Towards the use of asteroseismology to investigate the nature of dark matter
aa r X i v : . [ a s t r o - ph . C O ] O c t Mon. Not. R. Astron. Soc. , 1– ?? (2002) Printed 14 November 2018 (MN L A TEX style file v2.2)
Towards the use of asteroseismology to investigate thenature of dark matter
Jordi Casanellas ⋆ and Il´ıdio Lopes , † Centro Multidisciplinar de Astrof´ısica, Instituto Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Departamento de F´ısica, Universidade de ´Evora, Col´egio Luis Ant´onio Verney, 7002-554 ´Evora, Portugal
ABSTRACT
The annihilation of huge quantities of captured dark matter (DM) particles insidelow-mass stars has been shown to change some of the stellar properties, such as thestar’s effective temperature or the way the energy is transported throughout the star.While in the classical picture, without DM, a star of 1 M ⊙ is expected to have aradiative interior during the main sequence, the same star evolving in a halo of DMwith a density ρ χ > GeV cm − will develop a convective core in order to evacuatethe energy from DM annihilation in a more efficient way. This convective core leavesa discontinuity in the density and sound-speed profiles that can be detected by theanalysis of the stellar oscillations. In this paper we present an approach towards theuse of asteroseismology to detect the signature produced by the presence of DM insidea star, and we propose a new methodology to infer the properties of a DM halo fromthe stellar oscillations (such as the product of the DM density and the DM particle-nucleon scattering cross-section). Key words: dark matter - asteroseismology - stars: interiors - stars: low-mass - stars:fundamental parameters - Galaxy: centre
Different observations in a wide range of scales, from galac-tic to cosmological, suggest the existence of a new kind ofmatter, called Dark Matter (DM), formed by unknown par-ticles. Among the possible constituents of DM, the WIMPs,massive particles with non-negligible scattering cross-sectionwith baryons, are considered one of the best candidates(Bertone, Hooper & Silk 2005).Soon was realised that, if WIMPs exist, they willaccumulate inside stars (Press & Spergel 1985) andtheir annihilation may lead to significant changes inthe classical picture of stellar evolution if the halowhere the stars evolve has a very high density of DMparticles (Bouquet & Salati 1989; Salati & Silk 1989;Dearborn et al. 1990). In this context, the effects of thecapture of WIMPs by the Sun were studied, addressingthe prospects of helioseismology to test models thatsolved the old solar neutrino problem (Dappen et al. 1986;Faulkner et al. 1986), and to give constraints to the natureof DM particles (Lopes, Silk & Hansen 2002; Lopes & Silk2002; Lopes, Bertone & Silk 2002; Bottino et al. 2002;Cumberbatch et al. 2010; Frandsen & Sarkar 2010;Taoso et al. 2010). ⋆ E-mail: [email protected] † E-mail: [email protected]
Recently, particular attention has been given to the firststars formed in the early Universe due to the high DM con-tent in that epoch (Spolyar, Freese & Gondolo 2008; Iocco2008; Freese, Spolyar & Aguirre 2008; Taoso et al. 2008;Schleicher, Banerjee & Klessen 2009; Natarajan et al. 2009;Ripamonti et al. 2010; Sivertsson & Gondolo 2010), includ-ing the prospects for their detection with the
JWST tele-scope (Freese et al. 2010; Zackrisson et al. 2010). Similarly,other authors focused on the DM effects on stars in the lo-cal Universe, either on compact stars (Moskalenko & Wai2007; Bertone & Fairbairn 2008; Isern et al. 2008, 2010;de Lavallaz & Fairbairn 2010; Kouvaris & Tinyakov 2010;Perez-Garcia, Silk & Stone 2010) or on low-mass stars(Fairbairn, Scott & Edsj¨o 2008; Scott, Fairbairn & Edsj¨o2009; Casanellas & Lopes 2009).The purpose of this paper is to pave the way for the useof asteroseismology to provide an evidence of the footprintleft by DM annihilation on the stellar oscillations. To dothis, we will concentrate on solar-mass stars that evolve inhaloes with very high DM densities, and we will show howasteroseismology may tell us about the properties of suchDM haloes. c (cid:13) Jordi Casanellas and Il´ıdio Lopes
The evolution of a star within a halo of DM depends stronglyon the ability of the gravitational field of the star to capturethe DM particles that populate the halo. The rate at whichthe DM particles are captured is given by (Gould 1987) C χ ( t ) = X i Z R ⋆ πr Z ∞ f v ⋆ ( u ) u w Ω − v i ( w ) d u d r , (1)where f v ⋆ ( u ) is the velocity distribution of the DM parti-cles seen by the star (which is proportional to the densityof DM on the host halo ρ χ and inversely proportional tothe mass of the DM particles m χ ) and Ω v i is the probabil-ity of a DM particle to be captured after the collision withan element i (which is proportional to the scattering cross-section of the DM particle with the nucleus i , σ χ,i ). Thenumerical subroutines to calculate the capture rate (equa-tion 1) were adapted from the publicly available DARKSUSY code (Gondolo et al. 2004). Our assumptions regarding thiscalculation are described in Casanellas & Lopes (2010).Once DM particles are captured, they accumulate ina small region in the core of the star ( r χ ≃ .
