Chameleon mechanism in f(r) modified gravitation model of polynomial-exponential form

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS Chameleon mechanism in f(r) modified gravitation model of polynomial-exponential form To cite this article: Vo Van On and Nguyen Ngoc 018 J. Phys.: Conf. Ser Related content - Cosmic web and environmental dependence of screening: Vainshtein vs. chameleon Bridget Falck, Kazuya Koyama and Gong- Bo Zhao - Chameleon fields, wave function collapse and quantum gravity A Zanzi - Chameleons weigh in View the article online for updates and enhancements. This content was downloaded from IP address on 14/01/019 at 09:56

2 Chameleon mechanism in f(r) modified gravitation model of polynomial-exponential form Vo Van On and Nguyen Ngoc Group of Computational Physics, Faculty of Natural Sciences, University of Thu Dau Mot, Binh Duong, Viet Nam Abstract. In this paper, we first present briefly to Chameleon mechanism in modified gravity of f(r), then we apply it to modified gravity of polynomial exponential form to find constraints from solar system experiments to parameters of α and β of the model. Results show that the Chameleon mechanism sets strict constraints on α and β parameters as follows: 0 < β < and 0 < α < Key words: Chameleon mechanism; polynomial exponential modified gravity; solar system constraints. 1. Introduction In 1998, Astrophysicists uncovered a surprise that the universe was accelerating rather than slowing down as expected [1,,3]. To explain this event there are three main approaches. In the first approach, people think that the Einstein equation with cosmological constant is sufficient to present the acceleration of universe. However, detailed calculations show that the cosmological constant from theory is about 10 orders of magnitude greater than its experimental value.this is the problem of cosmological constant [4], so this approach is not supported by the majority of scientists. In the second approach, the acceleration of universe causes by fifth force field called Quintessence field [5]. However, to satisfy constraints from observations and experiments, artificial conditions are required on this force field, such as: the field potential must be very flat so that the field rolls very slowly into the potential hole; the mass of quintessence particle is very small about10 31 ev. In the third approach, people change Einstein s theory to describe the acceleration of universe today but do not need to dark energy. Modified gravity of f(r) belongs this class of theory. In the class of theories, action of Einstein-Hilbert R in general relativity theory is replaced by a function of f(r). Here f(r) is analytic function which satisfies some constraint conditions.the first authors to take this path are Soviet physicist Starobinski in the 1980s of the last century[6], followed by American physicist Sean Caroll[7]. These early approaches gave physicists more emotion and experience. We also use a theory model in the class to research the evolution of universe. In our model, f(r) takes of polynomial-exponential form[8,9] S = 1 κ f(r) gd 4 x + S M (g µυ, ψ); (1) Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd 1

3 where with f(r) = R + a + α ( R m 1 + br + cr 3) e βrn ; () α > 0, β > 0; m = n = 1; a = Λ; b = c = 1. (3) This is a small class in f(r) theory class but it also is quite general, and is also of interest to many cosmologists today. This class called the class of polynomial- exponential modified gravity. The paper is constructed as follows: in section, we introduce the Chameleon mechanism in the class of f(r) modified gravity; in section 3, we present the Chameleon mechanism in the f(r) modified gravity of polynomial exponential form, then we calculate constraints on parameters of the model; section 4 is discussions and conclusion.. The Chameleon mechanism in f(r) modified gravity We consider a static spherically symmetric compact object, it causes a perturbation on de Sitter background universe. The metric is as follows ds = [1 ϕ(r)]dt + [1 + ψ(r)]dr + r dω ; (4) where ϕ and ψ are post- Newtonian potentials satisfy ϕ(r), ψ(r) << 1. (5) We only note small scales such that H 0.r << 1; (6) we expand Ricci scalar about constant curvature of de Sitter universe R(r) = R 0 + R 1. (7) The parameter of γ in Parameterized Post-Newtonian formalism(ppn)[10], is introduced as follows γ = ϕ(r)/ψ(r); (8) three following assumptions are required f(r) is analytic at R 0 ; mr << 1, i.e the mass of scalar field is very small and it has acting scale is great than the scale of solar system the pressure inside the object can be ignored,i.e, T = T 0 + T 1 ρ; people prove that if the condition is satisfied, we have with an any f(r) theory while experiments in solar system give [11] γ = ϕ(r)/ψ(r) = 1/; (9) γ 1 < ; (10) It means γ = 1. This puts an end to f(r) modified theories if there is no Chameleon mechanism as follows: It is thought that the second request is not always satisfied because the scalar field s mass may be large due to it depends on the curvature of the universe i.e on matter density of the surrounding environment. This mechanism was first proposed in 004 by two American physicists, J. Khoury and A. Weltman, to overcome the above difficulty[1]. The oscillation

