aa r X i v : . [ a s t r o - ph ] A ug The effect of modified gravity on weak lensing
Shinji Tsujikawa and Takayuki Tatekawa
2, 3 Department of Physics, Faculty of Science, Tokyo University of Science,1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan ∗ Department of Computer Science, Kogakuin University,1-24-2 Nishi-shinjuku, Shinjuku, Tokyo, 163-8677 Japan † Research Institute for Science and Engineering,Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo, 169-8555 Japan (Dated: October 25, 2018)We study the effect of modified gravity on weak lensing in a class of scalar-tensor theory thatincludes f ( R ) gravity as a special case. These models are designed to satisfy local gravity constraintsby having a large scalar-field mass in a region of high curvature. Matter density perturbations inthese models are enhanced at small redshifts because of the presence of a coupling Q that charac-terizes the strength between dark energy and non-relativistic matter. We compute a convergencepower spectrum of weak lensing numerically and show that the spectral index and the amplitude ofthe spectrum in the linear regime can be significantly modified compared to the ΛCDM model forlarge values of | Q | of the order of unity. Thus weak lensing provides a powerful tool to constrainsuch large coupling scalar-tensor models including f ( R ) gravity. I. INTRODUCTION
The observations of the Supernovae Ia (SN Ia) in 1998[1] opened up a new research paradigm known as DarkEnergy (DE). In spite of the tremendous effort over thepast ten years, we have not yet identified the origin ofDE responsible for the late-time accelerated expansion.Many DE models have been proposed so far to allevi-ate the theoretical problem of the cosmological constantscenario [2, 3]. We can broadly classify these modelsinto two classes: (i) “changing gravity” models and (ii)“changing matter” models. The first class includes f ( R )gravity [4], scalar-tensor models [5] and braneworld mod-els [6], whereas scalar-field models such as quintessence[7] and k-essence [8] are categorized in the second class.While changing matter models lead to dynamical evo-lution for the equation of state of DE, it is not easy todistinguish them from the cosmological constant scenarioin current observations. Meanwhile, if we change gravityfrom General Relativity, the models need to pass localgravity tests as well as cosmological constraints. In thissense it is possible to place stringent experimental andobservational constraints on changing gravity models.In fact there have been a burst of activities to searchfor viable modified gravity DE models. In the so-called f ( R ) gravity where f is a function of the Ricci scalar R ,it was found that the model f ( R ) = R − α/R n ( α, n > f ,RR ≡ d f / d R >
0) for perturbations [9],cosmological viability [10] and local gravity constraints(LGC) [11]. Recently a number of authors proposed vi-able f ( R ) DE models that satisfy all these requirements[12, 13, 14, 15, 16, 17, 18, 19, 20]. For example, the model ∗ Electronic address: [email protected] † Electronic address: [email protected] f ( R ) = R − αR n with α > , < n < n < − [21] while at the same time satisfyingstability and cosmological constraints. However it is diffi-cult to distinguish this model from the ΛCDM cosmologybecause of the tight bound on the power n coming fromLGC.The f ( R ) models proposed by Hu and Sawicki [15] andStarobinsky [16] are designed to satisfy LGC in the re-gion of high density where local gravity experiments arecarried out. Moreover it is possible to find an appre-ciable deviation from the ΛCDM model as the Universeevolves from the matter-dominated epoch to the late-time accelerated era. In fact the equation of state of DEin these models exhibits peculiar evolution at small red-shifts [14, 18]. In addition, for the redshift smaller