Towards a cosmological neutrino mass detection
Rupert Allison, Paul Caucal, Erminia Calabrese, Joanna Dunkley, Thibaut Louis
TTowards a cosmological neutrino mass detection
R. Allison, P. Caucal,
2, 1
E. Calabrese,
1, 3
J. Dunkley, and T. Louis Sub-department of Astrophysics, University of Oxford,Denys Wilkinson Building, Oxford, OX1 3RH, UK Master ICFP, D´epartement de Physique, ´Ecole Normale Sup´erieure,24 rue Lhomond, 75005 Paris, France Department of Astrophysical Sciences, Peyton Hall,Princeton University, Princeton, NJ USA 08544
Future cosmological measurements should enable the sum of neutrino masses to be determinedindirectly through their effects on the expansion rate of the Universe and the clustering of matter.We consider prospects for the gravitationally lensed Cosmic Microwave Background anisotropies andBaryon Acoustic Oscillations in the galaxy distribution, examining how the projected uncertaintyof ≈
15 meV on the neutrino mass sum (a 4 σ detection of the minimal mass) might be reachedover the next decade. The current 1 σ uncertainty of ≈
103 meV (
Planck -2015+BAO-15) willbe improved by upcoming ‘Stage-3’ CMB experiments (S3+BAO-15: 44 meV), then upcomingBAO measurements (S3+DESI: 22 meV), and planned next-generation ‘Stage 4’ CMB experiments(S4+DESI: 15-19 meV, depending on angular range). An improved optical depth measurementis important: the projected neutrino mass uncertainty increases to 26 meV if S4 is limited to (cid:96) >
20 and combined with current large-scale polarization data. Looking beyond ΛCDM, includingcurvature uncertainty increases the forecast mass error by ≈
50% for S4+DESI, and more thandoubles the error with a two-parameter dark energy equation of state. Complementary low-redshiftprobes including galaxy lensing will play a role in distinguishing between massive neutrinos and adeparture from a w = −
1, flat geometry.
I. INTRODUCTION
A central goal in both cosmology and particle physicsis to measure the mass of the neutrino particles. Theneutrino sector is still poorly understood and the mecha-nism that gives rise to their mass is unknown. There arethought to be three active neutrino species, with massdifferences measured through solar, atmospheric, reactorand accelerator neutrino oscillation experiments (for re-views see e.g., Gonzalez-Garcia and Nir [1], Maltoni et al.[2], Smirnov [3], Feldman et al. [4]). The results implya minimum total mass of 60 meV in a normal hierarchywith two lighter and one heavier neutrino, or 100 meV inan inverted hierarchy with two massive neutrinos.Cosmology provides an indirect probe of massive neu-trinos [e.g., 5–17]. Massive neutrinos behave initially likenon-interacting relativistic particles, and then later likecold dark matter. As such they affect the expansion rateof the Universe, compared to a pure radiation or purematter component, as well as modifying the evolutionof perturbations at early times. They also modify thegrowth of structure through a suppression of the cluster-ing of matter on scales that entered the cosmic horizonwhile the neutrinos were relativistic.The current indirect 95% upper limit from cosmo-logical data on the sum of the neutrino masses isΣ m ν <
230 meV from the
Planck measurements of theCosmic Microwave Background (CMB), combined withBaryon Acoustic Oscillation (BAO) measurements fromthe Baryon Oscillation Spectroscopic Survey (BOSS)[18, 19]. The limit is Σ m ν <
680 meV from the CMBalone [19]. Tighter limits have been found includingLyman- α forest measurements from quasars in the BOSS survey (Σ m ν <
120 meV) [20, 21], but the result de-pends on numerical hydrodynamical simulations whichmay contribute additional systematic uncertainty.Recent forecasts of mass limits for upcoming cosmo-logical datasets, including galaxy lensing and cluster-ing, redshift-space distortions, the kinematic Sunyaev-Zel’dovich effect, and counts of galaxy clusters, havebeen studied extensively (e.g., Font-Ribera et al. [22],Villaescusa-Navarro et al. [23], Mueller et al. [24]), show-ing the promise of a wide range of future cosmologicaldata to target a neutrino mass measurement. In thispaper we focus on the combination of lensed CMB andBAO measurements, datasets which do not require de-tailed modelling of non-linear structure formation, or anunderstanding of galaxy bias. The gravitationally lensedCMB measures the growth of structure at times typi-cally before the Universe was half its current age, andon angular scales larger than ≈
100 Mpc, so is domi-nated by linear physics. Studies of this combination havebeen reported in Hall and Challinor [25], Abazajian et al.[26], Wu et al. [27] and Pan and Knox [28], with a 4 σ de-tection of neutrino mass forecast for the next generationof experiments. In this paper we investigate this further,exploring the dependence on experimental details and onparameter degeneracies.In § II we give a brief review of the cosmological ef-fects of neutrinos, and in § III study how the mass mea-surement may be reached step-wise using data collectedduring the coming decade. In § IV we investigate thedependence on experimental details, and in § V we ex-plore degeneracies with other cosmological parameters.We conclude in § VI. a r X i v : . [ a s t r o - ph . C O ] S e p II. COSMOLOGICAL EFFECTS OFNEUTRINOS
Standard Model neutrinos are initially relativistic, fol-lowing a thermal distribution after decoupling from theprimordial plasma when the Universe had a temperatureof around k B T ≈ m ν the tran-sition occurs at z ≈ m ν ) [10], so current limitsindicate a transition epoch of 120 (cid:46) z (cid:46)
460 for a nor-mal mass hierarchy.This limit implies that the neutrinos were still rel-ativistic when the CMB decoupled, so they would beindistinguishable from massless neutrinos in the pri-mary anisotropies. However, higher mass neutrinos be-come non-relativistic sooner, which reduces the early-time Integrated Sachs-Wolfe (ISW) effect. This gravi-tational redshift of the CMB photons arises while thenon-negligible radiation component causes the potentialsof the density fluctuations to evolve [5, 10] and affectsthe anisotropies on scales around the first acoustic peak[29].This effect is not sensitive to masses which remainrelativistic until well after decoupling [10, 30], but fur-ther information comes from probes of later-time largescale structure measurements. Massive neutrinos inter-act weakly, allowing them to free-stream out of overden-sities while relativistic, so the growth rate of matter per-turbations inside the horizon is suppressed compared toa universe with only cold dark matter. For comovingwavenumbers k (cid:29) k FS , Hu et al. [31] show that the sup-pression of the matter power spectrum today, P ( k ), isproportional to the sum of the neutrino masses: P Σ m ν ( k ) − P Σ m ν =0 ( k ) P Σ m ν =0 ( k ) ≈ − . (cid:18) Σ m ν . (cid:19) (cid:18) Ω m h . (cid:19) − , (1)where the comoving free-streaming scale is given by k FS = 0 . (cid:18) Σ m ν . (cid:19) / (cid:18) Ω m . (cid:19) / h Mpc − , (2)as illustrated in Fig. 1 and e.g., [26], for models with fixedtotal matter density. For current limits this scale is esti-mated to lie in the range 0 . (cid:46) k FS (cid:46) .
