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Scalar Field (Wave) Dark Matter Tonatiuh Matos∗ and Victor H. Robles∗∗ Departamento de F´ısica,Centro de Investigación y de Estudios Avanzados del IPN, AP 14-740, 0700 D.F., 6 1 México 0 2 n a J ABSTRACT 6 Recent high-quality observations of dwarf and low surface brightness (LSB) galaxies have shown that their dark matter (DM) halos prefer flat central density profiles. On the other hand ] A the standard cold dark matter model simulations predict a more cuspy behavior. Feedback G from star formation has been widely used to reconcile simulations with observations, this might be successful in field dwarf galaxies but its success in low mass galaxies remains uncertain. . h One model that have received much attention is the scalar field dark matter model. Here the p dark matter is a self-interacting ultra light scalar field that forms a cosmological Bose-Einstein - o condensate, a mass of 10−22eV/c2 is consistent with flat density profiles in the centers of dwarf r spheroidalgalaxies, reduces the abundance of small halos, might account for the rotation curves t s even to large radii in spiral galaxies and has an early galaxy formation. The next generation of a telescopeswillprovidebetterconstraintstothemodelthatwillhelptodistinguishthisparticular [ alternative to the standard model of cosmology shedding light into the nature of the mysterious 1 dark matter. v 0 Subject headings: dark matter: scalar field 5 3 1. INTRODUCTION expectations might require careful revision to our 1 understanding. 0 The standard model of cosmology assumes the . One of them is the longstanding core/cuspdis- 1 darkmatteriscoldandeffectivelycollisionless,the cussion, whether the central dark matter (DM) 0 galaxies are formed in a hierarchical way, and as 6 profilesindwarfsandlowsurfacebrightness(LSB) they evolve they are subject to frequent collisions 1 galaxies are more core-like and rounder than the and interactions with nearby galaxies that deter- : standard cold dark matter (CDM) model pre- v mined the properties that we observed today. i dicts (see van Eymeren et al. (2009); see de Blok X Thestandardmodel,alsocalledcolddarkmat- (2010) for a review). The core profiles most fre- r ter (CDM) model, is remarkably successful to de- quently used in the literature and that best fit a scribe the large scale structure of the universe, as the observations are empirical (Burkert 1995; well as large scale observations. Nowadays,galac- Kuzio de Naray et al. 2010). Albeit useful to tic observationsarebecoming moreprecisethatit characterize properties of galaxies, it is desir- is possible to assess some of the predictions from able to find a theoretical framework capable to the CDM model with more reliability. Moreover, produce the cores, since CDM suggest central thenumericalsimulationsarerapidlyreachingthe densities in small galaxies going as ρ ∼ r−1 at required resolution to study the inner parts of small r (Navarro et al. 2010) whereas observa- dwarf galaxies, that is within ∼500pc. Increasing tions of LSB galaxies suggest a core-like behavior the resolution has revealed that some discrepan- (ρ∼r−0.2)(Oh et al.2011;Robles & Matos2012; cies between the observations and the theoretical de Blok et al. 2001; Kuzio de Naray & Spekkens 2011;Kuzio de Naray & Kaufmann2011;Boylan-Kolchinet al. *Electronicaddress: [email protected] 2014). The trend to reduce the inner loga- **Electronicaddress: [email protected] 1 rithmic slopes invokes astrophyisical processes 2. SFDM: Previous work such as radiation wind, supernovae feedback, etc. The main idea in the scalar field dark mat- (Governato et al. 2010, 2012; Merrit et al. 2006; ter model (Sin 1994; Ji & Sin 1994; Lee & Koh Graham et al. 2006), although this seems possi- 1996; Guzmań & Matos 1998; Hu et al. 2000; ble in LSB galaxies, the question remains for the Matos & Urenã 2001) considers a self-interacting fainter galaxies where one supernovae could blow scalar field with a very small mass, typically out most of the gas due to the shallower gravita- of ∼ 10−22eV/c2, such that the quantum me- tional potential. chanical uncertainty principle and the inter- Another discrepancy possibly related the