01 R ⋆ for m χ = 100 GeV). There, assuming that they are Majoranaparticles, they annihilate providing a new source of energyfor the star. Capture and annihilation processes balance eachother in a short time-scale, and consequently almost all cap-tured particles will be converted to energy, contributing tothe total luminosity with L χ = f χ C χ m χ . The factor f χ ,which in this work we assumed to be 2/3 (Freese et al. 2008;Iocco et al. 2008; Yoon et al. 2008), accounts for the energythat escapes out of the star in the form of neutrinos. RecentMonte Carlo simulations suggest that the fraction of theenergy lost in neutrinos may be even smaller (Scott et al.2009).Due to this new source of energy, stars will evolve differ-ently from the classical picture if surrounded by a dense haloof DM. For very high DM densities ( ρ χ > × GeV cm − for a 1 M ⊙ star), the energy from DM annihilation pre-vents the gravitational collapse of the star, stopping its evo-lution in the pre-main-sequence phase, before the star couldreach enough central temperature to trigger hydrogen burn-ing (Casanellas & Lopes 2009).For lower DM densities (10 GeV cm − < ρ χ < × GeV cm − for a 1 M ⊙ star), DM burning is a com-plementary source of energy for the star. As it is produced ina region much more concentrated than the nuclear burning,which normally extends up to 0 . − . ⋆ , the radiative tem-perature gradient ( ▽ rad = d ln T /d ln P g ) is much steeper inthe core of the star. Consequently, as the radiative trans-port is not efficient enough to evacuate all the energy inthe central region, the star develops a convective core whichwas not present in the classical scenario without DM. Theradius and duration of the convective core increase whenmore energy from DM annihilation is produced (Scott et al.2009); therefore, they depend on the density of DM in theplace where the star evolves and on the properties of theDM particles. The balance between DM annihilation, nu-clear burning, and the gravitational energy leads to a newhydrostatic equilibrium with a lower central temperature.The star consumes its hydrogen at a lower rate, extendingthe time that it spends in the main sequence. These new properties allow us, as it will be shown, to provide a toolto infer the DM characteristics from the stellar oscillationsusing the seismological analysis. With the improvement on the quality of the data, asteroseis-mology is now becoming a precise tool to infer the propertiesof stars showing solar-like oscillations (Michel et al. 2008;Garc´ıa et al. 2009; Bedding et al. 2010), which are driven byturbulence in the superficial layers of the star. The eigenfre-quencies of solar-like oscillations can be approximated, for l/n → l and n are the degree and the radial orderof the modes), by the asymptotic expression ν n,l ≃ ( n + l ǫ ν ) ν + O ( ν − ) , (2)where ν = [2 R R drc ] − is the inverse of twice the time spentby the sound to travel between the centre and the acousticsurface of the star, and ǫ ν is determined by the propertiesof the surface layers. For a more in-depth explanation of thebasics of the seismological analysis, the reader is referredto Tassoul (1980); Gough (1985); Lopes & Turck-Chieze(1994). The value of ν can be estimated through the largeseparation ∆ ν n,l :∆ ν n,l = ν n,l − ν n − ,l ≃ ν . (3)This parameter is sensitive to the mean density of the star:∆ ν n,l ∝ ( M/R ) / (Cox 1980), while the small separation δν n,l , given by δν n,l = ν n,l − ν n − ,l +2 , (4)is sensitive to the temperature and chemical gradient in thedeep interior.In the last years, other relations between the frequenciesof the oscillation modes were proposed (for a recent review,see Christensen-Dalsgaard & Houdek (2009) or Aerts et al.(2010)), broadening the diagnostic potential of seismol-ogy. Among the possible diagnostic methods of convectivecores and envelopes (Monteiro et al. 1994; Lopes et al. 1997;Lopes & Gough 2001), we highlight the ratios between thesmall separations and the large separations developed byRoxburgh & Vorontsov (2003) in order to suppress the ef-fects of the modelling of the near-surface layers: r = d ∆ ν n, , r = d ∆ ν n +1 , , (5)where d = 18 ( ν n − , − ν n − , + 6 ν n, − ν n, + ν n +1 , ) , (6) d = −
