2.2 f (R) gravity
2.2.2 Generalized Friedmann and Raychaudhuri equations
gµν
[f (R) − Rf0(R)]
2 + ∇µ∇νf0(R) − gµν2f0(R)
(2.98) and
Gef f ≡ G/f0(R) (2.99)
can be regarded as an effective gravitational coupling constant. Positivity of Gef f, equivalent to the requirement that the graviton is not a ghost, therefore imposes that f0(R) > 0 [21]. Moreover, if f00(R) > 0 the Dolgov-Kawasaki instability is avoided. This stability condition, which expresses that the scalar degree of freedom is not a ghost, can be given a physical interpretation [29]. Assuming that f0(R) > 0 applies then, if the gravitational coupling decreases as R increases, as such
dGef f
dR = −f00(R)G
(f0(R))2 < 0, (2.100)
a mechanism operates which compensates for the increase of R that comes from eq. (2.96), which is a dynamical equation on R that shows that R generates larger curvature, the effect of which becomes stronger with increased Gef f. This can only be achieved if f00(R) > 0.
2.2.2 Generalized Friedmann and Raychaudhuri equations
In the context of GR, assuming a FLRW metric, we have obtained eq. (2.71) and eq. (2.75). We can consider the same metric and deduce the corresponding equations in f (R) gravity, in order to see how they differ from the previous ones. Using g00= −1, eq. (2.68) and eq. (2.67), the time-time component of eq. (2.95) is given by
1
2f (R) − 3¨a
af0(R) − [∇0∇0+2]f0(R) = κρ (2.101) Focusing on the last terms in the left-hand side of this equation, we have that
[∇0∇0+2]f0(R) = ∂0∂0f0(R) + gµν∇µ(∂νf0(R))
= ∂0∂0f0(R) + gµν∂µ∂νf0(R) − gµνΓνµη ∂ηf0(R)
= ∂0∂0f0(R) − ∂0∂0f0(R) − gµνΓνµ0 ∂0f0(R)
= −gµνΓνµ0 f˙0(R) , (2.102)
where we have used eq. (2.8), for Xν = ∂νf0(R) and the fact that the covariant derivative of the scalar f0(R) is the usual partial derivative, ∇νf0(R) = ∂νf0(R). Notice that, in the implicit summations in the third line of eq. (2.102), only the terms with partial derivatives with respect to time are relevant, since f0(R) only depends on time. Assuming a FLRW metric, using eq. (2.27), we can determine that
gµνΓνµ0 = 3˙a
a . (2.103)
We can express ˙f0(R) in terms of derivates with respect to R. As such, using the chain rule, we get f˙0(R) = d
dt d
dRf (R) = dR dt
d dR
df (R)
dR = ˙Rf00(R) . (2.104)
Using eq. (2.102), eq. (2.103) and eq. (2.104), eq. (2.101) becomes
The Ricci scalar, given by eq. (2.69), is given by
R = 6
which we can substitute in eq. (2.105), getting
˙a This is the modified Friedmann equation in the context of metric f (R) gravity. We can observe that, setting f (R) = R, we have f0(R) = 1 and f00(R) = 0 and eq. (2.108) reduces to the standard Friedmann equation, given by eq. (2.71), with k = Λ = 0. It is clear that eq. (2.108) is a higher than second order equation, given the presence of ˙R and eq.(2.106), which poses a problem that we will later discuss. In a similar way, we can determine the space-space components of eq. (2.95), which give [30]
2 ¨a Finally, if we introduce, in eq. (2.109), the expression we get for ( ˙a/a)2, from eq. (2.108), we obtain
¨a which is the modified Raychaudhuri equation in the context of metric f (R) gravity. This equation is also a higher than second order differential equation. Once again, setting f (R) = R, we have f0(R) = 1 and f00(R) = f000(R) = 0 and eq. (2.110) reduces to the standard Raychaudhuri equation, given by eq.
(2.75), with Λ = 0.
