Massive'neutrinos'and'the' CMB vstension'between'CMB'&'LSS'' LSS: Is there a tension?

AdamRichard'Ba8ye'' Moss UniversityJodrell'Bank'Centre'for'Astrophysics' of Nottingham In'collaboraAon'with'Adam'Moss,'Tom'Charnock,'Anna'Bonaldi,' In collaborationSam'Cusworth,'Sco8'Kay'&'the''Planck'CollaboraAon' with Richard Battye and Tom Charnock

Sunday, 23 March 14 Planck'CollaboraAon,'2013' Constraints'from'SZ+priorsConstraintsSZ ' Clusters

SZ+HST SZ+BAO 1Jb=0.8' ► Planck SZ (see talk by Anna Bonaldi) found ~1/2 the number of clusters predicted by Planck CMB cosmology

► Main uncertainty is thedegeneracy mass bias b - parameterizes extent in which (⌦ /0.29)0.32 X ray observations don’t8 m measure true mass ► Will consider fixed 1-b = 0.8, and 1-b = [0.7, 1.0] Planck'CollaboraAon,'2013' Comparison'with'CMB' ► Implication for fixed bias 1 b =0.59 0.05 0.9 − ± ns, Ωb, Y*, α, S marginalised0.3 over [+BBN prior & (1-b)=0.8] TT'from'Planck'+'WP' Ωm 0.88 M. Douspis, 03/04/2013, Cosmology fromσ8 Planck SZ cluster counts =0.782 0.010 9 !0.27" ± 0.86 0.84 Low'compared'' 0.82 to'expectaAons'' ► Discrepant with Planck 8 0.8 from'XJrays' CMB only at ~3σ 0.78 0.76

► Or the mass bias is 1-b 0.74 ~0.6 0.72 0.7 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.4 0.5 0.6 0.7 0.8 m 1−b

Sunday, 23 March 14 5

−4 We compute the usual CMB observables: the CMB 10 10 c2 = 1 temperature and polarization angular power spectra, as s c2 = 10−4 φφ well as the lensing potential power spectrum, C` .We ! s −6 C 4 + 5 ! also compute the galaxy lensing convergence power spec- ξ 10  7 trum, Pij(`), which in the absence of anisotropic stress 10 can be written in terms of the matter power spectrum, −8 0 P (k, z). However, since our models include anisotropic 10 0 2 stress, we use the more general form of the convergence 10 10 0 500 1000 θ [arcmin] ! power spectrum 5 (a) TDI (g)models L 4 2 −4

We compute the usual CMB observables: the CMB` 8⇡ 10d 10 2 −4 Pij(`) gi () gj () P (`/fK (), ) ,(22) c = 1 10 10 temperature and polarization angular power spectra,⇡ as 4 `3 s = 1 Z 2 c2 = 10−4 1

Galaxy Lensing φφ well as the lensing potential power spectrum, C` .We ! s = 1.5

C 1 φφ −6 ! 4 where we have made+ use of the Limber approximation,5 ! C also compute the galaxy lensing0.59 convergence power spec- ξ 10 7 −6 4 ing and the constraint of (⌦ /0.27) =0.79 0.03 + 5

8 m !  ξ 10 7

f () is the angular diameter distance, and10 P is the from thetrum, CFHTLenSPij(`), are which both symptoms in the absence of this tension.of± anisotropicK stress 10 A simple► Usecan illustration be data written from of in this terms CFHTLenS, point of is the to just matter compare154 power squarepower spectrum, degrees spectrum of the Weyl potential, defined in terms of −8 0 P (k, z). However, since our models include anisotropicthe Newtonian potentials10 and by =( + ) /2. For the expected lensing spectra for the best fitting mod- 0 2 −8 0 ► 10 10 0 500 101000 els to Planck2Dstress, dataCMB+WP+BAO we use (Kilbinger the more reported general et. inal.), [ form9]. Inshear of Fig. the1 convergencecorrelationa flat Universe functionfK ()= on, and angular we shall assume flatness 0 2 θ [arcmin] ! 10 10 0 500 1000 we havepower plotted spectrum the measurements of the CMB lensing from now on. The lensing eciency is given by θ [arcmin] ! scales 0.8 to 350 arcmin (a) TDI (g)models power spectrum, C` , and the galaxy lensing correlation L + (b) GSF models with ↵ =0, 2 =1 1 function, ⇠ (✓) (the Hankel4 2 transform of the convergence 1 0  ` 8⇡ d −4 gi()= d0ni (0) , (23) power spectrumPij(`) P), along with modelgi ( predictions) gj () P colour(`/fK (), ) ,(22) 10 10 ⇡ 4 `3 0 = 1 coded by their likelihood. ItZ is clear that, in both cases, Z 1 = 1.5 Figure1 2: Lensing observations in the TDI (g)andGSF those parameter combinations that are a good fit to the φφ where we have made use of the Limber approximation,where n () is the radial distribution of source! galaxies in L i Source galaxy−6 distribution, peaked at z~0.75C models with w = 0.8. On the left we plot the predictions 4 CMB+BAO data predict a higher level of lensing corre- + 5 ! ξ 10 + fK () is the angular2 diameter distance, andbinP i.is In the the case of no anisotropic stress, the7 convergence lations► than observed ( 20), indicating that there of the weak lensing correlation ⇠ spectra, as well as the

3D data (Heymans et. al.), shear 10 could bepower something spectrum missing of⇠ the within Weyl the potential, model. We defined will power in terms spectrum of can be written in the usual form data taken from CFHTLenS. On the right we plot the lens- 4 make thiscorrelationthe explicit Newtonian by performing potentialsfunction a full andon joint angularby likelihood =( + ) /102. For ing of the CMB C spectra, with observations from Planck −8 98350 ` 910 analysisa of flat the Universe publicly availablefK ()= lensing, and data we and shall com- assume flatness 2 0 4 2 (red/circles) and SPT (black/diamonds). Here we see that scales 1.5 to 35 arcmin,0.3 6 Pij(`)= ⌦10mH0 10 98320 500 1000 bining this with a prior on 8(⌦m/0.27) coming from 5 ! decreasing the sound speed decreases the large-scale cluster- from now on. The lensing eciency is given by 10 4 θ [arcmin] 9829 the SZredshift cluster counts, bins which will lead to a significant 1 gi()gj() ing of the dark energy. [fixed]

(b) GSF models with ↵ =0 , 2 =1 1 P (`/, )d9826 . (24) preference for such models. We1 note that this0 is not 2 log 2 g ()= d0n (0) , + (23)6 ⇥ 0 a () equivalent► In toboth performing casesi a fullobserved joint analysisi including 10 9823 Z L the SZ likelihood – which is not publicly available0 – but Z 9820 we havecorrelation tested that this function leads to similar is resultshigher to those thanThe convergenceFigure can also 2: Lensing be written observations in terms in of the the TDI cor- (g)andGSFIt is most interesting to note that some of the spectra 7 9817 L presentedPlanckwhere in [10n].i (cosmology) is the radial distribution of sourcerelation galaxies10 functions in models⇠± via with w = 0.8. On the left we plot themostly predictions lie inside the error bars from CMB lensing, but + Therebin arei. two In the separate case of analyses no anisotropic that we stress, have per- the convergence of the weak lensing correlation ⇠9814 spectra, as well as the 1 lie outside the error bars of galaxy weak lensing obser- formed:power a model spectrum with the can standard be written six parameters, in the usualp, form 8 data taken from CFHTLenS. On9811 the right we plot the lens- 10 ⇠± (✓)= d` Pij(`)J0/4(`✓), (25) vations: the CFHTLenS galaxy weak lensing data alone 100 i,j 101 102 and 1 extra parameter m⌫ with N⌫ =3(N⌫ is the ing of2 the⇡ CMB C` spectra, with observations from Planck [arcmin]Z could rule out a range of parameter space. number of massive neutrinos)9 and2 N4e↵ =3.046; a model (red/circles) and SPT (black/diamonds). Here we see that Pij(`)=P ⌦mH0 e↵ with a total of 8 parameters – p + m ,Ne↵ – and where J0,4 aredecreasing Bessel functions the sound of the speed zeroth decreases and the fourth large-scaleIn cluster- Fig. 3 we plot the fractional di↵erences in observa- 4 { ⌫, sterile } FIG. 1: The CMB lensing power spectrum (top) data points m⌫ =0.06 eV, N⌫ = 1. The first1 representsgi()gj( a) degener- order respectively.ing of the dark energy. [fixed] tional spectra for GSF models with a range of values of Sunday, 23 March 14 P (`/, )dfrom. Planck(24) (green squares) and SPT (blue squares) and ate active neutrino scenario, that is appropriate2 for large In Fig. 2 we plot the predictions of the CMB C spec- ↵, 1 and 2, relative to a fiducial quintessence scenario. P ⇥ 0 a () the shear correlation function ⇠+ from CFHTLenS (bottom),` values of m⌫ , whereas the secondZ is a sterile neutrino trum and galaxy correlation function ⇠+, for the TDI For all of the models shown, the di↵erences in the CTT scenario with active neutrinos in a standard hierarchy compared to predictions for parameters from samples of the ` (g) and GSF models (for simplicity we take ↵, to be spectra are negligible for angular scales ` > 50. However, that hasThe theP convergence lowest value of can alsom allowed be written by the in solar terms ofPlanck the cor-CMB+WP+BAOIt is most MCMC interesting chains with to non-linear note that cor-1 some of the spectra ⌫ Lrections [18, 19]. In both cases, the data is systematically¨ the di↵erences for the Cdd spectra remain roughly con- and atmosphericrelation functions constraints⇠ on± via the mass di↵erences. constants and mostly2 =1 lie inside1,whichremovesthe the error bars fromh-term). CMB lensing, but ` P lower than theory, although the significance is somewhat lower stant for all scales and, markedly, di↵erences in the ⇠+ In both cases we will follow the procedure out- For galaxy lensing,lie outside we use the the error single bars bin ⇠ of± galaxy⇠1±,1 distri- weak lensing obser- 1 than the eye would suggest in the case of CFHTLenS⌘ due to spectra increase with angular size. It is clear that mod- lined in [10] and use the Planck likelihood [20] that butioncorrelations of source between galaxies data points. from the Canada-France-Hawaii ⇠i,j± (✓)= d` Pij(`)J0/4(`✓), (25) vations: the CFHTLenS galaxy weak lensing data alone includes a number of nuisance2⇡ parameters describ- Telescope Lensing Survey (CFHTLenS)[12]. For illustra- els with an increased power in the spectrum of galaxy Z could rule out a range of parameter space. + ing the contamination from our own galaxy, extra- tion we have applied corrections from the Halofit fitting weak lensing, ⇠ , also have an increased power in the galacticwhere sourcesJ0 and,4 are the Bessel SZ e↵ functionsect. We will of the consider zeroth and fourth In Fig. 3 we plot the fractional di↵erences inspectrum observa- of CMB lensing, Cdd. This plot nicely illus- formulaeneutrino case. [57, 58 In] Fig.to the2 we lensing present potential the 1D likelihoodP . However, ` three dataorder combinations: respectively. (I) Planck CMB+WP+BAO; tional spectra for GSF models with a range oftrates values how of useful lensing data is, compared to the CMB sincefor wem⌫ dowhich not illustrates have a full that understanding the upper bound of dark of en- (II) Planck CMB+WP+BAO+lensing where lensing ↵, 1 and 2, relative to a fiducial quintessence scenario. In Fig. 2 we plot the predictions of the CMB C`m⌫spec-< 0.254 eV that we find in the case of (I) is weak- temperature data. The fundamental reason for this is is both the CMB lensing from Planck and SPT+ ergy perturbations on non-linear scales, we do not in- TT trum and galaxy correlation function ⇠ , forened theP by TDI the inclusionFor all of the of the lensing models data and shown, that the a peak di↵erences inthat the theC` lensing data is very sensitive to the evolution of and galaxy lensing from CFHTLenS; (III) Planck cludeP those scales to obtain constraints on the equations (g) and GSF models (for simplicity we take ↵develops, 1 to be in the likelihoodspectra are at non-zero negligiblem for⌫ .Byitselfthe angular scales ` > 50.perturbations However, of the dark sector theory. CMB+WP+BAO+lensing+SZL cluster counts imposed of state in the next section. We comparedd these predic- 0.3 lensing¨ data is notthe su di↵cientlyerences strong for the to induceC` spectra a strong remain roughly con- using aconstants prior in the and8 ⌦2 m=1plane of1,whichremovesthe8(⌦m/0.27) = tionsh-term). to CMB lensing data from Planck [7] and the South It is important to have a handle on which probes are preference, butstant the inclusion for all ofscales theP prior and, from markedly, the SZ di↵erences in the ⇠+ 0.78 For0.01. galaxy For CFHTLenS lensing, we we use use the the single⇠± correla- bin ⇠± Pole⇠1±,1 distri- Telescope (SPT)[6], and also plot the single-bin ob- sensitive to which types of theories, and so it is useful to ± ⌘ cluster catalogue leads to m⌫ =(0.320 0.081) eV, tion functionsbution and of source covariance galaxies matrix from as described the Canada-France-Hawaii in [17], spectra increase with angular± size. It is clear that mod- servationswhich corresponds from CFHTLenS to 4 detection. of m⌫ > 0. evaluate other probes of dark sector theories which are choosingTelescope the smallest Lensing and largest Survey angular (CFHTLenS scales to be)[12 0.9]. For illustra- els with⇡ anP increased power in the spectrum of galaxy ItWe is now clear consider that there the will sterile be neutrino+ ranges of model parameter which space rather prolific in the literature. We use our modified ver- and 300tion arcmin we haverespectively. applied To corrections compare the from shear the withHalofit fitting weak lensing, ⇠ , alsoP have an increased power in the that measured from large scale structure we correct the whichleads to yield a similar, observational but even spectra stronger which result. will Thedd sit re- inside sion of CAMB to compute the sum and di↵erence of the formulae [57, 58] to the lensing potential P . However, spectrum of CMB lensing, C` . This plot nicely illus- power spectrum on non-linear scales using the Halofit experimentalsults are present error in Fig. bars3. and For will (II) not we find be distinguishable. that there gravitational potentials, and ,thee↵ective gravita- since we do not have a full understanding ofis dark a 2.3 en-preferencetrates for howme↵ useful> lensing0 with m datae↵ is, compared= to the CMB fitting formulae [18, 19]. For SPT lensing data we follow temperature⌫, sterile data. The fundamental⌫, sterile reason for this is the sameergy procedure perturbations as in [16], on rescaling non-linear the diagonals scales, of we do(0.326 not in-0.143) eV although there is only an upper bound ± that the lensing data is verye↵ sensitive to the evolution of the covarianceclude those matrix scales according to obtain to sample constraints variance, andon theof equationsNe↵ < 0.96. This is strengthened to m⌫, sterile = addingof an state additional in the calibration-induced next section. We uncertainty compare to these(0.450 predic-0.124) eVperturbations and Ne↵ =0 of.45 the0 dark.23 for sector (III). theory. ± ± the covariance.tions to CMB lensing data from Planck [7] and theThe South sterile neutrinoIt is model important has the to added have feature a handle that on which probes are DetailedPole constraints Telescope on (SPT the)[ parameters6], and also are plotpresented the single-binit can be ob- made compatiblesensitive to with which the directtypes measurement of theories, and so it is useful to in tableservationsI. We first from turnCFHTLenS our attention. to the active of Hubble’s constantevaluate from other Cepheid probes variables of dark in nearby sector theories which are It is clear that there will be ranges of parameter space rather prolific in the literature. We use our modified ver- which yield observational spectra which will sit inside sion of CAMB to compute the sum and di↵erence of the experimental error bars and will not be distinguishable. gravitational potentials, and ,thee↵ective gravita- CMB Lensing