011 [ h Mpc − ].The suppression of small-scale power can be probedusing galaxy clustering and the gravitational lensing ofgalaxies. These are promising avenues for neutrino massmeasurements [e.g., 22], although these observables aresensitive to non-linearities in the matter power spectrumand scale-dependent galaxy and shape biases [32]. An al-ternative route is through the gravitational lensing of theCMB (see e.g., [33] for a review). Here the CMB photonsare deflected by the large-scale structure, integrated overthe photon path since decoupling.Following [33], the CMB convergence angular powerspectrum, C κκl , is a weighted projection of the matter power spectrum P ( k, χ ); under the Limber approxima-tion, C κκ(cid:96) = (cid:90) χ H dχ W ( χ ) f k ( χ ) P (cid:18) (cid:96)f k ( χ ) , χ (cid:19) , (3)where χ H is the comoving horizon size, f k ( χ ) relates line-of-sight comoving distances and transverse comoving dis-tances in a curved universe, and the window function W ( χ ) is W ( χ ) = 3Ω m H c f k ( χ ) f k ( χ ∗ − χ ) a ( χ ) f k ( χ ∗ ) (4)for χ < χ ∗ and zero otherwise. Here a ( χ ) is the scalefactor and χ ∗ is the radial comoving distance to the last-scattering surface. This power spectrum is sensitive toΣ m ν , as shown in Fig. 1, and does not depend on galaxybias or detailed non-linear modeling. In practice it isreconstructed from CMB temperature and polarizationmaps using a four-point function [e.g., 34].The CMB temperature and polarization angular powerspectra, { C T T(cid:96) , C
T E(cid:96) , C
EE(cid:96) , C
BB(cid:96) } , are also modified bylensing, which smears the acoustic peaks by adding vari-ance to the apparent scale of a mode, converts E -modepolarization into B -mode polarization, and adds small-scale power in T , E and B [e.g., 33]. The approximateeffect of massive neutrinos is shown in Fig. 1 for the E-mode polarization, where we artificially amplify the ef-fects of neutrinos on the CMB lensing, rather than theprimary CMB, by varying the amplitude of the lensingpotential. Increasing the neutrino mass has a similar ef-fect to decreasing the lensing amplitude. Compared tothe power spectra, the reconstructed convergence fieldcontains more information on the neutrino mass [35].Massive neutrinos also affect angular diameter dis-tances d A ( z ) and the expansion rate H ( z ), as their evolu-tion differs from a pure radiation or pure matter compo-nent [e.g., 28]. These can be measured using a ‘standardruler’ method that is relatively free of systematic uncer-tainties: the primordial oscillations in the photon-baryonfluid are imprinted in the galaxy distribution as BaryonAcoustic Oscillations (BAO). The comoving scale of theoscillations is fixed by the sound horizon at decoupling, r s , which is not significantly affected by neutrino massesgiven current limits and is in the linear regime of densityperturbations ( ≈
150 Mpc). The observed spherically-averaged BAO angular scale for galaxies at redshift z issensitive to the parameter combination r s /d V ( z ); d V isthe volume distance [36], d V ( z ) ≡ (cid:2) cz (1 + z ) d A ( z ) H − ( z ) (cid:3) / . (5)For fixed cold dark matter density, more massive neu-trinos increase the total late-time non-relativistic mattercontent, which increases the volume distance, as shownin Fig 1. − − − − k [ h Mpc − ] − − − − − − P Σ m ν ( k ) − P Σ m ν = ( k ) P Σ m ν = ( k ) Σ m ν = 0.1 eVΣ m ν = 0.2 eVΣ m ν = 0.3 eV ‘ C κκ ‘ × − Σ m ν = 0Σ m ν = 0.1 eVΣ m ν = 0.2 eVΣ m ν = 0.3 eV ‘ ‘ C EE ‘ × A lens = 1 A lens = 0.5 A lens = 0 z r s ( z ∗ ) / d V ( z ) Σ m ν = 0Σ m ν = 1 eVΣ m ν = 2 eV FIG. 1: Effect of neutrino mass on CMB power spectra and BAO distance scales.
Top:
Fractional change of the matter powerspectrum today P ( k ) (left) and CMB convergence power spectrum C κκl (right) with neutrino mass Σ m ν , for fixed physicaldark matter density, Ω c h + Ω ν h . Suppression of power is due to neutrino free-streaming. Bottom left:
Lensed CMB E -modepower spectrum with varying amplitudes of the lensing potential A lens , approximating and exaggerating the effect that massiveneutrinos have on the CMB polarization spectrum. Bottom right:
BAO distance ratio r s /d V for fixed θ A and Ω c h . Massiveneutrinos behave like additional matter in the BAO redshift range, decreasing H and increasing the volume distance d V . III. IMPROVEMENTS IN THE NEXT DECADE
We consider how upcoming and planned CMB andBAO experiments will improve the current limits on thesum of the neutrino masses, building on previous analy-ses [22, 25–28].