Chapter 3
Loop quantum cosmology and its modifications
3.1 Loop quantum gravity and loop quantum cosmology
At the order of the Planck length, where GR breaks down, a quantum theory of gravity is expected to provide insight on how gravity behaves at quantum scales, adressing many open questions, one of them being the initial singularity problem. LQG is one of the main candidate theories, which predicts that classical differential geometry, at small space-time curvatures, is replaced by a discrete quantum geometry at the Planck scale.
In the framework of GR, geometry is a manifestation of the gravitational field and, in quantum theory, we learn that fields have quantum properties. What quantum gravity is trying to understand is what are the quantum properties of geometrical quantities such as area and volume. In fact, in the same way that total angular momentum is quantized in quantum mechanics, for example, LQG is built into the notion of quanta of volume and area, using tetrahedrons and their geometrical properties. In this context, a region of physical space can be described by a set of interconnected grains of space and the word ”loops”, in
”loop quantum gravity”, refers to the loops formed by closed sequences of links between these grains of space [31].
Formally, in LQG, the elementary classical phase space variables for the gravitational sector are the Ashtekar-Barbero connection Aia and the conjugate triad Eia. When considering a spatially flat space-time, which is the case in standard LQC, the only relevant constraint is the Hamiltonian constraint, whose vanishing yields the equations of motion, as we will show later on. In terms of elementary classical phase space variables, the effective Hamiltonian is given by [32]-[33]
H = (Cgrav+ CM)/8πG , (3.1)
where CM is the matter sector constraint and Cgrav is the gravitational constraint, given by [32]-[33]
Cgrav = Cgrav(E) − (1 + γ2)Cgrav(L) , (3.2) where Cgrav(E) is the Euclidean term, Cgrav(L) is the Lorentzian term and γ is the Barbero-Immirzi parameter, not to be confused with the Lorentz factor, whose value is set to
γ ≈ 0.2375 (3.3)
using black hole thermodynamics in LQG [36]. The Euclidean part is given by [32]-[33]
Cgrav(E) = 1 2
Z
d3xijkFiab EajEbk
pdet(q) , (3.4)
where Fiab is the field strength of connection Aia and det(q) is the determinant of the spatial metric compatible with the triads. The Lorentzian part, on the other hand, is given by [32]-[33]
Cgrav(L) = Z
d3xK[ajKb]k EajEbk
pdet(q) , (3.5)
where Kai is the extrinsic curvature.
LQC is described as a symmetry-reduced model of LQG. Unlike in the framework of full LQG, in the standard LQC scenario, the Euclidean and Lorentzian terms are treated in the same way, due to a classical symmetry reduction, which makes the Lorentzian term a multiple of the Euclidean one. This reduction is possible if a spatially flat cosmological space-time is considered. The quantization is then performed on the combination of these two terms. This procedure is out of the scope of this work and it leads to the following effective Hamiltonian [33]
H = −3v sin2(λb)
8πGγ2λ2 + HM(v) , (3.6)
where λ is defined by the smallest nonzero eigenvalue of the area operator in LQG, which is given by [32]-[34]
∆ = λ2≡ 4√
3πγl2P l , (3.7)
and lP lis the Planck length, given by lP l=p
~G/c3or, in the geometrized unit system,
lP l= 1 . (3.8)
The Hamiltonian is a function of the gravitational phase space variables b and v, which in turn are given by [33], [35]
b = c/p1/2, v = p3/2= v0a3 , (3.9)
where c and p are the symmetry reduced connection and triad variables and v0 is the fiducial volume of the fiducial cell introduced on the spatial manifold R3. The Poisson brackets of these gravitational phase space variables satisfy [35]
{b, v} = 4πGγ . (3.10)
Taking that into account, and using the Hamiltonian formulation, one obtains the following Hamilton equations for the variables v and b [33], [35]
˙v = {v, H} = −∂H
∂b4πGγ = 3v
2λγsin(2λb) , (3.11)
˙b = {b, H} = ∂H
∂v4πGγ = −3 sin2(λb)
2γλ2 − 4πGγP , (3.12)
where P ≡ −∂HM/∂v is defined as the pressure [32]-[33] and, again, the curly brackets refer to the Poisson brackets. Once we establish the relation between the phase space variables and the cosmological ones, such as the energy density, ρ, and the scale factor of the universe, a, we will be able to uniquely
determine the evolution of the universe. We have already stated that v is related to the scale factor in eq.
