Late-time constraints on modified Gauss-Bonnet cosmology

Abstract

In this paper, we consider a gravitational action containing a combination of the Ricci scalar, R, and the topological Gauss-Bonnet term, G. Specifically, we study the cosmological features of a particular class of modified gravity theories selected by symmetry considerations, namely the $f(R,G)= R^n G^{1-n}$ model. In the context of a spatially flat, homogeneous and isotropic background, we show that the currently observed acceleration of the Universe can be addressed through geometry, hence avoiding de facto the shortcomings of the cosmological constant. We thus present a strategy to numerically solve the Friedmann equations in presence of pressureless matter and obtain the redshift behavior of the Hubble expansion rate. Then, to check the viability of the model, we place constraints on the free parameters of the theory by means of a Bayesian Monte Carlo method applied to late-time cosmic observations. Our results show that the f(R, G) model is capable of mimicking the low-redshift behavior of the standard $\Lambda$CDM model. Finally, we investigate the energy conditions and show that, under suitable choices for the values of the cosmographic parameters, they are all violated when considering the mean value of n obtained from our analysis, as occurs in the case of a dark fluid.

Appendices

Appendix A Useful relations

For the sake of completeness, we here report some useful relations for determining the cosmological dynamics in the case of $f(R,G) = R^n G^m$ gravity. Specifically, starting from the definitions given in Eqs. (14) and (15), the time derivatives of the Ricci scalar and the Gauss-Bonnet term take the form

$$\begin{aligned} \dot{R}&= 6(4H\dot{H}+\ddot{H})\,, \end{aligned}$$

(A1)

$$\begin{aligned} \dot{G}&=24H(4H^2\dot{H}+2\dot{H}^2+H\ddot{H}) \,. \end{aligned}$$

(A2)

Moreover, the time derivatives of the functions appearing in Eqs. (12) and (13) can be expressed in terms of the above equations and the derivatives with respect to R and G as follows:

$$\begin{aligned} \dot{f_R}=&\ \dot{R} f_{RR}+\dot{G} f_{RG}\,, \end{aligned}$$

(A3)

$$\begin{aligned} \dot{f_G}=&\ \dot{G} f_{GG} + \dot{R} f_{RG}\,, \end{aligned}$$

(A4)

$$\begin{aligned} \ddot{f_R}=&\ \ddot{R}f_{RR}+\ddot{G}f_{RG}+\dot{R}^2 f_{RRR}+\dot{G}^2 f_{RGG}+2\dot{R}\dot{G}f_{RRG}\,, \end{aligned}$$

(A5)

$$\begin{aligned} \ddot{f_G}=&\ \ddot{G} f_{GG} +\ddot{R} f_{RG}+\dot{G}^2 f_{GGG}+\dot{R}^2 f_{RRG}+2\dot{R}\dot{G}f_{RGG}\,. \end{aligned}$$

(A6)

Appendix B Effective dark energy pressure

In the case of $f(R,G) = R^n G^m$ gravity models, the dark energy pressure (13) can be written in the compact form (25), where the explicit expressions of the coefficients $c_k(j,s;n,m)$ are

$$\begin{aligned} c_0=&\ m \left( -m ^2+3 m -2\right) j^2\,, \end{aligned}$$

(B1)

$$\begin{aligned} c_1=&\ m (m -1) \left[ 3 j^2 (n +m -2)-j (4 n +6 m -6)+s\right] , \end{aligned}$$

(B2)

$$\begin{aligned} c_2=&\ m \left\{ 2 j \left[ 9 n m +n (4 n -11)+7 m ^2-16 m +9\right] \right. \nonumber \\&-(2 n +3 m -3) (2 n +3 m +s-2)\nonumber \\&\left. -3 j^2 (n +m-2) (n +m -1)\right\} , \end{aligned}$$

(B3)

$$\begin{aligned} c_3=&\ (m -1) \left[ m ^2 \left( j^2-6 j+15\right) -m \left( 2 j^2-18 j-3 s+15\right) \right. \nonumber \\&\left. +3\right] +n ^2 \left[ 12 m +3 (m -1) j^2-2 (5 m -7) j+s-10\right] \nonumber \\&+(j-2)^2 n^3+n \left[ 19 m ^2-25 m +\left( 3 m ^2-6 m +2\right) j^2\right. \nonumber \\&\left. -2 \left( 8 m ^2-13 m +5\right) j+4 m s-s+3\right] , \end{aligned}$$

(B4)

$$\begin{aligned} c_4=&\ n \left\{ 3 m -2 \left[ m (3 m +2) j+j+m (s-3 m )\right] +s+13\right\} \nonumber \\&+2 n ^3 (2-j)-n ^2 \left[ -5 m +2 (m -2) j+s+8\right] \nonumber \\&+(m -1) \left\{ 9+m \left[ 5 m -6 (m +1) j-s+11\right] \right\} , \end{aligned}$$

(B5)

$$\begin{aligned} c_5=&\ n ^2 \left[ m (4 j-9)-8\right] +(m -1) \left[ m ^2 (4 j-15)-3 m-9\right] \nonumber \\&+ n \left( m ^2 (8 j-17)+m (6-4 j)-2\right) +n^3\,, \end{aligned}$$

(B6)

$$\begin{aligned} c_6=&\ 4n m (2-n) -(n -4) n +3 m ^2-3\,, \end{aligned}$$

(B7)

$$\begin{aligned} c_7=&\ 2 m (2 m -1) (n +m -1) \,. \end{aligned}$$

(B8)

Appendix C Solutions to vacuum field equations

Making use of the relations reported in 1, it is possible to find analytic solutions for the scale factor of $f(R,G)=R^n G^m$ gravity in vacuum. In particular, it turns out that the theory under consideration admits two different sets of solutions. The first one is a time power-law scale factor of the form $a(t) = a_0 t^\ell$, with

$$\begin{aligned} \ell= & {} \left[ \frac{1}{{2 (2 m+n-2)}}\right] \Big \{-8 m^2-8 m n+11 m-2 n^2 \nonumber \\{} & {} +4 n-3 \pm \Big [\Big (8 m^2+8 m n-11 m+2 n^2 -4 n+3\Big )^2 \nonumber \\{} & {} +4 (2 m+n-2) \Big (4 m^2+6 m n-5 m +2 n^2-3 n+1 \Big ) \Big ]^{\frac{1}{2}} \Big \}. \end{aligned}$$

(C1)

Setting $m = 1-n$, the solution takes the form $a(t) = a_0 t^{2n-1}$, as written in Eq. (30). Another solution occurs when considering exponential scale factors of the form $a(t) = a_0 e^{s\, t}$, with s being a real number. However, in order for this scale factor to be the solution to the field equation, we must also have $m=1-n/2$.