Duality Cascade in Brane Inflation

TL;DR

This work investigates how tiny sharp features in extra dimensions—specifically steps in the warp factor from a Seiberg duality cascade—affect brane inflation. It develops a general framework for calculating the power spectrum and bispectrum in the presence of sharp features and applies it to two scenarios: slow-roll brane inflation with steps in the inflaton potential, and IR DBI inflation with warp-factor steps. The slow-roll case shows that the power spectrum can exhibit pronounced glitches and scale-dependent non-Gaussianity, with the amplitude $f_{NL}^{\rm feature} \sim 7 c^{3/2}/(d\epsilon)$ depending sensitively on step width $d$ and height $c$. In IR DBI, warp-factor steps induce small power-spectrum glitches but potentially large non-Gaussianities due to the DBI kinetic term, and they predict multiple, nearly equally spaced features in the data; collectively these stringy signatures offer a powerful cosmological test of the duality cascade and the underlying throat geometry.

Abstract

We show that brane inflation is very sensitive to tiny sharp features in extra dimensions, including those in the potential and in the warp factor. This can show up as observational signatures in the power spectrum and/or non-Gaussianities of the cosmic microwave background radiation (CMBR). One general example of such sharp features is a succession of small steps in a warped throat, caused by Seiberg duality cascade using gauge/gravity duality. We study the cosmological observational consequences of these steps in brane inflation. Since the steps come in a series, the prediction of other steps and their properties can be tested by future data and analysis. It is also possible that the steps are too close to be resolved in the power spectrum, in which case they may show up only in the non-Gaussianity of the CMB temperature fluctuations and/or EE polarization. We study two cases. In the slow-roll scenario where steps appear in the inflaton potential, the sensitivity of brane inflation to the height and width of the steps is increased by several orders of magnitude comparing to that in previously studied large field models. In the IR DBI scenario where steps appear in the warp factor, we find that the glitches in the power spectrum caused by these sharp features are generally small or even unobservable, but associated distinctive non-Gaussianity can be large. Together with its large negative running of the power spectrum index, this scenario clearly illustrates how rich and different a brane inflationary scenario can be when compared to generic slow-roll inflation. Such distinctive stringy features may provide a powerful probe of superstring theory.

Figures (7)

• Figure 1: The warp factor $h(r)$ in the KS throat, including the steps. Here $r$ is measured in $\sqrt{{\alpha'}}$ with $R_B \simeq 100$, $g_s=2$, M=20$, K=10$ and $N=200$. The width $d=10^{-3}$. In this figure, there are actually 4 steps, located at $r \simeq 73$, $r \simeq 82$, $r \simeq 89$ and $r \simeq 95$, although the step at $r \simeq 95$ is too small to show up. Here, the parameters (in particular, a large $g_s M$) are chosen so that at least 3 steps are big enough to show up in the figure. This leads to relatively large corrections to the positions of the steps. Other parameters should be used in more realistic situations and in comparison with data.
• Figure 2: In the small field case, we show how the step in the inflaton potential changes the power spectrum. We show the power spectrum in the left panel and the behavior of $\epsilon$ around the step in the right panel. For illustration, we use the KKLMMT scenario with only the Coulomb potential. We choose the background flux $N=2000$, so $\Delta\phi/M_\mathrm{pl} \lesssim 0.01$. The inflation scale $V_0 \sim 10^{-17}M_\mathrm{pl}^4$. $\epsilon$ is tiny, typically $\epsilon \sim 10^{-11}$ as shown in the right panel. We introduce a step with $c = 8\times10^{-12}$ and calculate the power spectrum for four different values of $d$: (1) $d=3\times 10^{-6} M_\mathrm{pl}$, green solid line. (2) $d=1.7\times 10^{-6} M_\mathrm{pl}$, black solid line. (3) $d=1.7\times 10^{-7}M_\mathrm{pl}$, red dashed line. (4) $d=1.7\times 10^{-8}M_\mathrm{pl}$, blue dotted line. Note that the dip in the power spectrum depends weakly on $d$. In case (1) and (2), $\Delta\epsilon$ has not saturated the bound Eq. \ref{['deltaepsilon']}, so decreasing $d$ will enhance the bump in $P_{\cal R}$ significantly. In case (3) and (4), where $\Delta\epsilon$ is maximized, the bump in $P_{\cal R}$ does not depend sensitively on $d$, but the range of the oscillations in $k$-space does. The black solid line is close to the best fit power spectrum given in Ref. Peiris:2003ffCovi:2006ci.
• Figure 3: Evolution of $c_s$. Parameters are $b=0.1$, step width $\Delta N_e=0.05$, $\beta=3$, $g_s m_s^{-4}=10^{39}$, $N_B=10^9$, $n_B=10^4$, $n_Ah_A^4=16$.
• Figure 4: The approximate power spectrum, $P_{{\cal R}}$, as a function of $x_{0}=kc_s\tau_0$, where $\tau_0$ is the time of the feature, due to a $\theta$ function step with $b=0.1$.
• Figure 5: In the IR DBI scenario, we show the power spectrum, $P_{{\cal R}}$, when there is a sharp step in the warp factor. For the same step width $\Delta N_e$, we show three cases with different step size $b$. The amplitude of the first dip and bump increases as we increase the step height $b$. A bump appears first for $b<0$ while a dip appears first for $b>0$.