RICCION AS A COSMIC DARK MATTER CANDIDATE AND LATE COSMIC ACCELERATION 8 0 0 2 S.K.Srivastava n a Department of Mathematics, North Eastern Hill University, J 4 NEHU Campus,Shillong - 793022 ( INDIA ) 1 e-mail:[email protected] 4 v and 0 4 K.P. Sinha 2 0 1 INSA Honorary Scientist, Department of Physics, 6 0 Indian Institute of Science , Bangalore - 560012 ( INDIA ) . / h t e-mail: [email protected] - p e h : v Abstract i X In the past few years, a posibility is investigated, where curvature r a itself behaves as a source of dark energy. So, it is natural to think whether curvature can produce dark matter too. It is found that, at classical level, higher-derivative gravity yields curvature inspired par- ticles namely riccions[31]. Here, it is probed whether riccion can be a possible source of dark matter. Further, it is found that the late universe accelerates. Here, it is interesting to see that acceleration is obtainedfromcurvaturewithout using anydarkenergy sourceofexotic matter. PACS nos.: 98.80 -k; 95.30 C. 1 2 S.K.SRIVASTAVAAND K.P.SINHA Key-words : Higher-derivative gravity, riccion and dark matter. By the end of the last century, cosmology got revolutionized due to Supernova observations at low red-shift, pointing towards very late cosmic acceleration and 73% content of the present universe as dark energy (DE)[1, 2, 3, 4, 5, 6]. Above all these experiments, WMAP data provide estimates of parameters of standard models of cosmology with unprecedented accuracy [7, 8]. These data suggest 73% content of the universe in the form of DE, 23% in the form of non-baryonic dark matter (DM) and the rest 4% in the form of baryonic matter as well as radiation. But identity of dark energy and dark matter is still in dark. For DE, many theoretical models were proposed in the past few years taking exotic matter ( quintessence, tachyon , phantom and k- essence )as a source [9, for detailed review]. Later on, higher-order curvature terms or functions of curvature were taken as gravitational alternative of dark energy [10]. In contrast to this approach of taking non-linear curvature terms as DE lagrangian, in [11], a different ap- proach is adapted where non-linear curvature terms are not taken as DE lagrangian. Rather,in [11], DE terms emerge spontaneously in the resulting Friedmann equations (giving cosmic dynamics), if non-linear curvature terms are added to Einstein - Hilbert term in the gravita- tional action. Here, we address to the identity of DM. In principle, DM can be hot as well as cold. The possibility of hot dark matter is ruled out by WMAP data due to re-organizationof the universe at red-shift z 20 . ≃ RICCION AS A COSMIC ...... 3 So,inthepresentuniverse, dominantcomponentofDMiscoldandnon- baryonic [8]. It is further characterized as a pressureless cosmic fluid. In the search of cold dark matter (CDM) candidate, some attempts have been made from time to time in the past. In [12], supersymmetric dark matter is suggested with particle mass . 