Measurement of Neutrino Masses from Relative Velocities
aa r X i v : . [ a s t r o - ph . C O ] M a y Measurement of Neutrino Masses from Relative Velocities
Hong-Ming Zhu, Ue-Li Pen,
2, 3
Xuelei Chen,
1, 4
Derek Inman, and Yu Yu Key Laboratory for Computational Astrophysics, National Astronomical Observatories,Chinese Academy of Sciences, 20A Datun Road, Beijing 100012, China Canadian Institute for Theoretical Astrophysics, 60 St. George Street, Toronto, Ontario M5S 3H8, Canada Canadian Institute for Advanced Research, CIFAR Program in Gravitation and Cosmology, Toronto, Ontario M5G 1Z8, Canada Center of High Energy Physics, Peking University, Beijing 100871, China Key laboratory for research in galaxies and cosmology, Shanghai Astronomical Observatory,Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China (Dated: May 11, 2016)We present a new technique to measure neutrino masses using their flow field relative to dark matter. Presentday streaming motions of neutrinos relative to dark matter and baryons are several hundred km/s, comparablewith their thermal velocity dispersion. This results in a unique dipole anisotropic distortion of the matter-neutrino cross power spectrum, which is observable through the dipole distortion in the cross correlation ofdifferent galaxy populations. Such a dipole vanishes if not for this relative velocity and so it is a clean signaturefor neutrino mass. We estimate the size of this effect and find that current and future galaxy surveys may besensitive to these signature distortions.
PACS numbers: 98.65.Dx, 14.60.Pq, 95.35.+d, 95.80.+p
Introduction. —Neutrinos are now established to be mas-sive, and the mass differences have been measured, but themass hierarchy and absolute mass values remain unknown [1].Precision large scale structure data can be used to measureor constrain the sum of neutrino masses, as cosmic neutrinoswith finite masses slightly suppress the growth of structure onscales below the neutrino thermal free-streaming scale [2–5].But the challenge of this method is to conclusively disentanglethe complex and poorly understood baryonic effects as manyprocesses can lead to power suppression on small scales. Inthis Letter, we present an astrophysical effect which providesa new way to measure the neutrino masses by using a distinctsignature in current or future galaxy surveys.We consider the relative velocity between cold dark mat-ter (CDM) and neutrinos. Neutrinos decoupled early in thehistory of the Universe when they were still relativistic, buttheir energy gradually decreased as the Universe expandeduntil they behaved as nonrelativistic particles. At this pointthey can cluster under the action of gravity. Nevertheless, dueto their low masses the neutrinos can travel relatively largedistances (even at low redshifts), and be perturbed by the un-derlying gravitational potential along their trajectories. Thelarge scale structures can induce a significant bulk relative ve-locity field between CDM and neutrinos, with typical veloc-ities comparable to the neutrino thermal velocity dispersion.As we shall show below, such a bulk relative velocity fieldwill cause a local dipole asymmetry in the CDM-neutrinocross-correlation function. The concept of dipole asymme-try in correlation functions was discussed in Ref. [6] recently.The CDM-neutrino cross correlation may be inferred from thecross-correlation of different galaxy populations, and such adipole asymmetry provides a distinctive and robust signatureof neutrino mass, since such dipole anisotropy would be ab-sent if not for this effect.In this Letter, we delineate the principle of this method, make an analytical estimate of the size of this effect, and thenforecast the detectability of this effect in a simplified galaxybias model.