over- actions prevent gravitational collapse in self- abundanceofsatellites(Klypin et al.1999;Moore et al. gravitating structures, thus the halos are char- 1999; Maciò et al. 2012; Garrison-Kimmel et al. acterized with homogeneous densities (usually 2014)istheToo-Big-to-Fail(Boylan-Kolchinet al. referred as a cores) in their centers, in general 2011; Garrison-Kimmel et al. 2014b) issue. The the core sizes depend on the values of the mass latter results from the higher number of massive and the self-interacting parameters (Colpi et al. dark matter halos around Milky-Way like host 1986) (for a review see Sua´rez, Robles & Matos with the most massive galaxies observed in our (2013); Rindler-Daller & Shapiro (2014)). From local neighborhood, assuming the most massive the particle physics point of view the most sim- galaxiesareinthemostmassivedarkmatterhalos, ple way to account for a scalar field with this there should be about ∼10 more around systems features is adding a Higgs-like term with a mass with virial masses comparable to our Milky Way ∼ 10−22eV/c2 to the standard model of particles or M31 (Andromeda) (Garrison-Kimmel et al. (Matos & Lo´pez-Fernańdez 2014). 2014b). There have been some possible solutions, Previous studies of the cosmological evolution mostofthemrelyingontidalstrippinginaddition of a scalar field with mass m ∼10−22eV/c2 have to supernovae feedback. shown that the cosmological density evolution However, some of these discrepancies might is reproduced and very similar to the one ob- also be solved assuming different properties for tainedfromCDM(Matos & Urenã2001;Chavanis the dark matter, such as scalar field dark matter 2011; Sua´rez & Matos 2011; Maganã et al. 2012; (SFDM)(Sin1994;Lee & Koh1996;Guzmań & Matos Schive, Chiueh, & Broadhurst2014),thereiscon- 1998;Matos & Urenã2001),stronglyself-interacting sistency with the acoustic peaks of the cosmic DM(Spergel & Steinhardt2000;Vogelsberger,Zavala& Loeb microwavebackgroundradiation(Matos & Urenã 2012; Rocha et al. 2013), warm dark matter 2001; Rodr´ıguez-Montoyaet al. 2010) and this (Maciò et al. 2012). small mass implies a sharp cut-off in the mass ItisofparticularinteresttoustheSFDMalter- power spectrum for halo masses below 108M⊙ native, here the massof the field is assumedto be suppressing structure formation of low mass dark verysmall(∼10−22eV/c2)suchthatitsdeBroglie matter halos(Matos & Urenã 2001;Marsh & Silk wavelengthis of order∼ kpc, relevantfor galactic 2014; Bozek et al. 2014; Hu et al. 2000). More- scales. Thequantumbehaviorofthe fieldhascre- over,there isparticularinterestinfindingequilib- atedmuchinterestin the modeldue to its success rium configurations of the system of equations toaccountforsomediscrepanciesmentionedabove that describe the field (Einstein-Klein-Gordon withdarkmatterpropertiesonly,forexample,the system) and of its weak field approximation smallmasskeepsthe centraldensityfromincreas- (Schrödinger-Poisson(SP) system), different au- ing indefinitely due to the uncertainty principle thors have obtained solutions interpreted as bo- in contrast to CDM simulations where supernova son stars or later as dark matter halos showing feedback is required(Governato et al. 2010, 2012; agreement with rotation curves in galaxies and Pontzen & Governato 2012; Scannapieco et al. velocity dispersion profiles in dwarf spheroidal 2012). galaxies(Lee & Koh1996;Böhmer & Harko2007; Robles & Matos 2012, 2013; Lora et al. 2012; Lora & Maganã2014;Martinez-Medina, Robles, & Matos 2015; Diez-Tejedor et al. 2014; Guzmań et al. 2 2014). So far the large and small scales obser- mass of the scalar field dark matter points to be vations are well described with the small mass close to ∼ 10−22eV/c2, consistent with the above and thus has been taken as a prefered value constraint, although there are still uncertainties but the precise values of the mass and self- onthemassparameter,inordertoavoidconfusion interaction parameters are still uncertain, tighter with the known QCD axion, we find it useful and constraints can come from numerical simulations appropriate to name the scalar field dark matter (Schive, Chiueh, & Broadhurst 2014) and model- candidate, from the above characteristics we can ing of