PoS(DSU2015)012. f (R) cosmology and dark matter

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1 f (R) cosmology ad dark matter Dipartimeto di Fisica E.R. Caiaiello, Uiveritá di Salero, Fisciao (SA), Italy INFN - Sezioe di Napoli, Gruppo Collegato di Salero, Italy. lambiase@sa.if.it I this paper is discussed the thermal abudace ad evolutio of relic particles (WIMPs) assumig that the backgroud is described by f (R) cosmology. As a reliable model, the margially deformed Starobisky model f (R) = R + αr, with 1 < 2 is cosidered. The 11th Iteratioal Workshop Dark Side of the Uiverse December 2015 Kyoto, Japa Speaker. A footote may follow. c Copyright owed by the author(s) uder the terms of the Creative Commos Attributio-NoCommercial-NoDerivatives 4.0 Iteratioal Licese (CC BY-NC-ND 4.0).

2 1. Itroductio Despite all fudametal results of Geeral Relativity (GR), the observatioal data of the preset Uiverse idicate that strog deviatios from the stadard Hilbert-Eistei picture [1] are edeed, ad that ew ukow form of (dark) matter ad eergy are ecessary. Several modified theories of gravity have bee proposed i last years, which try to address, at the same time, the shortcomigs of the Cosmological Stadard Model (for example, higher order curvature ivariats allow to get iflatioary-like solutios of early Uiverse, as well as to explai the flatess ad horizo problems) [2, 3, 4, 5]. I the framework of models that exted GR, f (R) gravity is oe of the favorite cadidate. It provides a uified descriptio of dark eergy ad dark matter, without ivokig exotic sources as dark matter, ad allows for the uificatio of the early-time (iflatio) ad the later-time acceleratio of the Uiverse [6, 7]. The gravitatioal actio for f (R) gravity is [8, 9] S = 1 2κ 2 d 4 x g f (R) + S m [g µν,ψ m ], (1.1) where S m is the actio of the stadard matter ad κ 2 = 8πG = 8π/M 2, with M GeV. Oe of the cosequeces of dealig with alterative cosmologies is that the thermal history of (relic) particles is modified. I these models the expasio rates H of the Uiverseca be writte as H(T ) = A(T )H GR (T ), where H GR is the expasio rate i GR, while the factor A(T ) ecodes the iformatio about the particular model of gravity extedig or modifyig GR. Usually, the factor A(T ) is defied i order that the successful predictios of the Big Bag Nucleosythesis (BBN) are preserved, that is A(T ) 1 at early time, ad A(T ) 1 before BBN begis (oe refers to the pre-bbn epoch sice it is ot directly costraied by cosmological observatios). O the other had, a ehaced (pre-bbn) expasio of the Uiverse ca recocile observed dark matter cosmic relic abudace with cotraits provided by idirect dark matter detectio experimets (such as PAMELA [10] ad the more recet AMS-02) [11]. These aspects will be studied i this paper assumig thta the evolutio of the Uiverse is govered by f (R) = R + αr, (1.2) where the case = 2 correspods to the Starobisky model. Models of the form (1.2) may geerate sizable primordial tesor modes provided 1 < < 2 [12]. Remarkably, these models might emerge from Supergravity [13, 14, 15] or dilato dyamics i brae cosmology scearios based o strig theory [17], ad are i agreemet with BICEP2 [14, 16] ad ack data [18] (R 2 -iflatio is ideed fully cosistet with observatios [2, 19, 20]). The paper is orgaized as follows. I Sectio 2 we derive the f (R) gravity field equatios ad solve them i the radiatio domiated era. Sectio 3 is devoted to the study of thermal relics abudace ad their evolutio. Coclusios are draw i Sectio Field equatios i f (R) gravity The field equatios for f (R) gravity follow by varyig the actio (1.1) with respect to the tesor metric g µν G c µν = κ 2 T m µν, G c µν f R µν f 2 g µν µ ν f + g µν f, (2.1) 2

