Challenging $\Lambda$CDM with Higher-Order GUP Corrections

TL;DR

This paper explores quantum corrections to the $ ext{Lambda}$CDM cosmological model using a higher-order Generalized Uncertainty Principle, leading to a modified cosmology that fits observational data as well as or better than standard models.

Contribution

It introduces a higher-order GUP framework with two free parameters, derives the modified Friedmann equations, and constrains the model using recent cosmological datasets.

Findings

The GUP-modified model fits observational data comparably to $ ext{Lambda}$CDM.

The deformation parameter is constrained to a negative value, indicating a phantom dark energy regime.

The model naturally reduces to $ ext{Lambda}$CDM in a specific limit.

Abstract

We study quantum corrections to the $\Lambda$CDM model model arising from a minimum measurable length in Heisenberg's uncertainty principle. We focus on a higher-order Generalized Uncertainty Principle, beyond the quadratic form. This generalized GUP introduces two free parameters, and we determine the modified Friedmann equation. This framework leads to a perturbative cosmological model that naturally reduces to $\Lambda$CDM in an appropriate limiting case of the deformation parameters. We construct the modified cosmological scenario, analyze its deviations from the standard case, and examine it as a mechanism for the description of dynamical dark energy. To constrain the model, we employ Cosmic Chronometers, the latest Baryon Acoustic Oscillations from the DESI DR2 release, and Supernova data from the PantheonPlus and Union3 catalogues. Our analysis indicates that the modified GUP model is statistically competitive with the $\Lambda$CDM scenario, providing comparable or even improved fits to some of the combined datasets. Moreover, the data constrain the deformation parameter of the GUP model, with the preferred value found to be negative, which corresponds to a phantom regime in the effective dynamical dark energy description.

Tables

Table 1. Table 1: Priors for the Free Parameters

Priors of Free Parameters	$\mathrm{𝐆𝐔𝐏}$	$𝚲$CDM
${𝐇}_0\left[km{s}^{-1}Mp{c}^{-1}\right]$	$[60,80]$	$[60,80]$
${𝛀}_{m0}$	$[0.01,0.99]$	$[0.01,0.99]$
${𝐫}_d\left[Mpc\right]$	$[120,170]$	$[120,170]$
$\beta$	$[-0.3,0.3]$	$\nexists$
$\gamma$	$[-10,10]$	$\nexists$

Table 2. Table 2: Best-Fit Parameters for the Higher-Order GUP theory

Best Fit/Data	PP&OHD	PP&OHD&BAO	U3&OHD	U3&OHD&BAO
${𝐇}_0$	${67.6}_{-1.8}^{+1.8}$	${68.1}_{-1.7}^{+1.7}$	${66.3}_{-1.9}^{+1.9}$	${66.4}_{-1.9}^{+1.9}$
${𝛀}_{m0}$	${0.319}_{-0.030}^{+0.047}$	${0.307}_{-0.014}^{+0.019}$	${0.333}_{-0.035}^{+0.051}$	${0.322}_{-0.017}^{+0.015}$
${𝐫}_d$	$\nexists$	${147.1}_{-3.4}^{+3.4}$	$\nexists$	${147.3}_{-3.8}^{+3.2}$
$\beta$	${0.002}_{-0.1}^{+0.1}$	$-{0.031}_{-0.030}^{+0.038}$	$-{0.034}_{-0.035}^{+0.046}$	$-{0.0564}_{-0.0025}^{+0.045}$
$\gamma$	${1.3}_{-3.5}^{+1.3}$	${1.8}_{-3.2}^{+1.1}$	$>0.254$	${3.7}_{-4.2}^{+2.0}$
${\chi }_{GUP\min }^2-{\chi }_{\Lambda \min }^2$	$-1.0$	$-3.33$	$-3.19$	$-7.61$
${\mathrm{𝐀𝐈𝐂}}_{GUP}-{\mathrm{𝐀𝐈𝐂}}_{\Lambda }$	$+3.0$	$+0.67$	$+0.81$	$-3.61$

