Newton law corrections and instabilities in $f(R)$ gravity with the effective cosmological constant epoch

We consider class of modified $f(R)$ gravities with the effective cosmological constant epoch at the early and late universe. Such models pass most of solar system tests as well they satisfy to cosmological bounds. Despite their very attractive properties, it is shown that one realistic class of such models may lead to significant Newton law corrections which become small at the future universe only. The model with acceptable Newton law regime shows the matter instability. This suggests that more complicated version of such theory (or extended parameters space) should be investigated.


II. THE NEWTON LAW CORRECTIONS IN F(R) GRAVITY WITH AN EFFECTIVE COSMOLOGICAL CONSTANT EPOCH
The action of general f (R) gravity (for a review, see [1]) is given by Here f (R) is an arbitrary function. The general equation of motion in f (R)-gravity with matter is given by Here F (R) = R + f (R) and T (m)µν is the matter energy-momentum tensor. By introducing the auxilliary field A one may rewrite the action (1) in the following form: From the equation of motion with respect to A, it follows A = R. By using the scale transformation g µν → e σ g µν with σ = − ln (1 + f ′ (A)), we obtain the Einstein frame action [4]: Here g (e −σ ) is given by solving σ = − ln (1 + f ′ (A)) = ln F ′ (A) as A = g (e −σ ). After the scale transformation g µν → e σ g µν , there appears a coupling of the scalar field σ with the matter. For example, if the matter is the scalar field Φ with mass M , whose action is given by there appears a coupling with σ in the Einstein frame: The strength of the coupling is of the gravitational coupling κ order. Unless the mass of σ, which is defined by is large, there appears the large correction to the Newton law. More exactly, in the Einstein frame, matter fields give a source term for the scalar field σ like Here ρ is the energy density (in the Jordan frame). Now we consider the fluctuations from the background of σ = σ 0 (σ 0 is not always a constant): For simplicity, we consider the limit where the spacetime is almost flat and consider the point like souces Then by the propagation of δσ, we find the following correlation function e a (1) σ(x (1) ) e a (1) σ(x (2) ) ∼ e (a (1) +a (2) )σ0+a (1) a (2) Gσ(x1,x2) .
Recently very interesting f (R) model has been proposed by Hu and Sawicki [11]. In the model f (R) is given by which satisfies the condition The estimation of ref. [11] suggests that R/m 2 is not so small but rather large even in the present universe and R/m 2 ∼ 41. Then we have which gives an "effective" cosmological constant −m 2 c 1 /c 2 . The effective cosmological constant generates the accelerating expansion in the present universe. Then In the intermediate epoch, where the matter density ρ is larger than the effective cosmological constant, there appears the matter dominated phase (such phase may occur for other modified f (R) gravity as well [7,8]) and the universe could expand with deceleration. Hence, above model leads to the effective ΛCDM cosmology like models [7,9]. Some remark is in order. The approximate expression for the Hu-Sawicky model should be taken with great care. The reason is that at very small curvatures where the (non-perturbative) function f (R) goes to zero, the approximation breaks down (the corresponding function f may become singular).
Due to the scalar field in (4), an extra (fifth) force could manifest itself. It could violate the Newton law. The Newton law is well understood and its correction should be very small at least in the present universe. If the mass of σ is large enough in the present universe, the problem could be avoided. We now investigate the model by assuming A/m 2 = R/m 2 ≫ 1 since R/m 2 ∼ 41 even in the present universe. Then one gets and (22) First we consider the universe at very large scales, where R ∼ 10 −33 eV −2 and therefore R/m 2 ∼ 41. If c 1 is not so small and/or n is not so large, R/m 2 ∼ 41, we find m σ should be very small m σ ∼ 10 −33 eV. Therefore, the correction to the Newton law is large. Note that σ 0 ∼ 0 in (15) for the model [11]. Since a 1,2 ∼ 1, the correction to the Newton law could be not so small.
