Constraints on neutrino mass from Cosmic Microwave Background and Large Scale Structure
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 3 August 2018 (MN L A TEX style file v2.2)
Constraints on neutrino mass from Cosmic MicrowaveBackground and Large Scale Structure
Z. Pan, (cid:63) L. Knox † Department of Physics, University of California, One Shields Avenue, Davis, CA, USA 95616
ABSTRACT
Our tightest upper limit on the sum of neutrino mass eigenvalues M ν comes fromcosmological observations that will improve substantially in the near future, enablinga detection. The combination of the Baryon Acoustic Oscillation feature measuredfrom the Dark Energy Spectroscopic Instrument and a Stage-IV Cosmic MicrowaveBackground experiment has been forecasted to achieve σ ( M ν ) < / M ν from atmospheric and solar neutrino oscillations (Abazajian et al. 2013;Fogli et al. 2012). Here we examine in detail the physical effects of neutrino mass oncosmological observables that make these constraints possible. We also consider howthese constraints would be improved to ensure at least a 5 σ detection. Key words: cosmology – cosmology:cosmic microwave background – cosmology:observations – large-scale structure of universe
Basic questions about the neutrino mass matrix remainunanswered, such as whether the CP-violating phase is non-zero, whether the neutrinos are Majorana or Dirac, andwhether the hierarchy of masses is normal or inverted (Fogliet al. 2012; Forero, T´ortola & Valle 2012; Barger, Marfatia& Whisnant 2002). Significant experimental and observa-tional efforts are underway and being planned to answerthese questions. Doing so may shed light on possible exten-sions beyond the standard model of particle physics.The question of the type of mass hierarchy may be set-tled by cosmological observations. The best lower limit onthe sum of the mass eigenstate masses, M ν , comes fromanalysis of solar and atmospheric neutrino oscillation data(Forero, T´ortola & Valle 2014). If cosmological determina-tions of this quantity tighten up near this lower bound, thenthe inverted hierarchy will be ruled out.Current data lead to 0 .
058 eV (cid:46) M ν (cid:46) .
21 eV, wherethe upper limit comes from cosmic microwave background(CMB) and baryon acoustic oscillation (BAO) data (PlanckCollaboration XIII 2015). Bringing this upper limit down isa major science goal of a Stage-IV (S4) cosmic microwavebackground project, CMB-S4, and also of the galaxy sur-vey project Dark Energy Spectroscopic Instrument (DESI).These projects are forecasted to determine M ν with a one-standard deviation of 45 meV (CMB-S4 alone) and 16 meV(CMB-S4 combined with DESI BAO) (Abazajian et al. (cid:63) Email: [email protected] † Email: [email protected] M ν (cid:54) = 0 at a 3 σ or greater confidence level.Of course any conclusions from such cosmological datawill be model dependent. How convincing will these databe that we are indeed seeing the impact of neutrino mass,and not misinterpreting some other signals? The forecastedprecision is also not quite as strong as one would like; is therea way to guarantee at least a 5 σ detection of M ν (cid:54) = 0? Herewe address these questions. To address the first questionwe examine the particular signatures of neutrino mass thatlead to the above forecasts. To address the second we lookat what additional types of data can further tighten theexpected uncertainties.The paper is organized as follows. In Section 2, webriefly introduce the cosmological signatures of massive neu-trinos. In Section 3, we focus on changes in the cosmic expan-sion rate and structure growth rate due to massive neutrinos.In Section 4, we analyze the influence of massive neutrinoson the CMB lensing potential power spectrum. Forecasts onthe constraints of total neutrino mass from CMB and LargeScale Structure (LSS) measurements are given in Section 5and conclusions are presented in Section 6. The cosmological signatures of massive neutrino have beeninvestigated since decades ago, e.g. (Doroshkevich et al.1980; Doroshkevich & Khlopov 1981; Doroshkevich et al.1981). We can more broadly view the cosmological neutrino c (cid:13) a r X i v : . [ a s t r o - ph . C O ] S e p Z. Pan and L. Knox program as a study of the dark radiation that we know existsas a thermal relic of the big bang. By dark radiation here wemean anything, other than photons, thermally produced inthe early universe that is relativistic at least through decou-pling. We know that such a background of nearly masslessnon-photon radiation exists with high confidence from lightelement abundances and the cosmic microwave backgrounddamping tail. Both are sensitive to the history of the expan-sion rate, which depends on the mean density via the Fried-man equation. Combining Helium abundance and CMB dataconstrains the effective number of relativistic species to be N eff = 2 . ± .
39 (Planck Collaboration XIII 2015).Is this background entirely that of the 3 active neu-trino species? Is any part of it from something else? Couldthere be a significant excess of neutrinos over anti-neutrinos?These are interesting questions, also to be addressed by fu-ture CMB observations that will significantly tighten up con-straints on N eff . Our confidence that the dark radiation isindeed that of cosmological neutrinos with phase-space dis-tributions as expected from the standard thermal historywill be greatly increased if the constraints on N eff tightenup to σ ( N eff ) = 0 .