The relative velocity. —We treat CDM and neutrinos as twofluids [7] interacting with each other through gravity. TheCDM particles and neutrinos are collisionless, neverthelessmuch of their behavior in gravitational fields can still be mod-eled with the introduction of an “effective pressure,” whichtakes into account the velocity dispersion or thermal motionof the particles [7]. In the fluid approximation, the effect ofthe thermal motion is included in this effective pressure andonly the bulk motion is considered. The two fluids have differ-ent effective pressure, so they acquire different densities andvelocities even though they are under the action of the samegravitational field. We use the moving background perturba-tion theory (MBPT) [8] to calculate analytically the evolutionof the density perturbations and velocities of the two fluids,the details of this calculation are given in the SupplementalMaterial [9]. The basic idea is to assume that within a cer-tain volume of radius R , each fluid has a coherent bulk ve-locity, which can be expanded around a background velocityas v i ( x , t ) = v ( bg ) i ( t ) + u i ( x , t ) , where i refers to neutrino ( ν ) or cold dark matter ( c ) . The background velocity v ( bg ) i is a slowly varying velocity long mode. Linear perturbativecalculation then can be applied within the region to obtain thecross-correlation of the two fluids.Starting at a high redshift (we use z = 15 in our calcula-tion) when the relative bulk mach number is small, we evolvethe MBPT equations down to lower redshifts, and obtain therelative velocity field v νc ( x , z ) . We estimate the variance ofthis relative velocity analytically by taking the ensemble aver-age for the given distribution of primordial fluctuations: h v νc ( z ) i = Z dkk ∆ ζ ( k ) (cid:20) θ ν ( k, z ) − θ c ( k, z ) k (cid:21) . (1) v [ k m / s ] σ ν ν ν ν rv rv rv rv FIG. 1: Redshift evolution of the neutrino velocity dispersion (thethin lines on top) and the neutrino-CDM relative velocity (thick linesat the bottom) for different neutrino masses. where ∆ ζ is the primordial curvature perturbation spectrum,and θ ≡ ∇· v is the velocity divergence. We plot the evolutionof p h v νc i ( σ rv ) and the neutrino thermal velocity dispersion σ ν for four neutrino masses in Fig. 1. The thermal velocitydispersion of the neutrinos decreases as the Universe expands.On the other hand, the bulk relative velocity as represented by p h v νc i grows to its maximum at < z < , then beginsto decay. At low redshifts it is comparable with the thermalvelocity dispersion.The relative velocity correlation function ξ vνc ( r ) ≡h v νc ( x ) v νc ( x + r ) i for four redshifts are shown in Fig.2.The bulk velocity correlation functions for different neutrinomasses are almost identical at very high redshifts, but becomeincreasingly differentiated at low redshifts, as the correlationfunctions of the lighter neutrinos have larger amplitudes andlonger correlation lengths. The coherent scales R , which isdefined as the scale at which the correlation function ξ vνc drops to half of its maximum value, are 14.5, 10.3, 7.0, and 4.6 Mpc /h , respectively, for the four neutrino masses at z = 0 .However, the neutrinos are not visible, so we cannot use thiscorrelation function to measure neutrino mass directly. Power spectra and correlation functions. —Because of thebulk relative velocity between the CDM and neutrinos, thereflection symmetry along the direction of the flow is bro-ken locally, and within a velocity coherent region the cross-correlation contains a dipole term, ξ cν ( r , v ( bg ) νc ) = ξ cν ( r, v ( bg ) νc ) + µξ cν ( r, v ( bg ) νc ) , (2)where µ = r · v ( bg ) νc . This also appears as an imaginary part inthe CDM-neutrino cross power spectrum: P cν ( k, v ( bg ) νc , µ ) = P cν ( k, v ( bg ) νc ) + iµP cν ( k, v ( bg ) νc ) . [We can see this by not-ing that when taking the Hermite conjugate of P cν , the imag-inary part changes sign and so the angular dependent part isantisymmetric in “ cν ,” i.e., ξ νc ( r , v ( bg ) νc ) = ξ cν ( r, v ( bg ) νc ) − µξ cν ( r, v ( bg ) νc ) .] This imaginary term would otherwise bezero if not for the relative flow between neutrinos and CDM. ξ v ν c [ ( k m / s ) ] z=1020 z=10 r [h −1 Mpc] ξ v ν c [ ( k m / s ) ] z=1 r [h −1 Mpc] z=0m ν =0.05eVm ν =0.10eVm ν =0.20eVm ν =0.40eV FIG. 2: The relative flow correlation function ξ vνc ( r ) at differentredshifts. The amplitude and scale of the relative flow depends onneutrino mass. The tick marks the correlation length. This effect is similar to gravitational redshift [10], whichbreaks the reflection symmetry along the line of sight, andcauses an imaginary part in the cross power spectrum betweentwo types of galaxies.Taking p h v νc i as the representative value for the back-ground velocity, we calculate the induced density correlationsusing MBPT. Figure 3 shows the monopole and the absolutevalue of the dipole (most parts of it are negative) terms of theCDM-neutrino cross power spectrum as well as the CDM au-topower spectrum for four different neutrino masses. The os-cillations in P cν (dotted line) are due to the sharp sound hori-zon which is an artifact of the fluid approximation of neutrinosin our calculation. (We have verified that the oscillation periodis inversely proportionate to the effective sound speed, so it isdue to the (false) acoustic oscillation in the fluid. Real neutri-nos are not a collisional fluid, and the effective sound speedis actually a superposition of different sound speeds, so we donot expect the true cross power spectrum to exhibit these oscil-lations.) We have thus smoothed the dipole power spectrumand obtained an average ¯ P cν , which is shown as the solidline, for the different neutrino masses the power spectra aredifferent and distinguishable. Figure 4 shows, respectively,the CDM autocorrelation function, the neutrino autocorrela-tion function, and the monopole and dipole part of CDM-neutrino cross correlation functions. We find that the neutrinoautocorrelation grows as the neutrino mass increases, sincethe more massive neutrinos tend to form more structures. Thedipole term of the cross power spectrum have a broad peak orhump, its amplitude also grows with the neutrino mass. Thescales of the peaks in the correlation function decrease withneutrino mass, and are located at 16, 11, 7, and 5 Mpc/ h , re-spectively, for the four neutrino masses.We have taken a single value of v ( bg ) = p h v νc i for eachneutrino mass. For a given background velocity value, thedipole correlation depends on the neutrino mass value, as isshown in the equations in the Supplemental Material [9]. But -3 -2 -1 P ( k ) [ ( M p c / h ) ] m ν =0.05eV -3 -2 -1 m ν =0.10eV -2 -1 k [h Mpc −1 ] -3 -2 -1 P ( k ) [ ( M p c / h ) ] m ν =0.20eV -2 -1 k [h Mpc −1 ] -3 -2 -1 m ν =0.40eVP c P cν0 |P cν1 |−¯P cν1 FIG. 3: The power spectra of CDM and neutrinos. The CDM auto-power P c , neutrino monopole P cν and dipole P cν , and smootheddipole term ¯ P cν are plotted. in fact the bulk relative velocity varies from point to point inspace. A more rigorous treatment would require a considera-tion of the distribution of the bulk velocity. The fact that boththe typical value of bulk velocity and the dipole correlation fora given background velocity depend on the neutrino mass en-hances the sensitivity for this technique. Below for simplicitywe will consider only the typical values. -3 -2 -1 C o rr e l a t i o n f un c t i o n m ν =0.05eV -3 -2 -1 m ν =0.10eV r [h −1 Mpc] -3 -2 -1 C o rr e l a t i o n f un c t i o n m ν =0.20eV r [h −1 Mpc] -3 -2 -1 m ν =0.40eVCDMmonopoleNeutrinodipole FIG. 4: The correlation functions, including CDM and neutrino au-tocorrelations, and the monopole and dipole part of their cross corre-lations. The dipole is ∆ corr ≡ ξ cν | µ =1 µ = − . Observability. —Neither the neutrinos nor the dark mattercan be observed directly, but as their densities affect galaxydensities, their cross power can be inferred from the crosspower of galaxies of different populations, provided that thebiases of the two populations