Equations

$$\Delta X\Delta P\ge \frac{1}{2}\left(1+{\beta }_0\Delta P^2+\hat{\gamma }\right),$$

$$[X_i,P_j]=i\left[{\delta }_{ij}(1-{\beta }_0{P}^2)-2{\beta }_0{P}_i{P}_j\right],$$

$$P_i=(1-{\beta }_0{p}^2)p_i,$$

$$P_i{P}^i\Phi -m^2\Phi =0.$$

$$\square \Phi +2{\beta }_0\square \left(\square \Phi \right)+m^2\Phi +O\left({\beta }_0^2\right)=0,$$

$${\dot{x}}_i=\{x_i,\mathcal{H}\},{\dot{p}}_i=\{p_i,\mathcal{H}\},$$

$${\dot{x}}^i=\left(1-{\beta }_0{p}^2\right)\frac{\partial \mathcal{H}}{\partial p_i},{\dot{p}}^i=\left(1-{\beta }_0{p}^2\right)\frac{\partial \mathcal{H}}{\partial x_i}.$$

$$\Delta X_i\Delta P_j⩾\frac{1}{2}[{\delta }_{ij}(1+\beta f\left(P\right))],$$

$${\dot{x}}^i=\left(1-\beta f(p)\right)\frac{\partial \mathcal{H}}{\partial p_i},{\dot{p}}^i=\left(1-\beta f(p)\right)\frac{\partial \mathcal{H}}{\partial x_i}.$$

$${\dot{\rho }}_m+3H{\rho }_m$$

$$\dot{H}+\frac{H^2}{9}+3\left(\frac{1}{2}{\rho }_m-\Lambda \right)$$

$$3H^2={\rho }_m+\Lambda ,$$

$$\mathcal{H}\left(t,{\rho }_m,P\right)=\frac{1}{2}{\rho }_m^{\frac{8}{3}}{P}^2-\frac{3}{2}{\rho }_m^{\frac{1}{3}}\left(1+\frac{\Lambda }{{\rho }_m}\right),$$

$$\dot{H}+H^2=\frac{1}{6}\left(2\Lambda -{\rho }_m\right)\left(1+2\beta f\left(t\right)\right)+\beta \dot{f}\left(t\right)H.$$

$${\Omega }_m=\frac{{\rho }_m}{3H^2},$$

$$\frac{d{\Omega }_m}{d\ln a}=3\left({\Omega }_m-1\right){\Omega }_m+2{\Omega }_m\beta \left(f(a)\left(3{\Omega }_m-2\right)-\frac{df(a)}{d\ln a}\right),$$

$$-\left(1+z\right)\frac{d{\Omega }_m}{dz}$$

$$+\left(1+z\right)\frac{df(z)}{dz}),$$

$$w_{eff}=-1+{\Omega }_m+\frac{2}{3}\beta \left(f\left(z\right)\left(3{\Omega }_m-2\right)+\left(1+z\right)\frac{df\left(z\right)}{dz}\right).$$

$${\left(\frac{H\left(z\right)}{H_0}\right)}^2=\frac{{\Omega }_{m0}{\left(1+z\right)}^3}{{\Omega }_m\left(z\right)},{\Omega }_m\left(0\right)\equiv {\Omega }_{m0},$$

$$\frac{d{\Omega }_m}{d\ln a}=3{\Omega }_m\left({\Omega }_m-1\right)+\left(4{\Omega }_m-\frac{10}{3}\right)\beta +O\left({\beta }^2\right),$$

$${\Omega }_m\left(a\right)=\frac{1}{6}\left(3-4\beta +\sqrt{9+16\beta }-\frac{2\sqrt{9+16\beta \left(1+\beta \right)}}{1+\frac{1-{\Omega }_{m0}{a}^{\sqrt{9+16\beta \left(1+\beta \right)}}}{{\Omega }_{m0}}}\right).$$

$$H^2\left(a\right)=H_{\Lambda CDM}^2\left(a\right)+\beta H_{cor}^B\left(a\right)+O\left({\beta }^2\right),$$

$$\frac{3H_{cor}^B\left(a\right)}{2H_{G0}^2}$$

$$+\frac{5{\left(1-{\Omega }_{m0}\right)}^2}{{\Omega }_{m0}}{a}^3+12\left(1-{\Omega }_{m0}\right)\ln a,$$

$$-\left(1+z\right)\frac{d{\Omega }_m}{dz}$$

$$+2{\left(3{\Omega }_m\right)}^{1-\gamma }\left((1+\gamma ){\Omega }_m-\frac{2}{3}-\gamma \right)\beta ,$$