A. New data
The current state-of-the-art for the CMB is the
Planck and
WMAP satellite data, including the first analysis ofthe full-mission
Planck data [19, 37]. Improved small-scale CMB measurements are currently being made bythe ‘Stage 2’ ground-based experiments: ACTPol, SPT-Pol and POLARBEAR [38–40]. These will soon be up-graded to ‘Stage 3’ (hereafter, S3) with new detectorsand sensitivity to multiple frequencies. Here we con-sider an S3 ‘wide’ experiment that maps 40% of the sky, and a ‘deep’ experiment that maps 6% of the sky. The‘wide’ experiment is similar to AdvACT specifications[41] and ‘deep’ to SPT-3G [42]. These experiments areexpected to take data during 2016-19, and the specifica-tions we adopt are given in Table I. We also anticipatedata from S3 experiments targeting larger angular scales(e.g., CLASS), but do not consider these specifically. Inaddition, we can expect a complete analysis of the
Planck polarization data, optimistically including reliable large-scale polarization data.Beyond S3, a ‘Stage-4’ (S4) experiment - or set of ex-periments - is being developed by the CMB communitythat may cover at least half the sky to typical noise levelsof 1 µ K-arcmin [26]. There are also proposed space-basedexperiments including LiteBIRD and PIXIE [43, 44],and we approximate their role with a cosmic variance-limited large-scale polarization measurement (‘CV-low’)that could supplement S3 ground-based data.On the BAO front, current state-of-the-art measure- ‘ − − C κκ ‘ ( Σ m ν ) − C κκ ‘ ( Σ m ν = ) C κκ ‘ ( Σ m ν = ) Σ m ν = 60 meVΣ m ν = 120 meVS3-wideS4
500 1000 1500 2000 2500 3000 3500 ‘ − − − − C EE ‘ ( Σ m ν ) − C EE ‘ ( Σ m ν = ) C EE ‘ ( Σ m ν = ) Σ m ν = 60 meVΣ m ν = 120 meVS3-wideS4 z − − − [ r s / d V ] ( Σ m ν ) − [ r s / d V ] ( Σ m ν = ) [ r s / d V ] ( Σ m ν = ) Σ m ν = 60 meVΣ m ν = 120 meVBAO-15BAO-DESI FIG. 2:
Top and middle : Fractional change in the conver-gence κ (top) and E -mode (middle) power spectrum withneutrino mass, for fixed Ω c h + Ω ν h , with expected uncer-tainties for S3-wide and S4 CMB data. A higher neutrinomass has less lensing, decreasing the E-mode peak smooth-ing. Bottom : Fractional change in distance ratio r s /d V , withuncertainties from current (BAO-15 [18]) and forecast (DESI,[22]) BAO data. Here Ω c h is fixed. Experiment f sky Beam ∆ T ∆ P (arcmin) ( µ K-arcmin) ( µ K-arcmin)S3-wide 0 . .
06 1 4.0 5.7S4 0 . . Planck . The
Planck specifica-tions we use are in the Appendix. ments come from the BOSS ‘LowZ’ and ‘C-MASS’ galaxysamples at z = 0 .
32 and z = 0 .
57 [18]. These are supple-mented by data from the Six Degree Field Galaxy Red-shift Survey at z = 0 .
11 [45] and the SDSS MGS sampleat z = 0 .
15 [46]. Improved measurements are being madeby the eBOSS survey which will survey a deeper sample[47]. A significant advance should be made with the DESIspectroscopic survey, due to begin in 2018, which is ex-pected to measure the BAO distance ratio from redshifts0 . < z < .
85 in bins of width ∆ z = 0 . B. Forecasting methods
We use a Fisher-matrix forecasting method to predictthe neutrino mass uncertainties. For a model defined byparameters θ the expected Fisher matrix is F ij ( θ ) = (cid:28) − ∂ ln p ( θ | d ) ∂θ i ∂θ j (cid:29) , (6)where p ( θ | d ) is the posterior distribution for θ given data d . The forecast parameter covariance is then given bythe inverse of the Fisher matrix, C ij = ( F − ) ij . Here ourdata are the lensed TT, TE, and EE CMB power spectra,reconstructed CMB convergence power spectrum κκ , andBAO distance ratio measurements r s /d V ( z ). Our param-eters are the standard six ΛCDM parameters, plus theneutrino mass sum, as well as possible extension param-eters including curvature and dark energy. Our methodsare summarized in the Appendix, including choices madeabout the fiducial model, choice of parameter basis, andstep-size for calculating derivatives. We also describe val-idation of our numerical code.We use the lensed CMB power spectra and the conver-gence power spectrum as our CMB observables, whichdiffers from the approach in [22, 25–27], but more closelyfollows the ‘real’ data analysis: the CMB sky we see islensed, and it is a difficult inverse problem to infer theunlensed sky [e.g., 49]. Using unlensed spectra in fore-casts removes information contained in the lensed tem-perature and polarization fields. However, it is challeng-ing to construct the full covariance matrix for the lensedpower spectra and convergence power spectrum: the T , Q and U fields are all lensed by the same lensing poten-tial, which correlates the power spectra and adds addi-tional non-Gaussian covariance. This is explored in de-tail in [35, 50]. In this analysis we make the approxima-tion of discarding BB information, and assuming Gaus-sian uncorrelated errors in TT, TE, EE, and κκ . Thisis likely a good approximation for S3 data, but couldunder-estimate certain parameter errors for S4-type databy up to ≈
20% [35]. We use camb for evaluation of allrelevant CMB power spectra [51].For the noise levels of
Planck , we consider two cases,‘
Planck -2015’ (P15) that produces cosmological con-straints which closely match the published results [19],and ‘
Planck -pol’ which includes TE and EE data com-ing from the polarization measurements of the High-Frequency Instrument (HFI), including large-scales. Thespecifications are given in the Appendix. For
Planck -pol,noise levels are approximated by taking temperature sen-sitivities from P15 and assuming the per-channel noisescalings from temperature to polarization in the
Planck
Blue Book [52]. This is likely to be over-optimistic at thelargest scales.For the CMB power spectra, we set a maximum mul-tipole for the recoverable information: (cid:96) T max = 3000, (cid:96) P max = 4000 and (cid:96) κ max = 3000 for the future S3 and S4 ex-periments. Smaller scales are likely hard to extract dueto extragalactic foreground contamination. We assumewhite noise, and do not include additional foregrounduncertainty beyond the multipole cuts outlined above,although the expected S3 white noise level includes someforeground inflation [41]. We also set a minimum multi-pole of (cid:96) min = 50 for S3 due to the challenge of recoveringlarge-scales from the ground, and consider two options forS4: (cid:96) min = 50 and (cid:96) min = 5. For S3 and S4 we include Planck data for 2 < (cid:96) < (cid:96) min , and our nominal analysisuses ‘
