The work of [25] characterises the apparent horizon of FLRW universe in analogy with the event horizon of a BH. The apparent horizon is a marginally trapped surface with vanishing expansion that always exists in FLRW universe (differently from the event and particle horizons) which makes it the best suited cosmological horizon for thermodynamical considerations, also in view of its trapping character [31]. To set our notation, we write the Friedmann equations as
$$\begin{aligned} H^2= & {} \frac{8\pi G}{3}\rho -\frac{kc^2}{a^2} \nonumber \\ \dot{H}= & {} -4\pi G\left( \rho +\frac{p}{c^2}\right) + \frac{kc^2}{a^2}. \end{aligned}$$
(3)
The continuity equation reads
$$\begin{aligned} \dot{\rho }+3H\left( \rho +\frac{p}{c^2}\right) =0 \end{aligned}$$
(4)
and will not be affected by the GUP corrections. Since we are interested in recovering the GUP-modified entropy-area law, we need to define the radius and the area of the apparent horizon. The radius is
$$\begin{aligned} \tilde{r}_{\mathrm {A}}=\frac{c}{\sqrt{H^2+\frac{kc^2}{a^2}}} \end{aligned}$$
(5)
and yields the area
$$\begin{aligned} A=4\pi \tilde{r}_{\mathrm {A}}^2=\frac{4\pi c^2}{H^2+\dfrac{kc^2}{a^2}}. \end{aligned}$$
(6)
The expressions for the entropy and the temperature associated to a BH horizon read
$$\begin{aligned} S=\frac{k_\mathrm{B}c^3 A}{4G\hbar }\quad T=\frac{\hbar c^3}{8\pi G k_\mathrm{B}M}, \end{aligned}$$
(7)
where A is the area of the event horizon, M the mass of the black hole and \(k_\mathrm{B}\) the Boltzmann constant. In the well-studied case of a Schwarzschild BH, the Schwarzschild radius \(r_\mathrm{S}=2GM/c^2\) is used to obtain
$$\begin{aligned} T=\frac{\hbar c}{4\pi k_\mathrm{B}r_\mathrm{S}}. \end{aligned}$$
(8)
In [32], this result has been generalized to the case of a de Sitter universe, and in [25] it is only assumed as a working hypothesis that the expressions above also work for the apparent horizon. However, since the authors do successfully recover the Friedmann equations, their assumption is justified a posteriori. Thus, for our purposes, we can simply assume [33]
$$\begin{aligned} T=\frac{\hbar c}{4\pi k_\mathrm{B}\tilde{r}_{\mathrm {A}}}. \end{aligned}$$
(9)
Quadratic GUP
We will now briefly review the treatment of [27], in which it is shown in an arbitrary number of dimensions that a modified form of the Friedmann equations can be recovered when the GUP corrects the standard entropy-area law (2). In the following, we will also restore all physical units which are generally omitted in theoretical studies, since our goal is to find a numerical estimate for the GUP parameter. The authors of [27] start from the following expression for the GUP, which includes a quadratic correction in momentum uncertainty (note that the dimensionless GUP parameter \(\beta \) in (1) is here \(\alpha _\mathrm{Q}^2\), where Q stands for “quadratic”)
$$\begin{aligned} \varDelta x \varDelta p \ge \frac{\hbar }{2}\left( 1 +\frac{\alpha _\mathrm{Q}^{2} \ell _\mathrm{p}^2}{\hbar ^2}\varDelta p^2 \right) . \end{aligned}$$
(10)
We can straightforwardly solve for the momentum uncertainty, obtaining
$$\begin{aligned} \varDelta p\ge \frac{\hbar \, \varDelta x}{\alpha _\mathrm{Q}^2\ell _\mathrm{p}^2}-\sqrt{\frac{\hbar ^2\,\varDelta x^2}{\alpha _\mathrm{Q}^4\ell _\mathrm{p}^4}-\frac{\hbar ^2}{\alpha _\mathrm{Q}^2\ell _\mathrm{p}^2}}, \end{aligned}$$