Although m σ could be very small at large scales since R 0 is very small, R 0 can be larger near or in the star. Since 1 g ∼ 6 × 10 32 eV and 1 cm ∼ 2 × 10 −5 eV −1 , the density is about ρ ∼ 1g/cm 3 ∼ 5 × 10 18 eV 4 inside the earth. This shows that the magnitude of the curvature could be R 0 ∼ κ 2 ρ ∼ 10 −19 eV 2 and therefore R 0 /m 2 ∼ 10 28 . Hence, in case n = 2, we find m σ ∼ 10 19 GeV, which is very large and the correction to the Newton law is very small. Even in air, one finds ρ ∼ 10 −6 g/cm 3 ∼ 10 12 eV 4 , which gives R 0 ∼ κ 2 ρ ∼ 10 −25 eV 2 and R 0 /m 2 ∼ 10 16 . In case n = 2, m σ ∼ 10 −1 eV, which gives a correlation length (Compton wave length) about 1 µm. Thus, the correction to the Newton law could not be observed on the earth for such a model. What happens in the solar system? In the solar system, there could be interstellar gas. Typically, in the interstellar gas, there is one proton (or hydrogen atom) per 1 cm 3 , which shows ρ ∼ 10 −5 eV 4 , R 0 ∼ 10 −61 eV 2 , and therefore R 0 /m 2 ∼ 10 4 . Then for n = 2, we find m σ ∼ 10 −25 eV, which corresponds to the correlation length of 10 18 m ∼ 100 pc. Then the correction to the Newton law could be observed. In case n = 8, however, we find m σ ∼ 10 −13 eV, which corresponds to the correlation length of 10 6 m, which is less than the radius of earth (∼ 10 7 m). Then the correction to the Newton law could not be observed. Hence, some sub-class of above theory may pass known solar sytem tests at the scales of the solar system order.
In (22), the Einstein frame was considered (4). However, similar conclusions may be made also in Jordan frame. By multipling (2) with g µν , one obtains Here T ≡ T ρ (m)ρ . The equation (23) corresponds to Eq.(39) in [11]. Now we consider the background where R is a constant R = R 0 , that is, (anti-)de Sitter space which can be obtained by solving the algebraic equation Since e −σ = F ′ (R), one gets Consider the fluctuation which leads to One may consider the point source Then the solution of (27) is given by Here If m 2 < 0, there appears tachyon and there could be some instability. Even if m 2 > 0, when m 2 is small, δR = 0 at long ranges, which generates the large correction to the Newton law. In case of [11], we find, when R/m 2 ≫ 1 as in the present universe, Compared this expression (31) with (22) by putting A = R 0 , we find m 2 ∼ m 2 σ . Then the correction to the Newton law is the same.
In [11], it is assumed R/m 2 ≫ 1 but it might be interesting to study the model assuming R/m 2 ≪ 1, which may correspond to the future universe. When A/m 2 = R/m 2 ≪ 1, the potential V (σ) (4) is given by and we find σ ∼ − ln 1 − nc 1 A/m 2 n−1 .
Let us consider the case n > 1 and 0 < n < 1 separately. In case n > 1, when A is small, (33) can be written as and therefore Then Note m σ > 0 if c 1 > 0. Eq.(34) shows that σ is small when A/m 2 is small. Then the mass m σ becomes large when the curvature R ∼ A. Therefore the scalar field does not propagate at large ranges and the Newton law could not be violated.
In order that the potential being real, c 1 should be negative. Since the squared mass m 2 σ is negative since c 1 < 0, which shows that σ is tachyon and unstable. Tachyon is inconsistent with quantum theory. Classically if we consider the perturbation with respect to σ, the perturbation becomes large. Since σ is related with the curvature by σ = − ln F ′ (A) = − ln F ′ (R), the instability may indicate the solution where by the perturbation, the curvature of the universe could become large.
Hence, it seems there may be significant correction to Newton law in the f (R) gravity model under consideration at cosmological scales. It is remarkable that such correction becomes negligible in the future, at least, for some range of parameters.