Planck -pol’ unless stated otherwise. We considerthe importance of the large-scale polarization measure-ments in § IV A.We use the quadratic-estimator formalism of Hu andOkamoto [53] to calculate the CMB convergence noisespectrum N κκ(cid:96) . This uses the coupling of otherwise un-correlated modes in temperature and polarization to re-construct the lensing potential. Iterative delensing pro-cedures are able to reduce the effective noise level ofthe estimated lensing field, particularly for the low-noise(∆ P (cid:46) µ K-arcmin) future experiments considered here[54]. We consider the impact of this process in § III C.For the BAO measurements we use the published un-certainties on the distance ratio r s /d V for the currentBAO data described in § III [18, 45, 46], labelled as ‘BAO-15’. For DESI we use the forecast uncertainties on d A ( z )and H ( z ) given in Font-Ribera et al. [22] to estimatethe expected r s /d V uncertainties, summarized in the Ap-pendix. We do not use broadband shape information inthe galaxy power spectra. Σ m ν [meV] p ( Σ m ν ) P15+BAO-15S3-wide+BAO-15S3-wide+DESIS4( ‘ > ‘ >
FIG. 3: Forecast marginal posterior constraints on the sum ofthe neutrino masses Σ m ν within a ΛCDM+Σ m ν model, as-suming Gaussian error distributions. The current uncertain-ties (P15+BAO-15) are expected to improve rapidly, with S3CMB data and DESI BAO data expected by ∼ Σ m ν [meV] Ω c h Planck-polPlanck-pol+BAO-15S3-wideS3-wide+DESIS4 ( ‘ > ‘ >
FIG. 4: Expected joint constraint (68% CL) on the neutrinomass sum Σ m ν and physical cold dark matter density Ω c h within a ΛCDM+Σ m ν model. The BAO constraint, sensi-tive to the total late-time cold dark matter density, is almostorthogonal to the CMB lensing constraint, breaking the de-generacy. C. Expected constraints
We find forecast marginal 1 σ uncertainties on the sumof the neutrino masses Σ m ν of σ (Σ m ν )meV =
103 (P15 + BAO-15)44 (S3-wide + BAO-15)22 (S3-wide + DESI)19 (S4 ( (cid:96) >
50) + DESI)15 (S4 ( (cid:96) >
5) + DESI) (7)where S3 and S4 include
Planck -pol at large-scales. Ifwe replace S3-wide with the S3-deep survey we find σ (Σ m ν ) = 53 meV combined with BAO-15, and 25 meVwith DESI. The expected constraints are summarized inFig. 3, and are consistent with findings in Wu et al. [27].These forecasts imply that if the neutrino hierarchy isinverted, with mass sum >
100 meV, we may have > σ evidence for non-zero mass in the next few years from S3data, and an almost 5 σ detection in ≈ − σ indi-rect measurement should be reachable in five years, withstronger evidence from the subsequent experiments.As illustrated in Fig. 4, there is a strong positive cor-relation between the neutrino mass and cold dark matterdensity in the CMB observables. This arises predomi-nantly from the competing influence of these parameterson the lensing signal. Increased neutrino mass suppressessmall-scale power, while increasing cold-dark matter den-sity boosts small-scale power by shifting matter-radiationequality to earlier times; this shortens the radiation-domination epoch in which sub-horizon modes of thegravitational potential decay, enhancing the small-scaleamplitude of structure [25, 55, 56]. Conversely, BAOdata constrain the sum of the CDM and massive neu-trino density, since it is this combination that affects theangular diameter distance and expansion rate.The power of the BAO and CMB data lie in their com-bination. BAO measurements alone cannot constrain theprimordial parameters A s and n s , nor the optical depthto reionisation τ ; the constraints lie in the 4-dimensionalsubspace spanned by Ω c h , Ω b h , Σ m ν and θ A (beingmost constraining in the directions corresponding ap-proximately to Ω m h and θ A ). S4-alone could achievea neutrino mass error of σ (Σ m ν ) = 53 meV; with DESIBAO this is expected to tighten to σ (Σ m ν ) = 19 meV,enough for a 3 σ detection of the minimal mass.Comparing to previous results, the S4( (cid:96) > σ (Σ m ν ) < .
23 eV at 95% confi-dence [19], with the forecast errors on other ΛCDM pa-rameters also matching closely. Here we chose a fiducialneutrino mass sum of 60 meV. We find a smaller uncer-tainty for higher neutrino mass (as in [25]), but the ef-fect is small when BAO data are included. For S4 alone,we find σ (Σ m ν ) = 53 (41) meV for a fiducial mass of 60 (120) meV, but with DESI this difference reduces to σ (Σ m ν ) = 19 (18) meV. We describe further tests of thecode in the Appendix.We consider the impact of using an improved,likelihood-based lensing estimator (going beyond thefirst-order quadratic estimator of Hu and Okamoto [53])to reduce the effective noise power N κκl of the lensing re-construction. Hirata and Seljak [57] show that N κκl canbe reduced by a factor of two for an S4-like experiment;under this modification, we find only a 3% tightening ofthe neutrino mass constraint for S4+DESI. The improve-ment is small due to the contribution of cosmic variance(CV) to the lensing power spectrum uncertainties, andthe degeneracy of neutrino mass with other ΛCDM pa-rameters; the S4 lensing power spectrum derived from thequadratic estimator is already CV-limited out to (cid:96) ≈ Planck -pol at large scales butno BAO). Lensing reduces the neutrino mass uncertaintyby a factor of ≈ A lens [58], would isolate theimpact of neutrinos on the 4-point function. Unlensed Lensed TT, TE, EE Unlensed+ κκ (2-point only) (4-point only)S3, σ (Σ m ν ): 435 75 61S4, σ (Σ m ν ): 363 64 53TABLE II: Impact of lensing on the neutrino mass constraint(in units of meV). Constraining power comes from both thelensed spectra (2-point) and the reconstructed lensing poten-tial (4-point). Gaussian uncorrelated errors are assumed. IV. DEPENDENCE ON EXPERIMENTALDETAILS
Since future CMB experiments are currently under de-velopment, we investigate the importance of certain ex-perimental details on the mass constraint.
A. Importance of the reionization bump
The amplitude of primordial power, A s , is partiallydegenerate with Σ m ν , since A s increases the amplitudeof clustering at small-scales, and Σ m ν decreases it. Theamplitude A s is not well determined by the primordialCMB temperature anisotropy; an increased optical depthto reionization lowers the signal such that the normaliza-tion of the anisotropy measures the combination A s e − τ [59]. This leads to a degeneracy between τ and Σ m ν which can be broken with precision measurements of thereionisation bump at multipoles (cid:96) <
20 in polarization[22, 60].Here we explore the importance of making a robustoptical depth measurement, considering three cases forS4: current
WMAP measurements [37, 61], optimisticfuture
Planck -pol measurements (see Appendix), and afuture S4 measurement that reaches the largest scales( (cid:96) min = 5). We find forecast constraints of σ (Σ m ν )meV =
27 (S4 ( (cid:96) >
50) +
WMAP -pol + DESI)19 (S4 ( (cid:96) >
50) +
Planck -pol + DESI)15 (S4 ( (cid:96) >
5) + DESI) (8)with the uncertainty on τ reducing from 0 .
008 to 0 . . .
