*3.1. The Cauchy Problem in Presence of a Perfect Fluid*

We discuss the Cauchy problem in presence of a perfect fluid. As mentioned above, implementing a conformal transformation (21), we show that the initial value problem can be analyzed by applying the same results stated in [23,24,46] for GR.

To start with, given the metric *gij* of signature (− + ++) in the Jordan frame, let us consider a perfect fluid having stress–energy tensor of the form:

$$\mathcal{T}\_{\text{ij}} = (\rho + p) \, \mathcal{U}\_{\text{i}} \mathcal{U}\_{\text{j}} + p \, \mathcal{g}\_{\text{ij}} \tag{23a}$$

and undergoing the usual conservation laws:

$$
\nabla\_j \mathcal{T}^{ij} = 0.\tag{23b}
$$

In Equation (23a), *ρ* and *p* are the matter–energy density and the pressure of the fluid, respectively, while *U<sup>i</sup>* denotes the components of the four velocity of the fluid (with *g ijUiU<sup>j</sup>* <sup>=</sup> <sup>−</sup>1). Performing the conformal transformation (21), we rewrite the field equations in the Einstein frame as:

$$
\tilde{\mathcal{R}}\_{ij} - \frac{1}{2} \tilde{\mathcal{R}} \mathfrak{g}\_{ij} = \mathcal{T}\_{ij\prime} \tag{24a}
$$

and:

$$\nabla\_{\dot{j}} \mathcal{T}^{ij} = 0,\tag{24b}$$

where:

$$\mathcal{T}\_{\rm ij} = \frac{1}{\varrho} (\varrho + p) \,\mathcal{U}\_i \mathcal{U}\_{\rm j} + \left( \frac{p}{\varrho^2} - \frac{V(\varrho)}{\varrho^3} \right) \,\mathbb{S}\_{\rm ij\prime} \tag{25}$$

plays the role of the effective stress–energy tensor.

In view of Proposition 2, Equation (24b) is equivalent to Equation (23b). This is a crucial aspect for our discussion, allowing us to apply to the present case the results stated in [23,24,46]. To see this point, we first suppose that the scalar field *ϕ* is positive, that is *ϕ* > 0. The opposite case *ϕ* < 0 differs from the former only for some technical aspects, and it will be briefly outlined after. Of course, it is implicitly assumed that *ϕ* 6= 0 at least in a neighborhood of the initial space-like surface.

Under the assumed conditions, the stress–energy tensor (25) can be rewritten in the form:

$$\bar{T}\_{\rm ij} = \frac{1}{\varrho^2} (\rho + p) \,\mathrm{\"\/} \mathrm{\/} \mathrm{\/} \mathrm{\/} + \left( \frac{p}{\varrho^2} - \frac{V(\varrho)}{\varrho^3} \right) \,\bar{\mathrm{g}}\_{\rm ij} \,\tag{26}$$

where *U*¯ *<sup>i</sup>* = <sup>√</sup>*ϕU<sup>i</sup>* is the four velocity of the fluid in the Einstein frame. Introducing the effective mass-energy density:

$$
\rho := \frac{\rho}{\varrho^2} + \frac{V(\varphi)}{\varrho^3},
\tag{27a}
$$

and the effective pressure:

$$
\bar{p} := \frac{p}{\varrho^2} - \frac{V(\varrho)}{\varrho^3},
\tag{27b}
$$

we can express the stress–energy tensor (26) in the standard form:

$$
\bar{\mathcal{T}}\_{\bar{\mathbf{i}}\mathbf{j}} = (\bar{\rho} + \bar{p}) \ \mathbf{\bar{U}}\_{\bar{\mathbf{i}}} \mathbf{\bar{U}}\_{\bar{\mathbf{j}}} + \bar{p} \,\bar{\mathbf{g}}\_{\bar{\mathbf{i}}\mathbf{j}}.\tag{28}
$$

We notice that, given an equation of state of the form *ρ* = *ρ*(*p*) and assuming that the relation (27b) is invertible (*p* = *p*(*p*¯)), by composition with Equation (27a), we obtain an effective equation of state *ρ*¯ = *ρ*¯(*p*¯). Moreover, we recall that the explicit expressions of the scalar field *ϕ* and the potential *V*(*ϕ*) depend on the specific form of the function *f*(*R*). As a consequence, the request that the relation (27b) is invertible together with the condition *ϕ* > 0 (or, equivalently, *ϕ* < 0) can be assumed as criteria for the viability of the functions *f*(*R*), providing us with suitable selection rules for admissible gravitational Lagrangian function *f*(*R*) (see also [9]).

After that, in order to discuss the Cauchy problem, we can follow step-by-step the Bruhat's arguments [23,24,46]. In particular, we recall that the Cauchy problem for the system of differential equations (24), with stress–energy tensor given by Equation (28) and equation of state *ρ*¯ = *ρ*¯(*p*¯), is well-posed if the condition:

$$\frac{d\bar{\rho}}{d\bar{p}} \ge 1,\tag{29}$$

is satisfied. The requirement (29) is easily verified by means of the relation:

$$\frac{d\overline{\rho}}{d\overline{p}} = \frac{d\overline{\rho}/dp}{d\overline{p}/dp} \ge 1,\tag{30}$$

together with expressions (27) and the equation of state *ρ* = *ρ*(*p*). Once again, condition (30) depends on the expressions of *ϕ* and *V*(*ϕ*); thus, it is strictly related to the form of the function *f*(*R*). Therefore, condition (30) represents a further criterion for the admissibility of *f*(*R*)-models.

