In the previous section, we found that somef(R) theories do not permit a Taylor expansion around their background value and others have large mass scales. Our analysis is not applicable to these
1This section contains previously unpublished work by the author.
theories and consequently, they may evade Solar System tests. Their interest, however, is limited, because theories of these types have not been shown to produce late-time cosmic acceleration, which was the motivation forf(R) gravity. Now we will consider theories that fail the condition specified by Eq. (3.57), which we will refer to as the linearity condition. This condition bears a strong resemblance to the “thin-shell” condition of chameleon gravity [133, 134], so it may hold the key to designing f(R) theories that cause cosmic acceleration and evade Solar System tests. Indeed, such models do exist [135, 79, 136, 137], and in this section we consider specifically thef(R) form proposed by Hu and Sawicki [79]:
f(R) =R−m2 c1(R/m2)n
c2(R/m2)n+ 1. (3.74)
In this expression,c1andc2are dimensionless constants,nis a positive number, andm2= 8πGρ0/3, whereρ0is the present-day matter density. This theory yields cosmic expansion that mimics a ΛCDM universe provided thatR≫m2 [79].
Before we examine how this theory violates the linearity condition and evades Solar System tests, we will briefly review the chameleon mechanism [133, 134], which provides a conceptual picture of the underlying physics. In chameleon gravity, there is a scalar fieldφwith a monotonically decreasing potential V(φ) that couples to matter and generates a fifth force. This fifth force is suppressed because the scalar field’s dynamics are governed by an effective potential that is density-dependent:
Veff(φ) =V(φ) +ρexp βφ
mPl
, (3.75)
where β is a coupling parameter between matter and the scalar field, ρis the density of the envi- ronment, and m2Pl = G−1. Since V(φ) is monotonically decreasing, this effective potential has a minimum ifρ is nonzero, as depicted in Fig. 3.1. The value ofφ at that minimum is also density- dependent, and the curvature of the effective potential around the minimum increases with the ambient density. Therefore, the scalar field is effectively more massive in a higher-density region, provided that the scalar field reaches the minimum of the effective potential in that region (φint).
The scalar field reaches this minimum only if the “thin-shell” condition is satisfied:
φext−φint≪βmPlΦ, (3.76)
where φext is the value of φ at the minimum of Veff outside the body, and Φ is the Newtonian gravitational potential of the massive body. In this case, the effects of the fifth force are suppressed because the scalar field is heavy inside the massive body and cannot propagate outside the body:
this is the essence of the “chameleon mechanism.”
We can already see two indications that an analogous effect could occur in f(R) gravity. In
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V
eff(φ, ρ
ext) V
eff(φ, ρ
int)
ρ
extexp
βφ
m
Plρ
intexp
βφ
m
PlV (φ) φ
extφ
intV
φ
Figure 3.1: The effective potential of chameleon gravity. The effective potential is the sum of a monotonically decreasing V(φ) (dotted curve) and ρexp[βφ/mPl] (solid curves). The effective potential is shown for two values of the density, with ρint > ρext. The short-dashed curve is the effective potential inside a massive body withρ=ρint. The minimum of this potential is atφ=φint. The long-dashed curve is the effective potential outside the massive body, whereρ=ρext, and its minimum is atφ=φext. Note that the curvature of the effective potential inside the massive body aroundφint is greater than the curvture of the effective potential outside the body aroundφext; this implies thatφis effectively more massive in the higher-density region, provided thatφ≃φint there.
Appendix A, we show that f(R) gravity is equivalent to a scalar-tensor theory, and we see that the effective mass of the scalar field, given by Eq. (3.38), contains a term that depends on the background density of matter. It therefore seems plausible that anf(R) theory could be designed so that the scalar field in the equivalent scalar-tensor theory behaves like the scalar field in chameleon gravity. In fact, such an f(R) theory, with an explicit connection to chameleon gravity, was first presented in Ref. [135], but this model was found to be indistinguishable from general relativity with a cosmological constant. Second, we note that the thin-shell condition is a lower bound on Φ, while our linearity condition is an upper bound on Φ =GM/Rs; it is therefore possible that the two conditions are related and mutually exclusive. We will see that this is indeed the case.