013 for
WMAP -pol from EE alone, i.e., improved CMB lensingdata helps constrain τ even when the neutrino mass isvaried. Fig. 5 shows the expected correlation between τ and neutrino mass.Fig. 5 also shows the impact of reducing the mini-mum multipole of the S4 experiment on the neutrinomass constraint, supplemented with Planck -pol or thecurrent
WMAP -pol at the largest scales. There is a lim-iting plateau for S4 at (cid:96) min >
20, and a clear improve-ment as the polarization is better measured at increas-ingly large scales. The S4( (cid:96) >
5) + DESI limit reachesthe cosmic-variance (CV) limit for CMB data .We then also consider the relative importance of mak-ing a higher sensitivity small-scale measurement, versus anew large-scale polarization measurement. We start withan S3-type (cid:96) >
50 experiment, and then either increasethe (cid:96) >
50 sensitivity, or supplement it with a new ‘CV-low’ large-scale measurement at (cid:96) <
50. We find forecastconstraints of σ (Σ m ν )meV =
22 (S3 ( (cid:96) >
50) +
Planck -pol + DESI)19 (S4 ( (cid:96) >
50) +
Planck -pol + DESI)17 (S3 ( (cid:96) >
50) + CV-low + DESI) (9)This indicates that a cosmic-variance-limited measure-ment of optical depth could be more valuable than moresensitive small-scale data, especially given that
Planck -pol large-scale polarization data is itself not yet demon-strated to be free of systematic errors.We note that 21cm experiments, which map the bright-ness temperature of neutral hydrogen as a function of Pan and Knox [28] found that an S4 experiment combined witha CV-limited BAO experiment could tighten the neutrino massconstraint further, to σ (Σ m ν ) = 11 meV. Σ m ν [meV] τ WMAP-pol+S4( ‘ ≥ ‘ ≥ ‘ ≥ ` min ( ⌃ m ⌫ ) [ m e V ] WMAP-pol + S4( `>` min ) + DESIPlanck-pol + S4( `>` min ) + DESI np FIG. 5:
Top:
The neutrino mass Σ m ν is correlated with theoptical depth to reionization τ (forecast 68% CL). Currentdata at (cid:96) <
20 (
WMAP -pol) would leave a degeneracy be-tween Σ m ν and τ that could be broken with improved large-scale polarization data. Bottom: the expected neutrino massconstraint as a function of the minimum multipole accessibleto S4, indicating the benefit of reaching large scales. redshift, will probe the epoch of reionization [62]; combi-nation of this information with CMB+BAO would breakthe Σ m ν - τ degeneracy and improve the neutrino massconstraint (investigated in Liu et al. [63]). B. Importance of sensitivity and angular range
For the particular case of an (cid:96) >
50 experiment cover-ing 40% of the sky at 3 arcmin resolution, combined with
Planck -pol, we vary the white-noise sensitivity. The fore-cast neutrino mass limits are shown in Fig. 6 for CMB-only, CMB+BAO-15, and CMB+BAO-DESI. There isclearly an improvement as the noise is reduced, and asignificant gain is expected over current
Planck measure-ments, but below white noise levels of ≈ µ K-arcminthere does not appear to be a substantial gain (as alsoseen in Wu et al. [27]). sensitivity[ µ K-arcmin] σ ( Σ m ν ) [ m e V ] CMBCMB+BAO-15CMB+DESI
FIG. 6: The dependence of the neutrino mass constraint σ (Σ m ν ) on CMB map sensitivity, for a 3 arcmin resolutionexperiment covering 40% of sky. It is not yet certain whether this ≈ µ K-arcmin noiselevel, over half the sky, will be achieved in practice fromthe upcoming S3 CMB experiments, or whether the lens-ing reconstruction will achieve the expected noise levels.Atmospheric, ground and foreground emission are typi-cal contaminants that would increase the effective noisein the maps and in the lensing reconstruction. New datafrom the current S2 experiments will help clarify the im-pact of non-white-noise on the lensing noise performance.Here we have continued to restrict our analysis to‘clean’ scales at (cid:96) < (cid:96) < (cid:96) >
T T spectrum is signal-dominated to small scales ( (cid:96) ≈ (cid:96) > ≈ (cid:96) > F Σ m ν (cid:96) in theFisher matrix, is concentrated in the multipole range100 (cid:46) (cid:96) (cid:46) (cid:46) (cid:96) (cid:46) z ≈ (cid:38)
200 Mpc.
V. HOW UNIQUE IS THE MASSIVENEUTRINO SIGNAL?
The measurable effect of neutrinos can be partly mim-icked by changes in other cosmological parameters, as ex-plored in e.g., Font-Ribera et al. [22], Benoit-L´evy et al. [35], Hamann et al. [65]. Massive neutrinos affect the ex-pansion rate and angular diameter distance, but changesin curvature, dark energy history or the Hubble parame-ter can compensate to keep the well-constrained acousticpeak positions and structure essentially unchanged.Here we consider changes in the spatial curvature, andchanges in the energy density of dark energy with time.Dark energy is invoked to explain the acceleration ofthe universe, and may take the form of a cosmologi-cal constant with constant energy per unit proper vol-ume, although an evolving dark energy equation of state( w (cid:54) = −
1) is not ruled out by current observations [19].We consider two parameterizations for dark energy: theseare the usual Taylor expansion in the scale factor for dy-namical dark energy [66]; w ( a ) = w + w a (1 − a ) , (10)with two free parameters w and w a , and also the Doranand Robbers [67] model for early dark energy,Ω Λ ( a ) = Ω Λ − Ω e (cid:0) − a − w (cid:1) Ω Λ + Ω m a w + Ω e (cid:0) − a − w (cid:1) , (11)with parameters Ω e and w . This has a background ex-pansion similar to a massive neutrino for periods of theevolution of the universe. Previous work has consideredneutrino mass constraints within this model [e.g., 68, 69].We use the camb PPF module [70] and a modified ver-sion of camb from Calabrese et al. [68] to compute thepower spectrum within these models.We take as our baseline the S4 ( (cid:96) > m ν model, we find σ (Σ m ν )meV =
19 (ΛCDM+Σ m ν )30 (ΛCDM+Σ m ν +Ω k )27 (ΛCDM+Σ m ν + w )46 (ΛCDM+Σ m ν + w + w a )37 (ΛCDM+Σ m ν +Ω e + w )64 (ΛCDM+Σ m ν + w + w a +Ω k ) (12)We discuss these parameter degeneracies in the remain-der of this section. We find that marginalizing over neu-trino number N eff (and other extension parameters thatmodify the primordial CMB spectrum such as a runningspectral index) have a <