large galaxy samples. define it as a particle with mass m<10−14eV/c2, Recently the idea of the scalar field has gained we name this DM candidate the psyon. interest, given the uncertainty in the param- It is worth emphasizing that despite the vari- eters the model has adopted different names ety of names given to the model the main idea in the literature depending on the regime that described above remains the same, it is the quan- is under discussion, for instance, if the inter- tum properties that arise due to the small mass actions are not present and the mass is ∼ of the boson that characterize and distinguishes 10−22eV/c2thislimitwascalledfuzzydarkmatter thisparadigm,analoguoustothestandardcosmo- (Hu et al. 2000) or more recently wave dark mat- logical model represented by the CDM paradigm ter (Schive, Chiueh, & Broadhurst2014), another whose preferred dark matter candidates are the limitiswhentheSFself-interactionsaredescribed WIMPs (weakly interacting massive particles), withaquarticterminthescalarfieldpotentialand one being the neutralino, we see that for all dominateoverthemass(quadratic)term,thiswas the above regimes SFDM, Repulsive DM, Ax- studied in (Goodman 2000; Slepian & Gooodman ion DM, or any other model assuming an ultra 2012) and called repulsive dark matter or fluid light bosonic particle comprise a single class of dark matter by (Peebles 2000). paradigm, which we categorized as the Quantum Notice that for a scalar field mass of ∼ Dark Matter (QDM) paradigm. As pointed be- 10−22eV/c2 the critical temperature of conden- fore, in the QDM paradigm the small mass of sation for the field is Tcrit ∼ m−5/3 ∼TeV, which the dark matter boson leads to the possibility is very high, if the temperature of the field is be- of forming cosmological condensates, even for low its critical temperature it can form a cosmo- axions which are non-thermally produced and logical Bose Einstein condensate, if it condenses havemassesin10−3−10−6eV/c2 (Sikivie & Yang it is called Bose-Einstein condensed(BEC) dark 2009),thisisacharacteristicpropertythatdistin- matter (Matos & Urenã 2001; Guzmań & Matos guishesthesedarkmattercandidatesfromWIMPs 1998; Bernal et al. 2010; Robles & Matos 2013; orneutrinos,namely,theexistenceofbosonsinthe Harko 2011; Chavanis 2011). Sikivie & Yang condensed state, or simply BICS, thus the axion (2009) mentioned that axions could also form and psyon are BICS. Bose-Einsteincondensateseventhoughtheirmass 3. Scalar field dark matter halos is larger than the previous preferred value, notice thattheresultwascontestedinDavidson & Elmer There has been considerable work in finding (2013), this suggestthatthe condensationprocess numerical solutions to the non-interacting SFDM should be study in more detail to confirm it can in the non-relativistic regime to model spheri- remain as BEC dark matter. In Urenã-Lopez cally symmetric haloes (Guzmań & Matos 1998; (2009), it was found that complex scalar field with m < 10−14eV/c2 that decoupled being still Guzmań & Urenã-López2004;Urenã-López & Bernal 2010; Bernal et al. 2010; Bray 2010; Kaup 1968), relativistic will always form a cosmological Bose- andalsofortheself-interactingSFDM(Böhmer & Harko Einsteincondensatedescribedbythegroundstate 2007; Robles & Matos 2012; Colpi et al. 1986; wavefunction,thisdoesnotprecludetheexistence Rindler-Daller & Shapiro 2012; Balakrishna et al. of bosons with higher energy, particularly in dark 1998; Goodman 2000), it is worth noting that matter halos. as mentioned in Guzmań & Urenã-López (2004) We see that the smallness of the boson mass is for the weak field limit of the system that de- its characteristic property and cosmological con- termines the evolution of a spherically symmetric densation is a likely consequence. The preferred 3 scalarfield,thatis,theEinsteinandKlein-Gordon Martinez-Medina, Robles, & Matos 2015). The equations,foracomplexandarealscalarfieldthe size of the MSH is determined by the most ex- system reduces to the Schrödinger-Poisson (SP) cited state that accurately fits the RC for large equations (Arbey et al. 2003). The contraints re- radii, excited states are distributed to larger radii ported in Li et al. (2014), obtained by imposing than the ground