thana critical value z k , the growth rate of matter density per-turbations is larger than in the case of General Relativity[16, 18].Recently the analysis in f ( R ) gravity was extended to aclass of scalar-tensor DE models, i.e., Brans-Dicke theorywith a scalar field potential V ( φ ) [22]. By introducing aconstant Q with the relation 1 / (2 Q ) = 3 + 2 ω BD ( ω BD is a Brans-Dicke parameter), one can reduce this the-ory to the one given by the action (2). The constant Q characterizes the coupling between dark energy and non-relativistic matter. If the scalar field φ is nearly massless,the coupling is constrained to be | Q | . − from solarsystem experiments [22]. However, if the field φ is mas-sive in the region of high density, it is possible to satisfyLGC even when | Q | is of the order of unity. In fact, in thecontext of f ( R ) gravity ( Q = − / √ R ≫ R ( R is the present cosmological Ricci scalar) andthat the mass becomes lighter as R approaches R . Forgeneral coupling Q , the potential given in Eq. (5) canbe compatible with both local gravity and cosmologicalconstraints.The scalar-tensor models mentioned above show devi-ations from the ΛCDM model at late times and hencethey can leave a number of interesting observational sig-natures. In Ref. [22] several bounds on the coupling Q and model parameters were derived by considering theevolution of matter density perturbations as well as LGC.It was found that there exists allowed parameter space ofmodel parameters even when | Q | is of the order of unity.In this paper we shall study the effect of such modifiedgravity models on weak lensing observations [23]. Sinceweak lensing carries the information of perturbations atlow redshifts, it is expected that this sheds light on reveal-ing the nature of DE [24, 25, 26, 27, 28, 29, 30, 31, 32].In Refs. [33, 34] a convergence power spectrum of weaklensing was derived in scalar-tensor theories with the La-grangian density L = F ( φ ) R/ − ( ∇ φ ) / − V ( φ ). Inthese theories a deflecting lensing potential Φ wl is modi-fied compared to General Relativity due to the differentevolution of gravitational potentials. This gives rise tothe change of the convergence power spectrum, whichprovides a powerful tool to distinguish modified gravityfrom the ΛCDM model.The lensing potential Φ wl is sourced by matter densityperturbations. The equation for matter perturbations inscalar-tensor models was derived in Ref. [35] under theapproximation on sub-horizon scales (see also Ref. [36]).This analysis can be generalized to the theories with theLagrangian density f ( R, φ, X ) (where X = − ( ∇ φ ) / wl was obtained analytically [37]. The DGPbraneworld model also leads to the modification to thelensing potential [38]. Thus the effect of modified gravitygenerally manifests itself in weak lensing observations.In this work we focus on scalar-tensor models (2) witha large coupling Q and evaluate the convergence powerspectrum to find signatures of the modification of gravityin weak lensing. This analysis is general in the sense that f ( R ) gravity is included as a special case. In Sec. II wereview our scalar-tensor models and present cosmologicalbackground equations to find dark energy dynamics. InSec. III we derive the form of the convergence power spec-trum as well as the equation for the deflecting potentialΦ wl . In Sec. IV we compute the convergence spectrumnumerically and estimate the effect of modified gravityon weak lensing. We conclude in Sec. V. II. MODIFIED GRAVITY MODELS
