Running-mass models of inflation, and their observational constraints

TL;DR

This paper develops a three-parameter (c, σ, τ) framework for inflation with a running inflaton mass arising from softly broken SUSY and unsuppressed couplings, using a linearized dependence of the mass on $\ln\phi$ to predict the spectrum via $\delta_H^2(k)$ and $n(k)$. It classifies four model types based on the position of $\phi_*$ (maximum or minimum) and the inflationary trajectory, deriving how $n(k)$ and the COBE normalization depend on the parameters and the end-of-inflation mechanism. Observational constraints are used to delimit viable regions in parameter space, with crude bounds indicating substantial allowed regions but potential detectability of scale dependence by Planck. The paper also analyzes the simpler case of a single gauge coupling, obtaining RG-evolved expressions for $m^2(\phi)$ and mapping them onto the four model types, thereby tying particle-physics parameters (masses, couplings) to cosmological observables. Overall, the work provides a concrete, testable bridge between microphysical running-mass dynamics and cosmological data, highlighting where future observations could decisively discriminate among running-mass scenarios.

Abstract

If the inflaton sector is described by softly broken supersymmetry, and the inflaton has unsuppressed couplings, the inflaton mass will run strongly with scale. Four types of model are possible. The prediction for the spectral index involves two parameters, while the COBE normalization involves a third, all of them calculable functions of the relevant masses and couplings. A crude estimate is made of the region of parameter space allowed by present observation.

Figures (11)

• Figure 1: A possible form for the renormalization-group-improved inflaton potential. In units of $M_{\rm P}$, the maximum is located roughly at $\phi\sim e^{-1/c}$. In the case illustrated, the minimum corresponds to the vacuum, where $V$ vanishes. This and the following figure are taken from p97toni.
• Figure 2: Alternatively, the true vacuum may lie in another field direction, 'out of the paper'.
• Figure 3: This and the following three figures show a crude observational constraint on the parameter space, obtained by requiring $|n-1|<0.2$ at both $N=N_{\rm COBE}$ and $N=N_{\rm COBE}-10$. These values of $N$ correspond respectively to the biggest and smallest scales on which $n$ can be observed. The first panel takes $N_{\rm COBE}=50$, while the second takes $N_{\rm COBE}=25$. In all cases, a full line corresponds to $n=0.8$ while a long--dashed line corresponds to $n=1.2$. The dotted straight line $|\sigma | = |c|$ is also shown in all cases; since the theoretically expected parameters satisfy $|\sigma | \geq |c|$, the allowed parameter space is bounded by these three lines. If $|\sigma |$ is bigger than the value indicated by the short--dashed line (labelled P-LB), the Planck satellite will measure the scale dependence of $n$.
• Figure 4: Model (ii), in which $n$ is less than 1, and decreases as the scale decreases. The full line corresponds to $n=0.8$ evaluated at $N=N_{\rm COBE}-10$. The allowed region lies above this line and below the dotted one. For parameters lying below the short--dashed line (P--LB) the spectral index scale dependence will be detected by the Planck satellite.
• Figure 5: Model (iii), in which $n-1$ increases as the scale decreases. The full line corresponds to $n=0.8$ evaluated at $N=N_{\rm COBE}$. The long--dashed line (which is invisible in the first panel) corresponds to $n=1.2$ evaluated at $N_{\rm COBE}-10$. The allowed region lies between the two lines and the axes and below the dotted line. For parameters lying below the short--dashed line (P--LB) the spectral index scale dependence will be detected by the Planck satellite.