The theory of quantum gravity [1] is one of the basic theories preferred by modern physics and astronomy in the study of physical problems, especially in explaining the information paradox of black holes [2, 3], the formation of molecular clouds and the instability of the collapsing of nebulae. Since the French scientist, Pierre Laplace, put forward the concept of the black hole in 1796 [4], various galaxies have gradually become known through the exploration of researchers. At present, the mass instability of galaxies composed of molecular clouds has been widely studied as a hot topic in many interdisciplinary fields. In the eyes of most people, molecular clouds are bound by gravity, and under the influence of gravity for a long time, a large number of molecular clouds come together to form the building blocks of stars. The studies [5–8] mention that although gas molecules are not bound by gravity in most cases, they will locally aggregate to form stars. This also happens when the density of the cloud reaches the limit at which the nebula collapses and stars form. Due to the efforts of the researchers, the theoretical value of nebula collapse mass (Jeans mass) was put forward in the literature [9]. However, [9, 10] show that the modified Jeans mass based on the generalized uncertainty principle (GUP) is smaller than the original condition, which is consistent with the conclusion that the observed value is smaller than the theoretical value.

Since string theory [11, 12], loop quantum theory [13, 14] and black hole theory [3, 4, 15] were proposed, researchers have been very interested in the GUP and related it to the minimum observable length [16]. The Gedanken [17] experiment confirmed the existence of minimum length and provided a theoretical basis for it. The original GUP was proposed by the famous scientist, Heisenberg, for the determination of the position and momentum of the electron, also known as the Heisenberg uncertainty principle (HUP) [18, 19]. The expression is ${\rm{\Delta }}x{\rm{\Delta }}p\geqslant \tfrac{{\hslash }}{2}$, where Δx is the position uncertainty (length), Δp is the momentum uncertainty, ℏ is the Planck constant, and the ratio is π. In addition, people have done more in-depth research and proposed different expressions of the GUP model based on HUP. At present, studies have shown that [10, 20, 21], combining the GUP with thermodynamics, the modification of Jeans mass is achieved, and the results show that GUP can reduce Jeans mass. For example, the literature [9, 10] explores the Jeans mass using the generalized and extended uncertainty principles and the higher-order GUP, respectively, and the Friedmann equation derived from Gauss-born gravity was used to modify the area law of entropy by GUP to realize the pair Jeans mass amendment in [10]. Most previous works modified the Jeans mass through a single different GUP form, and the calculated value of Jeans mass theory after modification was smaller than the original value. This conclusion provides a more novel research idea for Jeans mass to correct the problem. However, when the forerunners researched the instability of Jeans mass, there were still certain issues that needed to be resolved. For instance, on the one hand, the multiple GUP-modified Jeans masses were not employed at the same time, and the changed findings were not compared. On the other hand, the impact of temperature and GUP parameters on the mass of a molecular cloud collapse was not taken into account. Consequently, given the flaws in earlier research, this study first revised the gravitational potential and canonical energy of the ideal gas based on different physical contexts and realized the correction of Jeans mass using multiple GUP models in conjunction with thermodynamics. Second, we compared it with the modified Jeans mass. Furthermore, an image was created for comparative analysis, and the effect of temperature and GUP parameters on Jeans mass was taken into account.

This paper is organized as follows. In section 2, the theoretical values of Jeans mass that have not been modified by the GUP are specifically reviewed. In section 3, we mainly introduce three GUP models with different physical backgrounds. In section 4, based on the GUP models and thermodynamics, we have modified the entropy, entropy force, gravitational potential and canonical energy of an ideal gas to achieve the modification of Jeans mass. Then, in section 5, the influence of modified Jeans mass on the variation of temperature and GUP parameters and the degree of variation is thoroughly analyzed. The final summary of this paper is provided in section 6.

