν+Neff (PL): Planck TT+lowP When polarization measurements are added in the +MPkW dataanalyses,weobtaina95%CLrangeof2.5(cid:46)N (cid:46) eff +BAO 3.5, with very small differences in both the central val- +lensing uesandallowedrangesfortheseveraldatacombinations (PCHIP): Planck TT+lowP +MPkW explored here, see Tab. 7 in the Appendix. As in the +BAO ΛCDM+N model, thedatasetincludingBAOdatais eff +lensing the only one for which the mean value of N is larger (PL): Planck TT,TE,EE+lowP eff +MPkW than 3, while in all the other cases it lies between 2.9 +BAO and 3. Apart from these small differences, all the results +lensing are perfectly in agreement with the standard value 3.046 (PCHIP): Planck TT,TE,EE+lowP within the 68% CL range. Concerning the Σm param- +MPkW ν +BAO eter, the results are also very similar to those obtained +lensing in the ΛCDM + Σm model illustrated in Sec. V, with ν 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 only very small differences in the exact numerical values N ofthederivedbounds. Themostconstrainingresultsare eff always obtained with the inclusion of BAO data, from which we obtain Σm < 0.18 (0.24) eV when using the FIG. 8. As Fig. 2 but for the ΛCDM + N + Σm model. ν eff ν power-law (PCHIP) PPS, both really close to the values derived in the ΛCDM + Σm model. ν ΛCDM+Σmν+Neff For what concerns the remaining cosmological param- (PL): Planck TT+lowP eters, the differences between the power-law PPS and +MPkW the PCHIP PPS results are much less significant when +BAO +lensing the polarization spectra are considered in the analyses. (PCHIP): Planck TT+lowP We may notice that the predicted values of the Hub- +MPkW ble parameter H are lower than the CMB estimates in +BAO 0 +lensing the ΛCDM model, and consequently they show an even (PL): Planck TT,TE,EE+lowP stronger tension with local measurements of the Hubble +MPkW constant. Thisisduetothenegativecorrelationbetween +BAO H and Σm . On the other hand, the ΛCDM + N + +lensing 0 ν eff (PCHIP): Planck TT,TE,EE+lowP Σmν model predicts a σ8 smaller than what is obtained +MPkW in the ΛCDM model for most of the data combinations, +BAO partiallyreconcilingtheCMBandthelocalestimatesfor +lensing this parameter. 0.5 1.0 1.5 2.0 2.5 ThePCHIPnodesinthisextendedmodeldonotdeviate Σm [eV] ν significantly from the expected values corresponding to the power-law PPS. The small deviations driven by the FIG. 9. As Fig. 8, but for the Σmν parameter. degeneracieswiththeneutrinoparametersΣmν andNeff arecanceledbythestringentboundssetbythepolariza- tion spectra, that break these degeneracies. Deviations the Planck TT+lowP+BAO dataset, the largest allowed from the power-law expectations are still visible at small mean value for Neff is also obtained for this very same wavemodes, corresponding to the dip at (cid:96) (cid:39) 20 and to data combination (Neff = 3.94+−00..4627 at 68% CL in the thesmallbumpat(cid:96)(cid:39)40intheCMBtemperaturespec- PCHIP PPS analysis), showing the large degeneracy be- trum. tween Σm and N . However, when including the ν eff Planck CMB lensing measurements, the trend is oppo- sitetotheoneobservedwiththeBAOdataset,withthree VII. MASSIVE NEUTRINOS AND EXTRA timeslargerupperlimitsforΣm andlowermeanvalues ν MASSIVE STERILE NEUTRINO SPECIES forN . Asstatedbefore, thefactthatlensingdatapre- eff fer heavier neutrinos is well known (see e.g. Sec. V and Standard cosmology includes as hot thermal relics the Refs. [39, 65]). Notice, from Fig. 9, that the only combi- nation which shows a preference for Σm > 0.06 eV† at threelight,activeneutrinoflavorsoftheStandardModel ν 68% CL includes the lensing data (Σm =0.84+0.32 eV, of elementary particles. However, the existence of ex- ν −0.62 tra hot relic components, as dark radiation relics, sterile neutrino species and/or thermal axions is also possible. In their presence, the cosmological neutrino mass con- † Thisvalueroughlycorrespondstothelowerlimitallowedbyos- straints will be changed. The existence of extra sub- cillationmeasurementsifthetotalmassisdistributedamongthe eV massive sterile neutrino species is well motivated by massiveeigenstatesaccordingtothenormalhierarchyscenario. the so-called short-baseline neutrino oscillation anoma- 9 ΛCDM+Σmν+mseff+Neff 3.0 ΛCDM+mν+Neff+meff: Planck TT+lowP (PL): Planck TT+lowP ΛCDM+mν+Neff+meff: Planck TT+lowP+lensing +MPkW 2.5 ΛΛCCDDMM++mmνν++NNeeffff++mmeeffff:: PPllaanncckk TTTT+,TlEo,wEEP++lBoAwOP +BAO ΛCDM+mν+Neff+meff: Planck TT,TE,EE+lowP+BAO +lensing 2.0 (PCHIP): Planck TT+lowP +MPkW V] e ++BleAnOsing effm[s1.5 (PL): Planck TT,TE,EE+lowP +MPkW 1.0 +BAO +lensing 0.5 (PCHIP): Planck TT,TE,EE+lowP +MPkW +BAO 0.5 1.0 1.5 2.0 2.5 3.0 +lensing Σmν [eV] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 meff [eV] FIG.11. 68%and95%CLallowedregionsinthe(Σm ,meff) s ν s plane using the different combination of datasets, within the PCHIP PPS parameterization. FIG.10. 68%and95%CLconstraintsonmeff,obtainedinthe s ΛCDM + N +Σm +meffmodel. Different colors indicate eff ν s Planck TT+lowP with PL PPS (black), Planck TT+lowP with PCHIP PPS (red), Planck TT,TE,EE+lowP with PL Tab. 2 in the Appendix). As for the other extensions of PPS (blue) and Planck TT,TE,EE+lowP with PCHIP PPS the ΛCDM model we studied, the bounds on Σm , N ν eff (green). Foreachcolorweplot4differentdatasets: fromtop and meff are weaker when considering the PCHIP PPS s to bottom, we have CMB only, CMB+MPkW, CMB+BAO with respect to the ones obtained within the power-law and CMB+lensing. PPS canonical scenario. Notice that the bounds on meff s are not very stringent. This is due to the correlation between meff and N : sub-eV massive sterile neutrinos lies [52, 66–68]. These extra light species have an associ- s eff contributetothematterenergydensityatrecombination ated free streaming scale that will reduce the growth of and therefore a larger value of N will be required to eff matterfluctuationsatsmallscales. Theyalsocontribute leaveunchangedboththeangularlocationandtheheight to the effective number of relativistic degree of freedom of the first acoustic peak of the CMB. (i.e. to N ). eff Figure11illustratesthedegeneracybetweentheactive We explore in this section the ΛCDM scenario (in the and the sterile neutrino masses. Since both active and two PPS parameterizations, power-law and PCHIP) with sterile sub-eV massive neutrinos contribute to the mat- three active light massive neutrinos, plus one massive ter energy density at decoupling, an increase of meff can sterileneutrinospeciescharacterizedbyaneffectivemass s be compensated by lowering Σm , in order to keep fixed meff, that is defined by ν s the matter content of the universe. Notice that the most (cid:18) (cid:19)3 stringent 95% CL bounds on the three active and ster- T meff = s m =(∆N )3/4m , (6) ile neutrinos are obtained considering the BAO data in s T s eff s ν the two PPS cases. In particular, we find Σm < 0.481 ν eV, meff < 0.448 eV for the PCHIP parametrization and where T (T ) is the current temperature of the ster- s s ν Σm <0.263 eV, meff <0.449 eV for the power-law ap- ile (active) neutrino species, ∆N ≡ N − 3.046 = ν s eff eff proach. Furthermore, in general, when considering the (T /T )3 is the effective number of degrees of freedom s ν PCHIP