Exact solutions and cosmological constraints in fractional cosmology

This paper investigates exact solutions of cosmological interest in fractional cosmology. Given $\mu$, the order of Caputo's fractional derivative, and $w$, the matter equation of state, we present specific exact power-law solutions. We discuss the exact general solution of the Riccati Equation, where the solution for the scale factor is a combination of power laws. Using cosmological data, we estimate the free parameters. An analysis of type Ia supernovae (SNe Ia) data and the observational Hubble parameter data (OHD), also known as cosmic chronometers, and a joint analysis with data from SNe Ia + OHD leads to best-fit values for the free parameters calculated at $1\sigma$, $2\sigma$ and $3\sigma$ confidence levels (CLs). On the other hand, these best-fit values are used to calculate the age of the Universe, the current deceleration parameter (both at $3\sigma$ CL) and the current matter density parameter at $1\sigma$ CL. Finding a Universe roughly twice as old as the one of $\Lambda$CDM is a distinction of fractional cosmology. Focusing our analysis on these results, we can conclude that the region in which $\mu>2$ is not ruled out by observations. This parameter region is relevant because fractional cosmology gives a power-law solution without matter, which is accelerated for $\mu>2$. We present a fractional origin model that leads to an accelerated state without appealing to $\Lambda$ or dark energy.

where the solution for the scale factor is a combination of power laws. Using cosmological data, we estimate the free parameters. An analysis of type Ia supernovae (SNe Ia) data and the observational Hubble parameter data (OHD), also known as cosmic chronometers, and a joint analysis with data from SNe Ia + OHD leads to best-fit values for the free parameters calculated at 1σ, 2σ and 3σ confidence levels (CLs). On the other hand, these best-fit values are used to calculate the age of the Universe, the current deceleration parameter (both at 3σ CL) and the current matter density parameter at 1σ CL. Finding a Universe roughly twice as old as the one of ΛCDM is a distinction of fractional cosmology. Focusing our analysis on these results, we can conclude that the region in which µ > 2 is not ruled out by observations. This parameter region is relevant because fractional cosmology gives a power-law solution without matter, which is accelerated for µ > 2. We present a fractional origin model that leads to an accelerated state without appealing to Λ or dark energy. arXiv:2303.16409v2 [gr-qc] 9 May 2023 1. INTRODUCTION In fractional calculus, the classical derivatives and integrals of integer order are generalized to derivatives and integrals of arbitrary (real or complex) order [1][2][3][4][5][6][7][8][9]. Fractional derivatives have attracted increasing attention because they universally appear as empirical descriptions of complex social and physical phenomena. Fractional calculus applications have grown enormously in recent years because these operators have memory and are more flexible in describing the dynamic behavior of phenomena and systems using fractional differential equations, while the description with integer order differential equations uses local operators and they are limited in the order of differentiation to a constant. Consequently, the resulting models must be sufficiently precise in many cases [10]. Research into fractional differentiation is inherently multi-disciplinary and has applications across various disciplines, for example, fractional quantum mechanics and gravity for fractional spacetime [11,12] and fractional quantum field theory [13][14][15][16][17][18]. Such frameworks have been essential in understanding complex systems in classical and quantum regimes [19][20][21][22][23][24][25][26][27][28][29].
Regarding the classical regime, fractional derivative cosmology has been established by two methods: (i) The last-step modification method is the simplest one, in which the corresponding fractional field equations replace the given cosmological field equations for a specific model. (ii) The first-step modification method can be considered a more fundamental methodology. In this method, one starts by establishing a fractional derivative geometry.