For comparison with GUP-corrected Jeans mass, this section reviews the calculation of Jeans mass without GUP correction. According to the demonstration of the virial theorem in [9], molecular cloud collapse will occur when the following conditions are met: To obtain Jeans mass from the above equation, we need to solve the canonical energy (U) and gravitational potential energy (E_p). Space-time is typically thought of as being flat. Consequently, in light of Newton's law of gravitation [22–24], the formula of gravitation can be expressed as ${F}_{{\rm{N}}}=\tfrac{{GMm}}{{}^{{r}^{2}}}$, where G is the gravitation constant, r is the orbit radius, M is the mass of the central celestial body and m is the mass of the orbiting planet. Gravitational potential ${V}_{{\rm{N}}}\left(r\right)=-\tfrac{{GM}}{r}$, the corresponding potential energy in the framework of Newtonian gravity (NG):where the constant density of the cloud is ρ(r) in equation (2) (it is assumed that ρ(r) = ρ₀ is a constant), and the radius is $R={\left(\tfrac{3M}{4\pi {\rho }_{0}}\right)}^{\tfrac{1}{3}}$. Based on Liouville's theorem [20, 25, 26] and HUP, we assume that the chosen boundary is not an infinite extension but a finite region and ideally treat it as a closed surface. To characterize it more graphically, we approximate it as a sphere. Therefore, in the spherical coordinates, the density of states in momentum space can be expressed as φ(p)dp = 4πVp²dp/h³, and the initial partition function for an ideal gas is as follows:where N, T and μ respectively represent the number, temperature, and mass of non-interacting particles in an ideal gas in the equation above. k_B is the Boltzmann constant and h is the Planck constant. In addition, the total mass relation of an ideal gas is M = μN. According to equation (3), the equation of state of ideal gas PV = Nk_BT can be solved. The canonical energy of an ideal gas can be expressed as,Here, we observe that the original canonical energy is only related to the number of non-interacting particles N and the temperature T. Then, we assume that the ideal gas has a spherical distribution in space. By substituting equations (2) and (4) into inequality (1), we obtain the following inequality:equation (5) represents the lower limit of the mass of the molecular cloud to collapse, and the minimum value is Jeans mass:In a gravitational system, the lower limit of the collapsing mass of the nebula will have different scale effects on the formation of the structure of the nebula [27]. The usual approach is to test the formation of the nebula through the initial Jeans mass and its instability. In order to better explain this problem, we examine the modification of Jeans mass under different GUP frameworks.

This section focuses on three different GUP models. Whether studying physical problems, such as fundamental quantum mechanics, quantum gravity, or more complex black hole radiation, the GUP is one of the candidates used to explain the above physical problems. The GUP was expanded based on HUP, and it has been studied more deeply. Before that, some individuals [4, 9, 10, 28] proposed various GUP model expressions based on various physical contexts. They are the KMM model that only considers the minimum observable length, proposed by Kempf-Mangano-Mann [29], the Pedram model which considers the minimum observable length and maximum momentum, proposed by Pouria Pedram [30] and the Chung-Hassanabadi (CH) model with minimum observable length, maximum momentum and is non-perturbative, proposed by Won Sang Chung and Hassan Hassanabadi [31].

Kempf-Mangano-Mann (KMM) proposed a GUP form for quadratic forms based on minimum observable length, where the expression iswhere η denotes the GUP parameter, ${\rm{\Delta }}{x}_{\min }\approx {l}_{{\rm{p}}}\sqrt{\eta }$ is the minimum length it predicts, and l_p denotes the Planck length.

Pedram proposed a higher-order GUP form based on the minimum observable length combined with the idea of maximum momentum, where β denotes the GUP parameter. The expression is,inspired by these new forms of GUP being proposed one after another. Chung and Hassanabadi proposed a model with good non-perturbative properties:where γ denotes the GUP parameter, $\left|p\right|=\sqrt{\left|{p}^{2}\right|}$, set $\left\langle p\right\rangle =0$, $\left\langle \left|p\right|\right\rangle \geqslant 0$ and we use the constant equation $\left|\left\langle \left({ab}+{ba}\right)\right\rangle \right|\geqslant 2\sqrt{\left\langle {a}^{2}\right\rangle }\sqrt{\left\langle {b}^{2}\right\rangle }$ and then we have $\left\langle {\left({p}^{2}\right)}^{n}\right\rangle \,\geqslant {\left(\left\langle {p}^{2}\right\rangle \right)}^{n}$ [31]. The three different forms of the GUP described above can describe the minimum observable length and can be effectively combined with thermodynamics and applied to correction work.