$$\frac{D_M}{r_d}=\frac{D_L}{\left(1+z\right)r_d},$$

$$\frac{D_V}{r_d}=\frac{{\left(z{D}_H{D}_M^2\right)}^{1/3}}{r_d}.$$

$$\frac{D_H}{r_d}=\frac{c}{r_{d}H(z)},$$

$${\mu }^{th}=5{\log }_{10}{D}_L+25,$$

$$AIC\simeq -2\ln {\mathcal{L}}_{\max }+2\kappa ,$$

Full text

Observational Constraints; Generalized Uncertainty Principle; Dark Energy

I Introduction

The recent cosmological data support cosmological theories that deviate from the standard $\Lambda$-Cold Dark Matter ($\Lambda$CDM) model, with a dynamical dark energy component that can cross the phantom divide line union ; des4 ; des5 ; des6 ; ra1 ; ra2 ; ra3 ; ks1 ; ten1 ; your ; orl . There is a plethora of proposed models in the literature that attempt to explain the cosmological data from phenomenological and theoretical perspectives and elucidate the origin of the dark energy component. Parametric dark energy models were recently considered in a1 ; a2 ; a3 ; a4 ; a5 ; a6 ; a66 ; a666 ; a6b ; a6c , non-cold dark matter models were investigated in a77 ; a88 ; a99 , while scalar fields and modified theories of gravity were examined in a7 ; a8 ; a9 ; a10 ; a10a ; a10b ; a11c ; a11d . Cosmological models with interaction in the dark sector were studied in a11 ; a12 ; a13 ; a14 ; a15 ; a15a ; supr ; Pan:2025qwy , and entropic theories in Sheykhi:2018dpn ; Lymperis:2018iuz ; Saridakis:2020lrg ; Lymperis:2021qty ; Nojiri:2022dkr ; Gohar:2023hnb ; a16 ; a17 . On the other hand, quantum effects in the dark sector of the universe were examined in a18 ; a19 ; a20 ; angup2 .

In this context, the Generalized Uncertainty Principle (GUP) emerges as a quantum-gravity inspired modification of Heisenberg’s uncertainty relation, which incorporates a fundamental minimal length and gives rise to a deformed algebraic structure GUP1 ; GUP2 ; GUP3 ; GUP4 . Such a framework modifies the dynamical laws by embedding quantum effects into the equations of motion. In particular, as shown in angup2 , the quadratic form of the GUP can be phenomenologically recast within cosmology so as to yield semiclassical deformations of the Friedmann equations in the $\Lambda$CDM scenario. The resulting model admits a closed-form solution and provides an effective description of dynamical dark energy capable of crossing the phantom divide line.

Over the past decades, numerous studies have investigated the implications of a minimum length in gravitational theories de1 ; de2 ; de4 ; bh1 ; bh2 ; bh3 ; de4a ; de4b ; de4d ; de5 ; de10 ; de9 ; de8 ; bh5 ; bh6 ; bb1 ; Jizba:2022icu ; angup . It has been shown that the impact of the GUP on cosmological field equations exhibits dynamical similarities to those found in higher-order gravity theories f1 , as well as in scalar field models incorporating a Pais-Uhlenbeck oscillator term f2 . Additionally, various formulations of the GUP have been proposed in the literature, each motivated by different quantum-gravity considerations. These distinct versions provide alternative phenomenological frameworks, enriching the exploration of minimum-length effects in both gravitational and cosmological settings (see BossoRev for a recent review).

Starting from the above premises, in this work we extend the analysis presented in angup2 by considering a more general GUP. Specifically, we consider a two-parameter higher-order formulation and analyze its implications for the modifications of the $\Lambda$CDM cosmology. We apply late-time cosmological data to test the higher-order GUP as a potential mechanism for dynamical dark energy behavior. This analysis allows us to assess the implications of the GUP in cosmological studies and to identify the specific form of its correction to Heisenberg’s uncertainty relation. Finally, we use cosmological data to constrain the GUP model.