have different dependences onneutrinos and dark matter. Galaxies are known to be biasedrelative to each other [11]. The 21cm HIPASS galaxies typi-cally have a bias of b c ∼ . [12] relative to the dark matter,whereas the bias for luminous red galaxies is typically greater than 1. For a galaxy population, we assume its density con-trast is related to the dark matter and neutrino density contrasts δ c , δ ν as δ g = b c f c δ c + b ν f ν δ ν , where f c = Ω c / (Ω c + Ω ν ) and f ν = Ω ν / (Ω c + Ω ν ) [13]. Since the halo mass scale ∼ M ⊙ is smaller than the neutrino free streamingand coherent scales, we expect the neutrino bias to be insen-sitive to halo mass. This can also be seen by deriving the halobias via the peak-background split formalism or the extendedPress-Schechter formalism (see, e.g., Ref. [14]) with neutrinofluctuations only affecting the large scale background density.For the following calculations, we choose b ν to be 1 but em-phasize that an effect will be present as long as b ν is the samefor both galaxy populations, regardless of the particular value.The precise value could be calculated with the more elaboratetreatment as prescribed in Ref. [15].If we consider the cross-correlation of two galaxy popula-tions denoted by α, β , and use b α , b β to denote b c for α, β ,then ξ αβ = h δ α δ β i = b α b β f c ξ c + ( b α + b β ) f c f ν ξ cν + f ν ξ ν . Now consider the µ dependence of ξ αβ : because the crosscorrelation function is antisymmetric in “ cν ,” a dipole µ ( b α − b β ) f c f ν ξ cν appears. The observability of this dipole dependson the relative bias, ∆ b ≡ b α − b β . The known spread information bias provides a lower bound on ∆ b & . . Forsensitivity estimation, we will adopt ∆ b = 1 . The actual errorbar of the inferred neutrino mass will depend on the productof ∆ b and galaxy number density n g .For this measurement, the bulk velocity field can be recon-structed from the observed density field, v νc ( k ) = δ g ( k ) [ T θ,ν ( k ) − T θ,c ( k )] T δ,g ( k ) i k k . (3)Here T θ,ν ( k ) , T θ,c ( k ) are the velocity-divergence transferfunctions for neutrino and dark matter respectively, and T δ,g ( k ) is the density transfer function, which depends on theunknown neutrino mass. In practice, one can iterate the recon-struction with different trial masses m ν , until a self-consistentrelative velocity field v νc and dipole value is found. At thehigh sampling densities considered here, the fractional errorin v νc is comparable to the error in the CDM density field δ . The shot noise is much smaller than the sample variance,making the error on the velocity field negligible at the scalesof interest.The correlation function provides a local operational proce-dure to measure the dipole, ξ αβ ( r, µ ) = 1 N X x X | ∆ x |∼ r ˆ v νc · ˆ∆ x ∼ µ δ α ( x ) δ β ( x + ∆ x ) , (4)where N is appropriate normalization. The dipole term canbe extracted from this anisotropic correlation as in Eq.(2).Taking the Fourier transform then yields the power spectrumdipole. The error bar is easier to specify for the power spec-trum than the correlation function, since k bins are statistically TABLE I: The forecasted error on neutrino mass with a survey of V s = 1 . h − Gpc , n g = 2 . × − h Mpc − and with currentsurvey data, modeled with SDSS and 2dF as V s = 0 . h − Gpc , n g V s = 1 × . Note that substantial uncertainties exist due tounknown galaxy neutrino bias, which is a nuisance parameter thatwe marginalize over.current (SDSS) future m ν (eV) σ m ν relative error σ m ν relative error0.05 0.045 0.90 0.0042 0.0840.10 0.044 0.44 0.0041 0.0410.20 0.079 0.40 0.0074 0.0370.40 0.097 0.24 0.0091 0.023 independent. The transformation from real space to redshiftspace does not change our error estimate because the dipoleis orthogonal to the effect of redshift distortion, which is aquadrupole distortion.In Fig. 3, we plot the expected error bars of the angular-dependent CDM-neutrino cross power spectrum for a sur-vey with volume V s = 1 . h − Gpc and n g ∆ b = 2 . × − h Mpc − . This