Diego Saez-Gomez

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Moreover, lations along the the election cosmological of phantom evolution, conditions a fact atpointed z = 0 gives out before an anomalous for this behavior kind of models of the Scalar-tensor cosmological in Ref. [27]. parameters In general, for we both have models shown representation that whenthe z>0. transition to the phantom epoch occurs. Moreover, the election of phantom conditions z = 0 gives an anomalous behavior of the f(r) Then, Then, 4cosmological Scalar-tensor by bya a gravity: particular parameters f(R) f(R) representation for action, action, bothwe presence we models can can of reconstruct when f(R) z>0. gravity its its equivalent of scalar-tensor a sudden theory(4.1), so sothat thatwe wecan canstudy studythe theviable viablemodels modelsstudied studiedabove aboveby byanalyzing analyzingtheir their scalar-tensorcoun- terpart. terpart. It coun- singularity 4 is Scalar-tensor well Moreover, known that by byvarying representation f(R) gravity the the action(4.1) is equivalent of f(R) withrespecttothemetricg togravity a kind of Brans-Dicke µν µν theory , , field fieldequations with equations a null are are kinetic obtained, term, and a scalar potential, (see for example, [6, 23] and references therein), It is wellCosmological known that evolution f(R) gravity is equivalent to a kind of Brans-Dicke theory with a null kinetic term, and a scalarS potential, d 4 x √ (see −g for Rexample, − V (φ)+2κ [6, 23] 2 and references therein), Action, R L m . (4.1) µν 1 2 g µνR κ2 φ T µν (m) 1 S = d 4 x √ φ (∇ µ∇ ν φ − g µν φ) − 1 µν − 1 in viable modified 2 g µνR = κ2 φ T µν (m) + 1 gravity −g φ (∇ µ∇ ν φ − g µν φ) − 1 12 22 g µνV g 0.92 µνV (φ) (φ), , 2.0 Φz φ R − V (φ)+2κ 2 Φz By varying the action with respect to the scalar field φ, theLrelation m . between both theories 0.90 (4.1) is obtained, and f(R) action(2.1) 3φ 3φ = κisrecovered, κ 2 T 2 T (m) (m) φ dV dV (φ) (φ) 1.9 + φ −− 2V 2V (φ) (φ), , (4.4) (4.4) By varying the action with respect to the scalar dφ dφ field φ, the relation between both theories 0.88 R = V is obtained, and f(R) 1.8 (φ) → φ = φ(R) , ⇒ f(R) =φ(R)R − V (φ(R)) . (4.2) where wherethe thesecond secondequation equation action(2.1) is isthe thetrace trace isrecovered, of ofthe thefield fieldequations equationsin(4.4). in(4.4). Hence, Hence, for fora aflat flatFLRW metric metric While(2.3), the (2.3), scalar the the Friedmann field R = V and the equations equations potential yield, yield, 0.86 are related with 1.7 the particular f(R) action by, FLRW equations, (φ) → φ = φ(R) , ⇒ f(R) =φ(R)R − V (φ(R)) . (4.2) φ = f While the scalar field and the R (R) , V(φ(R)) = f potential are related R (R) R − f(R) . (4.3) 3H 2 1 κ 2 ρ m 3H with the particular f(R) action by, φ ˙φ + 1 3H 2 = 1 κ 2 ρ m − 3H φ ˙φ + 1 22 V V (φ) (φ) 0.84 1.6 , , φ = f R (R) , V(φ(R)) = f R (R) R − f(R) . (4.3) 3H 2 2Ḣ 1 κ 2 p m φ ¨φ +2H ˙φ − 1 − 3H 2 − 2Ḣ = 1 κ 2 p m + φ ¨φ +2H ˙φ − 1 1.5 1.0 0.5 0.0 0.5 1.0 1.0 0.5 –9– 22 V V (φ) (φ) 0.0 0.5 1.0 z z. . (4.5) (4.5) (a) (b) Trace while while equation, the thetrace traceequation equationin(4.4) in(4.4) is isgiven givenby, by, Figure 7: Evolution of the scalar field φ(z) –9– for the HS model (7a) and the NO model (7b). In both cases, the initial conditions 33 ¨φ ¨φ are: = κκ 2 (ρ 2 h(0) = 1, andh (ρ m m −p m p m ) ) − (0) φV φV = +2V 3Ω +2V 0 m/2. −− 9H 9H ˙φ ˙φ . . (4.6) (4.6) In vacuum, Let’s Let’s reconstruct the For phase simplicity, space the the corresponding can webe assume described the scalar-tensor FLRW by, equations theory theory (19) for forthe in the vacuum, particular particular so models by models combining considered the in inthe theprevious equations section. section. (19)-(20), By By the the following expressions(4.3), system results, the the correspondingrelation relationφ(R) φ(R) for forthe the models(2.10) are aregiven givenby, by, Ḣ = −2H 2 + 1 c 1 cc 1 2 c 2 R φφ HS =1+ 2 c 1 cR(aR − HS 1+ c 2R 1+ c , φ 2R NO =1− (1 + cR) 2 + aR 1+cR + aR − b HS =1+ 2 + c 6 V 1 (φ) , ˙φ = −3H 2 φ + 1 1 3H cR(aR −2b) V (φ) R 1+cR , (4.7) HS 1+ c 2R 1+ c , φ 2R NO =1− (1 + cR) 2 + aR . 