► CMB lensing probes to larger scales/distances than galaxy lensing 2 Cosmological prob►es.Use—Our SPTprimary (bluegoal squares)in this pap ander 1.0 is to constrain homogeneity over the largest volumes ac- Planck (green squares) = = = = cesible to us. To do this, we must use probes which reach 3 3 3 3 0 0 0 0 0 0 as far as possible. Ideally, we would also prefer meth- 0 lensing results 0.8 . ods sensitive to total matter (i.e., metric) fluctuations, 1 1 .3 ► Planck cosmology again . 0 0 to avoid bias uncertainties. The ISW, RS, and CMB and 0 1. galaxy lensing probes(marginally)satisfy these criteria highto various de- 2 CMB lensing grees. 0.6 0 . 3 0.59 O S

ing and the constraint of (⌦ /0.27) =0.79 0.03 10 A

8 m To characterize the usefulness of these various observa- L

± B from the CFHTLenS are both symptoms of this tension. 9835 0. / tions, we will express their power spectra as line-of-sight 1 q 3 e A simple illustration of this point is to just compare 8 9832 . 0.3 0 the expected lensing spectra for the bestintegrals fittingin mod-the Limber approximation [45, 46]. First, the 9829 0.4 els to Planck CMB+WP+BAO reportedISW in [9].sp Inectrum Fig. 1 can b6e written 1 .0

9826 log 2 we have plotted the measurements of the CMB lensing C 4 ISW 9823 r L power spectrum, C , and the galaxy lensing correlation 2 7 4LS ` 72π 2 ! ! 1 + 2 ISW 10 2 ! C drrg! (r) T . (1) 9820 . function, ⇠ (✓) (the Hankel transform of the convergence! 0.3 0 01.1.0 ! 25! 0 PR r r power spectrum P), along with model predictions colour 2 9817 0.2 L ! " # " # galaxy lensing N coded by their likelihood. It is clear that, in both cases, 0 9814 . 1 those parameter combinations that areHere a goodr fit tois thethe radius0 to the LSS, g is the Λ growth 0 1 3 LS 3 .3 . . 9811 .0 0 0 CMB+BAO data predict a higher levelsupression of lensing corre-factor, (k) is the (dimensionless) primor- 010.1 2 100 200 300 400 500 600 700 800 900 ..301 3.0 lations than observed ( 20), indicatingdial como that thereving curvPatureR perturbation spectrum, T (k) is 0.1 01..30 could be something missing⇠ within the model. We will 0.0 make this explicit by performing a fullthe joint(linear) likelihoodtransfer function accounting for the suppres- -4.0 -3.0 -2.0 -1.0 0.0 - 1 analysis of the publicly available lensingsion datadue andto com-radiation domination, and a prime indicates log 10( [Mpc ]) 0.3 bining this with a prior on 8(⌦m/0.27)a conformalcoming fromtime derivative. the SZ cluster counts, which will lead to a significant Sunday, 23 March 14 preference for such models. We note thatSimilarly this is not, we can write the CMB lensing potential FIG. 1: Limber approximation kernels for the CMB lensing equivalent to performing a full joint analysispower includingspectrum in the Limber approximation as [47] potential (solid black contours), ISW effect (dashed blue con- the SZ likelihood – which is not publicly available – but tours), and galaxy lensing (dot-dashed green contours), as we have tested that this leads to similar results to those 2 rLS 2 functions of the comoving scale, k, and comoving position, presented in [10]. 4 φφ 72π ! rLS r 2 ! 2 ! ! C! drr − g (r) T . r. Normalization is arbitrary. The dotted magenta curves There are two separate analyses that we have! per-25 r r PR r r !0 " LS # " # " # indicate the corresponding Limber multipole scale. Scales la- formed: a model with the standard six parameters, p, (2) belled aH, k , k , and k are, respectively, the comov- and 1 extra parameter m with N =3(N is the eq BAO NL ⌫ ⌫Finally,⌫ the galaxy lensing convergence power spectrum ing Hubble scale, the equality scale, the first baryon acoustic number of massive neutrinos) and Ne↵ =3.046; a model with a total of 8 parametersP – p + me↵ for sources,N – andat a single distance rs becomes (see, e.g., [48]) oscillation peak, and the nonlinearity scale, beyond which in- { ⌫, sterile e↵ } m⌫ =0.06 eV, N⌫ = 1. The first represents a degener- FIG. 1: The CMB lensing power spectrum (top) data pointsformation is harder to extract. Geometry prevents measure- ate active neutrino scenario, that is appropriate for large18π2!from3 rPlancks r (greenr squares)2 and SPT! (blue! squares) andments in the hatched region (delimited roughly by ! = 1). P 2 κκ the shear correlations function2 ⇠+ from CFHTLenS2 (bottom), values of m⌫ , whereas the second is a sterile! C! neutrino drr − g (r) T . The black box roughly indicates the range accessed by the ! 25 compared to predictionsr r for parametersPR r from samplesr of the scenario with active neutrinos in a standard hierarchy !0 " s # " # " # WiggleZ survey [25, 51], the grey box by the proposed Euclid that has theP lowest value of m allowed by the solar Planck CMB+WP+BAO MCMC chains with non-linear(3) cor- ⌫ rections [18, 19]. In both cases, the data is systematicallysurvey [52], and the blue box by primary CMB. and atmospheric constraints on the mass di↵erences. P To get an idealowerof the thansensitivit theory, althoughy of thethese significanceprobes is somewhatto the lower In both cases we will follow thescales procedureand out-redshiftsthanof thefluctuations, eye would suggestw ine theplot casein ofFig. CFHTLenS1 the due to lined in [10] and use the Planck likelihood [20] that correlations between data points. includes a number of nuisance parameterskernels describ-of Eqs. (1) – (3) in the k-r plane, together with lensing are the most promising probes of the region not ing the contamination from our ownthe galaxy,regions extra-sampled by galaxy surveys and measurements accessible to galaxy surveys [50]. galactic sources and the SZ e↵ect. We will consider of the primary CMBneutrino[49 case.]. For Ingalaxy Fig. 2 welensing presentw thee c 1Dhoose likelihood three data combinations: (I) Planck CMB+WP+BAO; Localized structures.—Next we consider the question r = r(z = 1). forIt is mclearwhichfrom illustratesthis plot thatthat the uppercurren boundt of (II) Planck CMB+WP+BAO+lensings where lensing ⌫ of how large localized structures may be “hiding” out- m < 0.254 eV that we find in the case of (I) is weak- is both the CMB lensing from Planckgalaxyandsurv SPTeys sample⌫ only a small fraction of the comov- enedP by the inclusion of the lensing data and that a peakside the reach of surveys but inside our LSS. We will see and galaxy lensing from CFHTLenS; (III) Planck ing distance to lastdevelopsP scattering, in the likelihoodand atalso non-zeroare insensitivm .Byitselfthee that even some nonlinear structures will be allowed, so CMB+WP+BAO+lensing+SZ cluster counts imposed ⌫ to the largest twlensingo decades data isof notlength sucientlyscales strongwhic toh induceare in a strong using a prior in the ⌦ plane of (⌦ /0.27)0.3 = we will need to treat the structure with exact GR. We 8 m principle8 m observable.preference,Similarly but the, while inclusionthe ofprimary theP priorCMB from the SZ 0.78 0.01. For CFHTLenS we use the ⇠± correla- use the spherically symmetric Λ-Lemaˆıtre-Tolman-Bondi ± cluster catalogue leads to m⌫ =(0.320 0.081) eV, tion functions and covariance matrix assamples described ina [17large], range of scales, it is restricted to±dis- (ΛLTB) spacetime [53], sourced by dust and Λ, to rep- which corresponds to 4 detection of m⌫ > 0. choosing the smallest and largest angulartances scales tovery be 0.9close to last scattering⇡(rLSP r < rLS/1000). We now consider the sterile neutrino model whichresent the standard ΛCDM background with superposed and 300 arcmin respectively. To compareTherefore, the shear withto the extent that galaxy surv− eys∼ haPve been that measured from large scale structure we correct the leads to a similar, but even stronger result. The re-spherical structure [54]. We choose a compensated un- power spectrum on non-linear scales usingutilized the Halofitto test homogeneitsults are presenty, there in Fig. is3.m Foruc (II)h more we findro thatom therederdensity with ΛLTB curvature function profile (in the is a 2.3 preference for me↵ > 0 with me↵ = fitting formulae [18, 19]. For SPT lensingfor datadepartures we follow from homogeneity to hide.⌫, sterileIn particular,⌫, sterile notation of [11]) the same procedure as in [16], rescaling the diagonals of (0.326 0.143) eV although there is only an upper bound modifications to the ±matter power spectrum on scalese↵ the covariance matrix according to sample variance,4 and1 of Ne↵2 < 0.96.1 This is strengthened to m⌫, sterile = 5 4 2 10− Mpc− < k < 10− Mpc− or localized structures of K(r) = K 2(r/R) 3(r/R) + (r/R) , (4) adding an additional calibration-induced uncertainty to (0.450 0.124) eV and Ne↵ =0.45 0.23 for (III). 0 ± ± > − the covariance. correspondingly largeThe sterilesizes, neutrinoor inhomogeneities model has the addedat z feature1 that Detailed constraints on the parametersare arepossible presentedwhileit canmain betaining made compatibleconsistency with thewith directgalaxy measurement∼ with amplitude K0$and radius R. The ISW, RS,%and SW in table I. We first turn our attentionsurv toeys. the activeOf course,of Hubble’sany suc constanth po fromwer Cepheidspectrum variablesmodifi- in nearbytemperature anisotropies caused by the structures are cations must be localized away from the last scattering calculated by ray tracing null geodesics from the observer surface for consistency with the primary CMB, so they to the LSS, as described in [11]. The deflection angles, would entail a breaking of statistical homogeneity. Fig- α, are translated into lensing potential, φ, via α = φ. ure 1 makes it apparent that ISW and especially CMB The ΛLTB solution is calculated by numerically evolving∇ 14 Florian Beutler et al. and S(2) are given by 6.2 The Alcock-Paczynski effect 2 (2) 5 q1 q2 q1 q2 2 q1 q2 If our fiducial cosmological parameters that we use to con- F (q1,q2)= + · + + · , S 7 2q q q q 7 q q vert galaxy redshifts into distances deviate from the true cos- 1 2 ! 2 1 " ! 1 2 " (48) mology, we artificially introduce an anisotropy in our cluster- 2 ing measurement, which is known as Alcock-Paczynski dis- (2) 3 q1 q2 q1 q1 4 q1 q2 G (q1,q2)= + · + + · , tortion (Alcock & Paczynski 1979). This effect can be used S 7 2q q q q 7 q q 1 2 ! 2 2 " ! 1 2 " to measure cosmological parameters (Matsubara & Suto (49) 20 Florian Beutler et al. 1996; Ballinger, Peacock & Heavens 1996). To account for 2 the Alcock-Paczynski effect and its different scaling along 14 Florian Beutler et al. (2) q1 q2 1 S (q1,q2)= · . (50) q1q2 − 3 and perpendicular to the line-of-sight direction, we can in- (2) ! " troduce the scaling factors and S are given by If we additionally6.2 define The Alcock-Paczynski effect 2 H fid(z)rfid(z ) (2) 5 q1 q2 q1 q2 2 q1 q2 If our fiducial2 cosmological parameters that we use to con- s d FS (q1,q2)= + · + + · , (2) (2) α! = , (55) 7 2q1q2 q2 q1 7 q1q2 D vert(q1 galaxy,q2)= redshiftsS into(q1 distances,q2) 1 deviate, from(51) the true0.55 cos- H(z)r (z ) ! " !0.8 " 7 − s d (48) mology, we artificially introduce an anisotropy in our cluster- fid 2 # $ DA(z)rs (zd) we can2 write downing measurement,σ3 (k)ofeq.41as which is known as Alcock-Paczynski dis- α⊥ = , (56) 3 q1 q2 q1 q1 4 q1 q2 fid (2) 0.5 D (z)rs(zd) GS (q1,q2)= + · + + · , tortion (Alcock & Paczynski 1979). This effect can be used A 7 2q1q2 q2 q2 7 0.75q1q2 3 ! " ! 2 " 105 tod measureq lin cosmological(2) parameters(2) (Matsubara & Suto fid fid σ3 (k)=(49) Pm (q)D ( q, k)S (q, k q). (52) where H (z)andD (z)arethefiducialvaluesforthe 16 1996;(2π) Ballinger,3 Peacock− & Heavens 1996).− To account for A 2 % 0.45 Hubble constant and angular diameter distance at z =0.57 (2) q1 q2 1 the Alcock-Paczynski effect and its different scaling along S (q1,q2)= · . 0.7As shown(50) in Chan, Scoccimarro & Sheth (2012) non-linear fid q1q2 − 3 and perpendicular to the line-of-sight direction, we can inand- rs (zd)isthefiducialsoundhorizonassumedinthe ! " (z = 0.57) # gravitational evolutiontroduce the naturally scaling factors induces such non-local bias 8 0.4 power spectrum template. The true wave-numbers k and

