The present paper is devoted to reconstruction and cosmological study in modified F(G) theory of gravity. Our reconstruction scheme is based on Friedmann metric induced equations in modified F(G) theory by supposing a power law form for the scale factor of Friedmann metric. Firstly, we deal with the stability study, the obtained model. This survey reveals that for appropriated choice of the reconstructed model parameter, this model is stable under two cosmological evolutions namely the de Sitter and the power law evolutions symbolized by the appropriated scale factor. Secondly, we investigate the inflationary survey by fitting the model with the inflation observables. These observables are determined and their comparison with Planck’s results leads to a special inflationary F(G) model. We prove that this model especially obtained for radiation domination evolution develops an instability, so can fall to ordinary matter domination era or dark energy domination era. This is the key of graceful exit from inflation.
Cosmology aims to study the universe as a whole [
G = R 2 − 4 R μ ν R μ ν + R μ ν ρ σ R μ ν ρ σ (1)
The Gauss-Bonnet term plays an important role because it makes it possible to avoid phantom contributions and helps to regulate the action of gravitation [
Several other interesting works have been done so far in this theory. The Gauss-Bonnet scalar and the F ( G ) theory were considered to reconstruct theories favorable to the expansion of the universe [
This success of the F ( G ) theory has motivated us to take an interest in it and solving some current riddles in cosmology. The present work constitutes a contribution going in the same direction as the points previously mentioned. In addition to the benefit of being able to explain the present Universe acceleration of the by modifying standard theories of gravity, another major issue which will hopefully be explained in the next two decades, is the primordial post-quantum gravity era of our Universe. To date there are two candidate descriptions for this primordial era, the inflationary scenario [
The paper is organized as follow. After providing the basic equations of the gravitational F ( G ) theory in Section 2, we deal with the reconstruction program through the Section 3. The stability of the reconstructed model is achieved in the Section 4. The model is put at the heat of cosmological inflationary survey via the Section 5. The final section 6is devoted to the conclusion.
The action in dimension 4 of F ( G ) gravity theory is given, as in [
S = 1 k ∫ − g [ R 2 + F ( G ) ] d 4 x + S m (2)
where R is the Ricci scalar curvature, F ( G ) is a generic function of the Gauss-Bonnet topological invariant G, k 2 = 8 π G et S m is the action of matter.
By varying this four-dimensional action with respect to the metric, we obtain the equation of the field which is written [
G μ ν + 8 [ R μ ρ ν σ + R ρ ν g σ μ + 1 2 ( g μ ν g σ ρ − g μ σ g ν ρ ) R − R ρ σ g ν μ − R μ ν g σ ρ + R μ σ g ν ρ ] ∇ ρ ∇ σ F ′ ( G ) + ( G F ′ ( G ) − F ) g μ ν = T μ ν m (3)
where G μ ν is Einstein’s tensor, T μ ν m the energy-momentum tensor of matter. In this work, we consider that k 2 = 8 π G = 1 and the prime represents the ordinary derivative with respect to G. For the Robertson-Walker flat space metric which is written as:
d s 2 = − d t 2 + a ( t ) 2 ∑ i 3 d x i 2 (4)
we have:
R = 6 ( H ˙ + 2 H 2 ) ; G = 24 H 2 ( H ˙ + H 2 ) (5)
where H is the Hubble parameter and the dot means () the time derivative.
Considering that the universe is flat and filled with a perfect fluid, Friedmann equations for the modified gravity f ( G ) are written:
3 H 2 = G F ′ ( G ) − F − 24 H 3 F ˙ ′ ( G ) + ρ m (6)
− 2 H ˙ = − 8 H 3 F ˙ ′ ( G ) + 16 H H ˙ F ˙ ′ ( G ) + 8 H 2 F ¨ ′ ( G ) + ρ m (7)
The continuity equation is given by:
ρ ˙ + 3 H ( ρ + P ) = 0 (8)
ρ ˙ + 3 H ρ ( 1 + ω ) = 0 (9)
where ω is the energy parameter.
The equations which constitute the basis of the theory being established, we will now design our model.