In this section, the entropy, entropy force, gravitational potential and canonical energy of an ideal gas are modified successively through the GUP to obtain the modified Jeans mass. Because the GUP has many forms, we consider three representative forms of GUPs and their modifications to Jeans mass for the sake of analytical integrity.

They are the previously mentioned KMM (considered only with minimum observable length), Pedram (with minimum observable length and maximum momentum) and CH (with minimum observable length, maximum momentum, and non-perturbative) models, respectively. To simplify the process, it takes the units of c = ℏ = k_B = 1 in the calculation. Under the constraints of the holographic principle and the homogeneity rule, use the entropic force theory to derive the modified gravitational potential energy. In the presence of KMM, Pedram and CH models, we obtain the following: where the parameters (η, β and γ) in the above equation are the dimensionless parameters of the above GUP. Equations (10)-(12) are obtained using Taylor expansion. Studies [32–36] point out that similar photons (massless particles) with available energy ϵ = pc(Δϵ = Δpc) can be used to determine the position of the quantum particles, to convert the Δp ≥ 1/(2Δx) to the lower bound of Δϵ ≥ 1/(2Δx). Thus, from equations (13)-(15), we obtain equations (16)-(18). As pointed out in [24, 37], when a system has a strong attraction (similar to a black hole), absorption, or launch energy for ϵ, and quantum particle radius R, the system of the minimum area is ${\rm{\Delta }}{A}_{\min }\geqslant 8\pi {\rm{\Delta }}\varepsilon {{Rl}}_{{\rm{p}}}^{2}$. Then, the arguments of [38, 39] that the uncertainty of the position of the quantum particle itself cannot be greater than the size of the quantum particle, so that there are Δx < R and ${\rm{\Delta }}{A}_{\min }\geqslant 8\pi {\rm{\Delta }}\varepsilon {\rm{\Delta }}{{xl}}_{{\rm{p}}}^{2}$, then this relation can be obtained by substituting it into equations (16), (17) and (18): From the analysis of a spherical gravitational system with Schwarzschild radius r, it can be seen that the area of the gravitational system is A = 4πr². In addition, near the event horizon of the system, Δx is approximately equal to the orbit radius r and Δx ≈ 2r [33, 40–42]. Thus, the relationship between Δx and A changes to Δx² = 4r² = A/π. By substituting this relation into equations (19)-(21), the minimum charge of the event horizon area of their gravitational system can be re-expressed as where λ is given by the Bekenstein-Hawking entropy [43] formula and l_p is the Planck length. In addition, based on information theory, it is believed that the area of a system is related to the increment of its minimum entropy [44]. Since the basic unit of a single information entropy is ${\rm{\Delta }}{S}_{\min }=b=\mathrm{ln}2$, we can obtain the following expression: when the GUP parameters approach zero (η → 0, β → 0, and γ → 0), b/λ = 1/4 satisfies the area law of the original entropy [37]. Considering that the GUP parameter in the above formula is small, by simplifying equations (25)-(27), three kinds of entropy modified by GUP can be obtained: In addition, the presence of logarithmic terms in the above equations is consistent with the requirement of quantum gravity [45–51]. Based on the holographic principle and the area law of entropy, the number of original information bits is N = 4S. Under different GUP frameworks, the number of bits can be expressed in the following form: Next, we express the total energy of the gravitational system, which is composed of N energies containing T/2. The expression is,where we consider E = Mc², ${l}_{{\rm{p}}}^{2}=G$ and A = 4πr². Verlinde's theory of entropic forces proves that entropic forces are more fundamental than gravity. We should consider a spherically symmetric gravitational system (such as a black hole) to calculate the modified entropic forces because it can absorb or emit particles within the event horizon. Thus, the entropic force [24] in a gravitational system can be written as,where T represents the temperature, ΔS is the change of entropy in the gravitational system, F represents entropy force, and Δx is the displacement of the motion in the gravitational system, which satisfies the expression ΔS = 2Δxπ [52, 53]. By substituting equations (28)-(34) into (35), the GUP-modified gravitation can be obtained: From the above formula, it can be seen that when the GUP parameters are equal to zero (η = 0, β = 0 and γ = 0), equations (36)-(38) are reduced to the original Newtonian gravity ${F}_{{\rm{N}}}=\tfrac{{GMm}}{{r}^{2}}$. Under the GUP-corrected Newtonian gravity (NG), the corresponding GUP-corrected gravitational potential is, Applying the above three expressions to the molecular clouds with radius R, mass M and density ρ₀, the modified potential energy expression can be obtained: From the above formula, it can be seen that the modified potential energy expression is related to the radius (R), the mass (M) and the parameters (η, β and γ) of GUP. In addition, at the limit η = 0, β = 0 and γ = 0, the modified potential energy results are the same as the original potential energy.

