Cosmological evolutions in Tsujikawa model of f(R) Gravity
Jian-Yong Cen, Shang-Yu Chien, Chao-Qiang Geng, Chung-Chi Lee
aa r X i v : . [ g r- q c ] A ug Cosmological evolutions in Tsujikawa model of f ( R ) Gravity
Jian-Yong Cen , Shang-Yu Chien , Chao-Qiang Geng , , and Chung-Chi Lee School of Physics and Information Engineering,Shanxi Normal University, Linfen 041004 Department of Physics, National Tsing Hua University, Hsinchu 300 Physics Division, National Center for Theoretical Sciences, Hsinchu 300 DAMTP, Centre for Mathematical Sciences,University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (Dated: August 28, 2019)
Abstract
We concentrate on the cosmological properties in the Tsujikawa model (TM) of viable f ( R )gravity with the dynamical background evolution and linear perturbation theory by using theCosmoMC package. We study the constraints of the cosmological variables along with the modelparameter from the current observational data. In particular, we show that the matter densityfluctuation is suppressed and the constraint of the neutrino mass sum is relaxed in the TM, whichare similar to other viable f ( R ) models. In addition, we discuss the parameters of the decelerationand equation of state for dark energy in the TM and compare them with those in the ΛCDMmodel. a [email protected] . INTRODUCTION To describe the accelerating expansion of the universe, the ΛCDM model [1] is the simplestcandidate. However, this simplest version has the so called “cosmological constant problem,”which is related to the “fine-tuning” [2, 3] and “coincidence” [4–6] problems. People havebeen motivated by these issues to explore new theories beyond ΛCDM, such as those with thedynamical dark energy [7, 8]. A typical way is to modify the standard general relativity (GR)by promoting the Ricci scalar of R in the Einstein-Hilbert action to an arbitrary function,i.e., f ( R ) [9]. In addition, many viable f ( R ) gravity models have been developed in theliterature [9] to satisfy the theoretical and observational constraints. The most popular onesare Hu-Sawicki [10], Starobinsky [11], Tsujikawa [12], and exponential [13–16] f ( R ) gravitymodels, in which the first three have been extensively examined in the literature, such asthe recent one in Ref. [17], whereas the last one, i.e. the Tsujikawa model (TM), has notbeen systematically explored yet, which is our concentration in this study.The TM of the viable f ( R ) models, first proposed by Tsujikawa in 2007 [12], is writtenas f ( R ) = R − λR ch tanh (cid:18) RR ch (cid:19) , (1)where λ is the dimensionless model-parameter and R ch corresponds to the constant charac-teristic curvature in the model. The TM has a simpler form than most of other f(R) modelsas it contains only one model-parameter beyond ΛCDM, which can be regarded as a similartype of the exponential f ( R ) model [18], but has a different form in the functional structure,which could result in some different cosmological behaviors in the numerical results.It is known that the viable f ( R ) gravity can well describe the power spectrum of the mat-ter density fluctuation [19–24] and the formation of the large scale structure (LSS) [25–28] inthe universe. To investigate the dynamical dark energy behaviors, we reply on the existingopen-source programs. However, most of them are written with either the parametrizationin term of the equation of state or the background evolution being the same as the ΛCDMmodel [23, 24]. Recently, the allowed parameter spaces of the cosmological observables inthe viable f ( R ) gravity models with the dynamical background evolution have been exploredin