σ ! If we additionally define # # (z=0.57) terms even starting from purely local biasfid atfid an initial time. k are then related to the observed wave-numbers by k = 0.65 H (z)rs (zd) ⊥ ! 2 AP (2) (2) Baldauf et al. (2012) shows thatα! the= 2nd-order bias, is im- (55) # D (q1,q2)= S (q1,q2) 1F , (51) H(z)r (z ) k!/α! and k = k⊥/α⊥.Transferringthisintoscalingsfor 7 − s d 0.35 ⊥ # $ portant to explain the large-scale bispectrumfid in simulations, 2 2 2 DA(z)rs (zd) the absolute wavenumber k = k + k⊥ and the cosine of we can write down σ3 (k)ofeq.41as α⊥ = , (56) ! while the 3rd-order non-local bias termsfid play a more impor-f(z = 0.57) 0.6 DA (z)rs(zd) 3 tant role in the power spectrum (Saito et al. in prep.). In 0.3 the angle to the line-of-sight µ&we can relate the true and 2 105 d q lin (2) (2) fid fid σ3 (k)= Pm (q)D ( q, k)S (q, k q). (52) where H (z)andD (z)arethefiducialvaluesfortheobserved values by (Ballinger, Peacock & Heavens 1996) 16 (2π)3 − the− case ofOur the analysis local (k = Lagrangian 0.01 - 0.20h/Mpc) biasA picture in which the Our analysis (k = 0.01 - 0.20h/Mpc) % Hubble constant and angular diameter distance at z =0.57 Our analysis (k = 0.01 - 0.15h/Mpc) Our analysis (k = 0.01 - 0.15h/Mpc) 1/2 As shown in Chan, Scoccimarro & Sheth (2012)0.55initial non-linear non-local biasfid is neglected, we can predict the ampli-0.25 k 1 Planck+WPand (rΛsCDM)(zd)isthefiducialsoundhorizonassumedinthe Samushiak #et =al. (2013) 1+µ2 1 , (57) gravitational evolution naturally induces such non-localtude ofbias theWMAP9 non-local ( CDM) bias as (Chan, Scoccimarro & Sheth # Planck+WP ( CDM) 2 powerΛ spectrum template. The true wave-numbers k! and Λ α⊥ F − terms even starting from purely local bias at an initial2012; time Baldauf. et# al. 2012; Saito et al. in prep.) # ' ! "( k⊥ are then related to the observed wave-numbers by k! = −1/2 Baldauf et al. (2012) shows that the 2nd-order bias is im-13 13.5# 14 14.5 1.1 1.15 1.2µ 1.25 1.31 1.35 k!/Dα!(z=0.57)/rand k⊥ =4k(z⊥/)α⊥.Transferringthisintoscalingsfor µ# = 1+µ2 1 (58) portant to explain the large-scale bispectrum in simulations, V s d b σ (z = 0.57) 2 the absolutebs2 = wavenumber(b1 1)k, = k2 + k2 and(53) the cosine of 1 F8 F − while the 3rd-order non-local bias terms play a more impor- − 7 − ! ⊥ BOSS:' Testing! Gravity"( with the power spectrum multipoles 21 tant role in the power spectrum (Saito et al. in prep.). In the angle to the32 line-of-sight µ&we can relate the true andwith F = α!/α⊥.Themultipolepowerspectrumincluding b3nl = (b1 1), (54) the case of the local Lagrangian bias picture in which the observed values315 by (Ballinger,− Peacock & Heavens 1996) the Alcock-Paczynski effect can then be written as initial non-local bias is neglected, we can predict the ampli- Our analysis (k = 0.01 - 0.20h/Mpc) 1/2 Our analysis (k = 0.01 - 0.20h/Mpc) which are in good agreement# withk the values2 1 measured in 1 Our analysis (k = 0.01 -k 0.15h/Mpc)= 1+µ 1 , (57) Our analysis (k(2 = 0.01% +1) - 0.15h/Mpc) # # tude of the non-local bias as (Chan, Scoccimarro0.55 & Sheth 2 0.6 simulations. In this work, we adoptα⊥ these relationsF − for sim- P!(k)= 2 dµ Pg k ,µ !(µ), (59) 2012; Baldauf et al. 2012; Saito et al. in prep.) Planck+WP ( ΛCDM) ' ! "( Our analysis (k = 0.01 - 0.20h/Mpc)Planck+WP ( ΛCDM)2α α L Beutler et al. (k = 0.01 - 0.20h/Mpc) 2 −1/2 ⊥ ! −1 Anderson et al. (2013b) (pre-recon) % Samushia et al. (2013) plicity, whileWMAP9 we (Λ floatCDM) b1, b#2 andµ N as2 free1 .Theimpact WMAP9 (ΛCDM) ) * # # 4 µ = 1+µ 2 1 Anderson et al.(58) (2013b) (post-recon) Chuang et al. (2013b) bs2 = (b1 1), of the 2nd-order(53) bias terms on theF power spectrumF − 105 is some-0.55 where we use the extended TNS model105 for Pg(k ,µ ). The − 7 − 0.5 ' ! "( Planck+WP (ΛCDM) Sanchez et al. (2013) what small. Figure 10 (left) shows the power spectrum multi-WMAP9 (ΛCDM) AP effect constrains the parameter combination Planck+WPFAP(z )=(ΛCDM) 32 with F = α!/α⊥.Themultipolepowerspectrumincluding b3nl = (b1 1), poles when(54) all higher order bias terms are set to zero (dash- (1 + z)DA(z)H(z)/c,whiletheBAOfeatureconstrainstheWMAP9 (ΛCDM) 315 − the Alcock-Paczynski effect can then be written as 0.5 combination D (z)/r (z ) D2 (z)/H(z). Together these which are in good agreement with the values0.45 measureddot green in line). The solid magenta line1 uses b2100=0.5and V s d A 100 (2% +1) # # ∝ (z=0.57) b =2.0. We can see thatP!( thek)= higher orderdµ bias Pg termsk ,µ mainly!(µ), (z=0.57) (59)two signals allow us to break the degeneracy between DA(z) simulations. In this work, we adopt these relations8 1 for sim- 2 8 2α α! L σ 2 ⊥ −1 σ 14 Florian Beutler et al. % 0.45 and H(z). We will include the scaling parameters α and α⊥ plicity, while we float b1, b2 and N as free .Theimpactaffect the monopole and while the effect is small,) [km/s/Mpc] it is sig- ) [km/s/Mpc] ! ) * # # of the 2nd-order bias terms on the power spectrum0.4nificant is some- when comparedwhere we use to the the extended measurement TNS modelfid s errors for P (greyg(k ,µ ). Thein our model parametrisation, which willfid s be discussed in the /r /r

s 95 s 95 14 Florian Beutlerwhat small.and et al.S Figure(2) are 10 given(left) shows by the power spectrumshaded multi- area). AP effect constrains6.2 the The parameter Alcock-Paczynski combination FAP( ez)=ffectnext section. poles when all higher order bias terms are set to zero (dash- (1 + z)DA(z)H(z)/c,whiletheBAOfeatureconstrainsthe0.4 f(z=0.57) We should also2 mention that the stochastic2 term,f(z=0.57) dot green line). The solid magenta line uses b2 =0.5and combination DV (Ifz)/r ours(z fiduciald) DA( cosmologicalz)/H(z). Together parameters these that we use to con- BOSS: Testing Gravity with the power spectrum multipoles 15 and S(2) are given by (2) 5 q1 q2 q1 0.35Nq2,caningeneraldependonscale(Dekel&Lahav6.22 Theq1 q Alcock-Paczynski2 eff∝ect b1 =2.0. WeFS can(q1 see,q2 that)= the+ higher· order bias+ terms mainly+ · two signals, allow us to break the degeneracy between DA(z6.3) Model parameterisation 7 2q1q2 q2 q1 7 q1q2 vert galaxy redshifts90 into distances0.35 deviate from the true cos- 90 affect the monopole and while the effect2 is small,1998; it is Baldauf sig- and et al.H(z 2012),). We will while include we the treat scaling it parameters as a con-α! and α⊥ ! " ! Redshift" mology, we artificially H(z=0.57)(r Space introduce an anisotropy Distortions in our cluster- H(z=0.57)(r (2) 5 nificantq1 q2 whenq1 comparedq2 2 to theq1 measurementq2 errorsstantIf (grey ourand fiducial freein parameter. our cosmological(48) model parametrisation, The parameters final ingredients which that will be we in discussed use our to con- in theWe parametrize our model using the scaling parameters α F (q1,q2)= + · + + · , F (z)istheparametercombinationwhichtheAPe! ffect is S 0.3 vert galaxy redshifts2 intoing distances measurement, deviate from which the is true known cos- as Alcock-Paczynski dis- AP 7 shaded2q1q2 area).q2 q1 73 qq1q2 q q modelq 4 of eq.q qnext 39 are section. the correction terms, A and B, and α⊥ introduced in the last section. Using these parame- ! (2) " ! 1 2" 1 1 1 2 tortion (Alcock & Paczynski 1979).0.3 This effect can be used We shouldGS (q1 also,q2)= mention+ that· the(48) stochastic+ whichmology,+ term, originate► we· artificially from, the higher-order introduce an correlation anisotropy between in our cluster-ters we can derive sensitive to (Padmanabhan & White 2009). Once such ge- N,caningeneraldependonscale(Dekel&Lahav7 2q1q2 q2 q2 7 BOSSq1q2 DR11 CMASS sample85 (z=0.43 to 0.7, zeff = 0.57) 85 1.3 N-body comparison !2 Kaiser"ing terms0.6 measurement,! and 0.65" velocity which 0.7 fieldsto is known measure 0.75 in mapping as Alcock-Paczynskicosmological 0.8 to redshift parameters dis- (Matsubara13 13.5 & Suto 14 14.5 1 6.3 Model parameterisation ometric parameters are3 constrained, the relative amplitude (2) 3 1998;q1 q2 Baldaufq1 etq al.1 2012),4 whileq1 q2 we treat it as a con- (49) 1300 1350 14002 1450 fid 2 czeff 1300 1350 1400 1450 F (z=0.57)1996; Ballinger, Peacock & Heavens 1996).fid To accountD (z=0.57)/r for (z ) fid G (q1,q2)= + · + + · , spacetortion (Taruya,► (Alcock Nishimichi &AP Paczynski & Saito 1979). 2010). This We effect refer can the be used αV⊥α! (1 + zeff )DA (zeff ) Hfid(z ) 1.2 S stant and free parameter. The final2 ingredients in our Monopole and quadrupole terms ofD (z=0.57)(rpowerDV (ze ffspectrum)/r s) [Mpc] (Beutlers d et. al. 2013)eff D (z=0.57)(r /r s) [Mpc] 7 2q1q2 q2 q2 7 q1q2 We parametrize ourthe model Alcock-Paczynski using the scaling eff parametersect andA itsα! differents = scaling alongof the monopole and quadrupoleA(60) s constrains the growth rate ! (2) " !q1 q2 " 1 readerto to measure Taruya, cosmological Nishimichi & parametersSaito (2010) (Matsubarafor the defini- & Suto + fid . model ofS eq.( 39q1,q are2)= the correction· terms,. A and B, and α(50)⊥ introduced in the last section. Using these parame- rs(zd) , rs (zd) - 1.1 q q (49)Figure3 14. Two► dimensionalSensitive likelihood to threeand distribution perpendicular combinations of DV (ze toff )/r thes(z line-of-sightd)and of parametersFAP(ze direction,ff )(topleft), web1 canσ8(ze inff )and- f(zeff )σ8(zeff )(topright),fid fid which originate from the higher-order1 2 − correlationtions between1996; of the Ballinger,A tersand weB Peacock canterms. derive Note & Heavens that these 1996).Figure terms To 15. areComparison account in for of the two dimensional(1) likelihoodf(z) distributionσ8(z). Besideof DA(zeff ) thers (zd three)/rs(zd)and mainH(ze parametersff )rs(zd)/rs (zd). Weabove show (α!, α⊥ 2 ! " FAP(z )andf(z )σ8(z )(bottomleft),2 DV (z )/rs(z )andf(z )σ8(z and)(bottomright).Weshowthe68%and95%confidence Kaiser terms and velocity fields in mappingefact toff redshift proportionaleff toeff b as physicallytroduce expected theeff scalingthe if 68% oned factors and takes 95% ac-eff confidence1 eff regions. The plot on the left compares our analysis (cyan contours) to the analysis by Anderson etal.(2013b) (2) q1 q2 1 the Alcock-Paczynski1 effect and its different2 scaling along3 1 If we additionally define regions. The plot on the top right also2 includes the resultfid of Samushiaczeff et al. (2013). All contours are directlyand derivedfσ8)wealsoincludefournuisanceparametersinour from the MCMC S (q1,q2)= space· (Taruya,. Nishimichi & Saito 2010).(50) We refer the α⊥α! (1 + zeff )DbeforeA (zeff ) applyingfid densityfid fieldfid reconstruction (greyα! contours) andfid after applyingfid density field reconstruction (blue contours). The plot on countand of perpendicularβ = f/bD1V.Alsonoticethatwedropthe2nd-order(zeff ) to the line-of-sight direction,H we(zeff ) can in- q1q2 − 3 chains and do not include the systematic uncertainties. The crossesH mark(z)r thes (z maximumd) FAP(ze likelihoodff )= (1 values + ze withff )DA colours(zeff )H corresponding(zeff )/c reader to Taruya, Nishimichi & Saito2 (2010) for the defini-(2) = fidthe right compares our analysis(60)(3) (cyan contours) to thepower analysis by spectrum Samushia et al. model: (2013) (grey The contours), power the analysis spectrum byChuangetal. amplitudes, 0.9