More concretely, the variational principle for fractional action is applied to establish a modified cosmological model. Fractional calculus has recently been explored to address problems related to cosmology in [27,. For example, in [30,34], the Riemann curvature and the Einstein tensor are defined as usual, but now with dependence on the µ fractional parameter. Then, it is possible to write a fractional analogous to the Einstein field equation through the expression γ ≈ 6/11. Here, γ is defined through the relation d ln δ m /d ln a ≈ Ω γ m , where δ m is the matter contrast and Ω m is the fractional energy density of matter. This result for the linear growth rate, d ln δ m /d ln a ≈ Ω γ m , was corrected to d ln δ m /d ln a ≈ Ω 6 11 m − 1 70 (1 − Ω m ) 5 2 in [84]. In fractional cosmology, the dimensionless energy density of dust matter depends on t through Ω m,µ = Ω m × t −(µ−1) ; thus, the growth index γ in the matter-dominated solution should be different to 6/11. In addition, it could be exciting to investigate and try fractional cosmology within a very early universe, fitting data and determining the impact of the fractional derivative term on primordial nucleosynthesis. Fractional cosmology ingredients may enhance inflation, raising the question whether it is possible that issues such as the cosmic no-hair conjecture, isotropization, etc., could be solved [85][86][87][88][89][90][91].
According to previous statements, it is essential to see if fractional calculus or fractional cosmology can well-describe the observational data. Then, we can perform more sophisticated calculations to describe the late-time Universe or the very early Universe. Therefore, one can argue that the Universe can be better described with a fractional derivative, not just to fit the data but also to describe the fundamental dynamics, highlighting the demand for new physics. These approaches can help to understand the Universe's acceleration with the mathematical background of fractional calculus. The mathematical richness generated by the corrections due to the fractional index µ of the fractional derivative can resolve the previous problems in future studies. Indeed, we can examine these topics in a forthcoming series of manuscripts with applications in inflation and dark energy models, investigating the physical implications and producing observational constraints.
In this paper, we investigate exact solutions of cosmological interest in fractional cosmology.
In particular, we study the cosmological applications of power-law solutions of the type a = (t/t 0 ) α 0 , where α 0 = t 0 H 0 is the current age parameter. Additionally, given µ, the order of Caputo's fractional derivative, and w, the equation of state (EoS) of matter, one must impose two compatibility conditions which allow particular solutions to (µ, w) Moreover, we are interested in an exact solution that gives the general solution of the system. For this purpose, one can solve the Riccati equation independent of the EoS, where the solution for the scale factor is a combination of power-law solutions. Additionally, we investigate if the solutions take account of the current late-time acceleration.
The paper is organized as follows. Section 2 discusses the basics of the fractional variational approach to cosmology and presents the cosmological equations for a perfect fluid. In Sec-tion 2.3, we comment on the crucial difference between fractional and standard cosmology; that is, we obtain late-time acceleration without adding a cosmological constant, quintessence scalar field or other exotic fluids as compared to standard cosmology. In Section 2.5, we consider a model with cold dark matter to present a specific realization of these possibilities and we interpret the fractional modification as dark energy in Section 2.6. Section 3 is devoted to finding exact solutions for the Hubble factor in this scenario. They correspond mainly to power-law solutions for the scale factor and a combination of power-law functions.
In Section 3.4, we provide a precise scheme to find approximated analytical solutions to aid in the asymptotic analysis. A discussion is presented in Section 3.5. We solve Bernoulli's equation using differential inequalities and asymptotic expansions to estimate H(z) in redshift.
A physical discussion of the results is presented in Section 3.6. In Section 4, a joint analysis using OHD and type Ia supernovae data is performed. This comparison with observational tests was used to find best-fit values for the fractional order of the derivative and the current age parameter α 0 . Section 5 is the conclusion.