The structure of the rest of the paper is as follows: in Sec. II we introduce the GUP and study the effects of the minimum measurable length in the classical field equations. The cosmological applications of the GUP are discussed in Sec. III, where we review previous results for the $\Lambda$CDM universe in a spatially flat Friedmann-Lemaître-Robertson-Walker(FLRW) spacetime, in which the second Friedmann equation is modified by the presence of the GUP. We then consider a higher-order GUP model and reconstruct the master equation governing cosmic evolution. This equation depends on two parameters: the deformation parameter of the GUP and the exponent characterizing the higher-order extension. The main results of this study are presented in Sec. IV, where we constrain our GUP-modified cosmological theory with late-time observational data and compare it with the $\Lambda$CDM model. For the statistical analysis, we consider Cosmic Chronometers, the recent data for the Baryon Acoustic Oscillations from the DESI DR2 collaboration and the Supernova observations of the PantheonPlus and Union3 catalogues. Finally, in Sec. V we draw our conclusions. Unless otherwise specified, Planck units are used throughout.

II Generalized Uncertainty Principle

The most widely employed realization of the GUP in the literature is the quadratic model considered in GUP3 , i.e.,

[TABLE]

where $\Delta X,\Delta P$ are the position and momentum uncertainties, respectively, and $\beta_{0}$ is a suitable parameter with dimension $1/P^{2}$.

A generalization of Eq. (1) can be formulated by introducing the following deformation of the canonical algebra mag1 ; mag2

[TABLE]

while the spatial coordinates commute, as do the momentum components111The range of variability of the indices $i,j$ depends on the specific GUP model considered. In Moayedi , for instance, a Lorentz-covariant deformed algebra is proposed, with the indices $i,j\in\{1,2,3,4\}$. A possible drawback of this model is that it also involves a commutator for the time operator. However, the definition of such operator in the quantum framework remains an open problem Bauer:2016aio . As our present goal is to investigate the phenomenological implications of the minimal length in cosmology, we postpone a detailed analysis of this conceptual aspect to future work.. Deformations of this type have been considered in the literature in the context of quantum gravity-motivated models Quesne2006 ; Vagenas ; Kemph1 ; Kemph2 ; Vag1 ; Vag2 ; Moayedi .

In both Eqs. (1) and (2) the parameter $\beta_{0}$ is the deformation parameter, which is directly related to the existence of a minimum measurable length. It is usually expressed as $\beta_{0}={\beta}/{M_{Pl}^{2}c^{2}=\beta}\ell_{Pl}^{2}/2\hbar^{2}$, where $M_{Pl}$ is the Planck mass, $\ell_{Pl}$ ($\approx 10^{-35}~m)$ is the Planck length, $M_{Pl}c^{2}$ ($\approx 1.2\times 10^{19}~GeV)$ is the Planck energy and $\beta$ is a dimensionless coupling parameter, often assumed to be of order unity in string theory GUP1 ; Amati:1987wq and extended field theoretical models hm5 (here we have temporarily restored the correct units). Although $\beta$ is commonly taken to be positive, a number of studies have considered the possibility of negative values of $\beta$ (see, e.g., nn1 ; nn2 ; nn3 ; nn4 ; Jizba:2022icu ). Clearly, in the limit $\beta\rightarrow 0$ the standard Heisenberg algebra, together with the usual structure of quantum mechanics, is recovered.

For computational purposes, we now adopt the coordinate representation for the position and momentum operators. We set $X_{i}=x_{i}$ and modify the momentum operator as Moayedi

[TABLE]

where $x_{i},~p_{i}$ denote the standard operators satisfying the canonical algebra $[x_{i},p_{j}]=i\hbar\delta_{ij}$ , and $p^{2}=p_{i}p^{i}$.

As a paradigmatic system to study the effects of the GUP, we consider a free spin-0 particle $\Phi$ with rest mass $m$. For such a system, the Klein-Gordon equation reads

[TABLE]

Within the coordinate representation, making use of (3) one obtains Moayedi

[TABLE]

where $\square=\partial_{i}\partial^{i}$ denotes the d’Alembertian operator, $\partial^{j}\equiv\frac{\partial}{\partial x_{j}}=-ip^{j}$ and $2\beta_{0}\square\left(\square\right)$ is a fourth-order operator which describes the quantum corrections to the GUP. Owing to the deformation of the algebra, the emergence of a minimum length gives rise to an additional quantum correction term in the classical field equations.

Let $\mathcal{H}=\mathcal{H}(x,p)$ be the Hamiltonian function describing the classical system. In the classical limit, the equations of motion follow from Hamilton’s equations and are given by

[TABLE]

where $\{\cdot,\cdot\}$ denotes the Poisson bracket. By expanding the bracket, one obtains the modified Hamilton’s equations hm1 ; hm2 ; hm3 :

[TABLE]

Up to this point, we have considered the quadratic form of the GUP. However, the notion of a minimum length can also arise within a more general framework, in which the uncertainty relation takes the form hm4 ; hm5 ; hm6 ; hm7 ; hm8

[TABLE]

for suitable choices of the dimensionless function $f(P)$ BossoPW .