4 of neutrino particles seems to support this mechanism because the mass of neutrino particles depends on the environment around them. Below we give a shortly review to this mechanism. We consider the following action S = 1 κ f(r) gd 4 x + S M (g µυ, ψ); (11) where g is the determinant of g µν, f(r) is an unknown function of the scalar curvature R and S m is the matter action. We use a conformal transformation g µν = pg µν ; (1) φ = 1 β 1 ln p; (13) where p df 1 dr = f (R), β 1 = 6. (14) This transforms the above action in Jordan frame to the Einstein frame[13] S = 1 d 4 x g[r g µν µ φ ν φ V (φ)] + S M (g µυ e β 1φ, ψ); (15) the scalar field φ in the Einstein frame has a self-interacting potential where r(p) is a solution of the equation V (φ) = 1 { } e β 1φ r [p(φ)] e β1φ f (r [p(φ)]) ; (16) f (r [p]) p = 0. (17) The conformal transformation makes the coupling of the scalar field with the matter sector. The strength β 1 of this coupling is the same for all types of matter field and it equals 1 6. In such case, the scalar potential has an important role for consistency with local gravity experiments in solar system. The potential of the scalar field has to attribute an effective mass to the field which has a strong dependence on material density of environment. A theory that has such dependence called a chameleon theory. In a chameleon theory of f(r), the scalar field can be heavy enough in the environment of laboratories on the Earth so that the local gravity constraints suppressed, while it can be light enough in the low density cosmological environment to makes the cosmic acceleration. Variation of the action (15) with respect to g µν and φ gives the field equations G µυ = µ φ υ φ 1 g µυ γ φ γ φ V (φ)g µυ + T µυ ; (18) where and φ dv dφ = β 1T ; (19) T µυ = δs m g δg µυ ; (0) T = g µυ T µυ. (1) 3

5 Covariant differentiation of (18) and Bianchi identities give µ T µυ = β 1 T υ φ; () Equation(0) indicates that the matter field is not generally conserved and feels a new force because of gradient of the scalar field. We now consider T µν as the stress-tensor of dust with energy density ρ in the Einstein frame. In a static spherically symmetric space-time region, equation (19) gives d φ dr + dφ r dr = dv eff (φ) ; (3) dφ where r is the distance from the center of symmetric body to point that we investigated in Einstein frame. We have V eff (φ) = V (φ) 1 4 ρe 4β 1φ ; (4) where ρ = e 4β1φ ρ is the relation that relates between the energy density in Jordan and Einstein frames. We now consider a spherically symmetric body which has the radius r c and assume that the energy density is constant ρ in (r < r c ) inside the body. We call the energy density outside the body by ρ out (r > r c ). We denote the values at two minimums of the effective potential V eff (φ) inside and outside by ϕ in and ϕ out, respectively. They satisfy relations V eff (ϕ in) = 0 and V eff (ϕ out) = 0, where prime indicates differentiation of V eff (φ) with respect to the argument of φ. Masses of small fluctuations about these minimums are given [ ] by m in = V 1 [ ] eff (ϕ in) and m out = V 1 eff (ϕ out). Which depend on matter density of surrounding environment. The scalar field can has shortrange effects in the solar system ( m > 10 3 ev for λ < 0.mm to avoid constraints by solar system experiments) due to the large material density in the solar system while it has a very small mass on cosmic scale due to very small density of universe and causes the current cosmic acceleration. For dense compact objects which satisfy the thin shell following r c r c = ϕ out ϕ in 6β 1 φ c << 1; (5) where φ c = Mc 8πr c is Newtonian potential at r = r c, with M c is the mass of the object. In this case, equation (3) with some appropriate boundary conditions gives the field profile outside the body[14] φ(r) = β 1 3 r c M c e mout(r rc) + ϕ out. (6) 4π r c r 3. The Chameleon mechanism in f(r) modified gravity of polynomial-exponential form We consider the above action S = 1 κ f(r) gd 4 x + S M (g µυ, ψ); (7) with f(r) = R Λ + α R (1 + R + R 3 )e βr ; (8) 4