18 ( ν n − , − ν n, + 6 ν n, − ν n +1 , + ν n +1 , ) . (7)The mixing of elements produced in the convective re-gions introduces a sharp structural variation in the borderwith the radiative regions that can be seen in the den-sity and sound-speed profiles. This sharp structural varia-tion produces an oscillatory signal in the frequency spec-trum (Gough 1990) whose period is related with the acous-tic depth of the discontinuity inside the star. Recently,Silva Aguirre et al. (2010) proposed the use of the ratios r and r to determine the size of a convective core by fitting c (cid:13) , 1– ?? steroseismology to investigate dark matter a sine wave to their oscillation pattern. Similarly, anothercombination of the small and large separations, dr ≡ D ∆ ν n − , − D ∆ ν n, , (8)where D l,l +2 ≡ δν n,l / (4 l + 6), was suggested byCunha & Metcalfe (2007) to measure the amplitude of thesound-speed discontinuity at the edge of a convective core.These seismic parameters (equations 5 and 8) are sensitiveto the presence of DM inside a star, given that they areuniquely dependent on the star’s core structure and almostindependent of the physical processes occurring in the sur-face layers. To grasp the signature that the annihilation of captured DMparticles leaves on low-mass stars we evolved a set of 1 M ⊙ stars, with the same initial conditions (Z=0.018), in haloes ofDM with different densities ρ χ and different spin-dependent(SD) WIMP-nucleon cross-sections σ χ,SD . Throughout ourwork, we considered fiducial values for the mass of the DMparticles, m χ = 100 GeV, and for their self-annihilationcross-section, < σ a v > = 3 · − cm s − . The evolutionof the stars was computed using a well-established stellarevolution code ( CESAM ; Morel (1997)) used to compute so-phisticated solar models for helioseismology (Couvidat et al.2003; Turck-Chi`eze et al. 2010) and more recently used inthe context of asteroseismic studies (Kervella et al. 2004;De Ridder et al. 2006; Su´arez et al. 2010). When the starsreached a luminosity equal to that of the Sun, a very precisemesh (with 1000 layers) was generated. Then, we calculatedthe frequencies of the oscillation modes of the stars usingthe
ADIPLS code (Christensen-Dalsgaard 2008). The char-acteristics of some of these stars are shown in Table 1, andtheir sound-speed and density profiles, in Fig. 1.The accretion and the annihilation of DM particles inthe core of the stars may change significantly their proper-ties. As expected, we found that the effective temperatureof the stars that evolved in haloes with high DM densities isshifted to lower values (see Table 1), due to the presence ofa convective core (see Fig. 2.a), in agreement with previousworks (Fairbairn et al. 2008; Casanellas & Lopes 2009). Thelower effective temperature and the larger radius lead to adecrease in the large separation ∆ ν n,l (see Fig. 2.b). For astar with a known mass, the drop in ∆ ν n,l , predicted by therelation ∆ ν n,l ∝ M / R − / , is unmistakably related withthe radius of the star. Furthermore, we also observed a dropon the small separation δν n,o (see Fig. 2.c), caused by a de-crease in the central density. The strong dependence of theglobal modes on the density profile of the star is responsiblefor that drop.In order to test the validity of our method, we checkedif classical stars with similar characteristics may mimic theproperties we described for stars evolving in DM haloes. Inparticular, we found that a star with a mass M ⋆ = 0 .
955 M ⊙ and a metallicity Z = 0 .
04 reaches, near the end of the mainsequence, the same luminosity and effective temperature asthe star (iv) in our set (see Table 1). At that moment, theradius of both stars is identical, leading to very similar great b) r (R ⋆ ) ρ ( g c m − ) · − (iv) 2 · − (iii) 10 − (ii) 10 − (i) 0a) c ( c m s − ) Figure 1.
Sound-speed (a) and density profiles (b) of 1 M ⊙ starsthat evolved in DM haloes with different densities ρ χ and SDWIMP-nucleon cross-sections σ χ,SD when they reached a lumi-nosity L = 1 L ⊙ (for each star, the product ρ χ σ χ is indicated inthe legend in GeV cm − ). d) log ( ρ χ σ χ / GeV cm − ) s l o p e o f d r -29-30-31-320-0.001-0.002-0.003 c) h δ ν n , i ( µ H z ) h ∆ ν n , l i ( µ H z ) r ( R ⋆ ) Figure 2. (a) Size of the convective core, and the calculatedseismologic parameters: (b) mean large separation (for l=0,1,2,3),(c) mean small separation (for l=0) and (d) slope of dr , for1 M ⊙ stars that evolved in DM haloes with different densities ρ χ and SD WIMP-nucleon cross-sections σ χ,SD , when the starsreached a luminosity L = 1 L ⊙ . separations ( < ∆ ν n,l > = 128 µ Hz for star (iv) and 126 µ Hzfor the other). However, as the star that evolved withoutDM is in a later stage of evolution ( X c = 0 .