In this framework, Hamilton’s equations governing the classical dynamics are accordingly modified as

[TABLE]

III $\Lambda$CDM with GUP Corrections

In angup , the effects of the GUP on the field equations of Szekeres spacetimes were investigated. In particular, when the field equations for the Szekeres system are expressed in terms of the propagation and constraint equations tsa1 , the propagation equations constitute a Hamiltonian system. This allows quantum correction terms to be introduced into the gravitational field equations.

The FLRW geometry is recovered in the limit where the Szekeres geometry becomes isotropic and homogeneous. The propagation equations for the $\Lambda$CDM model are

[TABLE]

with the constraint

[TABLE]

where $H=\dot{a}/a$ denotes the Hubble parameter, $a$ is the scale factor, $\rho_{m}$ is the energy density of cold dark matter (CDM) and $\Lambda$ represents the cosmological constant (for our purposes of studying the GUP as a potential mechanism for dynamical dark energy behavior, we can safely neglect the radiation component). An overdot indicates differentiation with respect to time.

Equations (10), (11) form a Hamiltonian system with Hamiltonian function

[TABLE]

where the momentum is defined as $P=\dot{\rho}_{m}\rho_{m}^{-\frac{8}{3}}$.

In the presence of the minimum length, the modified Hamilton’s equation lead to the modified second Friedmann’s equation

[TABLE]

Thus, the evolution of the fractional matter energy density,

[TABLE]

is governed by the nonlinear differential equation

[TABLE]

or equivalently

[TABLE]

where $z=a^{-1}-1$ denotes the redshift (with $a_{0}=1$, where the subscript [math] refers to the present epoch).

Moreover, the equation-of-state parameter of the effective fluid is defined as

[TABLE]

From Eq. (17), together with expression (15), we can construct the Hubble function

[TABLE]

where $\Omega_{m0}$ denotes the present-day matter energy density parameter.

Hence, the specification of the deformation function $f$ plays a central role in governing the dynamics and evolution of the cosmological model.

III.1 Quadratic GUP

The model $f=\tfrac{H^{2}}{\rho_{m}}$ was introduced in angup2 . The factor $\tfrac{1}{\rho_{m}}$ was included to ensure that $f$ is correctly dimensionless. Hence, when expressed in terms of the energy density parameter $\Omega_{m}$, the function $f$ takes the form $f=\frac{1}{3\Omega_{m}}$.

For this GUP, the dynamics in Eq. (16) becomes

[TABLE]

which admits the following closed-form solution for the energy density:

[TABLE]

The closed-form expression for the Hubble function is given as

[TABLE]

where $H_{cor}^{B}\left(a\right)$ encodes the quantum corrections to the Hubble function, namely

[TABLE]

and $H_{G0}^{2}$ is a normalization constant. Building on this model, it was shown in angup2 that GUP can serve as a mechanism for constructing dynamical dark energy models. In particular, it has been employed as a theoretical framework to realize a time-varying cosmological constant starting from the standard one. It was further demonstrated that the resulting GUP-modified $\Lambda$CDM model provides an improved fit to observational data compared with the undeformed theory.

For the quadratic GUP, a closed-form solution for the Hubble function was previously presented in vag01 . However, the solution obtained there differs from the one considered in our work because we adopt a different approach to formulating the GUP and modifying the gravitational field equations. In vag01 , the authors modify Einstein’s action integral by replacing the standard momentum with its deformed part, which introduces additional quantum-correction terms. In contrast, our analysis follows the approach described in BossoPW , where the deformed algebra is used to modify Hamilton’s equations, leading to new definitions for the observables, including the Hubble function. These different formulations lead to different reconstructed Hubble functions.

III.2 Beyond the quadratic GUP

In the present study, we propose an extension of the above framework by generalizing the functional dependence of the GUP model to $f=\left(\frac{H^{2}}{\rho_{m}}\right)^{\gamma}$, where $\gamma$ is an additional free parameter. The parameter $\gamma$ controls the strength and nonlinearity of quantum corrections to the Friedmann equations, with $\gamma=1$ reducing to the quadratic GUP case.