1+cR + aR − b (25) Note that this is a dynamical 1+cR , (4.7) R Rsystem R R HS whose critical points are given by Ḣ = ˙φ =0, which HS HS corresponds to HS de Sitter solutions, as mentioned in the previous sections as a feature of where wef(R) havegravity. assumed Then, a power the critical of n = points 1 for areboth the solutions models(2.10). of the algebraic Then, the equation, scalar poten- DSG, Class.Quant.Grav. 30 095008 (2013)
8 − 3H 2 − 2Ḣ = 1 κ 2 p m + φ ¨φ +2H ˙φ − 1 Let’s reconstruct the corresponding Cosmological 100 scalar-tensor 2 V (φ) evolution . theory in viable for the (4.5) particular modified models gravityconsi 6 in the previous section. By the expressions(4.3), the corresponding relation φ(R) fo while the 4 trace equation in(4.4) is given by, Scalar-tensor representation of models(2.10) are given by, f(r) gravity 3 ¨φ = κ 2 (ρ m − p m c) 1 −c 2 φV R +2V − 9H ˙φ . cR(aR(4.6) − b) 2 φ HS =1+ 2 + c 1 R Let’s reconstruct the corresponding scalar-tensor HS 1+ c 2R 1+ c , φ 2R NO =1− (1 + cR) 2 + aR 1+cR + aR − b Hence, 1+cR , 0 the R Rtheory for the HS scalar-tensor representation for both mod 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 HS particular models considered Φ potentials exhibit Φtwo branches due to the particu in the previous section. By the expressions(4.3), the corresponding relation φ(R) for the (a) where we have assumed models(2.10) are given by, considered a power of n here. = 1 for (b) Inboth addition, models(2.10). both models Then, the introduce scalar p tials(4.3) yield, c 1 c 2 R φ HS =1+ 2 + c 1 cR(aR − b) R HS 1+ c 2R 1+ c , φ 2R NO =1− V (1 + cR) 2 + aR 1+cR + aR − b HS (φ) = 1+c Figure 7: Scalar potentials V (φ) in terms of H0 2 ,oftheHSmodelwithn of the scalar 1 −field, φ =1,c ± 2being c 12 =2,c c1 φ (1 2 < = − 1+cR , 1φ) (4.7) (figure in the7a), HS model, an and of the NO model with n =1,a =0.1/H R HS , 0 2 , b =1,c =0.05/H limit where 0 2 ,fig.7b. both Both branches potentials c 2 of arethe not uniquely scalar potentials co defined, where each branch behaves veryR different, R HS converging to the boundary of the HS V where we have assumed a power of n = 1 for NO (φ) = 2 a + c (1 + b − φ) ± 2 scalar field φ. Moreover, the evolution(a of + the bc)(a scalar + c −field, φ) φ(z) =f both models(2.10). Then, the c 2 . scalar potentials(4.3) 0.92 yield, Hence, the scalar-tensor 16 V HS (φ) = 1+c representation 2.0 for both models is not uniquely defined, bu potentials exhibit 1 − φ ± two 2 cbranches, 2 c1 (114 −which φ) in principle do not affect V Φ the cosmological evolu Φz R HS Φz , V but it may influence cthe 2 V NO (φ) = 2 a + c (1 + b − φ) ± 2 behavior of the phase space. In addition, Φ 200 0.90 both models intro 1.9 12 a boundary condition on the value of the scalar field, being φ < 1 for the HS model (a + bc)(a + c − φ) φ
Page 1 and 2: Cosmological constraints on modifie
Page 3 and 4: Brief Article The Author April 6, 2
Page 5 and 6: FLRW cosmologies 12 3 No Big Bang (
Page 7 and 8: FLRW cosmologies Future singulariti
Page 9 and 10: Viable f(r) gravity Viability condi
Page 11 and 12: 3 Cosmological evolution in viable
Page 13 and 14: Cosmological evolution in viable f(
Page 15: Cosmological evolution in viable f(
Page 19 and 20: (z = 0) = H 0 = 100 hkms −1 Mpc
Page 21 and 22: (z = 0) = H 0 = 100 hkms −1 Mpc
Page 23 and 24: effective repulsive force 12h R ∼
Page 25 and 26: Testing F(R) gravity with Sne Ia da
Page 27 and 28: Testing F(R) gravity with Sne Ia da
Page 29 and 30: ere right lethand us remind side (r
Page 31 and 32: e previous section, we study two pa
Page 33 and 34: en at redshift z = 1000 where δ is

Diego Saez-Gomez

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