Some inequalities bridging stringy parameters and cosmological observables

TL;DR

The work tackles embedding inflation within string theory by deriving inequalities that connect cosmological observables to stringy parameters. By enforcing EFT validity through hierarchies $H_{inf} \ll m_{KK} < m_s < m_W < M_p$ and $V_{inf}^{1/4} \ll m_{KK}$, it translates Planck-scale observations of $r$ into constraints on the string coupling $g_s$ and compactification volume $\mathcal{V}$ (and its Einstein-frame counterpart $\mathcal{V}_E$) in type IIB orientifold setups. The authors derive explicit upper and lower bounds on $r$ (e.g., $1 \ll \frac{\mathcal{V}_s}{(2\pi)^6} < (\frac{g_s^4}{r})^{3/8} \times (1.4\times 10^3)$) and show how gravitino considerations can push $r$ to smaller values in LVS, with consistency checks in concrete models like KKLT/LVS and KNP-aligned natural inflation. These results provide a model-independent framework to constrain stringy parameters from cosmological data and guide the construction of viable string-inspired inflationary scenarios.

Abstract

By demanding the validity of an effective field theory description during inflation, in this note we derive some peculiar inequalities among the three interesting stringy and cosmological parameters, namely the tensor-to-scalar ratio ($r$), the string coupling ($g_s$) and the compactification volume (${\cal V}$). In deriving these inequalities, we explicitly demand that the inflationary scale and the Hubble parameter during inflation are well below the Kaluza-Klein (KK) mass scale, string scale, and the four dimensional Planck mass. For the inflationary models developed within the framework of type IIB orientifold comapctification, we investigate the regions of parameters space spanned by the three parameters $(r, g_s, {\cal V})$ by satisfying our inequalities, and we find that the same can reduce the size of available parameter space quite significantly. Moreover, we comment on obtaining further constraints on the parameters by comparing gravitino mass ($m_{3/2}$) with the Hubble scale ($H$), which also provides a lower bound on tensor-to-scalar ratio ($r$), for the cases when $m_{3/2} <H$. We also illustrate the outcome of our bounds in some specific class of string(-inspired) models.

Figures (4)

• Figure 1: Region plot of inequality given by Eq. (\ref{['eq:main1']}) showing the allowed region (shaded) in the parameter space $(v_s, g_s, r)$.
• Figure 2: Region of $(v_s,~g_s)$-parameter space allowed (shaded) for $r = 0.05$.
• Figure 3: Allowed shaded region for $(v_e,~r)$ bounded by Eq. (\ref{['rmax']}).
• Figure 4: Allowed lower bound on string coupling $g_s$ plotted for $r$ assuming that ${\cal V}_s/(2 \pi)^6 > 100$, i.e. $v_s >0.1$ in order to suppress $\alpha'$ and string loop corrections.