parametrization, the mean value on the Hubble associated to the massive sterile neutrino, and m is its s constant is smaller than the value obtained in the stan- physical mass. For the numerical analyses we use the dardpower-lawPPSframework,duetothestrongdegen- following set of parameters to describe the model with a eracy between Σm and H . The value of the clustering power-law PPS: ν 0 parameter σ is reduced in the two PPS parameteriza- 8 {Ω h2,Ω h2,θ,τ,n ,log[1010A ],Σm ,N ,meff} . (7) tions when comparing to the massless sterile neutrino b c s s ν eff s case. This occurs because the sterile neutrino mass is When considering the PCHIP PPS parameterization, n anothersourceofsuppressionofthelargescalestructure s and A are replaced by the twelve parameters P (with growth. s s,i i=1,...,12). The inclusion of the polarization data improves no- The 68% and 95% CL bounds on meff obtained with tably the constraints on the cosmological parameters in s differentdatasetsandPPScombinationsaresummarized the model with a PCHIP parametrization. In particular, in Fig. 10 and in Tabs. 8 and 9 in the Appendix. Notice theneutrinoconstraintsarestrongerthanthoseobtained that,ingeneral,thevalueofN islargerthaninthecase using only the temperature power spectrum at small an- eff inwhichthesterileneutrinosareconsideredmassless(see gular scales. This effect is related to the fact that many 10 thermalornon-thermal,theaxionisapossiblecandidate ΛCDM+ma foranextrahotthermalrelic,togetherwiththerelicneu- (PL): Planck TT+lowP +MPkW trinobackground,orforthecolddarkmattercomponent, +BAO respectively. In what follows, we shall focus on the ther- +lens mal axion scenario. The axion coupling constant f is (PCHIP): Planck TT+lowP a +MPkW related to the thermal axion mass via +BAO √ +lens f m R 107GeV (PL): Planck TT,TE,EE+lowP ma = πf π 1+R =0.6 eV f , (8) +MPkW a a +BAO +lens with R = 0.553±0.043, the up-to-down quark masses (PCHIP): Planck TT,TE,EE+lowP ratio, and f = 93 MeV, the pion decay constant. Con- π +MPkW sidering other values of R within the range 0.38−0.58 +BAO [73] does not affect in a significant way this relationship +lens [74]. 0.5 1.0 1.5 2.0 2.5 3.0 When the thermal axion is still a relativistic particle, m [eV] a it increases the effective number of relativistic degrees of freedom N , enhancing the amount of radiation in eff FIG. 12. As Fig. 2 but in the context of the ΛCDM + ma the early universe, see Eq. (4). It is possible to compute model, focusing on the thermal axion mass ma parameter. thecontributionofathermalaxionasanextraradiation component as: degeneracies are reduced by the high multipole polar- 4(cid:18)3n (cid:19)4/3 ization measurements (as, for example, the one between ∆Neff = 7 2na , (9) Σm and τ). Concerning the CMB measurements only, ν ν we find an upper limit on the three active and sterile with n the current axion number density and n the a ν neutrino masses of Σmν < 0.83 eV and mesff < 1.20 eV present neutrino plus antineutrino number density per at 95% CL, while for the effective number of relativistic flavor. When the thermal axion becomes a non rela- degrees of freedom we obtain Neff < 3.67 at 95% CL, tivistic particle, it increases the amount of the hot dark considering the PCHIP PPS approach. Also in this case matter density in the universe, contributing to the to- the most stringent constraints are obtained when adding tal mass-energy density of the universe. Thermal ax- the BAO datasets to the Planck TT,TE,EE+lowP data. ions promote clustering only at large scales, suppressing Finally, the addition of the lensing potential displaces the structure formation at scales smaller than their free- both the active and sterile neutrino mass constraints to streaming scale, once the axion is a non-relativistic par- higher values. ticle. Several papers in the literature provide bounds on Concerning the Ps,i parameters, we can notice the thermal axion mass, see for example Refs. [63, 75– that considering the Planck TT,TE,EE+lowP+BAO 80]. In this paper our purpose is to update the work datasets, the dip corresponding to the Ps,3 node is re- done in Ref. [35], in light of the recent Planck 2015 tem- duced with respect to the other possible data combina- perature and polarization data [32]. Therefore, in what tions. We have an upper bound for the Ps,12 node from follows,wepresentup-to-dateconstraintsonthethermal all the data combinations except for the CMB+lensing axion mass, relaxing the assumption of a power-law for dataset combination. In addition, as illustrated in the PPS of the scalar perturbations, assuming also the Secs. V and VI, a significant degeneracy between Neff PCHIP PPS scenario. and the nodes Ps,8 to Ps,10 and between Σmν and the TheboundsontheaxionmassarerelaxedinthePCHIP nodes Ps,5 and Ps,6 is also present in this ΛCDM exten- PPS scenario, as illustrated in Fig. 12 (see also Tabs. 10 sion. Finally, because of the correlation between Σmν and 11 in the Appendix). This effect is related to the and mesff, degeneracies between mesff and the nodes Ps,5 relaxed bound we have on Neff when letting it free to and Ps,6 will naturally appear. vary in an extended ΛCDM + Neff scenario. From the results presented in Tab. 2, we find N = 3.40+1.50 at eff −1.43 95% CL for the PCHIP PPS parameterization, implying VIII. THERMAL AXION that the PCHIP formalism favours extra dark radiation, and therefore a higher axion mass will be allowed. As a The axion field arises from the solution proposed by consequence, we find that the axion mass is totally un- Peccei and Quinn [69–72] to solve the strong CP prob- constrainedusingthePlanckTT+lowPdatainthePCHIP lem in Quantum Chromodynamics. They introduced a PPSapproach. Weinsteadfindtheboundm <1.97eV a new global Peccei-Quinn symmetry U(1) that, when at 95% CL for the standard power-law case. The most PQ spontaneously broken at an energy scale f , generates a stringent bounds arise when using the BAO data, since a Pseudo-Nambu-Goldstoneboson,theaxionparticle. De- they are directly sensitive to the free-streaming nature pending on the production process in the early universe, of the thermal axion. While the MPkW measurements

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Dark Radiation and Inflationary Freedom after Planck 2015 Eleonora Di Valentino,1 Stefano Gariazzo,2,3 Martina Gerbino,4,5,6 Elena Giusarma,6,7 and Olga Mena8 1Institut d’Astrophysique de Paris (UMR7095: CNRS & UPMC- Sorbonne Universities), F-75014, Paris, France 2Department of Physics, University of Torino, Via P. Giuria 1, I–10125 Torino, Italy 3INFN, Sezione di Torino, Via P. Giuria 1, I–10125 Torino, Italy 4The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden 5Nordita (Nordic Institute for Theoretical Physics), Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden 6Physics Department and INFN, Universit`a di Roma “La Sapienza”, P.le Aldo Moro 2, 00185, Rome, Italy 7McWilliams Center for Cosmology, Department of Physics, 6 Carnegie Mellon University, Pittsburgh, PA 15213, USA 1 8IFIC, Universidad de Valencia-CSIC, 46071, Valencia, Spain 0 2 The simplest inflationary models predict a primordial power spectrum (PPS) of the curvature fluctuationsthatcanbedescribedbyapower-lawfunctionthatisnearlyscale-invariant. Ithasbeen