III. THE ABSENCE OF MATTER INSTABILITY
There may exist another type of instability (so-called matter instability) in f (R) gravity [10]. The example of the model without such instability is given in [4] (for related discussions of matter instability, see, [13]). Let us show that current and related models are free from such instability. The instability might occur when the curvature is rather large, as on the planet, compared with the average curvature in the universe R ∼ 10 −33 eV 2 . By multipling Eq. (2) with g µν , one obtains Here T ≡ T ρ (m)ρ . We consider a perturbation from the solution of the Einstein gravity: Note that T is negative since |p| ≪ ρ on the earth and T = −ρ + 3p ∼ −ρ. Then we assume Now one can get If U (R 0 ) is positive, since R 1 ∼ −∂ 2 t R 1 , the perturbation R 1 is exponentially large and the system becomes unstable. One may regard ∇ ρ R 0 ∼ 0 if it is assumed the matter is almost uniform as inside the earth.
For the model (16), by assuming R 0 /m 2 ≫ 1, it follows which is large and negative if c 1 > 0. Hence, there is no instability in the sense of ref. [10]. When c 1 < 0, however, there could be an instability. In first ref. of [13], a simple condition for the stability in a sense of [10] was given, that Then F ′′ (R 0 ) ∼ −1/U (R 0 ) > 0 if c 1 is positive and theory is not stable. As one more example satisfing the conditions (17), we now consider The asymptotic behaviors of (46) are identical with the model (16) when R is large, and when R is small Then asymptotic behaviors of the universe does not change and the correction to the Newton law could be large when R is large and small when R is small. The instability is also absent, as one can reobtain the results identical with (32-44).
Another example is with a positive constants f 0 and m 4 . As in the model (16) in [11], we may assume R/m 2 ≫ 1 from the early universe to the present universe. Even in the model (47), we assume R 2 ≫m 4 . Then by expanding f B (R) with respect to m 4 /R 2 , we find By comparing (50) with (18), we may identify Hence, f 0 plays the role of the cosmological constant if f 0 > 0 Thus, the accelerated expansion of the present universe could be generated by the effective cosmological constant f 0 . As in (20), in the earlier but not primordial universe, the matter density ρ is larger than the effective cosmological constant f 0 . Hence, there occurs the matter dominated phase and the universe could have expanded with deceleration. The aymptotic behavior when the curvature is large is identical with the model (16), the correction to the Newton law could be not so small. We now investigate also the case that the curvature is small. Then for the model (47), we obtain and If f 0 is positive, m σ ≡ (1/2)(d 2 V /dσ 2 ) is positive and large when the curvature R = A is small and therefore there is no large corrrection to the Newton law. We should note, however, if f 0 is negative, which corresponds to the model in (16) m 2 σ becomes negative and there could occur an instability. On the other hand, when the curvature is large, U (R 0 ) in (43) has the following form: which is negative and large and therefore there is no instability. In fact, since the condition from first ref. of [13] is satisfied.
Recently another interesting f (R) model was proposed in [12], where with positive constants a and b (for first f (R) models with log-terms, see first ref. in [6]).
Since the correction to the Newton law has been studied in [12], we now investigate the possible instability for the model (57). Since it is positive. Then the condition [13] seems to be satisfied and therefore theory seems to be consistent. When the curvature R is large, one finds Then U (R 0 ) (43) has the following form: If 1 + 2 ln ((1 − tanh(b)) /2) > 0, U (R 0 ) is very large and negative and therefore there is no instability. In [12], b is choosen to be b 1.2, so 1 + 2 ln 1 − tanh(1.2) 2 = −3.97 < 0 , and therefore the matter instability seems to occur. This indicates that such model should be considered in the other range of parameters.

IV. DISCUSSION
In the present letter we considered some solar system tests for several modified gravities which satisfy to conditions (16). These theories show very realistic cosmological behaviour and may easily lead to ΛCDM cosmology. It is shown that the theory (15) passes known solar system tests as well as cosmological bounds. Signficant Newton law corrections appear only beyond the solar system scales as well as for specific values of curvature power which puts some bound for such theory. Theory (59) which has an acceptable Newton law regime shows the matter instability in the proposed range of the parameters. Thus, the suggested class of models seems to be very realistic and looks like the alternative for ΛCDM. More accurate and detailed check of cosmological bounds for such theories should be done but in any case it is expected that some (combination/extension) of such theories may fit with observable cosmological data.