03, while X c =0 .
38 for star (iv), the small separation, being very sensitiveto the chemical gradient in the deep interior, allows us todifferentiate both stars. In our case, star (iv), which evolvedin a dense halo of DM, has a < δν n,o > = 7 µ Hz. This isalmost double than that of the star with different M ⋆ and Z ( < δν n,o > = 4 µ Hz in that case). c (cid:13) , 1– ?? Jordi Casanellas and Il´ıdio Lopes
Table 1.
Characteristics of stars of 1 M ⊙ when they reached a luminosity L = 1 L ⊙ afterevolving in haloes of DM with different densities ρ χ and different SD WIMP-nucleon cross-sections σ χ,SD . The last two columns are the radius and the acoustic radius ( τ = R r drc ) ofthe convective core (CC). All the stars had the same initial conditions ( Z = 0 . ρ χ σ χ X c R ⋆ T eff T c ρ c r CC τ CC (GeV cm − ) (R ⊙ ) (K) (MK) (g cm − ) (R ⋆ ) (s)(i) 0 0.35 1.000 5777.5 15.52 148.7 No CC -(ii) 10 − − × − × − In addition, one of the most promising signatures ofannihilating DM in stars is the fact that it can originatethe formation of a convective core (unexpected in the clas-sical picture for stars with masses < . ⊙ ) whose radiusgrows with the DM density ρ χ . The convective core leavesa peculiar footprint in the profiles of the sound speed anddensity (see Fig. 1) characterized by a discontinuity in theedge of the core. The presence of the convective core can bedetected by the seismological analysis using a relation be-tween the small separation of modes with different degrees(and therefore with different depths of penetration insidethe star). For that purpose low-degree modes ( l = 0 , , , dr (see equation 8) is sensitive to the sound-speeddiscontinuity at the edge of the convective core and, conse-quently, to the characteristics of the DM halo. In Fig. 3.a)we show the behaviour of the parameter dr for starsthat evolve in DM haloes with different characteristics. Wefound that the absolute value of the slope of dr at highfrequencies increases with the amplitude of the sound-speeddiscontinuity caused by the convective core, as predicted byCunha and Metcalfe. Therefore, the slope of dr is di-rectly related with the amount of DM in the halo where thestar evolves (see also Fig. 2.d).We also tested the method recently proposed bySilva Aguirre et al. (2010) designed to estimate the size of aconvective core in 1.5 M ⊙ stars. We found that the period ofthe sinusoidal fit to the ratios r and r (see Fig. 3.b) doesnot match exactly the acoustic radius of the convective cores(see Table 1), most probably because we are applying thismethod to stars of mass 1 M ⊙ . However, the ratios r and r have a great sensitivity to the amplitude of the sharpvariation of the sound speed caused by the annihilation ofDM particles inside the star. We conclude that these ratiosmay be used in the future as a stellar probe to confirm thepresence of DM in the neighbourhood of low-mass stars.If enough radial modes are identified with the preci-sion presently achieved by space-based telescopes as CoRoT (a relative error on the individual frequencies of ∼ − (Deheuvels et al. 2010)), then our method will allow the dis-crimination between haloes of DM with different character-istics. To illustrate this point, we plotted in Fig. 3 the errorbars on dr , r and r for star (iii) derived from thementioned uncertainty (10 − ν ) on the determination of thefrequencies, as done by Cunha & Metcalfe (2007). T v = 174s T iv = 143sb) T iii = 108s ν ( µHz ) r , r · − (iv) 2 · − (iii) 10 − (ii) 10 − (i) 0a) ν ( µHz ) d r Figure 3. (a) The seismological parameter dr and (b) theratios r and r for stars that evolved in DM haloes with dif-ferent densities ρ χ and different SD WIMP-nucleon cross-sections σ χ,SD when they reached a luminosity L = 1 L ⊙ (for each star,the product ρ χ σ χ is indicated in the legend in GeV cm − ). Er-ror bars are shown for star (iii) assuming a relative error on theidentification of the frequencies of 10 − . In this paper, we have presented a new methodology to-wards the use of asteroseismology to prove the presenceof DM in the location where a star evolves. For a main-sequence star of 1 M ⊙ evolving in a DM halo with a density ρ χ > GeV cm − (assuming σ χ,SD = 10 − cm ), theannihilation of captured DM particles on its interior leadsto decreases in the large and small separations, when com-pared with the same star in the classical scenario withoutDM, which are related to changes in the global propertiesof the star. Furthermore, the highly concentrated produc-tion of energy by DM annihilation creates a convective corewhich is not present in the classical picture. This convectivecore leaves a discontinuity signature in the sound-speed anddensity profiles which can be detected by the analysis of thestellar oscillations.We have shown that seismological parameters such as c (cid:13) , 1– ?? steroseismology to investigate dark matter Figure 4.