02, as forecasted, consistent with the ex-pected value of 3.046. For the purposes of this paper, we willassume this is what will happen.If we assume that the dark radiation background is en-tirely the active neutrinos with the expected phase space dis-tributions, the assumption of non-zero neutrino mass leadsto very specific predictions for cosmological observables.First we consider the expansion rate as a function of red-shift. The rest-mass energy of the neutrinos begins to slowdown the decline of energy density with expansion as theybecome non-relativistic, leading to an increase in H ( z ) rel-ative to the M ν = 0 expectation. This increase would per-sist to z = 0 if we were holding the other contents of thelow-redshift universe constant. However, for our purposes ofexploring observable consequences of M ν (cid:54) = 0 it makes muchmore sense to hold the angular size of the sound horizon onthe last-scattering surface constant, since this quantity is sowell-determined from CMB observations (Planck Collabo-ration XIII 2015; Hinshaw et al. 2013). To do so one mustdecrease the density of dark energy. Assuming the dark en-ergy is a cosmological constant, the shape of ∆ H ( z ) has avery particular form, as shown in Fig. 1, changing sign verynear z = 1, with the onset of dark energy domination.Were we able to trace out this departure of H ( z ) fromthe M ν = 0 shape, it would contribute to our confidencewe are seeing the impact of non-zero neutrino mass. How-ever, as we will see, the DESI determinations of H ( z ) willbe insufficient to resolve this very small signal across red-shift. That is not to say the signal is altogether observablyinvisible. These changes to H ( z ) affect comoving angular di-ameter distance D A ( z ) in ways that are detectable by DESI.It is just that it will be difficult, if not impossible, to makethe case that there is the sign change in the H ( z ) correctionnear z (cid:39)
1. The changes to H ( z ) also directly impact thegrowth of structure, with observable consequences for theredshift-space distortion (RSD) and the CMB lensing po-tential power spectrum, which we will discuss in Section 3and 4 respectively. H ( z ) r s ( H ( z ) r s ) f i d DESI z D A ( z ) / r s ( D A ( z ) / r s ) f i d M ν =50 meV M ν =100 meV M ν =200 meV Figure 1.
The dependence of expansion rate H ( z ) and comovingangular diameter distance D A ( z ) on M ν , where we minimize the χ (Θ , M ν ) by adjust the 6 ΛCDM parameters Θ when increasing M ν from 0 to 50 , ,
200 meV.
To quantify the influence of massive neutrinos, we comparea fiducial cosmology with massless neutrinos and a cosmol-ogy with massive neutrinos. The fiducial cosmology is aflat ΛCDM universe with the
Planck best fit parameters(Planck Collaboration XVI 2014), i.e. ω b = 0 . , ω m =0 . , A s = 2 . × − , n s = 0 . , τ = 0 . , H =67 .
04 km / s / Mpc , M ν = 0 meV. The set of M ν (cid:54) = 0 cos-mologies have parameters θ = (Θ , M ν ), where Θ are ΛCDMparameters. Given a specific M ν , we choose Θ by minimizing χ (Θ , M ν ), with χ (Θ , M ν ) = F αβ λ α λ β = F ij λ i λ j + 2 F iν λ i M ν + F νν M ν , (1)where F is the Fisher matrix for the CMB observations, λ i = (Θ − Θ fid ) i with i indexing the 6 ΛCDM parametersand summation over repeated indexes α, β, i, j is implied.Minimizing χ (Θ , M ν ) requires0 = ∂χ /∂λ i → λ i = − ( G − ) ij F jν M ν , (2)where G is a subset of the Fisher matrix F , G ij ≡ F ij .In Fig. 1, we show the influence of M ν = 50 ,
100 and 200meV on expansion rate H ( z ) and comoving angular diameterdistance D A ( z ). We see that H ( z (cid:46)
1) decreases and H ( z (cid:38)
1) increases compared to the fiducial cosmology, and thecomving distance D A ( z ) increases accordingly. Though thedeparture of H ( z ) from the M ν = 0 shape is undetectableby DESI BAO, the changes in D A ( z ) are readily detectable(Font-Ribera et al. 2014).One of the observable consequences of these changes to H ( z ) is the impact on the structure growth rate. How onedescribes this impact depends on what one is using for acomparison model. We use, as a comparison model, a cos-mology with massless neutrinos in place of the massive ones.One could also use as a comparison model one with addi-tional cold dark matter in place of the neutrinos. We usethe former, consistent with our underlying assumption that c (cid:13) , 000–000 onstraints on neutrino mass from Cosmic Microwave Background and Large Scale Structure z f ( a ) σ ( a ) [ f ( a ) σ ( a ) ] f i d M ν =20 meV σ f σ ( a ) /f σ ( a ) =0 . Figure 2.
