In this paper we will consider the “Running FLRW” model introduced in (Grande 2011). In this model the value of the Cosmological Constant (CC) is seen as an effective quantity whose value can evolve with the expansion of the universe. This model also enforces the conservation of matter by allowing a cosmological evolution for the gravitational coupling.

In this model we retain the standard model’s interpretation of the CC as vacuum energy while considering the reasonable possibility that its energy density might be related to other time-varying cosmological quantities. This idea has solid roots in fundamental physics but in this paper we will limit our discussion to the introduction the main equations and notation necessary for our analysis. We refer the reader to the aforementioned literature for a thorough description of the underlying Quantum Field Theory background necessary to justify some choices used in literature and also in this paper.

To describe this model we start with the standard General Relativity description of the interaction of the curvature of space-time with its matter contents described by the Einsten’s equation: \[R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = 8 \pi G T_{\mu\nu} + \Lambda g_{\mu\nu}\] where the CC term \(\Lambda\) is interpreted as a source and can be incorporated in the modified energy-momentum tensor \[\tilde{T}_{\mu\nu} \equiv T_{\mu\nu} = (\rho_\Lambda - \rho_m) g_{\mu\nu} + (\rho_m + p_m)U_\mu U_\nu\] whose form arises from the description of the matter contents of the universe as perfect fluids with velocity 4-vector \(U_\mu\) and the inclusion of the vacuum energy density \(\rho_\Lambda = \frac{\Lambda}{8 \pi G}\) associated with the CC term.

We also assume a spatially flat Friedmann-Lemaïtre-Robertson-Walker (FLRW) metric \[ds^2 = dt^2 - a^2(t)d\vec{x}^2\] with scale factor \(a(t)\).

This general framework is also the basis of the standard cosmological model (hereafter \(\Lambda CDM\) model), but the Cosmological Principle embodied by the FLRW metric allows \(\rho_\Lambda\) and \(G\) to be functions of time without losing the covariance of the theory. In fact the Bianchi identities imply that \[\nabla^{\mu}(G \tilde{T}_{\mu\nu}) = 0\] that in our case becomes \[\frac{d}{dt}[G (\rho_\Lambda + \rho_m)] + 3 G H (\rho_m + p_m) = 0\] where \(H\) is the Hubble rate \(H = \frac{\dot{a}}{a}\) , and this equation implies the local conservation of matter \[\label{matterconservation} \frac{d}{dt}\rho_m + 3 G H (\rho_m + p_m) = 0\] only if \(G\) and \(\rho_\Lambda\) are constants, as in the standard model, or are both functions of time and satisfy the constraint \[\label{bianchiconstraint} (\rho_m + \rho_\Lambda) \frac{dG}{dt} + G \frac{d\rho_\lambda}{dt} = 0\] where \(\rho_m\) is given by (\ref{matterconservation}).

The background expansion of the universe is so described by a set of functions \(H(t)\), \(\rho_m(t)\), \(p_m(t)\), \(G(t)\) and \(\rho_\Lambda(t)\) and to obtain a closed system of equations that describes the evolution of these quantities we need to specify the functional form for the evolution of \(G\) and \(\rho_\Lambda\).

According to (Grande 2011) we can start by considering that the gravitational coupling and the vaccum energy density might evolve as a power series of some energy scale \(\mu\) with rates given by \[\label{variationrates} \frac{d\rho_\Lambda(\mu)}{d \text{ ln } \mu} = \sum\limits_{k =0,1,2,...} A_{2k} \mu^{2k}, ~~ \frac{d}{d \text{ ln } \mu}\left( \frac{1}{G(\mu)} \right) = \sum\limits_{k =0,1,2,...} B_{2k} \mu^{2k}.\] We can regard this choice as purely phenomenological, and in this context a sensibile choice for the energy scale can be \(H\). In this way we associate the running of the cosmological quantities to the typical energy scale of the gravitational field associated with the FLRW metric.

