Observational Constraints on Exponential Gravity
aa r X i v : . [ a s t r o - ph . C O ] O c t Observational Constraints on Exponential Gravity
Louis Yang, ∗ Chung-Chi Lee, † Ling-Wei Luo, ‡ and Chao-Qiang Geng § Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan (Dated: October 29, 2018)
Abstract
We study the observational constraints on the exponential gravity model of f ( R ) = − βR s (1 − e − R/R s ). We use the latest observational data including Supernova Cosmology Project (SCP)Union2 compilation, Two-Degree Field Galaxy Redshift Survey (2dFGRS), Sloan Digital SkySurvey Data Release 7 (SDSS DR7) and Seven-Year Wilkinson Microwave Anisotropy Probe(WMAP7) in our analysis. From these observations, we obtain a lower bound on the modelparameter β at 1.27 (95% CL) but no appreciable upper bound. The constraint on the presentmatter density parameter is 0 . < Ω m < .
311 (95% CL). We also find out the best-fit value ofmodel parameters on several cases.
PACS numbers: 98.80.-k, 04.50.Kd, 95.36.-x ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] . INTRODUCTION Cosmic observations from type Ia supernovae (SNe Ia) [1, 2], large scale structure (LSS)[3, 4], baryon acoustic oscillations (BAO) [5] and cosmic microwave background (CMB)[6, 7] indicate that our universe is undergoing an accelerating expansion. The reason for thisacceleration, the so-called dark energy problem, remains a fascinating question today. Thesimplest model to explain this problem is the ΛCDM model, in which a time independentenergy density is added to the universe. However, the ΛCDM model suffers from bothfine-tuning and coincidence problems [8–13]. In general, the ways to understand the cosmicacceleration can be separated into two branches. One is to modify the matter by introducingsome kind of “dark energy”. The other one is to modify Einstein’s general relativity – themodification of gravity.In modified gravity, one of the popular approaches is to promote the Ricci scalar R inthe Einstein-Hibert action to a function, f ( R ). Although there are several viable f ( R )models, many of them are restricted to the regimes to be effectively identical to the ΛCDMby the observational constraints. Recently, Linder [14] has explored an f ( R ) theory named“exponential gravity”, which has also been discussed in Refs. [15–17]. The exponentialgravity has the feature that it allows the relaxation of fine-tuning and it has only onemore parameter than the ΛCDM model. In addition, the exponential gravity satisfies allconditions for the viability [18] such as the local gravity constraint, stability of the late-time de Sitter point, constraints from the violation of the equivalence principle, stability ofcosmological perturbations, positivity of the effective gravitational coupling, and asymptoticbehavior to the ΛCDM model in the high curvature regime. In this paper, we will studythe constraints given by latest observational data, reexamine the alleviation of the fine-tuning problem, and find the possibility of the derivation from ΛCDM. We use units of k B = c = ~ = 1 and the gravitational constant is given by G = M − with the Planck massof M Pl = 1 . × GeV.The paper is organized as follows. In Sec. II, we review equations of motion and theasymptotic behavior at the high redshift regime in the exponential gravity model. In Sec.III, we discuss the observations and methods. We show our results in Sec. IV. Finally,conclusions are given in Sec. V. 2
I. EXPONENTIAL GRAVITY