y shown, however, that the low-multipole spectrum of the CMB anisotropies may hint the presence a of some features in the shape of the scalar PPS, which could deviate from its canonical power- M law form. We study the possible degeneracies of this non-standard PPS with the active neutrino masses, the effective number of relativistic species and a sterile neutrino or a thermal axion mass. 4 The limits on these additional parameters are less constraining in a model with a non-standard 2 PPS when including only the temperature auto-correlation spectrum measurements in the data analyses. The inclusion of the polarization spectra noticeably helps in reducing the degeneracies, ] O leadingtoresultsthattypicallyshownodeviationfromtheΛCDMmodelwithastandardpower-law PPS. These findings are robust against changes in the function describing the non-canonical PPS. C Albeit current cosmological measurements seem to prefer the simple power-law PPS description, . h the statistical significance to rule out other possible parameterizations is still very poor. Future p cosmological measurements are crucial to improve the present PPS uncertainties. - o r I. INTRODUCTION as a power-law. While the significance of the deviations t s is small for some cases, it is interesting to note that the a CMB temperature power spectra as measured by both [ Inflationisoneofthemostsuccessfultheoriesthatex- WMAP[30]andPlanck[31,32]showsimilarresults: the 2 plains the so-called horizon and flatness problems, pro- differencesfromthepower-lawarelocatedinthelowmul- v viding an origin for the primordial density perturbations tipole region. These deviations could arise from some 7 that evolved to form the structures we observe today [1– statistical fluctuations, or, instead, result from a non- 5 11]. The standard inflationary paradigm predicts a sim- standard inflationary mechanism. 5 ple shape for the primordial power spectrum (PPS) of 7 Ifthefeaturesweobservearetheresultofanon-standard scalarperturbations: inthiscontext,thePPScanbede- 0 inflationary mechanism, we may be using an incomplete . scribed by a power-law expression. However, there also parameterization for the PPS in our cosmological anal- 1 exist more complicated inflationary scenarios which can 0 yses. It has been shown that this could lead to biased give rise to non-standard PPS forms, with possible fea- 6 results in the cosmological constraints of different quan- 1 tures at different scales, see e.g. Refs. [12, 13] and the tities. Namely, the constraints on the dark radiation : reviews [14, 15]. properties [33–35] or on non-Gaussianities [36] can be v i The usual procedure to reconstruct the underlying distorted, leading to spurious conclusions. In this work X PPS is to assume a model for the evolution of the Uni- we aim to study the impact of a general PPS form in r verse and calculate the transfer function, and then use the constraints obtained for the properties of dark ra- a different techniques to constrain a completely unknown diation candidates, such as the active neutrino masses PPS,comparingthetheoreticalpredictionwiththemea- and their effective number, sterile neutrino species and sured power spectrum of the Cosmic Microwave Back- thermal axion properties. The outline of the Paper is as ground radiation (CMB). Among the methods devel- follows: wepresentthebaselinestandardΛCDM cosmo- oped in the past, we can list regularization methods as logical model, the PPS parameterization and the cosmo- the Richardson-Lucy iteration [16–19], truncated singu- logical data in Sec. II. The results obtained within the larvaluedecomposition[20]andTikhonovregularization ΛCDM framework are presented in Sec. III. Concerning [21, 22], or methods as the maximum entropy deconvo- possible extensions of the ΛCDM scenario, we study the lution [23] or the cosmic inversion methods [24–28]. Re- constraints on the effective number of relativistic species cently, the Planck collaboration presented a wide discus- in Sec. IV, on the neutrino masses in Sec. V, on massive sion about constraints on inflation [29]. All these meth- neutrinos with a varying effective number of relativistic ods provide hint for a PPS which may not be as simple speciesinSec.VI,onmassivesterileneutrinosinSec.VII, 2 andonthethermalaxionpropertiesinSec.VIII.Finally, the assumed PPS. Several cosmological parameters are in Sec. IX we show the reconstructed PPS shape, com- known to present degeneracies with the standard PPS paring different possible approaches, and we draw our parameters, as, for example, the existing one between conclusions in Sec. X. effective number of relativistic species N and the tilt eff of the power-law PPS n . These degeneracies could be s even stronger when more freedom is allowed for the PPS II. BASELINE MODEL AND COSMOLOGICAL shape. We adopt here a non-parametric description for DATA thePPSofscalarperturbations: wedescribethefunction P (k)astheinterpolationamongaseriesofnodesatfixed s In this Section we outline the baseline theoretical wavemodes k. Unless otherwise stated, we shall consider model that will be extended to study the dark radiation twelve nodes kj (j ∈ [1,12]) that cover a wide range of properties. For our analyses we use the numerical Boltz- values of k: the most interesting range is explored be- mann solver CAMB [37] for the theoretical spectra calcu- tween k2 = 0.001Mpc−1 and k11 = 0.35Mpc−1, that is lation, and the Markov Chain Monte Carlo (MCMC) al- approximately the range of wavemodes probed by CMB gorithm CosmoMC [38] to sample the parameter space. experiments. In this range we use equally spaced nodes in logk. Additionally, we consider k =5×10−6Mpc−1 1 and k =10Mpc−1 in order to ensure that all the PPS 12 evaluationsareinsidethecoveredrange. Weexpectthat A. Standard Cosmological Model thenodesattheseextremewavemodesareunconstrained by the data. The baseline model that we will extend to study vari- Having fixed the position of all the nodes, the free pa- ous dark radiation properties is the ΛCDM model, de- rametersthatareinvolvedinourMCMCanalysesarethe scribed by the six usual parameters: the current en- values of the PPS at each node, P =P (k )/P , where ergydensityofbaryonsandofColdDarkMatter(CDM) s,j s j 0 P istheoverallnormalization, P =2.2×10−9 [39]. We (Ω h2, Ω h2), the ratio between the sound horizon and 0 0 b c use a flat prior in the interval [0.01,10] for each P , for the angular diameter distance at decoupling (θ), the op- s,j which the expected value will be close to 1. tical depth to reionization (τ), plus two parameters that ThecompleteP (k)isthendescribedastheinterpola- describe the PPS of scalar perturbations, P (k). The s s tion among the points P : simplestmodelsofinflationpredictapower-lawformfor s,j the PPS: P (k)=P ×PCHIP(k;P ,...,P ), (2) s 0 s,1 s,12 P (k)=A (k/k )ns−1 , (1) s s ∗ wherePCHIPisthepiecewise cubic Hermite interpolating where k =0.05Mpc−1 is the pivot scale, while the am- polynomial [41, 42] (see also Ref. [34] for a detailed de- ∗ scription∗). Inthefollowing,whenpresentingourresults, plitude (A ) and the scalar spectral index (n ) are free s s we will compare the constraints obtained in the context parameters in the ΛCDM model. From these fundamen- of the standard ΛCDM model with