We start with the following action S = Z d x √− g (cid:20) χR − ω BD χ ( ∇ χ ) − V ( χ ) (cid:21) + S m ( g µν , Ψ m ) , (1)where χ is a scalar field coupled to the Ricci scalar R , ω BD is a constant parameter, V ( χ ) is a field potential,and S m is a matter action that depends on the metric g µν and matter fields Ψ m . The action (1) corresponds to Brans-Dicke theory [39] with a potential V ( χ ). In thefollowing we use the unit 8 πG = 1, but we restore thebare gravitational constant G when it is required.Setting χ = F = e − Qφ , where Q is a constant and φ = − / (2 Q ) ln χ is a new scalar field, we find that theaction (1) is equivalent to S = Z d x √− g (cid:20) F R −
12 (1 − Q ) F ( ∇ φ ) − V (cid:21) + S m ( g µν , Ψ m ) , (2)where Q is related with the Brans-Dicke parameter ω BD via the relation 1 / (2 Q ) = 3 + 2 ω BD [22]. The f ( R )gravity corresponds to the coupling Q = − / √
6, i.e., ω BD = 0 [40].In the absence of the potential V the Brans-Dickeparameter is constrained to be ω BD > . × fromsolar system experiments [41], which gives the bound | Q | < . × − . If the potential V is present, it ispossible to satisfy solar system constraints even when | Q | is of the order of unity by having a large mass in ahigh-curvature region. In the context of f ( R ) gravity,the following model is designed to satisfy LGC [18]: f ( R ) = R − µR c [1 − ( R/R c ) − n ] , (3)where µ , R c , n are positive constants, and R c is roughlyof the order of the present cosmological Ricci scalar R .Note that this satisfies the stability condition f ,RR > R ≥ R ( R is a Ricci scalar at a late-time de-Sitterpoint) unlike the model f ( R ) = R − α/R n ( α, n >
0) [16,18]. In the limit R ≫ R c the above model approaches theΛCDM model, which allows a possibility to be consistentwith LGC in the region of high density.In fact, the model (3) satisfies LGC for n > . V = ( RF − f ) / φ = ( √ /
2) ln F . The field potential inthis case is given by V ( φ ) = µR c (cid:20) − n + 1(2 nµ ) n/ (2 n +1) (1 − e φ/ √ ) n n +1 (cid:21) . (4)The models proposed by Hu and Sawicki [15] and byStarobinsky [16] reduce to this form of the potentialin the high-curvature region ( R ≫ R c ) where localgravity experiments are carried out. When R ≫ R c the field φ is almost frozen at instantaneous minimaaround φ = 0 characterized by the condition e φ/ √ =1 − nµ ( R/R c ) − (2 n +1) with a large mass squared M ≡ V ,φφ ∝ φ − n +22 n +1 . These minima are sustained by an effec-tive coupling Q between non-relativistic matter and thefield φ [21].For arbitrary coupling Q with the action (2), one canalso construct viable models by generalizing the analysisof f ( R ) gravity. An explicit example of the potentialconsistent with LGC is given by [22] V ( φ ) = V (cid:2) − C (1 − e − Qφ ) p (cid:3) , (5)where V > , C > , < p <
1. This is motivatedby the potential (4), which means that the f ( R ) model(3) is recovered by setting p = 2 n/ (2 n + 1). The anal-ysis using the potential (5) with the action (2) is suffi-ciently general to understand essential features of mod-ifed gravity models that satisfy local gravity and cosmo-logical constraints. As p gets closer to 1, the field mass inthe region of high-curvature tends to be heavier so thatthe models are consistent with LGC. In Ref. [22] it wasfound that the constraints coming from solar system testsand the violation of equivalence principle give the bounds p > − / (9 . − ln | Q | ) and p > − / (13 . − ln | Q | ),respectively. In f ( R ) gravity with the potential (4) thesebounds translate into the conditions n > . n > . s = − d t + a ( t )d x , where t is cosmic time and a ( t ) is the scalefactor. As a source term for the matter action S m , wetake into account a non-relativistic fluid with energy den-sity ρ m and a radiation with energy density ρ rad . Theseobey the usual conservation equations ˙ ρ m + 3 Hρ m = 0and ˙ ρ rad + 4 Hρ rad = 0, where H ≡ ˙ a/a . The variation ofthe action (2) leads to the following equations of motion:3 F H = 12 (1 − Q ) F ˙ φ + V − H ˙ F + ρ m + ρ rad , (6)2 F ˙ H = − (1 − Q ) F ˙ φ − ¨ F + H ˙ F − ρ m − ρ rad , (7)(1 − Q ) F ¨ φ + 3 H ˙ φ + ˙ F F ˙ φ ! + V ,φ + QF R = 0 , (8)where R = 6(2 