FRACTIONAL ACTION INTEGRAL
Recently, a wide range of definitions of fractional derivatives [92], such as the Riemann-Liouville derivative (RLD) and the Caputo derivative (CD), among others, have been used in many applications.
Note that the main parameter of fractional calculus is given by µ, recovering standard calculus when µ → 1.
The Caputo left derivative is defined as In fractional calculus, we now have the following relation (see [92]) for the case of more than one derivative: at c. Additionally, the fractional derivative of the Leibniz rule [92] reads as having the usual when µ = n ∈ N.

Frational Action-like Variational Problems
Within the first-step modification method, one procedure uses the fractional variational approach developed in [30,[93][94][95][96][97] with the following fractional action integral, where Γ(µ) is the Gamma function, L is the Lagrangian, µ is the constant fractional parameter and τ and θ are the observers and intrinsic time, respectively, and the action integral depends on second order derivatives of the generalized coordinates q i .
Variation in (5) with respect to q i leads to the Euler-Poisson equations [97]:

Applications to Cosmology
In cosmology, it is assumed that the flat Friedmann provides the geometry of spacetime Lemaître-Robertson-Walker (FLRW) metric: where a(t) denotes the scale factor and N (t) is the lapse function. This result is based on Planck's observations [80].
For the metric (7), the Ricci' scalar depends on the second derivatives of a and first derivatives of N and reads Consider the point-like action integral where R(θ) is the Ricci scalar (8). In cosmology, the Einstein-Hilbert Lagrangian density is related to the Ricci scalar. Generically, one takes integration by parts, such that a total derivative is removed for the action, and the derivativesä(t) andṄ (t) are eliminated. We will use a fractional version of the Lagrangian (9); thus, we do not follow the standard procedure and keep the higher order derivatives and use formulation (5) leading to the Euler-Poisson Equation (6). We use fractional variational calculus with classical and Caputo derivatives.
For a fixed τ , the expressions and define the energy density and the isotropic pressure of the matter fields. Then, Defining q i ∈ {N, a } in (5) for a fixed τ , we have the Lagrangian The Euler-Poisson equations (6) obtained after varying the action (10) for q i ∈ {N, a } lead to the field equations a(θ) a(θ) Here, we have substituted the lapse function N = 1 after the variation.
To designate the temporary independent variables, the rule (τ, θ) → (2t, t) is applied, where new cosmological time t [34] is used, where the dots denote these derivatives. Furthermore, the Hubble parameter is H ≡ȧ/a. Hence, Equations (15) and (16) and the conservation Equation (13) can be written aṡ where expressions (11) and (12) are transformed to which defines the energy density and the isotropic pressure of the matter fields in cosmological time.

Some Cosmological Solutions
From (18) and assuming p = 0 (w = 0), the Hubble parameter is We have considered the positive root because we are interested in expanding universes.
To understand the self-accelerating behavior of H, let us assume that there is no matter, i.e., ρ = 0 and µ > 1. Then, from (18), Henceforth, and the deceleration parameter can be expressed as 1 + q = −Ḣ/H 2 . Therefore, and where the usual case q = −1, corresponding to a cosmological constant Λ, is excluded.
The case ρ = 0 can also be interpreted as a fluid whose energy density quickly vanishes with evolution. The asymptotic solution (22) was examined in detail using a dynamical systems analysis in reference [61], and its properties are summarized in Table I. The table summarizes the asymptotic behavior for µ > 1 when the energy density of matter tends to zero. Hence, even in the absence of matter, fractional cosmology gives a power-law solution a(t) = (t/t 0 ) µ−1 , which is accelerated for µ > 2. This is a crucial difference to standard cosmology, where we must add a cosmological constant, quintessence scalar field, or other exotic fluids to accelerate the expansion. Label Ω m H q Acceleration? Stability Scale Factor Solving algebraically Equations (17)- (19) forḢ,ρ and ρ, we obtaiṅ In General Relativity (GR), we have the flat Friedmann-Lemaître-Robertson-Walker metric; the main equations are the Friedmann constraint and conservation equation, Using (29) and (30), we obtain Now, using (29) and (31) we re-obtain (30). That is, we have three equations, two of them independent. However, as we discussed before, [62] studied Equations (17)- (19), and using a similar procedure as in GR, we obtain a new equation (see Equation (34)) instead of showing that two out of three equations are independent.
By demanding that (18) is conserved in time, i.e., we calculate the corresponding derivatives and substitute them into (26)-(28) to obtain This equation is an identity for µ = 1 as expected in standard cosmology. However, for µ = 1, we acquire the new relation for the pressure of the fluid: Using a similar procedure as in GR, we obtain a new Equation (34) instead of showing that two out of three equations are independent. This characteristic of fractional cosmology leads to some restrictions of the matter fields in the Universe that were explored in [62] for different matter fields.
Replacing the expression of p defined by (34) into (26) and (27), we obtaiṅ The previous results are valid for any ideal gas source.
Moreover, following references [61,62], the system can be extended by including several matter sources in Equations (15) and (16). After performing algebra, and using 8πG = 1, the following Raychaudhuri equation (with N (t) = 1) is obtained: along with the Friedmann equation Furthermore, the continuity equation leads to where ρ i and p i are the density and pressure of the ith matter component, respectively, and the sum is over all species, e.g., matter, radiation, etc. Note that when µ = 1 in Formula (38) and Formula (39), the standard cosmology without Λ is recovered, which by itself does not produce an accelerated expanding universe.
Using the equation of state p i = w i ρ i , where w i = −1 and are constants, we have Assuming separate conservation equations for each species and integrating for each ρ i , we have the following solution: where a(t 0 ) = 1, t 0 is the age of the universe and ρ 0i is the current value of the energy density of the ith species. Therefore, by substituting (41) into (38), we have Note that for µ = 1, the modified continuity Equation (39) provides the condition Combining these results with (37) and (38), we have the Riccati equation (35). This equation generically appears in fractional cosmology, independent of the matter content.
Therefore, in the following, we consider only one matter source.
Comparing with other fractional formulations, according to [42,44], and assuming Λ = 0 and using 8πGΓ(µ) = 1 for simplicity, we obtain the field equations where and P are the bare dark matter energy density and pressure, respectively. Equations Now, we consider a constant EoS, Then, one obtains, Then, for µ = 1 we have and Furthermore, we recover GR.