a standard power- tal cosmological parameters we will compute other de- rivedquantities,suchastheHubbleparametertodayH law PPS to those obtained with the free PCHIP PPS, de- 0 and the clustering parameter σ , defined as the mean of scribedby(atleast)sixteenfreeparameters(Ωbh2,Ωch2, 8 matter fluctuations inside a sphere of 8h Mpc radius. θ, τ, Ps,1,...,Ps,12). This minimal model will be ex- tended to include the dark radiation properties we shall From what concerns the remaining cosmological pa- study in the various analyses. rameters, we follow the values of Ref. [39]. In particular, unlesstheyarefreelyvarying,weconsiderthesumofthe The impact of the assumptions on the PPS param- active neutrino masses to be Σm =0.06 eV, and the ef- eterization will also be tested. We shall compare the ν fective number of relativistic species to be N = 3.046 results obtained with twelve nodes to the ones derived eff [40]. using a PCHIP PPS described by eight nodes. The posi- tion of these eight nodes k(8) is selected with the same j rules as above: equally spaced nodes in logk between B. Primordial Power Spectrum of Scalar k(8) = k = 0.001Mpc−1 and k(8) = k = 0.35Mpc−1, 2 2 7 11 Perturbations plus the external nodes k(8) =k =5×10−6Mpc−1 and 1 1 k(8) =k =10Mpc−1. Asstatedbefore,possiblehintsofanon-standardPPS 8 12 of scalar perturbations were found in several analyses, including both the WMAP and the Planck CMB spec- tra [16–29, 34, 35]. From the theoretical point of view, ∗ The PCHIP method is similar to the natural cubic spline, but there are plenty of well-motivated inflationary models it has the advantage of avoiding the introduction of spurious that can give rise to non-standard PPS forms. Our ma- oscillationsintheinterpolation: thisisobtainedwithacondition jorgoalhereistostudytherobustnessoftheconstraints onthefirstderivativeinthenodes,thatisnullifthereisachange on different cosmological quantities versus a change in inthemonotonicityofthepointseries. 3 To ease comparison bewteen the power-law and the PCHIP PPS approaches, we list in all the tables the re- 6000 ΛCDM (PL): Planck TT+lowP (PCHIP): Planck TT+lowP sults obtained for these two schemes. When considering 5000 (PL): Planck TT,TE,EE+lowP a power-law PPS model, we show the constraints on n (PCHIP): Planck TT,TE,EE+lowP s and A , together with the values of the nodes Pbf to 4000 Planck2015 Pansb,df12Atsha(tnwbfoualnddcAorbrfe)s.poInndottohetrhewboersdts-,fitinvaelaucehssot,1afbnles TT2D[µK]‘3000 s s s presenting the marginalized constraints for the different 2000 cosmological parameters, in the columns corresponding 1000 to the analysis involving a power-law PPS, we shall list the values 0.004 5 1015202530 TTD‘ 0.02 Abf (cid:18)k (cid:19)nbsf−1 TTD/‘ 00..0002 Pbf ≡ s j with j ∈[1,...,12], (3) ∆ 0.04 s,j P0 k∗ 5 1015202530 500 1000 1500 2000 ‘ thatcanbeexploitedforcomparisonpurposesamongthe two PPS approaches. ΛCDM (PL): Planck TT+lowP 150 (PCHIP): Planck TT+lowP (PL): Planck TT,TE,EE+lowP 100 (PCHIP): Planck TT,TE,EE+lowP Planck2015 C. Cosmological data 2K] 50 TED[µ‘ 0 We base our analyses on the recent release from the 50 Planck Collaboration [32], that obtained the most pre- 100 cise CMB determinations in a very wide range of multi- poles. We consider the full temperature power spectrum 150 2 5 1015202530 athtempuoltlaipriozlaetsio2n≤p(cid:96)ow≤er25s0p0ec(t“rPalainnctkheTTra”ngheer2ea≤fte(cid:96)r)≤an2d9 TE∆D‘ 011 (“lowP”). We shall also include the polarization data at 2 5 1015202530 500 1000 1500 2000 30 ≤ (cid:96) ≤ 2500 (“TE, EE”) [43]. Since the polarization ‘ spectra at high multipoles are still under discussion and some residual systematics were detected by the Planck 70 ΛCDM (PL): Planck TT+lowP Collaboration [39, 43], we shall use as baseline dataset (PCHIP): Planck TT+lowP 60 (PL): Planck TT,TE,EE+lowP thecombination“PlanckTT+lowP”.Theimpactofpo- (PCHIP): Planck TT,TE,EE+lowP larizationmeasurementswillbeseparatelystudiedinthe 50 Planck2015 dataset “Planck TT,TE,EE+lowP”. 2K]40 Additionally, we will consider the two CMB datasets EED[µ‘30 above in combination with the following cosmological 20 measurements: 10 • BAO: Baryon Acoustic Oscillations data as ob- 0 tained by 6dFGS [44] at redshift z = 0.1, by the EED‘ 00..0150 5 1015202530 SDSS Main Galaxy Sample (MGS) [45] at redshift EED/‘ 00..0005 zeff = 0.15 and by the BOSS experiment in the ∆ 0.10 DR11 release, both from the LOWZ and CMASS 5 1015202530 500 1000 1500 2000 ‘ samples [46] at redshift z =0.32 and z =0.57, eff eff respectively; FIG. 1. Comparison of the Planck 2015 data [32] with the TT, TE and EE spectra obtained using the marginal- • MPkW: the matter power spectrum as measured ized best-fit values from the analyses of Planck TT+lowP by the WiggleZ Dark Energy Survey [47], from (black) and Planck TT,TE,EE+lowP (blue) in the ΛCDM measurements at four different redshifts (z =0.22, modelwiththepower-law(PL)PPS,andfromtheanalysesof z = 0.41, z = 0.60 and z = 0.78) for the scales PlanckTT+lowP(red)andPlanckTT,TE,EE+lowP(green) 0.02hMpc−1 <k <0.2hMpc−1; in the ΛCDM model with the PCHIP PPS. The adopted val- ues for each spectrum are reported in Tab. 1. We plot the D = (cid:96)((cid:96)+1)C /(2π) spectra and the relative (absolute for (cid:96) (cid:96) • lensing: the reconstruction of the lensing poten- thecaseoftheTEspectra)differencebetweeneachspectrum tial obtained by the Planck collaboration with the and the one obtained in the ΛCDM (power-law PPS) model CMB trispectrum analysis [48]. from the Planck TT+lowP data (black line). 4 III. THE ΛCDM MODEL ΛCDM+Neff (PL): Planck TT+lowP +MPkW In this section we shall consider a limited number +BAO of data combinations, including exclusively the datasets +lensing that can improve the constraints on the PCHIP PPS at (PCHIP): Planck TT+lowP +MPkW small scales, namely, the Planck polarization measure- +BAO mentsathigh-(cid:96)andtheMPkWconstraintsonthematter +lensing power spectrum. (PL): Planck TT,TE,EE+lowP +MPkW The results we obtain for the ΛCDM model are re- +BAO ported in Tab. 1 in the Appendix. In general, in the ab- +lensing sence of high multipole polarization or large scale struc- (PCHIP): Planck TT,TE,EE+lowP +MPkW ture data, the parameter errors are increased. Those +BAO associated to Ω h2, Ω h2, H and σ show a larger +lensing b c 0 8 (PCHIP 8 nodes): Planck TT,TE,EE+lowP+MPkW difference, with deviations of the order of 1σ in the PCHIPPPScasewithrespecttothepower-lawPPScase. 2.0 2.5 3.0 3.5 4.0 4.5 5.0 The differences between the PCHIP and the power-law Neff PPSparameterizationsaremuchsmallerforthe“Planck TT,TE,EE+lowP+MPkW” dataset, and the two de- FIG. 2. 