This modification is motivated by both theoretical and phenomenological considerations. On the theoretical side, the power-law form represents the most natural generalization of the linear dependence previously assumed, with the case $\gamma=1$ recovered as a special limit. It also allows for a broader class of deformations that could emerge from different realizations of quantum gravity, where non-linear corrections to the uncertainty principle may be expected. On the phenomenological side, the inclusion of $\gamma$ provides enhanced flexibility in modeling dynamical dark energy, as the ratio $H^{2}/\rho_{m}$ encodes the interplay between the cosmic expansion rate and the matter sector. By varying $\gamma$, one can interpolate between different scaling behaviors of the effective cosmological term, thereby capturing a richer range of cosmological evolutions.

In this framework, the evolution of the energy density is given by the differential equation

[TABLE]

It is worth noting that for $\gamma=1$, Eq. (III.2) consistently reproduces the dynamics of Eq. (20). Moreover, in the absence of a minimal length, i.e. for $\beta=0$, the standard $\Lambda$CDM limit is recovered.

For arbitrary values of the parameter $\gamma$, however, Eq. (III.2) does not admit a closed-form analytic solution. In this case, we solve the differential equation numerically with the initial condition $\Omega_{m}(0)=\Omega_{m0}$, and, using the definition in Eq. (19), we subsequently compute the Hubble function $H(z)$.

IV Observational Constraints

We investigate the higher-order GUP cosmology (III.2) as a possible mechanism for describing late-time cosmic acceleration. In particular, we employ observational data to constrain the free parameters $\{H_{0},\Omega_{m0},\beta,\gamma\}$ of the model (III.2), and we compare its performance with that of the standard $\Lambda$CDM scenario.

IV.1 Observational Data

In the following, we present the observational datasets employed in this study.

• •

Observational Hubble Data (OHD): We consider the Cosmic Chronometers (CC), which provide direct measurements of the Hubble parameter without relying on any cosmological assumptions. Cosmic Chronometers correspond to passively evolving galaxies with synchronous stellar populations and similar cosmic evolution co01 . This dataset is therefore model independent. In the present analysis, we use the 31 direct measurements of the Hubble parameter in the redshift range $0.09\leq z\leq 1.965$ reported in cc1 .

• •

Baryonic acoustic oscillations (BAO): We employ the recent release of the Dark Energy Spectroscopic Instrument (DESI DR2) baryon acoustic oscillation (BAO) observations des4 ; des5 ; des6 . This dataset provides measurements of the transverse comoving angular distance ratio,

[TABLE]

the volume-averaged distance ratio,

[TABLE]

and the Hubble distance ratio,

[TABLE]

at seven distinct redshifts, where $D_{L}$ refers to the luminosity distance and $r_{d}$ denotes the sound horizon at the drag epoch, which in our analysis is treated as a free parameter (we have restored $c$ for consistency with the notation in the literature).

• •

Supernova of PantheonPlus (PP): This catalogue includes 1701 light curves of 1550 spectroscopically confirmed supernova events. The data provide measurements of the distance modulus $\mu^{{obs}}$ at redshifts in the range $10^{-3}<z<2.27$ pan . The theoretical distance modulus is defined as

[TABLE]

where, in a spatially flat FLRW geometry, the luminosity distance is expressed in terms of the Hubble function as $D_{L}(z)=c\,(1+z)\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})}$. We employ the PantheonPlus catalogue without the SH0ES Cepheid calibration.

• •

Supernova of Union3 (U3): This Supernova catalogue includes 2087 events within the same redshift range as the PP data, of which 1363 are shared with the PantheonPlus catalogue union .

IV.2 Methodology

For the analysis of our cosmological theory, we employ the Bayesian inference framework COBAYA222https://cobaya.readthedocs.io/ cob1 ; cob2 , using a custom theory implementation together with the MCMC sampler mcmc1 ; mcmc2 . Furthermore, for the analysis of the results, we make use of the GetDist library333https://getdist.readthedocs.io/ getd .

We also apply the same observational tests to the $\Lambda$CDM model. Given the different number of free parameters, we use the Akaike Information Criterion (AIC) AIC to assess which model is statistically favored. If $\chi_{\min}^{2}=-2\ln\mathcal{L}_{\max}$ denotes the minimum chi-squared value corresponding to the maximum likelihood, then the AIC is defined as

[TABLE]

where $\kappa$ represents the number of free parameters of the model.