In this section, we will reconstruct a F ( G ) model. To do this, we choose a scale factor a ( t ) obeying the power law and which is in the form:
a ( t ) = a * ( t t * ) p (10)
where a * is the value that takes a at the time t * and p is a constant. In this case, the Hubble parameter satisfies the following relations:
H = p t ; H ˙ = − p t 2 ; H 2 = p 2 t 2 (11)
Then, the GB invariant takes the form:
G = 24 p 3 ( p − 1 ) t 4 , (12)
leading to
t = [ 24 p 3 ( p − 1 ) G ] 1 4 (13)
From the continuity Equation (9), we obtain:
ρ = ρ 0 [ 24 p 3 ( p − 1 ) G ] − 3 4 p ( 1 + ω ) (14)
Taking into account the Equations (11)-(13), the first Friedmann equation becomes:
4 p − 1 G 2 F ″ ( G ) + G F ′ ( G ) − F ( G ) − [ 3 p G 8 ( p − 1 ) ] 1 2 + ρ 0 [ G 24 p 3 ( p − 1 ) ] 3 4 p ( 1 + ω ) = 0 (15)
This equation takes the form:
C G F ″ ( G ) + F ′ ( G ) − F ( G ) − A G 1 2 + ρ 0 G α B α = 0 (16)
with: C = 4 p − 1 ; A = 3 p 8 ( p − 1 ) ; B = 24 p 3 ( p − 1 ) et α = 3 4 p ( 1 + ω ) .
It is a differential equation in term of G which, after solving, gives the solution:
F ( G ) = − 4 A G 1 2 2 + C − ρ 0 G α B α ( − 1 + α ) ( 1 + α C ) + C 1 G − 1 C + C 2 G (17)
that we write more simply:
F ( G ) = D G 1 2 + E G α + C 1 G β + C 2 G (18)
where: D = − 4 A 2 + C ; E = − ρ 0 B α ( − 1 + α ) ( 1 − α C ) ; β = − 1 C ; C 1 and C 2 being
integration constants. The model (18) is a general model. The reason is that for appropriate choice of its parameters namely the parameters C , p , ω , ρ 0 , C 1 et C 2 , the viable F ( G ) models investigated in [
In this section, it is about the study of the stability of the F ( G ) model constructed. To achieve this, we use de Sitter solutions and power law solutions which are techniques generally used in cosmology as shown by the work [
H ( t ) = H b ( t ) ( 1 + δ ( t ) ) , (19)
ρ ( t ) = ρ b ( t ) ( 1 + δ m ( t ) ) (20)
where H b ( t ) and ρ b ( t ) correspond respectively to the Hubble parameter and to the energy density of basic ordinary matter. By making an analogy with the continuity Equation (9), we can write:
ρ ˙ b ( t ) + 3 H b ( t ) ρ b ( t ) ( 1 + ω ) = 0 (21)
whose solution is:
ρ b ( t ) = ρ 0 e − 3 ( 1 + ω ) ∫ H b ( t ) d t (22)
whith ρ 0 , an intégration constant.
In order to study the linear perturbation of H ( t ) and ρ ( t ) , we carry out a development of the model F ( G ) in series of G b = 24 H b 2 ( H ˙ + H b 2 ) as following:
F ( G ) = F b + F G b ( G − G b ) + O 2 (23)
On the one hand, we substitute the Equations (19), (20) and (23) in Equation (6) which is the first Friedmann equation, and we obtain after simplification:
δ ˙ ( t ) [ 24 H b 3 ( 24 H b 2 H ˙ b + 24 H b 4 ) F G G b − 24 H b 3 F G b − 576 H b 6 ( 48 H b H ˙ b 2 + 24 H b 2 H ¨ b + 96 H b 3 H ˙ b ) F G G G b ] + δ ( t ) [ ( 24 H b 2 H ˙ b + 24 H b 4 ) ( 72 H b 2 H ˙ b + 96 H b 4 ) F G G b − ( 72 H b 2 H ˙ b + 96 H b 4 ) F G b − 72 H b 3 ( 48 H b H ˙ b 2 + 24 H b 2 H ¨ b + 96 H b 3 H ˙ b ) F G G b − 24 H b 3 ( 48 H b H ˙ b 2 + 24 H b 2 H ¨ b + 96 H b 3 H ˙ b ) ( 72 H b 2 H ˙ b + 96 H b 4 ) F G G G b − 6 H b 2 ] + ρ b δ m ( t ) = 0 (24)
On the other hand, the substitution of the Equations (19) et (20) in Equation (9) leads to:
δ ( t ) = − δ ˙ m ( t ) 3 ( 1 + ω ) H b ( t ) (25)
By eliminating δ ( t ) from combination of Equations (24) and (25), we get the differential equation:
d δ m ( t ) δ m ( t ) = 3 ( 1 + ω ) ρ b [ 24 H b H ˙ b F G b − 24 H b H ˙ b ( 24 H b 2 H ˙ b + 24 H b 4 ) F G G b + 576 H b 4 H ˙ b ( 48 H b H ˙ b 2 + 24 H b 2 H ¨ b + 96 H b 3 H ˙ ) F G G G b + ( 24 H b H ˙ b + 24 H b 3 ) ( 72 H b 2 H ˙ b + 96 H b 4 ) F G G b − ( 72 H b H ˙ b + 92 H b 3 ) F G b − 72 H b 2 ( 48 H b H ˙ b 2 + 24 H b 2 H ¨ + 96 H b 3 H ˙ b ) F G G b − 24 H b 2 ( 48 H b H ˙ b 2 + 24 H b 2 H ¨ b + 96 H b 3 H ˙ b ) ( 72 H b 2 H ˙ b + 96 H b 4 ) F G G G b − 6 H b ] − 1 d t (26)
whose general solution is written in the form:
δ m ( t ) = C 0 exp { 3 ( 1 + ω ) ∫ C H d t } (27)
where C 0 is an intégration constant and C H has for expression:
C H = ρ b [ 24 H b H ˙ b F G b − 24 H b H ˙ b ( 24 H b 2 H ˙ b + 24 H b 4 ) F G G b + 576 H b 4 H ˙ b ( 48 H b H ˙ b 2 + 24 H b 2 H ¨ b + 96 H b 3 H ˙ ) F G G G b + ( 24 H b H ˙ b
+ 24 H b 3 ) ( 72 H b 2 H ˙ + 96 H b 4 ) F G G b − ( 72 H b H ˙ b + 92 H b 3 ) F G b − 72 H b 2 ( 48 H b H ˙ b 2 + 24 H b 2 H ¨ + 96 H b 3 H ˙ b ) F G G b − 24 H b 2 ( 48 H b H ˙ b 2 + 24 H b 2 H ¨ b + 96 H b 3 H ˙ b ) ( 72 H b 2 H ˙ b + 96 H b 4 ) F G G G b − 6 H b ] − 1 (28)
From the Equations (25) et (27), we obtain:
δ ( t ) = − C 0 C H H b exp { 3 ( 1 + ω ) ∫ C H d t } (29)
In the rest of our work, we will use the stability of de Sitter solutions as well as that of power law solutions in order to appreciate the convergence of each of the perturbation terms.
De Sitter’s solutions are well known in Cosmology because they constitute a perfect approximation of the exponential expansion of the Universe during its primordial inflation [
H b ( t ) = H 0 ⇒ a ( t ) = a 0 e H 0 t (30)
So the expression (22) becomes:
ρ b ( t ) = ρ 0 e − 3 ( 1 + ω ) H 0 t (31)
And C H takes the form:
C H = ρ b [ − 3 D ( 24 ) 1 2 H 0 + 4 α ( − 2 + α ) E ( 24 ) α H 0 4 α − 1 + 4 β ( − 2 + β ) C 1 ( 24 ) β H 0 4 β − 1 − 96 C 2 H 0 3 − 6 H 0 ] − 1 (32)
Moreover, from the differentiation of each member of the Equation (31), we express dt which allows us to write:
∫ C H d t = − 1 3 ( 1 + ω ) H 0 ∫ C H ρ b d ρ b (33)
which give:
∫ C H d t = − ρ b 3 ( 1 + ω ) [ − 3 D ( 24 ) 1 2 H 0 2 + 4 α ( − 2 + α ) E ( 24 ) α H 0 4 α + 4 β ( − 2 + β ) C 1 ( 24 ) β H 0 4 β − 96 C 2 H 0 4 − 6 H 0 2 ] − 1 (34)
By substitution of this previous expression in the relations (27) et (29), we obtain respectively:
δ m ( t ) = C 0 exp { − ρ b [ − 3 D ( 24 ) 1 2 H 0 2 + 4 α ( − 2 + α ) E ( 24 ) α H 0 4 α + 4 β ( − 2 + β ) C 1 ( 24 ) β H 0 4 β − 96 C 2 H 0 4 − 6 H 0 2 ] − 1 } (35)
and
δ ( t ) = − C 0 ρ b [ − 3 D ( 24 ) 1 2 H 0 2 + 4 α ( − 2 + α ) E ( 24 ) α H 0 4 α + 4 β ( − 2 + β ) C 1 ( 24 ) β H 0 4 β − 96 C 2 H 0 4 − 6 H 0 2 ] − 1 × exp { − ρ b [ − 3 D ( 24 ) 1 2 H 0 2 + 4 α ( − 2 + α ) E ( 24 ) α H 0 4 α + 4 β ( − 2 + β ) C 1 ( 24 ) β H 0 4 β − 96 C 2 H 0 4 − 6 H 0 2 ] − 1 } (36)
In this study, we are interested in the solutions described by the power law of the scale factor. They are also initiated in Cosmology to describe a power law type inflation [
a ( t ) ∝ t n ⇒ H b ( t ) = n t (37)
Note that by posing n = p , the scale factor of (37) corresponds to that of the Equation (10). The density of ordinary energy given by the relation (22) and the Gauss-Bonnet invariant given by (5) take the respective forms:
ρ b ( t ) = ρ 0 t − 3 n ( 1 + ω ) (38)
and
G b = 24 n 3 ( n − 1 ) t 4 (39)
Then the relation (28) expressing the magnitude C H becomes here:
C H = ρ 0 { − 48 n 2 ( 2 n − 1 ) [ 1 2 D k − 1 2 g b 1 + q 4 + α E k α − 1 g b α + q − 1 4 + β C 1 k β − 1 g b β + q − 1 4 + C 2 g b 3 + q 4 ] + 2 ( 24 ) 2 n 5 ( n − 1 ) ( 2 n − 7 ) [ − 1 4 D k − 3 2 g b 1 + q 4 + α ( α − 1 ) E k α − 2 g b α + q − 1 4 + β ( β − 1 ) C 1 k β − 2 g b β + q − 1 4 ] + 8 ( 24 ) 3 n 8 ( n − 1 ) ( 2 n − 1 ) [ 3 8 D k − 5 2 g b 1 + q 4 + α ( α − 1 ) ( α − 2 ) E k α − 3 g b α + q − 1 4 + β ( β − 1 ) ( β − 2 ) C 1 k β − 3 g b β + q − 1 4 ] − 6 n g b 1 + q 4 } − 1 (40)
avec q = − 3 n ( 1 + ω ) ; k = 24 n 3 ( n − 1 ) et g b ( t ) = G b ( t ) k
Taking into account the Equation (39), we deduce: d t = − 1 4 g b − 5 4 d g b .
It follows:
∫ C H d t = − ρ 0 4 ∫ { − 48 n 2 ( 2 n − 1 ) [ 1 2 D k − 1 2 g b q + 6 4 + α E k α − 1 g b α + q + 4 4 + β C 1 k β − 1 g b β + q + 4 4 + C 2 g b q + 8 4 ] + 2 ( 24 ) 2 n 5 ( n − 1 ) ( 2 n − 7 ) [ − 1 4 D k − 3 2 g b q + 6 4 + α ( α − 1 ) E k α − 2 g b α + q + 4 4 + β ( β − 1 ) C 1 k β − 2 g b β + q + 4 4 ] + 8 ( 24 ) 3 n 8 ( n − 1 ) ( 2 n − 1 ) [ 3 8 D k − 5 2 g b q + 6 4 + α ( α − 1 ) ( α − 2 ) E k α − 3 g b α + q + 4 4 + β ( β − 1 ) ( β − 2 ) C 1 k β − 3 g b β + q + 4 4 ] − 6 n g b q + 6 4 } − 1 d g b (41)
Note that we have not found an analytical solution for the integral of the relation (41), then a numerical study will be carried out in order to obtain the perturbation functions and to discuss their convergence.