H + ˙ H ).In order to solve the background equations (6)-(8) nu-merically, we introduce the dimensionless variables x = ˙ φ √ H , x = 1 H r V F , x = 1 H r ρ rad F . (9)We also defineΩ DE ≡ (1 − Q ) x + x + 2 √ Qx , (10)Ω rad ≡ x , (11)Ω m ≡ − (1 − Q ) x − x − √ Qx − x , (12)which satisfy the relation Ω DE + Ω rad + Ω m = 1. UsingEqs. (7) and (8) we find˙ HH = − − Q (cid:20) x − x + x − Q x +2 √ Qx (cid:21) + 3 Q ( λx − Q ) , (13)where λ = − V ,φ /V . For the potential (5) we have λ = 2 C p Qe − Qφ (1 − e − Qφ ) p − − C (1 − e − Qφ ) p . (14) The effective equation of the system is defined by w eff ≡ − − H/ (3 H ) . (15)Using Eqs. (6)-(8), we obtain the following equationsd x d N = √
62 ( λx − √ x ) + √ Q (cid:20) (5 − Q ) x +2 √ Qx − x + x − (cid:21) − x ˙ HH , (16)d x d N = √
62 (2 Q − λ ) x x − x ˙ HH , (17)d x d N = √ Qx x − x − x ˙ HH , (18)where N ≡ ln ( a ) is the number of e-foldings. We notethat the variable F satisfies the equation of motion:d F/ d N = − √ Qx F .There exists a radiation fixed point: ( x , x , x ) =(0 , ,
1) for this system. During radiation and mattereras, the field φ is stuck around the “instantaneous” min-ima characterized by the condition V ,φ + QF R = 0, i.e.,2 Qφ m ≃ (cid:18) V pCρ m (cid:19) − p ≪ , (19)where we used the fact that V is of the order of thesquared of the present Hubble parameter H so thatthe potential (5) is responsible for the accelerated ex-pansion today. Note that we have F = e − Qφ m ≃ | λ | defined in Eq. (14) is much larger than unity.The field value | φ m | increases as the system enters theepoch of an accelerated expansion, which leads to thedecrease of | λ | . The matter-dominated epoch is real-ized by the instantaneous fixed point characterized by( x , x , x ) = ( √ / (2 λ ) , [(3+2 Qλ − Q ) / λ ] / ,
0) withΩ m = 1 − (3 − Q +7 Qλ ) /λ ≃ w eff = − Q/λ ≃ | λ | ≫ Q there exists a de-Sitter point character-ized by ( x , x , x ) = (0 , , m = 0 and w eff = − λ = 4 Q . This solution is stable ford λ/ d φ < M = V ,φφ , is given by M = 4 V CpQ (1 − pe − Qφ )(1 − e − Qφ ) p − e − Qφ . (20)Plugging the field value φ m into Eq. (20), we find M ≃ − p (2 p pC ) / (1 − p ) Q (cid:18) ρ m V (cid:19) − p − p V . (21)Since the energy density ρ m is much larger than V during the radiation and matter eras, we have that M ≫ V ∼ H . The mass squared M decreases to theorder of V after the system enters the accelerated epoch.This evolution of the field mass leads to an interesting ob-servational signature in weak lensing observations, as wewill see in subsequent sections. III. WEAK LENSING
Let us consider a perturbed metric about the flatFLRW background in the longitudinal gauge:d s = − (1 + 2Φ)d t + a ( t )(1 − δ ij d x i d x j , (22)where scalar metric perturbations Φ and Ψ do not co-incide with each other in the absence of an anisotropicstress. Matter density perturbations δ m in the pressure-less matter contribute to the source term for the gravi-tational potentials Φ and Ψ. The equation of δ m for theaction (2) was derived in Ref. [22] under an approxima-tion on sub-horizon scales [3, 35, 37]. Provided that theoscillating mode of the field perturbation δφ does notdominate over the matter-induced mode at the initialstage of the matter era, we obtain the following approx-imate equation¨ δ m + 2 H ˙ δ m − πG eff ρ m δ m ≃ , (23)where the effective gravitational “constant” is given by G eff = 18 πF ( k /a )(1 + 2 Q ) F + M ( k /a ) F + M . (24)Here k is a comoving wavenumber and M is given inEq. (20) for the potential (5). Using the derivative withrespect to N , Eq. (23) can be written asd δ m d N + (cid:18) − w eff (cid:19) d δ m d N −
