4.2 Reduced equations in f (R) gravity
4.2.1 Time-time component of the reduced equations
−1
2gµνϕ(R) + ϕ0(R)Rµν− [∇µ∇ν− gµν2]ϕ0(R)
= κTµν . (4.9) We will now be applying the order reduction technique to eq. (4.9). In the present case, this technique amounts to just replacing the Ricci scalar and the Ricci tensor, in the terms of order , by the expression we get for them, from the = 0 version of the same equations. As such, when we set = 0 in eq. (4.9), we get
RTµν− 1
2gµνRT − gµνΛ = κTµν ⇔ RTµν = 1
2gµν(RT + 2Λ) + κTµν (4.10) as the reduced expression of the Ricci tensor, where RT is the reduced-order Ricci scalar. Calculating the trace of eq. (4.10), we are able to determine an expression for the reduced-order Ricci scalar, like so
gµν
RTµν−1
2gµν(RT + 2Λ)
= κgµνTµν ⇔ RT − 2(RT + 2Λ) = κT
⇔ RT = −κT − 4Λ . (4.11)
Substituting eq. (4.11) in eq. (4.10), we get
RTµν = κTµν−1
2gµνκT − Λgµν . (4.12)
Replacing R and Rµν, in the order terms in eq. (4.9), by eq. (4.11) and eq. (4.12), we get Gµν− gµνΛ +
−1
2gµνϕ(RT) + ϕ0(RT)
κTµν− 1
2gµνκT − Λgµν
− [∇µ∇ν − gµν2]ϕ0(RT)
= κTµν (4.13)
These are reduced field equations in the sense that we are not using the complete field equations to determine expressions for R and Rµν. We should notice that, in theory, this approximation is only valid when
|ϕ(R)| |R| , (4.14)
which has to be verified after an expression for ϕ(R) is determined.
4.2.1 Time-time component of the reduced equations
Since, in GR, the time-time component of the field equation is the Friedmann eq. (2.71), in the context of f (R) gravity, the time-time component of eq. (4.13) gives a modified Friedmann equation, which is
given by calculation, in eq. (2.102) and eq. (2.103). As such, we have that
[∇0∇0+2]ϕ0(RT) = −3˙a such, using the chain rule, we get
˙ Substituting eq. (2.76) in eq. (2.74) we get
T = ρ(3w − 1) . (4.19)
We calculate the derivative of eq. (4.11), getting ∂RT/∂T = −κ. The derivative of eq. (4.19) can also be determined and it gives ˙T = ˙ρ(3w − 1). Putting everything together, while using eq. (2.77), we end up with [25]
˙
ϕ0(RT) = ϕ00(RT)κ3H(1 + w)ρ(3w − 1) . (4.20) Substituting this expression in equation (4.17), yields
H2 = 1 Notice how we recover the usual Friedmann equation when = 0. In the context of the order reduction method, we substitute the H2 term on the term on the right hand side of eq. (4.21) by the usual Friedmann eq. (2.71), getting [25]
H2= 1 as our final form of the reduced modified Fridmann equations in f (R) gravity. It is clear now, that the reduced differential equation is effectively a second-order one, contrary to eq. (2.108). With this result, we would like to find a function ϕ(RT), such that eq. (4.22) is the same as some Friedmann equation with a modified source, compared to GR, namely the ones discussed in chapter 3. This exercise is explored in the next chapter.
Chapter 5
Effective actions for loop quantum cosmology models
This chapter consists on presenting the main results of our work. Following the procedure mentioned at the end of the previous chapter, we begin by specifying the conditions in which we will be using eq.
