A&A 540, A70 (2012) Astronomy DOI: 10.1051/0004-6361/201118543 & c ESO 2012 ! Astrophysics
Analytical properties of Einasto dark matter haloes
E. Retana-Montenegro1 ,E.VanHese2,G.Gentile2,M.Baes2,andF.Frutos-Alfaro1
1 Escuela de Física, Universidad de Costa Rica, 11501 San Pedro, Costa Rica e-mail: [email protected] 2 Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281-S9, 9000 Gent, Belgium e-mail: [email protected] Received 29 November 2011 / Accepted 19 January 2012
ABSTRACT
Recent high-resolution N-body CDM simulations indicate that nonsingular three-parameter models such as the Einasto profile perform better than the singular two-parameter models, e.g. the Navarro, Frenk and White, in fitting a wide range of dark matter haloes. While many of the basic properties of the Einasto profile have been discussed in previous studies, a number of analytical properties are still not investigated. In particular, a general analytical formula for the surface density, an important quantity that defines the lensing properties of a dark matter halo, is still lacking to date. To this aim, we used a Mellin integral transform formalism to derive a closed expression for the Einasto surface density and related properties in terms of the Fox H and Meijer G functions, which can be written as series expansions. This enables arbitrary-precision calculations of the surface density and the lensing properties of realistic dark matter halo models. Furthermore, we compared the Sérsic and Einasto surface mass densities and found differences between them, which implies that the lensing properties for both profiles differ. Key words. gravitational lensing: weak – galaxies: clusters: general – dark matter – methods: analytical – galaxies: halos – gravitational lensing: strong
1. Introduction obtain the best fit to observational data from strong- and weak- lensing studies. The Λ cold dark matter (CDM) model, with (Ωm, ΩΛ) = Recently, N-body CDM simulations (Navarro et al. 2004; (0.3, 0.7),hasbecomethestandardtheoryofcosmologicalstruc- Merritt et al. 2006; Gao et al. 2008; Hayashi & White 2008; ture formation. ΛCDM seems to agree with the observations Stadel et al. 2009; Navarro et al. 2010)havefoundthatcertain on cluster-sized scales (Primack 2003); however, on galaxy/sub- three-parameter profiles provide an excellent fit to a wide range galaxy scales there appears to be a discrepancy between obser- of dark matter haloes. One of these is the Einasto (1965)pro- vations and numerical simulations. High-resolution observations file, a three-dimensional version of the two-dimensional Sérsic of rotation curves, in particular of low surface brightness (LSB) (1968)profileusedtodescribethesurfacebrightnessofearly- and dark matter dominated dwarf galaxies (de Blok et al. 2001; type galaxies and the bulges of spiral galaxies (e.g. Davies van den Bosch & Swaters 2001; Swaters et al. 2003; Weldrake et al. 1988; Caon et al. 1993; D’Onofrio et al. 1994; Cellone et al. 2003; Donato et al. 2004; Gentile et al. 2005; Simon et al. et al. 1994; Andredakis et al. 1995; Prugniel & Simien 1997; 2005; Gentile et al. 2007; Banerjee et al. 2010)favourdensity Möllenhoff &Heidt2001; Graham & Guzmán 2003; Graham profiles with a flat central core (e.g. Burkert 1995; Salucci & et al. 2006; Gadotti 2009). The Sérsic profile can be written as: Burkert 2000; Gentile et al. 2004; Li & Chen 2009). In con- trast, N-body (dark matter only) CDM simulations predict galac- R 1/m tic density profiles that are too high in the centre (e.g. Navarro Σ (R) =ΥI exp b 1 , (1) S e − m R − et al. 1996, 1997,NFW;Moore et al. 1999). This discrepancy is ( e ) called the cusp-core problem; a complete review can be found in de Blok (2010). where R is the distance in the sky plane, m the Sérsic index, Gravitational lensing is one of the most powerful tools in Υ is the mass-to-light ratio, Ie is the luminosity density at the observational cosmology for probing the matter distribution of effective radius Re,andbm is a dimensionless function of m that galaxies and clusters in the strong regime (Kochanek et al. can be determined from the condition that the luminosity inside 1989; Wambsganss & Paczynski 1994; Bartelmann 1996; Chae Re equals half of the total luminosity. Numerical solutions for et al. 1998; Kochanek et al. 2000; Keeton & Madau 2001; Sand bm are given by Ciotti (1991), Moriondo et al. (1998), Prugniel &Simien(1997)andanasymptoticexpansionb = 2m 1/3 + et al. 2002; Keeton 2002, 2003; Keeton & Zabludoff 2004; m − Limousin et al. 2008; Anguita et al. 2009; Zitrin et al. 2011a,b) 4/405m+ m2 using analytical methods was obtained by Ciotti O and the weak regime (Kaiser & Squires 1993; Mellier 1999; &Bertin(1999). 