32 Ω m ( k /a )(1 + 2 Q ) F + M ( k /a ) F + M δ m ≃ . (25)The gravitational potentials Φ and Ψ satisfy k a Φ ≃ − ρ m F ( k /a )(1 + 2 Q ) F + M ( k /a ) F + M δ m , (26) k a Ψ ≃ − ρ m F ( k /a )(1 − Q ) F + M ( k /a ) F + M δ m . (27)In order to confront our model with weak lensing obser-vations, we define the so-called deflecting potential [34]Φ wl ≡ Φ + Ψ , (28)together with the effective density field δ eff ≡ − a H Ω m, k Φ wl , (29)where the subscript “0” represents the present values andwe set a = 1. Using the relation ρ m = 3 F H Ω m, /a , (30)together with Eqs. (26) and (27), we getΦ wl = − a k ρ m F δ m , δ eff = F F δ m . (31) We write the angular position of a source to be ~θ S andthe direction of weak lensing observation to be ~θ I . Thedeformation of the shape of galaxies is characterized bythe amplification matrix A = d ~θ S / d ~θ I . The componentsof A are given by [23, 34] A µν = I µν − Z χ χ ′ ( χ − χ ′ ) χ ∂ µν Φ wl [ χ ′ ~θ, χ ′ ]d χ ′ , (32)where χ is the comoving radial distance satisfying therelation d χ = − d t/a ( t ) along the geodesic. In terms ofthe redshift defined by z = 1 /a −
1, we have that χ ( z ) = Z z d z ′ H ( z ′ ) . (33)The convergence κ and the shear ~γ = ( γ , γ ) can bederived from the components of the 2 × A , as κ = 1 −
12 Tr A , ~γ = ([ A − A ] / , A ) . (34)If we consider a redshift distribution p ( χ )d χ of the source,the convergence is given by κ ( ~θ ) = R p ( χ ) κ ( ~θ, χ )d χ . Us-ing Eqs. (29), (32) and (34) we obtain κ ( ~θ ) = 32 H Ω m, Z χ H g ( χ ) χ δ eff [ χ ~θ, χ ] a d χ , (35)where χ H is the maximum distance to the source and g ( χ ) ≡ Z χ H χ p ( χ ′ ) χ ′ − χχ ′ d χ ′ . (36)Since the convergence is a function on the 2-sphere itcan be expanded in the form κ ( ~θ ) = R ˆ κ ( ~ℓ ) e i~ℓ · ~θ d ~ℓ π , where ~ℓ = ( ℓ , ℓ ) with ℓ and ℓ integers. Defining the powerspectrum of the shear to be h ˆ κ ( ~ℓ )ˆ κ ∗ ( ~ℓ ′ ) i = P κ ( ~ℓ ) δ (2) ( ~ℓ − ~ℓ ′ ), one can show that the convergence has a same powerspectrum as P κ [23]. It is given by [34] P κ ( ℓ ) = 9 H Ω m, Z χ H (cid:20) g ( χ ) a ( χ ) (cid:21) P δ eff (cid:20) ℓχ , χ (cid:21) d χ . (37)In our scalar-tensor theory we have P δ eff =( F /F ) P δ m from Eq. (31), where P δ m is the matterpower spectrum. In the following we assume that thesources are located at the distance χ s (corresponding tothe redshift z s ), which then gives p ( χ ) = δ ( χ − χ s ) and g ( χ ) = ( χ s − χ ) /χ s . This leads to the following conver-gence spectrum P κ ( ℓ ) = 9 H Ω m, Z χ s (cid:18) χ s − χχ s a F F (cid:19) P δ m (cid:20) ℓχ , χ (cid:21) d χ. (38)Let us consider the action (2) with the potential (5).In the deep matter era where the Ricci scalar R is muchlarger than H , we have M /F ≫ k /a and F ≃ k relevant to the matter power spectrum[22]. Since G eff ≃ G in this regime from Eq. (24), theperturbations evolve in a standard way: δ m ∝ t / andΦ wl = constant. Meanwhile, at the late epoch of thematter era, the system can enter a stage characterizedby the condition M /F ≪ k /a . Since G eff ≃ (1 +2 Q ) / πF during this stage, the perturbations evolve ina non-standard way: δ m ∝ t ( √ Q − / , Φ wl ∝ t ( √ Q − / . (39)The critical redshift z k at M /F = k /a can be esti-mated as z k ≃ "(cid:18) k H Q (1 − p ) (cid:19) − p p pC (3 F Ω m, ) − p V H − p − . (40)As long as z k & z ≫ z k , the deflecting potential Φ wl at latetimes is given by [43]Φ wl ( k, a ) = 910 Φ wl ( k, a i ) T ( k ) D ( k, a ) a , (41)where Φ wl ( k, a i ) ≃ k, a i ) corresponds to the initialdeflecting potential generated during inflation, T ( k ) isa transfer function that describes the epochs of horizoncrossing and radiation/matter transition (50 . z . ),and D ( k, a ) is the growth function at late times definedby D ( k, a ) /a = Φ wl ( a ) / Φ wl ( a I ) ( a I corresponds to thescale factor at a redshift 1 ≪ z I < z k is smaller than 50, we can use the standardtransfer function of Bardeen et al. [44]: T ( x ) = ln(1 + 0 . x )0 . x (cid:20) . . x + (1 . x ) +(0 . x ) + (0 . x ) (cid:21) − . , (42)where x ≡ k/k EQ and k EQ = 0 .