The equation for H becomeṡ
The solution to H (t) turns out to be According to (64), For 0 < µ < 3, we have Hence, we obtain a late-time de Sitter solution without including a cosmological constant.
In summary, fractional cosmology allows for an accelerated expansion without adding exotic fluids to the model. Therefore, we now consider a model with cold dark matter to present a specific realization of these possibilities.
The conservation equation for matter (19), for cold dark matter (p CDM = w CDM ρ CDM and w DM = 0), takes the formρ when then reduces toρ Hence, we have for the matter energy density, Choosing we obtain Then, from Equation (28), we obtain Using the redshift parameter 1 + z = 1/a, we obtain then, where α is defined by (71). Comparing with GR, where the EoS w eff is defined through we have Similarly to GR, we have the usual relation q = 1 2 (1 + 3w eff ). Therefore, in fractional cosmology, we have an acceleration (ä > 0, q < 0) as is present in GR when the effective fluid Finally, we have an accelerated expansion if µ > 2, caused by the fractional derivative correction and not by the matter content. That is the powerful advantage of fractional cosmology over GR. This is consistent as ρ CDM → 0 with the asymptotic solution H(t) = µ−1 t , where q = − µ−2 µ−1 , which is a power-law solution a(t) = (t/t 0 ) µ−1 . It is accelerated if µ > 2 and decelerated if 1 < µ < 2, as proven in [61].

First Exact Solution
From Equations (28) and (34) and defining the effective equation of state w := p/ρ, we Assuming w = −1 and is a constant, and by solving (93) algebraically for H, we obtain The deceleration parameter for each algebraic solution is a constant such that the solutions for the scale factor are power laws.
Therefore, upon physical consideration, we select the one that gives an accelerated Universe.
Substituting (94), (28) and (34) into (26) and (27), we obtain the compatibility conditions where α ± is defined by (95) for the existence of an exact solution. We define the current value of H(t 0 ) = H 0 through H 0 t 0 = α ± , and α ± is interpreted as the age parameter α = tH evaluated at t 0 (the current time).
Therefore, to obtain solutions, we solve (97) and (98) simultaneously for µ and w.
From (94), the definition of q and Equations (28), (34) and (93), we have where we set the conditions a (t 0 ) = H 0 , and fix the current value of the scale factor to a(t 0 ) = 1.
For simplicity, let us assume that the source is dust, with p = 0 (w = 0). Then, we have (99) an identity.
That means if we fix the equation of state w, there are specific values that µ has to satisfy to obtain an exact solution.

Second Exact Solution
One can also solve a Riccati equation for µ = 1: which follows from substituting p = wρ into (26) and removing ρ using (28). We obtain the exact solution where is an integration constant, H 0 is the current value of H at t = t 0 and Substituting in (26) and (27), we obtain the compatibility conditions These compatibility conditions have to be satisfied for all t, such that b 1 = 0. Then, (106) becomes and the compatibility conditions are Therefore, to obtain solutions, we solve (112) and (113) simultaneously for µ and w. Upon physical consideration, we remove the cases with w = −1 and µ ∈ {1, 2}, and we assume −1 < w < 1.
For b 1 = 0 and µ = 5/2, we have the physical solution For b 1 = 0 and µ = 7/2, we have the nonphysical solution As in the previous section, if we impose the equation of the state of the fluid as dust, this fixes the values of µ to 5/2.

General Solution
In this section, we are interested in an exact solution that gives the general solution of the system. For this purpose, one can solve the Riccati equation (35) independent of the EoS. That has the H(t) solution defined by where c = −2µ + r − 6α 0 + 9 2µ + r + 6α 0 − 9 , and r = 8µ(2µ − 9) + 105, and for the current time α 0 = H 0 t 0 , where t 0 is the value of t today. H 0 is the current value of the Hubble factor, α 0 , for which we obtain the best-fit values.
That is the exact solution for H(t) studied in [62] (see an analogous case in [36], Equation (36), and in [61], Equation (24)). In this case, expressions (28) and ( This result is generic since it does not require specifying the EoS. Hence, Equation (117) gives a one-parameter family of solutions that gives a complete solution and is independent of the matter content.
Taking the limit ξ → ∞, we have where α(t) = tH is the age parameter.
The main difficulty of this approach is the need to invert (120) to obtain ξ as a function of z because the data are in terms of redshift, which is impossible using analytical tools.
After all, the equation is a rational one. However, the variable ξ can be used as a parameter instead of z in the parametric representation.