(2) (2) α = , α (55) (k) ! " biastroduce terms in the the scalingrAs(zandd) factorsB +correction, terms.rs (zd) - ! . ⊥ (61) D (q1,q2)= toS the(q1 contours.,q2) 1 In, all plots(51) we also compare to Planck+WP(2013b) (blue within contours)HΛ(CDMz)rs and(z (greend Sanchez) contours) et al. (2013) and (orange WMAP9 contours). within InΛCDM both plots (magenta we also compare to Planck+WP within ΛCDM (green If we additionally definetions of the A and B terms. Note that7 these terms are− in EH 0 2 =(1+zbeff1)σD8A((zzeff ))andH(zeff )/c.b2σ8(z ), the velocity dispersion σv and the 0.8 best fit (eTNS) fact proportional to b as physically expectedcontours). if one takes ac- and fid fid contours) and WMAP9fid within ΛCDMscale (magenta fiducial contours). cosmologyeff (flat eff 1 2 # $ H (z)rs (zd) DA(z)rs (zd) we2 can write down σ (k)ofeq.41as α! fid fid linear fiducial model 3 α⊥ = , (56) (k)/P (2) count of β =(2)f/b1.Alsonoticethatwedropthe2nd-order whereF (zα!)== (1 + z )D (z, )H (z )/cfid (55)The parameter combinationshotD noiseV (z)/rs component(zd)representstheac-N. 0.7 D (q1,q2)= S (q1,q2) 1 , (51) AP eff eff A eff effD (z)r (z ) L α⊥H(z)rs(zd) A s d LCDM, h=0.7, ΩM = 0.3) bias terms7 in the A and−B correction3 terms. 2 We actually vary b σ , b σ and N,seesection6.3. (61)tual quantity which is constrained by the BAO signal, while P 2 105 d q lin (2) (2) 1 8 2 8 fid BOSS: TestingAny Gravity use of the with parameter the power constraints spectrum from this multipoles analy- 15 σ# (k)= $ P (q)D ( q, k)S (q, k q). (52) =(1+DA(zzeff)r)DfidA((zzeff))H(zeff )/c.fid 0.6 we can write down σ2(k)ofeq.41as3 3 m amultivariateGaussianlikelihoodwith► We makeα =usewhere of Hthes (zd )andleadingcovariance, D toA (z)arethefiducialvaluesfortheand matrix(56) the symmetric estimated covariance with matrix kmax is given= 0.2leading by h/Mpc to 3 16 (2π) − − ⊥ fid sis should take into account the underlying assumption of % The parameter combinationHubbleDA (z) constantrsD(zVd()z)/rs and(zd)representstheac- angular diameter distance at z =0.57 c 2013 RAS, MNRAS 000,1–30 0.5 3 2 DV (zeff )/rs(zd) fid13.88 31.032 77.773 16.796 ! 14.877 57.455 22.825 2 105 d q Welin actuallyAs shown(2) vary b in1σ8 Chan,, b2(2)σ8 and ScoccimarroN,seesection6.3. & Sheth (2012) non-lineartual quantity whichand is constrainedr (zd)isthefiducialsoundhorizonassumedinthe by the− BAO1 signal, while −36.400 (NB:2.0636 rs from CAMB1.8398 not− Hu1 and − data Ffid (z)istheparametercombinationwhichtheAPefid s 3 2687.7 1475.9ffectour− is analysis.− We assume3 that the1267. measured7 787.74 Planck cosmol- σ3 (k)= Pm (q)D ( q, k)S (q, k q). (52) V where=H AP(z)andF (zeff )D (z)arethefiducialvaluesforthe= 0.683 C(70)kmax=0.20 = # . (72) Ckmax=0.15 = , (75) 3 gravitational evolution naturally induceskmax=0 such.20 non-local bias A 10 Ckmax=0.20 = −  1Eisenstein).0773 1.1755  −  0.4 16 (2π) − −  power spectrum template. The true wave-numbers1323k.!0 and  841.62 % Hubble constantf(zeff )σ and8(zeff angular) # diameter0.422c 2013 distance RAS, MNRAS at z 000=0,1–30.57 2 # ogy at very high redshift can be used to build the “initial terms even starting from purely local bias at an initialsensitive time. tok (Padmanabhanare then! relatedζ0 to & the Whiteζ observed1 ρχ 2009). wave-numbers1-PTE Once by suchk = ge- 1.8478 + 0.196 fid ⊥  !   0.3 As shown in Chan, Scoccimarro & Sheth (2012) non-linear -2.94±1.94 0.32±0.13 -0.72 1.34 0.99 1.3 N-body comparison Baldauf et al. (2012) shows that the 2nd-orderand rs ( biaszd)isthefiducialsoundhorizonassumedinthe is im-   #For fσ8 we included the systematic error of 3condition”.1% (see where for the no systematic(71) linear error clustering is included. amplitude Note that the on values which our ometric parametersk!/α! and arek⊥ =-2.07 constrained,k⊥±1.88/α⊥.Transferringthisintoscalingsfor 0.28±0.10# -0.70 the 3.31 relative 0.86 amplitud0.55 e gravitational evolution naturally induces such non-local bias section 7), where we assumed uncorrelated systematic er- above are based on the sound horizon, rs,calculatedfrom 0.2 portant to explain the large-scale bispectrumpower in► spectrum simulatioBOSS ns, template.results consistent The true wave-numbers with otherk! and RSD2 2 data 1.2 (z) power spectrum model, including all non-linear corrections, 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 # of the monopolethe absolute and quadrupole wavenumberrors. The diagonalk constrains= elements# k! + k⊥ ofand the inverse the growth cosine covariance8 0.5 of rate ma- CAMB (Lewis et al. 2000), while the equivalent values using terms even starting from purely local bias at an initial time. σ while the 3rd-order non-local bias termsk play⊥ are a more then impor-related to the observed wave-numbersTABLE II: Results from by fitsk to! the= RSD data. The first line k [h/Mpc] # the angle toof thetrix results line-of-sight represent is for the LRG the60 dataµ errorwe set, and can on the the secondrelate di linefferent the is for true parameters andis based. when rs calculated1.1 from Eisenstein & Hu (1998) are given in ap- Baldauf et al. (2012) showstant that role the in 2nd-order the power bias spectrum is im- (Saitok! et/α al.!►and in f prep.).( k z⊥ ) σ = 8 Ink( z⊥ )./ α ⊥ Besideconsistently.Transferringthisintoscalingsfor the three lower main than parameters& Planck above (α f(z) 0.45!, α⊥ LRGnot200 marginalising.Foreachset,wepresentthebest-fitvaluesofthe over the other parameters. For example pendix C. The likelihood for any cosmological model using portant to explain the large-scale bispectrum in simulations, observed valuesgravitational by (Ballinger, slip at redshift Peacock 0 and 1 (ζ0 & &ζ1). Heavens The uncer- 1996) the case of the local Lagrangian bias picturethe absolute inand which wavenumberf theσ8)wealsoincludefournuisanceparametersinourk = k2 + k2 and the cosine of 1 predicts ! taintiesfor⊥ the are growth at the one-standard rate we deviation find level.f(zeff The)σ fiduci8(zealff )=0.4220.4 0.027. our constraints can then be calculated as Figure 11. The power spectrum monopole (top) and quadrupole while the 3rd-order non-localinitial bias non-local terms play bias a more is neglected, impor- we can predict the ampli- value of both parameters in General Relativity is1/ 0.2 We also ± Note, thatk this constraint1 assumes that we know the geome- data m T −1 data m the anglepower to the line-of-sight spectrumµ model:we canindicate# relate The the correlation the power true coe2fficient and spectrumρ of the distribution of amplitudes, the 0.9 Planck ΛCDM & 0.35 RSD fit Growth Rate, tant role in the power spectrum (Saito et al. in prep.). In k = 1+µ 1 2 , (57) (k) exp (V V ) C (V V )/2 , (76) (bottom) measured in a set of N-body simulations (black data tude of the non-local bias as (Chan, Scoccimarro & Sheth fit to these two parameters, the minimum2 χ of the fit and c 2013 RAS, MNRAS 0006dFGS,1–30 try of the Universe exactly and neglects the large correlation L ∝ LRG − − −