0 1 Bartelmann & Schneider 2001; Hoekstra et al. 2004; Clowe The Einasto profile (model) is characterised by a power-law et al. 2006; Mahdavi et al. 2007; Jee et al. 2009; Huang et al. logarithmic slope, 2011). Comparing these observations to theoretical models pro- vides key information to help resolve the cusp-core problem. dln ρ 1/n Evidently, one must use the most accurate density profile to γ(r) (r) r , (2) ≡−dln r ∝ Article published by EDP Sciences A70, page 1 of 10 A&A 540, A70 (2012) with n,whichwecalltheEinastoindex,apositivenumberdefin- of MR. Chemin et al. (2011)modelledtherotationcurvesofa ing the steepness of the power law. Integrating leads to the gen- spiral galaxies subsample from THINGS (The HI Nearby Galaxy eral density profile Survey, de Blok et al. 2008), using the Einasto profile, and found that n tends to have lower values by a factor of two or more r 1/n compared to those predicted by N-body simulations. Dhar & ρ(r) = ρs exp dn 1 , (3) − rs − Williams (2012)fittedthesurfacebrightnessprofilesofalarge ( ) sample of elliptical galaxies in the Virgo cluster, using a multi- component Einasto profile consisting of two or three compo- where rs represents the radius of the sphere that contains half of the total mass, ρs is the mass density at r = rs,anddn is a nents for each galaxy with different n for each individual com- numerical constant that ensures that rs is indeed the half-mass ponent. For the central component, they found values of n ! 1 radius. In the context of dark matter haloes, the density can also for the most massive and shallow-cusp galaxies and n < 2for be expressed as steep-cusp galaxies, while 5 ! n ! 8fortheoutercomponentof massive galaxies. These values for the outer component are con- r 1/n sistent with the results from N-body simulations for Einasto dark ρ (r) = ρ 2 exp 2n 1 , (4) matter haloes, and the authors argued that Einasto components − − r 2 − ( − ) with an n in this range could be dark matter dominated. In the light of its increasing popularity to describe the den- where ρ and r are the density and radius at which ρ (r) r 2. 2 2 − sity of simulated dark matter haloes, a detailed investigation of In the remainder− − of this paper, we will use yet another, equivalent∝ the properties of the Einasto model is of paramount importance. version, Some aspects of the Einasto model have been presented by sev- r 1/n eral authors (Mamon & Łokas 2005; Cardone et al. 2005; Merritt ρ(r) = ρ0 exp , (5) et al. 2006; Dhar & Williams 2010). The most complete study of − h 2 3 4 5 the properties of the Einasto model is the work by Cardone et al. where we introduced the central density (2005), who provided a set of analytical expressions for quanti- ties such as the mass profile and gravitational potential and dis- dn 2n ρ0 = ρs e = ρ 2 e , (6) cussed the dynamical structure for both isotropic and anisotropic − cases. Nevertheless, the Einasto model has not been studied ana- and the scale length lytically as extensively as the Sérsic models, and several proper- ties still have to be further investigated in more detail. One area rs r 2 − h = n = n (7) where progress is still to be made is on the value of the dimen- dn (2n) · sionless parameter dn. If a model is to describe real galactic systems, several condi- The most important lacuna, however, concerns the surface tions must be imposed on the model description functions, as density on the plane of the sky, an important quantity that de- discussed by Einasto (1969a). When a model is constructed, an fines the lensing properties of a dark matter halo. The surface initial descriptive function is chosen; the most practical choice mass density has an importance in theoretical predictions and is the density profile, because themaindescriptivefunctions observations. On the observational side, based mostly on the (cumulative mass profile, gravitational potential, surface mass mass decomposition of rotation curves using cored haloes, var- density) are integrals of the density profile. Furthermore, a phys- ious studies (Kormendy & Freeman 2004; Spano et al. 2008; ical model has to satisfy several conditions: i) the density pro- Donato et al. 2009; Gentile