According to [54, 55] and equations (10)-(12), when the partition function of an ideal gas with temperature T, mass m, and N non-interacting particles is expressed as,the integral form is as follows:By using the spherical coordinates and Gaussian integral to calculate the above expression, the GUP-modified partition function for the cloud obtains,For different forms of GUP, ${\rm{\Theta }}\left(\beta \right)$ is not the same, but in the form of the GUP revised expression can be written as, The original canonical energy is expressed as,The canonical energy after GUP correction is shown as follows: To simplify the calculation, three constant variables ${\mathfrak{R}}=\mu \sqrt{T}$, λ = μT and $\unicode{x00141}=\sqrt{\mu T}$ are respectively defined. It is worth noting that the modified canonical energy is not the same as that without GUP modification. It is not only related to the original U₀, but also to the GUP parameters, the mass μ of the non-interacting particle and the temperature T.

With the above-revised results, we can then combine equations (42)-(44) with equations (52)-(54) and (1). We can smoothly derive the modified Jeans mass as follows:Here, the radius $R={\left(3M/4\pi {\rho }_{0}\right)}^{\tfrac{1}{3}}$ and equation (47) can be rewritten as, where ${X}_{\mathrm{KMM}}=1+\tfrac{5\mathrm{ln}\left[4\pi {\left(\tfrac{3M}{4\pi {\rho }_{0}}\right)}^{2}\right]\eta }{144{\left(\tfrac{3M}{4\pi {\rho }_{0}}\right)}^{2}}+\tfrac{{\eta }^{2}}{384{\left(\tfrac{3M}{4\pi {\rho }_{0}}\right)}^{4}}+o\left({\eta }^{3}\right)$. If β = 0, the lower limit of mass is Jeans mass $({M}_{\mathrm{HUP}}^{J}={\left(\tfrac{5{{\rm{K}}}_{{\rm{B}}}T}{\mu G}\right)}^{\tfrac{3}{2}}{\left(\tfrac{3}{4\pi {\rho }_{0}}\right)}^{\tfrac{1}{2}})$. Moreover, due to the limited mass of M ≫ 0 and η, Jeans mass modified by KMM can be obtained through equation (58).

