Cosmological constraints on general, single field inflation

Inflation is now an accepted paradigm in standard cosmology, with its predictions consistent with observations of the cosmic microwave background. It lacks, however, a firm physical theory, with many possible theoretical origins beyond the simplest, canonical, slow-roll inflation, including Dirac-Born-Infeld inflation and k-inflation. We discuss how a hierarchy of Hubble flow parameters, extended to include the evolution of the inflationary sound speed, can be applied to compare a general, single field inflationary action with cosmological observational data. We show that it is important to calculate the precise scalar and tensor primordial power spectra by integrating the full flow and perturbation equations, since values of observables can deviate appreciably from those obtained using typical second-order Taylor expanded approximations in flow parameters. As part of this, we find that a commonly applied approximation for the tensor-to-scalar ratio, $r\approx 16c_sϵ$, becomes poor (deviating by as much as 50%) as $c_s$ deviates from 1 and hence the Taylor expansion including next-to-leading order contribution terms involving $c_s$ is required. By integrating the full flow equations, we use a Monte-Carlo-Markov-Chain approach to impose constraints on the parameter space of general single field inflation, and reconstruct the properties of such an underlying theory in light of recent cosmic microwave background and large-scale structure observations.

Figure 1

Monte Carlo sampling of canonical ($c_s=1$, $\kappa ={}^l\alpha =0$) [upper panels] and general [lower panels] inflation models plotted in the $(r,n_s)$ (left) and $(n_{\mathrm{run}},n_s)$ (right) planes, truncated at $l=5$ in the ${}^l\lambda$ and ${}^l\alpha$ flow hierarchies using linear priors (black points) and log priors (blue points) on $a(t_{\mathrm{en}})$, $c_s$, and flow parameters, in the ranges specified in (65). For general inflation we plot models that have $c_s(k_{*s})\in [0,1]$.

Figure 2

The difference between $r_{\mathrm{exact}}$ and $r_{\mathrm{approx}}$ [left panel] and the dependence on $c_s(k_{*s})$ [right panel] for general inflation models with $c_s(k_{*s})\in [0,1]$ (blue points), and $c_s(k_{*s})\in [1,2]$ (red points), for an order 5 Monte-Carlo simulation with ranges as given in (65).

Figure 3

Evolution of $c_s$ [left panel], $\kappa$ [center panel] and $c_{s}k/aH$ [right panel] for $k=0.01{\mathrm{Mpc}}^{-1}$, for two example trajectories with $\kappa >0$ (black) and $\kappa <0$ (red) at the epoch when current cosmological scales exit the horizon. We find that for all viable trajectories $c_{s}k/aH$ is monotonically decreasing over the course of inflation so that one does not need to be concerned about the prospect of multiple horizon crossings, and freezing and thawing of scalar perturbations.

Figure 4

Comparison of the 1D marginalized posterior probability distributions for the flow parameters and observables, for the cases of power law inflation with scale independent running (blue full line) and our models C1 (black dotted), C2 (black dashed), G2 (red dot-dashed) and G3 (red triple dot-dashed). The constraints for model G1 are the same as those for G2, with just $c_s$ cutting off at 1, so we do not show them here.

Figure 5

The 68% and 95% confidence regions of the $({}^1\lambda ,{}^2\lambda )$ [left], $(r,n_s)$ [center], and $(n_{\mathrm{run}},n_s)$ [right] parameter spaces. Constraints for a standard power law spectrum with constant running are shown in dark blue and light blue, as well as canonical inflation models C1 (black dotted), C2 (black dashed) and general inflation model G2 (red dot-dashed).

Figure 6

Comparison of the 1D marginalized posterior probability distributions, showing 68% (dark blue) and 95% (pale blue) confidence limits, for the primordial scalar power spectrum for models C1 [left], C2 [center] and G2 [right], in comparison to a “standard” power law spectrum with scale independent running (black dashed lines). Increasing the number of flow parameters increases the freedom with which the spectrum is reconstructed. In particular we can get significantly greater or smaller power at small and large scales, least well measured by the CMB and large-scale structure data, respectively. The bounds on the higher-order parameters in model C2 directly arise from constraining this greater or lesser power at the extreme ends of the observed scales.

Figure 7

While models with large $ϵ$ and small $c_s$ are consistent with observational constraints from the bound on $r$, imposing the condition that inflation persists while the observable scales exit the horizon places an additional restriction on $ϵ(k_{*s})$. The left panel shows $c_s(k_{*s})$ vs $ϵ(k_{*s})$ for models which satisfy observational constraints. Models which additionally satisfy $ϵ(k)<1$ (red crosses) have an upper limit on the value of $ϵ(k_{*s})$, while for larger $ϵ(k_{*s})$ (blue squares) the $ϵ(k)<1$ condition is not met (these models are rejected in the MCMC analysis). The right panel shows the evolution of $ϵ(k)$ for observable modes, for example models which give constraints at the pivot point in agreement with the data, and which satisfy (red, lower two lines) or break (blue, upper two curves that reach $ϵ=1$).

Figure 8

Comparison of the 1D marginalized posterior probability distributions for Model C1 [top panels] and C2 [lower panels], showing 68% (dark blue) and 95% (pale blue) confidence limits, for the flow parameters and observables as each observed comoving mode $k$ exits the horizon, where $k(N)$ is given by (70).

Figure 9

Comparison of the 1D marginalized posterior probability distributions for Model G1, showing 68% (dark blue) and 95% (pale blue) confidence limits, for the flow parameters and observables as each observed comoving mode $k$ exits the horizon.

Figure 10

Reconstructed energy density $\rho$ [upper panels] and kinetic term $X{\mathcal{L}}_X$ [lower panels] for canonical models C1 [left panel], C2 [center] and general inflationary model G1 [right] which satisfy both conditions 1 and 2 from Sec. 4. Each has $A_s$, $n_s$, $n_{\mathrm{run}}$ at $k=k_{*}$ and the tensor-to-scalar ratio consistent with observations at the $2\sigma$ level, and $ϵ=1$ at the end of inflation with $N_{*}\sim 50–80$. The range of $\varphi$ represents 100 e-foldings of evolution, with $\varphi =0$ at scalar horizon crossing for $k_{*}$. (For canonical models with the usual scalar field definition, ${\mathcal{L}}_X=c_s^{-1}$, $X{\mathcal{L}}_X=X$.)