et al. 2009)haveshownthattheprod- file must be non-negative and finite; ii) the density must be a uct of the central density ρ0 and the core radius r0 is consistent smoothly decreasing function that approaches zero at large radii; with a universal value, independent of galaxy mass. The prod- iii) some moments of the mass function must be finite, in partic- uct ρ0r0 is directly proportional to the average surface density ular moments that define the central gravitational potential, the within r0,andtothegravitationalaccelerationofdarkmatterat total mass, and the effective radius of the system; and iv) the de- r0.Inviewofadetailedcomparisonoftheseresultswiththeout- scriptive functions must not have jump discontinuities. Einasto come of the most recent dark matter simulations, it is therefore (1969a)presentedseveralfamiliesofvaliddescriptivefunctions, important to study the analytical properties of the surface den- among which the so-called Einasto profile is a special case that sity distribution of Einasto haloes. Recent works studying theo- agrees best with observations. retical and observational aspects of dark matter surface densities The Einasto profile was used by Einasto (1969b)toobtaina include Boyarsky et al. (2010), Walker et al. (2010), Napolitano model of M31. Later, this model was applied by Einasto (1974) et al. (2010), and Cardone et al. (2011): efforts are being made to to several nearby galaxies, including M32, M87, Fornax and confirm or call into question the universality of the dark matter Sculptor dwarfs, and also M31 and the Milky Way. These mod- surface density within one dark halo scale length. els were multi-component ones, and each component has its own Cardone et al. (2005)showedthatthesurfacedensityof parameter set ρ0, h, n representing certain physically homo- the Einasto model cannot be expressed in terms of elementary geneous stellar{ populations.} functions and discussed the general properties using numeri- Navarro et al. (2004)foundthatforhaloeswithmassesfrom cal integration. Dhar & Williams (2010)presentedananalyti- dwarfs to clusters, 4.54 ! n ! 8.33 with an average value cal approximation for the Einasto surface brightness profile and of n = 5.88. Hayashi & White (2008)andGao et al. (2008) demonstrated that this surface density profile is not Sérsic-like. found that n tends to decrease with mass and redshift, with Ageneralanalyticalformula,which would enable an arbitrary- n 5.88 for galaxy-sized and n 4.35 for cluster-sized haloes precision calculation and an analytical study of the asymptotic in∼ the Millennium Run (MR) (Springel∼ et al. 2005). Navarro behaviour, is still lacking to date. et al. (2010)obtainedsimilarresultsforgalaxy-sizedhaloes The most recent studies in strong and weak lensing (e.g. in the Aquarius simulation (Springel et al. 2008). Also, Gao Donnarumma et al. 2011; Sereno & Umetsu 2011; Morandi et al. et al. (2008)showedthatn 3.33 for the most massive haloes 2011; Umetsu et al. 2011; Oguri et al. 2012)usetheNFWprofile ∼ A70, page 2 of 10 E. Retana-Montenegro et al.: Analytical properties of Einasto dark matter haloes or its generalization (Zhao 1996; Jing & Suto 2000)tomodelthe where Γ(α, x)istheincompletegammafunction, dark matter halo instead of the Einasto profile. One of the factors that limits its adoption is the absence of analytical formulas for ∞ α 1 t its lensing properties. Therefore, its application in lensing stud- Γ(α, x) = t − e− dt. (14) ies is not as wide as for other profiles. A complete general set 6x of analytical formulas for the lensing properties of the Einasto Given that this is the radius of the sphere that encloses half of profile would help to increase its use in lensing studies. the total mass, we find that d is the solution of the equation In this paper, we extend the analytical study of the Einasto n model using some of the techniques also employed in the exten- sive literature on the analytical properties of the Sérsic model 2 Γ(3n, dn) =Γ(3n). (15) (e.g. Ciotti 1991; Ciotti & Lanzoni 1997; Ciotti & Bertin 1999; Trujillo et al. 2001; Mazure & Capelato 2002; Cardone 2004; This equation cannot be solved in a closed form; one option is Graham & Driver 2005; Elíasdóttir & Möller 2007; Baes & to solve it numerically or use interpolation formulae. For exam- Gentile 2011; Baes & van Hese 2011). In Sect. 2,wediscuss ple, Merritt et al. (2006), quoting G. A. Mamon (priv. comm.), propose d 3n 1 + 