Modeling Wave Dark Matter in Dwarf Spheroidal Galaxies Hubert L. Bray∗, Alan R. Parry† 3 December 11, 2013 1 0 2 n a Abstract J 2 This paper studies a model of dark matter called wave dark matter (also known as scalar field dark matter and boson stars) which has recently also been motivated by a new ] A geometric perspective by Bray [9]. Wave dark matter describes dark matter as a scalar field which satisfies the Einstein-Klein-Gordon equations. These equations rely on a fundamental G constantΥ(alsoknownasthe“massterm”oftheKlein-Gordonequation). Inthiswork, we . h compare the wave dark matter model to observations to obtain a working value of Υ. p Specifically, we compare the mass profiles of spherically symmetric static states of wave - o dark matter to the Burkert mass profiles that have been shown by Salucci et al. [31] to r predict well the velocity dispersion profiles of the eight classical dwarf spheroidal galaxies. t s We show that a reasonable working value for the fundamental constant in the wave dark a matter model is Υ=50 yr−1. We also show that under precise assumptions the value of Υ [ can be bounded above by 1000 yr−1. 1 v 5 1 Introduction 5 2 0 Ever since the first postulation of dark matter in the 1930’s by Zwicky [40], much evidence . for the existence of dark matter has accumulated including the unexpected behavior in the 1 0 rotationcurvesofspiralgalaxies[2,8],thevelocitydispersionprofilesofdwarfspheroidalgalaxies 3 [31,37–39], and gravitational lensing [13]. These and other observations support the idea that 1 : mostofthematterintheuniverseisnotbaryonic,butis,infact,someformofexoticdarkmatter v and that almost all astronomical objects from the galactic scale and up contain a significant i X amount of this dark matter. Describing this dark matter is currently one of the biggest open r problems in astrophysics [5–7,17,27,30,36]. a In the last two decades, there has been substantial progress on describing the distribution of dark matter on the galactic scale. Navarro, Frenk, and White’s popular model [26], which resulted from detailed N-body simulations, has been shown to agree well with observations outside the centers of galaxies [18,38]. However, the Navarro-Frenk-White dark matter energy density profile also exhibits an infinite cusp at the origin, while observations favor a bounded value of the dark matter energy density at the centers of galaxies [14,15]. This has prompted many astrophysicists to employ a cored profile, such as the Burkert profile [11], which “cores” ∗Mathematics and Physics Departments, Duke University, Box 90320, Durham, NC 27708, USA, [email protected] †Mathematics Department, Duke University, Box 90320, Durham, NC 27708, USA, [email protected] 1 H. Bray and A. Parry Modeling Wave Dark Matter in dSph Galaxies out the infinite cusp, to model the dark matter energy density in a galaxy. Resolving this “core-cusp problem” remains an important open problem in the study of dark matter. One possible solution to this problem, and potentially other astrophysical problems, is the introduction of a new model for dark matter. Recently, Bray has geometrically motivated the study of a scalar field satisfying the Einstein-Klein-Gordon equations as a viable dark matter candidate via constructing axioms for general relativity [9]. The idea of using a scalar field to describe dark matter is not new. In fact, such a model has been seriously considered as a candidate for dark matter for more than two decades and has been shown to be in agreement with many cosmological observations [1,3,4,9,16,19,22–25,32–35]. In most of these settings, these scalar fields are considered from a quantum mechanical motivation for the same Einstein- Klein-Gordon equations and go by the name scalar field dark matter or boson stars. However, due to the fact that the Klein-Gordon equation is a wave-type partial differential equation, we prefer the name wave dark matter. In this paper, we begin to test wave dark matter against observations at the galactic level. In particular, we seek a working estimate of the fundamental constant in the wave dark matter model, Υ, to be used in future comparisons to data. To do so, we will compare the simplest model defined by wave dark matter to models of dark matter that are already known to fit observations well. Salucci et al. recently used the Burkert profile to model the dark matter energy density profiles of the eight classical dwarf spheroidal galaxies orbiting the Milky Way. They found excellent agreement between the observed velocity dispersion profiles of these galaxies and those velocity dispersion profiles predicted by the Burkert profile [31]. This can be seen in Figure 1, which we have reproduced exactly as it appears in the paper by Salucci et al. In what follows, we will show that a value of Υ = 50 yr−1 produces wave dark matter mass models that are qualitatively similar to the Burkert mass models found by Salucci et al. We will also show that underpreciseassumptions, comparisonstotheseBurkertprofilescanbeusedtoboundthevalue of Υ above by 1000 yr−1. 