Ref. [17]. In this work, we will use the same method to examine the TM. In particu-lar, we will show the allowed windows for the active neutrino masses, dark energy densityand Hubble parameter as well as other cosmological parameters, such as the deceleration2nd equation of state for dark energy. In addition, since the first detection (GW150914) ofgravitational waves by the LIGO Collaboration [29], the gravitational radiation has been be-lieved to be a new tool to test GR and search for new physics. Beyond GR, the gravitationalwaveforms of the modified gravity theories have been discussed in the literature [30–32]. Itis possible that the TM and other f(R) models may be stringently constrained by the futuregravitational wave detectors.In this paper, we take the open source program of the Modification of Growth (MG) withCode for Anisotropies in the Microwave Background (CAMB) [33, 34], which is designedto examine the dynamical dark energy model. In order to put the TM into the programof MGCAMB, we modify the growth equations of the scalar perturbations and densityfluctuations in the Newtonian gauge. We also include the dynamical background evolutionof dark energy [23, 24] instead of the ΛCDM one, and use the MG Cosmological MonteCarlo(MGCosmoMC) [35, 36] together with the latest data from the cosmological observationas.The paper is organized as follows. In Sec. II, we present the TM of viable f ( R ) gravity.In Sec. III, we show the cosmological evolutions in the TM. In particular, we include theperturbation equations of the dynamical background evolution. In Sec. IV, we show theconstraints from the cosmological observational data. Finally, our conclusions are given inthe Sec. V. II. TSUJIKAWA MODEL OF VIABLE f ( R ) GRAVITY
The modified Einstein Hilbert action of the f ( R ) gravity models is given by S = Z d x √− g κ f ( R ) + S M , (2)where κ = 8 πG with G the Newton’s constant, g stands for the determinant of the metrictensor g µν , f ( R ) is an arbitrary function of the Ricci scalar R , and S M corresponds to theaction of the relativistic and non-relativistic matter. After the variation of g µν in the action,we obtain the modified field equation: f R R µν − f g µν − ( ∇ µ ∇ ν − g µν (cid:3) ) f R = κ T ( M ) µν , (3)where f R ≡ df ( R ) /dR , ∇ µ is the covariant derivative, (cid:3) ≡ g µν ∇ µ ∇ ν represents thed’Alembert operator, and T ( M ) µν denotes the energy momentum tensor. To describe our3niverse, we use the Friedmann-Lema¨ıtre-Robertson-Walker (FLRW) metric, given by ds = g µν dx µ dx ν = − dt + a ( t ) d~x , (4)where a ( t ) is the scale factor. The 00 component of Eq. (3) gives the modified Friedmannequation, 3 f R H = 12 ( f R R − f ) − H ˙ f R + κ ρ M , (5)while the trace of the linear combination of Eq. (3) leads to the modified Friedmann accel-eration equation, 2 f R ˙ H = − ¨ f R + H ˙ f R − κ ( ρ M + P M ) , (6)where the dot “ · ” stands for the derivative with respect to the cosmic time t , H ≡ ˙ a/a isthe Hubble parameter, and ρ M = ρ r + ρ m ( P M = P r + P m ) represent the energy density(pressure) of relativistic ( r ) and non-relativistic ( m ) fluids. Comparing with the originFriedmann equations, we can get the dark energy density and pressure as follows: ρ DE = κ − (cid:18)
12 ( f R R − f ( R )) − H ˙ f R + 3 (1 − f R ) H (cid:19) , (7) P DE = κ − (cid:18) −