500 GeV. In [13], wino- like neutrilo, with mass of the order 100 GeV, is suggested as a CDM candidate. In [14], it is shown that pseudo-Nambu-Godostone boson can be a possible source of DEand its supersymmetric partner can give dark matter. The possibility of Kaluza-Klein dark matter is discussed in [15, 16, 17, 18, 19, 20]. In this letter, there is a digression from so far attempts to identify CDM.It is mentioned above that DE canbe obtained fromthe gravita- tional sector also using higher-order curvature terms without resorting to exotic matter. So, it is natural to think whether it is possible to obtain CDM too from gravity with non-linear curvature terms. In [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], it is shown that when higher-order curvature terms are added to Einstein-Hilbert lagrangian, Ricci scalar manifests itself in dual manner (i) as a geometrical field and (ii) as a physical field. Its physical aspect is given by a particle called riccion with (mass)2 inversely proportional to the gravitational constantG. Higher-drivative gravityhasunitarityproblematquantum level, but, at classical level, it does not face any problem if coupling constants are taken properly. In what follows as well as in [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], it is found that the scalar field, namely riccion, is different from the scalar mode of graviton as graviton is massless, whereas riccionisaparticlewithmassproportionaltoinverse of square root of the gravitational constant. A detailed discussion on 4 S.K.SRIVASTAVAAND K.P.SINHA the difference between riccion and graviton is given in [31, Appendix A]. In what follows, to get curvature inspired DM, we probe an answer to the question,“Is riccion a cosmic dark matter candidate too ?” Here, natural units (k = ~ = c = 1) are used with GeV as a B fundamental unit with k being the Boltzman constant, ~ being the B Planck’s constant h divide by 2π and c being the speed of light. We take R2-gravity being the simplest higher-derivative theory of gravity with the action R S = d4x√ g[ +αR2], (1) g − 16πG Z where α is a dimensionless coupling constant. This action yields the gravitational field equations 1 1 1 [R g R]+2α[R g 2R+RR ] g R2 = 0, (2) µν µν ;µν µν µν µν 16πG − 2 − − 2 wheresemicolon (;) standsforthecovariant derivative andtheoperator 2 = (1/√ g)∂ [√ ggµν∂ ].Traceofthesefieldequationsareobtained µ ν − − as 1 2R+ R = 0 (3) 96πGα with α > 0 to avoid the ghost problem. This is the Klein-Gordon equation for R. It shows that when higher- order terms of curvature are added to Einstein- Hilbert lagrangian R/16πG in S ,R behaves as a physical field also [21, 22, 23, 24, 25, g 26, 27, 28, 29, 30, 31]. R has mass dimension 2, being combination of of second order de- rivative and squares of the first order derivatives of g (which is di- µν mensionless). In a scalar field theory, scalars satisfying the Klein-Gordon equation have mass dimension 1. So, to have consistency with other scalar fields, RICCION AS A COSMIC ...... 5 we multiply (3) by η, which is a constant having (mass)−1 dimension and measured in GeV−1. ηR is recognized as R˜, which is called riccion having mass dimension 1. It satifies the equation 2R˜ +m2R˜ = 0 (4) with 1 m2 = . (5) 96πGα If R˜ is a physical field (as mentioned above), there should be an action S yielding (4) on using its variation with respect to R˜. Here, R˜ we find S as given below. According to these requirements for S , we R˜ R˜ have [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] δS = d4x√ g[2R˜ +m2R˜]δR˜, R˜ − − Z subject