The stochastic gravitational wave background from QCD phase transition in the framework of higher-order GUP

TL;DR

This work investigates how the Du–Long higher-order GUP modifies the stochastic gravitational wave background from a phenomenological strongly first-order QCD-scale phase transition. By deriving GUP-corrected photon gas thermodynamics and embedding them in a lattice-QCD based expansion history, the authors quantify the impact on the SGWB through BC, SW, and MHD sources, highlighting the crucial role of the sign and magnitude of $β_0$. They find that $β_0>0$ yields a physically consistent thermodynamic framework and shifts the SGWB peak to lower frequencies with modest amplitude changes, while $β_0<0$ induces a maximal temperature and unphysical behavior. The results suggest that future pulsar timing array observations could constrain the higher-order GUP parameter, giving an indirect probe of quantum-gravity effects at the Planck scale.

Abstract

This work studies the impact of a new higher-order generalized uncertainty principle (GUP) on the stochastic gravitational wave background (SGWB) associated with a QCD-scale first-order phase transition. Assuming a strongly first-order transition at the QCD-scale as a phenomenological benchmark, the analysis shows that the sign and magnitude of the dimensionless deformation parameter $β_0$ play a crucial role. For negative $β_0$, the thermodynamic quantities of the radiation fluid develop a maximal temperature beyond which entropy and pressure vanish, and the SGWB spectrum exhibits divergent behavior at high temperatures, so this branch is discarded as phenomenologically inconsistent. For positive $β_0$, the higher-order GUP shifts the SGWB peak frequency towards lower values and slightly enhances the peak energy density, with the size of the effect controlled by $β_0$. For natural values $β_0=\mathcal{O}\left( 1 \right)$ the corrections at QCD temperatures are strongly suppressed, whereas larger benchmark values still compatible with existing experimental and cosmological bounds can induce appreciable shifts in the SGWB spectrum. A future detection of a QCD-scale first-order SGWB would therefore allow the framework developed here to be used to translate the measured signal into constraints on the higher-order GUP parameter, providing an indirect probe of quantum gravity effects.

Figures (6)

• Figure 1: The ratio ${\nu_{0{\rm peak}}}/{\nu_*}$ as a function of the transition temperature $T_*$ (in GeV) for different values of the dimensionless GUP parameter ${\beta_0}$. The present temperature is fixed at $T_0 = 2.725~{\rm K} = 2.348\times10^{-13}~{\rm GeV}$, with $g_s \left( {{T_0}} \right)=3.4$ and $g_s \left( {{T_*}} \right)=35$. (a) Positive values of ${\beta_0}$. (b) Negative values of ${\beta_0}$.
• Figure 2: The relationship between the effective EOS $\omega_{\rm{eff}}$ and transition temperature $T_*$.
• Figure 3: The relationship between the ${{{H_*}} \mathord{\left/ {\newline} \right. \nulldelimiterspace} {{H_0}}}$ and transition temperature $T_*$. We set $T_r=10^4~\rm{GeV}$ and ${g_s}\left( {{T_r}} \right) = 106$. (a) Without the effect of GUP. (b) With different values of GUP parameter.
• Figure 4: The relationship between ${{{\Omega _{{\rm{gw}}}}} \mathord{\left/ {\newline} \right. \nulldelimiterspace} {{\Omega _{{\rm{gw*}}}}}}$ and the function of transition temperature $T_*$ with different values of GUP parameter ${\beta_0}$.
• Figure 5: The total peak frequency $\nu_{\rm{total}}$ of SGWB at the epoch of phase transition with different values of deformation parameter ${\beta_0}$.