(11)
which can be recast as
$$\begin{aligned} \varDelta p\ge \frac{\hbar }{2 \varDelta x}\left[ \frac{2\varDelta x^2}{\alpha _\mathrm{Q}^2\ell _\mathrm{p}^2}-\frac{2 \varDelta x^2}{\alpha _\mathrm{Q}^2\ell _\mathrm{p}^2}\sqrt{1-\frac{\alpha _\mathrm{Q}^2\ell _\mathrm{p}^2}{\varDelta x^2}}\right] . \end{aligned}$$
(12)
The expression inside square brackets is the function characterizing the departure of the GUP from the HUP, which we define as
$$\begin{aligned} f_\mathrm{G}(\varDelta x^2)=\frac{2\varDelta x^2}{\alpha _\mathrm{Q}^2\ell _\mathrm{p}^2}-\frac{2\varDelta x^2}{\alpha _\mathrm{Q}^2\ell _\mathrm{p}^2}\sqrt{1-\frac{\alpha _\mathrm{Q}^2\ell _\mathrm{p}^2}{\varDelta x^2}}. \end{aligned}$$
(13)
In order to study the GUP effects on the thermodynamics of FLRW universe, the authors of [27] (relying on [34]) consider the following picture: if the apparent horizon has absorbed or radiated a particle with energy dE, this energy can be identified with the uncertainty in momentum, \(dE\simeq c\varDelta p\). Therefore, the HUP \(\varDelta p\ge \hbar /(2\varDelta x)\) yields the corresponding increase or decrease in the area of the apparent horizon, due to (7):
$$\begin{aligned} dA=\frac{4G\hbar }{k_\mathrm{B}c^3 T}dE\simeq \frac{2G\hbar ^2}{k_\mathrm{B}c^2 T\varDelta x}. \end{aligned}$$
(14)
However, in the case where the GUP is taken into account, this relationship becomes
$$\begin{aligned} dA_\mathrm{G}\simeq \frac{2G\hbar ^2}{k_\mathrm{B}c^2 T}\frac{f_\mathrm{G}(\varDelta x^2)}{\varDelta x}, \end{aligned}$$
(15)
with
$$\begin{aligned} dA_\mathrm{G}=f_\mathrm{G}(\varDelta x^2)dA. \end{aligned}$$
(16)
The uncertainty in position of the absorbed or radiated particle is reasonably considered of the order of its Compton length \(\lambda \), which is approximately the inverse of the Hawking temperature in natural units. In physical units, as noted in [22], it is customary to consider \(\varDelta x\simeq \lambda \). For a Schwarzschild BH, the particle has a wavelength of the order of the inverse Hawking temperature (for an asymptotic observer) or, more generally, of the inverse of the surface gravity \(\kappa ^{-1}=2r_\mathrm{S}\) (since \(T=\kappa /2\pi \) in natural units). As previously noted, it seems sensible to extend the argument from the context of BHs to that of the apparent cosmological horizon, thus assuming \(\varDelta x\simeq 2\tilde{r}_{\mathrm {A}}=\sqrt{A/\pi }\).
We can express the departure function (13) in terms of the area of the apparent horizon and subsequently in terms of the entropy. If we expand \(f_\mathrm{G}(A)\) around \(\alpha _\mathrm{Q}\, l_p =0\), we obtain
$$\begin{aligned} f_\mathrm{G}(A)=1+\frac{\pi \alpha _\mathrm{Q}^2\ell _\mathrm{p}^2}{4}\frac{1}{A}+\frac{\pi ^2\alpha _\mathrm{Q}^4\ell _\mathrm{p}^4}{8}\frac{1}{A^2}+{\mathcal {O}}(\alpha _\mathrm{Q}^6) \end{aligned}$$
(17)
up to second order, which is sufficient for the purposes of this work. Substituting (17) in (16) and integrating, we find the expression
$$\begin{aligned} A_\mathrm{G}=A+\frac{\pi \alpha _\mathrm{Q}^2\ell _\mathrm{p}^2}{4} \ln {A}-\frac{\pi ^2\alpha _\mathrm{Q}^4\ell _\mathrm{p}^4}{8}\frac{1}{A}, \end{aligned}$$
(18)
where in \(\ln {A}\) we have included the integration constant \(A_0\).