α⊥ F − 6.4 EffectiveEH 0 wave-number observed values by (Ballinger, Peacockcorresponding & Heavens Probability 1996) To Exceed (PTE). ! BOSS the case of the local Lagrangian2012; Baldauf bias picture et al. in2012; which Saito the et al. in prep.) b1σ8(zeff )andb2σ8(zeff ), the' velocity! dispersion"( σv and the 0.8mWiggleZ% & best fit (eTNS) between fσ8 and FAP.Werecommenduseingthefullmul-0.3 VIPERS points) plotted relative to the fiducial Eisenstein & Hu (1998) no- −1/2 where V is a vector with model predictions for the three initial non-local bias is neglected, we can predict the ampli- tivariate# (Macaulay1µ/2 Gaussian et.2 al., for 2013,1 any PRL, cosmological inc BOSS modelDR9) constraints0 0.2 . 0.4 0.6 0.8 1 linear fiducial model 4 shot# noisek component2 1 µN=. 1+µ2 1 (58) Redshift,cosmological z (k)/P 0.7 parameters. BAO monopole power spectrum. The solid black line represents We consider the χ statistic2 for the fits, given by In our introductionL we advertised RSD as one probe which bs2 = (b1 1), k (53)= 1+µ 1 F , F(57) tude of the non-local bias as (Chan, Scoccimarro & Sheth Sunday, 23 March 14 2 − P − 7 − α⊥ F − ' 2 ! −1 "( 2012; Baldauf et al. 2012; Saito et al. in prep.) Any use' of the! parameter"( χ =( constraintsx − x¯) C (x − x¯)(2) from this analy-is able to test GR0.6 on very large scales. So what is the scale of the best fitting model. The fitting range is k =0.01 - 0.20h/Mpc. 32 with F = α!/α⊥−.Themultipolepowerspectrumincluding1/2 FIG. 1: Comparing models to recent measurements of −1 b3nl = (b1 1), sis should#(54)µ take2 into1 accountwhere x is a vector the of observed underlying values,x ¯ is a vector assumption of f(z)σ8(z). We of are plotting results8.4 for the Comparison LRG200 data set. to other measurements The error at each data point is the variation between the 20 sim- 315 the Alcock-Paczynskicorresponding e valuesffect from can a then model for bex,and writtenC is the as The openour markers measurement? are the original published values The from the information covariance, C can 4 − µ = 1+µ 2 1 (58) 0.5 ij,info 3 bs2 = (b1 1), (53) F F − covariance matrix for the data. We note that for both RSD measurements, and the filledIn markers Figure are after 15 account- we show the constraints on H(z )r /rfid ulation boxes covering a total volume of 67.5[Gpc/h] . − 7 − our analysis.' We! assume"( that2 the1 measured Plancking cosmol- for the Alcock-Paczynski effect in going from WMAP to eff s s which are in good agreement with the values measured in data sets,We the encourageχ is substantially the less use than of the our 7 degrees results forbekmax calculated= as fid (2% +1) # # Planck cosmology. The measurement error0.4 bars are at the 32 P (kof)= freedom in the fit. Wedµ calculate Pg k the,µ Probability(µ To), (59) and DA(zeff )rs /rs from different CMASS analyses as well simulations. In this work, we adopt thesewith relationsF =ogyα for!/α at sim-⊥.Themultipolepowerspectrumincluding very high redshift! 0.20h/ canMpc,2 bebut2 we used also toprovide build the! results the1standarddeviationuncertaintylevel.Thedashedredline “initial using kmax = b3nl = (b1 1), (54) Exceed2 (PTE)α α this χ , under the assumptionL that the ! 2 ⊥ ! −1 illustrates the expected growth rateas from theΛCDM Planck withd ln Planck predictionP (k ) within ΛCDM.d Ourln analysisP (k us-) 315 − the Alcock-Paczynski effect can then beuncertainties0.15 writtenh/Mpc. are asThe indeed% maximum correctly estimated. likelihood The very values for the fitting−1 0.3 ! i −1 ! j plicity, while we float b1, b2 and N as free .Theimpact parameters, with the 1 and 2 standard deviation uncertainty condition” for the linearlow PTE clustering values suggest that amplitude either the) uncertainties* on have# which# our Cij,infoing= the fitting range k =0Cij,.01Hartlap - 0.20h/Mpc is included, where we userange the extendedk =0.01 - TNS 0.15h/ modelMpc are for P (k ,µillustrated). The with the shaded bands. The solid blue line and (62) which are in good agreementof the with 2nd-order the values bias measured terms on the in power spectrum is some- 1 been over estimated, or genuine scatter in the measure-g as the0.2 cyan! contours.df σ8 Anderson et al. (2013b)df updatedσ8 the (2% +1) # # corresponding blue shaded regions illustrates!! the0.02 best fit t0.04o 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 what small. Figure 10 (left) shows the power spectrumpowerP!(k multi-)= spectrumAP effdµ model,ect P constrainsg mentsk ,µ including is being the!( systematicallyµ) parameter, all(59) suppressed. non-linear combination While only ad-correctionFAPthe(z RSD)= datas, with the gravitational slip model. We note simulations. In this work, we adopt these relations for sim- 2 ditional observations will determineDV (z whethereff )/rs this(zd) trend is 13.83 CMASS-DR9! analysis published in Anderson et al. (2013), for the rest of this paper. We calculate the monopole and 2α⊥α! −1 dataL that almost all the measurements include our best fit model k [h/Mpc] poles when all higher2 order bias terms are set to zero (dash- (1 + z)DA(z)trulyH(z statistically)/c,whiletheBAOfeatureconstrainsthe significant, the results already in hand plicity, while we float b1, b2 and N as free .Theimpact is based. % Vkmax=0.15 = F (zeff ) = 0at.684 the 1 standard(73) deviation uncertaintywhere level, only which the is BAO re- information is exploited, while the )appear to* suggest that either 2 the quoted uncertainties  flected inwhere the low χ2 in TableP! II.is The the one standard extended deviation TNS model power spectrum we quadrupole power spectrum for these simulations and per- dot green line). The solid magenta line uses b2 =0.5and combination DV (z)/rs(zd#) # DAf((z)/H)σ ((zz).) Together0.420 these RSD signal and broadband shape is marginalised out. of the 2nd-order bias terms on the power spectrum is some- where we use the extended TNS modelhave for been overestimated,Pg(k ,µ∝). or The the analysiseff 8 ise suppressingff range of the model (the darker blue band) is narrower than −1 b =2.0. We can see that the higher order bias terms mainly two signals allow us to break the degeneracy between theDA typical(zintroduced) one standard deviationThe in uncertainty BAO section on constraint any of the 6.1 can and be improvedC substantiallyis the by us- covariance form a fit using our power spectrum model. When using the what small. Figure 10 (left) shows1 the power spectrum multi- AP effect constrains the parameter combinationgenuine scatter in theFAP measurements.(z)=    Figure 11. The power spectrumij,Hartlap monopole (top) and quadrupole andWe note the that symmetric the PTE decreases covariance with the matrix LRG200 data is givenmeasurements by because the fit has been calculated from the affect the monopole and while the effect is small, it is sig- and H(z). We will include the scaling parameters α! andseveralα independent⊥ measurements.ing density field reconstruction. We compare our results poles when all higher order bias terms are set to zero (dash- (1 + z)DA(z)H(z)/c,whiletheBAOfeatureconstrainstheset, since the LRG200 measurements have a larger scatter matrix we(bottom) derived measured in section in 4. a We set ofnow N-body can calculate simulations the (black datafitting range k =0.01 - 0.20h/Mpc, the best fitting value 84.732 5.7656 3.0985 with Anderson et al. (2013b) in Figure 15 (left), before re- nificant when compared to the measurementcombination errors6.4D (grey E(z)ff/rective(zin) ourD wave-number model2 (z)/Hthan parametrisation,(z the). LRG Together60 measurements. which these This will is likely be due discussed to the in the dot green line). The solid magenta line uses b2 =0.5and V s d A fact that3 most of the coherent clustering signal− is due − effective wave-numberpoints)construction plotted (grey as contours) relative and to after the reconstruction fiducial Eisenstein (blue & Hu (1998)of no-f(z )σ8(z )deviatesfromthefiducialvalueofthesim- next∝ section. 10 Ckmax=0.15 = −1 2.2777 1.9755 (74) eff eff shaded area). to correlations on scales less than 100 h Mpc, so the with a range of models for the RSD effect, and found b1 =2.0. We can see that the higher order bias terms mainly two signals allow us to break the degeneracy between DA(z) 2.9532 contours). Samushia et al. (2013) (grey contours on the We should also mention that the stochasticIn our term, introduction weadditional advertised correlations are RSD effectively as adding one noise to probe the that which measurements of Ωm (whichBAO is directly monopole sensitive to power spectrum. The solid black line representulations by 3.1%, while we cannot find any significant de- affect the monopole and while the effect is small, it is sig- and H(z). We will include the scaling parameterssignal. α! andα⊥ the growth rate) were highly dependent on the model 1 −1 N,caningeneraldependonscale(Dekel&Lahav In most recent results, the uncertainties have been es- used. In particular, the modelthe of a HALOFITbestke fittingff [27]=P (k) model. ThekiC fittingkj range. is k =0.01(63) - 0.20h/Mpc. nificant when compared to the measurement errors (grey in our modelis able parametrisation, to test6.3 GR which Model on very willc parameterisation2013 be large discussed RAS, scales. MNRAS in the000 So,1–30 what is thewith scale a linear of model for the redshift space distortion re- ij,info viation for α! and α⊥.Acomparisonbetweenthemodel 1998; Baldauf et al. 2012), while we treat it as a con- timated! from several hundred simulated realisations of −1 A shaded area). next section.our measurement? Thethe survey,information from which the uncertainty covariance, (and the covari-C covered acan lower Ωm comparedThe to the error fiducial valueat each on" datai,j point is the variation between the 20and sim- the measured power spectrum in these N-body simu- stant and free parameter. The final ingredients in our We parametrizeance ourbetween model measurements, using in the the casescaling of several parameters red- ij,whichinfoα the! simulation was based. ! 3 We should also mention that the stochastic term, shift bins) can be deduced from the scatter in the re- The preference for a lowerulation growth rate or boxesσ8 appears covering a total volume of 67.5[Gpc/h] . model of eq. 39 are the correction terms, Abeand calculatedB, and asα⊥ introduced in the last section. Using these parame- −1 lations can be seen in Figure 11. Using kmax =0.15h/Mpc N,caningeneraldependonscale(Dekel&Lahav alisations. Although it may appear that the uncertain- to agreeHere with recent the results from normalisation [20], studying Sunyaev- A is given by A = C . which originate from the higher-order correlation between ters we can deriveties on the measurements have been overestimated, good Zeldovich (SZ) cluster counts, who find σ8 =0.77 ± 0.02 ij ij,info 6.3 Model parameterisation ± 1998; Baldauf et al. 2012), while we treat it as a con- agreement between the quoted values and Fisher! fore- and Ωm =0.29 0.02. Collectively, these results may we find deviations of 0.1%, 0.1% and 0.7% for α!, α⊥ Kaiser terms and velocity fields in mapping to redshift−1 d ln P!(ki) −1 d ln P! (kj ) 1 Using kmax =0.20h/Mpc we get keff =0.178h/Mpc, casts [e.g., 30] of the minimum intrinsic statistical uncer- 3be suggesting that ΛCDM does not fully model simulta- − − − C = 2 C fid 2 czeff , stant and free parameter. The final ingredients in our We parametrize ourij,info model using the scalingtainties suggests parametersij, thatHartlap this is notα! the case, although [21] neously(62) the Cosmic Microwave Background and the Uni- # and f(z )σ8(z ), respectively. We include these values in space (Taruya, Nishimichi & Saito 2010). We refer the α⊥α! (1 + zeff )DA (zeff ) Hfid(z ) eff eff DV (!zeff ) dfnoteσ that8 the uncertainties in the BOSS growthdf σ rate8 mea-eff verse at whichz<1. However, can future work be will require related detailed to a real-space scale by s = model of eq. 39 are the correction terms, A and B, and α⊥ introduced in the last!! section.= Using these parame- (60) reader to Taruya, Nishimichi & Saito (2010) for the defini- surements are around 40% largerfid than the Fisher matrix work with simulated catalogues for a range of cosmolog- r!s(zd) + , rs (zd) - . for the rest of this paper. We calculate the monopoleTable and 1 and Figure 12. which originate from the higher-ordertions of the A correlationand B terms. between Note thatters these we terms can derive are in predictions. ical models1.15 [e.g.,π 11,/k 12]e andff an improved20 understanding.3Mpc/h (Reid & White 2011). The ef- Perhaps the stage of an RSD analysis most likely to of the relationship between the observed≈ galaxies, the pe- Kaiser terms and velocity fields in mapping2 to redshift where P! isand the extended TNS model1 power spectrum we quadrupole power spectrum for these simulations and per- Several authors have recently performed similar stud- fact proportional to b1 as physically expected if one takes ac- introduce2 a systematic3 shift, and artificially reduce the culiar velocityfective field, and wave-numberthe underlying dark matter [e.g., of our measurement using kmax = 2 fid czeff −1 space (Taruya, Nishimichi & Saito 2010). We refer the α⊥α! (1 + zeff )DA (zescatter,ff ) α may!fid be in fitting a modelfid to the two-dimensionalfid 22, 26], before we can more robustly use RSD measure- count of β = f/b1.Alsonoticethatwedropthe2nd-orderDV (zeffintroduced) in section 6.1H and(zeffC) is the covariance form a fit using our power spectrum model. When usingies the to what we have done here (Nishimichi & Taruya 2011; = FAP(zefftwo-point)= correlation(1 + ze functionffij,)DHartlap(60)A (or(z powereff )H spectrum).(zeff ) [17]/c ments to0 study.15 departuresh/Mpc from isΛCDM.keff =0.132h/Mpc. reader to Taruya, Nishimichibias & terms Saito in (2010) the A forand theB defini-correction terms. + fid analysedα⊥ simulated catalogues. for the WiggleZ survey We(61) thank the two anonymous referees for useful com- rs(zd)matrix we, derivedrs ( inzd) section- 4. We now can calculate the fitting range k =0.01 - 0.20h/Mpc, the best fitting valuede la Torre & Guzzo 2012; Ishikawa et al. 2013; Oka et al. tions of the A and B terms. Note that these terms are in =(1+zeff )DA(zeff )H(zeff )/c. 2 and effective wave-number as 2013). They studied the systematic uncertainty against ha- fact proportional to b1 as physically expected if one takes ac- of f(zeff )σ8(zeff )deviatesfromthefiducialvalueofthesim- α! The parameter combination DV (z)/rs(zd)representstheac- count of β = f/b1.Alsonoticethatwedropthe2nd-order fid fid los (or sub-halos) in N-body simulations using the TNS 2 FAP(zeff )= (1 + zeff )D (zeff )H (zeff )/c ulation by 3.1%, while we cannot find any significant de- We actually vary b1σ8, b2σ8 and N,seesection6.3. tual quantityA which1 is constrained by the BAO signal, while bias terms in the A and B correction terms. α⊥ −(61)1 7 TESTING FOR SYSTEMATIC keff = kiCij,infokj . (63) viation for α! and α⊥.Acomparisonbetweenthemodelmodel. Although some of these studies ignore the Alcock- =(1+zeff )DA(zeff )H(zeff )/c.A " i,j c 2013 RAS, MNRAS 000,1–30 UNCERTAINTIES AND DETERMINING Paczynski effect, which is degenerate with fσ and use a ! ! and the measured power spectrum in these N-body simu- 8 The parameter combination DV (z)/rs(zd)representstheac- THE MAXIMUM WAVENUMBER, kmax phenomenological treatment of the galaxy/halo bias, they 2 −1 lations can be seen in Figure 11. Using kmax =0.15h/Mpc We actually vary b1σ8, b2σ8 and N,seesection6.3. tual quantityHere which the is normalisationconstrained by the BAOA is signal, given while by A = ij Cij,info. The questionwe find of the deviations maximum of 0 wavenumber,.1%, 0.1% andk 0up.7% to for α!,reachα⊥ very similar conclusions. Using kmax =0.20h/Mpc we get keff =0.178h/Mpc, − − max− c 2013 RAS, MNRAS 000,1–30 # and f(z )σ (z ), respectively. We include these values in which can be! related to a real-space scale by swhich= we can trusteff our8 powereff spectrum model, is directly Table 1 and Figure 12. 1.15π/keff 20.3Mpc/h (Reid & White 2011). Thelinked ef- to the question of possible systematic uncertainties. ≈ Several authors have recently performed similar stud-7.2 Uncertainties from perturbation theory fective wave-number of our measurement using kmaxWe= would like to make use of as much data as possible, but there are significanties to what power we have spectrum done here modelling (Nishimichi issues & given Taruya 2011; 0.15h/Mpc is keff =0.132h/Mpc. Because we want to make use of the power spectrum be- de la Torre & Guzzo 2012; Ishikawa et al. 2013; Oka et al. the small error bars of our measurement. yond k =0.10h/Mpc we cannot rely on standard perturba- 2013). They studied the systematic uncertainty against ha- tion theory (SPT) which seems to break down at low red- los (or sub-halos) in N-body simulations using the TNS shift for k>0.10h/Mpc, where the 2-loop term turns out 7 TESTING FOR SYSTEMATIC 7.1 Testmodel. with Although N-body simulation some of these studies ignore the Alcock- to be larger than the 1-loop term (Crocce & Scoccimarro UNCERTAINTIES AND DETERMINING Paczynski effect, which is degenerate with fσ and use a To test whether our power spectrum model can extract8 2006; Taruya et al. 2009; Carlson, White & Padmanabhan THE MAXIMUM WAVENUMBER, kmax phenomenological treatment of the galaxy/halo bias, they the correct cosmological parameters from a power spec- 2009). We therefore use re-normalised perturbation theory reach very similar conclusions. The question of the maximum wavenumber, kmax uptrum to measurement, we use a set of 20 N-body simulations to calculate Pδδ, Pθθ and Pδθ (Taruya & Hiramatsu 2008; which we can trust our power spectrum model, is directlydescribed in White et al. (2011) that were generated us- Taruya et al. 2009; Taruya, Nishimichi & Bernardeau 2013) linked to the question of possible systematic uncertaintieings. a TreePM code. The simulations cover a total volume and include corrections up to 2-loop order. We make use of We would like to make use of as much data as possible,of but 67.5[Gpc7.2/h]3 Uncertainties.Note,thatweusetheseN-bodysimula- from perturbation theory the publicly available RegPT code (Taruya et al. 2012). there are significant power spectrum modelling issues giventions onlyBecause for this sub-section we want to and make use use the ofQPM the simulations power spectrum be- The authors of this code suggested a phenomenological the small error bars of our measurement. yond k =0.10h/Mpc we cannot rely on standard perturba- c 2013 RAS, MNRAS 000,1–30 ! tion theory (SPT) which seems to break down at low red- shift for k>0.10h/Mpc, where the 2-loop term turns out 7.1 Test with N-body simulation to be larger than the 1-loop term (Crocce & Scoccimarro To test whether our power spectrum model can extract 2006; Taruya et al. 2009; Carlson, White & Padmanabhan the correct cosmological parameters from a power spec- 2009). We therefore use re-normalised perturbation theory trum measurement, we use a set of 20 N-body simulations to calculate Pδδ, Pθθ and Pδθ (Taruya & Hiramatsu 2008; described in White et al. (2011) that were generated us- Taruya et al. 2009; Taruya, Nishimichi & Bernardeau 2013) ing a TreePM code. The simulations cover a total volume and include corrections up to 2-loop order. We make use of of 67.5[Gpc/h]3.Note,thatweusetheseN-bodysimula- the publicly available RegPT code (Taruya et al. 2012). tions only for this sub-section and use the QPM simulations The authors of this code suggested a phenomenological

c 2013 RAS, MNRAS 000,1–30 ! The Problem

Fixed b Marg b

► Planck (light blue), WMAP9 + high l (red), SZ (dark blue), RSD (dark green), lensing (3D galaxy, light green), combined LSS (grey)

► LSS probes have Planck ns and θMC priors ► Note discrepancy also with WMAP9 and LSS - not a Planck issue

Sunday, 23 March 14 Evidence for massive neutrinos from CMB and lensing observations

Richard A. Battye⇤ Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, U.K.

Adam Moss† Centre for Astronomy & Particle Theory, University of Nottingham, University Park, Nottingham, NG7 2RD, U.K.

We discuss whether massive neutrinos (either active or sterile) can reconcile some of the tensions within cosmological data that have been brought into focus by the recently released Planck data. We point out that a discrepancy is present when comparing the primary CMB and lensing measurements both from the CMB and galaxy lensing data using CFHTLenS, similar to that which arises when 4 comparing CMB measurements and SZ cluster counts. A consistent picture emerges and including a prior for the cluster constraints and BAOs we find that: for an active neutrino model with 3 degenerate neutrinos, m⌫ =(0.320 0.081)eV, whereas for a sterile neutrino – in addition to ± e↵ We also note that there are published limits on the 3 neutrinos with a standard hierarchy and m⌫ =0.06 eV – m⌫, sterile =(0.450 0.124) eV and P ± Ne↵ =0.45 0.23. In both cases there is a significant detection of modification to them neutrino⌫ that are contrary to the arguments presented ± P sector from the standard model and in the case of the sterile neutrino it is possible tohere reconcile [23 the, 24]. These are based on the shape of the power BAO and local H0 measurements. spectrumP of LSS as opposed to its amplitude. We believe