The dependence of structure growth rate on M ν , wherewe minimize the χ (Θ , M ν ) by adjust the 6 ΛCDM parametersΘ when increasing M ν from 0 to 20 meV. we have 3 neutrino species with phase-space distributions asexpected from the standard thermal history.The structure growth rate that is usually quantified by d ln σ ( a ) /d ln a , can be determined by RSD from galaxy sur-veys, where σ ( a ) is the amplitude of mass fluctuations σ R on scale of 8 h − Mpc, i.e., σ ( a ) ≡ σ ( R = 8 h − Mpc; a ).Here σ R ( a ) ≡ (cid:90) ∞ k π P δ ( k ; a ) W ( kR ) d ln k, (3)where P δ ( k, a ) is the matter power spectrum defined by, (cid:104) δ ( k ; a ) δ (cid:63) ( k (cid:48) ; a ) (cid:105) = P δ ( k ; a ) δ (3) ( k − k (cid:48) ), and W ( kR ) =3 j ( kR ) /kR is the window function. Introducing the growthfunction D ( a ), D ( a ) ≡ δ ( a ) δ ( a = 1) = σ ( a ) σ ( a = 1) , (4)where δ ( a ) is the matter overdensity at redshift z =1 /a −
1, and defining perturbation growth rate f ( a ) ≡ d ln D ( a ) /d ln a (Pierpaoli, Scott & White 2001), we mayrewrite the structure growth rate as d ln σ ( a ) /d ln a = f ( a ) σ ( a ).In the above definition, the growth function D ( a ) andthe structure growth rate f ( a ) σ ( a ) depend on σ ( a = 1)which varies for our fiducial M ν = 0 model and the modelwith massive neutrinos. For easier comparison, we introducethe growth function D e ( a ) normalized at early time, say at a e = 1 / D e ( a ) ≡ δ ( a ) δ ( a e ) = σ ( a ) σ ( a e ) , (5)and the structure growth rate can be rewritten as f ( a ) σ ( a ) = dD e ( a ) d ln a σ ( a e ) , (6)where σ ( a e ) is the same for the model with massive neu-trinos and the fiducial model. So the variation of f ( a ) σ ( a )only depends on the growth rate dD e ( a ) /da .In Fig. 2, we show the impact of massive neutrinos withmass 20 meV on the structure growth rate. The structuregrowth rate is decreased at high redshift because of the en-hanced expansion rate (Fig. 1). Note though that the tran- sition of structure growth rate from suppressed to enhancedis delayed to z (cid:39) . z (cid:39) dD e ( a ) /da is determined by two factors: the expan-sion rate and gravitational attraction. The slower growthat z (cid:46) z (cid:39)
1, the expansion rate is the same for the two models,but the gravitational potential is weaker for the M ν (cid:54) = 0model. Therefore the growth rate remains suppressed,untilsome later time z (cid:39) . . < z < . z = 0 . σ fσ ( a ) /fσ ( a ) = 0 .
35% from 0 . < z < . M ν = 20 meV onthe structure growth rate f ( a ) σ ( a ), which is detectable byDESI RSD. We begin with a brief review of gravitational lensing of theCMB. For details see e.g. the review by Lewis & Challinor(2006). Gradients in the gravitational potential, Φ, distortthe trajectories of photons traveling to us from the last scat-tering surface. The deflection angles, in Born approximation,are d = ∇ φ , where the lensing potential, φ , is a weightedradial projection of Φ. The key quantity for calculating theimpact of lensing on the temperature power spectrum is theangular power spectrum of the projected potential, C φφ(cid:96) ,which we also call the lensing power spectrum. Taking ad-vantage of the Limber approximation (Limber 1953), it canbe written as a radial integral over the three dimensionalgravitational potential power spectrum P Φ (cid:96) C φφ(cid:96) (cid:39) (cid:90) χ (cid:63) d χ ( k P Φ ) (cid:18) (cid:96)χ ; a (cid:19) (cid:20) − χχ (cid:63) (cid:21) , (7)where χ is the comoving distance from the observer, a = a ( χ ), a (cid:63) subscript indicates the last scattering surface, (1 − χ/χ (cid:63) ) is the lensing kernel, and the power spectrum P Φ isdefined as (cid:10) Φ( k ; a )Φ (cid:63) ( k (cid:48) ; a ) (cid:11) = P Φ ( k ; a ) δ (3) ( k − k (cid:48) ) . (8)To calculate P Φ we assume a power-law primordial powerspectrum P p Φ ( k ), k π P p Φ ( k ) = A s (cid:18) kk (cid:19) n s − , (9)where k is an arbitrary pivot point, A s and n s are theprimordial amplitude and power law index respectively.The gravitational potential at late times, Φ( k, a ), is re-lated to the primordial potential Φ p ( k ) by (e.g. Kodama &Sasaki 1984)Φ( k, a ) = 910 Φ p ( k ) T ( k ) s ( k ; a ) g ( a ) , (10)where the potential on very large scales is suppressed by a c (cid:13)000