For the entropy, due to (16) and (7), we obtain \(dS_\mathrm{G}=f_\mathrm{G}(A)\,dS\). After integration, we find the modified entropy-area relation to be
$$\begin{aligned} S_\mathrm{G}=\frac{k_\mathrm{B}c^3}{4G\hbar }\left[ A+\frac{\pi \alpha ^2\ell _\mathrm{p}^2}{4} \ln {A}-\frac{\pi ^2\alpha ^4\ell _\mathrm{p}^4}{8}\frac{1}{A}\right] . \end{aligned}$$
(19)
The apparent horizon approach to find the Friedmann equations devised in [25] consists in the application of the first law of thermodynamics to the apparent horizon of a FLRW universe, with the additional assumption (7). This procedure involves the definition of a work density and an energy supply vector, respectively regarded as the work done by a change of the apparent horizon and the total energy flow through it, which is associated to the entropy. Such definitions yield a specific form of the first law of thermodynamics for cosmological horizons. The first time derivative of (5) is
$$\begin{aligned} \dot{\tilde{r}}_{\mathrm {A}}=-\frac{1}{c^2}\tilde{r}_{\mathrm {A}}^3 H\left( \dot{H}-\frac{kc^2}{a^2}\right) , \end{aligned}$$
(20)
which can be rewritten as
$$\begin{aligned} \frac{d\tilde{r}_{\mathrm {A}}}{\tilde{r}_{\mathrm {A}}^3}=-\frac{1}{c^2}H\left( \dot{H}-\frac{kc^2}{a^2}\right) dt. \end{aligned}$$
(21)
Taking into account (13) and (19), it is straightforward to see that
$$\begin{aligned} S_\mathrm{G}'(A)=\frac{k_\mathrm{B}c^3}{4G\hbar }f_\mathrm{G}(A), \end{aligned}$$
(22)
where \(S_\mathrm{G}'\) is a first derivative with respect to A.
Following [26, 35], we can use
$$\begin{aligned} f_\mathrm{G}(A)\frac{d\tilde{r}_{\mathrm {A}}}{\tilde{r}_{\mathrm {A}}^3}=\frac{4\pi G}{c^2}\left( \rho +\frac{p}{c^2}\right) H dt, \end{aligned}$$
(23)
as the starting point for finding the Friedmann equations.
On the one hand, using (21), we can find the dynamical Friedmann equation
$$\begin{aligned} f_\mathrm{G}(A)\left( \dot{H}-\frac{kc^2}{a^2}\right) =-4\pi G \left( \rho +\frac{p}{c^2}\right) . \end{aligned}$$
(24)
On the other hand, taking into account the continuity equation (4), (23) can be recast as
$$\begin{aligned} \frac{8\pi G}{3}d\rho =-4\pi c^2 f_\mathrm{G}(A)\frac{dA}{A^2}. \end{aligned}$$
(25)
Integrating this equation yields the Friedmann constraint, so that, in summary, the two GUP-modified Friedmann equations (up to second order in \(\alpha _\mathrm{Q}^2\)) are given by
$$\begin{aligned}&\frac{8\pi G\rho }{3}=4\pi c^2\left[ \frac{1}{A}+\frac{\pi \alpha _\mathrm{Q}^2\ell _\mathrm{p}^2}{8}\frac{1}{A^2}+{\mathcal {O}}\left( \alpha _\mathrm{Q}^4\right) \right] \end{aligned}$$
(26)
$$\begin{aligned}&-4\pi G\left( \rho +\frac{p}{c^2}\right) = \left( \dot{H}-\frac{kc^2}{a^2}\right) \nonumber \\&\quad \cdot \left[ 1+\frac{\pi \alpha _\mathrm{Q}^2\ell _\mathrm{p}^2}{4}\frac{1}{A}+\frac{\pi ^2\alpha _\mathrm{Q}^4\ell _\mathrm{p}^4}{8}\frac{1}{A^2}+{\mathcal {O}}\left( \alpha _\mathrm{Q}^4\right) \right] . \end{aligned}$$
(27)
Implementing the Friedmann equations
In order to compare the model arising from these modified Friedmann equations with cosmological data, we need an expression for H(z) in terms of the cosmological parameters. Our cosmological background will be a standard \(\varLambda \)CMD model [36], fully characterized by the Hubble constant \(H_0=100 \cdot h\), the dimensionless density parameters for matter \(\varOmega _{m}\), for radiation \(\varOmega _{r}\) and for a cosmological constant as dark energy component \(\varOmega _{\varLambda }\), where \(\varOmega _i=8\pi G\rho _i/3H_0^2\) and the curvature k is expressed as \(\varOmega _{ k}=-kc^2/H_0^2\).