0.0079/n,butdonotmentiontheorigin an analytical expansion for the dimensionless parameter dn.In n ≈ − 3 Sect. 3,wederiveananalyticalexpressionfortheEinastosur- of this approximation or quote its accuracy. An elegant way to face mass density in terms of the Fox H function, using the determine an approximation for dn builds on the work by Ciotti Mellin transform-method. We then use the result for the pro- &Bertin(1999), who used an asymptotic expansion method to jected surface mass density to calculate the cumulative surface solve the general relation Γ(α, x) =Γ(α)/2. If we apply the re- mass, lens equation, deflection angle, deflection potential, mag- sulting expansion to our problem, we obtain for the parameter dn nification, shear and the critical curves for a spherically symmet- from the Einasto model ric lens described by the Einasto profile in terms of this func- tion. We also calculate explicit series expansions for the surface 1 8 184 d 3n + + mass density and all the lensing properties. In Sect. 4,wede- n ≈ − 3 1215 n 229635 n2 rive some special cases for the lensing properties for integer 1048 17557576 1 and half-integer values of n in terms of the Meijer G function. + + (16) 31000725 n3 − 1242974068875 n4 O n5 · Furthermore, we compare the Einasto and Sérsic surface mass ( ) densities for the same values of their respective indices. Finally, in Sect. 5 we discuss the asymptotic behaviour of the surface The gravitational potential of a spherically symmetric mass dis- mass density and cumulative surface mass at small and large tribution ρ(r)canbefoundthroughtheformula radii. We present our conclusions in Sect. 6. r 1 2 ∞ Ψ(r) = 4πG ρ(r&) r& dr& + ρ(r&) r& dr& , (17) r 0 r 2. Spatial properties 2 6 6 5 The total mass of an Einasto model with a mass density profile or equivalently given by (5), is ∞ M(r&)dr& 3 Ψ(r) = (18) M = 4πρ0 h n Γ(3n). (8) r 2 · 6r & If we use this formula to replace the central density ρ0 by the to- tal mass M as a parameter in the definition of the Einasto models, For the Einasto model, we find we obtain 1/n 1/n GM 1 Γ(3n, s ) s Γ(2n, s ) Ψ(r) = s− 1 + (19) M s1/n ρ(r) = e− , (9) h − Γ(3n) Γ(3n) · 4π h3 n Γ(3n) 2 5 where we have introduced the reduced radius The Einasto model has a finite potential well, given by n (dn) r GM Γ(2n) s = (10) Ψ0 = (20) rs · h Γ(3n)· At small radii, the density profile behaves as At small radii, the potential decreases parabolically M ρ(r) = 1 s1/n + ... . (11) 3 GM Γ(2n) s2 4π h n Γ(3n) − Ψ(r) + ... , (21) 0 1 ∼ h Γ(3n) − 6n Γ(3n) The cumulative mass profile M(r)ofasphericalmassdistribu- 2 5 tion is found through the equation which is not surprising, given the finite density core of the r Einasto density profile. At large radii, we obtain a Keplerian fall- 2 M(r) = 4π ρ(r&) r& dr&. (12) off, 60 GM For the Einasto model, we find Ψ(r) + ..., (22) ∼ r Γ(3n, s1/n) M(r) = M 1 , (13) − Γ(3n) which is also expected for a model with a finite total mass. 2 5 A70, page 3 of 10 A&A 540, A70 (2012)
3. Lensing properties transforms of f1 and f2.TheMellintransformanditsinverseare defined as 3.1. Surface mass density and cumulative surface mass density ∞ u 1 M f (u) = φ(u) = f (z) z − dz, (29) The surface mass density of a spherically symmetric lens is given 60 by integrating along the line of sight of the 3D density profile 1 1 u M− (z) = f (z) = φ(u) z− du, (30) φ 2πi + L ∞ 6 Σ (ξ) = ρ (ξ, r) dz, (23) with a line integral over a vertical line in the complex plane. L 6−∞ This implies that where ξ is the radius measured from the centre of the lens and 1 u r = ξ2 + z2.ThisexpressioncanalsobewrittenasanAbel f (z) = M (u) M (u) z− du. (31) 2πi f1 f2 transform (Binney & Tremaine 1987) 6L 7 It now turns out that the integral (31)isaMellin-Barnesintegral ∞ ρ (r) r dr Σ (ξ) = 2 (24) for large classes of the functions f1 and f2,andthatthisintegral 2 2 · 6ξ r ξ can be evaluated as a Fox H function or a Meijer G function in − many cases. Inserting Eq. (57)intotheaboveexpression,weobtain We can immediately apply this formalism to the inte- gral (25), with z = 1and s1/n ∞ e− s ds Σ (x) = 2 ρ0 h , (25) t1/n 2 x √s2 x2 f1(t) = 2 ρ0 h e− t , (32) 6 − where we have introduced the quantities x = ξ/h and s = r/h. and As discussed by Cardone et al. (2005)andDhar & Williams t 1 (2010), the integral (25)cannotbeexpressedintermsofele- if 0 t x− , f (t) = √ 2 2 ≤ ≤ (33) mentary or even the most regular functions for all the values of 2 1 x t 