2 Burkert Mass Profiles The Burkert energy density profile models the energy density of a spherically symmetric dark matter halo using the function ρ r3 µ (r) = 0 c (1) B (r+r )(r2+r2) c c where ρ is the central density and r is the core radius. Integrating this function over the ball 0 c of radius r with respect to the standard spherical volume form yields the Burkert mass profile as follows. (cid:90) M (r) = µ (s)dV B B R3 Br(0) (cid:90) r = 4π s2µ (s)ds B 0 2 H. Bray and A. Parry Modeling Wave Dark Matter in dSph Galaxies Figure1: Observedvelocitydispersionprofilesoftheeightclassicaldwarfspheroidalgalaxiesare denoted by the points on each plot with its associated error bars. The solid lines overlayed on these profiles are the best fit velocity dispersion profiles predicted by the Burkert mass profile. This figure is directly reproduced from the paper by Salucci et al. [31] and the reader is referred to their paper for a complete description of how these models were computed. 3 H. Bray and A. Parry Modeling Wave Dark Matter in dSph Galaxies Figure 2: Plot of a Burkert mass profile. The inflection point is marked with an ×. (cid:90) r s2ρ r3 = 4π 0 c ds (s+r )(s2+r2) 0 c c (cid:18) (cid:18)r+r (cid:19) 1 (cid:18)r2+r2(cid:19) (cid:18) r (cid:19)(cid:19) M (r) = 2πρ r3 ln c + ln c −arctan (2) B 0 c r 2 r2 r c c c A generic plot of a Burkert mass profile, M (r), defined to be the dark matter mass in the B ball of radius r, is shown in Figure 2. We make a few remarks about the behavior of this mass function. Note that the behavior of the graph changes concavity at the inflection point r = r , which ip we have marked on the plot in Figure 2 with an ×. Recalling from equation (2) the fact that M (r) is the integral over the interval [0,r] of the function 4πr2µ (r), we can compute this B B inflection point as follows. 4πρ r3r2 M(cid:48) (r) = 4πr2µ (r) = 0 c (3) B B (r+r )(r2+r2) c c Differentiating again yields −4πρ r3(r4−r2r2−2rr3) M(cid:48)(cid:48)(r) = 0 c c c , (4) B (r+r )2(r2+r2)2 c c which has two complex zeros and two real zeros. The two real zeros are r = 0 and (cid:32) √ (cid:33) 3+(27+3 78)2/3 r = √ r ≈ 1.52r , (5) ip c c 3(27+3 78)1/3 the latter being the inflection point of the mass model. Forr (cid:29) r ,theplotgrowslogarithmicallyduetothefactthatthearctanterminequation(2) ip approaches a constant value as r → ∞. To describe the behavior when r (cid:28) r , we note that ip the Taylor expansion of M (r) centered at r = 0 is as follows, B 4 M (r) = πρ r3+O(r4). (6) B 0 3 4 H. Bray and A. Parry Modeling Wave Dark Matter in dSph Galaxies Figure 3: Left: Plot of the Burkert mass profile for the Carina galaxy found by Salucci et al. [31] along with a mass plot of a wave dark matter static ground state, the cubic function which is the leading term of the Taylor expansion of the Burkert mass profile, and the quadratic power M (r ) function B c r2 where r is the core radius of the Carina galaxy. The × marks the location r2 c c of the inflection point of the Burkert mass profile, while the vertical line denotes the location of the outermost data point for the Carina galaxy and is presented for reference purposes only. Right: Closeup of the plot on the left over the r interval [0,r ]. ip Thus for r (cid:28) r , M (r) is dominated by an r3 term making the initial behavior cubic. ip B In fact, several other models for dark matter mass profiles have similar initial behavior to the Burkert profile including a quadratic mass profile (which is not physical and is only included for the sake of comparisons) and wave dark matter mass profiles. In Figure 3, we have collected several mass models that have similar behavior inside r = r to the Burkert mass profile ip computed by Salucci et al. for the Carina galaxy [31]. While these models have similar behavior inside r = r , they are very different outside r = r . ip ip We have computed the inflection points of each of the Burkert mass profiles computed by Salucci et al. for the eight classical dwarf spheroidal galaxies [31] and have marked these points on a plot of each Burkert mass profile in Figure 4. We have constrained the viewing window of each plot to the range of data points collected. That is, we plot the Burkert mass profiles on the interval [0,r ], where r is the radius of the outermost data point given by Walker et last last al. [38,39] for the observed velocity dispersion