073 Ω m, h Mpc − .In the ΛCDM model the growth function during thematter-dominated epoch (Ω m = 1) is scale-independent: D ( k, a ) = a . In our scalar-tensor model the masssquared M given in Eq. (21) evolves as M ∝ t − − p ) / (1 − p ) , which implies that the transition time t k at M /F = k /a has a scale-dependence t k ∝ k − − p )4 − p [22]. This leads to the scale-dependent growth of metricperturbations. Note that in the late-time accelerated epoch the growth of matterpertubations is no longer described by D ( a ) = a . Using Eqs. (29) and (41) we obtain the matter pertur-bation δ m at the redshift z < z I : δ m ( k, a ) = − FF k Ω m, H Φ wl ( k, a i ) T ( k ) D ( k, a ) . (43)The initial power spectrum generated during inflation is P Φ wl ≡ | Φ | = (200 π / k )( k/H ) n s − δ H , where n s isthe spectral index and δ H is the amplitude of Φ wl . Thenthe power spectrum, P δ m ≡ | δ m | , is given by P δ m ( k, a ) = 2 π (cid:18) FF (cid:19) k n s Ω m, H n s +30 δ H T ( k ) D ( k, a ) . (44)From Eqs. (38) and (44) we get P κ ( ℓ ) = 9 π Z z s (cid:18) − XX s (cid:19) E ( z ) δ H × (cid:18) ℓX (cid:19) n s T ( x ) (cid:18) Φ wl ( z )Φ wl ( z I ) (cid:19) d z , (45)where E ( z ) = H ( z ) H , X = H χ , x = H k EQ ℓX . (46)From Eq. (33) the quantity X satisfies the differentialequation d X/ d z = 1 /E ( z ). In the following we use thevalue z s = 1 in our numerical simulations. IV. OBSERVATIONAL SIGNATURES OFMODIFIED GRAVITY
When Q = 0 the evolution of δ m during the time-interval t k < t < t Λ (where t Λ is the time at ¨ a = 0)is given by Eq. (39), whereas δ m ∝ t / in the ΛCDMmodel ( Q = 0). Hence, at time t Λ , the power spectrumfor Q = 0 exhibits a difference compared to the ΛCDMmodel [22]: P δ m ( t Λ ) P ΛCDM δ m ( t Λ ) = (cid:18) t Λ t k (cid:19) „ √ Q − − « ∝ k ∆ n ( t Λ ) , (47)where ∆ n ( t Λ ) = (1 − p )( p
25 + 48 Q − − p . (48)In order to derive the difference ∆ n ( t ) at the presentepoch, we need to solve perturbation equations numer-ically by the time t . However, as long as z k is largerthan the order of unity, the growth rate of δ m during thetime-interval t Λ < t < t hardly depends on k for fixed Q . Hence it is expected that the analytic estimation (48)does not differ much from ∆ n ( t ) provided z k ≫ P δ m [ h - M p c ] k [h Mpc -1 ] (a) Q=0.7, p=0.6(b) Q=-(1/6) , p=0.6(c) Λ CDM, linear(d) Λ CDM, nonlinear
Figure 1: The matter power spectra P δ m ( k ) at the presentepoch for (a) Q = 0 . p = 0 . C = 0 .
9, (b) Q = − / √ p = 0 . C = 0 .
9, (c) the ΛCDM model, and (d) the ΛCDMmodel with a nonlinear halo-fitting ( σ = 0 .
78 and shapeparameter Γ = 0 . m, = 0 . H = 3 . × − h Mpc − , n s = 1 and δ H = 3 . × − . Inthe cases (a) and (b) we start integrating Eqs. (16)-(18) withinitial conditions ( x , x , x ) = (0 , [(3+2 Qλ − Q ) / λ ] / , F − − − . epoch by the condition Ω m = 0 .
28. We then run thecode again from z = z I ( <
50) to z = 0 in order to solvethe perturbation equations (25) and (31). Since we areconsidering the case in which z k is smaller than z I , theinitial conditions for matter perturbations are chosen tobe d δ m d N = δ m (i.e., those for the ΛCDM model).In Fig. 1 we plot the matter power spectra at thepresent epoch for (a) Q = 0 . p = 0 . C = 0 .
9, (b) Q = − / √ p = 0 . C = 0 .
9, (c) the ΛCDM model,and (d) the ΛCDM model with a nonlinear halo-fitting[45]. Since we do not take into account nonlinear effectsin the cases (a)-(c), these results are trustable in the lin-ear regime k . . h Mpc − .In the case (b), which corresponds to f ( R ) gravity with n = 0 .