12 ( f R R − f ( R )) + ¨ f R + 2 H ˙ f R − (1 − f R ) (cid:16) H + 3 H (cid:17)(cid:19) . (8)The equation of state of dark energy is defined by w DE = ρ DE P DE . (9)Following the same procedures in Refs. [10, 16], we can simplify Eqs. (5) and (6) to a secondorder differential equation, y ′′ H + J y ′ H + J y H + J = 0 , (10)with y H ≡ ρ DE ρ (0) m = H m − a − − χa − ,J = 4 + 1 y H + a − + χa − − f R m f RR , J = 1 y H + a − + χa − − f R m f RR ,J = − a − − (1 − f R ) ( a − + 2 χa − ) + ( R − f ) / m y H + a − + χa − m f RR , (11)where m ≡ κ ρ m / ρ i ≡ ρ i ( z = 0), and χ ≡ ρ r /ρ m , with ρ m ( r ) being the energy density ofthe relativistic (non-relativistic) fluid at the present time. Here, the prime “ ′ ” in Eq. (10)4enotes the derivative with respect to ln a . Using the differential equation in Eq. (10),the cosmological evolutions can be calculated through the various existing programs in theliterature. Consequently, the deceleration parameter q is found to be q ≡ − HH ! = 12 a − + 2 χa − + (1 + 3 w DE ) y H a − + χa − + y H . (12)As the TM is one of the popular viable f ( R ) gravity models, the conditions for the via-bility must be satisfied. For example, the TM has the following viable properties: (a) when λ < cosh ( R/R ch ), f R = 1 − λcosh − ( R/R ch ) >
0, leading to a positive effective gravitationalcoupling; (b) when λ > f RR >
0, resulting in a stable cosmological perturbation and apositivity of the gravitational wave for the scalar; (c) when R → ∞ , f ( R ) → R −
2Λ withΛ = λR ch /
2, showing an asymptotic behavior to the ΛCDM model in the large curvatureregion; and (d) when λ > . < m ( R = R d ) <
1, indicating the existence of a late-timestable de-Sitter solution, where m = Rf RR /f R . III. COSMOLOGICAL EVOLUTIONS IN TSUJIKAWA MODEL
To explore the expansion history and the linear perturbation of the universe in the TM,we use the MGCAMB program. In particular, we examine the cosmological parameters inthe evolutions of the universe with the TM of viable f ( R ) gravity. The initial conditionsfor the model are from the MGCosmoMC fitting, in which the input parameters have beenchosen as the mean values. In Figs. 1 and 2, we show Hubble and deceleration parametersfor the TM and ΛCDM, respectively. We see that the difference between the two models inFig. 1 is less than 1% in the whole expansion history of the universe. There are two reasons.The first one is that the initial energy density ratios of matter and dark energy in the twomodels are close to each other. The second one is that the TM is a ΛCDM-like theory, inwhich it gives only a tiny contribution to the total energy density before the dark energydominated era. In Fig. 2, the behaves of the deceleration parameter in the two models aresimilar when z > .
2. The TM starts to have an accelerated expansion of the universe at z = 0 . z = 0 .
649 in the ΛCDM model. In the present time, the differenceis within 0 . . Clearly, it is hard to distinguish these two models by either H or q .As one of the characteristics in the viable f ( R ) models, the behavior of the TM approachesthe cosmological constant when z is large. In Fig. 3, the effective energy density is almost5 .5 FIG. 1. Hubble parameter H ( z ) as a function of the redshift z , where the dashed (red) and dotted(blue) lines represent the TM and ΛCDM for λ − = 0 . H ,T M , H , Λ CDM ) = (67 . , . km/s · M pc , and the initial conditions are given by (Ω m , Ω r , Ω DE ) T M = (0 . , . × − , . m , Ω r , Ω DE ) Λ CDM = (0 . , . × − , . H = ( H T M − H Λ CDM ) /H Λ CDM . FIG. 2. Deceleration parameter q ( z ) as a function of the redshift z , where the legend is the sameas Fig. 1. FIG. 3. Evolutions of the normalized effective dark energy density ρ DE ( z ) /ρ DE (0) in the TM andΛCDM.FIG. 4. Equation of state w ( z ) for dark energy as a function of z in the TM and ΛCDM. constant in the early time, which is smaller than the present dark energy density. When z < .