The action of f ( R ) gravity with matter is given by S = 12 κ Z d x √− g [ R + f ( R )] + S m , (2.1)where κ ≡ πG and f ( R ) is a function of the Ricci scalar curvature R . In this paper, wefocus on the exponential gravity model [14], given by f ( R ) = − βR s (1 − e − R/R s ) , (2.2)where R s is related to the characteristic curvature modification scale. Since the product of β and R s can be determined by the present matter density Ω m [14], we can choose β andΩ m as the free parameters in the model.We use the standard metric formalism. From the action (2.1), the modified Friedmannequation of motion becomes [19] H = κ ρ M f R R − f ) − H ( f R + f RR R ′ ) , (2.3)where H ≡ ˙ a/a is the Hubble parameter, a subscript R denotes the derivative with respectto R, a prime represents d/d ln a , and ρ M = ρ m + ρ r is the energy density of all perfectfluids of generic matter including (non-relativistic) matter, denoted by m , and relativisticparticles, denoted by r . Here, we only consider the matter density. Since the modification bythe exponential gravity only happens at the low redshift, the contributions from relativisticparticles are negligible. In a flat spacetime, the Ricci scalar is given by R = 12 H + 6 HH ′ . Following Hu and Sawicki’s parameterization [20], we define y H ≡ ρ DE ρ m = H m − a − , y R ≡ Rm − a − , (2.4)where m ≡ κ ρ m / ρ DE is the effective dark energy density, and ρ m is the present matterdensity. Then, Eqs. (2.3) and (2.4) can be rewritten as two coupled differential equations, y ′ H = y R − y H (2.5)and 3 ′ R = 9 a − − H f RR (cid:20) y H + f R (cid:18) H m − R m (cid:19) + f m (cid:21) , (2.6)where R and H can be further replaced by y R and y H from equations in (2.4). CombiningEqs. (2.5) and (2.6), we obtain a second order differential equation of y H , y ′′ H + J y ′ H + J y H + J = 0 , (2.7)where J = 4 − y H + a − f R m f RR ,J = − y H + a − f R − m f RR ,J = − a − + f R a − + f / m y H + a − m f RR , (2.8)with R = m (cid:2) y ′ H + 4 y H ) + 3 a − (cid:3) . (2.9)Solving Eq. (2.7) numerically, we can get the evolution of the Hubble parameter in the lowredshift regime ( z = 0 ∼ w DE is given by w DE = − − y ′ H y H . (2.10)In the high redshift regime ( z & e − R/R S of f ( R ) in Eq. (2.2)becomes negligible ( e − R/R S < − ). The exponential gravity model behaves essentially likea cosmological constant model with the dark energy density parameter Ω Λ = βR S / H ∼ =Ω m y H ( z high ). Thus, the Hubble parameter as a function of z in this regime can be expressedas H ( z ) = H s Ω m (1 + z ) + Ω r (1 + z ) + βR S H , (2.11)where Ω r is the density parameter of relativistic particles including photons and neutrinos .The equation (2.11) will be used in the data fitting of CMB and the high redshift part ofBAO in section III. Ω r = Ω γ (1 + 0 . N eff ), where Ω γ is the present fractional photon energy density and N eff = 3 .
04 isthe effective number of neutrino species [21]. II. OBSERVATIONAL CONSTRAINTS
To constrain the free parameters of β and Ω m in the exponential gravity model, we usethree kinds of the observational data including SNe Ia, BAO and CMB. The SNe Ia andCMB data lead to constraints at the low and high redshift regimes, respectively, while theBAO data provide constraints at the both regimes. A. Type Ia Supernovae (SNe Ia)
The observations of SNe Ia, known as “standard candles”, give us the information aboutthe luminosity distance D L as a function of the redshift z . The distance modulus µ is definedas µ th ( z i ) ≡ D L ( z i ) + µ , (3.1)where µ ≡ . − h with H = h · km/s/M pc is the present value of the Hubbleparameter. The Hubble-free luminosity distance for the flat universe is D L ( z ) = (1 + z ) Z z dz ′ E ( z ′ ) , (3.2)where E ( z ) = H ( z ) /H . The χ of the SNe Ia data is χ SN = X i [ µ obs ( z i ) − µ th ( z i )] σ i , (3.3)where µ obs is the observed value of the distance modulus. Since the absolute magnitudeof SNe Ia is unknown, we should minimize χ SN with respect to µ , which relates to theabsolute magnitude, and expand it to be [22, 23] χ SN = A − µ B + µ C, (3.4)where A = X i [ µ obs ( z i ) − µ th ( z i ; µ = 0)] σ i ,B = X i µ obs ( z i ) − µ th ( z i ; µ = 0) σ i , C = X i σ i . (3.5)The minimum of χ SN with respect to µ is˜ χ SN = A − B C . (3.6)5e adopt this ˜ χ SN for our later χ minimization. We will use the data from the SupernovaCosmology Project (SCP) Union2 compilation, which contains 557 supernovae [24], rangingfrom z = 0 .