35% from 0 . < z < . M ν = 20 meV onthe structure growth rate f ( a ) σ ( a ), which is detectable byDESI RSD. We begin with a brief review of gravitational lensing of theCMB. For details see e.g. the review by Lewis & Challinor(2006). Gradients in the gravitational potential, Φ, distortthe trajectories of photons traveling to us from the last scat-tering surface. The deflection angles, in Born approximation,are d = ∇ φ , where the lensing potential, φ , is a weightedradial projection of Φ. The key quantity for calculating theimpact of lensing on the temperature power spectrum is theangular power spectrum of the projected potential, C φφ(cid:96) ,which we also call the lensing power spectrum. Taking ad-vantage of the Limber approximation (Limber 1953), it canbe written as a radial integral over the three dimensionalgravitational potential power spectrum P Φ (cid:96) C φφ(cid:96) (cid:39) (cid:90) χ (cid:63) d χ ( k P Φ ) (cid:18) (cid:96)χ ; a (cid:19) (cid:20) − χχ (cid:63) (cid:21) , (7)where χ is the comoving distance from the observer, a = a ( χ ), a (cid:63) subscript indicates the last scattering surface, (1 − χ/χ (cid:63) ) is the lensing kernel, and the power spectrum P Φ isdefined as (cid:10) Φ( k ; a )Φ (cid:63) ( k (cid:48) ; a ) (cid:11) = P Φ ( k ; a ) δ (3) ( k − k (cid:48) ) . (8)To calculate P Φ we assume a power-law primordial powerspectrum P p Φ ( k ), k π P p Φ ( k ) = A s (cid:18) kk (cid:19) n s − , (9)where k is an arbitrary pivot point, A s and n s are theprimordial amplitude and power law index respectively.The gravitational potential at late times, Φ( k, a ), is re-lated to the primordial potential Φ p ( k ) by (e.g. Kodama &Sasaki 1984)Φ( k, a ) = 910 Φ p ( k ) T ( k ) s ( k ; a ) g ( a ) , (10)where the potential on very large scales is suppressed by a c (cid:13)000 , 000–000 Z. Pan and L. Knox ‘ -9 -8 -7 ‘ ( ‘ + ) C φφ ‘ / π CLASSLimberLimber with g ( a ) =1 Figure 3.
The lensing power spectrum calculated from
CLASS (solid line), calculated with Limber approximation (dashed line)and calculated with Limber approximation and setting g ( a ) = 1(dashed-dotted line). factor 9 /
10 through the transition from radiation domina-tion to matter domination. For modes which enter the hori-zon during radiation domination when the dominant com-ponent has significant pressure ( p ≈ ρ/
3) the amplitude ofperturbations cannot grow and the expansion of the Uni-verse forces the potentials to decay. In a cosmology withmassless neutrinos, for modes which enter the horizon aftermatter-radiation equality (but before dark energy domina-tion) the potentials remain constant. The transfer function T ( k ) takes this into account, being unity for very large scalemodes and falling approximately as k − for small scales. Thetransfer function T ( k ) is independent of scale factor a be-cause in a cosmology with massless neutrinos, potentials onall scales keep constant in matter domination. The impactof massive neutrinos on the potentials is described by thefunction, s ( k ; a ) , which is unity on scales above the free-streaming scale and decreases with time on scales below.Once the cosmological constant starts to become importantthe potentials on all scales begin to decay. This effect is cap-tured by the growth function, g ( a ), which is unity duringmatter domination. With these definitions we have( k P Φ )( k ; a ) = 8150 π g ( a ) s ( k ; a ) kT ( k ) A s (cid:18) kk (cid:19) n s − . (11)With these pieces in place, we now examine the accu-racy of the Limber approximation. According to Loverde& Afshordi (2008), it is a better approximation to replace k = (cid:96)/χ with (cid:112) (cid:96) ( (cid:96) + 1) /χ (cid:39) ( (cid:96) +0 . /χ in the original Lim-ber approximation Eq.(7). With the replacement and defin-ing x = χ/χ (cid:63) , the lensing power spectrum can be writtenas (cid:96) ( (cid:96) + 1) C φφ(cid:96) (cid:39) χ (cid:63) (cid:90) d x ( k P Φ ) (cid:18) (cid:96) + 0 . xχ (cid:63) ; a (cid:19) (1 − x ) , (12)where a = a ( x ). Fig. 3 shows the lensing power spectrumcalculated from CLASS (Lesgourgues 2011a,b; Blas, Lesgour-gues & Tram 2011; Lesgourgues & Tram 2011) compared tothe lensing power spectrum calculated from the Limber ap-proximation. We see the Limber approximation reproduces ‘ R ‘ Numerical result from CLASSLimber with fixed χ Limber with g ( a ) =1Limber with fixed ω m , g ( a ) =1Limber with fixed A s ,n s ,ω m ,ω b , g ( a ) =1 Figure 4.