Asymptotic Analysis
The following is a precise scheme which does not require inverting (120).
By introducing the logarithmic independent variable s = − ln(1 + z), with s → −∞ as z → ∞, s → 0 as z → 0 and s → ∞ as z → −1, and defining the age parameter as α = tH, we obtain the initial value problem Equation (128) gives a one-dimensional dynamical system. The equilibrium points are Hence, T 1 is a source whenever it exists.

Approximated Analytical Solution
Substituting α = e −3s y 1 2 into (128), it is transformed into the following equation: Let m be the solution of the following Bernoulli equation The solutions of Bernoulli's Equation (137) are three: Assuming y = m + n, where n ≥ 0 is the remainder in the approximation of y by m, and considering the following inequality for m ≥ 0 and n ≥ 0, from (139), we then obtain the differential inequality dn ds ≤ 2(µ − 2)(µ − 1)e 6s + 2(9 − 2µ)e 3s n 1 2 .
Suppose that n = A 2 e 6s , where A is to be determined. Then, Equation (141) leads to The equality occurs at A values of which are the α values of the equilibrium points T 1 and T 2 of the one-dimensional dynamical system (128).
For Equation (129), we obtain Consequently, We have substituted an asymptotic expansion of the integral for large s given by (151). Finally, and as z → −1, where α 0 is defined by (132), A is defined by (147) and m 0 = α 2 0 − A 2 .

Discussion
Our analysis shows differences between standard ΛCDM cosmology-based GR and the fractional version. Given the energy density expression ρ in GR, one can calculate the Hubble parameter through the Friedmann equation. Therefore, we deduce H and we investigate the cosmological history. Finally, we consider the existence of dark matter (w = 0), as w = −1 for Λ and −1 < w < −1/3 for quintessence, as suggested from the observations.
If we proceed as before, to give w, we use w = p/ρ and then use the equations for ρ and p, i.e., Equations (28) and (34). With these expressions, one can calculate H (t) and q (t).
In fractional cosmology, the asymptotic behavior H (t) ∼ 1/t is a characteristic due to the fractional parameter µ > 1.
We related the possible matter scenarios, and according to the relevant discussion on the EoS w, we have two regimes of interest.
First, consider ρ-like dark matter with the behavior of dark energy, −1 < w < −1/3, i.e., quintessence. This case was investigated and the relevant results are summarized in the following.
In this case, the equation of state for radiation is not recovered. if µ < 1 or 1 < µ < 2 or 2 < µ < 5 2 or µ > 5 2 . In this case, the equation of state for radiation is recovered for µ = 41/8.
The second approach consisted of solving the Riccati equation (105) for µ = 1 following substitution of p = wρ in (26) and removing ρ using (28). The exact solution is (106), where H 0 is the current value of H at t = t 0 . From (26) and (27), we found that compatibility conditions have to be satisfied for all t. It is necessary that b 1 = 0. Therefore, to obtain solutions, we solved (112) and (113) simultaneously for µ and w. We have the following solutions: Upon physical consideration, we remove the cases with w = −1 and µ ∈ {1, 2} and we assume −1 < w < 1.
Using the two previous approaches, one obtains power-law solutions of the type a = (t/t 0 ) α .