10% effect on the predicted neu-trino mass uncertainties. These findings are summarizedin Fig. 7, which also includes the corresponding S3 fore-casts.At first sight, the degradation of the neutrino massestimate in the case of varying w , w a and Ω k simulta-neously appears severe, more than tripling the error bar.However, this model has three extra parameters com-pared to ΛCDM, and within the Bayesian framework,one can rigorously ask whether additional parameters arerequired by the data, quantifying the trade-off betweenimproving the fit against an increased complexity of the ΛCDM+Σ m ν + N eff + Ω k + w + { w , w a } + { w , Ω e } σ ( Σ m ν ) [ m e V ] σ σ σ S3(wide)+BOSSS3(wide)+DESIS4( ‘ ≥ ‘ ≥ ΛCDM+Σ m ν + N eff + Ω k + w + { w , w a } + { w , Ω e } np FIG. 7: Neutrino mass constraints forecast for different data combinations and simple one- or two-parameter extensions to theΛCDM+Σ m ν model (all except S4( (cid:96) ≥
5) include
Planck -pol information at low- (cid:96) ). With CMB+BAO data, neutrino mass isnot correlated with the number of neutrino species, N eff , but is partly correlated with spatial curvature, Ω k , and with the darkenergy equation of state, w . Expected confidence levels are shown assuming the minimal total neutrino mass Σ m ν = 60 meV. model [e.g., 71, for discussion in the context of cosmolog-ical data]. This model selection approach would quantifythe need for additional extension parameters, and woulddisfavor an over-parameterized model if it is not requiredby the data. In practice our challenge is likely to liein distinguishing between different one-parameter exten-sions to ΛCDM: are we seeing non-zero neutrino mass, orcould it be similarly well-explained by a small amount ofcurvature, or a small deviation from a w = − A. Physical degeneracies
To help understand these degeneracies, we note theHubble parameter H ( z ) is given by (cid:20) H ( z ) H (cid:21) = Ω r (1 + z ) + Ω m (1 + z ) +Ω k (1 + z ) + Ω Λ ( z ) , at times after the neutrinos become non-relativistic,where Ω Λ ( z ) is the dark energy density, constant for w = −
1, Ω r is the radiation density today (e.g., pho-tons and massless neutrinos), Ω m = Ω c + Ω b + Ω ν is thematter density today (CDM, baryons and massive neu-trinos) and Ω ν h = Σ m ν /
93 eV is the physical massive neutrino density today. The angular diameter distanceis given by d A ( z ) = cH (1 + z ) √− Ω k sin (cid:16) √− Ω k (cid:82) z H dz (cid:48) H ( z (cid:48) ) (cid:17) Ω k < , (cid:82) z H dz (cid:48) H ( z (cid:48) ) Ω k = 0 , √ Ω k sinh (cid:16) √ Ω k (cid:82) z H dz (cid:48) H ( z (cid:48) ) (cid:17) Ω k > . (13)Considering curvature, photons propagating in a non-flat universe follow curved geodesics, changing the angu-lar diameter distance to an object of fixed proper size,at a given comoving distance, relative to a flat universe.Varying curvature shifts the angular scale of the acous-tic peaks, so to remain consistent with the CMB data,the matter density and Hubble constant H must varyto keep the peak structure unchanged. This is the well-known geometric degeneracy [72].This degeneracy is partially broken by CMB lensingmeasurements [e.g., 73], which are sensitive to the growthof structure in the late-time universe and therefore to thematter density Ω m and dark energy density Ω Λ . For afixed CMB acoustic peak scale, the effect of decreasingthe curvature parameter Ω k , moving to a more closeduniverse, is to decrease the Hubble constant and increaseΩ m . This enhances the amplitude of the CMB lensingpower spectrum, as illustrated in Fig. 8, which can becompensated by increasing Σ m ν . This leads to an anti-0 ` C ` ⇥ ⌦ k = 0⌦ k = +0.01⌦ k = FIG. 8: CMB convergence power spectrum for varying Ω k ,with other parameters holding the primary CMB fixed. De-creasing Ω k requires a smaller Hubble constant H and in-creased growth rate. This has a similar effect to decreasingthe neutrino mass. correlation between Σ m ν and Ω k when using CMB mea-surements alone, as shown in Fig. 9.The BAO constraint in the Σ m ν -Ω k plane is approxi-mately orthogonal to the CMB-only constraint, becausedecreasing Ω k decreases the volume distance to a givenredshift (Eq. 5). This can be compensated by a smallermatter density, lowering the neutrino mass. These dataare powerful in combination, and for S3+BOSS the neu-trino mass constraint is independent of curvature. How-ever, the neutrino mass constraint from S3+DESI orS4+DESI is expected to degrade by ≈
50% when allow-ing for curvature, as illustrated in Fig. 7.Similar arguments apply to dark energy, which mod-ifies the background evolution according to its equationof state, but does not contribute to clustering. Increas-ing the dark energy equation of state parameter w leadsto an increased expansion rate, shifting the angular scaleof the acoustic peaks, so to remain consistent with theCMB data the Hubble constant H decreases to keep thepeak structure unchanged. Similar to the curved model,this increases the clustering, which can be compensatedwith larger neutrino masses. This gives the positive cor-relation between w and Σ m ν in CMB data, illustratedin Fig. 9 and reported in e.g., [35, 74]. The BAO degener-acy is also positively correlated however, so the neutrinomass uncertainty is inflated more than when allowing forcurvature. Increasing Σ m ν increases the contribution ofneutrinos to Ω m , requiring a smaller Ω Λ (in a flat uni-verse); the volume distance to a given redshift, and hencethe BAO peak position, can then be preserved by increas-ing w . The early dark energy density parameter Ω e , isanti-correlated with neutrino mass, due to their similar Σ m ν [meV] − − − − Ω k Planck-polPlanck-pol+BAO-15S3-wideS3-wide+DESIS4 ( ‘ > ‘ > Σ m ν [meV] − − − w Planck-polPlanck-pol+BAOS3-wideS3-wide+DESIS4S4+DESI
FIG. 9:
Top:
Forecast joint constraint on the neutrino mass,Σ m ν , and spatial curvature, Ω k , within a ΛCDM+Σ m ν +Ω k model. The BAO data breaks the anti-correlated degeneracyin the CMB data. Bottom:
Forecast constraint on Σ m ν andthe dark energy equation of state, w , marginalized over theΛCDM and w a parameters. effects on the background expansion [68, 69]. We find that the neutrino mass constraints fromS3+DESI or S4+DESI are degraded by more than a fac-tor of two when allowing for a non-minimal ( w (cid:54) = − m ν +Ω k + w + w a model) degrades the con-straint further to σ (Σ m ν ) = 64 meV for S4+DESI, butwould include three new parameters. Here we adopted a fiducial Ω e = 0 . e would improve the neutrino mass constraint,as the parameters are anti-correlated and Ω e cannot be negative.