The dependence of the lensing power spectrum on totalneutrino mass, ( C φφ(cid:96) − C φφ(cid:96), fid ) /C φφ(cid:96), fid = R (cid:96) × ( M ν / eV). The blackline is the numerical result from CLASS , the red line is the result ofLimber approximation setting χ (cid:63) = 1 . × Mpc, the blue lineis the result of Limber approximation with g ( a ) = 1, the magentaline is the result of Limber approximation fixing both ω m and g ( a ), and green line is the result of Limber approximation with A s , n s , ω m , ω b and g ( a ) fixed. the accurate numerical result even for small (cid:96) . In order tounderstand the the influence of the growth function, we alsocalculated the lensing power spectrum by setting g ( a ) ≡ (cid:96) (cid:46)
50 (Pan,Knox & White 2014). Now we are to understand the impactof massive neutrinos on C φφ(cid:96) . According to Eq.(11) and Eq.(12), it is clear that C φφ(cid:96) is de-termined by the primordial perturbation A s , n s , the transferfunction T ( k ), the impact of massive neutrinos s ( k ; a ), thegrowth factor g ( a ), and the comoving distance to the lastscattering surface χ (cid:63) .In order to quantify the dependence of the lensing poweron total neutrino mass M ν , we take samples from a Planck
ΛCDM + M ν chain, and fit the linear relation C φφ(cid:96) − C φφ(cid:96), fid C φφ(cid:96), fid = R (cid:96) × M ν (eV) , (13)The linear fitting result is shown as the thin solid line in Fig.4. To understand the contribution of the various parametervariations and effects, we also plot R (cid:96) in Fig. 4 for the casesshown.We now work our way towards an understanding of thefull response, the thin solid curve, in stages. We begin withthe case where we fix most of the parameters other than M ν , and turn off the impact of dark energy on the growthfactor, g ( a ), fixing it to unity. In this case, increasing M ν has two effects: 1) it decreases the free-streaming length ∼ ( T ν /M ν ) /H ( z ) (Abazajian et al. 2011) , and 2) increasesthe expansion rate once the neutrinos start to become non-relativistic. The increased expansion rate acts to suppressthe growth of structure on all length scales. However, the de- c (cid:13) , 000–000 onstraints on neutrino mass from Cosmic Microwave Background and Large Scale Structure creased free-streaming length acts to boost structure growthon scales above the free-streaming length, nearly exactlycanceling the suppression. The result is the bottom-mostcurve of Fig. 4: nearly no effect at low (cid:96) , a constant suppres-sion of power at high (cid:96) , and a smooth transition betweenthese two regimes.The difference between the bottom-most curve and thedot-dashed curve (the one labeled, ‘Limber with g ( a ) = 1’)is due to how other parameters adjust as M ν varies. We canisolate these changes as almost all due (at least at (cid:96) (cid:38) M ν and ω m , as demonstrated bythe following: if we fix ω m , and let A s , n s , and ω b vary,we get the second curve from the bottom which differs verylittle from the bottom-most curve. Once we let ω m vary aswell, we get the dot-dashed curve. Letting ω m vary leads tovariation in n s and A s as well, but these changes all flowfrom the correlation between ω m and M ν .Once we let g ( a ) vary as well we get a curve that is in-distinguishable from the thin solid curve, which is boostedeverywhere, and especially at (cid:96) (cid:46)
50. The contributions tothese large angular scales come predominantly from modesto the left of the peak in the matter power spectrum. Atfixed (large) angular scale, structures that are nearer by, andtherefore on smaller length scales, are closer to the peak ofthe matter power spectrum. Thus the large angular scalesare weighted toward later times, and therefore more influ-enced by g ( a ) than the smaller angular scales. The growthfactor increases with increasing M ν because to keep the an-gular size of the sound horizon fixed Ω Λ must decrease.The lensing kernel’s dependence on cosmological param-eters comes entirely via its dependence on χ ∗ . To see howmuch of the variation in the lensing power spectrum is dueto the lensing kernel, we fix χ ∗ = 1 . × Mpc (top-mostdashed curve). By examining the difference between the top-most dashed curve, for which χ ∗ is fixed, and the thin solidcurve, which is the full numerical result, one can see thiseffect is very small This result is in contrast to the case of tomographic cosmicshear as a probe of dark energy. In this case the sensitivity of thedata to variations in the dark energy equation-of-state parameterlargely