In this part of the work, we will deal with the evolution survey of the solutions obtained as a time function. We emphasize here that in the case of the study of de Sitter’s solutions, an explicit analytical expression of each of the perturbation functions δ m ( t ) and δ ( t ) is obtained respectively in (35) and (36) respectively. And we can easily notice that for an appropriate choice of parameters, the quantities δ m ( t ) and δ ( t ) converge. Indeed, when t straights forward to infinity, the fonction ρ b ( t ) = ρ 0 e − 3 ( 1 + ω ) H 0 t tends to 0 for positive or zero ω . Then, for ρ b ( t ) tending to 0, we see that δ m ( t ) tends to C 0 (which is an arbitrary constant) and δ ( t ) tends to 0. Hence these two functions converge. In the case the power law solutions, the idea was to calculate the integral given by the Equation (41) then introduce the result in each of the Equations (27) and (29) in order to obtain the respective expressions of δ m ( t ) and δ ( t ) . But, it turns out that the analytical solution of the Equation (41) is not easy to get, so we proceeded to a numerical study. Thus, our approach at this level consists to plot the perturbation functions δ m ( t ) and δ ( t ) without having obtained their analytical expressions; which can permit us to appreciate their convergence. These curves are those of
The convergence of these quantities for different values of the parameters of the reconstructed model, according to the de Sitter approach and the power law approach, proves that this model is stable. It is therefore conducive to the reproduction of the stage dominated by matter, radiation and also the phase characterized by de Sitter inflation [
In this section of our work, we study cosmological inflation. In particular, we will determine the values of the observables of the inflation phenomenon within the framework of a scale factor obeying the power law. Taking into consideration the Planck’s results, we will deduce a family of models f ( G ) capable of describing inflation.
In cosmological inflation survey, several works have always been based on the study of the inflationary observables [
the curvature perturbations n s , the running α s ≡ d n s d ln k of the spectral index
n s where k is the absolute value of the wave number k, the spectral index n T of the tensor and the ratior.
From the scalar potential denoted V ( ϕ ) characterizing inflation and its derivatives, the parameters of the slow-roll are defined as follows [
ε ≡ M p 2 2 ( 1 V d V d ϕ ) 2 (42)
η ≡ M p 2 2 d 2 V d ϕ 2 (43)
ξ 2 ≡ M p 4 V 2 d V d ϕ d 3 V d ϕ 3 (44)
Note that the inflation ends when ε = 1 .
We have the approximate expressions of the inflation observables as a function of the slow-roll parameters relative to the potential. They are written according to [
r ≈ 16 ε (45)
n s ≈ 1 − 16 ε + 2 η (46)
α s ≈ 16 ε η − 24 ε 2 − 2 ξ 2 (47)
n T ≈ 16 − 2 ε (48)
Let us specify that in modified theory of gravity, it is not possible to exploit the conformal transformation of the Einstein theory because one cannot define neither a scalar potential, nor the parameters of the solw-rool relating to it. We then introduce the Hubble slow-roll parameters ε n which are defined by:
ε n + 1 ≡ d ln | ε n | d N (49)
with ε 0 ≡ H i n i / H and N ≡ ln ( a / a i n i ) the e-folding number and where a i n i is the scale factor at the start of inflation and H i n i is the corresponding Hubble parameter. It follows:
ε 1 ≡ H ˙ H 2 (50)
ε 2 ≡ H ¨ H ˙ H − 2 H ˙ H 2 (51)
ε 3 ≡ ( H ¨ H − 2 H ˙ 2 ) − 1 [ H ⃛ H ˙ H − H ¨ ( H ˙ 2 + H ¨ H ) H ˙ H − 2 H ˙ H 2 ( H ¨ H − 2 H ˙ 2 ) ] (52)
Thus, we obtain as in [
r ≈ 16 ε 1 (53)
n s ≈ 1 − 2 ε 1 − 2 ε 2 (54)
α s ≈ − 2 ε 1 ε 2 − ε 2 ε 3 (55)
n T ≈ − 2 ε 1 (56)
These observables thus expressed can be easily determined. Particularly within the framework of our study where we considered a scale factor evolving according to the power law.