For the sake of completeness, we conclude by outlining the case *ϕ* < 0. Supposing again that the signature of the metric in the Jordan frame is (− + ++), the signature of the conformal metric is now (+ − −−), and the components of the four velocity of the fluid in the Einstein frame are *<sup>U</sup>*¯ *<sup>i</sup>* = √ −*ϕU<sup>i</sup>* . The effective stress–energy tensor is expressed as:

$$\mathcal{T}\_{\rm ij} = -\frac{1}{\varrho^2} (\rho + p) \,\,\tilde{\mathcal{U}}\_i \tilde{\mathcal{U}}\_j + \left(\frac{p}{\varrho^2} - \frac{V(\varrho)}{\varrho^3}\right) \,\,\tilde{\varrho}\_{\rm ij} = (\mathfrak{p} + \mathfrak{p}) \,\,\,\tilde{\mathcal{U}}\_i \tilde{\mathcal{U}}\_j - \mathfrak{p} \,\,\tilde{\varrho}\_{\rm ij} \tag{31}$$

where, as above, the quantities:

$$\Phi := -\frac{\rho}{\varrho^2} - \frac{V(\varphi)}{\varrho^3},\tag{32a}$$

and:

$$\vec{p} := -\frac{p}{\varrho^2} + \frac{V(\varrho)}{\varrho^3},\tag{32b}$$

represent the effective mass-energy and the effective pressure.

After that, everything proceeds again as in [23,24,46], with the exception of a technical aspect: if *ρ* and *p* are positive, the quantity *r* := *ρ*¯ + *p*¯ = − *ρ* + *p ϕ*2 is now negative. About this, the reader can easily verify that, with the choice log(−*f* −2 *r*) instead of log(*f* −2 *r*) as in [23,24,46], the Bruhat's arguments apply equally well.

As a simple example, we consider the model *f*(*R*) = *R* + *αR* 2 coupled with dust. In the Jordan frame, the matter stress–energy tensor is given by T*ij* = *ρUiU<sup>j</sup>* , and the trace of the Einstein-like Equation (6a) yields the relation:

$$(1 + 2aR)R - 2R - 2aR^2 = -\rho \qquad \Longleftrightarrow \qquad R = \rho. \tag{33}$$

The scalar field (9) assumes the form:

$$
\varphi(\rho) = f'(R(\rho)) = 1 + 2a\rho. \tag{34}
$$

Taking into account small values of the density *ρ* << 1 (for instance, the present cosmological baryonic matter density) and choosing values of |*α*| not comparable with 1/*ρ*, we can reasonably suppose *ϕ* > 0, independently of the sign of the parameter *α*. We have to calculate the potential (14):

$$V(\varphi) = \frac{1}{4} \left[ \varphi F^{-1}((f')^{-1}(\varphi)) + \varphi^2 (f')^{-1}(\varphi) \right]. \tag{35}$$

To this end, since (*f* ′ ) −1 (*ϕ*) = *ρ*, from Equation (34) we get the relation:

1 4 *ϕ* 2 (*f* ′ ) −1 (*ϕ*) = <sup>1</sup> 4 (1 + 2*αρ*) 2 *ρ* , (36)

and considering that *F* −1 (*Y*) = *f* ′ (*Y*)*Y* − 2 *f*(*Y*), we have the identities:

$$\frac{1}{4}F^{-1}((f')^{-1}(\rho)) = \frac{1}{4}F^{-1}(\rho) = -\rho\tag{37}$$

and:

$$\frac{1}{4}\rho F^{-1}((f')^{-1}(\rho)) = -\frac{(1+2a\rho)\rho}{4}.\tag{38}$$

We conclude that:

$$V(\varphi(\rho)) = \frac{\mathfrak{a}\rho^2(1+2\mathfrak{a}\rho)}{2}.\tag{39}$$

In the Einstein frame, the stress–energy tensor (26) is expressed as:

$$\mathcal{T}\_{\rm ij} = \frac{\rho}{\varrho^2} \mathcal{U}\_i \mathcal{U}\_{\rm j} - \frac{V(\varphi)}{\varrho^3} \mathfrak{g}\_{\rm ij}. \tag{40}$$

Tensor (40) can be considered as the stress–energy tensor of a perfect fluid with density and pressure given, respectively, by:

$$\rho := \frac{\rho}{\varrho^2} + \frac{V(\rho)}{\varrho^3} = \frac{2\rho + \alpha \rho^2}{2(1 + 2\alpha \rho)^2} \tag{41a}$$

and:

$$\mathfrak{p} := -\frac{V(\mathfrak{p})}{\mathfrak{q}^3} = -\frac{\mathfrak{a}\mathfrak{p}^2}{2(1+2\mathfrak{a}\mathfrak{p})^2}.\tag{41b}$$

It is an easy matter to verify that the function (41b) is invertible. Indeed, for *ρ* > 0, one has:

$$\frac{d\vec{p}}{d\rho} = -\frac{4\alpha\rho}{4(1+2\alpha\rho)^3} \neq 0.\tag{42}$$

In addition, we have:

$$\frac{d\overline{\rho}}{d\rho} = \frac{4 - 4\alpha\rho}{4(1 + 2\alpha\rho)^3},\tag{43}$$

so that:

$$\frac{d\overline{\rho}}{d\overline{p}} = \frac{d\overline{\rho}/dp}{d\overline{p}/dp} = \frac{-1 + \alpha\rho}{\alpha\rho} \ge 1 \quad \Longleftrightarrow \quad \alpha < 0 \tag{44}$$

With condition (29) satisfied, it is then proved that the model *f*(*R*) = *R* + *αR* 2 , with *α* < 0, possesses a well-posed Cauchy problem when coupled with dust.