It is possible for anf(R) gravity theory to giveγ≃1 in the Solar System ifR≃8πGρinside the Sun [79]. However, in Section 3.2 we stressed that this relation is not necessarily true inf(R) gravity.
On the contrary, we found that R ≪ 8πGρ both inside and outside the Sun in 1/Rgravity. The
form off(R) given by Eq. (3.74) is special because it does permit solutions of the field equation in whichR≃8πGρfor certain values ofn, c1andc2and certain values of the density inside and outside the Sun, as demonstrated numerically in Ref. [79]. The connection to the chameleon mechanism gives insight into what is happening: R≃8πGρ inside the Sun corresponds to φ=φint inside the massive body. The background value of the Ricci scalar,R =R0, corresponds to φ =φext. If R can smoothly transition from R0 far from the Sun to R ≃8πGρ ≫ R0 inside the Sun, then the chameleon mechanism hides the deviation from general relativity andγ≃1. However, if the Sun’s potential is too small to causeR to deviate significantly fromR0 inside the Sun, then the linearity condition given by Eq. (3.57) is satisfied, andγ= 1/2.
Hu and Sawicki [79] define a “thin-shell” condition for their model: R≃8πGρ inside a massive body only if (fR−fR0)≪Φ inside the Sun. Clearly, this condition is identical in form to Eq. (3.76), which further illustrates the connection between chameleon gravity and thisf(R) model. We will now show how this condition is mutually exclusive with our linearity condition. With the assumption thatR≫m2, Eq. (3.74) implies
fR≃1−nc1
c22 m2
R n+1
, (3.77)
and sinceR0≪Ris implied by the solution R≃8πGρ, we see that
fR−fR0≃nc1
c22 m2
R0
n+1
. (3.78)
Furthermore, Eq. (3.74), withR0≫m2, implies that
fRR0≃n(n+ 1)c1
c22 m2
R0
n+1
1 R0
. (3.79)
As long as n is not much greater than unity, we see that R0fRR0 ∼ (fR−fR0). The thin-shell condition is therefore equivalent to 1/fRR0 ≫R0Φ−1. When we note thatfR0≃1 for R0 ≫m2, we see that the thin-shell condition implies fR0/fRR0≫R0Φ−1, which is the exact opposite of our linearity condition given by Eq. (3.57).
Our linearity condition provides some insight into the conditions necessary for the Hu-Sawicki f(R) theory to evade Solar System tests. A key parameter combination in this model is
I0≡nc1
c22 1
41 n+1
, (3.80)
which was called|fR0|in Ref. [79]. In terms ofI0, the linearity condition is
I0≫ R0
41m2 n+1
Φ. (3.81)
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Thus we see that the nonlinear R≃8πGρ solution will not exist ifI0 is too large, and we expect that the maximum value will increase with increasingn; both of these features are confirmed by the numerical analysis presented in Ref. [79].
So what value of I0 is required to violate the linearity condition, thus giving thisf(R) theory a chance of evading Solar System tests? We first consider a star surrounded by a background cosmological density. To match the observed expansion of the Universe, the cosmological background value of the Ricci scalar must beR0= 41m2[79]. The potential of the Sun is Φ≃10−6. With these values, the linearity condition is I0 ≫ 10−6. Therefore, I0 must be very small if we are to avoid γ= 1/2. This is not a very realistic description of our Solar System, however; the Sun is surrounded by interstellar matter, and the density of this matter is much higher than the cosmological average.
If we assume thatR0= 8πGρ, withρ= 10−24 g cm−3, throughout the Galaxy, thenR0/m2≃106. The linearity condition is thenI0 ≫106n/(41n+1), which is much easier to violate. However, this solution is only valid if the nonlinear R = 8πGρ solution is stable inside the Galaxy. Since the potential of the Galaxy is also Φ ≃ 10−6, the linearity condition for the Galaxy imbedded in a background with R= 41m2 isI0≫10−6. This bound matches the numerical results presented in Ref. [79]; Hu and Sawicki find thatR= 8πGρin the Galaxy only if I0∼<2×10−6.