Let’s first consider the evolution of \(\rho_\Lambda\): this type of expansion has been widely discussed in the literature and according to the results in (Basilakos 2009),(Babić 2002) and (Borges 2008) we can keep only the zeroth and second orders to prevent deviations from the \(\Lambda CDM\) model that are too big to be reconciled with current observations. After integration we obtain the functional form: \[\label{firstlambdalaw} \rho_\Lambda (H) = n_0 + n_2 H^2\] with the coefficient given by \[n_0 = \rho_\Lambda^0 - \frac{3 \nu}{8 \pi} M_P^2 H_0^2, ~~~ n_2 = \frac{3 \nu}{8 \pi}M_P^2\] with \[\nu = \frac{1}{6 \pi } \sum_i B_i \frac{M_i^2}{M_P^2}.\] Here \(H_0\) and \(\rho_\Lambda^2\) are respectively the the value of the Hubble rate and of the energy density of vacuum at the present time and the \(M_i\) are the masses associated to every term in the expansion (\ref{variationrates}) in the underlying QFT derivation of the theory.

The parameter \(\nu\) is the most import in this framework: for \(\nu = 0\) the vacuum energy remains constant with \(\rho_\Lambda = \rho_\Lambda^0\) but for \(\nu \not= 0\) the evolution law (\ref{firstlambdalaw}) can be rewritten as \[\label{secondlambdalaw} \rho_\Lambda(H) = \rho_\Lambda^0 + \frac{3 \nu}{8 \pi} M_P^2 (H^2 - H_0^2).\]

In (Grande 2011) the \(\nu\) parameter as been considered a free parameter with natural range \(|\nu| \ll 1\) and has been constrained against Supernovae, CMB and BAO observations to be in the \(1\sigma\) range \(|\nu| < 0.004\).

The evolution equation for \(G^{-1}\) can now be obtained by again keeping only the dominant terms in (\ref{variationrates}) and by combining the equation (\ref{secondlambdalaw}) for \(\rho_\lambda\) with the constraint (\ref{bianchiconstraint}) imposed by the Bianchi identity, after integration the solution reads: \[g(H) \equiv \frac{G(H)}{G_0} = \frac{1}{1+\nu \text{ ln } (H^2/H_0^2)}\] we note here how the sign of \(\nu\) determines the increase (\(\nu < 0\)) or decrease (\(\nu > 0\)) of the gravitational coupling with the expansion of the universe with an overall slow convergence to the present day value \(G_0 \equiv G(H_0)\).

To fully determine the background evolution for this model we need to rewrite the Friedmann equations in therm of the density parameters, these are the energy densities for matter and vacuum normalized to the current critical density \(\rho_c^0 = \frac{3H_0^2}{8 \pi G_0}\): \[\Omega_i(z) \equiv \frac{\rho_i(z)}{\rho_c^0}\] where the time-dependence is expressed trough the redshift \(z\) (or equivalently the scale factor \(a\)) because this will be useful in our analysis. We can also define the energy densities normalized to the critical density at an arbitrary redshift \(\rho_c(z) = \frac{3 H^2(z)}{8 \pi G(z)}\): \[\tilde{\Omega}_i(z) \equiv \frac{\rho_i (z)}{\rho_c(z)} = \frac{g(z)}{E^2(z)} \Omega_i(z)\] where \(E(z)\) is the Hubble rate normalized to its current value \(H_0\): \[E(z) = \frac{H(z)}{H_0} = \sqrt{g(z)} [ \Omega_m(z) + \Omega_\Lambda (z)]^{\frac{1}{2}}.\] It is important to note that the running FLRW model only the tilded parameters satisfy the flat space cosmic sum rule \[\tilde{\Omega}_m (z) + \tilde{\Omega}_\Lambda(z) = 1\] at all times, while the non-tilded parameters satisfy it only at the present time.