0, it starts to rise and fall slightly. The equation of state evolution is shown in Fig. 4.For the TM, it indeed oscillates and crosses the phantom divide line as mentioned inRef. [37].In Fig. 5, we show the cosmological evolutions of the normalized Ricci scalar
R/m IG. 5. Normalized Ricci Scalar
R/m and scalaron mass m s /m as the functions of z in the TM. and scalaron mass m s /m as the functions of z in the TM with m ≡ κ ( ρ m / / . In thecosmological background evolution, the singularity problem is not avoidable because of thegeneric property of the viable f ( R ) models. As z gets larger, the scalaron mass becomesmuch heavier so that the Ricci scalar strongly oscillates, which causes the program to faileasily. By considering the asymptotic behavior of the ΛCDM model in the viable conditionsand solving the differential equation in the z decreasing direction, the numerical error can behandled in some code technique. The other way is to add the R term in the f ( R ) modelsas mentioned in Ref. [38]. This term can also help the models to have a steady performancein the R evolution.From Ref. [39], we know that the scalaron mass also affects the propagation of the scalarmode of the gravitational waves. It will make the mode to decay so fast below the cutofffrequency in the viable f ( R ) models. From the numerical result, we can obtain the cutofffrequency in the background around 10 − Hz now in the TM. If the wave propagates inthe more dense region, such as the inner galaxy with the density around 10 − g/cm , thefrequency will rise to infinity. This may be tested in the future stochastic gravitational wavedetection.The TM can be seen as the same branch of the exponential model of viable f ( R ) gravity.Compared with the TM, the background evolution in the exponential model illustrates amore sharp variation. The related work has been done in Ref. [17]. We can compare thesetwo models in the linear perturbation theory. In the original CAMB program, we choosethe synchronous gauge to do the simulation. But in the open source of MGCAMB, the8ewtonian gauge is used to do the calculation, in which the metric is given by [40, 41] ds = − (1 + 2Ψ) dt + a ( t ) (1 − d~x . (13)Under the subhorizon limit, one has that [17] k a Ψ = − πGµ ( k, a ) ρ M ∆ M (14)with µ ( k, a ) = 1 f R k a f RR f R k a f RR f R and ΦΨ = γ ( k, a ) = 1 + 2 k a f RR f R k a f RR f R , (15)where k is the comoving wavenumber and ∆ M ≡ δ M + 3 H (1 + ω M ) v M /k is the gauge-invariant matter density perturbation with w M = P M /ρ M the equation of state and v M thevelocity for matter. The growth equation for the matter density perturbation at the matterdominated epoch with v m = 0 is given by¨ δ m + 2 H ˙ δ m − πGµ ( k, a ) ρ m δ m = 0 . (16)As shown in Sec. II, the TM satisfies the viable conditions of 0 < f R < f RR >
0, which imply two scenarios. Firstly, if k increases, it will cause a larger value of µ ( k, a ). Secondly, the matter density fluctuations are enhanced due to a larger value of µ ( k, a ), which is regarded as the scale independent and dependent factors of f − R and(1 + 4 k f RR / ( a f R )) / (1 + 3 k f RR / ( a f R )) for k ≪ f RR / ( a f R ) and k ≫ f RR / ( a f R ),respectively. Besides, if we consider the wavenumber outside the Hubble radius, i.e. k → ′′ + (cid:18) − H ′′ H ′ + B ′ − B + B H ′ H (cid:19) Φ ′ + (cid:18) H ′ H − H ′′ H ′ + B ′ − B (cid:19) Φ = 0 , ( k = 0) . (17)2Φ eff + B E ′ E E ′ E ′ + E ′′ S = − f R κ a ρ M k ∆ M (18)where B = f RR f R R ′ HH ′ , E = H H , Φ eff = 12 (Φ + Ψ) , S = −
2Φ + Ψ . (19)In the TM and ΛCDM, in terms of Eq. (17), Φ grows as a increases. However, the change ratein the TM is higher than that in the ΛCDM model. With the relation Ψ = ( B Φ ′ +Φ) / (1 − B ),one obtains a smaller value for the effective gravitational potential [42]. Due to the negative9 FIG. 6. Spectra of the matter power perturbation in the TM and ΛCDM, where δ m is the matterdensity of the perturbation and ∆ δ m = ( δ T Mm − δ Λ CDMm ) /δ Λ CDMm , . sign of the second term in the LHS of the modified Poisson equation in Eq. (18), the smallernegative φ eff would finally enhance the matter density perturbation in the TM. In Fig. 6,we concentrate on the sub-horizon regime, which is more relevant to the observation. Here, k should be larger than 0.001 to satisfy the sub-horizon limit of k/aH ≫