Massive neutrinos are now part of the standard mod- AS, and spectral index, nS.that The optical these depth constraints to the could easily be ignored if there els of particle physics and cosmology. Solar and atmo- epoch of reionization is ⌧. were significant scale dependent bias in the galaxy pop- spheric neutrino experiments have measured two di↵er- In addition, the results of Planck have highlighted a ences between the masses squared and from this it can possible discrepancy betweenulations the cosmological detected parame- by the redshift surveys. For example, be inferred thatPossible the sum of the active Solutions neutrino masses, ters preferred by CMB data andit BAOs, has been and those show which [25] that di↵ering amounts of red and m⌫ , must be at least 0.06 eV [1]. This is the quantity come from fitting the counts ofblue galaxy galaxies clusters selected in surveys can make it dicult to use the that► canMassive be constrained neutrinos - 3 byactive cosmological neutrinos, involved observations. in weak In interactionusing the Sunyaev Zeldovich (SZ) e↵ect [10]. This is best P addition,➡ Assume some for experiments cosmology degenerate suggest that hierarchy there could be quantified in terms of derivedshape parameters to⌦m determine= ⌦b + ⌦c cosmological parameters. While we a sterile neutrino that does not interact with the stan- and , which are the total matter density relative to ► Sterile neutrinos - assume 1 massive active neutrino with m = 0.06 eV8 acknowledge the existence of these limits, our opinion is dard model [2], but in the context of cosmology still con- critical and the amplitude of fluctuations on 8h 1 Mpc ➡ Parameterize sterile neutrino by effective number of additional tributes a mass, me↵ , and in a model dependent way scales, respectively. Using athat bias between they are the much hydro- less reliable than the arguments that degrees of freedom⌫, sterile and effective mass an increase in the number of e↵ective relativistic0.4 degrees static mass and the true mass of 20% (1 b =0.8 e we have put forward. of freedom, Ne↵ =3.046 + Ne↵ . m, sterile [eV] in the parlance of [10]) the SZ cluster counts require (⌦ /0.27)0.3 =0.78 0.01, which is lower than pre- Using► Modified observations gravity - need of theory the angular which suppresses power spectrum growth of 8 m ± of temperaturestructure (less anisotropies common) in the Cosmic Microwave ferred by CMB data. A similar discrepancy can be in- ferred from other measurements of cluster number counts Background► Modified initial (CMB) power from spectrum the Planck - might satellitenot be possible [3], po- given larisationprimary measurements CMB from the Wilkinson Microwave using the SZ [11, 12], X-rays [13] and optical richness [14]. This could be due to a number of incorrect assumptions Anisotropy► Optical depth Probe to (WMAP) reionization [4 ]assumptions and observations of Bary- onic Acoustic Oscillations (BAOs) [5–8], a constraint of in calculation of the cluster number counts which are in ► Are the constraints from all three LSS probes wrong? common between the di↵erent analyses, for example, the m⌫ < 0.248 eV (95% Confidence Level - CL) has been Note added: As we were preparing to submit this achieved in the case of active neutrinos [9], whereas in relationship between the observable and the true mass P e↵ or mass function. However,paper, it could also a preprint be as a re- appeared on the arXiv [26] closely re- the sterile case Ne↵ < 3.80 and m⌫, sterile < 0.42 eV (95%Sunday, 23 March CL) 14 for the case of a thermal sterile neutrino sult of additional physics thatlated is missing to from this the work. stan- Despite the di↵erent treatment of dard 6 parameter model and in [10] it was suggested with mass < 10 eV. This analysis, which will be re- CFHTLens data, the results are in excellent agreement. ferred to as Planck CMB+WP+BAO below, was per- that massive active neutrinos could lead to an improved arXiv:1308.5870v1 [astro-ph.CO] 27 Aug 2013 formed by adding m in the active case, or me↵ fit, with m⌫ =(0.22 0.09)We eV had from only an analysis just of submitted our paper to the Planck ed- ⌫ ⌫, sterile CMB+SZ+BAO. Although± not explicitly discussed there and Ne↵ in the sterile case to the standard 6 parame- P itorial board for review, which took 8 days, explaining ter, p = ⌦ h2, ⌦ hP2, ✓ ,A ,n , ⌧ , ⇤CDM model. ⌦ a similar e↵ect could be achieved from the inclusion of { b c MC S S } b sterile neutrinos. the delay in this work appearing. and ⌦c are the baryonic and cold dark matter densities relative to the critical density. The Hubble constant is In this letter we will make the case that this expla- 1 1 nation of the discrepancy between the CMB and cluster 100h km sec Mpc , which is a derived parameter; the counts is also favoured by lensing data. This data comes parameter used in the fit is the acoustic scale, ✓MC.The primordial power spectrum is described by an amplitude, from CMB lensing as detected by Planck [15] and the South Pole Telescope (SPT) [16], and also from galaxy lensing detected by the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS) [17]. A careful reading of

⇤Electronic address: [email protected] [15] and [17] might already suggestAcknowledgements: this: the increase This research was supported by †Electronic address: [email protected] in the limit m⌫ for active neutrinosSTFC. from We CMB thank lens- Jim Zibin for useful discussions. We P acknowledge the use of the CAMB [27] and COSMOMC [28] codes, and the use of Planck data. The develop- ment of Planck has been supported by: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN, JA and RES (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). A de- scription of the Planck Collaboration and a list of its FIG. 3: Marginalized likelihoods for the sterile neutrino mass members, including the technical or scientific activi- and the extra e↵ective degrees of freedom (top and middle ties in which they have been involved, can be found panels, labelling as in Fig. 2), together with the 2D joint at http://www.sciops.esa.int/index.php?project= likelihood (bottom panel). planck&page=Planck_Collaboration

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We have seen that in perfect fluids, sound waves can propagate at the sound speed on scales smaller than the sound horizon. Sound waves cannot prop- agate in a collisionless fluid, but the individual particles free-stream with a characteristic velocity –for neutrinos, in average, the thermal velocity vth.So, it is possible to define an horizon as the typical distance on which particles travel between time t and t.DuringMDandRDandfort t ,thishorizon i ! i is, as usual, asymptotically equal to vth/H,uptoanumericalfactoroforder one (see section 4.4.2). Exactly as we defined the Jeans length, we can define the free-streaming length by taking Eq. (48) and replacing cs by vth

2 1/2 4πGρ¯(t)a (t) a(t) 2 vth(t) kFS(t)= 2 , λFS(t)=2π =2π . (93) ! vth(t) " kFS(t) #3 H(t)

As long as neutrinos are relativistic, they travel at the speed of light and their free-streaming length is simply equal to the Hubble radius. When they become non-relativistic, their thermal velocity decays like

0 p 3Tν 3Tν a0 1eV −1 vth # $ = 150(1 + z) km s , (94) ≡ m % m m $ a % % $ m %

0 1/3 0 where we used for the present neutrino temperature Tν (4/11) Tγ and 0 % Tγ 2.726 K. This gives for the free-streaming wavelength and wavenumber during% matter or Λ domination

6000 no ν’s 4 1+z 1eV −1 fν=0 10 λFS(t)=7.7 h Mpc , (95) 3 5000 fν=0.1 ΩΛ + Ωm(1 + z) $ m % 2 3 & K)

3 µ ΩΛ + Ωm(1 + z) 4000 m ( −1 3 k (t)=0.82 h Mpc ,π (96) 10 FS & 2 / 2 (1 + z) $1eV% l 3000

where Ω (resp. Ω )isthecosmologicalconstant(resp.matter)densityfrac-Massive Neutrinos P(k) (Mpc/h) Λ m l(l+1) C 2000 tion evaluated today. So, after the non-relativistic transition and during mat- 102 no ν’s f =0 ter domination, the free-streaming length continues to inc1.21000rease, but only like ν −1 1/3 2/3 f =0.1 ►(aHEffect) t on,i.e.moreslowlythanthescalefactor massive neutrinos a t .Therefore,theknr ν ∝ ∝ 0 2 −1 −1/3 comoving free-streaming length λFS/a actually decreases like1 2 (a H200) 400t 600. 800 1000Cluster1200 scales1400 10-3 10-2 10-1 on matter power spectrum ∝ ∑mν = 0.046 eV As a consequence, for neutrinos becoming non-relativistic duringCMB scales matter dom- l k (h/Mpc) ination, the comoving free-streaming wavenumber passes 0.8 through a minimum fν = 0.01 =0 ν

► f k Powerat the time suppressed of the transition, on i.e. when scalesm = p =3T and a /a =(1+z)= T nr Fig.ν 14.0 CMB temperature anisotropy spectrum Cl and matter power spectrum 3 # $ 0.6

2.0 10 (m/1eV).Thisminimumvalueisfoundtobe / P(k)

k > kNL ν × Pf (k)forthreemodels:theneutrinolessΛCDM model of section 4.4.6, a more re-

1/2 P(k) 1/2 m −1 alistic 0.4 ΛCDM model with three massless neutrinos (fν 0), and finally a ΛMDM knr 0.018 Ω h Mpc . (97) m ∑mν = 0.46 eV ! % $1eV% model with three massive degenerate neutrinos and a total density fraction f =0.1. 0.2 ν fν = 0.1 In all models, the values of (ωb, ωm, ΩΛ, As, n, τ)havebeenkeptfixed. 41 0 10-4 10-3 10-2 10-1 1 Lesgourgues and Pastor k (h/Mpc)

Fig. 13. Ratio of the matter power spectrum including three degenerate massive 6000 fν=0 ► Relative shift on CMB smallerneutrinos with density fraction f to that with three massless neutrinos. The pa-4 ν fν=0.1, fixed (ωc, ΩΛ) 10 rameters (ω , Ω )=(0.147, 0.70) are kept fixed,f =0.1, andfixed from( , h) top to bottom the curves but important! m 5000Λ ν ωc

correspond to2 fν =0.01, 0.02, 0.03,...,0.10. The individual masses mν range from 3 K) −3 −1 −3 −1 0.046 eV to 0.46µ 4000 eV, and the scale knr from 2.1 10 h Mpc to 6.7 10 h Mpc

► ( × × −2 −1 Show fixed CDM and baryonas shown on theπ top of the figure. k is approximately equal to 1.5 10 h Mpc 3. eq × 10 / 2 density - change equality l 3000

Looking now at all wavenumbers, we plot in Fig. 12 the ratio of theP(k) (Mpc/h) matter epoch, neutrino fluctuations l(l+1) C 2000 2 power spectrum for ΛMDM over that of ΛCDM, for different valuesof10fν , fν=0 but for fixed parameters1000 (ωm, ΩΛ). Here again, the ΛMDM model has three fν=0.1, fixed (ωc, ΩΛ) degenerate massive neutrinos. As expected from the analytical results, this fν=0.1, fixed (ωc, h) ratio is a step-like0 function, equal to one for k

4.6.1 Effects on CMB and LSS power spectra for fixed (ωm, ΩΛ) and degen- erate masses models, the values of (ωb, ωm, ΩΛ, As, n, τ)havebeenkeptfixed,withthe T In Fig. 14,increase we show Cl inandωPν (kbeing)fortwomodels: compensatedΛCDM with byfν a=0and decrease in ωcdm.Thereisaclear ΛMDM withdiffNerenceν =3massiveneutrinosandatotaldensityfraction between the neutrinoless andfν massless=0.1. neutrino cases, caused by a We also display for comparison the neutrinoless model of Sec.4.4.6.Inall large change in the time of equality and by the role of the neutrino energy-

momentum fluctuations60 in the perturbed Einstein equation [91]. However our purpose is to focus on the impact of the mass, i.e. on the difference between the solid (red) and thick dashed (green) curves in Fig. 14.

Impact on the CMB temperature spectrum. For fν 0.1, the three neutrino species are still relativistic at the time of decoupling,≤ and the di- rect effect of free-streaming neutrinos on the evolution of the baryon-photon

61 Massive Neutrinos 3

Active neutrinos Sterile neutrinos Parameter I II III I II III 2 ⌦bh 0.02218 0.00025 0.02231 0.00024 0.02234 0.00024 0.02244 0.00029 0.02256 0.00028 0.02258 0.00027 2 ± ± ± ± ± ± ⌦ch 0.1184 0.0018 0.1162 0.0013 0.1152 0.0013 0.1244 0.0051 0.1221 0.0041 0.1206 0.0040 ± ± ± ± ± ± 100✓MC 1.04151 0.00056 1.04163 0.00056 1.04170 0.00056 1.04086 0.00072 1.04106 0.00065 1.04117 0.00065 ± ± ± ± ± ± ⌧R 0.092 0.013 0.093 0.013 0.096 0.014 0.096 0.014 0.099 0.014 0.097 0.014 ± ± ± ± ± ± nS 0.9643 0.0059 0.9685 0.0052 0.9701 0.0056 0.9775 0.0106 0.9792 0.0106 0.9772 0.0104 10 ± ± ± ± ± ± log(10 AS) 3.091 0.025 3.088 0.024 3.091 0.026 3.115 0.030 3.116 0.031 3.109 0.030 ± ± ± ± ± ± m⌫ [eV] < 0.254 < 0.358 0.320 0.081 - - - ± me↵ [eV] - - - < 0.479 0.326 0.143 0.450 0.124 P⌫, sterile ± ± Ne↵ - - - < 0.98 < 0.96 0.45 0.23 ± H0 67.65 0.90 67.80 1.08 67.00 1.07 69.69 1.68 69.51 1.41 69.02 1.21 ± ± ± ± ± ± ⌦m 0.310 0.12 0.306 0.13 0.314 0.13 0.308 0.12 0.308 0.12 0.312 0.12 ± ± ± ± ± ± 8 0.818 0.023 0.789 0.020 0.757 0.014 0.813 0.032 0.779 0.020 0.756 0.012 ± ± ± ± ± ± 2ln CMB 9804.96 9808.41 9811.35 9804.69 9809.15 9809.09 L 2ln BAO 1.38 3.09 1.29 1.62 1.61 1.99 L ? ? 2ln Lensing 1009.56 -1030.12 -1030.05 1018.68 -1031.76 -1031.43 L ? ? ? ? 2ln SZ 92.49 5.61 2.19 59.62 5.74 0.37 L 2ln 9806.34 8781.37 8784.78 9806.31 8779.00 8780.02 L

TABLE I: Summary of parameter constraints for both the active and sterile neutrino analyses discussed in the text. Likelihoods denoted(2D galaxy by ? arelensing, not included results use in the fixed total bias) likelihood for that particular dataset.(From'Ba8ye'&'Moss,'PRL'2014)'

