Thermodynamics Positivity Bound from 3-Form Black Holes and Inflation with Higher-Derivative Corrections

Abstract

We investigate the interplay between the thermodynamics positivity bounds and slow-roll inflation within a framework governed by a 3-form gauge field. Starting from classical considerations, we derive an upper bound on the mass of extremal charged black holes in dS spacetime which constrains the admissible parameter space. To incorporate quantum gravity effects, we introduce higher-derivative corrections to the 3-form action and obtain additional bounds on these terms, ensuring consistency with swampland criteria. We further analyze these corrections from a thermodynamic perspective, confirming that the Wald entropy remains compatible with the classical extremality bound. Extending this setup to cosmological inflation, we examine the scalar dual of the 3-form in both large-field and small-field regimes. In the large-field limit, the potential acquires a Higgs-like structure that supports slow-roll inflation. In contrast, the small-field limit leads to an effective potential with an AdS minimum, rendering it inconsistent with the dS swampland constraints. Notably, we find that thermodynamic consistency can impose constraints more stringent than those derived from inflationary dynamics alone. These results underscore the utility of swampland-inspired principles in shaping viable models of early universe cosmology.

Figures (7)

• Figure 1: The plot illustrates the dependence of the event horizon radius $r_H$ (green curve) and the cosmological horizon radius $r_c$ (red curve) on the black hole mass. The vertical dashed lines represent specific values of the black hole mass expressed as fractions of the extremal mass $M_{\text{ext}}$: $M = 0.01\,M_{\text{ext}}$ (green), $M = 0.5\,M_{\text{ext}}$ (blue), and $M = M_{\text{ext}}$ (orange). As the mass increases toward $M_{\text{ext}}$, the event and cosmological horizons approach each other, eventually coinciding at the extremal limit.
• Figure 2: Radial profile of the weak energy condition (WEC), expressed as $\rho + p$, for different mass values: $M = M_{\text{ext}}$ (orange), $0.5 M_{\text{ext}}$ (blue), and $0.01 M_{\text{ext}}$ (green). The solid lines show that the WEC is satisfied ($\rho + p \geq 0$) in the regions of interest. Vertical dashed lines indicate the event horizons $r_H$, while dash-dotted lines represent the corresponding cosmological horizons $r_c$. The WEC holds between the horizons for all considered masses.
• Figure 3: Radial profile of the strong energy condition (SEC), expressed as $\rho + 3p$, for different mass values: $M = M_{\text{ext}}$ (orange), $0.5 M_{\text{ext}}$ (blue), and $0.01 M_{\text{ext}}$ (green). The curves indicate that the SEC is violated ($\rho + 3p < 0$) in the region near the black hole for all mass configurations. Vertical dashed lines denote the event horizons $r_H$, while dash-dotted lines indicate the corresponding cosmological horizons $r_c$.
• Figure 4: Contour plot of e-folding number $N_{\chi}$ as a function of $c_{7}$ and the 3-fomr coupling ${g_{3}}$, with fixed parameters $c_{6}$ and $12 c_4 + 3 c_5$. The shaded region indicates the range consistent with the positivity bound in Eq.\ref{['sbound']}. The e-folding number increases with both $g_{3}$ and $c_{7}$, reaching $N_{\chi} \sim 45.0-57.5$ within the viable inflationary regime, beyond which the parameters asymptotically approach the bound.
• Figure 5: Contour plot of the e-folding number $N_{\chi}$ as a function of $c_{6}$ and the coupling combination $12 c_4 + 3 c_5$, with fixed parameter $c_7$ and 3-form coupling $g_3$. The blue-shaded region satisfies the positivity bound, while the green linear line marks the transition between de Sitter (dS) and anti–de Sitter (AdS) vacua. The consistent inflationary region lies near the intersection of these bounds, corresponding to $N_{\chi}\sim 44-64$.