We are now able to obtain the full system of equations the govern the background expansion in this model: \[\begin{aligned} & E^2(z) = g(z)[\Omega_m(z) + \Omega_\Lambda(z)], \label{syshubble}\\ & (\Omega_m + \Omega_\Lambda) dg + g d \Omega_\Lambda = 0, \label{sysbianchi} \\ & \Omega_\Lambda(z) = \Omega_\Lambda^0 + \nu [E^2(z) - 1], \label{syslambda} \\ & \Omega_m(z) = \Omega_m^0 (1+z)^{3(1+w_n)}, \label{sysom}\end{aligned}\] The first equation is the equivalent of the Friedmann equation in the \(\Lambda CDM\) model, the second is the diffrerential form of the Bianchi equation (\ref{bianchiconstraint}), the third is just a rewrite of (\ref{secondlambdalaw}) using the density parameter and the last equation is a rewrite of the standard equation for \(\rho_m\) generalized to include relativistic (\(w_m = \frac{1}{3}\)) and nonrelativistic matter (\(w_m = 0\)).

The linear perturbations for this model where thoroughly studied in (Grande 2010) where the perturbations of the matter components of the universe where considered alongside the perturbations for \(\rho_\Lambda\) and \(G\). This is necessary to grasp the different dynamics present in this model compared to the \(\Lambda CDM\) model. If fact in both these models matter is covariantly conserved and the density contrast \(D \equiv \delta\rho_m/\rho_m\) satisfies the following second order differential equation: \[\label{scalefactorfirst} D''(a) + \left( \frac{3}{a} + \frac{H'(a)}{H(a)} \right) D'(a) = \frac{3 \tilde{\Omega}_m(a)}{2 a^2} \left( D(a) + \frac{\delta G}{G} \right )\] where a prime denotes differentiation with respect to the scale factor. This equation reduces to the standard \(\Lambda CDM\) equation for the scale factor if we assume a vanishing \(\delta G\), but to follow the true evolution of the perturbations in this model we will assume \(\delta G \not= 0\). In this case the perturbations for \(\rho_\Lambda\) and \(\rho_m\) are related to \(\delta G\) due to the constraint imposed by the Bianchi identity (\ref{bianchiconstraint}): \[\label{perturbationrelations} \frac{\delta \rho_\Lambda}{\rho_\Lambda} = - \frac{\delta G}{G}, ~~~ \frac{\delta \rho_m}{\rho_m} = - \frac{(\delta G(a))'}{G'(a)}\] we can now use these relations to substitute the \(\delta G\) terms in (\ref{scalefactorfirst}) after another differentiation to obtain a third order differential equation for the growth factor that will depend only on the cosmological quantities already introduced: \[\label{scalefactorsecond} \begin{split} &D'''(a)+\frac{1}{2}\left( 16-9\tilde{\Omega}(a) \right) \frac{D''(a)}{a} + \\ &\frac{3}{2} \left( 8 - 11 \tilde{\Omega}_m(a) + 3 \tilde{\Omega}_m^2(a) - a \tilde{\Omega}_m'(a) \right) \frac{D'(a)}{a^2} = 0 \end{split}\] This equation lends itself to a numerical solution whose behavior is shown in [FIGURE] where the growth factor is shown as a function of redshift alongside with the \(\Lambda CDM\) solution. It is possible to see how the model predicts and enhancement of the growth for the \(\nu < 0\) solution, this is due to the strengthening of the gravitational coupling at high redshift that allows to overcome the “repulsion” associated with the expansion of cause by the vacuum energy density. The opposite situation is evident for \(\nu > 0\) where the higher value of \(\rho_\Lambda\) and the weakening of the gravitational coupling at high redshift hinders the early growth of the perturbations.

This is also the main distinguishing feature of the linear analysis of the model, in fact as was demonstrated in (Grande 2010) the shape of the matter power spectrum is the same as that of the \(\Lambda CDM\) model due to the perturbations of \(G\) and \(\rho_\Lambda\) being negligible at early times with the result that the only signature in this regime is in the different spectrum amplitude.