COSMOLOGICAL CONSTRAINTS
In this section, to study the capability of the models obtained in fractional cosmology to describe the late-time accelerated Universe expansion, we shall constrain the free parameters with the SNe Ia data and OHD. In particular, for the first one, we consider the Pantheon sample [101], which consists of 1048 supernovae data points in the redshift range 0.01 ≤ z ≤ 2.3. On the other hand, we consider the OHD compiled by Magaña et al. [102], which consists of 51 data points in the redshift range 0.07 ≤ z ≤ 2.36.
For the constraints, we compute the best-fit parameters and their respective confidence regions at 1σ(68.3%), 2σ(95.5%) and 3σ(99.7%) confidence levels (CLs) with the affineinvariant Markov chain Monte Carlo (MCMC) method [103], implemented in the pure-Python code emcee [104] by setting 35 chains or "walkers". As a convergence test, we computed the autocorrelation time τ corr of the chains provided by the emcee module at every 50th step. Hence, if the current step is larger than 50τ corr and the values of τ corr changed by less than 1%, then we will consider that the chains are converged and the constraint is stopped. The first 5τ corr steps are thus discarded as "burn-in" steps. This convergence test was complemented with the calculation of the mean acceptance fraction, which must have a value between 0.2 and 0.5 [104] and can be modified by the stretch move provided by the emcee module.
For this Bayesian statistical analysis, we need to construct the following Gaussian likelihood: where N is a normalization constant, which does not influence the MCMC analysis, and χ 2 I is the merit function of each dataset considered, i.e., I stands for SNe Ia, OHD and their joint analysis. In the following subsections, we will briefly describe the construction of the merit function of each dataset, and we will present the main results and discussions.

Observational Hubble Parameter Data
The merit function for the OHD is constructed as

Type Ia Supernovae Data
Similarly to the OHD, the merit function for the SNe Ia data is constructed as where µ i is the observational distance modulus of each SNe Ia at redshift z i with an associated error σ µ,i , µ th is the theoretical distance modulus for each SNe Ia at the same redshift and θ encompasses the free parameters of the model under study. Following this line, for a spatially flat FLRW spacetime, the theoretical distance modulus is given by On the other hand, the distance estimator used in the Pantheon sample is obtained by a modified version of Tripp's formula [105], with two of the three nuisance parameters calibrated to zero with the BEAMS with bias correction (BBC) method [106]. Hence, the observational distance modulus for each SNe Ia reads and M is a nuisance parameter which must be jointly estimated with the free parameters θ of the theoretical model. Furthermore, the Pantheon sample provides the systematic uncertainties in the BBC approach, C sys (currently available online in the GitHub repository https://github.com/dscolnic/Pantheon (accessed on 28 April 2023) in the document sys_full_long.txt). Therefore, we can rewrite the merit function (160) in matrix notation denoted by bold symbols as Then, the merit function (164) can be expanded as [107] where Therefore, by minimizing the expanded merit function (165) with respect toM,M = B(z, θ)/C is obtained and the expanded merit is function reduced to which depends only on the free parameters of the theoretical model.
It is essential to mention that the expanded and minimized merit function (169) provides the same information as the merit function (164). This is a consequence of the fact that the best-fit parameters minimize the merit function. Therefore, the evaluation of the best-fit parameters in the merit function can be used as an indicator of the goodness of the fit independently of the dataset used; the smaller the value of χ 2 min , the better the fit.

Joint Analysis and Theoretical Hubble Parameter Integration
The merit function for the joint analysis is constructed directly as with χ 2 OHD and χ 2 SN e given by Equations (159) and (169), respectively. Following this line, note how in the merit function of the two datasets, the respective model is considered through the (theoretical) Hubble parameter as a function of the redshift (see Equations (159) and (162)). Hence, for the constraint, we numerically integrate the system given by Equations (128) and (129), which represents a system for the variables (α, t) as a function of s = − ln (1 + z), and for which we consider the initial conditions α(s = 0) ≡ α 0 = t 0 H 0 and t(s = 0) ≡ t 0 = α 0 /H 0 . Then, the Hubble parameter is obtained numerically by H th (z) = α(z)/t(z). For this integration, we consider the NumbaLSODA code, a python wrapper of the LSODA method in ODEPACK to C+ (currently available online in the GitHub repository https://github.com/Nicholaswogan/numbalsoda (accessed on 28 April 2023)).
Furthermore, for further comparison, we also constrain the free parameters of the ΛCDM model, whose respective Hubble parameter as a function of the redshift is given by Finally, based on the analysis made in Section 3.5, we consider the parameterization given by Equation (132)  and Ω m,0 ∈ F (0, 1). It is important to mention that due to a degeneracy between H 0 and M, the SNe Ia data are not able to constrain the free parameter h (as a reminder, H 0 = 100 km/s Mpc h), contrary to the case for the OHD and, consequently, in the joint analysis. Thus, the posterior distribution of h for the SNe Ia data is expected to cover all the prior distributions. On the other hand, the prior is chosen as 0 because 0 is a measure of the limiting value of the relative error in the age parameter tH when it is approximated by t 0 H 0 as given by Equation (132). For the mean value 0 = 0, we acquire α 0 = 1 6 (−2µ + r + 9), which implies c = 0. Then, we have the leading term for E(z) defined by (133). The lower prior of µ is because the Hubble parameter (22) becomes negative when µ < 1 in the absence of matter, as we can see from Section 2.3.

