Strong Upper Limits on Sterile Neutrino Warm Dark Matter
aa r X i v : . [ a s t r o - ph ] S e p Strong Upper Limits on Sterile Neutrino Warm Dark Matter
Hasan Y¨uksel,
1, 2
John F. Beacom,
1, 2, 3 and Casey R. Watson Department of Physics, Ohio State University, Columbus, Ohio 43210 Center for Cosmology and Astro-Particle Physics, Ohio State University, Columbus, Ohio 43210 Department of Astronomy, Ohio State University, Columbus, Ohio 43210 Department of Physics and Astronomy, Millikin University, Decatur, Illinois 62522 (Dated: 27 June 2007)Sterile neutrinos are attractive dark matter candidates. Their parameter space of mass andmixing angle has not yet been fully tested despite intensive efforts that exploit their gravitationalclustering properties and radiative decays. We use the limits on gamma-ray line emission from theGalactic Center region obtained with the SPI spectrometer on the INTEGRAL satellite to set newconstraints, which improve on the earlier bounds on mixing by more than two orders of magnitude,and thus strongly restrict a wide and interesting range of models.
PACS numbers: 95.35.+d, 13.35.Hb, 14.60.St, 14.60.Pq
Introduction.—
The existence of dark matter is cer-tain, but the properties of the dark matter particlesare only poorly constrained, with several attractive butrather different candidates. One of these, sterile neutri-nos, would be a very plausible addition to the StandardModel [1, 2, 3, 4]. If their masses were in the range ∼ . −
100 keV, they would also act as “warm” darkmatter [2, 3, 4], which could be in better accord with ob-servations than standard “cold” dark matter candidates.Even if sterile neutrinos are not a dominant componentof the dark matter, they may still exist and cause otherinteresting effects [5, 6], such as pulsar kicks [7], and mayaffect reionization [8]. It is therefore important to deeplyprobe the sterile neutrino parameter space, as definedby the mass m s and mixing sin θ with ordinary activeneutrinos, and shown in Fig. 1.One means of testing sterile neutrino dark matter mod-els is through cosmological searches, which rely on theeffects of sterile neutrino dark matter on the large-scalestructure of gravitationally-collapsed objects. While re-cent results based on the clustering of the Lyman- α forestand on other data have been interpreted as lower limitson the sterile neutrino mass of up to about 10–13 keV, in-dependent of the mixing angle [9], these constraints maybe weakened depending on the sterile neutrino produc-tion model (e.g., Ref. [10]).Another means of constraining sterile neutrino darkmatter is through their radiative decay to active neutri-nos, ν s → ν a + γ . These decays produce mono-energeticphotons with E γ = m s /
2. While the decay rate is exceed-ingly slow due to the tiny active-sterile mixing, modernsatellite experiments can detect even these very smallx-ray/gamma-ray fluxes, and such a signal could specif-ically identify a sterile neutrino dark matter candidate.The signal from nearby dark matter halos is line emissionand the cosmic signal from all distant halos is broad-ened in energy by the integration over redshift. There are limits obtained using the Cosmic X-ray Background(CXB) data [4, 11] and, at lower masses, stronger lim-its using data from a variety of nearby sources (see e.g.,Refs. [12, 13] and references therein).It is important to improve on both the cosmologicaland radiative decay constraints; despite intensive efforts,viable models that match the observed dark matter den-sity still remain. In fact, it has recently been empha-sized [2, 4] that some models may extend to regions ofthe parameter space far from the earliest and simplestmodels [3] to much smaller mixing angles. We calculatethe gamma-ray flux from dark matter decays around theMilky Way center and compare this to the limits on the -20 -18 -16 -14 -12 -10 -8 -6 sin θ m s [ k e V ] CXB
Milky Way Constraints
Limits from γ -ray line search Lyman- α Limits X-ray