0else− . n.Onlythecentralsurfacemassdensitycanbeevaluatedanalyt- ically as The Mellin transforms of these functions are readily calculated Σ(0) = 2n ρ0 h Γ (n) . (26) M f1 (u) = 2 ρ0 hnΓ [n(2 + u)], (34) 1+u This situation is very similar to the deprojection of the Sérsic √π Γ 2 1 surface brightness profile (1). Indeed, the deprojection formula M f2 (u) = (35) 0u 1 ux1+u · leads to an integral that is an inverse Abel transform, and also in Γ 2 this case it was long thought that no further analytical progress Combining these0 1 results, we obtain was possible. However, Mazure & Capelato (2002)usedthe computer package Mathematica to demonstrate that it is possi- Σ(x) = 2n √πρ0 hx ble to write the Sérsic luminosity density in terms of the Meijer Γ (2ny) Γ 1 + y G function for all integer Sérsic indices m. Baes & Gentile 1 2 y − x2 − dy. (36) (2011)andBaes & van Hese (2011)tookthisanalysisonestep × 2πi Γ (y0) 1 further and showed that the deprojection of the Sérsic surface 6L 8 9 brightness profile for general values of m can be solved elegantly This Mellin-Barnes integral may be recognized as a Fox H func- using Mellin integral transforms and gives rise to a Mellin- tion, which is generally defined as the inverse Mellin transform Barnes integral. The result is that the Sérsic luminosity density of a product of gamma-functions, can be written compactly in terms of a Fox H function, which (a, A) reduces to a Meijer G function for all rational values of m. Hm,n z = The obvious similarity between these two cases invites us p,q (b, B) 2 : 5 to apply the same Mellin transform technique (Marichev 1982; : m Γ(b + B s) n Γ(1 a A s) 1 : j=1 j j j=1 j j s Adamchick 1996; Fikioris 2007)totheintegral(25). The basic : q p − − z− ds. 2πi Γ(1 b j B js) Γ(a j + A j s) idea behind this technique is that any definite integral 6L ;j=m+1 − − ; j=n+1 (37) ∞ ; ; f (z) = g(t, z)dt, (27) Using this definition, we can write the surface density of the 60 Einasto model in the following compact form, can be written as (0, 1) Σ (x) = 2n √πρ hxH2,0 x2 . (38) ∞ z dt 0 1,2 (0, 2n), ( 1 , 1) f (z) = f1(t) f2 (28) 2 − 2 : 5 0 t t · : 6 The Fox H function is a general analytical: function that is 3 4 : This expression is exactly a Mellin convolution of two functions becoming more and more used both in mathematics and ap- f1 and f2.NowwecanapplytheMellinconvolutiontheorem, plied sciences; in fact, it is scheduled for inclusion in the which states that the Mellin transform of a Mellin convolution Mathematica numerical library. While not the most mainstream is equal to the products of the Mellin transforms of the origi- special function, the Fox H is an extremely flexible function that nal functions. As a result, the definite integral (27)canbewrit- contains very broad classes of elementary and special functions ten as the inverse Mellin transform of the product of the Mellin as particular cases. It is indeed a very powerful tool for analytical
A70, page 4 of 10 E. Retana-Montenegro et al.: Analytical properties of Einasto dark matter haloes work. It has many general properties that allow one to manipu- Case 2: if n is integer or a rational number p/q with an odd late expressions to equivalent forms, reduce the order for certain denominator, some poles are of second order, and the expansion valuesof the parameters, etc. For details on its many useful prop- q 1 is a logarithmic-power series. If we define k0 = −2 ,thenwe erties, we refer the interested reader to Mathai (1978), Kilbas & obtain after some algebra Saigo (2004), or Mathai et al. (2009)andthereferencestherein. As an illustration of the power of the Fox H function as an 1 k k k/n+1 ∞ Γ 2 2n ( 1) x analytical tool, and as a sanity check on the formula, we can Σ(x) = 2n √πρ0 h − − − calculate the total mass of the Einasto model by integrating the 0Γ k 1 k! 2n k=1 2n surface mass density over the plane of the sky, k mod