As a first step, we solve Eq. (26) for A, choosing the positive and real solution
$$\begin{aligned} A=\frac{3 c^2+\sqrt{3} \sqrt{3 c^4+ \pi c^2 \alpha _\mathrm{Q}^2\ell _\mathrm{p}^2 G \rho }}{4 G \rho }. \end{aligned}$$
(28)
Due to (6), we can invert to find
$$\begin{aligned} H^2(z)=\frac{16 \pi G \rho }{3 +\sqrt{3} \sqrt{3 + \pi G\rho \dfrac{\alpha _\mathrm{Q}^2 \ell _\mathrm{p}^2}{c^2}}} + \varOmega _{ k} (1+z)^2, \end{aligned}$$
(29)
where \(a=1/(z+1)\) and \(\rho \) is the total energy-matter density of the universe.
Given the continuity equation (4), each component of the energy density is \(\rho _{i}=\rho _{0,i}a^{-3(1+w_i)}\), where the equation of state parameter is \(w_i=p_i/\rho _i\), \(\varOmega _i=\rho _{0,i}/\rho _{c,0}\) and \(\rho _{c,0}=3H_0^2/(8\pi G)\). Therefore, it is possible to write the total energy density \(\rho \) in (29) as \(\rho =\sum _i\varOmega _i\rho _{c}a^{-3(1+w_i)}\) and express the dimensionless Hubble parameter \(E(z)=H(z)/H_0\) as
$$\begin{aligned} E(z)=\sqrt{\frac{2 X(z)}{1+\sqrt{1+ \dfrac{H_0^2 \alpha ^2 \ell _\mathrm{p}^2}{8c^2} X(z)}}+\varOmega _{ k} (1+z)^2}, \end{aligned}$$
(30)
where
$$\begin{aligned} X(z) = \varOmega _{\varLambda }+\varOmega _{ m} (1+z)^3+\varOmega _{ r} (1+z)^4. \end{aligned}$$
(31)
Additionally, after ensuring the normalization condition \(E(z=0)=1\), we can define \(\varOmega _{\varLambda }\) in terms of all other parameters, inverting (30) at \(z=0\), which yields
$$\begin{aligned} \varOmega _{\varLambda }= (1-\varOmega _k-\varOmega _m-\varOmega _r) + \frac{H_0^2 \alpha ^2 \ell _\mathrm{p}^2 \left( 1-\varOmega _{k}\right) ^2}{32\, c^2} . \end{aligned}$$
(32)
Linear GUP
Making use of an alternative formulation of the GUP proposed in [12], the author of [30] also performed the computation of the modified Friedmann equations with the apparent horizon formalism. More specifically, this work deals with
$$\begin{aligned} \varDelta x\varDelta p\ge \frac{\hbar }{2}\left( 1+\frac{\alpha _\mathrm{L}\ell _\mathrm{p}}{\hbar }\varDelta p +\frac{\alpha _\mathrm{L}^2\ell _\mathrm{p}^2}{\hbar ^2}\varDelta p^2\right) , \end{aligned}$$
(33)
where, in addition to the quadratic term in the momentum, a linear term appears (the GUP parameter is named \(\alpha _\mathrm{L}\) here, where L stands for “linear”, to avoid any ambiguities with the quadratic case). Following the same procedure of Sect. 2, the resulting modified Friedmann equations read
$$\begin{aligned}&\frac{8\pi G\rho }{3}=4\pi c^2\left[ \frac{1}{A}+\sqrt{\pi }\frac{\alpha _\mathrm{L}\ell _\mathrm{p}}{3}\frac{1}{A^{3/2}} + {\mathcal {O}}(\alpha _\mathrm{L}^2) \right] \end{aligned}$$
(34)