68% and 95% CL constraints on N , obtained eff scriptionsofthePPSgiveboundsfortheΛCDMparam- in the ΛCDM + N model. Different colors indicate eff eters tha fully agree. Therefore, the addition of the high Planck TT+lowP with PL PPS (black), Planck TT+lowP multipole polarization spectra has a profound impact in with PCHIP PPS (red), Planck TT,TE,EE+lowP with our analyses, as we carefully explain in what follows. PL PPS (blue) and Planck TT,TE,EE+lowP with PCHIP Figure 1 depicts the CMB spectra measured by Planck PPS (green). For each color we plot 4 different datasets: [32], together with the theoretical spectra obtained from from top to bottom, we have CMB only, CMB+MPkW, CMB+BAO and CMB+lensing. We also illustrate the re- the best-fit values arising from our analyses. More con- sults, in the context of the 8-nodes parameterization, for cretely, we use the marginalized best-fit values reported the Planck TT,TE,EE+lowP+MPkW dataset (last point in in Tab. 1 for the ΛCDM model with a power-law PPS black). obtained from the analyses of the Planck TT+lowP (in black) and Planck TT,TE,EE+lowP (in blue) datasets, plus the best-fit values in the ΛCDM model with whereN =3.046[40]forthethreeactiveneutrinostan- eff a PCHIP PPS, from the Planck TT+lowP (red) and dardscenario. DeviationsofN fromitsstandardvalue eff Planck TT,TE,EE+lowP (green) datasets. We plot the may indicate that the thermal history of the active neu- D = (cid:96)((cid:96)+1)C /(2π) spectra of the TT, TE and EE (cid:96) (cid:96) trinosisdifferentfromwhatweexpect,orthatadditional anisotropies, as well as the relative (absolute for the TE relativistic particles are present in the Universe, as addi- spectra) difference between each spectrum and the one tional sterile neutrinos or thermal axions. obtained from the Planck TT+lowP data in the ΛCDM A non-standard value of N may affect the Big Bang eff model with the power-law PPS. Notice that, in the case Nucleosynthesisera,andalsothematter-radiationequal- oftheTTandEEspectra,thebest-fitspectraareingood ity. A shift in the matter-radiation equality would cause agreement with the observational data, even if there are a change in the expansion rate at decoupling, affecting variations among the ΛCDM parameters, as they can be the sound horizon and the angular scale of the peaks of compensatedbythefreedominthePPS.However,inthe the CMB spectrum, as well as in the contribution of the TEcross-correlationspectrumcase,suchacompensation early Integrated Sachs Wolfe (ISW) effect to the CMB isnolongerpossible: theinclusionoftheTEspectrumin spectrum. To avoid such a shift and its consequences, theanalysesisthereforeexpectedtohaveastrongimpact it is possible to change simultaneously the energy den- on the derived bounds. sities of matter and dark energy, in order to keep fixed all the relevant scales in the Universe. In this case, the CMB spectrum will only be altered by an increased Silk IV. EFFECTIVE NUMBER OF RELATIVISTIC damping at small scales (see e.g. Refs. [49–52]). SPECIES The constraints on N are summarized in Fig. 2, eff where we plot the 68% and 95% CL constraints on N eff The amount of energy density of relativistic species in obtained with different datasets and PPS combinations the Universe is usually defined as the sum of the photon for the ΛCDM + N model. eff contributionργ plusthecontributionofalltheotherrela- The introduction of N as a free parameter does not eff tivisticspecies. Thisisdescribedbytheeffectivenumber change significantly the results for the ΛCDM parame- of relativistic degrees of freedom Neff: ters if a power-law PPS is considered. However, once (cid:34) 7(cid:18) 4 (cid:19)4/3 (cid:35) thefreedominthePPSisintroduced, somedegeneracies ρrad = 1+ 8 11 Neff ργ, (4) btheetwleesse,netvheenPiCfHtIhPe ncoondsetsraPins,tjsaonndNNeffaarpepleoaors.enNedevfeorr- eff 5 Planck TT+lowP Planck TT+lowP+MPkW Planck TT,TE,EE+lowP Planck TT,TE,EE+lowP+MPkW 5 4 Neff3 2 1 2 4 6 8 0.8 1.2 1.6 2.0 0.3 0.6 0.9 1.2 1.5 0.75 1.00 1.25 1.50 1.75 0.9 1.0 1.1 1.2 1.3 0.96 1.04 1.12 1.20 P P P P P P s,1 s,2 s,3 s,4 s,5 s,6 5 4 Neff3 2 1 0.88 0.96 1.04 1.12 1.20 0.88 0.96 1.04 1.12 1.20 0.8 0.9 1.0 1.1 1.2 1.3 0.75 0.90 1.05 1.20 1.35 1.5 3.0 4.5 6.0 2 4 6 8 P P P P P P s,7 s,8 s,9 s,10 s,11 s,12 FIG. 3. 68% and 95% CL constraints in the (N , P ) planes, obtained in the ΛCDM + N model. We eff s,j eff show the results for Planck TT+lowP (gray), Planck TT+lowP+MPkW (red), Planck TT,TE,EE+lowP (blue) and Planck TT,TE,EE+lowP+MPkW (green). the PCHIP PPS case, all the dataset combinations give ΛCDM+Σmν constraints on Neff that are compatible with the stan- (PL): Planck TT+lowP dard value 3.046 at 95% CL, as we notice from Fig. 2 +MPkW +BAO and Tab. 2 in the Appendix. The mild preference for +lensing Neff > 3.046 arises mainly as a volume effect in the (PCHIP): Planck TT+lowP Bayesian analysis, since the PCHIP PPS parameters can +MPkW +BAO be tuned to reproduce the observed CMB temperature +lensing spectrumforawiderangeofvaluesofN . Asexpected, (PL): Planck TT,TE,EE+lowP eff the degeneracy between the nodes P and N shows +MPkW s,j eff +BAO up at high wavemodes, where the Silk damping effect is +lensing dominant, see Fig. 3. As a consequence of this correla- (PCHIP): Planck TT,TE,EE+lowP tion, the values preferred for the nodes P to P are +MPkW s,6 s,10 +BAO slightly larger than the best-fit values in the power-law +lensing PPS at the same wavemodes. The cosmological limits (PCHIP 8 nodes): Planck TT,TE,EE+lowP+MPkW for a number of parameters change as a consequence of 0.5 1.0 1.5 2.0 thevariousdegeneracieswithN . Forexample,tocom- Σm [eV] eff ν pensatetheshiftofthematter-radiationequalityredshift due to the increased radiation energy density, the CDM (cid:80) FIG. 4. As Fig. 2 but for the ΛCDM plus m case. energy density Ω h2 mean value is slightly shifted and ν c its constraints are weakened. At the same time, the un- certaintyontheHubbleparameterH isconsiderablyre- 0 with 3.046, the ΛCDM + N model gives results that laxed,becauseH mustbealsochangedaccordingly. The eff 0 are very close to those obtained in the simple ΛCDM introduction of the polarization data helps in improving model, but with slightly larger parameter uncertainties, theconstraintsinthemodelswithaPCHIPPPS,sincethe in particular for H and Ω h2. effectsofincreasingN andchangingthePPSarediffer- 0 c eff ent for the temperature-temperature, the temperature- polarizationandthepolarization-polarizationcorrelation V. MASSIVE NEUTRINOS spectra, as previously discussed in the context of the ΛCDM model (see Tab. 3 in the Appendix): the pre- Neutrinososcillationshaverobustlyestablishedtheex- ferred value of N is very close to the standard value eff istence of neutrino masses. However, neutrino mixing 3.046. Apparently, the Planck polarization data seem to data only provide information on the squared mass dif- prefer a value of N slightly smaller than 3.046 for all eff ferencesandnotontheabsolutescaleofneutrinomasses. the datasets except those including the BAO data, but Cosmology provides an independent tool to test it, as the effect is not statistically significant (see the blue and massiveneutrinosleaveanonnegligibleimprintindiffer- green points in Fig. 2). ent cosmological observables [53–64]. The primary effect In conclusion, as the bounds for N are compatible of neutrino masses in the CMB temperature spectrum eff 6 is due to the early ISW effect. The neutrino transition shift in the Hubble constant toward lower values. This from the relativistic to the non-relativistic regime affects occurs because there exists a strong, well-known degen- thedecayofthegravitationalpotentialsatthedecoupling eracy between the neutrino mass and the Hubble con- period,producinganenhancementofthesmall-scaleper- stant, see Fig. 6. In particular, considering CMB data turbations,especiallynearthefirstacousticpeak. Anon- only, a