4. Integer and half-integer values of n the average convergence can be calculated by inserting Eq. (67) into Eq. (61), and then integrating, we find The expressions for the Einasto surface mass density (38)andits lensing properties (42), (54), (56), (62)–(64)intermsoftheFox κ 1 x2 H function for case n is a rational number, can be reduced to a c 2n,1 2 κ¯ (x) = xG − 3 (72) Meijer G function, 2(2π)n 1 √n Γ (n) 1,2n+1 b, (2n)2n · − 2 2 5 − : : m,n a : Gp,q z = 1 b 4.1. Simple cases: n = 1 and n = 2 2 5 : m n 1 : Γ(b j + s) Γ(1 a j s) The Einasto profile corresponding to n = 1isasimpleexponen- : j=1 j=1 − − s : q p z− ds. (65) tial model, characterised by the density profile 2πi Γ(1 b j s) Γ(a j + s) 6L ;j=m+1 − −; j=n+1 Meijer G function; numerical routin; es had been implemented r ρ (r) = ρ exp , (73) in computer algebraic systems, such as the commercial Maple, 0 − h Mathematica and the free open-source Sage and mpmath li- ? 3 4@ brary; in contrast, there is no Fox H function implementation the surface density profile and lensing properties are readily available. found by setting n = 1inEqs.(67), (68), (69), (70), (71)and(72) For case n integer or half-integer, we can write Σ (x) as a
Meijer G function. Inserting the Gauss multiplication formula 2 (Abramowitz & Stegun 1970) 2,0 x Σ(x) = ρ0 hxG0,2 1 − 1 , (74) 2 , 2 4 2n 1 2 − : 5 1 2 1 +2ny 1 n − j : x 2 2 3 3 2,1 : 2 Γ(2ny) = (2n)− (2π) − Γ(y) Γ + y , (66) M (x) = πρ0 h x G : − , (75) 2n 1,3 1 ,: 1 , 3 4 j=1 3 4 2 2 − 2 − 2 : 5 > 1 2 : κc x : into Eq. (36)andcomparingwiththedefinition(65), we obtain α (x) = x2 G2,1 − 2 : , (76) 2 1,3 1 , 1 , 3 4 : an expression for the surface mass density of the Einasto profile 2 2 − 2 − 2 : 5 in terms of the Meijer G function κ 1 , 1: x2 c 3 2,2 2 2: ψ (x) = x G2,4 1 −1 −3 : 3 , (77) 2 4 2 , 2 , 2 , 2 4 √n ρ0 h x 2 − − − : 5 Σ(x) = xG2n,0 , (67a) 2 : n 1 0,2n −b 2n κc 2,0 x : (2π) − (2n) κ (x) = xG − , : (78) 2 : 5 2 0,2 1 , 1 4 : : 2 2 − 2 : 5 where b is a vector of size: 2n given by 1: 2 : κc : x κ¯ (x) = xG2,1 − 2: (79) 1 2 2n 1 1 2 1,3 1 , 1 ,: 3 4 · b = , ,..., − , . (67b) 2 2 − 2 − 2 : 5 2n 2n 2n −2 : ? @ The specific Meijer G functions: in these expressions can be re- Integrating the above expression for surface mass density ac- duced to more standard special functions. We obtain the equiva- cording to (41)andusingtheintegralpropertiesoftheMeijerG lent expressions function (Eq. 07.34.21.0003.01 on the Wolfram Functions Site1), we obtain Σ (x) = 2ρ0 hxK1 (x) , (80) √n ρ h3 1 x2 2 0 3 2n,1 2 3 x M (x) = x G − 3 (68) M (x) = 8πρ h 1 K (x) , (81) 2(2π)n 2 1,2n+1 b, (2n)2n · 0 − 2 2 − 2 − 2 : 5 2 5 : 2 Inserting Eq. (67)into(49)andperformingtheintegralofthe: 4κc x : α (x) = 1 K2 (x) , (82) Meijer G function, we find x − 2 2 5 1 2 x x 1 κc 2 2n,1 2 x ψ (x) = 4κc ln + K1 (x) + K0 (x) + γ , (83) α (x) = x G − 3 (69) 2 2 − 2 2(2π)n 1 √n Γ (n) 1,2n+1 b, (2n)2n · 2 5 − 2 − 2 5 3 4 : κ (x) = κc xK1 (x) , (84) We obtain the deflection potential in terms of: the Meijer G func- 2 : 4κc x tion combining Eqs. (55), (67)andagainweintegrate,applying κ¯ (x) = 1 K2 (x) , (85) x2 − 2 the Meijer G function properties. We obtain 2 5
1 1 2 where Kν(x)isthemodifiedBesselfunctionofthesecondkind κc 3 2n,2 2 , 2 x ψ (x) = x G − 3− 3 (70) of order ν,andγ 0.57721566 the Euler-Mascheroni constant. 4(2π)n 1 √n Γ (n) 2,2n+2 b, , (2n)2n · − 2 − 2 − 2 : 5 The expressions (≈80)–(85)canalsobereadilycalculatedbyin- : The convergence can be simply found dividing Eq.: (67)byΣcrit serting the density profile (73)intotheformula(24)andlater : carrying out all corresponding calculations for the lensing prop- 2 κc 2n,0 x erties. κ (x) = xG − , (71) 1 2(2π)n 1 √n Γ (n) 0,2n b (2n)2n For n = ,thedensityprofile(4)becomesaGaussian, − 2 5 2 : : 1 http://functions.wolfram.com/: r 2 ρ (r) = ρ exp , (86) HypergeometricFunctions/MeijerG/ 0 − h A 3 4 B A70, page 7 of 10 A&A 540, A70 (2012) the surface density and lensing properties can be found through The Sérsic Profile 1 substitution of n = 2 in Eqs. (67), (69), (68), (70), (71)and(72), 106 1,0 2 Σ(x) = √πρ0 hxG0,1 −1 x , (87) 2 2 − : 5 105 : 1 M (x) = π3/2 ρ h3 x3 G1,1 : 2 x2 , (88) 0 1,2 : 1−, 3 2 − 2 − 2 : 5 1 : 104 2 1,1 2 2 : α (x) = κc x G1,2 1− 3 x , : (89) 2 , 2 s 2 − − : 5 ! 1 : 1 κc , :
log 1000 ψ (x) = x3 G1,2 − 2 −: 2 x2 , (90) 2 2,3 1 , 3:, 3 2 − 2 − 2 − 2 : 5 : 4 1,0 : 2 : 100 κ (x) = κc xG0,1 −1 x , : (91) 2 2 2 − : 5 1 :1 κ¯ (x) = κ xG1,1 :2 x2 . (92) c 1,2 1−,: 3 10 2 − 2 − 2 5 1 : m= These Meijer G function: can also be written in terms of elemen- 2 : tary and special functions, 1 0 20 40 60 80 100 x2 Σ (x) = √πρ0 h e− , (93) Rad iu s 3/2 3 x2 M (x) = π ρ0 h 1 e− , (94) − Fig. 1. Sérsic profile, in dimensionless units, where ΥIe and Re are held κc x2 fixed for different values of the Sérsic index m. α (x) = 1 e− 0 , 1 (95) x − κ ψ (x) = c 0ln x2 +1E (x2) + γ , (96) The Projected Einasto Profile 2 1 x2 κ (x) = κc e−8 ,0 1 9 (97) 2 κc x 6 κ¯ (x) = 1 e− , (98) 10 x2 − 0 1 with Eν(x)theexponentialintegraloforderν.Theaboveresults can be easily checked by substituting Eq. (86)intorecipes(24), (41), (49), (50), (55)and(61). 104 4 ! 4.2. Profile comparison