The same can be said for Pedram and CH modified Jeans mass:where ${X}_{\mathrm{Pedram}}=1+\tfrac{5\mathrm{ln}\left[16\pi {\left(\tfrac{3M}{4\pi {\rho }_{0}}\right)}^{2}\right]\beta }{144{\left(\tfrac{3M}{4\pi {\rho }_{0}}\right)}^{2}}+\tfrac{{\beta }^{2}}{256{\left(\tfrac{3M}{4\pi {\rho }_{0}}\right)}^{4}}+o\left({\beta }^{3}\right),$ here ${X}_{\mathrm{CH}}=1+\tfrac{5{\gamma }^{2}}{144{\left(\tfrac{3M}{4\pi {\rho }_{0}}\right)}^{2}}\mathrm{ln}\left[16\pi {\left(\tfrac{3M}{4\pi {\rho }_{0}}\right)}^{2}\right]+o\left({\gamma }^{3}\right),$

To sum up, through the above calculation, we find that the quality modified by GUP is smaller than the original Jeans mass. It is worth noting that in addition to the constants in the modified expression, Jeans mass after modification is related to the GUP parameters and temperature, which will be discussed in the next section.

The impact of Jeans mass modified by GUP on temperature and GUP parameters can be more clearly recognized in light of the previous calculation findings. For further analysis, we will draw the images using equations (57), (59) and (61). To make the drawing of the image simpler, it takes G = μ = ρ₀ = 1. It is worth noting that we restrict the value range of the temperature and GUP parameters to avoid the generation of imaginary numbers.

In general, the GUPs have positive parameters, which lead to the existence of a minimum length. However, [19, 56, 57] show that the parameters of the GUP should be negative rather than positive in some cases. In particular, [56, 58–60] show that in the case of high energy or large mass, the GUP principle will have negative parameters, which will lead to the removal of the minimum length, and physics will return to the classical state. Since Jeans mass represents the collapsing mass of a galaxy, it faces a massive system. Therefore, to complete the analysis, we will discuss the positive and negative parameters of the GUP. It is also important to note that to better analyze the differences between the three GUP corrections, for the degree of change (the mean curvature of the change) of the modified mass curve, we only compare the mass curves given by the three GUPs, without referring to the original mass curve ${M}_{\mathrm{HUP}}^{J}$ given by HUP.

First, we consider the case where the parameters are positive. In figure 1, it can be seen that there is a change in Jeans mass with temperature T after three different forms of GUP modification. These corrected masses are significantly smaller than the original mass ${M}_{\mathrm{HUP}}^{J}$, which is consistent with the observation conclusion [9, 10].

On the other hand, the mass after GUP correction depends on the degree of temperature change in a different way; the mean curvature of mass modified by Pedram with temperature changes is the smallest and gentlest. However, as the temperature increases, it increases and then decreases, compared with the other two curves, and represents a much smaller value than the value of the original mass. The average curvature of mass modified by KMM with temperature changes is large and obvious. However, with the increase in temperature, the quality of its correction gradually becomes stable. The average curvature of mass with temperature modified by CH is the largest, and the corrected mass also continues to rise with the increase in temperature. It is closer to the original mass than the other two curves.

Second, it can be seen in figure 2 that at constant temperature T there is a change in Jeans mass with the positive parameters after three different forms of GUP modification. These corrected masses are all significantly less than the original mass, which is consistent with the observation conclusion. The quality of the modified GUP depends on the positive GUP parameters to different degrees; the mass modified by KMM has the largest and most obvious variation with the mean curvature of parameter η, and with the increase in parameter η value, the quality of its correction decreases rapidly. The average curvature of the mass modified by Pedram with parameter β is larger. However, with the increase in parameter β value, the declining trend of its modified mass becomes much slower. The mean curvature of mass change with parameter γ modified by CH is minimal. As parameter γ increases, it corrects the curvature of the mass most gently and the value represented by the curve is closer to the value of the original mass than the other two curves. As the limit of the GUP parameters approaches zero from positive, the mass of the three GUP corrections gradually approaches the original mass given by HUP. This is consistent with the conclusion that the GUP degenerates to HUP when the parameter is zero.