profiles. We have presented them in order from greatest to least according to the ratio of r /r . last ip In Table 1, we have collected the defining parameters, ρ and r , computed by Salucci et al. 0 c for the Burkert mass profiles which best predict the velocity dispersion profiles of each galaxy [31]. We have also collected the outermost data point, r , of these velocity dispersion profiles last [38,39], as well as our computations of the inflection point, r , and the ratio r /r for each of ip last ip the classical dwarf spheroidal galaxies. All quantities have been converted to geometrized units (the universal gravitational constant and the speed of light set to one) of (light)years for mass, length, and time. 5 H. Bray and A. Parry Modeling Wave Dark Matter in dSph Galaxies Figure 4: Plots of the Burkert mass profiles computed by Salucci et al. of the eight classical dwarf spheroidal galaxies within the range of observable data. The inflection point is marked on each plot by an ×. Carina and Draco have no inflection point marked because the inflection point for their Burkert mass profiles occurs outside the range of observable data. 6 H. Bray and A. Parry Modeling Wave Dark Matter in dSph Galaxies Galaxy Name ρ (yr−2) r (yr) r (yr) r (yr) r /r 0 c last ip last ip Sextans 2.47×10−14 1.53×102 3.26×103 2.32×102 14.05 Leo II 1.83×10−14 1.88×102 1.37×103 2.86×102 4.80 Fornax 8.57×10−16 1.21×103 5.54×103 1.84×103 3.01 Leo I 1.83×10−15 9.19×102 3.03×103 1.40×103 2.17 Sculptor 1.10×10−15 1.16×103 3.59×103 1.76×103 2.04 Ursa Minor 1.83×10−15 8.01×102 2.41×103 1.22×103 1.98 Carina 2.90×10−16 1.97×103 2.84×103 2.99×103 0.95 Draco 8.19×10−16 2.11×103 3.00×103 3.20×103 0.94 Table 1: Burkert mass profile data for the eight classical dwarf spheroidal galaxies converted to units of years for mass, length, and time. The parameters ρ and r are those found by Salucci 0 c et al. for the best fit Burkert profiles [31], and r is the radius of the outermost data point last given by Walker et al. [38,39]. Also included is the value of the inflection point, r , of the ip Burkert mass profile for each galaxy and the ratio of r to r . last ip 3 Static States of Wave Dark Matter Now that we have presented mass profiles which model actual data well, we need to describe the wave dark matter models we will use to make our comparison. In the following, we present only the basic background information required to understand the model we use and refer the reader to [9,10] for more discussion on its motivation and successes thus far. Let (N,g) be a spacetime whose metric has signature (− + ++). Let f : N → C be a smooth complex-valued scalar field defined on the spacetime. Finally, let f and g satisfy the Einstein-Klein-Gordon equations, (cid:32) (cid:32) (cid:33) (cid:33) df ⊗df¯+df¯⊗df |df|2 G = 8πµ − +|f|2 g (7a) 0 Υ2 Υ2 (cid:50) f = Υ2f (7b) g where (cid:50) is the Laplacian with respect to the metric g. The parameter Υ is a fundamental g constant of this system and its value must be determined in order to use these equations to model dark matter in the universe. On the other hand, the parameter µ is not fundamental to 0 the system and can be completely absorbed into f if desired. In [9], wave dark matter may be modeled as a real scalar field or as a complex scalar field. For the Einstein-Klein-Gordon system, a complex scalar field is equivalent to two real scalar fields. Complex scalar fields are more convenient because they have static spacetime solutions. Analogous “nearly static” real solutions can be achieved by redefining f for a static spacetime √ solution as 2 Re(f) whose spacetime metrics are very similar except for a high frequency oscillating pressure term which averages out to zero. Preliminary estimates suggest that this oscillating pressure effect might not be significant enough to make it measurable on physically relevant time scales. Hence, there may not be a testable difference between the predictions of real and complex scalar field dark matter at this time, though more thought on this question is well deserved. In either case, we will call these scalar field models “wave dark matter.” 