log 2 We compared the Einasto and Sérsic surface mass densities for 1 the same values of the Sérsic index m and Einasto index n,in- cluding the exponential and Gaussian cases. For this comparison 100 1 we used the Mathematica implementation of the Meijer G func- n= 2 tion and Eq. (67). Figure 1 shows ΣS(R)fordifferent values of m, while ΥIe and Re are held fixed; Fig. 2 displays Σ(x)fordifferent values of n,whileρ0rs and rs are held fixed. In both figures, it can be clearly seen that the respective index is very important in 1 determining the overall behaviour of the curves. The Sérsic profile is characterised by a steeper central core 0 20 40 60 80 100 and extended external wing for higher values of the Sérsic in- Rad iu s dex m.Forlowvaluesofm the central core is flatter and the ex- Fig. 2. Projected Einasto profile, where ρ0rs and rs are held fixed for ternal wing is sharply truncated.TheEinastoprofilehasasimilar different values of the Einasto index n. behaviour, with the difference that the external wings are more spread out. Also in the inner region for both profiles with low values of the respectively index we obtain higher values of Σ S differ, which agrees with previous work of Cardone et al. (2005) and Σ.Additionally,theEinastoprofilehashighervaluesofthe and Dhar & Williams (2010). Studies of the lensing properties central surface mass density than the Sérsic profile, comparing of the Sérsic profile have been made by Cardone (2004)and both profiles for the same index. However, the Einasto profile Elíasdóttir & Möller (2007). seems to be less sensitive to the value of the surface mass den- sity for a given n and radius in the inner region than the Sérsic profile. It is in this region that the lensing effect is more impor- 5. Asymptotic behaviour tant and the surface mass density characteristics determine the lensing properties of the respective profiles. The series expansions allow us to directly investigate the be- Given these differences between the two profiles, we clearly haviour of Einasto models at small radii. It follows that the cen- see that the lensing properties of Sérsic and Einasto profiles also tral asymptotic behaviour of the surface density Σ(x)dependson
A70, page 8 of 10 E. Retana-Montenegro et al.: Analytical properties of Einasto dark matter haloes the value of n.Ifn < 1, we find the following expansion at small Einasto n indices for fixed values of ΥIe, Re, ρ0rs and rs,showing radii (x 1) that both profiles have a similar behaviour. However, we noted , that for the Einasto profile the external wings are more spread 2 Σ(x) ρ0 h 2Γ(n + 1) Γ(1 n) x . (99) out and it seems to be less sensitive than the Sérsic profile to ∼ − − the value of the surface mass density for a given Einasto index 8 9 If n = 1, then the expansion has the form and radius in the inner region. Also, we find that the Einasto profile is more “cuspy” than the Sérsic profile: the former has Σ(x) ρ h 2 + 2ln 1 1 x2 . (100) ∼ 0 2 − higher values of the central surface mass density. These features Finally, if n8> 1,0 the central0 1 surface1 9 density behaves as are of key importance, because it is in this region that the lensing effect is more important, and these dissimilarities in the surface n 1 mass densities imply a difference in the lensing properties of Γ − √π 2n 1+ 1 Σ(x) ρ0 h 2Γ(n + 1) x n . (101) both profiles. This result agrees with previous work of Cardone ∼ − n + 1 2n 1 Γ 0 2−n 1 et al. (2005)andDhar & Williams (2010). 0 1 Our results are the first step in studying the properties of the The behaviour of the cumulative surface mass M(x)anddeflec- Einasto profile using analytical means. The constant increase of tion potential ψ (x) are more straightforward. At small radii, we computational power opens the possibility of using more real- find the asymptotic expressions istic and sophisticated profiles like the Einasto profile in cos- 3 2 mological studies, where our results may apply. For example, M(x) 2πρ0 h Γ(n + 1) x , (102) ∼ they can be used in strong- and weak-lensing studies of galax- and ies and clusters, where dark matter is believed to be the main