75 in the model (3), the spectrum shows a devi-ation from the ΛCDM model for k > . h Mpc − . Onthe scales k = 0 . h Mpc − and k = 0 . h Mpc − the crit-ical redshifts at M /F = k /a are given by z k = 2 . z k = 5 . n ( t ) = 0 .
017 and ∆ n ( t ) = 0 .
119 for k = 0 . h Mpc − and k = 0 . h Mpc − respectively, whereas the estimation(48) gives the value ∆ n ( t Λ ) = 0 . z k decreasesfor smaller k , the analytic estimation (39) obtained byusing the condition z k ≫ n ( t ) and ∆ n ( t Λ ) found for k < . h Mpc − .We checked that ∆ n ( t ) approaches the analytic value∆ n ( t Λ ) = 0 .
088 on smaller scales, e.g., ∆ n ( t ) = 0 . -9 -8 -7
10 100 P κ l (a) p=0.5, C=0.9(b) p=0.7, C=0.9(c) Λ CDM
Figure 2: The convergence power spectrum P κ ( ℓ ) in f ( R )gravity ( Q = − / √
6) for the cases: (a) p = 0 . C = 0 . p = 0 . C = 0 .
9. We also show the spectrum in theΛCDM model. Other model parameters are chosen similarlyas in the case of Fig. 1. for k = 4 . h Mpc − .For larger | Q | the growth rate of δ m increases in theregime z Λ < z < z k , which alters the shape of the matterpower spectrum. In the case (a) of Fig. 1 we numericallyfind that ∆ n ( t ) = 0 .
323 on the scale k = 0 . h Mpc − ,while the estimation (48) gives ∆ n ( t Λ ) = 0 . n ( t ) on smaller scales, e.g., ∆ n ( t ) = 0 .
244 for k =4 . h Mpc − . In Fig. 1 we also show the matter powerspectrum in the ΛCDM model derived by using the non-linear halo-fit [45]. This gives rise to an enhancementof the power in the nonlinear regime ( k > . h Mpc − ).The spectrum in the case (a) exhibits a significant differ-ence compared to this halo-fit ΛCDM spectrum even for k < . h Mpc − , which implies that our linear analysis isenough to place stringent constraints on model parame-ters Q and p from observations of galaxy clustering.Let us next proceed to the convergence power spectrumof weak lensing. Compared to the matter power spec-trum the wavenumber k is replaced by k = ℓ/χ . In thedeep matter era the evolution of the Hubble parametercan be approximated as H ( z ) ≃ H Ω m, (1 + z ) , whichgives χ ≃ / ( H Ω / m, ) = constant. Hence the time t ℓ at M /F = ( ℓ/χ ) /a has an ℓ -dependence t ℓ ∝ ℓ − − p )4 − p ,provided this transition occurs at the redshift z ℓ ≫ wl ≃ constant for t I < t < t ℓ and Φ wl ∝ t ( √ Q − / for t ℓ < t < t Λ , we have thatΦ wl ( z Λ )Φ wl ( z I ) ≃ (cid:18) t Λ t ℓ (cid:19) ( √ Q − / . (49) -9 -8 -7
10 100 P κ l (a) Q=1, C=0.9(b) Q=0.5, C=0.9(c) Λ CDM
Figure 3: The convergence power spectrum P κ ( ℓ ) for p = 0 . Q = 1, C = 0 . Q = 0 . C = 0 . As long as z ℓ ≫
1, the evolution of Φ wl during the time-interval t Λ < t < t is almost independent of ℓ for a fixedvalue of Q . Then we obtain the following ℓ -dependencefor 0 < z < z Λ ∼ z s : (cid:18) Φ wl ( z )Φ wl ( z I ) (cid:19) ∝ ℓ (1 − p )( √ Q − − p . (50)From Eq. (45) this leads to a difference of the spectral in-dex of the convergence spectrum compared to the ΛCDMmodel: P κ ( ℓ ) P ΛCDM κ ( ℓ ) ∝ ℓ ∆ n , (51)where ∆ n is the same as ∆ n ( t Λ ) given in Eq. (48). Wecaution again that the estimation (51) is valid for z ℓ ≫ f ( R )gravity for two different values of p together with theΛCDM spectrum. We focus on the linear regime charac-terized by ℓ . n → ∞ in Eq. (3), the power p = 2 n/ (2 n + 1)approaches 1 in this limit. The deviation from the ΛCDMmodel becomes important for smaller p away from 1.When p = 0 .