higher value of Σm will shift the location of the ν zero value of the neutrino mass also induces a higher ex- angular diameter distance to the last scattering surface, pansion rate, which suppresses the lensing potential and change that can be compensated with a smaller value the clustering on scales smaller than the horizon when of the Hubble constant H . The mean values of the 0 neutrinos become non-relativistic. However, the largest clustering parameter σ are also displaced by ∼ 2σ (ex- 8 effect of neutrino masses on the different cosmological cept for the BAO case) toward lower values in the PCHIP observables comes from the suppression of galaxy clus- PPS approach with respect to the mean values obtained tering at small scales. After becoming non-relativistic, when using the power-law PPS, as can be noticed from the neutrino hot dark matter relics possess large veloc- Fig. 7. Concerning the P parameters, the bounds on s,i itydispersions, suppressingthegrowthofmatterdensity P with i ≥ 5 are weaker with respect to the ΛCDM s,i fluctuationsatsmallscales. Thebaselinescenariowean- case(seeTab.1intheAppendix),andonlythecombina- alyzeherehasthreeactivemassiveneutrinospecieswith tionof PlanckTT+lowP datawiththe MPkWmeasure- degeneratemasses. Inaddition,weconsiderthePPSap- ments provides an upper limit for the P (concretely, s,12 proach outlined in Sec. II. For the numerical analyses, P <3.89 at 95% CL). s,12 when considering the power-law PPS, we use the follow- Alsowhenconsideringthehigh-(cid:96)polarizationmeasure- ing set of parameters: ments,theboundsonthesumoftheneutrinomassesare larger when using the PCHIP parameterization with re- {Ωbh2,Ωch2,θ,τ,ns,log[1010As],Σmν} . (5) spect to the ones obtained with the power-law approach. However,theseboundsaremorestringentthanthoseob- We then replace the n and A parameters with the s s tained using the Planck TT+lowP data only (see Tab. 5 othertwelveextraparameters(P withi=1,...,12)re- s,i in the Appendix). The reason for this improvement is lated to the PCHIP PPS parameterization. The 68% and duetothefactthattheinclusionofthepolarizationmea- 95% CL bounds on Σm obtained with different dataset ν surements removes many of the degeneracies among the and PPS combinations are summarized in Fig. 4. parameters. Concerning the CMB measurements only, Notice that, when considering Planck TT+lowP CMB we find an upper limit Σm < 0.880 eV at 95% CL in measurements plus other external datasets, for all the ν the PCHIP approach. The addition of the matter power data combinations, the bounds on neutrino masses are spectrummeasurementsleadstoavalueofΣm <0.458 weaker when considering the PCHIP PPS with respect ν eV at 95% CL in the PCHIP parameterization, improv- to the power-law PPS case (see also Tab. 4 in Ap- ing the Planck TT,TE,EE+lowP constraint by a factor pendix). Concerning CMB data only, the bound we find of two. Notice that, as in the Planck TT+lowP results, in the PCHIP approach is Σm < 2.16 eV at 95% CL, ν the data combination that gives the most stringent con- muchlessconstrainingthantheboundΣm <0.75eVat ν straintsistheoneinvolvingthePlanckTT,TE,EE+lowP 95%CLobtainedinthepower-lawapproach. Thislarger and BAO datasets, since it provides a 95% CL upper value is due to the degeneracy between Σm and the ν bound on Σm of 0.218 eV in the PCHIP PPS case. Fi- nodesP andP ,asillustratedinFig.5. Inparticular, ν s,5 s,6 nally,whenthelensingmeasurementsareadded,thecon- these two nodes correspond to the wavenumbers where straintontheneutrinomassesisshiftedtoahighervalue thecontributionoftheearlyISWeffectislocated. There- (agreeing with previous findings from the Planck collab- fore, the change induced on these angular scales by a oration), being Σm <1.17 eV at 95% CL for the PCHIP largerneutrinomasscouldbecompensatedbyincreasing ν case. The degeneracies between Σm and H , σ , even P andP . Theadditionofthematterpowerspectrum ν 0 8 s,5 s,6 if milder than those without high multipole polarization measurements,MPkW,leadstoanupperboundonΣm ν data,arestillpresent(seeFigs.6and7). Theconstraints of 1.15 eV at 95% CL in the PCHIP parameterization, ontheP parametersdonotdiffermuchfromthoseob- which is twice the value obtained when considering the s,i tained with the Planck TT+lowP data. power-law PPS with the same dataset. The most strin- gent constraints on the sum of the three active neutrino masses are obtained when we use the BAO data, since the geometrical information they provide helps breaking VI. EFFECTIVE NUMBER OF RELATIVISTIC SPECIES AND NEUTRINO MASSES degeneracies among cosmological parameters. In partic- ular,wehaveΣm <0.261eV(Σm <0.220eV)at95% ν ν CLwhenconsideringthePCHIP(power-law)PPSparam- AfterhavinganalyzedtheconstraintsonNeff andΣmν eterization. Finally,thecombinationofPlanckTT+lowP separately,westudyinthissectiontheirjointconstraints datawiththePlanckCMBlensingmeasurementsprovide in the context of the ΛCDM + N + Σm extended eff ν a bound on neutrino masses of Σm < 1.64 eV at 95% cosmological model, focusing mainly on the differences ν CL in the PCHIP case. with the results presented in the two previous sections. It can be noticed that in the PCHIP PPS there is a The68%and95%CLconstraintsonN andΣm are eff ν 7 ΛCDM+Σmν:PlanckTT+lowP ΛCDM+Σmν:PlanckTT+lowP+BAO ΛCDM+Σmν:PlanckTT,TE,EE+lowP ΛCDM+Σmν:PlanckTT,TE,EE+lowP+BAO 2.5 V]2.0 m,[eν1.5 Σ1.0 0.5 2 4 6 8 0.8 1.2 1.6 2.0 0.3 0.6 0.9 1.2 1.5 0.75 1.00 1.25 1.50 1.75 0.90 1.05 1.20 1.35 0.96 1.04 1.12 1.20 1.28 Ps,1 Ps,2 Ps,3 Ps,4 Ps,5 Ps,6 2.5 V]2.0 m,[eν1.5 Σ1.0 0.5 0.96 1.04 1.12 1.20 0.88 0.96 1.04 1.12 1.20 0.88 0.96 1.04 1.12 0.88 0.96 1.04 1.12 1.20 1.5 3.0 4.5 6.0 2 4 6 8 Ps,7 Ps,8 Ps,9 Ps,10 Ps,11 Ps,12 (cid:80) FIG. 5. As Fig. 3 but for the ΛCDM plus m case. ν 70 1.0 ΛCDM+mν: Planck TT+lowP ΛCDM+mν: Planck TT+lowP ΛCDM+mν: Planck TT+lowP+BAO ΛCDM+mν: Planck TT+lowP+BAO ] 65 ΛCDM+mν: Planck TT,TE,EE+lowP 0.9 ΛCDM+mν: Planck TT,TE,EE+lowP pc ΛCDM+mν: Planck TT,TE,EE+lowP+BAO ΛCDM+mν: Planck TT,TE,EE+lowP+BAO M 0.8 / 60 s m/ σ80.7 55 k [ 0.6 0 H 50 0.5 45 0.5 1.0 1.5 2.0 2.5 0.4 0.5 1.0 1.5 2.0 2.5 Σm [eV] Σm [eV] ν ν FIG. 6. 68% and 95% CL allowed regions in the (Σmν, H0) FIG. 7. 