(3.9). In order to obtain a correspondence, regarding variable b, one needs an additional equation, which is the, previously mentioned, vanishing of the Hamiltonian constraint. In order to compute it, we need to take into account that the energy density is defined as ρ ≡ HM/v [32]-[33]. As such, the vanishing of the Hamiltonian constraint (H = 0), given by eq. (3.6), gives [33]
ρ = 3 sin2(λb)
8πGγ2λ2 = ρcsin2(λb) , (3.13)
where
ρc≡ 3/(8πGγ2λ2) (3.14)
is the critical energy density in LQC. This expression can be inverted in order to obtain sin2(λb) = ρ
ρc
. (3.15)
This is the relation between the variables b and ρ. One is finally able to determine the equations of motion. Recalling eq. (3.9), one has that H ≡ ˙a/a = ˙v/(3v) and, therefore,
H = sin(2λb)
2λγ . (3.16)
Using the trigonometric identity sin2(2θ) = 4 sin2(θ)(1 − sin2(θ)), we can write H2 in such a way that allows us to identify ρ/ρcin the expression
H2= sin2(λb)(1 − sin2(λb))
λ2γ2 = ρ
ρcλ2γ2
1 − ρ
ρc
= 8πG 3 ρ
1 − ρ
ρc
. (3.17)
Finally, in order to be consistent with the notation that we are using throughout this work, we write the previous expression in terms of the coupling constant, given by eq. (2.37), as such [33]
H2 = 1 3κρ
1 − ρ
ρc
. (3.18)
The resulting equation of motion is just the usual Friedmann eq. (2.71), for a spatially flat universe with a cosmological constant equal to zero, with a modified source, having no extra degrees of freedom compared to GR. As such, we state that LQC leads to an effective Friedmann equation. Moreover, from the modified Friedmann equation, one finds that the big bang singularity is replaced by a quantum bounce that occurs at the critical density ρc, determined by the underlying quantum geometry, independently of the matter content of the universe, when H = 0 and ¨a > 0.
With a similar procedure, one can determine the modified Raychaudhuri equation. Differentiating the Hubble parameter with respect to time, one finds that
¨ a
a = ˙H + H2 . (3.19)
Having already presented the expression for H2, one also needs to determine the derivative of eq. (3.16) with respect to time,
H =˙ ˙b
γ cos(2λb) , (3.20)
and substitute everything in eq. (3.19), finding that
¨ a a = 1
γ
−3 sin2(λb)
2γλ2 − 4πGγP
cos(2λb) + 8πG 3 ρ
1 − ρ
ρc
. (3.21)
Finally, recalling eq. (3.15) and using the trigonometric identity cos(2θ) = 1 − sin2(θ), one obtains [33]
¨ a
a = −4πG 3 ρ
1 − 4ρ
ρc
− 4πGP
1 − 2ρ
ρc
. (3.22)
The resulting equation is an effective Raychaudhuri equation in the sense that it resembles eq. (2.75), found in GR, in the same way eq. (3.18) resembles eq. (2.71). In fact, the classical limits, when ρ/ρc 1, of eq. (3.18) and eq. (3.22) are, respectively, given by
H2 ≈ κ
3ρ , (3.23)
¨ a a ≈ −κ
6(ρ + 3P ) , (3.24)
which are the equations found in GR. As a final remark, it is also pertinent to mention that the classical energy-momentum conservation law, given by eq. (2.77), is recovered from eq. (3.18) and eq. (3.22), and they form a consistent set of equations.