7, for example, Fig. 2 shows that such adeviation becomes significant for ℓ &
10. Numerically weget ∆ n = 0 .
056 at ℓ = 200, which is slightly smaller thanthe analytic value ∆ n = 0 .
068 estimated by Eq. (51).The main reason for this difference is that the criticalredshift z ℓ = 3 .
258 at ℓ = 200 is not very much largerthan unity.When p = 0 . wl is amplifiedeven for small ℓ ( . z ℓ is greater than 1 even for ℓ >
2. For example we find that z ℓ = 1 .
386 for ℓ = 5. In this case the systementers the non-standard regime ( z < z ℓ ) before enteringthe epoch of an accelerated expansion ( z < z Λ ∼ wl . This changes thetotal amplitude of P κ ( ℓ ) relative to the ΛCDM model.The numerical value of ∆ n at ℓ = 200 is found to be∆ n = 0 .
084 for p = 0 .
5. Since ∆ n increases for smaller p , this information is useful to place a lower bound on p in f ( R ) gravity from weak lensing observations.In Fig. 3 the convergence spectrum for p = 0 . Q together with the ΛCDMspectrum. We note that the transition redshift z ℓ de-creases for larger | Q | , see Eq. (40). Hence the deviationfrom the ΛCDM model is insignificant for small ℓ , unlesswe choose smaller values of p . However the spectrum isstrongly modified for ℓ &
10 with the increase of | Q | .The numerical values of ∆ n at ℓ = 200 are found to be∆ n = 0 .
084 and ∆ n = 0 .
311 for Q = 0 . Q = 1,respectively. Hence it should be possible to derive an up-per bound on the strength of the coupling Q by usingobservational data of weak lensing. V. CONCLUSIONS
We have discussed the signature of modified gravityin weak lensing observations. Our model is describedby the action (2) with a constant coupling Q , which isequivalent to Brans-Dicke theory with a field potential V . This theory includes f ( R ) gravity as a special case( Q = − / √ V ( φ ) can bedesigned to satisfy local gravity constraints through achameleon mechanism. The representative potential thatsatisfies LGC is given in Eq. (5), which is motivated byviable f ( R ) models proposed by Hu and Sawicki [15] andby Starobinsky [16]. Note that most of past works inscalar-tensor dark energy models restricted the analysisin the small coupling region ( | Q | . − ). In this paperwe focused on the large | Q | region in which a signifi-cant difference from the ΛCDM model can be expectedin weak lensing observations.Cosmologically these models can show deviations fromthe ΛCDM model at late epochs of the matter-dominatedera. The growth rate of matter density perturbations getslarger for redshifts smaller than a critical value z k . Since z k increases for larger k , the matter power spectrum issubject to change on smaller scales. We evaluated thematter power spectrum P δ m ( k ) numerically and showedthat the spectral index and the amplitude of P δ m ( k ) canbe significantly modified for larger values of | Q | .The non-standard evolution of matter perturbationsaffects the convergence power spectrum P κ ( ℓ ) of weaklensing. As long as the transition redshift z ℓ is largerthan the order of unity, one can estimate the difference∆ n of spectral indices between modified gravity and theΛCDM cosmology to be ∆ n ≃ (1 − p )( p
25 + 48 Q − / (4 − p ) with 0 < p <
1. In f ( R ) gravity the parameter n for the model (3) is linked with the parameter p viathe relation p = 2 n/ (2 n + 1). The limit p → n → ∞ ) corresponds to the ΛCDM model, in which casewe have ∆ n →
0. The difference of the convergencespectrum relative to the ΛCDM case is significant for p away from 1. As seen in Fig. 2 (which corresponds to thecase Q = − / √ P κ ( ℓ ) are modified for smaller values of p .If we take larger values of | Q | , the convergence spec-trum deviates from that in the ΛCDM model more sig-nificantly. This situation is clearly seen in the numericalsimulation of Fig. 3. It should be possible to place strongobservational constraints on the parameters Q and p by using observational data of weak lensing and the mat-ter power spectrum, which we leave for future work. Wehope that some signatures of modified gravity can be de-tected in future high-precision observations to reveal theorigin of dark energy. ACKNOWLEDGEMENTS
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