For the higher-order GUP cosmology, $\kappa=5$, and for the $\Lambda$CDM model is $\kappa=3$. Consequently,

[TABLE]

which can be equivalently expressed as

[TABLE]

According to Akaike’s scale, the value of $\Delta AIC$ provides information on which model offers a better fit to the data. Specifically, for $\lvert\Delta{AIC}\rvert<2$, the two models are statistically equivalent; for $2<\lvert\Delta{AIC}\rvert<6$, there is weak evidence in favor of the model with the smaller AIC value; if $6<\lvert\Delta{AIC}\rvert<10$, the evidence is strong; and for $\lvert\Delta{AIC}\rvert>10$, there is clear evidence supporting the preference for the model with the lower AIC.

At this stage, it is important to note that, in order to eliminate the influence of systematic errors in the comparison between the two models, the Hubble function for the $\Lambda$CDM model has also been derived numerically, using the same procedure as for Eq. (III.2).

IV.3 Results

For the MCMC sampling, we adopt the priors listed in Table 1. The results obtained for each dataset are presented below.

For the combined PP&OHD dataset, we obtain the best-fit parameters $H_{0}=67.6_{-1.8}^{+1.8}\hskip 2.84526ptkm\hskip 1.70717pts^{-1}Mpc^{-1}$, $\Omega_{m0}=0.319_{-0.030}^{+0.047}$, $\beta=0.002_{-0.1}^{+0.1}$, and $\gamma=1.3_{-3.5}^{+1.3}$. The comparison with the $\Lambda$CDM model yields $\chi_{{GUP},\,\min}^{2}-\chi_{\Lambda,\,\min}^{2}=-1.0$ and $\Delta{AIC}=+3.0$, which indicates weak evidence in favor of the $\Lambda$CDM scenario.

When the BAO data are included, i.e., for the combined dataset PP&OHD&BAO, the best-fit parameters are $H_{0}=68.1_{-1.7}^{+1.7}\hskip 2.84526ptkm\hskip 1.70717pts^{-1}Mpc^{-1}$, $\Omega_{m0}=0.307_{-0.014}^{+0.019}$, $\beta=-0.031_{-0.030}^{+0.038}$ and $\gamma=1.8_{-3.2}^{+1.1}$. In this case, $\chi_{GUP\min}^{2}-\chi_{\Lambda\min}^{2}=-3.33$ and $\Delta AIC=+0.67$. Thus, the GUP model yields a slightly better fit to the data than $\Lambda$CDM. However, once the larger number of free parameters is accounted for, the AIC indicates that the two model are statistical equivalent.

For the U3&OHD dataset, the best-fit parameters are $H_{0}=66.3_{-1.9}^{+1.9}\hskip 2.84526ptkm\hskip 1.70717pts^{-1}Mpc^{-1}$, $\Omega_{m0}=0.333_{-0.035}^{+0.051}$, $\beta=-0.034_{-0.035}^{+0.046}$ and $\gamma>0.254$, with $\chi_{GUP\min}^{2}-\chi_{\Lambda\min}^{2}=-3.19$ and $\Delta AIC=+0.81$. GUP model provides a slightly lower $\chi^{2}$ than $\Lambda$CDM, the AIC once again indicates that there is not any preferred model by the data.

Finally, for the combined U3&OHD&BAO data, the best-fit parameters are $H_{0}=66.4_{-1.9}^{+1.9}\hskip 2.84526ptkm\hskip 1.70717pts^{-1}Mpc^{-1}$, $\Omega_{m0}=0.322_{-0.017}^{+0.015}$, $\beta=-0.0564_{-0.0025}^{+0.045}$, and $\gamma=3.7_{-4.2}^{+2.0}$. The corresponding statistical indicators are $\chi_{GUP\min}^{2}-\chi_{\Lambda\min}^{2}=-7.61$ and $\Delta AIC=-3.61$. In this case, the GUP model provides a noticeably better fit to the data, and the AIC indicates that GUP model has a week support over the $\Lambda$CDM from this dataset.