arises from the lensing kernel (Simpson & Bridle 2005;Zhang, Hui & Stebbins 2005).To summarize, there are three main effects of massive neutri-nos on the lensing power: 1) increased expansion rate suppressespower, 2) decreased free-streaming length compensates for thesuppressed power at scales above the free-streaming length, 3)other parameter variations due to partial degeneracies in C TTl (most notably an increase in ω m ) boost the power on all scales.The net result is increased power at large scales and a decrease inpower at small scales. One might potentially include the growthfactor here as the fourth-most important effect, somewhat in-creasing the power at large angular scales.The origin of the degeneracy in C TTl between ω m and M ν isactually due to lensing itself. Planck Collaboration XVI (2014)demonstrated that the dominant effect leading to the constraintof neutrino mass from the CMB temperature anisotropy powerspectrum is gravitational lensing. As shown in Fig. 4, increasing M ν suppresses the lensing power, while increasing ω m increasesthe lensing power. The lensing power suppression by massive neu-trinos can be compensated by the enhancement from increasing ω m , so uncertainties in M ν and ω m are expected to be positivelycorrelated (Namikawa, Saito & Taruya 2010). We use the Fisher matrix formalism to forecast constraintson neutrino mass from future CMB and LSS experiments.The fiducial cosmology used here is the same as the one usedin Section 3 except with a different value of total neutrinomass, M ν = 85 meV. Following Wu et al. (2014) and Dodelson (2003), the Fishermatrix for cosmological parameters constrained by CMBspectra is written as F αβ = (cid:96) max (cid:88) (cid:96) (cid:96) + 12 f sky Tr (cid:18) C − (cid:96) ∂ C (cid:96) ∂θ α C − (cid:96) ∂ C (cid:96) ∂θ β (cid:19) , (14)and it is related to the expected uncertainty of a parameter θ α by σ ( θ α ) = (cid:112) ( F − ) αα , where C (cid:96) = C TT(cid:96) + N TT(cid:96) C TE(cid:96) C Td(cid:96) C TE(cid:96) C EE(cid:96) + N EE(cid:96) C Td(cid:96) C dd(cid:96) + N dd(cid:96) , (15)and C dd(cid:96) is the angular power spectrum of the deflectionfield d , which is related to the lensing power spectrum by C dd(cid:96) = (cid:96) ( (cid:96) + 1) C φφ(cid:96) . The Gaussian noise N XX(cid:96) is defined as N XX(cid:96) = ∆ X exp (cid:18) (cid:96) ( (cid:96) + 1) θ (cid:19) , (16)where ∆ X ( X = T, E, B ) is the pixel noise level of the ex-periment and θ FWHM is the full-width-half-maximum beamsize in radians (Knox 1995; Zaldarriaga, Spergel & Seljak1997). The noise power spectrum of deflection field N dd(cid:96) iscalculated assuming a lensing reconstruction that uses thequadratic EB estimator(Okamoto & Hu 2003). We use theiterative method proposed by Smith et al. (2012), whichperforms significantly better than the uniterated quadraticestimators (Hirata & Seljak 2003).For the CMB-S4 experiment, we assume the temper-ature noise level ∆ T = 1 . µ K-arcmin, the polarizationnoise level ∆ E = ∆ B = √ T , the fraction of coveredsky f sky = 0 . θ FWHM = 1 (cid:48) . With thesegiven experiment sensitivities, we obtain a constraint fromCMB with σ ( M ν ) = 38 meV. The 1 σ and 2 σ constraint areshown in Fig. 5.According to the analysis in Section 3 and 4 , DESIBAO are helpful to break the degeneracy between M ν and ω m . BAO uncertainties are independent from CMB experi-ments, so the total Fisher matrix is simply given by addition F CMB+BAO = F CMB + F BAO , (17)where the DESI sensitivities of BAO signal can be found inFont-Ribera et al. (2014) and shown in Fig. 1. It is foundthat, adding the DESI BAO data greatly improves the con-straint to σ ( M ν ) = 15 meV (similar forecasts were also con-ducted by Abazajian et al. (2013); Wu et al. (2014)). The largest signal of massive neutrinos on H ( z ) and D ( z ) isfound at low redshifts (see Fig. 1), where BAO has inevitably c (cid:13)000
