Brane inflation and the WMAP data: a Bayesian analysis

TL;DR

<3-5 sentence high-level summary>The paper tests a string-theory-based brane inflation model (KKLMMT) against WMAP3 data by solving the full background and perturbation dynamics exactly and exploring the parameter space with Bayesian MCMC, without relying on slow-roll approximations for the perturbations. It shows that the data favor inflation ending by slow-roll violation before brane annihilation, and places quantitative constraints on throat geometry (e.g., log(v) > -10) and reheating (e.g., ln R > -38), while revealing correlations among string parameters such as g_s, N, and v. The analysis finds negligible tensor modes and a mildly red scalar spectrum, and discusses the limits of the perturbative stochastic approach for quantum effects in this setup. The work also provides a framework for interpreting future Planck-level data and for incorporating richer stringy corrections in a controlled Bayesian context.

Abstract

The Wilkinson Microwave Anisotropy Probe (WMAP) constraints on string inspired ''brane inflation'' are investigated. Here, the inflaton field is interpreted as the distance between two branes placed in a flux-enriched background geometry and has a Dirac-Born-Infeld (DBI) kinetic term. Our method relies on an exact numerical integration of the inflationary power spectra coupled to a Markov-Chain Monte-Carlo exploration of the parameter space. This analysis is valid for any perturbative value of the string coupling constant and of the string length, and includes a phenomenological modelling of the reheating era to describe the post-inflationary evolution. It is found that the data favour a scenario where inflation stops by violation of the slow-roll conditions well before brane annihilation, rather than by tachyonic instability. Concerning the background geometry, it is established that log(v) > -10 at 95% confidence level (CL), where "v" is the dimensionless ratio of the five-dimensional sub-manifold at the base of the six-dimensional warped conifold geometry to the volume of the unit five-sphere. The reheating energy scale remains poorly constrained, Treh > 20 GeV at 95% CL, for an extreme equation of state (wreh ~ -1/3) only. Assuming the string length is known, the favoured values of the string coupling and of the Ramond-Ramond total background charge appear to be correlated. Finally, the stochastic regime (without and with volume effects) is studied using a perturbative treatment of the Langevin equation. The validity of such an approximate scheme is discussed and shown to be too limited for a full characterisation of the quantum effects.

Figures (15)