$$\begin{aligned}&-4\pi G\left( \rho +\frac{p}{c^2}\right) = \left( \dot{H}-\frac{kc^2}{a^2}\right) \nonumber \\&\quad \cdot \left[ 1+\sqrt{\pi }\frac{\alpha \ell _\mathrm{p}}{2}\frac{1}{A^{1/2}} + \frac{\pi \alpha _\mathrm{L}^2 \ell _\mathrm{p}^2}{2}\frac{1}{A} + {\mathcal {O}}\left( \alpha ^{3/2}\right) \right] \end{aligned}$$
(35)
where terms containing higher orders of 1/A have been neglected. Given the similarity of the Friedmann equations (34) and (35) with (26) and (27) of [27], it should be straightforward to compare them with cosmological data, adopting the method in Sect. 2.2.
However, this task proved substantially more challenging with the inclusion of a linear term in the GUP, due to the fractional exponent of A involved in (34). If we assume spatial flatness, neglecting the curvature k, (34) can be recast as
$$\begin{aligned} \frac{8\pi G\rho }{3} = H^{2} + \frac{\alpha _\mathrm{L}\ell _\mathrm{p}}{6c} H^{3}, \end{aligned}$$
(36)
i.e. a third order equation in H(z). It admits one single real solution, which reads (in terms of the dimensionless Hubble parameter E(z))
$$\begin{aligned} E(z)=\frac{2\,c}{\alpha _\mathrm{L}\ell _\mathrm{p}\,H_0}\left[ \frac{F^{2}(z)-F(z)+1}{F(z)}\right] , \end{aligned}$$
(37)
where
$$\begin{aligned} F(z)=\left[ \dfrac{2}{X(z)+\sqrt{-4+X^2(z)}}\right] ^{1/3} \end{aligned}$$
(38)
and
$$\begin{aligned} X(z)= & {} -2+ \frac{3\alpha _\mathrm{L}^2 \ell _\mathrm{p}^2\, H_0^2}{4c^2}\nonumber \\&\cdot \,\left( \varOmega _{m}(1+z)^3+\varOmega _{r}(1+z)^4+\varOmega _{\varLambda }\right) . \end{aligned}$$
(39)
Expression (37) contains a square root, which has profound implications for our goal of finding an upper bound on the GUP parameter. Indeed, an additional condition needs to be satisfied to guarantee that E(z) is real, namely
$$\begin{aligned} \alpha _\mathrm{L}^2>\frac{16\, c^2}{3\, \ell _\mathrm{p}^2 H_0^2 \left[ \varOmega _{m}(1+z)^3+\varOmega _{r}(1+z)^4+\varOmega _{\varLambda }\right] }. \end{aligned}$$
(40)
If we restrict to positive values of the GUP parameter only, this requirement imposes a lower bound on \(\alpha _\mathrm{L}\), which proves incompatible with the notion that General Relativity and standard Quantum Mechanics are to be recovered in the limit \(\alpha _\mathrm{L}\rightarrow 0\).
An upper bound would be theoretically possible if negative values of the GUP parameter were allowed. However, this would contrast with the idea of a minimum length \(\varDelta x_\mathrm{min}\approx \sqrt{\beta } \ell _\mathrm{p}\) (for the GUP formulation (1)) which cannot be imaginary [37]. Nonetheless, a negative GUP parameter has interesting implications, see e.g. [28, 38,39,40,41]. For the purposes of this work, we found that allowing \(\alpha _\mathrm{L}\) to be negative leads to the unphysical result \(H(z)<0\). This is the reason why we only consider \(\alpha _\mathrm{L}>0\) in order to test the model with cosmological data.