015 to z = 1 . B. Baryon Acoustic Oscillations (BAO)
The observation of BAO measures the distance ratios of d z ≡ r s ( z d ) /D V ( z ), where D V is the volume-averaged distance, r s is the comoving sound horizon and z d is the redshift atthe drag epoch [25]. The volume-averaged distance D V ( z ) is defined as [5] D V ( z ) ≡ (cid:20) (1 + z ) D A ( z ) zH ( z ) (cid:21) / , (3.7)where D A ( z ) is the proper angular diameter distance: D A ( z ) = 11 + z Z z dz ′ H ( z ′ ) , (for flat universe) . (3.8)The comoving sound horizon r s ( z ) is given by r s ( z ) = 1 √ Z / (1+ z )0 daa H ( z ′ = a − ) q b / γ ) a , (3.9)where Ω b and Ω γ are the present values of baryon and photon density parameters, respec-tively. We use Ω b = 0 . h − and Ω γ = 2 . × − h − [21]. The fitting formula for z d is given by [26] z d = 1291(Ω m h ) . . m h ) . (cid:2) b (Ω b h ) b (cid:3) , (3.10)where b = 0 . m h ) − . (cid:2) . m h ) . (cid:3) ,b = 0 . m h ) . . (3.11)The typical value of z d is about 1021 with Ω m = 0 .
276 and h = 0 . z d is in thehigh redshift regime, we use Eq. (2.11) to calculate r s ( z d ). On the other hand, D V ( z ) isevaluated by the numerical result of Eq. (2.7) as it is in the low redshift regime.The BAO data from the Two-Degree Field Galaxy Redshift Survey (2dFGRS) and theSloan Digital Sky Survey Data Release 7 (SDSS DR7) [25] measured the distance ratio d z at6wo redshifts z = 0 . z = 0 .
35 to be d obsz =0 . = 0 . ± . d obsz =0 . = 0 . ± . C − BAO = − − . (3.12)The χ for the BAO data is χ BAO = ( x thi,BAO − x obsi,BAO )( C − BAO ) ij ( x thj,BAO − x obsj,BAO ) , (3.13)where x i,BAO ≡ ( d . , d . ). C. Cosmic Microwave Background (CMB)
The CMB is sensitive to the distance to the decoupling epoch z ∗ [27]. It can give con-straints on the model in the high redshift regime ( z ∼ l A , l A ( z ∗ ) ≡ (1 + z ∗ ) πD A ( z ∗ ) r S ( z ∗ ) , (3.14)(ii) the shift parameter R [28], R ( z ∗ ) ≡ p Ω m H (1 + z ∗ ) D A ( z ∗ ) , (3.15)and (iii) the redshift of the decoupling epoch z ∗ . The fitting function of z ∗ is given by [29] z ∗ = 1048 (cid:2) . b h ) − . (cid:3) (cid:2) g (Ω m h ) g (cid:3) , (3.16)where g = 0 . b h ) − . . b h ) . , g = 0 . . b h ) . . (3.17)The χ of the CMB data is χ CMB = ( x thi,CMB − x obsi,CMB )( C − CMB ) ij ( x thj,CMB − x obsj,CMB ) , (3.18)where x i,CMB ≡ ( l A ( z ∗ ) , R ( z ∗ ) , z ∗ ) and C − CMB is the inverse covariance matrix. The datafrom Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP7) observations [21] leadto l A ( z ∗ ) = 302 . R ( z ∗ ) = 1 .
725 and z ∗ = 1091 . C − CMB = .
305 29 . − . .
698 6825 . − . − . − .
180 3 . . (3.19)7inally, the χ of all the observational data is χ = ˜ χ SN + χ BAO + χ CMB . (3.20)In our fitting process, we did not use the Markov chain Monte Carlo (MCMC) approachbecause the numerical calculation for each solution of f ( R ) theory is very time-consuming,and the necessary change to the code like CosmoMC [30] is very extensive with no obviousbenefit in our study of the exponential gravity. Therefore, we take the simple χ methodas our main fitting procedure. The ΛCDM result obtained from SNe Ia, BAO and CMBconstraints with this χ method is Ω m = 0 . +0 . − . , while that with the MCMC method isΩ m = 0 . +0 . − . [31]. We note that the fitting in Ref. [31] has also included the observa-tional constraints from the radial BAO and Hubble parameter H(z). TABLE I. The best-fit values of the matter density parameter Ω m (68% CL) and χ for theexponential gravity model with β = 2 , , m isobtained when β is fixed.Model Ω m χ β = 2 0 . +0 . − . β = 3 0 . +0 . − . β = 4 0 . +0 . − . . +0 . − . IV. RESULTS
Based on the methods described in Sec. III, we now examine the parameter space ofthe exponential gravity model. In Fig. 1, we present likelihood contour plots at 68.3, 95.4and 99.7% confidence levels obtained from the SNe Ia, BAO and CMB constraints. Theresults show that the observational data give no upper bound on the model parameter β ,making it a free parameter. Hence, there is no fine-tuning problem. However, a larger valueof β , which is closer to the ΛCDM model, is slightly preferred by the observational dataas expected. The lower bound on β is β > .