FIG. 1: The sterile neutrino dark matter mass m s and mixingsin θ parameter space, with shaded regions excluded. Thestrongest radiative decay bounds are shown, labeled as MilkyWay (this paper), CXB [11], and X-ray Limits (summarizedusing Ref. [12]; the others [13] are comparable). The strongestcosmological bounds [9] are shown by the horizontal band(see caveats in the text). The excluded Dodelson-Widrow [3]model is shown by the solid line; rightward, the dark matterdensity is too high (stripes). The dotted lines are models fromRef. [14], now truncated by our constraints. line emission flux from the INTEGRAL satellite. Thehigh sensitivity and spectral resolution of the availabledata enable us to derive new and very stringent con-straints. For masses above 40 keV, this improves on theCXB constraints [11] on the mixing angle by more thantwo orders of magnitude. INTEGRAL Gamma-Ray Line Search.—
Tee-garden and Watanabe have reported results from asearch for gamma-ray line emission from point and dif-fuse sources in the Milky Way [15], using the SPI spec-trometer on the INTEGRAL satellite [16]. In the en-ergy range 20–8000 keV, they tested for lines of intrin-sic width 0, 10, 100, and 1000 keV. The additional linewidth due to instrumental resolution increases over theabove energy range from ∼ ∼ − , which is therefore small enoughto be neglected.Two large-scale regions around the Galactic Centerwere considered, with angular radii of 13 ◦ and 30 ◦ , andexposures of 1.9 and 3.6 Ms, respectively. The 24 ◦ col-limated field of view was used without the coded maskimage reconstruction and the corresponding limits on theflux from an unknown line emission were derived by de-convolving an assumed sky brightness distribution (ei-ther a Gaussian with 10 ◦ FWHM for the former or flatfor the latter region) and the wide angular response ofthe collimator. To improve the sensitivity to line emis-sion specifically from these regions, the average flux awayfrom the Galactic Center Region (angular radii of > ◦ )was subtracted from the flux from inside the GalacticCenter Region. This procedure cancels almost all of theinstrumental backgrounds. This also cancels all of thecosmic signal and part of the halo signal, and a carefulcalculation of the latter effect is taken into account in ouranalysis. For the Galactic Center Region, the 3 . σ limitson narrow line emission are < ∼ − photons cm − s − forthe full range of energies. The actual energy dependenceof the limiting flux, F lim ( E ), is more complicated, andwe took this into account (leading to the slightly jaggededge of our exclusion region). Milky Way Dark Matter Decay Flux.—
To turnthe INTEGRAL limits on generic line emission into con-straints on sterile neutrino dark matter, we calculatedthe expected gamma-ray emission from the decay of ster-ile neutrinos in the Milky Way (the INTEGRAL limitsalso strongly constrain certain decays of GeV-mass darkmatter models [19]). For a long-lived decaying sterile neutrino with lifetime τ and mass density ρ = m s n , theintensity [20] (number flux per solid angle) of the decayphotons coming from an angle ψ relative to the GalacticCenter direction is I ( ψ ) = ρ sc R sc πm s τ J ( ψ ) , (1)where the dimensionless line of sight integral, J ( ψ ) = 1 ρ sc R sc Z ℓ max dℓ ρ (cid:16)p R sc − ℓ R sc cos ψ + ℓ (cid:17) , (2)is normalized at the solar circle, with R sc = 8 . ρ sc = 0 . − (these cancel later). While ℓ max depends on the adopted size of the halo, contributionsbeyond the scale radius of the density profile, typicallyabout 20–30 kpc, are negligible.The sterile neutrino radiative lifetime τ is1 τ = (cid:0) . × − s − (cid:1) (cid:20) sin θ − (cid:21) h m s keV i , (3)where we have chosen the Dirac neutrino decay life-time [21]; for the Majorana case, which may be favored,the lifetime is 2 times shorter, which would lead to morerestrictive constraints. The prefactor in Eq. (1) can thenbe expressed in terms of the mass and mixing of the ster-ile neutrino, ρ sc R sc πm s τ = (cid:0) . × − cm − s − sr − (cid:1) (cid:20) sin θ − (cid:21) h m s keV i . (4)The number flux of photons at energy E γ = m s / F s = Z ∆Ω d Ω I ( ψ ) = ρ sc R sc πm s τ Z ∆Ω d Ω J ( ψ ) , (5)where the solid angle is ∆Ω = 2 π (1 − cos ψ ).The dark matter distribution of the Milky Way is notperfectly known [22], though the variations between mod-els make little difference for dark matter decay, since thedensity appears only linearly in the calculations (unlikefor dark matter annihilation, where it appears quadrat-ically). A trivial lower bound for the integral in Eq. (5)can be obtained by taking the dark matter density to beconstant within some radius from the Galactic Center,which we take to be R sc . Then the line of sight and fieldof view integrals are just multiplications: using Eq. (2),the former is ≃
2, and since ∆Ω ≃ .