3 where f f R, ad T m µν is the eergy-mometum tesor for matter. The trace equatio is 3 f + f R 2 f = κ 2 T m, T m = ρ 3p. (2.2) The Biachi s idetities are satisfied: µ G c µν = 0 = µ T m µν = 0. We work i the regime αr > R, with 1 < < 2 accordig to Ref. [12]. We look for solutios of the form a(t) = a 0 t β. I a (spatially flat) Friedma-Robertso-Walker (FRW) metric ds 2 = dt 2 a 2 (t)[dx 2 + dy 2 + dz 2 ], (2.3) we fid that the 0 0 field equatio ad the trace equatio read (i the very early Uiverse t 0) where ρ is the eergy desity ad Ω β, 1 [ ] (β + 2 3) 1, Γ 2 2β 1 β, 2 αω β, R = κ 2 ρ, αγ β, R = κ 2 T m, (2.4) ( 1)(2 1) β(2β 1) + 3( 1) 2β 1. (2.5) Durig the radiatio domiated era ρ = π2 g 30 T 4 (g 10 2 ) ad T m = 0, hece Γ β, = 0. Oe therefore gets β = Ω 2 β= 2, = (2.6) 4( 1) The fuctio Ω β= 2, is positive for It is worth to ote that oe ca also cosider scearios i which T m 0, which are related, for example, to bulk viscosity effects [21]. From (2.4) oe gets the relatio betwee the cosmic time t ad the temperature T ( ) T 2 t(t ) = Σ M 1, M ( ) 1 15 αωβ, Σ [6 β(1 2β) ] 1/2 2 α, α = 4π 3 g M 2(1 ). (2.7) We assume that at the istat t (the trasitio time) the Uiverse passes from the cosmic evolutio described by f (R) cosmology to the cosmic evolutio described by the stadard cosmological model. The trasitio time t or the trasitio temperature T are determied via the relatio αω β, R (t ) = H 2 GR (t ), yieldig t = [ 4 αω β, [6 β(2β 1) ] ] 1 2( 1) M 1 T M [6( β(1 2β) )] The relatio (2.7) ca be the cast i the form 4( 1) [, (2.8) ] 1 4 [ ] 1 4 αωβ, 4( 1). (2.9) 15 16π 3 g ( ) T 2 t T t = t, with 2 = T M The expasio rate of the Uiverse i f (R) cosmology is H(T ) = A(T )H GR (T ), A(T ) 2 ( T 3β where the factor A(T ) is the so called ehacemet factor π 3 g. (2.10) T ) p, p 2 2 (2.11) 3

4 3. Relic abudace ad WIMP particles Curret observatios idicate that ot oly our Uiverse is domiated by dark matter, resposible of galactic ad extragalactic dyamics, but also by dark eergy, resposible of the accelerated expasio of the Uiverse [24] Ω CDM h , 0.30 Ω DE h , (3.1) where h = 100Km s 1 Mpc 1 is the Hubble costat. Favorite cadidates for o-baryoic cold dark matter are the WIMPs (weakly iteractig massive particles). The iterest about these particles as dark matter follows from the fact that WIMPs i chemical equilibrium i the early Uiverse have the abudace which agrees with the expected oe i the cotext of cold dark matter. This kid of studies are motivated by recet astrophysical results which ivolve cosmic ray electro ad positros, atiprotos, ad γ-rays. Particular attetio is devoted to the risig behavior of the positro fractio observed i PAMELA experimet [10] (the astrophysical iterpretatio of this pheomeo is discussed i [22], while the possibility of dark matter aihilatio ito lepto is discussed i [23]). The geeral aalysis that accouts for the ehacemet of the expasio rates i alterative cosmological models has bee performed i [11] (see also [25]). Typically the expasio rate is writte i the for H = A(T )H GR, where the fuctio A(T ) = η(t /T f ) ν ecodes parameters characterizig a particular model (here η ad ν are free parameters, while T f 10GeV is the decouplig temperature). I our f (R) model, we have ν = 2/ 2, so that 1 ν 0 for 1 2. To compute the DM abudace, oe has to umerically solve the Boltzma equatio for the umber desity of thermal relic Y. Fixig η = { 1,0,1,2}, σ a v cm 3 sec 1, ad m χ = [10,10 4 ]TeV, oe determies the values of parameter η vs the DM masses [11] required to ifer the correct relic abudace of DM particles Ω χ h 2 = Ωh 2 WMAP CDM relic abudace is give by Ω χ h 2 = m χs 0 Y 0 ρ c, = ± [26]. The where ρ c = 3H0 2M2 /8π is the critical desity of the Uiverse, s 0 is the preset value of the etropy desity s = 2π2 45 g χ(t )T 3, ad Y 0 is the preset value of the WIMP abudace for comovig volume 1 Y 0 = 1 Y f + πm 2 45 m g χ (x) σ a v χ x f g (x)a(x)x dx, 2 i which Y f is the value of the WIMP abudace for comovig volume at the freeze-out, x = m χ /T, [ {g χ (T ),g (T )} couts the effective ] umber of degrees of freedom at temperature T, ad, M m χ σ a v f x f = l g χ A(x f ) (computatios refer to o relativistic DM particles). Fially x f g (x f ) oe gets [11] η 0.1 for m χ 10 2 GeV. (3.2) More precisely, the aalysis shows that DM masses are i the rage [ ]GeV, ad the allowed regio for the parameter η is 0.1 η , where the upper bouds o η vary for the differet cosmological models labelled by ν [11]. 4