Regarding the deformation parameter $\beta$, in most combined datasets the value $\beta=0$ lies within $1\sigma$, except for U3&OHD&BAO, where it is consistent only within $2\sigma$. For PP&OHD, the best-fit value is $\beta>0$, while for the other datasets it is $\beta<0$. According to Ref. angup2 , a negative $\beta$ is associated with phantom-like behavior in the effective dark energy sector, leading to a more rapid cosmic expansion. This interpretation is consistent with the U3&OHD&BAO results, which strongly favor an accelerated expansion.

As for the parameter $\gamma$, the best-fit values are constrained to be positive, with the case $\gamma=1$ lying within $1\sigma$. Nevertheless, for most datasets the best-fit value is greater than unity, which points towards a preference for the generalized GUP framework introduced in this work.

In Figs. 1 and 2, we present the $2\sigma$ confidence regions for the posterior parameters. In all cases, the best-fit values lie within the $1\sigma$ contours. A summary of these results is provided in Table 2.

V Conclusions

We examined the GUP as a possible mechanism to explain the late-time acceleration of the universe. Focusing on a spatially flat FLRW geometry with cold dark matter and a cosmological constant, we examined the modifications induced by the existence of a minimal length scale in the classical gravitational equations. We found that the second Friedmann equation - namely, the Raychaudhuri equation - is altered in this framework, leading to a modified $\Lambda$CDM cosmology that can naturally realize a dynamical dark energy behavior.

For the GUP relation, we considered a two-parameter extension beyond the quadratic (i.e., $\gamma=1$) case, incorporating higher-order derivative corrections. The resulting cosmological field equations were solved numerically and confronted with late-time observations, namely the Observational Hubble Data from Cosmic Chronometers, the Baryon Acoustic Oscillations from the DESI DR2 release and the Supernova samples from the PantheonPlus and Union3 catalogues.

We fitted our modified cosmological model to different combinations of observational data and found that the GUP-modified theory can reproduce the observations with a quality comparable to, and in some cases slightly better than, the standard $\Lambda$CDM scenario. Nevertheless, when model selection criteria are applied, $\Lambda$CDM remains favored: owing to its smaller number of free parameters, it achieves a lower AIC value. Consequently, the Akaike Information Criterion does not permit us to draw firm conclusions regarding a preferred model.

For all data combinations, the quadratic GUP limit is recovered within the $1\sigma$ range of the parameter $\gamma$. However, the best-fit values of $\gamma$ are greater than unity, lending support to the higher-order GUP framework. In particular, when the Union3 data are included, the best-fit value of $\gamma$ is further increased. Regarding the deformation parameter $\beta$, the value $\beta=0$, which corresponds to the standard $\Lambda$CDM model, lies within $1\sigma$ for the combined datasets PP&OHD, PP&OHD&BAO, and U3&OHD, and within $2\sigma$ for the U3&OHD&BAO dataset. In addition, our analysis reveals a tendency toward negative values, $\beta<0$. As discussed in Ref. angup2 , a negative deformation parameter $\beta$ leads the effective dynamical dark energy sector to exhibit phantom-like behavior.

In summary, the GUP offers a simple and natural mechanism through which quantum effects can induce dynamical behavior in the dark energy sector. Moreover, we have shown that cosmological observations of the expansion history can be employed to constrain the free parameters of the deformation algebra and, in particular, to place bounds on the deformation parameter $\beta$.

In a future work, we plan to extend the present analysis by incorporating Cosmic Microwave Background (CMB) data. While in this study we have focused on low- and intermediate-redshift probes, the inclusion of CMB measurements will provide tight constraints on the model parameters across the entire expansion history. This will allow us to examine more rigorously whether the higher-order GUP framework can alleviate existing cosmological tensions, such as the discrepancy between local and early-universe determinations of the Hubble constant, as well as possible deviations in the growth of cosmic structures.

Acknowledgements.

AP & GL authors thanks the support of VRIDT through Resolución VRIDT No. 096/2022 and Resolución VRIDT No. 098/2022. Part of this study was supported by FONDECYT 1240514. The research of GGL is supported by the postdoctoral fellowship program of the University of Lleida. GGL gratefully acknowledges the contribution of the LISA Cosmology Working Group (CosWG), as well as support from the COST Actions CA21136 - Addressing observational tensions in cosmology with systematics and fundamental physics (CosmoVerse) - CA23130, Bridging high and low energies in search of quantum gravity (BridgeQG) and CA21106 - COSMIC WISPers in the Dark Universe: Theory, astrophysics and experiments (CosmicWISPers).