100 200 300 400 500 ‘ ‘ C κ g κ g ‘ × − FIG. 10: Galaxy lensing power spectra for two models degen-erate in CMB spectra and BAO distance ratios. They couldbe distinguished using a future galaxy weak lensing survey( f sky = 0 . n gal = 10 arcmin − , and source redshift distribu-tion n ( z ) as for the LSST ‘gold sample’ [76]). B. Breaking degeneracies with complementarymeasurements
We have focused so far on the minimal combinationof future CMB and BAO data. To break these re-maining degeneracies between neutrino mass and curva-ture/dark energy parameters, we would turn to otherlarge-scale structure probes such as the galaxy powerspectrum, redshift-space distortions, galaxy weak lens-ing, the kinematic Sunyaev-Zel’dovich effect and galaxycluster counts, which measure the growth of structure atlater times, and Type Ia supernovae which better con-strain the expansion rate.We have not included these and other data in our base-line forecasts as they are, arguably, more prone to sys-tematic uncertainties such as tracer bias and the shapeof the non-linear power spectrum, which contains un-known baryonic feedback effects, source distribution un-certainties and multiplicative bias [19, 32, 77]. However,a promising path would be to examine a suite of differ-ent complementary probes, each in combination with theCMB and BAO data, to distinguish neutrino mass froma non-flat or non-Λ model.For example, in Fig. 10 we illustrate schematically howgalaxy weak lensing can break remaining degeneraciesbetween two example models. Here the CMB spectraand BAO distances are indistinguishable with S4+DESIdata (see parameters in the Appendix).Their galaxy weak lensing signals, which probe thegrowth of structure during the dark energy dominatedera, are distinct, and should be distinguishable with aplausible future weak lensing survey. With a 2% differ-ence in σ , robust galaxy cluster measurements shouldalso help discriminate between these models [e.g., 78, 79]. VI. DISCUSSION AND CONCLUSIONS
We have demonstrated that an indirect detection of thesum of the neutrino masses should be possible with up-coming CMB and BAO surveys. In the next decade a 4 σ detection should be reachable, within the ΛCDM+Σ m ν model, even in the minimal mass (Σ m ν = 60 meV) sce-nario. We have found that this is contingent on obtain-ing improved large-scale polarization measurements fromthe CMB, which may be the hardest experimental chal-lenge. It will also be necessary to exclude degeneracieswith other plausible extensions such as curvature anddark energy. We find that allowing for these extensionsdegrades the expected neutrino mass constraint, and thatuse of other large-scale structure probes will be necessaryto rule out other departures from ΛCDM.Our forecasts make a number of assumptions. We haveneglected non-Gaussian terms in the CMB covariance dueto correlation of the temperature, polarization, and lens-ing fields, which should have a small impact on immi-nent data but will get more important as noise levelsreduce. We assume Gaussianity of the posterior distri-bution and rely on data-independence of the covariancematrix. These assumptions become increasingly accuratefor the precision future experiments considered here, butremain an approximation. Our analysis also assumes lit-tle foreground contamination. This will be most impor-tant for the large-scale CMB signal, but is not expectedto significantly degrade the smaller-scale lensing signal.We have assumed white noise in the new CMB data.This has not yet typically been achieved in practice fromthe ground, due to atmospheric and scan-synchronousemission which induces additional variance at largescales. However, polarization measurements are cleanerthan temperature in this respect, and new experimentshave sophisticated designs to modulate the polarizationsignal. Performance of current ‘Stage-2’ experiments willhelp refine noise projections for the future experiments.We have also assumed ideal performance of the futureDESI BAO experiment.In terms of theoretical scope, beyond an evolvingdark energy equation of state, we have not consideredother possible new physics. For instance, theoretically-motivated axion contributions to the background expan-sions and perturbations might offer an alternative expla-nation for a neutrino mass detection. The cosmologicaleffects are not identical [80], but further investigation willbe useful.Finally, neutrino mass is the next beyond-ΛCDM pa-rameter that we know will be needed to fit data, soit is valuable that competitive constraints are expectedto come from different combinations of cosmologicaldatasets beyond the CMB/BAO considered here. Theircomplementarity will aid in convincingly excluding sys-tematic effects and alternative cosmological models.2 Acknowledgments
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We forecast posterior parameter constraints and de-generacy directions by evaluating the relevant Fisher ma-trix as defined in Eq. 6. We assume uniform priors on allparameters, allowing the posterior p ( θ ) to be replaced4 Parameter Fiducial value Step sizeΩ b h c h θ A A s n s τ m ν (meV) 60 20Ω k w -1 0.3 w a N eff e camb powerspectra. The fiducial value of Ω e = 0 .