7 H. Bray and A. Parry Modeling Wave Dark Matter in dSph Galaxies Since we wish to compare solutions of (7) to the spherically symmetric Burkert mass profiles computed by Salucci et al. [31], we choose to work in spherical symmetry. In a recent paper [29], Parry surveyed the well-known form of the metric of a general spherically symmetric spacetime in polar-areal coordinates, namely, (cid:18) 2M(t,r)(cid:19)−1 g = −e2V(t,r)dt2+ 1− dr2+r2dσ2, (8) r for real valued functions V and M and where dσ2 = dθ2 + sin2θdϕ2 is the standard metric on the unit sphere. This metric has the following useful properties. The function M(t,r) is the Hawking mass of the metric sphere of radius r and time t. Under the Einstein equation, G = 8πT, M is also the flat volume integral of the energy density term in the stress energy tensor. This motivates interpreting the function M(t,r) as the mass inside the metric sphere of radiusr attimet. Finally, giventheEinsteinequation, inthelowfieldlimit, V isapproximately the gravitational potential of the system. We refer the reader to [29] for detailed proofs of these facts. It is also shown in [29] that in spherical symmetry and using the metric (8), solving the Einstein-Klein-Gordon system (7) reduces to solving the system (cid:32) (cid:33) (cid:18) 2M(cid:19) |f |2+|p|2 M = 4πr2µ |f|2+ 1− r (9a) r 0 r Υ2 (cid:32) (cid:32) (cid:33)(cid:33) (cid:18) 2M(cid:19)−1 M (cid:18) 2M(cid:19) |f |2+|p|2 V = 1− −4πrµ |f|2− 1− r (9b) r r r2 0 r Υ2 (cid:114) 2M f = peV 1− (9c) t r (cid:32) (cid:18) 2M(cid:19)−1/2 2f (cid:114) 2M(cid:33) (cid:32) (cid:114) 2M(cid:33) p = eV −Υ2f 1− + r 1− +∂ eVf 1− . (9d) t r r r r r r To solve this system, we need boundary conditions. At the central value, we require all of the functions to be smooth. Since all of the functions are spherically symmetric, this implies that M , V , f , and p all vanish at r = 0 for all t. We will also require that the spacetime r r r r be asymptotically Schwarzschild, that is, it approaches a Schwarzschild metric as r → ∞. Specifically, this implies that (cid:18) (cid:19) 2M e2V → κ2 1− as r → ∞ (10) r (cid:50) f → Υ2f and f → 0 as r → ∞ (11) gS where κ > 0 and g is the appropriate Schwarzschild metric. Since f → 0 as r → ∞, M S approaches a constant value m, which is the total mass of the system. Note that these boundary conditions ensure that as r → ∞, the metric g in equation (8) becomes the Schwarzschild metric (cid:18) 2m(cid:19) (cid:18) 2m(cid:19)−1 g = −κ2 1− dt2+ 1− dr2+r2dσ2. (12) S r r 8 H. Bray and A. Parry Modeling Wave Dark Matter in dSph Galaxies Thus κ represents a scaling of the t coordinate in the standard Schwarzschild metric. The effect of this on our discussion is that V → lnκ as r → ∞. We have to numerically solve these equations and so in practice, we will impose these boundary conditions at an artificial right hand boundary point r and solve the system on the r-interval [0,r ]. max max One of the simplest solutions to this system are those where the scalar field is of the form f(t,r) = eiωtF(r) (13) where ω ∈ R is a constant and F is real valued. Note that for f of this form, solving equation (11) for large r and requiring the solution to decay to 0 yields that, for large r, F must satisfy (cid:32)(cid:114) (cid:33) ω2 1 F(cid:48)+ Υ2− + F ≈ 0. (14) κ2 r Requiring this condition on our system ensures that f appropriately decays to 0 as r → ∞ [28]. Solutions of the form in equation (13) produce static metrics and, once substituted into the system (9), yield the following set of ODEs [28], (cid:34) (cid:35) (cid:18) ω2 (cid:19) (cid:18) 2M(cid:19) |H|2 M(cid:48) = 4πr2µ 1+ e−2V |F|2+ 1− (15a) 0 Υ2 r Υ2 (cid:40) (cid:34) (cid:35)(cid:41) (cid:18) 2M(cid:19)−1 M (cid:18) ω2 (cid:19) (cid:18) 2M(cid:19) |H|2 V(cid:48) = 1− −4πrµ 1− e−2V |F|2− 1− (15b) r r2 0 Υ2 r Υ2 F(cid:48) = H (15c) (cid:18) 2M(cid:19)−1(cid:20)(cid:18) ω2 (cid:19) (cid:18)M 1(cid:19)(cid:21) H(cid:48) = 1− Υ2− F +2H +4πrµ |F|2− (15d) r e2V r2 0 r with boundary conditions F(0) = 1, H(0) = 0, M(0) = 0, V(0) = V , (16) 0 (cid:32)(cid:114) (cid:33) ω2 1 F(cid:48)(r )+ Υ2− + F(r ) ≈ 0, (17) max κ2 r max max (cid:18) (cid:19) 1 2M(r ) max V(r )− ln 1− −lnκ ≈ 0, (18) max 2 r max by equations (10) and (11). For simplicity, we set κ = 1, which corresponds to the assumption on our choice of t coordinate that V goes to zero at infinity. A solution to these equations depends on the choice of the parameters Υ, µ , ω, and V . We solve a shooting problem for ω 0 0 and V to satisfy (17) and (18) leaving Υ and µ freely selectable. 0 0 For each choice of Υ and µ , there are an infinite number of discrete finite mass solutions 0 characterized by the number of zeros that F exhibits [28]. These are called static states. A static state with no zeros is called a ground state. With n zeros for n > 0, it is called an nth excited state. In Figure 5, we have plotted examples of F for a ground through third excited state. In Figure 6, we have presented the plots of the mass, M, corresponding to the plots of F in Figure 5. 9 H. Bray and A. Parry Modeling Wave Dark Matter in dSph Galaxies Figure 5: Plots of spherically symmetric static state scalar fields (specifically the function F(r) in (13)) in the ground state and first, second, and third excited states. Note the number of nodes (zeros) of each function. 10