16 for ψ = 13 ◦ , thelatter is R ∆Ω d Ω J ( ψ ) ≃ . ρ ( R sc ) = 0.30, 0.27, and 0.37 GeV cm − , respec-tively. These slight differences in normalization compen-sate the different slopes at inner radii so that the massesenclosed at outer radii are the same [22]. In the left panelof Fig. 2, the thin lines show J ( ψ ) as a function of theangle ψ for each profile; in the right panel, the corre-sponding thin lines show these integrated over the fieldof view (up to the angle ψ ), as in Eq. (5). These resultstake into account the variation of density with position,and also the contribution from halo dark matter beyondthe solar circle on the other side of the Milky Way. Notethat all three profiles have similar values of R ∆Ω d Ω J ( ψ ),since the large field of view de-emphasizes the inner radiiwhere the differences between the profiles are the largest. Constraints on Sterile Neutrinos.—
As notedabove, the INTEGRAL limits on line emission from theGalactic Center region are obtained by subtracting theaverage flux outside this region ( ψ > ◦ ) from the fluxinside this region ( ψ < ◦ ), which must be taken intoaccount in our analysis. To be conservative, we consid-ered the maximum effect of this subtraction by fixing theintensity outside the Galactic Center region to its valueat ψ = 30 ◦ . (In fact, it is smaller at larger angles.) Interms of our equations, this is∆ F s = ρ sc R sc πm s τ Z ∆Ω d Ω [ J ( ψ ) − J (30 ◦ )] . (6)In the right panel of Fig. 2, our results for the integrated J ( ψ ) −J (30 ◦ ) are shown by the thick lines. The effect ofthis subtraction correction is not large, less than a factorof 3 at ψ = 13 ◦ for all three profiles. In addition, theINTEGRAL flux limits of Ref. [15] for an angular region o o o o ψ ∫ d Ω [ J ( ψ ) - J ( o )] ∫ d Ω J ( ψ ) o o o o ψ J ( ψ ) FIG. 2:
Left panel:
The line of sight integral J ( ψ ) as afunction of the pointing angle ψ with respect to the Galac-tic Center direction for the three different profiles considered(Kravtsov, NFW, and Moore, in order of solid, dashed anddotted lines). Right panel:
Integrals up to the angle ψ of J ( ψ )(thin upper lines) and J ( ψ ) −J (30 ◦ ) (thick lower lines). Thegrey line at 13 ◦ marks the field of view for the INTEGRALflux limit, and we chose R ∆Ω d Ω [ J ( ψ ) − J (30 ◦ )] ≃ . of ψ < ◦ assume that the line emission intensity followsa two-dimensional Gaussian with FWHM of 10 ◦ , while aflat-source profile would yield somewhat weaker limits.To shield our results from such uncertainties associatedwith the distribution of dark matter in the Milky Way,including whether warm dark matter profiles are less cen-trally concentrated than cold dark matter profiles, we usea rather conservative value, R ∆Ω d Ω [ J ( ψ ) − J (30 ◦ )] ≃ .
5, in our subsequent calculations. Our results can beeasily rescaled for a different value and our limits shouldimprove as the amount of data increases in time.While we have presented our results for the regionwithin 13 ◦ of the Galactic Center, there are also fluxlimits for an angular region of ψ < ◦ and an assump-tion that the intensity is constant in angle [15]. The fluxlimits for ψ < ◦ are ≃ ψ < ◦ [15]. However, as shown in the right panel ofFig. 2, the sterile neutrino decay flux, which is propor-tional to R ∆Ω d Ω [ J ( ψ ) − J (30 ◦ )], is ≃ ψ < ◦ than for ψ < ◦ , compensating the lowersensitivity. Thus our results are rather robust against thechoice of angular region used and other assumptions foranalyzing the INTEGRAL limits.With these detailed results on the sterile neutrino darkmatter distribution, we define constraints in the param-eter space of mass and mixing. The expected line flux at E γ = m s / m s and sin θ , should not exceed the INTEGRAL limits(for 3 . σ ), i.e., F lim > ∆ F s , or F lim ( E ) > ρ sc R sc πm s τ Z ∆Ω d Ω [ J ( ψ ) − J (30 ◦ )] . (7)Substituting Eq. (4) and R ∆Ω d Ω [ J ( ψ ) −J (30 ◦ )] ≃ . F lim ( E ) (see Fig. 9 ofRef. [15]). The energy range available with the SPI in-strument causes the sharp cut-off at m s = 40 keV. Ourconstraint is coincidentally in line with prior constraintsat lower masses using the x-ray emission from nearbysources. There is only a narrow gap, m s ≃ Conclusions.—
Sterile neutrinos require only a min-imal and plausible extension of the Standard Model [1,2, 3, 4] and can solve problems in reconciling the ob-servations and predictions of large-scale structure [2, 4].Despite intensive efforts on setting constraints, there arestill viable sterile neutrino dark matter models over awide range of mass m s and mixing sin θ ; the focusis now at larger mass and smaller mixing than consid-ered in the earliest and simplest models [3]. In thisregion, the models are very challenging to test, eitherthrough their differences in clustering with respect tocold dark matter candidates [9] or their astrophysical ef-fects [5, 7, 8]), or through their very small radiative decayrates [4, 11, 12, 13] or laboratory tests [6].Teegarden and Watanabe [15] presented the results ofa sensitive search for line emission in the Galactic Cen-ter Region, using data from the SPI spectrometer on theINTEGRAL satellite [16]. Based on a simple and conser-vative calculation of the expected gamma-ray flux fromsterile neutrino dark matter decays, we have used theselimits to set new and very strong constraints on sterileneutrino parameters, as shown in Fig. 1. The large-massregion is now very strongly excluded, improving on theprevious CXB mixing constraints [11] by more than twoorders of magnitude. At fixed m s , the boundary in sin θ is defined by the 3 . σ exclusion; using Eqs. (7) and (4),it is easy to see that points with sin θ values ten timeslarger than at the boundary are excluded by a nominal35 σ , and so on. On the scale of the figure, any reasonablefurther degradations in the conservatively-chosen inputswould not be visible. We anticipate that it will be possi-ble to extend our constraints, in particular going to lowermasses, by dedicated analyses of the INTEGRAL data,which we strongly encourage. 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