state, and in contrast to the that the SF behaves cosmologically as pressure- halo with single state there are MSHs that are less matter (dust), imply that the interacting pa- stable under finite perturbations provided the rameter would be extremely small for the typical ground state in the final halo configuration has mass of ∼10−22eV/c2, therefore we expect that enough mass to stabilize the coexisting state solutions to the SP system with no interactions (Urenã-López & Bernal 2010; Bernal et al. 2010). would behave qualitatively similar to those when Althoughtherearestilluncertaintiesinthesta- self-interactionsareincluded, assupported by the bility of the MSHs, the appearance of bosons in similarity in the solutions of the non-iteracting excited states seems to be a straightforward con- case and those with a small self-coupling found in sequences of quantum interference triggered by other works (Balakrishna et al. 1998; Colpi et al. halomergersasconfirmedrecentlyinSchive et al. 1986; Briscese 2011). (2014b), and possibly the internal evolution of One characteristic feature of stationary solu- the halo. Moreover, initial fluctuations that grow tions of the form ψ(x,t) = e−iEntφ(r) for the SP due to the cosmologicalexpansion of the universe system is the appearance of nodes in the spatial eventually separate from it and start collapsing function φ(r), these nodes are associated to dif- due to its own gravity, at this time (known as ferent energy states of the SF, the zero node so- turnaround)thehalocanhaveanumberofpsyons lution corresponds to the ground state, one node thatareindifferentstateswhichdependonthethe to the first excited state, and so on. These ex- localenvironment. Dependingonthenumberden- cited states solutions fit rotation curves (RCs) of sityofbosonspopulatingtheexcitedstateswecan large galaxies up to the outermost measured data have different fates for the halos (Robles et al. and can even reproduce the wiggles seen at large 2015). radii in high-resolution observations (Sin 1994; On the other hand, including rotation in the Colpi et al. 1986; Robles & Matos 2013). How- halo might be needed, in fact, (Guzmań et al. ever,halosthatarepurelyinasingleexcitedstate 2014) have included rotation to axis-symmetric seem to be unstable when the number of parti- halos in the condensed state and show that it can clesisnotconserved(finiteperturbations)andde- leadtotheflatteningoftheRCs,otherworkshave cay to the ground state with different decay rates alsoincluded rotationbut in the contextof MSHs (Guzmań & Urenã-López2004;Balakrishna et al. in asymmetric configurations (Bray 2012), both 1998), though they are stable when the num- studies suggest that rotation is a relevant ingre- ber of particles is conserved(infinitesimal pertur- dient in halo modeling, in fact it should be, in bations). Thegroundstatesolutionisstableunder the end we observed rotation in galaxies embed- finite perturbations and infinitesimal perturba- ded in dark halos. However, we require more de- tions(Bernal et al.2010;Seidel & Suen1990),but tailstudiestoassessthegoodnessoftheagreement has difficulties to correctly fit the rotation curves withalargesampleofgalaxies,especiallybecause in large galaxies because its associated RC has there ara several surveys underway (e.g. GAIA, a fast keplerian behavior after reaching its max- MANGA) that will provide precise data to test imum value, hence unable to remain flat enough the viability of the standard and alternative dark at large radii. matter models. One way to keep the flatness of the RC to large radii is to consider that bosons are not 4. Conclusions fully in one state, but instead coexist in differ- There are several DM models in the literature ent states within the dark halo, these multistate that are addressing some discrepancies found in halos (MSHs) have been studied in some works the standard model of cosmology, one of them is (Bernal et al. 2010; Urenã-López & Bernal 2010; the scalar field dark matter. The quantum prop- Matos & Urenã-López2007;Robles & Matos2013,b; 4 ertiesofthefieldaffectkpcscalesduetothesmal- Davé,R.,Spergel,D.N., Steinhardt,P.J.,&Wan- leness of the mass. 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