1. The result in theTM has a more enhancement than that in the ΛCDM model for the higher value of k . It alsorelaxes the limit of the neutrino mass sum a little because there is more freedom for a largervalue of Σ m ν resulting from the suppressed effect. On the other hand, the viable modifiedgravity models also affect the CMB spectrum through the late-time integrated Sachs-Wolfeeffect as shown in Fig. 7. As the gravitational potential in the TM evolves in different wayscomparing with ΛCDM, the figure in the TM declines slightly but still approaches to theΛCDM result when ℓ <
10. However, for the observational data in this regime, the errorsare still large. Strict constraints on the TM of modified gravity can be given from the CMBwhen more precise measurements of the low- ℓ regime are available. IV. CONSTRAINTS FROM COSMOLOGICAL OBSERVATIONS
We have used the best fitted values from the MGCosmoMC to evaluate the backgroundevolutions in the previous section. Now we would study the constraints from the cosmological10 l FIG. 7. Spectra for the cosmic microwave background in the TM and ΛCDM. observations. With the MGCosmoMC, the input parameters are given in Table. I and thefitting results are shown in Table. II. In the following, we compare the results in the TMand ΛCDM. First, the best fitted parameters in the TM are close to those in the ΛCDMmodel. For example, Ω b h bestfit and Ω c h bestfit are almost the same ( < . m ν in the TMis about 20% larger than that in the ΛCDM one. This result extends the discuss in thematter power spectrum for Σ m ν . Finally, for the contour plots in Fig. 8, we can see thatmost of the results in the TM are similar to those in the exponential model mentioned inRef. [17], except the model parameter. Note that the model parameter is more sensitive inthe exponential model because its 1 σ range is obviously smaller than that in the TM. V. CONCLUSIONS
We have studied the cosmological evolutions of the universe in the TM of viable f(R)gravity, which have also been compared with those in the ΛCDM model. We have foundthat the results in the TM are not much different from the corresponding ones in the ΛCDM.We have demonstrated that the transition point from the deceleration to acceleration in ouruniverse is z = 0 .
688 in the TM, which is higher than z = 0 .
649 in the ΛCDM model. As a11
ABLE I. List of priorsParameters PriorsModel parameter 10 − < λ − < × − < Ω b h < . − < Ω c h < . < Σ m ν < . < n s < . < ln(10 A s ) < . < τ < . θ MC . < θ MC < · Mpc) 20 < H < λ −1 Ω c h Σ m ν Ω b h λ − Ω c h Σm ν FIG. 8. Two-dimensional contour plots of Ω b , Ω c , λ − and Σ m ν in the TM. ABLE II. Fitting results in TM and ΛCDMParameters TM ΛCDM λ − . +0 . − . -Ω b h . +0 . − . . +0 . − . Ω c h . +0 . − . . +0 . − . Σ m ν . +0 . − . . +0 . − . n s . +0 . − . . +0 . − . ln(10 A s ) 3 . +0 . − . . +0 . − . τ . +0 . − . . +0 . − . θ MC . +0 . − . . +0 . − . H ( km/s · M pc ) 67 . +1 . − . . +1 . − . Age / Gyr 13 . +0 . − . . +0 . − . σ . +0 . − . . +0 . − . χ best − fit result, the dark energy dominance is slightly pushed up in the TM.For the large scale structure, the amplitude of the matter power spectrum in the TM isstrenghen in k > . k = 0 . .
9% larger than that in the ΛCDM. On the other hand, the TM of viable f ( R ) gravityaffects the gravitational potential evolution through the modified Possion equation. In theCMB spectrum, when ℓ <
10, it is sensitive to the change of gravitational potential in theuniverse history. There is only slightly difference between the TM and ΛCDM in this region,whereby both of them fit very well in other regions. As the current observational data arenot accurate enough, it is still possible to test GR and modified gravity models when morefuture measurements are available. From the contour plots for the parameter fittings, wehave displayed that the TM also gives a relaxed constraint on the neutrino mass sum asthe other viable f ( R ) gravity models. In addition, the model parameter in the TM is moresensitive than that in the exponential model. Our numerical results have demonstrated somedifferent features among the viable f ( R ) models. If we fully clarify the characters of thesemodels and estimate their cosmological evolutions, they can potentially hint about what isnext to do in the dark energy research in the future.13 CKNOWLEDGMENTS
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