68% and 95% CL allowed regions in the (Σmν, σ8) plane, using different combinations of datasets, within the plane, using different combinations of datasets, within the PCHIP PPS parameterization. PCHIP PPS parameterization. reportedinFigs.8and9respectively,fordifferentdataset model, but we have verified that they are qualitatively combinations and PPS choices, see also Tabs. 6 and 7 in similar to those depicted in Fig. 3 (Fig. 5) for the Neff the Appendix. Notice that the qualitative conclusions (Σmν) parameter. When considering CMB data only, drawn in the previous sections do not change here. The theconstraintsareslightlyloosenedwithrespecttothose PCHIP PPS parameterization still allows for a significant obtained when the Neff and Σmν parameters are freely freedom in the values of N and Σm , as these param- variedseparatelyandnotsimultaneously. Whencompar- eff ν eters have an impact on the CMB spectrum that can be ing the power-law PPS and the PCHIP PPS models, we easilymimickedbysomevariationsinthePPSnodes. In cannoticethatthevariationsoftheneutrinoparameters particular,asignificantdegeneracybetweenN andthe lead to several variations in other cosmological parame- eff nodes P to P appears, in analogy to what happens ters. Such is the case of the baryon and CDM densities s,6 s,10 intheΛCDM+N model(seeFig.3andthediscussion andtheangularscaleofthepeaks,whichareshiftedbya eff in Sec. IV). At the same time, the strongest degeneracy significant amount, as a consequence of the degeneracies involving the total neutrino mass appears between Σmν withbothNeff andΣmν. AstheeffectsofNeff andΣmν and Ps,5. This corresponds to a rescaling of the PPS on the Hubble parameter H0 and the clustering param- that compensates the change in the early ISW contri- eter σ8 are opposite, we find an increased uncertainty in bution driven by massive neutrinos (see Fig. 5 and the theseparameters,beingtheirallowedrangessignificantly discussion in Sec. V). We do not show here the degen- enlarged. eracieswiththeP nodesfortheΛCDM+N +Σm While the tightest neutrino mass bound arises from s,i eff ν 8 for the PCHIP PPS). ΛCDM+Σmν+Neff (PL): Planck TT+lowP When polarization measurements are added in the +MPkW dataanalyses,weobtaina95%CLrangeof2.5(cid:46)N (cid:46) eff +BAO 3.5, with very small differences in both the central val- +lensing uesandallowedrangesfortheseveraldatacombinations (PCHIP): Planck TT+lowP +MPkW explored here, see Tab. 7 in the Appendix. As in the +BAO ΛCDM+N model, thedatasetincludingBAOdatais eff +lensing the only one for which the mean value of N is larger (PL): Planck TT,TE,EE+lowP eff +MPkW than 3, while in all the other cases it lies between 2.9 +BAO and 3. Apart from these small differences, all the results +lensing are perfectly in agreement with the standard value 3.046 (PCHIP): Planck TT,TE,EE+lowP within the 68% CL range. Concerning the Σm param- +MPkW ν +BAO eter, the results are also very similar to those obtained +lensing in the ΛCDM + Σm model illustrated in Sec. V, with ν 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 only very small differences in the exact numerical values N ofthederivedbounds. Themostconstrainingresultsare eff always obtained with the inclusion of BAO data, from which we obtain Σm < 0.18 (0.24) eV when using the FIG. 8. As Fig. 2 but for the ΛCDM + N + Σm model. ν eff ν power-law (PCHIP) PPS, both really close to the values derived in the ΛCDM + Σm model. ν ΛCDM+Σmν+Neff For what concerns the remaining cosmological param- (PL): Planck TT+lowP eters, the differences between the power-law PPS and +MPkW the PCHIP PPS results are much less significant when +BAO +lensing the polarization spectra are considered in the analyses. (PCHIP): Planck TT+lowP We may notice that the predicted values of the Hub- +MPkW ble parameter H are lower than the CMB estimates in +BAO 0 +lensing the ΛCDM model, and consequently they show an even (PL): Planck TT,TE,EE+lowP stronger tension with local measurements of the Hubble +MPkW constant. Thisisduetothenegativecorrelationbetween +BAO H and Σm . On the other hand, the ΛCDM + N + +lensing 0 ν eff (PCHIP): Planck TT,TE,EE+lowP Σmν model predicts a σ8 smaller than what is obtained +MPkW in the ΛCDM model for most of the data combinations, +BAO partiallyreconcilingtheCMBandthelocalestimatesfor +lensing this parameter. 0.5 1.0 1.5 2.0 2.5 ThePCHIPnodesinthisextendedmodeldonotdeviate Σm [eV] ν significantly from the expected values corresponding to the power-law PPS. The small deviations driven by the FIG. 9. As Fig. 8, but for the Σmν parameter. degeneracieswiththeneutrinoparametersΣmν andNeff arecanceledbythestringentboundssetbythepolariza- tion spectra, that break these degeneracies. Deviations the Planck TT+lowP+BAO dataset, the largest allowed from the power-law expectations are still visible at small mean value for Neff is also obtained for this very same wavemodes, corresponding to the dip at (cid:96) (cid:39) 20 and to data combination (Neff = 3.94+−00..4627 at 68% CL in the thesmallbumpat(cid:96)(cid:39)40intheCMBtemperaturespec- PCHIP PPS analysis), showing the large degeneracy be- trum. tween Σm and N . However, when including the ν eff Planck CMB lensing measurements, the trend is oppo- sitetotheoneobservedwiththeBAOdataset,withthree VII. MASSIVE NEUTRINOS AND EXTRA timeslargerupperlimitsforΣm andlowermeanvalues ν MASSIVE STERILE NEUTRINO SPECIES forN . Asstatedbefore, thefactthatlensingdatapre- eff fer heavier neutrinos is well known (see e.g. Sec. V and Standard cosmology includes as hot thermal relics the Refs. [39, 65]). Notice, from Fig. 9, that the only combi- nation which shows a preference for Σm > 0.06 eV† at threelight,activeneutrinoflavorsoftheStandardModel ν 68% CL includes the lensing data (Σm =0.84+0.32 eV, of elementary particles. However, the existence of ex- ν −0.62 tra hot relic components, as dark radiation relics, sterile neutrino species and/or thermal axions is also possible. In their presence, the cosmological neutrino mass con- † Thisvalueroughlycorrespondstothelowerlimitallowedbyos- straints will be changed. The existence of extra sub- cillationmeasurementsifthetotalmassisdistributedamongthe eV massive sterile neutrino species is well motivated by massiveeigenstatesaccordingtothenormalhierarchyscenario. the so-called short-baseline neutrino oscillation anoma- 9 ΛCDM+Σmν+mseff+Neff 3.0 ΛCDM+mν+Neff+meff: Planck TT+lowP (PL): Planck TT+lowP ΛCDM+mν+Neff+meff: Planck TT+lowP+lensing +MPkW 2.5 ΛΛCCDDMM++mmνν++NNeeffff++mmeeffff:: PPllaanncckk TTTT+,TlEo,wEEP++lBoAwOP +BAO ΛCDM+mν+Neff+meff: Planck TT,TE,EE+lowP+BAO +lensing 2.0 (PCHIP): Planck TT+lowP +MPkW V] e ++BleAnOsing effm[s1.5 (PL): Planck TT,TE,EE+lowP +MPkW 1.0 +BAO +lensing 0.5 (PCHIP): Planck TT,TE,EE+lowP +MPkW +BAO 0.5 1.0 1.5 2.0 2.5 3.0 +lensing Σmν [eV] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 meff [eV] FIG.11. 68%and95%CLallowedregionsinthe(Σm ,meff) s ν s plane using the different combination of datasets, within the PCHIP PPS parameterization. FIG.10. 68%and95%CLconstraintsonmeff,obtainedinthe s ΛCDM + N +Σm +meffmodel. Different colors indicate eff ν s Planck TT+lowP with PL PPS (black), Planck TT+lowP with PCHIP PPS (red), Planck TT,TE,EE+lowP with PL Tab. 2 in the Appendix). As for the other extensions of PPS (blue) and Planck TT,TE,EE+lowP with PCHIP PPS the ΛCDM model we studied, the bounds on Σm , N ν eff (green). Foreachcolorweplot4differentdatasets: fromtop and meff are weaker when considering the PCHIP PPS s to bottom, we have CMB only, CMB+MPkW, CMB+BAO with respect to the ones obtained within the power-law and CMB+lensing. PPS canonical scenario. Notice that the bounds on meff s are not very stringent. This is due to the correlation between meff and N : sub-eV massive sterile neutrinos lies [52, 66–68]. These extra light species have an associ- s eff contributetothematterenergydensityatrecombination ated free streaming scale that will reduce the growth of and therefore a larger value of N will be required to eff matterfluctuationsatsmallscales. Theyalsocontribute leaveunchangedboththeangularlocationandtheheight to the effective number of relativistic degree of freedom of the first acoustic peak of the CMB. (i.e. to N ). eff Figure11illustratesthedegeneracybetweentheactive We explore in this section the ΛCDM scenario (in the