κc mass component and the mass distribution can be assumed to be ψ(x) x2. (103) given by an Einasto profile. The better performance in cosmo- ∼ 2 logical N-body simulations of the Einasto profile (Navarro et al. The x2 slope for M(x)isnotunexpected,giventhattheEinasto 2004; Merritt et al. 2006; Gao et al. 2008; Hayashi & White models have a finite central surface mass density. The asymp- 2008; Stadel et al. 2009; Navarro et al. 2010) makes its inclusion totic behaviour of the Fox H function at large radii is described in strong and weak lensing studies very promising. Recently, in Kilbas & Saigo (1999). We obtain the following expansions Chemin et al. (2011)analysedtherotationcurves(RC)ofspiral (x 1) galaxies from THINGS (The HI Nearby Galaxy Survey, de Blok - et al. 2008)andfoundthattheEinastoprofileprovidesabet- x1/n 1 1 Σ(x) √8n πρ0 h e− x − 2n , (104) ter match to the observed RC than the NFW profile (Navarro ∼ 3 3/2 3 x1/n 3 3 et al. 1996, 1997)andthecoredpseudo-isothermalprofile.Also, M(x) 4πρ h n Γ(3n) 2(2πn) ρ h e− x − 2n , (105) ∼ 0 − 0 Dhar & Williams (2012)modelledthesurfacebrightnesspro- and files of a sample of elliptical galaxies of the Virgo cluster using a multi-component Einasto profile based on an analytical approxi- Γ (3n) x mation, and obtained a good fit for shallow-cusp and steep-cusp ψ(x) 2 κ ln n ψ (3n) + 1 ∼ Γ (n) c 2 − galaxies with fit residual errors lower in comparison to measure- = 3 4 3/2C ment errors in a wide dynamical radial range. Additionally, in √π (2n) x1/n 3 5 + κc e− x − 2n . (106) their models of the most massive galaxies, the outer components Γ (n) are characterised by being in the range 5 ! n ! 8, and a compa- rable range is obtained from N-body simulations for the Einasto 6. Summary and conclusions profile. Our exact analytical results for the spatial and lensing We studied the spatial and lensing properties of the Einasto pro- properties of the Einasto may be used to constrain the value of file by analytical means. For the spatial properties we applied the Einasto index and determine if the galaxy or cluster studied the method used by Ciotti & Bertin (1999)toderiveananalyti- is dark matter dominated or not. More studies like Chemin et al. cal expansion for the dimensionless parameter dn of the Einasto (2011)andDhar & Williams (2012)couldhelptostrengthenthe model. We also derived analytical expressions for the cumula- position of the Einasto profile as a new standard model for dark tive mass profile M (r) and the gravitational potential Ψ (r).For matter haloes. Also, increasing the use of the Einasto profile in the lensing properties, we used the Mellin integral transform new cosmological studies could provide progress towards a so- formalism to derive closed, analytical expressions for the sur- lution to the cusp-core problem. face mass density Σ (x),cumulativesurfacemassM (x),deflec- This paper continues the effort initiated by Baes & Gentile tion angle α (x),deflectionpotentialψ (x),magnificationµ (x) (2011)andBaes & van Hese (2011)toadvocatetheuseofthe and shear γ (x) . For general values of the Einasto index n,these Fox H and Meijer G functions in theoretical astrophysics, in par- are expressed in terms of the Fox H function. Using the prop- ticular for studying the analytical properties of density models erties of the Fox H function, we calculated explicit power and like the Sérsic model, where no additional analytical progress logarithmic-power expansions for these lensing quantities, and could be made until the Mellin integral transform formalism we obtained simplified expressions for integer and half-integer was applied. We hope that our work has again demonstrated values of n.Theseseriesexpansionsallowtoperformarbitrary- the usefulness of the Fox H and Meijer G functions as tools for precision calculations of the surface mass density and lensing analytical work. properties of Einasto dark matter haloes. We also studied the asymptotic behaviour of the surface mass density, cumulative Acknowledgements. E.R.M. and F.F.A. wish to thank H. Morales and R. Carboni surface mass and deflection potential at small and large radii us- for critical reading. This research has made use of NASA’s Astrophysics Data ing the series expansions. System Bibliographic Services. G.G. is a postdoctoral researcher of the FWO- Furthermore, we compared the Sérsic and Einasto surface Vlaanderen (Belgium). Moreover, we wouldliketothanktherefereeforvaluable mass densities using the equivalent values for the Sérsic m and suggestions on the manuscript.
A70, page 9 of 10 A&A 540, A70 (2012)
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