Sunday, 23 March 14 MCMC'results'of Hubble’s constant from Cepheid variables in nearby galaxies which appears to be at odds with the values of I':'Planck'+WP'+'BAO''II':'Planck'+WP'+'BAO'+'Lensinginferred''III':'Planck'+'WP'+'BAO''+' by CMB analyses [21]. ThisLensing is illustrated'+'SZ3 by'' preferred values of H0 in these models presented in ta- Active neutrinos ble I being significantlySterile larger neutrinos than in the active neutrino model without leading to an increased 2ln .In- Parameter I II III I II LIIIBAO 2 cluding the prior h =0.738 0.024 from [22]to(III)mod- ⌦bh 0.02218 0.00025 0.02231 0.00024 0.02234 0.00024 0.02244 0.00029 0.02256± 0.00028 0.02258 0.00027 2 ± ± ± ifies the constraints± to m⌫±=(0.246 0.077)± eV in the ⌦ch 0.1184 0.0018 0.1162 0.0013 0.1152 0.0013 0.1244 0.0051 0.1221 e↵0.0041 ±0.1206 0.0040 ± ± ± active neutrino± model and m±⌫, sterile =(0.425 0±.122) eV, P ± 100✓MC 1.04151 0.00056 1.04163 0.00056 1.04170 0.000561N.04086e↵ =0.5920.000720.2751.04106 in the sterile0.00065 neutrino1.04117 model,0.00065 with ± ± ± 2 ± ± ± ± ⌧R 0.092 0.013 0.093 0.013 0.096 0.014 0.=6096.30 between.014 the0.099 two.0 The.014 larger0. change097 0. in014 the ± ± ± mean value± in the active case± arises from the± degener- nS 0.9643 0.0059 0.9685 0.0052 0.9701 0.0056 0.9775 0.0106 0.9792 0.0106 0.9772 0.0104 10 ± ± ± acy between± H0, ⌦m and ±m⌫ , with increased± H0 cor- log(10 AS) 3.091 0.025 3.088 0.024 3.091 0.026 3.115 0.030 3.116 0.031 3.109 0.030 ± ± ± responding± to lower ⌦m and± m⌫ . ± m⌫ [eV] < 0.254 < 0.358 0.320 0.081 - P - - ± It is also instructive to perform the same analysis with e↵ FIG. 2: Marginalized likelihoods for m⌫ . The datasets are P m⌫, sterile [eV] - - - < 0.479 0.326 0.143 0.450 0.124 P colour coded in the legend, but the solid line is for (I), the WMAP 9-year plus high-` data± in the place of Planck± for Ne↵ - -P - (III) (see< 0 [9.98] for details of< the0.96 high-` analysis).0.45 We0.23 find dashed line is for (II) and the dotted line is for (III). It is clear ± that inclusion of lensing leads to a preference for m⌫ > 0 m⌫ =(0.297 0.084) eV in the active neutrino model H0 67.65 0.90 67.80 1.08 67.00 1.07 69e.↵69 1.68± 69.51 1.41 69.02 1.21 which is compatible± with that coming± from the SZ cluster± and m⌫, sterile± =(0.367 0.156)± eV, Ne↵ =0.276± 0.203 ⌦m 0.310 0.12 0.306 0.13 0.314 0.13 0.308 0.12 ±0.308 0.12 0.312 ±0.12 counts and that± there is a strong preference± ( 4P)inthe± inP the sterile± neutrino model.± This increases our± confi- 8 0.818 0.023 0.789 0.020 ⇡ 0.757 0.014 0.813 0.032 0.779 0.020 0.756 0.012 case of dataset (III).± ± ± dence that±Planck results are± consistent with WMAP,± but 2ln CMB 9804.96 9808.41 9811.35 at higher9804.69 significance. 9809.15 9809.09 L 2ln BAO 1.38 3.09 1.29 The main1.62 argument that1.61 we have presented1.99 in this L sults are present in? Fig. 3. For (II) we find that there paper is that amplitude? of Large-Scale Structure (LSS) 2ln Lensing,Planck 1009.56 e↵ -1030.12 e↵-1030.05 1018.68 -1031.76 -1031.43 L is a 2.3 preference ? for m⌫, sterile > 0 with m⌫, sterile = when normalized? to the amplitude of CMB fluctuations 2ln Lensing(0,SPT.326 0.143)1009 eV.56 although there-1030.12 is only an upper-1030.05 bound 1018.68 -1031.76 -1031.43 L ? ? are in excess? of that inferred? by lensing and cluster 2ln SZ ± 92.49 5.61 e↵ 2.19 59.62 5.74 0.37 L of Ne↵ < 0.96. This is strengthened to m⌫, sterile = counts, and indeed that these two measures of the am- 2ln (0.450 0.124)9806.34 eV and Ne↵ =08781.37.45 0.23 for (III).8784.78 plitude9806.31 of the power spectrum8779.00 are consistent.8780.02 If we add L ± ± The sterile neutrino model has the added feature that massive neutrinos – either active or sterile – to the cos- it can beNB'WMAP'+ made compatiblehighL with'+'BAO'+'SZ'+' the direct measurementLensingmological':'0.297'+/J'0.084''J'3.5 model then we get significant'' detections that TABLE I: Summary of parameter constraints for both the active and sterile neutrino analyses discussed in theσ text. Likelihoods denoted by ? are not included in the total likelihood for that particular dataset.

ifies the constraints to m⌫ =(0.246 0.077) eV in the active neutrino model and me↵ =(0±.425 0.122) eV, P ⌫, sterile ± Ne↵ =0.592 0.275 in the sterile neutrino model, with 2 =6.3 between± the two. The main argument that we have presented in this paper is that amplitude of Large-Scale Structure (LSS) when normalized to the amplitude of CMB fluctuations are in excess of that inferred by lensing and cluster counts, and indeed that these two measures of the am- plitude of the power spectrum are consistent. If we add massive neutrinos – either active or sterile – to the cos- mological model then we get significant detections that are due to the decrease in small- relative to large-scale power in such models. These measures of the amplitude of LSS are not without their modelling diculties, but FIG. 2: Marginalized likelihoods for m . The datasets are ⌫ the fact that they appear to agree is encouraging. There colour coded in the legend, but the solid line is for (I), the dashed line is for (II) and the dotted lineP is for (III). It is clear are, however, caveats to what we have said. that inclusion of lensing leads to a preference for m⌫ > 0 Firstly, we note that the improved global fit when in- which is compatible with that coming from the SZ cluster cluding massive neutrinos is usually at the expense of an counts and that there is a strong preference ( 4P)inthe ⇡ increase in 2ln CMB. This increase is 6.3 for the case of dataset (III). best-fitting modelL in the active neutrino case⇡ and 4.4 for sterile neutrinos. This is outweighed by the signifi-⇡ cant reductions in 2ln Lensing and 2ln SZ (see ta- galaxies which appears to be at odds with the values of ble I), but is reflected byL the fact that preferredL values inferred by CMB analyses [21]. This is illustrated by in the case of detections overlap somewhat the 95% CL preferred values of H0 in these models presented in ta- limits in the case of (I). It could be that there exists a ble I being significantly larger than in the active neutrino variant of, for example, the sterile neutrino model that model without leading to an increased 2ln BAO.In- leads to a better fit to the CMB data while preserving cluding the prior h =0.738 0.024 from [22]to(III)mod-L the positive impact on the amplitude of LSS. ± Massive Neutrinos

► “It’s Planck”, “The cluster result is wrong” common arguments but aren’t good explanations

WMAP+high `+RSD+Lensing 1.0 Planck+WP+RSD+Lensing

0.8

max 0.6 P / P 0.4

0.2

0.0 0.00 0.15 0.30 0.45 0.60 0.75 ⌃m⌫ [eV] (From'Ba8ye,'Charnock'&'Moss,'to'appear)'

Sunday, 23 March 14

*'

PRELIMINARY'!'

*' *' Massive Neutrinos

Combined result ~ 4.5 σ

(From'Ba8ye,'Charnock'&'Moss,'to'appear)'

Sunday, 23 March 14

*'

PRELIMINARY'!'

*' *' Implications ImplicaAons'for'the'mass'hierarchy'?'

m3' m3' m3' m ' m2' 2 m1'

m2'

m ' m ' 0' 1 1 Normal' Inverted' Degenerate'' m1 m2 m3 m1 m2 m3 m m m ! ! ! ∼ 1 ∼ 2 ∼ 3

mν ! ► 2 2 If ∑∆mmν =sol 0.3=(8 eV. 9meV)degenerate hierarchy favoured, if ∑mν = 0.2 eV inverted hierarchy favoured (72.6,'72.0,'55)' 2 2 +' ∆matm =(47meV)

Sunday, 23 March 14 CaveatsCaveats' • (From'Ba8ye'&'Moss,'PRL'2014)' Tension'between'CMB'&'LSS'Caveats'Caveats' ► Internal tension between CMB and LSS for massive neutrino model • Tension'between'CMB'&'LSS'• Tension'between'CMB'&'LSS'(From'Ba8ye'&'Moss,'PRL'2014)'(From'Ba8ye'&'Moss,'PRL'2014)' • Shape'of'the'ma8er' (2.5σ) '•'power'spectrum' Shape'of'the'ma8er' ► Shape of the• matter power spectrum ''power'spectrum'Shape'of'the'ma8er' ''power'spectrum'

WiggleZ (Riemer-Sorenson et. al.) Using'Planck'+'BAO'+'WiggleZ' (RiemerJSorenson,'Parkinson'&'Davis)' Using'Planck'+'BAO'+'WiggleZ' ► Are other(Riemer consistentJSorenson,'Parkinson'&'Davis)' LSS probes discussed moreUsing'Planck'+'BAO'+' robust thanWiggleZ the' shape of P(k)?(Riemer JSorenson,'Parkinson'&'Davis)'

WiggleZ (Riemer-Sorenson et. al.)

Sunday, 23 March 14 4

We also note that there are published limits on the m⌫ that are contrary to the arguments presented here [23, 24]. These are based on the shape of the power spectrumP of LSS as opposed to its amplitude. We believe that these constraints could easily be ignored if there were significant scale dependent bias in the galaxy pop- ulations detected by the redshift surveys. For example, it has been show [25] that di↵ering amounts of red and blue galaxies in surveys can make it dicult to use the shape to determine cosmological parameters. While we Sterile Neutrinos acknowledge the existence of these limits, our opinion is that they are much less reliable than the arguments that 4 we have put forward.

Planck+WP+BAO+Lensing+SZ We also note thatPlanck there+WP+BAO+Lensing+SZ are published limits on the 1.0 Planck+WP+BAO+Lensing 1.0 Planck+WP+BAO+Lensing m⌫ that are contrary to the arguments presented Planck+WP+BAO Planck+WP+BAO here [23, 24]. These are based on the shape of the power 0.8 0.8spectrumP of LSS as opposed to its amplitude. We believe that these constraints could easily be ignored if thereNote added: As we were preparing to submit this paper, a preprint appeared on the arXiv [26] closely re-

max 0.6 max 0.6 were significant scale dependent bias in the galaxy pop- P P / lated to this work. Despite the di↵erent treatment of / P

P ulations detected by the redshift surveys. For example, 0.4 CFHTLens data, the results are in excellent agreement. 0.4 it has been show [25] that di↵ering amounts of redWe and had only just submitted our paper to the Planck ed- blue galaxies in surveys can make it dicult to useitorial the board for review, which took 8 days, explaining 0.2 0.2 shape to determine cosmological parameters. Whilethe delay we in this work appearing. acknowledge the existence of these limits, our opinion is 0.0 0.0that they are much less reliable than the arguments that 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.25 0.50 0.75 1.00 1.25 e we have put forward.Ne m, sterile [eV]

► Acknowledgements: This research was supported by Additional degrees of freedom can help alleviate local H0 tension STFC. We thank Jim Zibin for useful discussions. We (see also Wyman et. al. PRL 2014) acknowledge the use of the CAMB [27] and COSMOMC [28] Note added: As we were preparing to submitcodes, this and the use of Planck data. The develop- ► Planck + WP + BAO + Lensing + SZ: H0paper, = 69.0 a preprint ± 1.2 appeared km/s/Mpc on the arXiv [26] closelyment re- of Planck has been supported by: ESA; CNES lated to this work. Despite the di↵erent treatmentand CNRS/INSU-IN2P3-INP of (France); ASI, CNR, and (Planck CMB result H0 = 67.4 ± 1.4 km/s/Mpc) CFHTLens data, the results are in excellent agreement.INAF (Italy); NASA and DoE (USA); STFC and We had only just submitted our paper to the PlanckUKSAed- (UK); CSIC, MICINN, JA and RES (Spain); itorial board(From'Ba8ye'&'Moss,'PRL'2014)' for review, which took 8 days, explainingTekes, AoF and CSC (Finland); DLR and MPG the delay in this work appearing. (Germany); CSA (Canada); DTU Space (Denmark); Sunday, 23 March 14 SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); MCMC'results' FCT/MCTES (Portugal); and PRACE (EU). A de- I':'Planck'+WP'+'BAO''II':'Planck'+WP'+'BAO'+'Lensing''III':'Planck'+'WP'+'BAO''+'Lensing'+'SZ3 '' scription of the Planck Collaboration and a list of its FIG. 3: Marginalized likelihoods for the sterile neutrino mass members, including the technical or scientific activi- Active neutrinos and the extra e↵ective degreesSterile of freedom neutrinos (top and middle ties in which they have been involved, can be found panels, labelling as in Fig. 2), together with the 2D joint Parameter I II III Acknowledgements:I ThisII research wasIII supportedat http://www.sciops.esa.int/index.php?project= by 2 likelihood (bottom panel). ⌦bh 0.02218 0.00025 0.02231 0.00024 0.02234 0.00024 0.02244 0.00029 0.02256 0.00028 0.02258 0.00027 planck&page=Planck_Collaboration 2 ± ± ± STFC. We± thank Jim Zibin± for useful± discussions. We ⌦ch 0.1184 0.0018 0.1162 0.0013 0.1152 0.0013 0.1244 0.0051 0.1221 0.0041 0.1206 0.0040 ± ± ± acknowledge± the use of the± CAMB [27] and± COSMOMC [28] 100✓MC 1.04151 0.00056 1.04163 0.00056 1.04170 0.00056 1.04086 0.00072 1.04106 0.00065 1.04117 0.00065 ± ± ± codes, and± the use of ±Planck data.± The develop- ⌧R 0.092 0.013 0.093 0.013 0.096 0.014 0.096 0.014 0.099 0.014 0.097 0.014 ± ± ± ± ± ± nS 0.9643 0.0059 0.9685 0.0052 0.9701 0.0056ment0 of.9775Planck0.0106has0.9792 been0. supported0106 0.9772 by:0.0104 ESA; CNES 10 ± ± ± ± ± ± log(10 AS) 3.091 0.025 3.088 0.024 3[1].091M.0 C..026and Gonzalez-Garcia, CNRS/INSU-IN2P3-INP3.115 0.030 M.3.116 Maltoni,0.031 (France); J. Salvado,3.109 ASI,0. and030 CNR, andArXiv e-prints (2012), arXiv:1212.5225. ± ± ± ± ± ± m⌫ [eV] < 0.254 < 0.358 0.320T.0 Schwetz,.081INAF Journal(Italy);- of NASA High Energy and- DoE Physics (USA);12- ,123 STFC[5] andF. Beutler, C. Blake, M. Colless, D. H. Jones, et al., ± e↵ (2012), arXiv:1209.3023. MNRAS 416,3017(2011),arXiv:1106.3366. m⌫, sterile [eV] - - - UKSA (UK);< 0.479 CSIC,0.326 MICINN,0.143 JA0.450 and0.124 RES (Spain); P [2] J. M. Conrad, W. C. Louis, and± M. H. Shaevitz,± ArXiv [6] C. Blake, S. Brough, M. Colless, C. Contreras, et al., Ne↵ - - - Tekes, AoF< 0.98 and CSC< 0. (Finland);96 0.45 DLR0.23 and MPG e-prints (2013), arXiv:1306.6494. ± MNRAS 425,405(2012),arXiv:1204.3674. H0 67.65 0.90 67.80 1.08 67.00 1.(Germany);07 69.69 1 CSA.68 (Canada);69.51 1.41 DTU69.02 Space1.21 (Denmark); ± ± [3] Planck± Collaboration,± P. A. R.± Ade, N. Aghanim,± [7] N. Padmanabhan, X. Xu, D. J. Eisenstein, R. Scalzo, ⌦m 0.310 0.12 0.306 0.13 0.314 0.SER/SSO13 0.308 (Switzerland);0.12 0.308 RCN0.12 (Norway);0.312 0. SFI12 (Ireland); ± ± C.± Armitage-Caplan,± et al., ArXiv± e-prints (2013),± et al., MNRAS 427,2132(2012),arXiv:1202.0090. 8 0.818 0.023 0.789 0.020 0.757 0.014FCT/MCTES0.813 0.032 (Portugal);0.779 0.020 and PRACE0.756 0.012 (EU). A de- ± ± arXiv:1303.5062± . ± ± ± [8] L. Anderson, E. Aubourg, S. Bailey, D. Bizyaev, et al., 2ln CMB 9804.96 9808.41 9811.35scription9804.69 of the Planck9809.15Collaboration9809.09 and a list of its L [4] C. L. Bennett, D. Larson, J. L. Weiland, N. Jarosik, et al., MNRAS 427,3435(2012),arXiv:1203.6594. FIG. 3: Marginalized2ln BAO likelihoods1.38 for the sterile3.09 neutrino mass1.29 members,1.62 including the1.61 technical or1.99 scientific activi- L ? ? and the extra2ln Lensing e↵ective,Planck degrees1009 of.56 freedom-1030.12 (top and middle-1030.05ties in which1018.68 they have-1031.76 been involved,-1031.43 can be found L ? ? panels, labelling2ln Lensing as,SPT in Fig. 21009),. together56 with-1030.12 the 2D joint-1030.05at http://www.sciops.esa.int/index.php?project=1018.68 -1031.76 -1031.43 L ? ? ? ? likelihood (bottom2ln SZ panel). 92.49 5.61 2.19 59.62 5.74 0.37 L planck&page=Planck_Collaboration 2ln 9806.34 8781.37 8784.78 9806.31 8779.00 8780.02 L NB'WMAP'+highL'+'BAO'+'SZ'+'Lensing':'0.297'+/J'0.084''J'3.5 '' TABLE I: Summary of parameter constraints for both the active and sterile neutrino analyses discussed in theσ text. Likelihoods denoted by ? are not included in the total likelihood for that particular dataset. [1] M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado, and ArXiv e-prints (2012), arXiv:1212.5225. T. Schwetz, Journal of High Energy Physics 12,123 [5] F. Beutler, C. Blake, M. Colless, D. H. Jones, et al., ifies the constraints to m⌫ =(0.246 0.077) eV in the e↵ ± (2012), arXiv:1209.3023. active neutrinoMNRAS model416 and,3017(2011),m⌫, sterile =(0.425arXiv:1106.33660.122) eV, . [2] J. M. Conrad, W. C. Louis, and M. H. Shaevitz, ArXivN [6]=0.C.592 Blake,0.275 inS.P the Brough, sterile neutrino M. Colless,± model, C. with Contreras, et al., e↵ ± e-prints (2013), arXiv:1306.6494. 2 =6.3MNRAS between the425 two.,405(2012),arXiv:1204.3674. [3] Planck Collaboration, P. A. R. Ade, N. Aghanim, The[7] mainN. argument Padmanabhan, that we X. have Xu, presented D. J. Eisenstein, in this R. Scalzo, C. Armitage-Caplan, et al., ArXiv e-prints (2013),paper is thatet al., amplitude MNRAS of Large-Scale427,2132(2012), StructurearXiv:1202.0090 (LSS) . arXiv:1303.5062. when normalized[8] L. Anderson, to the amplitude E. Aubourg, of CMB S. Bailey,fluctuations D. Bizyaev, et al., [4] C. L. Bennett, D. Larson, J. L. Weiland, N. Jarosik, et al.,are in excessMNRAS of that427 inferred,3435(2012), by lensingarXiv:1203.6594 and cluster . counts, and indeed that these two measures of the am- plitude of the power spectrum are consistent. If we add massive neutrinos – either active or sterile – to the cos- mological model then we get significant detections that are due to the decrease in small- relative to large-scale power in such models. These measures of the amplitude of LSS are not without their modelling diculties, but FIG. 2: Marginalized likelihoods for m . The datasets are ⌫ the fact that they appear to agree is encouraging. There colour coded in the legend, but the solid line is for (I), the dashed line is for (II) and the dotted lineP is for (III). It is clear are, however, caveats to what we have said. that inclusion of lensing leads to a preference for m⌫ > 0 Firstly, we note that the improved global fit when in- which is compatible with that coming from the SZ cluster cluding massive neutrinos is usually at the expense of an counts and that there is a strong preference ( 4P)inthe ⇡ increase in 2ln CMB. This increase is 6.3 for the case of dataset (III). best-fitting modelL in the active neutrino case⇡ and 4.4 for sterile neutrinos. This is outweighed by the signifi-⇡ cant reductions in 2ln Lensing and 2ln SZ (see ta- galaxies which appears to be at odds with the values of ble I), but is reflected byL the fact that preferredL values inferred by CMB analyses [21]. This is illustrated by in the case of detections overlap somewhat the 95% CL preferred values of H0 in these models presented in ta- limits in the case of (I). It could be that there exists a ble I being significantly larger than in the active neutrino variant of, for example, the sterile neutrino model that model without leading to an increased 2ln BAO.In- leads to a better fit to the CMB data while preserving cluding the prior h =0.738 0.024 from [22]to(III)mod-L the positive impact on the amplitude of LSS. ± Sterile Neutrinos