Based on the relevant literature [19, 56–60], we consider the case where the GUP parameters are negative. In figure 3, it can be seen that there is a change in Jeans mass with temperature after three different forms of GUP modification. Among these corrected masses, only the Pedram corrected mass curve is always below the original mass curve. This indicates that the Jeans mass modified by Pedram is always smaller than the original mass ${M}_{\mathrm{HUP}}^{J}$. This is consistent with the observation. The corrected mass of KMM and CH is always greater than the original mass, which is inconsistent with the observation conclusion.

On the other hand, the mass corrected by different GUPs depends on different degrees of temperature change. The average curvature of mass modified by KMM with temperature is the largest and shows a gradually increasing trend. The average curvature of CH-modified mass with temperature is larger than that of KMM-modified mass, which also shows an upward trend. The average curvature of mass modified by Pedram with temperature changes is the smallest and gentlest. With the increase in temperature, it increases slowly at first and then decreases. However, as the temperature increases it compares to the other two curves and represents a much smaller number than the original number.

Second, it can be seen in figure 4 that at a constant temperature T, three different forms of GUP-modified Jeans mass change with the negative parameters. As in figure 3, it can be seen that the Jeans mass modified by Pedram is always smaller than the original mass ${M}_{\mathrm{HUP}}^{J}$, which is consistent with the observation conclusion. However, the corrected mass of KMM and CH is always greater than the original mass, which contradicts the observation conclusion. The quality of the modified GUP depends on the negative parameters to different degrees; the mass modified by KMM has the largest and most obvious variation with the mean curvature of parameter η, and the quality of its correction decreases rapidly with the increase in parameter η value. The average curvature of the mass modified by Pedram is larger since the parameter β changes and the result ranks only second to KMM correction. With the increase in parameter value, there is a positive increase trend. The mean curvature of the mass modified by CH with the change in parameter γ is the smallest, and the curve showed a slow decreasing trend with the increase in parameter γ. When the limit of parameters starts from a negative value and approaches zero, the mass of the three kinds of GUP correction gradually approaches the original mass given by HUP. This is consistent with the conclusion that the GUP degenerates to HUP when the parameter is zero.

In summary, we give ${M}_{\mathrm{HUP}}^{J}$ and modify Jeans mass in the form of an image, which is affected by temperature and positive and negative parameters, respectively. Here, the average curvature of the curve represents the sensitivity of the Jeans mass modified by three GUPs to temperature and the positive and negative parameters. Remarkably, we found that at a constant temperature, for Jeans mass modified by Pedram, no matter whether the β parameter is positive or negative, the corrected mass is always smaller than the original mass ${M}_{\mathrm{HUP}}^{J}$, which accords with the observed results. In contrast, the Jeans mass modified by KMM and CH is larger than the original mass when the η and γ parameters are negative, thus contradicting the observed results.

In this paper, the theoretical values of Jeans mass are larger than the observed values, so in order to achieve a better match between them, we have corrected the Jeans mass using three GUPs, which are KMM, Pedram and CH. It can be seen that the GUPs can lower the Jeans mass to correct results to be more consistent with the observed values than the original results. In addition, the impact of temperature and GUP parameters on the modified Jeans mass is analyzed by drawing images.

First, we analyzed the effect of temperature on the corrected results. The sensitivity of the corrected results is related to the sign of the GUP parameter. For the case of the positive GUP parameter, the sensitivity degree order of the corrected results from three GUPs to temperature is CH > KMM > Pedram. This means that the CH model is more sensitive to temperature than the KMM and Pedram models. For the case of the negative GUP parameter, the sensitivity degree order of the corrected results from three GUPs to temperature is KMM > CH > Pedram. This means that the KMM model is more sensitive to temperature than the CH and Pedram models.

Second, we analyzed the effect of GUP parameter on the corrected results. When the temperature T remains constant, no matter whether the GUP parameters are positive or negative, the sensitivity degree order of the corrected results from three GUPs to GUP parameter is KMM > Pedram > CH. This means that the KMM model is more sensitive to the GUP parameters than the Pedram and CH models. Finally, we consider that our research can contribute to exploring the range of GUP parameters with observational data.