to the condition δS /δR˜ = 0. This equation reduces to R˜ δS = d4x√ g[gµν∂ R˜∂ (δR˜) m2R˜δR˜], R˜ − µ ν − Z which is covariant. So, using the principle of equivalence, we have δS = d4X[ηµν∂ R˜(X)∂ (δR˜)(X) m2R˜(X)δR˜(X)] (6a) R˜ µ ν − Z in a locally inertial co-ordinate system (X) having g (X) = η and Γµ (X) = 0. µν µν νρ Now, (6a) integrates to 1 S = d4X[ηµν∂ R˜(X)∂ R˜(X) m2R˜2(X)] R˜ 2 µ ν − Z 1 = d4x√ g[gµν∂µR˜∂νR˜ m2R˜2] (6b) 2 − − Z due to principle of equivalence and principle of covariance [31]. Like other physical fields, energy-momentum tensor components for R˜ can be obtained varying S with respect to gµν. This variation is R˜ 6 S.K.SRIVASTAVAAND K.P.SINHA obtained from (6b) as 1 1 δS = d4√ g δgµν∂ R˜∂ R˜ +gµν∂ R˜∂ δR˜ m2R˜δR˜ R˜ − 2 µ ν µ ν − 2 Z h 1 1 g δgµν ∂ρR˜∂ R˜ m2R˜2 µν ρ −2 2 − n1 oi = d4√ g δgµν∂ R˜∂ R˜ 2R˜δR˜ m2R˜δR˜ µ ν − 2 − − Z h 1 1 1 g δgµν ∂ρR˜∂ R˜ m2R˜2 . −2 µν 2 ρ − 2 (7a) n oi On using (4) in (7a), 1 1 1 1 δS = d4x√ g ∂ R˜∂ R˜ g ∂ρR˜∂ R˜ m2R˜2 δgµν, (7b) R˜ − 2 µ ν −2 µν 2 ρ −2 Z h n oi which yields 2 δS T(R˜) = R˜ µν √ gδgµν − 1 1 = ∂ R˜∂ R˜ g ∂ρR˜∂ R˜ m2R˜2 . µ ν − µν 2 ρ − 2 (8) n o Experimental observations [32, 33, 34, 35, 36] support spatially ho- mogeneous and isotropic cosmological model for the universe, given by the metric ds2 = dt2 a2(t)[dx2 +dy2+dz2], (9) − where a(t) is the scale factor. So, due to homogeneity of space-time (9), R˜ ,being ηR,depends on cosmic time t only. In this space-time, energy density and pressure of the cosmic fluid, constituted by spinless riccions (represented by R˜ scalars) are obtained from (8) as ρ(R˜) = T0 = 1R˜˙2 + 1m2R˜2 (10) 0 2 2 and p(R˜) = T1 = 1R˜˙2 1m2R˜2. (11) − 1 2 − 2 RICCION AS A COSMIC ...... 7 (10) and (11) satisfy the conservation equation a˙ ρ˙(R˜) +3 (ρ(R˜) +p(R˜)) = 0. (12) a Correctness of (12) can be verified easily. On using (10) and (11) in (12), it is obtained that R¨˜ +3a˙R˜˙ +m2R˜2 = 0, (13) a which is (4) (derived above from the action (1)), giving field equation for R˜ or riccion equation of motion in the space-time (9) using (10) and R˜˙2 = ρ(R˜) +p(R˜) obtained from (10) and (11). From (5), we find riccion as a massive particle with very high mass. So, it is reasonable to think for riccions to behave like a gas of heavy massive and very weakily interacting particles with negligibly small velocity distribution. So, these particles are non-relativistic. Moreover, in the late universe, temperature is low. Pressure density , due to non- relativistic particles , are obtained as [37] p(R˜) = (m/2π)3/2T5/2exp[ m/T] 0 (14) − ≃ using Bose-Einstein distribution. It yields w = p(R˜)/ρ(R˜) = 0. Here, m is given by (5). Moreover, T T∗ = T0 a0/a∗ = 3.44 10−13GeV ≤ × as it is found below that riccion fluid, b(cid:16)ehaving(cid:17)as CDM, dominates at the red-shift z∗ = 0.46 and a0/a∗ = 1.46. from (30) given below. Here T = 2.730K = 2.35 10−13GeV is the present temperature of 0 × the background radiation in the universe. Though, even visible matter is pressureless with equation of state parameter w = 0, riccion fluid can not behave as visible matter. It is becausevisiblematterisbaryonic, whereas thericcionfluid, originating 8 S.K.SRIVASTAVAAND K.P.SINHA from curvature, is non-baryonic. So, it is possible for riccion to be a source of