50. The contributions tothese large angular scales come predominantly from modesto the left of the peak in the matter power spectrum. Atfixed (large) angular scale, structures that are nearer by, andtherefore on smaller length scales, are closer to the peak ofthe matter power spectrum. Thus the large angular scalesare weighted toward later times, and therefore more influ-enced by g ( a ) than the smaller angular scales. The growthfactor increases with increasing M ν because to keep the an-gular size of the sound horizon fixed Ω Λ must decrease.The lensing kernel’s dependence on cosmological param-eters comes entirely via its dependence on χ ∗ . To see howmuch of the variation in the lensing power spectrum is dueto the lensing kernel, we fix χ ∗ = 1 . × Mpc (top-mostdashed curve). By examining the difference between the top-most dashed curve, for which χ ∗ is fixed, and the thin solidcurve, which is the full numerical result, one can see thiseffect is very small This result is in contrast to the case of tomographic cosmicshear as a probe of dark energy. In this case the sensitivity of thedata to variations in the dark energy equation-of-state parameterlargely arises from the lensing kernel (Simpson & Bridle 2005;Zhang, Hui & Stebbins 2005).To summarize, there are three main effects of massive neutri-nos on the lensing power: 1) increased expansion rate suppressespower, 2) decreased free-streaming length compensates for thesuppressed power at scales above the free-streaming length, 3)other parameter variations due to partial degeneracies in C TTl (most notably an increase in ω m ) boost the power on all scales.The net result is increased power at large scales and a decrease inpower at small scales. One might potentially include the growthfactor here as the fourth-most important effect, somewhat in-creasing the power at large angular scales.The origin of the degeneracy in C TTl between ω m and M ν isactually due to lensing itself. Planck Collaboration XVI (2014)demonstrated that the dominant effect leading to the constraintof neutrino mass from the CMB temperature anisotropy powerspectrum is gravitational lensing. As shown in Fig. 4, increasing M ν suppresses the lensing power, while increasing ω m increasesthe lensing power. The lensing power suppression by massive neu-trinos can be compensated by the enhancement from increasing ω m , so uncertainties in M ν and ω m are expected to be positivelycorrelated (Namikawa, Saito & Taruya 2010). We use the Fisher matrix formalism to forecast constraintson neutrino mass from future CMB and LSS experiments.The fiducial cosmology used here is the same as the one usedin Section 3 except with a different value of total neutrinomass, M ν = 85 meV. Following Wu et al. (2014) and Dodelson (2003), the Fishermatrix for cosmological parameters constrained by CMBspectra is written as F αβ = (cid:96) max (cid:88) (cid:96) (cid:96) + 12 f sky Tr (cid:18) C − (cid:96) ∂ C (cid:96) ∂θ α C − (cid:96) ∂ C (cid:96) ∂θ β (cid:19) , (14)and it is related to the expected uncertainty of a parameter θ α by σ ( θ α ) = (cid:112) ( F − ) αα , where C (cid:96) = C TT(cid:96) + N TT(cid:96) C TE(cid:96) C Td(cid:96) C TE(cid:96) C EE(cid:96) + N EE(cid:96) C Td(cid:96) C dd(cid:96) + N dd(cid:96) , (15)and C dd(cid:96) is the angular power spectrum of the deflectionfield d , which is related to the lensing power spectrum by C dd(cid:96) = (cid:96) ( (cid:96) + 1) C φφ(cid:96) . The Gaussian noise N XX(cid:96) is defined as N XX(cid:96) = ∆ X exp (cid:18) (cid:96) ( (cid:96) + 1) θ (cid:19) , (16)where ∆ X ( X = T, E, B ) is the pixel noise level of the ex-periment and θ FWHM is the full-width-half-maximum beamsize in radians (Knox 1995; Zaldarriaga, Spergel & Seljak1997). The noise power spectrum of deflection field N dd(cid:96) iscalculated assuming a lensing reconstruction that uses thequadratic EB estimator(Okamoto & Hu 2003). We use theiterative method proposed by Smith et al. (2012), whichperforms significantly better than the uniterated quadraticestimators (Hirata & Seljak 2003).For the CMB-S4 experiment, we assume the temper-ature noise level ∆ T = 1 . µ K-arcmin, the polarizationnoise level ∆ E = ∆ B = √ T , the fraction of coveredsky f sky = 0 . θ FWHM = 1 (cid:48) . With thesegiven experiment sensitivities, we obtain a constraint fromCMB with σ ( M ν ) = 38 meV. The 1 σ and 2 σ constraint areshown in Fig. 5.According to the analysis in Section 3 and 4 , DESIBAO are helpful to break the degeneracy between M ν and ω m . BAO uncertainties are independent from CMB experi-ments, so the total Fisher matrix is simply given by addition F CMB+BAO = F CMB + F BAO , (17)where the DESI sensitivities of BAO signal can be found inFont-Ribera et al. (2014) and shown in Fig. 1. It is foundthat, adding the DESI BAO data greatly improves the con-straint to σ ( M ν ) = 15 meV (similar forecasts were also con-ducted by Abazajian et al. (2013); Wu et al. (2014)). The largest signal of massive neutrinos on H ( z ) and D ( z ) isfound at low redshifts (see Fig. 1), where BAO has inevitably c (cid:13)000 , 000–000 Z. Pan and L. Knox M ν [eV] ω m CMB − S4CMB − S4+DESI BAOCMB − S4+DESI BAO+DESI RSD
Figure 5.