(4.22). In this context, we will be considering a flat universe (k = 0) with a cosmological constant equal to zero (Λ = 0). Taking these aspects into account eq. (4.22) becomes
H2= 1 3κρ −
3
1
2(3w + 1)ϕ0(RT)κρ +1
2ϕ(RT) − 3κ2ρ2ϕ00(RT)(1 + w)(1 − 3w)
, (5.1)
Substituting eq. (4.19) in eq. (4.11), with Λ = 0, we get
RT = −κρ(3w − 1) . (5.2)
In the following sections, we will be requiring that eq. (5.1) is the same as the modified Friedmann equations considered on chapter 3. Here, we reinforce the idea that there are two kinds of modified Friedmann equations at play. One of them is eq. (5.1), which was derived in the context of metric f (R) gravity, and the other one was derived in the context of LQC. Thus, our main goal is to determine ϕ(RT) such that eq. (5.1) is the same as eq. (3.34), eq. (3.38) and eq. (3.51). In doing so, we will be determining an effective action that leads to the quantum bounce that is caracteristic of those models. The modified Friedmann equation, in the context of LQC and its modifications, can be written in a general way as
H2= 1
3κρ + Ψ (ρ) , (5.3)
where Ψ (ρ) is some algebraic function, which will depend on the model. Comparing with eq. (5.1), the requirement is that ϕ(RT) satisfies
− 3
1
2(3w + 1)ϕ0(RT)κρ +1
2ϕ(RT) − 3κ2ρ2ϕ00(RT)(1 + w)(1 − 3w)
= Ψ (ρ) , (5.4) for a given Ψ (ρ). The same equation, writen in terms of RT, using eq. (5.2), is given by
− 3
1
2ϕ(RT) − (3w + 1)
2(3w − 1)RTϕ0(RT) − 3(1 + w)(1 − 3w)
(3w − 1)2 R2Tϕ00(RT)
= Ψ (RT) . (5.5) For each model, mLQC-I and mLQC-II, we will be considering the scenario of matter as a scalar
field (w = 1) and the general w scenario. An effective action for LQC was already determined in [25], in the context of matter as a scalar field. We will, nonetheless, present the main steps to get this result and, later on, we will also generalize it.
5.1 Matter as a scalar field
In order to be in agreement with the approach leading to eq. (3.18), we have to consider matter as a scalar field or, in other words, we have to set the EoS parameter to be w = 1. Therefore, eq. (5.1) simplifies to
H2 = 1 3κρ −
3
2ϕ0(RT)κρ +1
2ϕ(RT) + 12κ2ρ2ϕ00(RT)
(5.6) and, as a consequence, eq. (5.4) becomes
− 3
2ϕ0(RT)κρ +1
2ϕ(RT) + 12κ2ρ2ϕ00(RT)
= Ψ (ρ) (5.7)
and eq. (5.2) can be written as
RT = −2κρ . (5.8)
Here, we should point out that, in this case, RT is negative, since it only makes physical sense for the density ρ to be positive. Finally, with eq. (5.8), we can write eq. (5.7) in terms of RT as
− 3
1
2ϕ(RT) − RTϕ0(RT) + 3RT2ϕ00(RT)
= Ψ (RT) . (5.9)
Independently of the model of LQC that we are considering, at a given moment, the solution to the non-homogeneous eq. (5.9) consists on the sum of the solution to the corresponding homogeneous equation and its particular solution,
ϕ(RT) = ϕh(RT) + ϕp(RT) . (5.10)
As such, before specifying any of the models, by substituting Ψ (RT) in eq. (5.9) for each case, let us consider the homogeneous equation first, which is common to all cases and is given by
1
2ϕ(RT) − RTϕ0(RT) + 3R2Tϕ00(RT) = 0 . (5.11) This equation is a second order Cauchy-Euler equation, having the form
x2d2y(x)
dx2 + axdy(x)
dx + by(x) = 0 . (5.12)
Assuming a trial solution y(x) = xm, we can arrive at a characteristic equation for m, given by
m2+ (a − 1)m + b = 0 . (5.13)
Through the roots of eq. (5.13), we obtain a solution for eq. (5.12) whose form depends on the number of roots, and if they are real or complex. In the present case, the characteristic equation for the homogeneous eq. (5.11) is
m2−4 3m +1
6 = 0 (5.14)
and, solving for m, one gets two real roots, m = 16(4 ±√
10), meaning that the solution to eq. (5.11) is given by
ϕh(R) = c1R16(4−
√10)+ c2R16(4+
√10) . (5.15)
This solution does not contain analytic functions of R, meaning it is not locally given by a convergent power series. For this reason, and also for the fact that ϕh(R) does not contribute to the full eq. (5.9), we can set c1 = c2 = 0, without loss of generality [25]. As such, the solutions to eq. (5.9) are simply given by the particular solution in question
ϕ(RT) = ϕp(RT) , (5.16)
depending on Ψ (RT).