CDM. Further, (11) and (14) imply 1R˜˙2 = 1m2R˜2. (15) 2 2 Connecting (10) and (15), it is obtaind that ρ(R˜) = m2R˜2 (16) Moreover, using (14) and (16) in (12) and integrating, it is obtained that A ρ(R˜) = m2R˜2 = , (17) a3 AccordingtoWMAP1[8],currentvalueofCDMdensityρ isρ0 = dm dm 0.23ρ0 , where ρ0 = 3H2/8πG,H = 100hkm/Mpcsec = 2.33 0.68 cr cr 0 0 × × 10−42GeV using h = 0.68 ( a value having the maximum likelihood ). So, ρ0 = 0.69 10−47GeV4 . dm × Using WMAP1 results, integration constant A is evaluated as A = 0.23ρ0 a3, so from (17) cr 0 R˜ a3/2 R = = 0.23ρ0 0 . (18) η ± crηma3/2 p In the space-time, given by the metric (9) a¨ a˙ 2 R = 6 + . (19) a a h (cid:16) (cid:17) i From (18) and (19), we obtain 3/2 a¨ a˙ 2 a + = 0.23ρ0 0 . (20) a a ± cr6ηma3/2 (cid:16) (cid:17) p The first integral of (20) is obtained as a˙ 2 C2 2 0.23ρ0 a 3/2 = cr 0 (21) a a4 ± 15mη a p (cid:16) (cid:17) (cid:16) (cid:17) where C is an integration constant. This is the Friedmann equation giving dynamics of the universe. RICCION AS A COSMIC ...... 9 If we take ( ) sign in (18), a˙/a is complex, when − C2 2 0.23ρ0 a 3/2 < cr 0 . a4 15mη a p (cid:16) (cid:17) So, from (21), we have a˙ 2 C2 2 0.23ρ0 a 3/2 = + cr 0 . (22) a a4 15mη a p (cid:16) (cid:17) (cid:16) (cid:17) In case, C2 2 0.23ρ0 a 3/2 > cr 0 , (23) a4 15mη a p (cid:16) (cid:17) (22) reduces to a˙ 2 C2 . (24) a ≃ a4 (cid:16) (cid:17) This equation is integrated to a(t) = [D +2Ct]1/2 (25) showing deceleration as it yields a¨ < 0. Here D is an integration con- stant. When C2 2 0.23ρ0 a 3/2 < cr 0 , (26) a4 15mη a p (cid:16) (cid:17) (22) is approximated to a˙ 2 2 0.23ρ0 a 3/2 cr 0 . (27) a ≃ 15mη a p (cid:16) (cid:17) (cid:16) (cid:17) (27) yields the solution 2 0.23ρ0 4/3 a(t) = E +0.75 cra 3/4t (28) 0 s 15mη p h i showing acceleration as it yields a¨ > 0. Here E is an integration con- stant. Thus, it is obtained that universe decelerates when the inequality (23) holds and accelerates when the inequality (26) holds. It means 10 S.K.SRIVASTAVAAND K.P.SINHA that transition from deceleration to acceleration takes place at a = a∗, where a∗ = a(t∗) with t∗ being the transition time. So, at t = t∗, C2 2 0.23ρ0 a 3/2 = cr 0 , (29) a4∗ p15mη a∗ (cid:16) (cid:17) 16 Type Supernova observations [38] have conclusive evidence that, in the late universe, acceleration begins at small red-shift z∗ 0.46 ≃ with a jerk giving a transition from deceleration to acceleration. So, we have a 0 = 1+z∗ = 1.46, (30) a∗ where a = a(t ) with t = 13.6Gyr = 6.6 1041GeV−1 being the 0 0 0 × present time. Connecting (29) and (30), C is pbtained as 2 0.23ρ0 C = 1.33a2 cr. ∗ s 15mη p Using this value of C and a = a∗ at t = t∗, (25) looks like 2 0.23ρ0 1/2 a(t) = a∗ 1+1.33a2∗ cr(t∗ t) (31) s 15mη − p h i showing deceleration when t < t∗. Moreover, using (30) in (28), it is obtained that 2 0.23ρ0 3/4 a(t) = a∗ 1+0.996 cr(t t∗) (32) s 15mη − p h i showing acceleration when t > t∗. Using a0 = a(t0) in (32), t∗ is ob- tained as 2 0.23ρ0 −1 t∗ = t0 0.65 cr . (33) − s 15mη p h i In what follows, results are summarized. According to above in- vestigations, R2-gravity suggests that Ricci scalar manifests itself as a physical field too in addition to its usual nature as a geometri- cal object. The dual nature of R has been found in the past in