e↵ ► Combined LSS constraint m ⌫ , sterile > 0 at ~4.5σ

Planck+WP+RSD+WL+SZ(1-b=0.8)

1.0 Planck+WP+RSD+WL+SZ(1-b=[0.7,1])

0.8

max 0.6 P / P 0.4

0.2

0.0 0.00 0.25 0.50 0.75 1.00 1.25 e↵ m⌫, sterile [eV] (From'Ba8ye,'Charnock'&'Moss,'to'appear)'

Sunday, 23 March 14

*'

PRELIMINARY'!'

*' *' Reionization!

► Results assume WMAP TE polarization. ► Low l polarization is typically hard (foregounds etc.)

► Instead let LSS constrain the optical depth. Implies zreion ~ 7 (assumed instantaneous)

Planck+WP+RSD+WL+SZ(1-b=0.8) 1.0 Planck+WP+RSD+WL+SZ(1-b=[0.7,1]) ► Find τ = 0.039 ± 0.018 (fixed bias), τ = 0.049 ± 0.8 0.021 (marginalized bias)

max 0.6 P / P 0.4

0.2

0.0 0.000 0.025 0.050 0.075 0.100 0.125 0.150 ⌧

Sunday, 23 March 14 Advertisment

► Are results from BICEP really a smoking gun of inflation? Paper out Tuesday by AM and Levon Pogosian 2

0 sourced by vector modes and has two peaks. The less 10 Inflation prominent peak at ` 10 is due to rescattering of pho- Inflation + strings tons during reionization,⇠ while the main peak, at higher

1 Strings `, is the contribution from last scattering. Both peaks 2 BICEP2 are quite broad because a string network seeds fluctua- K 10 µ

PolarBear tions over a wide range of scales at any given time. The

2 position of the main peak is determined by the most dom- /

2 inant Fourier mode stimulated at last scattering, which BB C

10 is set by the values of ⇠ and v [50]. The power tends to move to lower multipoles (larger angular scales) when + 1) `

( either v or ⇠ are increased. Increasing v also increases ` 3

the width of the peak. In fact, because v<1 sets a

10 maximum scale, it takes a large increase in ⇠ to move the peak to the right (to lower `) even by a small amount. Let us briefly comment on how we quantity the string 1 2 3 10 10 10 contribution to CMB. Bounds on cosmic string are often ` Sunday, 23 March 14 quoted solely in terms of Gµ. Such bounds implicitly assume the scaling configuration in the Abelian Higgs FIG. 1. The thick blue long dashed line is the best fit lens- ing+strings model (r=0), with the thin blue long dashed line model, where at any time there is roughly one Hubble showing the corresponding string contribution alone. The length string per Hubble volume. More generally, the thick red short dash is the best fit lensing+strings+inflation bound on strings depends on the combination of Gµ and 2 model (r=0.16), with the corresponding string contribution the string number density Ns ⇠ . In the Abelian plotted as a thin red short dashed line. The lensing contri- Higgs model, ⇠ (1), but can/ be much smaller in bution is shown separately with a thin black dot-dashed line. models with lower⇠ intercommutingO probabilities. More- The BICEP2 best fit inflationary model (r=0.2) contribution over, di↵erent types of observations probe di↵erent com- is shown with a thin black dotted line, and the solid thin black binations of ⇠ and µ. As shown in [42], CMB power line is the sum of r=0.2 and lensing contributions. The circles spectra (and other two-point correlation functions) con- show the band powers measured by BICEP2 and the triangles strain µpNs µ/⇠, while gravity wave probes essentially are the POLARBEAR data (the third band is negative with ⇠ 2 its absolute value plotted as an inverted triangle). constrain the string energy density given by µ/⇠ .To avoid the model-dependence when interpreting the CMB bounds in terms of Gµ, we follow the Planck Collabo- the CMB spectra from di↵erent types of strings [36]by ration [67] and quantify the amount of the anisotropy adjusting its parameters. The USM model was intro- contributed by strings in terms of f10, which is the frac- duced in [29, 63], based on the approach suggested in tional contribution of strings to the CMB temperature spectrum at ` = 10, f Cstr/Ctot. The first year [62], developed into its present form in [64], and imple- 10 ⌘ 10 10 mented in a publicly available code CMBACT [65]. The Planck data constrains it at f10 . 0.03 [67]. In this work string unequal time correlators of the USM model can we do not fit to Planck data, instead focusing on the im- be derived using analytical expressions developed in [66], plications of the B-mode data alone. Values of f10 that which we use in this work. exceed Planck bounds can be disregarded. We also ignore In the USM, in addition to the dimensionless string the small theoretical uncertainty involved in calculating tension Gµ, there are two important parameters – the the lensing contribution to the B-mode spectrum. scaling parameter ⇠,whichsetsthee↵ective inter-string We first discuss how the string only model compares to distance1, and the root-mean-square (RMS) velocity v. inflation. The blue solid lines in Fig. 2 show the marginal- On cosmological scales, probed by the CMB measure- ized likelihoods of f10, ⇠ and v obtained by fitting the ments, the fine details of the string evolution do not play string B-mode spectra, combined with lensing, to the BI- a major role. It is the large-scale properties, such as the CEP2 and POLARBEAR data. There is a well-defined scaling distance and the rms velocity, that determine the peak with f10 =0.036 0.008, with preference for larger ⇠ ± shape of the string-induced spectra. These e↵ects are values. The overall 2 is only slightly worse (2 =2.65) accounted for in the USM. The overall normalization of compared to inflation, although the string model has 2 the spectrum depends on Gµ as well as the string number additional parameters. The corresponding string contri- density controlled by ⇠. bution to the B-mode spectrum is shown with a thin The thin red short dashed line in Fig. 1 shows a typical blue long dash line in Fig. 1, and strings+lensing with B-mode spectrum generated by strings. It is primarily a thick blue long dash line. Not surprisingly, the data, which has a bump at ` 100, favours a spectrum with a peak at a lower `, which⇠ is at ` 250 for the best fit model. A model with such large values⇠ of ⇠ corresponds 1 ⇠ `/⌘,where` is the mean comoving inter-string distance and to rare and heavy strings – the implied value of Gµ in ⌘ ⌘is the conformal time 6 this model is 10 LP: Check and correct!,buttheir 3

global phase transitions, of the kind discussed in [73, 74], would provide a much better fit than local strings. We now consider the model in which both strings and inflation generate B-modes. The red dotted lines in Fig. 2 show the marginalized likelihoods of r and the string pa- rameters in a model with an additional inflationary ten- sor mode contribution. In this case, the fit is improved relative toAdvertisment the model with no strings, with 2 = 6.06 and 3 additional parameters. The marginalized string fraction is f =0.025 0.014 with slight preference ► Are results from BICEP10 really a smoking± gun of inflation? Paper out Tuesday forby lowerAM and values Levon of ⇠ Pogosian, characteristic for local strings, and r =0.14 0.05. ±

0.32 BICEP2+PolarBear

0.24 r 0.16

FIG. 2. Marginalized likelihoods derived from the BICEP2 and POLARBEAR data for the scalar-to-tensor ratio r,the 0.08 strength of the string contribution f10, the inter-string dis- tance ⇠,andtheRMSvelocityv. The red dotted lines are for the lensing+strings+inflation model, while the blue solid 0.00 lines are for the lensing+strings fit only. 0.000 0.015 0.030 0.045 0.060 f10

Sunday, 23 March 14 be disregarded. We also ignore the small theoretical un- FIG. 3. The marginalized joint likelihood for the tensor-to- scalar ratio r and the strength of the string contribution f . certainty involved in calculating the lensing contribution 10 The two di↵erent shades indicate the 68% and the 95% confi- to the B-mode spectrum. dence regions. The vertical dashed line indicates the approx- We first discuss how the string only model compares to imate bound on f10 from Planck. inflation. The blue solid lines in Fig. 2 show the marginal- ized likelihoods of f10, ⇠ and v obtained by fitting the Fig. 3 shows the marginalized joint likelihood for r and string B-mode spectra, combined with lensing, to the BI- the strength of the string contribution f10. It clearly CEP2 and POLARBEAR data. There is a well-defined shows that a combination of the two contributions fits peak at f10 =0.036 0.008, with preference for larger ⇠ ±2 2 the data better than when either of them is zero. The values. The overall is only slightly worse ( =2.65) thin vertical line indicates the approximate upper bound compared to inflation, although the string model has 2 3 on f10 from Planck . It should be noted that the im- additional parameters. The corresponding string contri- provement in the fit comes primarily from data points at bution to the B-mode spectrum is shown with a thin blue higher `, while the BICEP2 collaboration warns [24] that long dash line in Fig. 1, and strings+lensing with a thick points at ` > 150 should be considered as preliminary. blue long dash line. Not surprisingly, the data, which has Our findings carry implications for models that predict a bump at ` 100, favours a spectrum with a peak at defects. The inability of local strings to fit the B-mode a lower `, which⇠ is at ` 250 for the best fit model. A ⇠ spectrum on their own, poses a problem for brane in- model with such large values of ⇠ corresponds to rare and flation models in which inflation ends with a production heavy strings – the implied value of Gµ in this model is 6 of cosmic superstrings [41–43]. Such models predict tiny 5 10 , but their number density is low, which allows it ⇥ values of r, and the only observable B-modes could come to remain consistent with Planck bounds. The peak po- from strings, which are e↵ectively of local type. The fact sition for this model is closer to that of global strings and that local strings do not fit the BICEP2 data puts these textures [51, 55], and certainly not representative of local strings [36]. This is also clear from the likelihood plot for ⇠,whiche↵ectively rules out models with ⇠ < 1.8(2) as the only primordial source of B-modes. For reference, the 3 The Planck Collaboration did not scan over all values of ⇠ and B-mode spectra from the local string simulation of [49] v,insteaditprovidedtwoseparateboundsonf10 corresponding correspond to the USM with ⇠ 0.4[36]. We can fore- to two di↵erent string models. We quote the weaker of the two see that models with global strings,⇡ textures [51, 55] or bounds. Conclusions

► There is a tension between CMB and LSS ➡ Based on published results ➡ Interpretation issues ► Not reliant on SZ - lensing and RSD combined are >3σ discrepant ► Not reliant on Planck - WMAP is also discrepant ► Massive neutrinos provide a explanation - “detection” at ~4.5σ

► Sterile neutrinos also at ~4.5σ, can alleviate tension with local H0 ► A couple of caveats ► If WMAP low l polarization is wrong, can also solve these problems

Sunday, 23 March 14