• Figure 1: Left panel: The dotted blue line represents the potential given by equation (\ref{['eq:Vofphifull']}). The solid green line shows the approximate potential for $\phi\gg\mu$, see equation (\ref{['eq:Vofphiapprox']}). The conventional slow-roll phase occurs while the field is rolling on the extremely flat region to the right of the plot. Right panel: Same as the left panel, but in logarithmic unit for the potential.
• Figure 2: Left panel: Sketch of the expected dynamical regimes according to the vev of the inflaton field for the potential (\ref{['eq:Vofphifull']}). The field starts out in the flat (no hatched) region of the potential and rolls towards smaller field values. Eventually, it will reach the light green hatched region where the slope of the potential becomes noticeable and where the DBI effects can no longer be neglected. This regime may or may not be reached according to the value of $\phi_\mathrm{strg}$ for which the derivation of equation (\ref{['eq:Vofphifull']}) breaks down. On the far right (dark green hatched region), the potential is extremely flat and quantum fluctuations are expected to dominate over the field classical evolution. Right panel: The slow-roll parameters $\epsilon _1$ (green line) and $\epsilon _2$ (blue line) for the KKLMMT potential. The solid curves have been obtained for $\mu/m_{{\mathrm{Pl}}}=0.1$ whereas the dotted ones for $\mu/m_{{\mathrm{Pl}}}=0.01$. Decreasing $\mu/m_{{\mathrm{Pl}}}$ increases the differences between $\phi_{{\epsilon_1}}$ and $\phi_{{\epsilon_2}}$, the field values for which ${\epsilon_1}=1$ and ${\epsilon_2}=1$, respectively. Notice that we always have $\phi_{{\epsilon_2}}>\phi_{{\epsilon_1}}$.
• Figure 3: Left panel: The contour $\phi_{{\epsilon_2}}=\phi_\mathrm{strg}$ in the plane $(\ln v,\ln\mathcal{N})$, obtained from equations (\ref{['eq:phi2eqstrg']}) and (\ref{['eq:phi2eqstrgnorm']}) using the normalisation given by the CMB quadrupole with $N_*=50$ [see equation (\ref{['eq:scaleMclas']})]. The dotted line corresponds to $g_\mathrm{s}=0.1$, the dashed line to $g_\mathrm{s}=10^{-3}$ and the dotted-dashed one to $g_\mathrm{s}=10^{-5}$. In each case, the area enclosed by the contour is the part of the parameter space where the slow-roll conditions are violated before brane annihilation and $\phi_\mathrm{strg}$ does not play an important role for the end of inflation. It is clear from this plot that the contour sensitively depends on the value of $g_\mathrm{s}$. However, such a dependence can be absorbed by an appropriate rescaling of the parameters as shown in the right panel. The same contour $\phi_{{\epsilon_2}}=\phi_\mathrm{strg}$ is represented in the plane $(\ln x,\ln\bar{v})$, these parameters being defined in equation (\ref{['eq:rescaledparam']}). It is universal for all values of the string coupling $g_\mathrm{s}$.
• Figure 4: Evolution of the field $\phi$, the Hubble-flow functions ${\epsilon_1}$, ${\epsilon_2}$ and the DBI parameter $\gamma$ in the last e-foldings of an extreme model living close to the throat edge ($\mu=m_{{\mathrm{Pl}}}/\sqrt{32 \pi}$). This model has been chosen to emphasise the DBI effects: the solid lines correspond to an exact numerical integration of the full action (\ref{['eq:generalaction2']}) whereas the dashed lines are obtained by using the slow-roll approximations (\ref{['eq:eps1kklt']}), (\ref{['eq:eps2kklt']}) and (\ref{['eq:trajectory']}). The DBI regime smoothly connects the slow-roll evolution ($\gamma\simeq 1$) to the ultra-relativistic matter-like expansion ($\gamma \gg 1$). Note that in the model at hand, brane annihilation occurs at $\phi_\mathrm{strg}>\mu$ preventing any observable effects coming from the DBI evolution.
• Figure 5: Allowed regions (ticks in) in the rescaled parameter plane $(\ln x,\ln\bar{v})$ for $\alpha' m_{{\mathrm{Pl}}}^{2}=1000$ (left panel) and $\alpha' m_{{\mathrm{Pl}}}^{2}=10$ (right panel) with the fiducial values $N_*=50$ and $N_{_{\rm T}}=60$. The black dotted curve is the contour $\phi_{{\epsilon_2}}=\phi_\mathrm{strg}$ of figure \ref{['fig:rapp']}. For all points located inside this contour, the slow-roll conditions are violated before brane annihilation. The green dashed line (with ticks down) represents the volume ratio constraint (\ref{['eq:throatsize-xvbar']}), whose slope is universal but whose offset depends on $\alpha'm_{{\mathrm{Pl}}}^{2}$. Regions above this line are therefore excluded. The solid green curve (with ticks up) represents the condition that all of the brane evolution occurs within one throat, and has been obtained through a numerical integration. All points below that curve would not satisfy this condition. Its shape can be piecewise analysed. In the region $\phi_\mathrm{strg} < \phi_{{\epsilon_2}}$ (inside the black dotted contour), this is a straight line given by equation (\ref{['eq:onethroat-phieps2']}). The slope of this line is universal, but the offset again depends on $\alpha'm_{{\mathrm{Pl}}}^{2}$. Outside the dotted contour, namely for $\phi_\mathrm{strg}>\phi_{{\epsilon_2}}$, and in the limit $\phi_\mathrm{strg} \gg \phi_{{\epsilon_2}}$, this boundary is described by equation (\ref{['eq:onethroat-phistrings']}). Combined, these pieces lead to the solid green curve with ticks up; to make the shape of both pieces visible individually, they have been extended outside their respective domains of validity (dotted-dashed black curve).