What does inflation really predict?

TL;DR

This work investigates what inflationary cosmology predicts when the inflaton potential exhibits multiple minima, as suggested by the string theory landscape, and how measure choices alter those predictions. By Monte Carlo sampling Gaussian-random potentials with varying horizontal and vertical scales and comparing two distinct measures, the authors demonstrate that observable predictions are driven not only by V(φ) but aggressively by the chosen measure, leading to three qualitative prediction regimes. In the high mh limit they obtain sharp predictions for ns≈0.963 and r≈0.15, while in the low mh limit ns and Q become broad and often conflict with observations; Planck-scale mh ≈ m_Pl restores sensitivity to the full potential and yields prospects for detectable gravitational waves around r∼0.03. Conditioning on reference objects (galaxies, halos, protons) reveals a pronounced smoothness problem: the required selection effects couple Q and ρ_Λ in ways that typically overproduce ρ_Λ or fail to reproduce the observed Q, challenging simple anthropic explanations. Overall, the paper argues that the measure problem is as crucial as the potential in shaping observable predictions and calls for principled resolutions to render inflation a falsifiable, testable framework in light of upcoming cosmological data.

Abstract

If the inflaton potential has multiple minima, as may be expected in, e.g., the string theory "landscape", inflation predicts a probability distribution for the cosmological parameters describing spatial curvature (Omega_tot), dark energy (rho_Lambda, w, etc.), the primordial density fluctuations (Omega_tot, dark energy (rho_Lambda, w, etc.). We compute this multivariate probability distribution for various classes of single-field slow-roll models, exploring its dependence on the characteristic inflationary energy scales, the shape of the potential V and and the choice of measure underlying the calculation. We find that unless the characteristic scale Delta-phi on which V varies happens to be near the Planck scale, the only aspect of V that matters observationally is the statistical distribution of its peaks and troughs. For all energy scales and plausible measures considered, we obtain the predictions Omega_tot ~ 1+-0.00001, w=-1 and rho_Lambda in the observed ballpark but uncomfortably high. The high energy limit predicts n_s ~ 0.96, dn_s/dlnk ~ -0.0006, r ~ 0.15 and n_t ~ -0.02, consistent with observational data and indistinguishable from eternal phi^2-inflation. The low-energy limit predicts 5 parameters but prefers larger Q and redder n_s than observed. We discuss the coolness problem, the smoothness problem and the pothole paradox, which severely limit the viable class of models and measures. Our findings bode well for detecting an inflationary gravitational wave signature with future CMB polarization experiments, with the arguably best-motivated single-field models favoring the detectable level r ~ 0.03. (Abridged)

Figures (17)

• Figure 1: Yellow/light grey cosmological parameter distributions show inflationary predictions for our § 5 example with characteristic $\phi$-scale ${m_h}\approx 2m_{\rm Pl}$, $V$-scale ${m_v}=0.002m_{\rm Pl}$ and uniform measure over initial $\phi$-values. Green/dark grey regions show observational constraints ($1\sigma$) from Table 1 and sdssparssdsslyaf. $\rho_\Lambda$ is in Planck units.
• Figure 2: A small segment of the inflaton potential $V(\phi)$ for simulation number 26140213 from § 5, with vertical and horizontal energy scales ${m_h}=3{\bar{m}}$ and ${m_h}=0.007{\bar{m}}$, respectively (${\bar{m}}=m_{\rm Pl}/\sqrt{8\pi}$). A half-basin of attraction stretches from a local maximum (circle) to a local minimum (triangle). The thicker curve indicates regions where the slow-roll approximation is valid, with squares indicating inflation endpoints. Many starting points $\phi$ fail to produce galaxies, either because the slow-roll approximation in invalid and there is no inflation (three-pointed star), because inflation is rapidly followed by $\rho_\Lambda<0$ recollapse (five-pointed star) or because inflation never ends (six-pointed star). Only a small fraction of starting points (like the four-pointed star) give inflation followed by a vacuum density $|\rho_\Lambda|$ near zero.
• Figure 3: The spatial curvature parameter $\Omega_{\rm tot}$, the primordial gravitational wave power spectrum $\delta_H^T(k)$ and the dark energy function $\rho_\Lambda(a)$ are all determined by the single curve $\rho(a)$ above, as is the primordial density fluctuation spectrum $\delta_H(k)$ in many cases. Dotted diagonals of slope $-2$ are lines of constant comoving horizon size $a/H^{-1}=\dot a$, governing when fluctuations exit end reenter the horizon. Inflation $(\ddot a>0)$ is when the cosmic density history $\rho(a)$ has a shallower slope than these diagonals, $d\ln\rho/d\ln a>-2$, and this logarithmic slope normally becomes $-3$ during reheating, $-4$ during radiation domination, $-3$ during matter domination and $>-2$ now during dark energy domination. When two points lie on the same diagonal (like the two triangles), it means that that the horizon volume at the two epochs is the same comoving spatial region. $\Omega_{\rm tot}$ is constant on these diagonals, approaching unity towards the upper right, and unless the $\rho(a)$ curve crosses the leftmost heavy diagonal, we should observe curvature $|\Omega_{\rm tot}-1|>10^{-5}$. The dark energy function $\rho_\Lambda(a)$ is simply the curve $\rho(a)$ at late times minus the matter contribution. The primordial gravitational wave power spectrum is simply the curve $\rho(a)$ at early times, rescaled vertically (by a factor of two since $\delta_H^T(k)\propto\rho^{1/2}$) and horizontally (mapping $a$ into $k$ matching horizon exit and reentry with the diagonal lines, as with the pairs of triangles, squares and circles). For single-field inflation and many multi-field cases, the primordial density fluctuation spectrum is determined from the curve $\rho(a)$ and its derivative by $\delta_H(k)\propto\rho^{1/2} (-d\ln\rho/d\ln a)^{-1/2}$.
• Figure 4: Spacetime foliation examples. Solid curves (top and middle panels) show infinite hypersurfaces corresponding to the end of inflation, starting at $t=0$, $a=1$ with an inflaton field $\phi({\bf r})=\varepsilon x$ with $\varepsilon$ so small that spatial gradients are negligible. Dashed curve (middle panel) corresponds to the present density. The four shaded regions of comoving space fail to produce any Hubble volumes resembling ours, either because inflation never ends (yellow/light grey) or because inflation never starts or lasts less than 55 e-foldings (cyan/grey). Lower panel shows the inflaton potential used, with vertical dotted lines showing maxima, minima and inflation endpoints.
• Figure 5: The cosmic density history $\rho(a)$ is shown in the three panels with labeling on the right side for a handful of simulations with Measure A and inflation mass scales $({m_h},{m_v})$=$(0.2{\bar{m}},10^{-6}{\bar{m}})$, $(2{\bar{m}},0.0004{\bar{m}})$ and $(100{\bar{m}},0.02{\bar{m}})$. The corresponding scalar and tensor power spectra are shown with labeling on the left side. The shaded vertical band indicates the range 43-55 e-foldings from the end of inflation where there is current hope to measure these primordial fluctuationspwindowsspacetime.