Forecasted 1 σ and 2 σ constraints in the M ν − ω m plane, where the CMB-S4 experiment results in a σ ( M ν ) = 38meV constraint, the combination of CMB-S4 and DESI BAOyield a σ ( M ν ) = 15 meV constraint. and adding measurementsof the structure growth rate by DESI RSD further improves theconstraint to σ ( M ν ) = 9 meV. large noise because of small amount of survey volume andlarge cosmic variance. Other than DESI BAO, we also inves-tigate other low-redshift tracers of H ( z ) and D ( z ) which arepossible to tighten the uncertainty of total neutrino mass.DESI RSD: similar to BAO, RSD uncertainties are alsoindependent from those of CMB observations, so the totalFisher matrix of CMB+BAO+RSD is also approximatelygiven by addition F CMB+BAO+RSD = F CMB + F BAO + F RSD , (18)where we use the RSD sensitivities from DESI survey whichcan be found in Huterer et al. (2013) and shown in Fig. 2.Here we use the approximation that uncertainties in BAOand RSD are uncorrelated, due to they are sensitive to differ-ent aspects of the matter power spectrum: BAO is sensitiveto its characteristic length scale r s while RSD is sensitiveto its amplitude. In fact, our result is insensitive to theapproximation because we find that both CMB-S4+DESIBAO+DESI RSD and CMB-S4+DESI RSD yield the same σ ( M ν ) = 9 meV uncertainty. Better
BAO: DESI survey cover 14 ,
000 squared degrees(about 1 / 0. Constraints on D A ( z ) and H ( z ) from this BAOexperiment are shown in Fig. 6. It is found that CMB-S4and the cosmic variance limited BAO constrain the totalneutrino mass with uncertainty σ ( M ν ) = 11 meV. So weconclude that 11 meV is a lower limit of σ ( M ν ) we couldmeasure from CMB-S4+BAO, where the limit mainly comesfrom noise level of the CMB lensing signal. Supernovae : The constraining power of BAO is limitedby its large cosmic variance at low redshifts (Fig. 6), so su-pernovae distance measurements which do not suffer fromthe cosmic variance problem may be effective complementsif their systematic errors are well controlled. Supernovae per-form better in relative distance measurements than in abso-lute distance measurements. However for the ΛCDM + M ν H ( z ) r s ( H ( z ) r s ) f i d BAO Cosmic Variance Limit z D A ( z ) / r s ( D A ( z ) / r s ) f i d M ν =20 meV M ν =50 meV M ν =100 meV Figure 6. Same as Fig. 1, but with suppressed errorbars of D A ( z )and H ( z ) coming from CMB-S4 and a cosmic-variance-limitedBAO experiment. z σ [ D ( z ) / D ( z f i d ) ] D ( z ) / D ( z f i d ) × z fid =1 .z fid =0 . Figure 7. The uncertainties in relative distances from CMB-S4 +DESI BAO. Note that the uncertainties is multiplied by a factorof 10 in the plot. model, the uncertainties in relative distances from CMB-S4+ DESI BAO are very small (see Fig. 7) . We conclude thatsupernova observations must result in relative distance de-terminations with systematic errors less than about 0 . 05% ifthey are to tighten the constraints on neutrino mass. Com-pared to systematic errors from current supernova observa-tions (e.g., Suzuki et al. (2012)) this would be a reductionby a factor of ∼ 20 . This paper is motivated by our desire to better understandthe origin of current and forecasted cosmological constraintson the sum of neutrino masses. We took as a given that de-termination of N eff will solidify the predicted value of 3.046,increasing our confidence that the phase-space distributionof the cosmic neutrino background is what we expect basedon the standard thermal history. With that as a given, themost important aspect of increased neutrino mass (relative c (cid:13) , 000–000 onstraints on neutrino mass from Cosmic Microwave Background and Large Scale Structure to some reference model) is an increased neutrino energydensity. If the model with increased mass is to remain consis-tent with CMB observations, the distance to last-scatteringmust be preserved and so the total energy density, and there-fore the expansion rate, cannot increase at all redshifts. Tocompensate for the increase in neutrino energy density, thecosmological constant must decrease in value. Thus varyingneutrino mass leads to changes in H ( z ) with a very partic-ular shape: a mild increase at high redshifts, a larger de-crease in low redshifts, with a transition near the onset ofΛ − domination at z (cid:39) 1. Unfortunately this very specificprediction of the shape for H ( z ) is difficult to verify in de-tail because of how small the departures from the referencemodel are at z > 1. We see that this difficulty persists evenfor a cosmic-variance limited all-sky ( z < 4) BAO experi-ment (see Fig. 6).The sensitivity of CMB-S4 to neutrino mass comes viathe impact of this increased expansion rate on the growthof structure. At scales below the neutrino free-streaminglength, this increased expansion rate suppresses the growthof structure. Above the free-streaming length, the ability ofmassive neutrinos to cluster compensates for the increasedexpansion and there is no net suppression. Because increas-ing matter density increases lensing power amplitude, theCMB lensing-derived constraints on neutrino mass have un-certainties positively correlated with the matter density un-certainties. This correlation with dark matter density leadsto secondary correlations of neutrino mass uncertainty withuncertainties in n s and A s . We disentangled all these variouseffects in Fig. 4.The correlation between M ν and ω m has the oppositesign as that from BAO, since increasing neutrino mass andincreasing ω m both increase the expansion rate at z > 1, andlead to a compensating decrease at z < 1. Thus the com-bination of CMB-S4 and DESI-BAO leads to improvementsin the determination of both quantities.Finally, we briefly investigated how constraints mightbe improved beyond the ∼ σ to ∼ σ detection expectedfrom CMB-S4 + DESI BAO in the case of the lowest possi-ble neutrino mass of 58 meV. 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