007 is at the upper 95%confidence level given current data [75]. with the likelihood L ( θ ) = P ( d | θ , M ) in Eq. 6. TheFisher matrix is evaluated at fiducial parameters ; thechoice is unimportant under the assumption of a Gaus-sian posterior and data-independent covariance, althoughin practice they are chosen to match the current best-fitparameters of the model, as the real covariance does havea term that scales with the signal. Our choices are shownin Table III.To compute the Fisher matrix we use as observablesthe CMB power spectra, and the BAO distance measure-ments. For the CMB, we use the lensed power spectrumbetween each pair of (assumed Gaussian) fields X, Y fromtheir spherical harmonic coefficients:ˆ C XY(cid:96) = 12 (cid:96) + 1 m = (cid:96) (cid:88) m = − (cid:96) x ∗ (cid:96)m y (cid:96)m . (A.1)This formula omits beam smoothing effects and the sub-traction of a noise term, which we account for below.The estimated power spectrum is a sum of many ran-dom variables of finite variance, and to good approxima-tion follows a Gaussian distribution. This approximationbreaks down at large scales but does not have a signifi-cant impact on expected errors. For a full-sky survey, wehave − L ( θ ) = − (cid:88) (cid:96) ln p ( ˆ C (cid:96) | θ )= (cid:88) (cid:96) (cid:104) ( ˆ C (cid:96) − C (cid:96) ( θ )) (cid:62) C − (cid:96) ( θ ) (cid:0) ˆ C (cid:96) − C (cid:96) ( θ ))+ln det(2 π C (cid:96) ( θ )) (cid:105) (A.2)where ˆ C (cid:96) = ( ˆ C T T(cid:96) , ˆ C T E(cid:96) , ... ) contains auto- and cross-spectra and C (cid:96) is their covariance matrix. Inserting this likelihood into Eq. 6 and neglecting parameter depen-dence in the power spectrum covariance matrix one ob-tains F ij = (cid:88) (cid:96) ∂C (cid:62) l ∂θ i C − (cid:96) ∂C l ∂θ j . (A.3)From Eq. A.1, and applying Wick’s theorem, the covari-ance matrix for the power spectra has elements C ( ˆ C αβl , ˆ C γδl ) = 1(2 l + 1) f sky (cid:2) ( C αγl + N αγl )( C βδl + N βδl )+ ( C αδl + N αδl )( C βγl + N βγl ) (cid:3) , (A.4)where α, β, γ, δ ∈ { T, E, B, κ c } and f sky accounts for theloss of information due to partial sky coverage [81, 82].Noise spectra are generated for each observable given in-put noise properties such as CMB map sensitivities. Weassume additive white-noise for the CMB: N αα(cid:96) = (∆ T ) exp (cid:18) (cid:96) ( (cid:96) + 1) θ (cid:19) (A.5)for α ∈ { T, E, B } , where ∆ T (∆ P for polarization) is themap sensitivity in µ K-arcmin and θ FWHM is the beamwidth. This as an optimistic approximation: real noise-spectra from ground-based experiments have a dominantcontribution from atmospheric variance at large scales(see e.g. Fig. 4 of Das et al. [77]). The atmosphereis weakly polarized, and hence the white-noise approxi-mation is better in E and B than T . The CMB lensingreconstruction noise is calculated using the [5] quadratic-estimator formalism. As described in the main text, weneglect non-Gaussian terms in the power spectrum co-variance, and also neglect the BB spectrum as it doesnot contribute significantly to upcoming constraints andhas a highly non-Gaussian covariance [35].We add information from Baryon Acoustic Oscillation(BAO) experiments by computing the BAO Fisher ma-trix: F BAO ij = (cid:88) k σ f,k ∂f k ∂θ i ∂f k ∂θ j (A.6)where f k ≡ f ( z k ) = r s /d V ( z k ) is the sound horizon atphoton-baryon decoupling r s over the volume distance d V to the source galaxies at redshift z k . These real andforecast data are reported in Table V.The total Fisher information matrix is given by thesum of the CMB and BAO Fisher matrices, and is in-verted to forecast parameter covariances. An alternativeMCMC approach using simulated data can be taken toaccount for non-Gaussianity of the posterior [e.g., 25],but the Gaussian approximation is likely increasinglygood as the data quality improve from Planck throughS3 to S4.Our forecasting code,
OxFish , has been developed forthis analysis and is used to forecast parameter covariance5
Experiment f sky ν /GHz l min l max FWHM/arcmin ∆ T / µ K-arcmin ∆ P / µ K-arcmin
Planck -2015 0 .
44 30,44,70,100,143,217,353 2 2500 33,23,14,10,7,5,5,5 145,149,137,65,43,66,200 -,-,450,-,-,-,-
Planck -pol ” -,-,450,103,81,134,406
WMAP -pol 0.74 33,41,64,94 2 1000 41,28,21,13 -,-,298,296 425,420,424,-TABLE IV: Specification for the
Planck and
WMAP experiments used in the analysis, assuming white noise properties. Wedefine ‘
Planck -2015’ to reproduce the constraints from
Planck
Planck -pol’ we use the
Planck
Blue Book scalingfactors to convert to polarization. For
WMAP -pol we recover an optical depth uncertainty that matches the WMAP9 data.When combining with S3 and S4, we include
Planck -pol data across the full f sky = 0 .
44 at large scales ( (cid:96) < (cid:96) S3 / S4min ) and across f sky = 0 . (cid:96) S3 / S4min < (cid:96) <
Planck data and S3/S4 will likely not overlap completely. Ourresults are insensitive to the exact choice of this non-overlapping region size. When using
WMAP -pol data we substitute it inat large scales ( (cid:96) < (cid:96) S3 / S4min ) over f sky = 0 . σ ( r s /d V )( r s /d V ) σ ( r s /d V ) Ref(%)6dFGRS 0.106 4.83 0.0084 [45]SDSS MGS 0.15 3.87 0.015 [46]LOW-Z 0.32 2.35 0.0023 [18]C-MASS 0.57 1.33 0.00071 [18]DESI 0.15 1.89 0.0041 [22]0.25 1.26 0.00170.35 0.98 0.000880.45 0.80 0.000550.55 0.68 0.000380.65 0.60 0.000280.75 0.52 0.000210.85 0.51 0.000180.95 0.56 0.000181.05 0.59 0.000171.15 0.60 0.000161.25 0.57 0.000141.35 0.66 0.000151.45 0.75 0.000161.55 0.95 0.000191.65 1.48 0.000281.75 2.28 0.000411.85 3.03 0.00052TABLE V: Specification for current BAO-15 data (top), andforecast DESI data (bottom). We derive the expected frac-tional uncertainties on r s /d V for DESI from the fractionalerrors on D A /r s and H ( z ) forecast in [22]. The absolute val-ues correspond to a ΛCDM model with Σ m ν = 60 meV. matrices in one coherent python package. The code inter-faces with the camb code for evaluation of power spectra.We compare with real data or previous work where possi-ble. We construct the ‘Planck-2015’ (P15) specification,given in Table IV, to produce constraints which matchthe ΛCDM uncertainties in Planck Collaboration et al.[19], with the beam sizes and noise levels matching thedetector sensitivities in [83]. We also forecast the neutrino mass constraint fromP15+BAO, finding σ (Σ m ν ) = 103 meV. Placing the peakof the posterior at the fiducial Σ m ν = 60 meV, this cor-responds to Σ m ν <
245 meV at 95% confidence, compa-rable to the actual result of Σ m ν <
230 meV [19]. Thisincludes JLA SNe data and an H prior, but these are ex-pected to have a small impact. We also agree with [26–28]on the neutrino mass constraint for the S4+DESI datacombination, finding σ (Σ m ν ) = 15 meV if we assumethat the reionization bump is measured.We use the following parameters for the curves inFig. 10: solid curve; { Ω b h = 0 . c h = 0 . m ν = 3 meV, τ = 0 . θ A = 1 . A s = 2 . n s = 0 . σ = 0 . H = 68 . } , dashed curve; { Ω b h = 0 . c h = 0 . m ν = 117 meV, τ = 0 . θ A = 1 . A s = 2 . n s = 0 . σ = 0 . H = 66 . }}