6 where g is determinant of g µν, S m is the action of matter which depends on the metric g µν and an any scalar field ψ. Transforming this action from Jordan frame to Einstein frame as equations(1),(13),(14),(15),(16),(17) and assuming that φ << 1, one can find the solution of V eff (φ) = 0, We have V eff = V (φ) 1 4 ρe 4β 1φ ; = 1 (1+4β 1 φ) { } (8αββ1 φ 4αβ β 1 φ) 8β β1 φ +8Λαβ +6αββ 1 φ+0α ββ 1 φ+α β α ; 4αβ (9) from V eff = 0; (30) we have αβ + 7α 33α φ = ± 16ραβ 8ββ 1 64β β1. (31) In section 3, we shall consider the thin-shell condition together constraints from the equivalence principle and fifth force experiments on the model 3.1. The thin shell condition For a heavy and large object, the chameleon field is trapped inside the object so that its effect on other objects is only due to a thin shell near the outer surface of the object. The criterion for thin shell is given by equation of (5). When combining (5) and(31), we have r c r c r c r c = = ϕout ϕ in ( ± 6β 1 Φ c << 1; 33α 16ρ outαβ 64β β 1 ) 33α 16ρ in αβ 64β β1 1 6β 1 Φ c << 1; (3) where ρ in and ρ out are the energy densities inside and outside of the object in Jordan frame. In the weak approximation for a spherically symmetric object, the metric in Jordan frame has the form ds = [1 X(r)]dt + [1 + Y (r)]dr + r dω ; (33) where X(r) and Y (r) are functions of the variable of r. The relation of r and r is r = p 1/ r. We assume that m out.r << 1 i.e, the Compton wave length m 1 out is much large than solar system scales, then we have r r. In this case, Chameleon mechanism gives the parameter γ in Post-Newtonian formalism as follows γ = 3 rc r c 1 r c. (34) 3 + rc 3 r r c c We obtain the thin shell condition for the Earth when applying (9). We consider the Earth as a solid sphere with the radius of R e = cm and mean density ρ out 10g/cm 3. We also consider the Earth is surrounded by an atmosphere layer with homogeneous density ρ a 10g/cm 3 and thickness of 100km. When applying equation (3) and the constraint comes from Cassini tracking[15] 1 γ < ; (35) we have ( R e 33α = 16ρ out αβ R e 64β β1 33α 16ρ in αβ 64β β 1 ) 1 6β 1 Φ e < ; (36) 5

7 or 33α 16αβ ρ 33α out 16αβ ρ in 165.6β 1 Φ e < 0; (37) α set a = 33α 16αβ, b = 165.6β 1 Φe α ; therefore a ρout a ρ in b < 0; (38) when maintaining only first order of small parameter β, we have 56β 33 (165.6) β 4 1 Φ e < 4 (ρ in ρ out ) β < β < ( 4.33(165.6) 1 36 ( ) (10 4 ) ( 4.33(165.6) β 4 1 Φ e (ρ in ρ out ) ) 1 ) 1 ; (39) = 1.3. (40) This is the constraint condition on the parameter β of the model. We see that this condition is not too strict on β due to in the model we have assumed from the beginning that 0 < β < The equivalence principle In this section, we consider constraints on the parameter β coming from possible violation of the equivalence principle. We assume that the Earth together its atmosphere is an isolated system. Far away from the Earth, the matter density is assumed to be homogeneous with the density ρ G 10 4 gr/cm 3. We now consider conditions for the Earth to satisfy the thin shell condition. We take the thickness of atmosphere to be the thickness of thin shell about R a 100km and R a 4000km, we obtain Ra R a < We also have relations R e R e = ϕ a ϕ e 6βϕ e ; (41) and R a R a = ϕ G ϕ a 6βϕ a ; (4) where ϕ a, ϕ e and ϕ G are field values at local minimums of effective potential in regions r < R e, R a > r > R e and r > R a respectively. Because of the Newtonian potential inside a spherically symmetric object with the mass density ρ is Φ ρr, we have Φ e = 10 4 Φ a. Where Φ a and Φ e are Newtonian potentials on the surface of the Earth and atmosphere, respectively. This gives R e R e such that, the condition for atmosphere to have a thin shell is 10 4 R a R a ; (43) R e R e < (44) The tests of the equivalence principle measure the difference of free fall accelerations of the Earth and the Moon towards to the Sun. The constraint on this difference of two accelerations is given by the relation[16] a m a e a N < ; (45) 6