corresponds to an all-sky survey out toredshift z < . , comparable to the sloan digital sky survey(SDSS) main sample volume, but with a tenfold higher galaxysampling density, about the density of HIPASS galaxies [16].The two populations of galaxies could be, for example, a deepoptical survey and an HI survey at low redshifts. Alternatively,the second tracer might be obtained by a nonlinear weightingof the same density field such as the cosmic tide field [17].We proceed to calculate the error on the neutrino mass mea-surement using a Fisher matrix estimate. We use five k bins( k = 0 . , 0.12, 0.24, 0.47, 0.94 h /Mpc) in Fig. 3. Modeswith smaller k are not used because MBPT is not a very goodapproximation unless the background velocity comes fromscales larger than the k mode. We fit for two parameters: amultiplicative (relative) galaxy bias ∆ b , treated as a nuisanceparameter, and a neutrino mass, and marginalize the resultover the relative bias. The result is given in Table I for thefour different neutrino masses. Existing galaxy redshift datamay result in a detection for optimistic neutrino mass and biasparameters. Future surveys can measure the neutrino massesprecisely. Discussions. —The neutrino mass measurement methodproposed here differs from the one based on small scale powerspectrum suppression, and it is more robust to scale-dependentgalaxy biasing. In the approach based on power suppression,if for some reason there is a weak scale-dependent variationof bias at the level of ∼ , it can completely swamp theneutrino signal. In our dipole cross correlation approach, themeasured signal arises only from the relative velocity effect.If the galaxy bias were to depend on scale, the impact on theinferred neutrino mass would only be proportionate to anysuch changes, unlike for total power measurements where anyuncertainty in bias is amplified by 2 orders of magnitude ormore.For the cases we considered, the correlation function peaksoccur at scales ( , , , Mpc/h) comparable to the relative velocity field coherency scales ( . , . , . , . Mpc/h);this is not unexpected as it is the coherence of the bulk velocitywhich induces such correlation. However, for the analyticalMBPT calculation we used here, it does pose a problem, be-cause strictly speaking the MBPT approximation is valid onlyfor scales below the coherence scale. The nonlinear effects be-come significant for k & . h /Mpc. Nevertheless, the essenceof large scale velocity modulation and the expected physicaleffect (the dipole structure) is still captured in the calculation,though quantitatively it may not be very accurate at the largestscales. This can be remedied with numerical simulations. Wewill study this in a future paper; preliminary results, however,show that the result is generally consistent with the analyticalone.In our Fisher analysis, we have treated the galaxy relativebias as a nuisance parameter. As described above, the sensi-tivity to this effect depends on n g ∆ b and so the galaxy densityneeded to detect this dipole depends on the bias. In any givendetection of the dipole, ∆ b is immediately known, and thusthe error on the neutrino mass would also be known. The un-certainty in the bias, and thus the error, is proportionate to thesignificance of the detection, i.e. for a σ detection, there isan additional uncertainty in the error itself.In the above we have considered a single neutrino mass. Infact, unlike the power spectrum suppression effect, which issensitive only to the sum of the neutrino masses, the dipoleeffect discussed here can in principle be used to measure themass of a single neutrino. For multiple neutrinos, the differentmass eigenstates will have different bulk velocity directionsfor each of them, which at least in theory can be solved inde-pendently by repeating this procedure once for each mass. Inpractice this may be difficult, but if one or two neutrino massesare dominant and degenerate, then the procedure discussed inthis Letter is already sufficient. 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