and the sterile neutrino masses. Since both active and two PPS parameterizations, power-law and PCHIP) with sterile sub-eV massive neutrinos contribute to the mat- three active light massive neutrinos, plus one massive ter energy density at decoupling, an increase of meff can sterileneutrinospeciescharacterizedbyaneffectivemass s be compensated by lowering Σm , in order to keep fixed meff, that is defined by ν s the matter content of the universe. Notice that the most (cid:18) (cid:19)3 stringent 95% CL bounds on the three active and ster- T meff = s m =(∆N )3/4m , (6) ile neutrinos are obtained considering the BAO data in s T s eff s ν the two PPS cases. In particular, we find Σm < 0.481 ν eV, meff < 0.448 eV for the PCHIP parametrization and where T (T ) is the current temperature of the ster- s s ν Σm <0.263 eV, meff <0.449 eV for the power-law ap- ile (active) neutrino species, ∆N ≡ N − 3.046 = ν s eff eff proach. Furthermore, in general, when considering the (T /T )3 is the effective number of degrees of freedom s ν PCHIP parametrization, the mean value on the Hubble associated to the massive sterile neutrino, and m is its s constant is smaller than the value obtained in the stan- physical mass. For the numerical analyses we use the dardpower-lawPPSframework,duetothestrongdegen- following set of parameters to describe the model with a eracy between Σm and H . The value of the clustering power-law PPS: ν 0 parameter σ is reduced in the two PPS parameteriza- 8 {Ω h2,Ω h2,θ,τ,n ,log[1010A ],Σm ,N ,meff} . (7) tions when comparing to the massless sterile neutrino b c s s ν eff s case. This occurs because the sterile neutrino mass is When considering the PCHIP PPS parameterization, n anothersourceofsuppressionofthelargescalestructure s and A are replaced by the twelve parameters P (with growth. s s,i i=1,...,12). The inclusion of the polarization data improves no- The 68% and 95% CL bounds on meff obtained with tably the constraints on the cosmological parameters in s differentdatasetsandPPScombinationsaresummarized the model with a PCHIP parametrization. In particular, in Fig. 10 and in Tabs. 8 and 9 in the Appendix. Notice theneutrinoconstraintsarestrongerthanthoseobtained that,ingeneral,thevalueofN islargerthaninthecase using only the temperature power spectrum at small an- eff inwhichthesterileneutrinosareconsideredmassless(see gular scales. This effect is related to the fact that many 10 thermalornon-thermal,theaxionisapossiblecandidate ΛCDM+ma foranextrahotthermalrelic,togetherwiththerelicneu- (PL): Planck TT+lowP +MPkW trinobackground,orforthecolddarkmattercomponent, +BAO respectively. In what follows, we shall focus on the ther- +lens mal axion scenario. The axion coupling constant f is (PCHIP): Planck TT+lowP a +MPkW related to the thermal axion mass via +BAO √ +lens f m R 107GeV (PL): Planck TT,TE,EE+lowP ma = πf π 1+R =0.6 eV f , (8) +MPkW a a +BAO +lens with R = 0.553±0.043, the up-to-down quark masses (PCHIP): Planck TT,TE,EE+lowP ratio, and f = 93 MeV, the pion decay constant. Con- π +MPkW sidering other values of R within the range 0.38−0.58 +BAO [73] does not affect in a significant way this relationship +lens [74]. 0.5 1.0 1.5 2.0 2.5 3.0 When the thermal axion is still a relativistic particle, m [eV] a it increases the effective number of relativistic degrees of freedom N , enhancing the amount of radiation in eff FIG. 12. As Fig. 2 but in the context of the ΛCDM + ma the early universe, see Eq. (4). It is possible to compute model, focusing on the thermal axion mass ma parameter. thecontributionofathermalaxionasanextraradiation component as: degeneracies are reduced by the high multipole polar- 4(cid:18)3n (cid:19)4/3 ization measurements (as, for example, the one between ∆Neff = 7 2na , (9) Σm and τ). Concerning the CMB measurements only, ν ν we find an upper limit on the three active and sterile with n the current axion number density and n the a ν neutrino masses of Σmν < 0.83 eV and mesff < 1.20 eV present neutrino plus antineutrino number density per at 95% CL, while for the effective number of relativistic flavor. When the thermal axion becomes a non rela- degrees of freedom we obtain Neff < 3.67 at 95% CL, tivistic particle, it increases the amount of the hot dark considering the PCHIP PPS approach. Also in this case matter density in the universe, contributing to the to- the most stringent constraints are obtained when adding tal mass-energy density of the universe. Thermal ax- the BAO datasets to the Planck TT,TE,EE+lowP data. ions promote clustering only at large scales, suppressing Finally, the addition of the lensing potential displaces the structure formation at scales smaller than their free- both the active and sterile neutrino mass constraints to streaming scale, once the axion is a non-relativistic par- higher values. ticle. Several papers in the literature provide bounds on Concerning the Ps,i parameters, we can notice the thermal axion mass, see for example Refs. [63, 75– that considering the Planck TT,TE,EE+lowP+BAO 80]. In this paper our purpose is to update the work datasets, the dip corresponding to the Ps,3 node is re- done in Ref. [35], in light of the recent Planck 2015 tem- duced with respect to the other possible data combina- perature and polarization data [32]. Therefore, in what tions. We have an upper bound for the Ps,12 node from follows,wepresentup-to-dateconstraintsonthethermal all the data combinations except for the CMB+lensing axion mass, relaxing the assumption of a power-law for dataset combination. In addition, as illustrated in the PPS of the scalar perturbations, assuming also the Secs. V and VI, a significant degeneracy between Neff PCHIP PPS scenario. and the nodes Ps,8 to Ps,10 and between Σmν and the TheboundsontheaxionmassarerelaxedinthePCHIP nodes Ps,5 and Ps,6 is also present in this ΛCDM exten- PPS scenario, as illustrated in Fig. 12 (see also Tabs. 10 sion. Finally, because of the correlation between Σmν and 11 in the Appendix). This effect is related to the and mesff, degeneracies between mesff and the nodes Ps,5 relaxed bound we have on Neff when letting it free to and Ps,6 will naturally appear. vary in an extended ΛCDM + Neff scenario. From the results presented in Tab. 2, we find N = 3.40+1.50 at eff −1.43 95% CL for the PCHIP PPS parameterization, implying VIII. THERMAL AXION that the PCHIP formalism favours extra dark radiation, and therefore a higher axion mass will be allowed. As a The axion field arises from the solution proposed by consequence, we find that the axion mass is totally un- Peccei and Quinn [69–72] to solve the strong CP prob- constrainedusingthePlanckTT+lowPdatainthePCHIP lem in Quantum Chromodynamics. They introduced a PPSapproach. Weinsteadfindtheboundm <1.97eV a new global Peccei-Quinn symmetry U(1) that, when at 95% CL for the standard power-law case. The most PQ spontaneously broken at an energy scale f , generates a stringent bounds arise when using the BAO data, since a Pseudo-Nambu-Goldstoneboson,theaxionparticle. De- they are directly sensitive to the free-streaming nature pending on the production process in the early universe, of the thermal axion. While the MPkW measurements

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