27 (95% CL). The present matter densityparameter Ω m is constrained to 0 . < Ω m < .
311 (95% CL), which agrees with the8 + W m Β FIG. 1. The 68.3%, 95.4% and 99.7% confidence intervals for the exponential gravity model,constrained by the SNe Ia, BAO, and CMB data. The best-fit point in this parameter region ismarked with a plus sign. current observations. The best-fit value (smallest χ ) in the parameter space between β = 1and 4 is χ = 545 . β = 4 and Ω m = 0 . m and χ for the model with β = 2 , , We only concentrate on the region of 1 < β <
4. For β >
4, it is almost the ΛCDM model. For β <
1, itis ruled out by the local gravity constraints and the stability of the de-Sitter phase.
9n Fig. 2, we illustrate the evolution of the effective dark energy equation of state w DE for β = 2 , , m , which is given in Table I. We can see that, for every valueof β , the effective dark energy equation of state w DE starts at the phase of a cosmologicalconstant w DE = − w DE < −
1) to the non-phantomphase ( w DE > − β , the deviation from cosmological constantphase ( w DE = −
1) become smaller. For β = 2, there is still another small oscillation afterthe main phantom phase crossing. Negative z means the future evolution. It is clear thatthe exponential gravity model has the feature of crossing the phantom phase in the past aswell as the future [32].In Fig. 3, we depict the effective dark energy density Ω DE and non-relativistic matterdensity Ω m vs. the redshift z . Β = 2 Β = 3 Β = 4 - - - - - - - - - w D E FIG. 2. Evolution of the effective dark energy equation of state w DE corresponding to β = 2 , , m given in Table I. DE W m - - FIG. 3. The evolutions of the effective dark energy density parameter Ω DE and non-relativisticmatter density parameter Ω m as functions of z , where the solid lines indicate the exponentialgravity model with β = 1 .
27 and the best-fit Ω m = 0 .
270 and the dashed lines represent theΛCDM model with Ω m = 0 . β , the evolution becomes closer to that inΛCDM. V. CONCLUSION
We have studied the exponential gravity model. In the low redshift regime, we follow Huand Sawicki’s parameterization to form the differential equation for the exponential gravityand solve it numerically. In the high redshift regime, we take advantage of the asymptoticbehavior of the exponential gravity toward an effective cosmological constant. The analyticalform of the Hubble parameter H as a function of the redshift z can be expressed in the highredshift limit. We have constrained the parameter space of the model by the SNe Ia, BAOand CMB data. We have found that there is a lower bound on the model parameter β at 1.27 but no upper limit, and Ω m is constrained to the concordance value. This means11hat the exponential gravity model shows no need of fine-tuning. Nevertheless, the ΛCDMmodel is still included by the observational constraints since β → ∞ corresponds to themodel. Current observational data still lack the ability to distinguish between the ΛCDMand exponential gravity models.Finally, we remark that as seen from Fig. 3, the noticeable difference between the ex-ponential gravity and ΛCDM models lies in the regime 0 . < z <
1, and is maximizedat z = 0 . f g ( z ) = d ln δ m /d ln a has the potential to distinguish the models with thesame expansion history but different physics. In the exponential gravity case, the growthindex is γ = 0 .
540 for β = 2. It is clear that if those surveys such as WiggleZ, EUCLID,BigBOSS and JDEM/Omega can measure the growth rate with a high accuracy, they willbe able to discriminate the exponential gravity from the ΛCDM model. ACKNOWLEDGMENTS
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