Recall that our model (18) was reconstructed from the first Friedmann equation. By ejecting this model into the second Friedmann equation, we get the following differential equation
2 H ˙ ( 1 + ω ) [ ( − 1 + p ) p 3 H 2 + ( H 2 + H ˙ ) ] − 3 4 p ( 1 + ω ) + 4608 H 4 { − ( ( 1 + ω ) [ − 2 + 3 4 p ( 1 + ω ) ] × [ H 2 + ( H 2 + H ˙ ) ( − 1 + p ) p 3 ] 3 4 ( − 4 + p + p ω ) ) [ 18432 ( − 1 + p ) 2 p 8 ( − 1 + p ( 4 + 3 ω ) ) ] − 1 − p [ 6144 ( 1 + p ) H 2 ( H 2 + H ˙ ) 2 p H 2 ( H 2 + H ˙ ) − 1 + p ] − 1 } [ 2 H ˙ 2 ( 2 H 2 + H ˙ ) + H H ¨ ] 2 + { p ( [ 2 − 2 p ( 4 + 3 ω ) ] H 4 − 2 [ − 1 + p ( 4 + 3 ω ) ] H 2 H ˙ + ( − 1 + p 2 ) ( 1 + ω ) × [ H 2 ( H 2 + H ˙ ) ( − 1 + p ) p 3 ] 3 4 p ( 1 + ω ) p H 2 ( H 2 + H ˙ ) − 1 + p ) [ 4 H 4 H ˙ − 6 H ˙ 3 H ¨ − 8 H H ˙ H ¨ − H 2 ( 18 H ˙ 2 + H 3 ) ] } [ 4 ( 1 + P ) [ − 1 + p ( 4 + 3 ω ) ] H 2 ( H 2 + H ˙ ) 2 p H 2 ( H 2 + H ˙ ) − 1 + p ] − 1 = 0 (57)
The Equation (57) is a differential equation in H ( t ) which admits for solution H ( t ) = p t . Thus, from relations (50)-(52), we obtain:
ε 1 = 1 p (58)
ε 2 = 0 (59)
ε 3 = 1 p (60)
then, the relations (53)-(56) lead to:
r ≈ 16 p (61)
n s ≈ 1 − 2 p (62)
α s ≈ 0 (63)
n T ≈ − 2 p (64)
These are the values of the observables of power law inflation. We will now determine a F ( G ) model capable of describing such inflation and which is in agreement with Planck’s results.
According to Planck’s results, we have: 0.962 ≤ n s ≤ 0.974 and by virtue of equality (62), we obtain: 52.632 ≤ p ≤ 76.923 .
By posing for example p = 60 , we obtain: r = 0.266 , n s = 0.966 , α s = 0 and n T = − 0.033 which are values actually in agreement with Planck’s results. Considering that just after inflation, the universe is essentially dominated by radiation, we have ω = 1 / 3 and the model becomes:
F ( G ) = C 2 G − 59 4 − 3 590 61 G + C 1 G − ρ 0 4.042.10 511 G 60 (65)
Note that it is possible to derive a F ( G ) model family describing inflation and in perfect agreement with Planck’s results by suitably choosing the values of the parameter p of our model.
The question that is actually to be asked consists in checking if the inflationary model (65) extols the exit from inflation.
Several approaches are used to study the exit from inflation. Based on the dynamic of the Hubble parameter, the exit from inflation are succesfully performed through the type VI singularity survey [
Recently, the authors in [
m ( r ) satisfies the condition m ′ ( − 1 / 2 ) > − 1 . The parameter m and r are given by
m ( G ) = G f ″ ( G ) f ′ ( G ) and r ( G ) = − G f ′ ( G ) f ( G ) (66)
without providing the functional form of m ( r ) and m ′ ( r ) in the framework of the inflationary model (65), their parametric plot leads to the
During this study, we devoted to the reconstruction of a model in the modified theory of gravity F ( G ) . From well elucidated processes, we have conducted the study of its stability. The results obtained clearly show that the reconstructed model is stable for the both de Sitter and the power-law evolutions. Therefore, this model is potentially suitable for dealing with cosmological questions such as the inflation of the Universe, the explanation of the different phases of the evolution of the Universe, and many others. In this work, the question of the Universe inflation was also approached. The study made it possible to find the observables within the framework of a scale factor evolving according to the power law. This study also made it possible to determine a model describing the inflation of the universe and which is in agreement with Planck’s results. Our analysis shows that the obtained top model develops clearly the graceful exit from the inflation.
The authors declare no conflicts of interest regarding the publication of this paper.
Aïnamon, C., Houndjo, M.J.S., Ayivi, A.A.L., Ganiou. M.G. and Kanfon, A. (2021) Reconstruction Method in F(G) Gravity: Stability Study and Inflationary Survey. Journal of Modern Physics, 12, 781-797. https://doi.org/10.4236/jmp.2021.126050