Results and Discussion
In Table II, we present the total steps, the mean acceptance fraction and the   The best-fit values of the free parameters space for the ΛCDM model and the fractional cosmological model, obtained for the SNe Ia data, OHD and in their joint analysis, with their corresponding χ 2 min criteria, are presented in Table III. The uncertainties correspond to 1σ, 2σ and 3σ CL. In Figures 2 and 3, we depict the posterior distribution and joint admissible regions of the free parameter space of the ΛCDM model and the fractional cosmological model, respectively. The joint admissible regions correspond to 1σ, 2σ and 3σ CL. Due to the degeneracy between H 0 and M, the distribution of h for the SNe Ia data was not represented in its full parameter space.   Table III.
From the values for the χ 2 min criteria presented in Table III, it is possible to see that the ΛCDM model is the best model to constrain the SNe Ia data, OHD and SNe Ia + OHD data. Nevertheless, the fractional cosmological model studied in this paper exhibits values of the χ 2 min criteria close to the values of the ΛCDM model, with differences of 1.2 for the SNe Ia data, 2.2 for the OHD data and 4.8 for their joint analysis. Thus, this fractional cosmological model is suitable for describing the SNe Ia and OHD data, as can be seen from Figures 4 and 5, which are characterized by accounting for a universe that experiences a transition between a deceleration expansion phase and an accelerated one. Therefore, fractional cosmology can be considered an alternative valid cosmological model to describe the late-time Universe. It is essential to mention that the core of this work is to probe this possibility by studying a particular model; the ΛCDM model is used only as a reference model for this aim.
The analysis of the SNe Ia data leads to h = 0.696 +0.302 −0.295 , µ = 1.340 +2.651 −0.339 and 0 = 1.976 +1.709 −2.067 × 10 −2 , which are the best-fit values at 3σ CL. In this case, the value obtained for h cannot be considered as a best fit due to the degeneracy between H 0 and M. On the other hand, the lower limit of the best fit for µ is very close to 1. That is because the posterior distribution for this parameter is close to this value, as seen from Figure 3. This indicates that a value of the SNe Ia data prefers µ < 1, but, as a reminder, this value leads to a negative Hubble parameter in the absence of matter. However, as can be seen from the same Figure 3, the posterior distribution for these parameters is multi-modal (this explains the large value of τ corr presented in Table II) and, therefore, it is possible to obtain a best-fit value that satisfies µ > 1. It is important to mention that the OHD and the joint analysis do not experience this issue, which allows us to maintain the validity of the prior used for µ.  Table III. On the other hand, the analysis from OHD leads to h = 0.675 +0.041 −0.021 , µ = 2.239 +1.386 −1.190 and 0 = 0.865 +0.793 −0.773 × 10 −2 , which are the best-fit values at 3σ CL. In this case, note how the OHD can properly constrain the free parameters h, µ and 0 , i.e., we obtain the best fit for the priors considered in our MCMC analysis. Furthermore, note how the posterior distribution of µ includes the value of 1, as seen from Figure 3, but for a CL greater than 3σ.
Finally, the joint analysis with data from SNe Ia + OHD leads to h = 0.684 +0.031 −0.027 , µ = 1.840 +1.446 −0.773 and 0 = 1.213 +0.482 −1.057 × 10 −2 , which are the best-fit values at 3σ CL. Focusing our analysis on these results, we can conclude that the region in which µ > 2 is not ruled out by observations. On the other hand, these best-fit values lead to an age of the Universe with a value of t 0 = α 0 /H 0 = 25.62 +6. 89 −4.46 Gyrs at 3σ CL. Universe age is roughly double the one of the ΛCDM models, and is also in disagreement with the value obtained with globular clusters, with a value of t 0 = 13.5 +0. 