5 Table 1: Estimatios of α for fixed values of the trasitio temperature T = ( )MeV. T (GeV) α GeV GeV GeV GeV 2 Applicatios to f (R) cosmology Accordig to the above results, we rewrite the factor A(T ) (see Eqs. (2.9) ad (2.11)) i the form ( ) T ν A(T ) = η, η 2 ( ) ν Tf 3β, ν = 2 2. (3.3) T f T The trasitio temperature T is fixed for values greater tha the free-out temperature T f. Therefore we set T = ( )T BBN. From (2.9) we get ( ) 15 1 [6 β(1 2β) ] α = 16π 3 g 4Ω β= 2, ( M T ) 4( 1) M 2(1 ). The order of magitudes of α are reported i Table I for Ω β= 2, give i (2.6). The fuctio η vs is plotted i Fig. 1. As we ca see, the parameter η assumes values of the order O(0.1 1), so that the mass of WIMPs particles is of the order 10 2 GeV. Figure 1: η vs for β = /2 ad trasitio temperatures T = {1,10,10 2 }MeV. T f = 10GeV is the freeze-out temperature, while η = 0.1 is the lower boud o η (see (3.2)). 4. Coclusios I this paper we have studied the evolutio of thermal relic particles i f (R) cosmology, assumig f (R) = R+αR ( = 2 correspods to Starobisky s model, while the so called margially 5

6 deformed model ( 1) produces sizable primordial tesor modes provided the expoet falls dow i the rage [1,2]). As we have show, if the cosmic evolutio of the early Uiverse is described by modified field equatios, as provided ideed by f (R) gravity, the the expasio rate gets modified by a factor A(T ) (H(T ) = A(T )H GR (T ), see Eq. (2.11)). This quatity essetially weights how much the expasio rate of the Uiverse i f (R) cosmology deviates from the expasio rate derived i the stadard cosmology, ad affects, i tur, the productio of relic particles (thermal relics decouple with larger relic abudaces). As a cosequece, the latter is obtaied for larger aihilatio cross sectio, ad therefore also the idirect detectio rates get ehaced. This effect may have its imprit o supersymmetric cadidates for DM. For a power law scale factor, ( solutios ) of the modified field equatios, ad parameterizig the T ν ehacemet factor as A(T ) = η, we fid that the f (R) model is cosistet with PAMELA T f costraits, ad the abudace of relic DM Ω χ h , provided that η O(0.1 1) (ad 1 ν 0.46). Accordig to (3.2), the correspodig WIMPs masses are m χ 10 2 GeV. Refereces [1] S. Capozziello ad M. De Lauretis Phys. Rept. 509, 167 (2011). [2] A. A. Starobisky, Phys. Lett. B 91, 99 (1980). [3] N.D. Birrell ad P.C.W. Davies, Quatum Fields i Curved Space, Cambridge Uiversity Press, Cambridge, UK (1982). [4] I.L. Buchbider, S.D. Oditsov, I.L. Shapiro, Effective Actio i Quatum Gravity, IOP, Bristol, UK (1992). [5] N.H. Barth ad S. Christese, Phys. Rev. D 28, 8 (1983). [6] A.G. Reiss et al., Astro. J. 116, 1009 (1998). S. Perlmutter et al., Nature (Lodo) 391, 51 (1998). [7] S. Nojiri ad S.D. Oditsov, Phys. Rep. 505, 59 (2011). A. Silvestri ad M. Trodde, Rep. Prog. Phys. 72, (2009). J.A. Friema, M.S. Turer, ad D. Huterer, Au. Rev. Astro. Astrophys. 46, 385 (2008). S. Capozziello ad, e-prit: arxiv: [astro-ph.co]. M. Sami, Lect.Notes Phys. 720, 219 (2007). E.J. Copelad, M. Sami, ad Sh. Tsujikawa, It. J. Mod. Phys. D 15, 1753 (2006). [8] L. Amedola ad S. Tsujikawa, Dark Eergy: Theory ad Observatios, Cambridge Uiversity Press, [9] S. Nojiri, S.D. Oditsov, Phys. Rev. D 77, (2008). L. Pogosia ad A. Silvestri, Phys. Rev. D 77, (2008). I. Sawicki ad W. Hu, Phys. Rev. D 75, (2007). B. Li ad J.D. Barrow, Phys. Rev. D 75, (2007). S. Capozziello ad, Ge. Relat. Grav. 32, 295 (2000); Ge. Relat. Grav. 31, 1005 (1999); Phys. Lett. B 750, 344 (2015). S. Capozziello,, ad H.J. Schmidt, Aale Phys. 9, 39 (2000). SI. Brevik, E. Elizalde, S. Nojiri, ad S.D. Oditsov, Phys. Rev. D 84, (2011). S. Nojiri ad S.D. Oditsov, Prog. Theor. Phys. Suppl. 190, 155 (2011). T.P. Sotiriou ad V. Faraoi, Rev. Mod. Phys. 82, 451 (2010). A. De Felice ad S. Tsujikawa, Livig Rev. Rel. 13, 3 (2010)., Phys. Rev. D 90, (2014)., S. Mohaty, ad A.R. Prasaa, It. J. Mod. Phys. D 22, (2013). ad G. Scarpetta, Phys. Rev. D 74, (2006). B. Jai, V. Vikram, ad J. Sakstei, Astrophys. J. 779, 39 (2013).. L. Lombriser, A. Slosar, U. Seljak, ad W. Hu, Phys. Rev. D 85, (2012). S. Derraro, F. Schmidt, ad W. Hu, 6

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