DM densities at which 1 M ⊙ stars with a luminos-ity 1 L ⊙ are expected to show strong signatures on the seis-mological parameters ∆ ν, δν, dr , r and r (see text) dueto the annihilation of DM particles with different characteristics( m χ , σ χ,SD ) in their interior. In the particular case of the GalacticCentre (GC), the DM densities in the Figure (from top to bottom)are expected at a distance from the GC of 0.1 pc, 0.04 pc and0.02 pc, following the adiabatically contracted profile of Bertone& Merritt. The grey lines are the present limits from direct detec-tion experiments: XENON10 (dotted), PICASSO (dashed) andCOUPP (solid), and the grey region is the DAMA/LIBRA al-lowed region. dr and the ratios r and r are very sensitive to the sizeof the convective core, which is determined by the density ofDM ρ χ where the star evolved and by the scattering cross-section of the DM particles off nuclei σ χ . Consequently, thisrelationship may be used in the future to help in the deter-mination of these parameters (or at least to their product, ρ χ σ χ ) and to provide a stellar probe that identifies the pres-ence of self-annihilating DM.The method presented in this paper is valid for haloeswith very high DM densities. In Fig. 4 we show the DMdensities at which a 1 M ⊙ star with a luminosity 1 L ⊙ is expected to have a small separation 25% smaller thanthat in the classical picture, because of the annihilation ofDM particles with different characteristics ( m χ , σ χ → p,SD )in its interior. If found, this kind of star will have alsostrong signatures on the seismic parameters ∆ ν , dr , r and r when compared with a star with the same lumi-nosity that evolved without DM. In the same Figure arealso shown the current limits on σ χ → p,SD from the directdetection experiments XENON10 (Angle et al. 2008), PI-CASSO (Archambault et al. 2009), COUPP (Behnke et al.2008) and the allowed region from the DAMA/LIBRA ex-periment (Savage et al. 2009).The extreme DM densities shown in Figure 4 may bepresent within the inner parsec of our Galaxy, accordingto models that account for the effect of the baryons onthe DM halo via adiabatic contraction (Blumenthal et al.1986; Gnedin et al. 2004). For instance, following the adi-abatically contracted profile of Bertone & Merritt (2005),DM densities as high as ρ χ = 10 GeV cm − are ex- pected at 0.1 pc from the Galactic Centre (GC). Even higherDM densities may be present at the GC if a hypotheticalspike is formed due to the influence of the central blackhole (Gondolo & Silk 1999). However, as other models pre-dict lower central DM densities (the so-called core models (Burkert 1995)), the open questions about the DM halo pro-file at the inner parsec of our Galaxy are still far from beingsolved (for a recent review on this topic, see Merritt (2010)and de Blok (2010)). In this sense, the method proposedhere may provide a complementary tool to help in the dis-crimination of different models. Other possible locations ofenvironments with such high DM densities are the dwarfspheroidal galaxies around the Milky Way (Dekel & Silk1986; Kormendy & Freeman 2004; Diemand et al. 2007).The precision required for our analysis is similar to theone achieved by present asteroseismic missions in observa-tions of one hundred days. Nevertheless, the most likelyplace to find the kind of stars described here is near thecentre of our Galaxy, where the distance and the presenceof dust makes the observations difficult. These difficultiesencourage us to extend our study to more massive and lu-minous stars in a future work. Future technical improve-ments in the observations of the GC and of the Milky Waydwarf galaxies may open the possibility of using the methodproposed here to investigate the nature of DM. Acknowledgements
We acknowledge the anonymous referee for his use-ful comments, as well as the authors of
CESAM, ADIPLS and
DARKSUSY , and the Brown University’s Particle As-trophysics Group, which maintains the DM tools website,used for the σ χ → p,SD limits in Figure 4. This work wassupported by grants from “Funda¸c˜ao para a Ciˆencia e Tec-nologia” (SFRH/BD/44321/2008) and “Funda¸c˜ao CalousteGulbenkian”. REFERENCES
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