16 −0.14 ± 0.23 [70]. This discrepancy is a distinction of fractional cosmology. This result also agrees with the analysis made in Section 8 of [61], where the best-fit µ-value was obtained from the reconstruction of H(z) for different priors of µ. The results are summarized in Table IV. In [61], a set of 31 points obtained by differential age tools was considered, namely cosmic chronometers (CC), to represent the measurements of the Hubble parameter, which is cosmologically independent [69] (in the present research we consider the datasets from [102], which consists of 51 data points in the redshift range 0.07 ≤ z ≤ 2.36, 20 more points as compared with [69]). The 1048 luminosity modulus measurements, known as the Pantheon sample, from Type Ia Supernovae cover the region 0.01 < z < 2.3 [101]. In [61], it is unclear if the different priors used for µ lead to properly constraining µ. Their analysis is inconclusive because of their present different values of µ for the different priors used. In order to establish that this fractional cosmological model can describe a universe that experiences a transition from a decelerated expansion phase to an accelerated one, we computed the deceleration parameter q = −1 −Ḣ/H 2 , using the Riccati equation (35), which leads to Following this line, in Figure 6, we depict the deceleration parameter for the fractional cosmological model as a function of the redshift z obtained from the best-fit values for the SNe Ia+OHD data presented in Table III, with an error band at 3σ CL. We also depict the deceleration parameter for the ΛCDM model as a reference model. From this figure, we can conclude that the fractional cosmological model effectively experiences this transition at z t 1, with the characteristic that z t > z t,ΛCDM , where z t,ΛCDM is the transition redshift of the ΛCDM model. Furthermore, the current deceleration parameter of the fractional cosmological model is q 0 = −0.37 +0.08 −0.11 at 3σ CL. On the other hand, in Figures 7 and 8, we depict the matter density and fractional density parameters for the fractional cosmological model (the last one interpreted as dark energy), respectively, as a function of the redshift z for the best-fit values for the SNe Ia+OHD data presented in Table III, with an error band at 1σ CL. We depict the matter density and dark energy density parameters in both figures for the ΛCDM model. From Figure 7, we can see that the matter density parameter for the fractional cosmological model, obtained from Equation (126), presents significant uncertainties, which could be a consequence of their reconstruction from a Hubble parameter that does not take into account any EoS. In this sense, the current value of this matter density parameter at 1σ CL is Ω m,0 = 0.531 +0.195 −0.260 , a value that is in agreement with the asymptotic value obtained from Equation (155) of Ω m,t→∞ = 0.519 +0.199 −0.262 , computed at 1σ CL for the best-fit values for the SNe Ia+OHD data presented in Table III Table III.
Finally, we compute the cosmographic parameter known as the jerk, which quantifies if the fractional cosmological model tends to Λ or if it another kind of DE, which can be written as j(s) = q(s)(2q(s) + 1) − dq(s) ds , where q is given by Equation (172). Hence, j(α(s)) = 12(µ − 4) α(s)  Table III with an error band at 3σ CL, represented by a shaded region. A departure of more than a 3σ CL for the current value for ΛCDM shows an alternative cosmology with an effective dynamical equation of state for the Universe for late times in contrast to ΛCDM.
On the other hand, for the reconstruction of the H0(z) diagnostic [108] for the fractional cosmology, we define where the Hubble parameter is obtained numerically by H(z) = H th (z) as we explained before. Therefore, in Figure 10 Table III, with an error band at 3σ CL, represented by a shaded region. As a reminder, in both Figures 9 and 10, we also depict the jerk and the H0 diagnostic for the ΛCDM model as a reference model.  Table III.

CONCLUSIONS
In this paper, we investigated the cosmological applications of power-law solutions of the type a = (t/t 0 ) α 0 in fractional cosmology, where α 0 = t 0 H 0 is the current age parameter.
Additionally, given µ, the order of the fractional derivative, and w, the matter equation of state, we have imposed compatibility conditions which allow particular solutions to (µ, w).
Finally, we estimated the free parameters (α 0 , µ) using cosmological data and the reparameterization H 0 = 100 km/s Mpc h, α 0 = 1 6 9 − 2µ + 8µ(2µ − 9) + 105 (1 + 2 0 ). Separate analyses of the SNe Ia data and OHD, and the joint analysis with SNe Ia data + OHD, led, respectively, to h = 0.696 +0.302 Focusing our analysis on these results, we can conclude that the region in which µ > 2 is not ruled out by observations. This region of a parameter is relevant because, in the absence of matter, fractional cosmology gives a power-law solution a(t) = (t/t 0 ) µ−1 , which is accelerated for µ > 2. We presented a fractional origin model that leads to an accelerated state without appealing to Λ or dark energy.

Data availability
The data supporting this article can be found in Section 4.