8 where a m and a e are the accelerations of the Moon and the Earth, and a N is Newtonian acceleration. The Sun and the Moon all have the thin shell and the field profile outside of these objects are given by formula (6) with respective quantities.the accelerations a e and a m are given by a e a N {1 + 18β1( R e ) Φ e }; (46) R e Φ s a m a N {1 + 18β1( R e ) Φ e }; (47) R e Φ s Φ m where Φ e = , Φ m = and Φ s = are Newtonian potentials on the surface of the Earth, the Moon, and the Sun respectively. When differentiating two free - fall accelerations, we have ( ) a m a e = (0.13)β1 Re ; (48) a N combining with (45)and(48), we obtain result therefore R e R e R e < ; (49) β < (50) This value is smaller than that in equation(40) about two orders of magnitude Fifth force A fifth force is parameterized by a Yukawa potential following U(r) = ε m 1m 8π e r/λ r ; (51) where m 1, m are masses of two bodies, r is distance between them, ε is the strength of the interaction and λ is the range which is of the order of the size of vacuum chamber[17], namely λ R vac. The tightest bound on the strength of interaction is ε < 10 3 [18]. Inside the chamber, one considers two identical bodies with uniform densities ρ c, radius r c and masses m c. If two these bodies satisfy thin shell condition, their field profile outside these bodies are given by φ(r) = β 1 3 r e m c e r/rvac + ϕ vac ; (5) 4π r e r the interaction potential is the bound on the strength of interaction becomes V (r) = β1( 3 r c ) m c e r/rvac ; (53) r c 8πr β 1( 3 r e r e ) < 10 3 ; (54) combining(54) and (3), we have ( ) R e 33α = 16ρ out αβ 33α R e 64β β1 16ρ in αβ 1 64β β1 < 10 3/ 6β 1 Φ e 3 ; (55) β 1 7

9 β < ( 4 33β 1 Φ e 10 3.(ρ in ρ out ) ) 1 ; (56). β < ( ( ) 10 3 ( ) ) 1 = (57) 3.4. The constraint on the parameter of alpha From experiments in solar system, Gu and Lin[19] have obtained the constraints on Chameleon theories of f(r) as follows F R 0; (58) with R H 0 and 0 RF RR 0.4; (59) for R H 0 ; (60) where F (R) = f(r) R = Λ + α R (1 + R + R 3 )e βr and F R = df/dr; F RR = d F/dR ; with H m R m. (61). In equation (8), we have approximatelly therefore e βr = 1 βr; (6) F R = α [ 3βR 1 ] R (β 1) R + 1 ; (63) F RR = α [ β + 1 ] R 3 3βR + 1. (64) Since R << 1, therefore in expressions(63),(64), we have F R = α R ; (65) F RR = + α R 3 ; (66) when combining (65),(61) and (58), we have from (66),(61),(60) and (59), we have thus, the constraint on the parameter of alpha is α ; (67) α ; (68) α (69) From the constraints on beta parameter in formula (57) and on the alpha parameter in the formula (69), we see that these parameters are very small. Therefore we expect that the difference between this model and General relativity theory is observed only in regions that cosmic curvature is very large as in very early time of universe[0] and in regions of edge of black holes. In solar system, where the cosmic curvature is not large, predictive results from this model and Einstein s theory are indistinguishable. 8

10 4. Conclusion In this paper, we have obtained constraints on the parameters of α and β of the model based on solar system experiments. The constraints on the parameter β required by the experiments of the thin shell condition of the Earth and the Equivalence principle are not too rigorous. The requirement from the absence of the fifth force caused by the scalar field puts a quite strict constraint on the parameter β and this is also the general status of many f(r) gravity modified models for this experiment. The constraint on the parameter α from experiments in solar system is rigorous. The above constraints are only based on experiments in the solar system. There are indications that the constraints from the inflation of the universe to these parameters are even more severe, which will be examined again in a near future article. References [1] Perlmutter S et al 1997 Bull. Am. Astron. Soc [] Riess A G et al 1998 Astron. J [3] Perlmutter S et al 1999 Astrophys. J [4] Weinberg S 1989 Rev. Mod. Phys [5] Zlatev I, Wang L, Steinhardt P 1999 Phys. Rev. Lett [6] Starobinsky A A 1980 Phys. Lett. B [7] Carroll S M, De Felice A, Duvvuri V, Easson D A, Trodden M, Turner M S 005 Phys. Rev. D [8] Vo Van On, Tran Trong Nguyen 01 Journal of Thu Dau Mot University [9] Vo Van On, Tran Trong Nguyen 015 Chinese Journal of Physics [10] Will C M 1971 Astrophys. J [11] Faulkner T, Tegmark M, Bunn E F and Mao Y 007 Phys. Rev. D [1] Khoury J and Weltman A 004 Phys. Rev.D [13] Flanagan E E 004 Class. Q. Grav [14] Bisabr Y 010 Phys. Lett. B [15] Will C M 005 Liv. Rev. Rel. 9 3 [16] Weinberg S 197 Gravitation and Cosmology (John Wiley and Sons, New York) [17] Khoury J and Weltman A 004 Phys. Rev. Lett [18] Hoskins J K, Newman R D, Spero R and Schultz J 1985 Phys. Rev. D [19] Lin W T, Gu J A, and Chen P 010 Cosmological and Solar-System Tests of f(r) Modified Gravity (Preprint astro-phy.co/ ) [0] Vo Van On, Truong Huu Nghi 017 Scientific Journal of Thu Dau Mot University