Phenomenology of D-Brane Inflation with General Speed of Sound

TL;DR

This work investigates D-brane inflation with a general speed of sound by extending the inflationary flow formalism to DBI-like models parameterized by $H(\varphi)$ and $\gamma(\varphi)$. It combines a Hamilton-Jacobi treatment with Monte-Carlo sampling to map warp factors, potentials, and observables to data, while enforcing microscopic constraints such as the field-range bound. The analysis reveals that most phenomenological models conflict with the compactification bound, and those that survive predict either ultra-small tensor amplitudes ($r\lesssim 10^{-15}$) or a blue scalar spectrum ($n_s>1$) when $f_{NL}$ is large, challenging the viability of relativistic DBI inflation for red-tilted spectra. Consequently, only slow-roll DBI with negligible non-Gaussianity or relativistic DBI with large $f_{NL}$ and $n_s>1$ remain consistent with current constraints, highlighting the falsifiability of DBI inflation and guiding future observational tests including non-Gaussianity and tensor modes.

Abstract

A characteristic of D-brane inflation is that fluctuations in the inflaton field can propagate at a speed significantly less than the speed of light. This yields observable effects that are distinct from those of single-field slow roll inflation, such as a modification of the inflationary consistency relation and a potentially large level of non-Gaussianities. We present a numerical algorithm that extends the inflationary flow formalism to models with general speed of sound. For an ensemble of D-brane inflation models parameterized by the Hubble parameter and the speed of sound as polynomial functions of the inflaton field, we give qualitative predictions for the key inflationary observables. We discuss various consistency relations for D-brane inflation, and compare the qualitative shapes of the warp factors we derive from the numerical models with analytical warp factors considered in the literature. Finally, we derive and apply a generalized microphysical bound on the inflaton field variation during brane inflation. While a large number of models are consistent with current cosmological constraints, almost all of these models violate the compactification constraint on the field range in four-dimensional Planck units. If the field range bound is to hold, then models with a detectable level of non-Gaussianity predict a blue scalar spectral index, and a tensor component that is far below the detection limit of any future experiment.

Figures (11)

• Figure 1: The $r-n_s$ plane populated by numerical models of DBI inflation with $M=5$ and $M'=1$ in equations (\ref{['eq:h']}) and (\ref{['eq:gamma']}), combining simulations with general and small $\gamma_0$ priors (left panel) and by numerical models of standard slow-roll inflation from Kinney Kinney:2002qn (right panel).
• Figure 2: Cosmological observables evaluated at $k_{\rm CMB}$ for a simulation applying the general $\gamma_0$ prior with $M=5$ and $M'=1$ in equations (\ref{['eq:h']}) and (\ref{['eq:gamma']}). The three regimes of DBI inflation are shown with color-coded points: slow roll DBI (black), intermediate DBI (orange) and ultra-relativistic DBI (blue). The individual plots from top-left in clockwise direction are: (a) the scalar-to-tensor ratio $r$ vs. scalar spectral index $n_s$, (b) spectral index $n_s$ vs. running of the spectral index $\alpha_s=dn_s/d\ln k$, (c) $r$ vs. $\Delta \phi= \phi_0-\phi_{\rm end}$, and (d) $r$ vs. non-Gaussianity parameter $f_{\rm NL}$. In (a), we show the lower limit on $r$ for ultra-relativistic DBI models with $r > (1-n_s)/8$ from Lidsey as a dashed line. In (d), the limit from WMAP, $|f_{\rm NL}| <332$fNL is indicated by a vertical dashed line. In (c), the relation between $r$ and $\Delta \phi$ can be described approximately as $r \propto (\Delta \phi/M_P)^2$. This follows from equation (\ref{['equ:lyth']}); integration from $N_0$ to $N_{\rm end}$ instead of from $N_{\rm CMB}$ to $N_{\rm end}$ just results in a different normalization $N_{\rm eff}$.
• Figure 3: Same as Fig. \ref{['fig:2']}, but with a small $\gamma_0$ prior. Note that while in Fig. \ref{['fig:2']}(a), the tensor/scalar ratio for ultra-relativistic DBI models satisfied a general lower bound, this is not the case for non-relativistic models.
• Figure 4: Slow-roll inflation (left) and DBI (right) consistency relations, combining the simulations with general and small $\gamma_0$ priors shown in Figs. \ref{['fig:2']} and \ref{['fig:3']}, respectively. The left panel shows $r$ vs. $n_t$. In standard slow-roll inflation $r=-8n_t$ (dashed line), but as shown, DBI inflationary models depart significantly from this relation except in the case of slow roll DBI models (black points). In the right panel, we show $r \left[1+ \frac{108}{35} f_{NL} \right]^{1/2}$ vs. $-8 n_t$ describing the DBI consistency relation; all models satisfy this relation.
• Figure 5: A consistency check of the field range bound (\ref{['equ:dphi2']}) for the general $\gamma_0$ simulation shown in Fig. \ref{['fig:2']} (left) and the small $\gamma_0$ simulation shown in Fig. \ref{['fig:3']} (right), with $\Delta \phi^2/M_P^2$ vs. the expression on the right hand side of Eq. (\ref{['equ:dphi2']}), assuming ${\rm Vol}(X_5) = \pi^3$. The microscopic bound requires that $\Delta \phi^2/M_P^2$ be smaller than the right side of eq. (\ref{['equ:dphi2']}), and all models in these simulations, which had a general $\epsilon_0$ prior applied, violate this bound.