Constraining quantum initial conditions before inflation

We theoretically and observationally investigate different choices of initial conditions for the primordial mode function that are imposed during an epoch preceding inflation. By deriving predictions for the observables resulting from several alternate quantum vacuum prescriptions we show some choices of vacua are theoretically observationally distinguishable from others. Comparing these predictions to the Planck 2018 observations via a Bayesian analysis shows no significant evidence to favor any of the quantum vacuum prescriptions over the others. In addition we consider frozen initial conditions, representing a white-noise initial state at the big-bang singularity. Under certain assumptions the cosmological concordance model and frozen initial conditions are found to produce identical predictions for the cosmic microwave background anisotropies. Frozen initial conditions may thus provide an alternative theoretic paradigm to explain observations that were previously understood in terms of the inflation of a quantum vacuum.

Figure 1

Phase portrait of scalar field inflation with a Starobinsky potential $V(\varphi )={\mathrm{\Lambda }}^4{(1-\exp [-\sqrt{2/3}\varphi ])}^2$ [21] where ${\mathrm{\Lambda }}^2=0.1$. Arrows denote the direction of system evolution in phase space. Kinetic dominance ($w_{\varphi }>0.9$), inflation ($w_{\varphi }<-\frac{1}{3}$), and slow-roll inflation ($w_{\varphi }<-0.9$) regimes are highlighted. The plot visualizes the evolution of possible universes under this model; universes beginning with $\varphi >0$ typically start in kinetic dominance and then move onto the attractor solution inside slow-roll inflation. Plot is based upon Fig. 3 in [13].

Figure 2

Evolution of the comoving horizon and ${\mathcal{R}}_k$ modes. For illustrative purposes the $R_k$ modes are shifted in the vertical direction to be about the corresponding value of $k$. The modes oscillate when they enter the horizon during kinetic dominance, and then freeze out again once they leave during inflation. Some modes at low $k$ however always remain far above the horizon and so stay frozen throughout. Figure is based upon Fig. 1 in [17].

Figure 3

Comoving horizon in the Contaldi approximation. Note how, unlike in Fig. 2, the behavior of the comoving horizon changes discontinuously at a well-defined transition point. At this transition point the comoving horizon takes its maximum value $1/k_t$.

Figure 4

Approximations of the primordial power spectra produced by BD, RHM [19], HD, and RSET initial conditions. BD, RHM, and HD all produce very similar spectra, with a cutoff when $k<2k_t$, and a flat spectra when $k>10k_t$, with a small amount of oscillation in between. In RSET however the oscillations are far more prominent and die away far more slowly.

Figure 5

Comparison of numerical evaluation and two approximations for the primordial power spectra produced by frozen initial conditions. Numerical evaluation took the scalar field potential as $V(\varphi )=\frac{1}{2}{m}^2{\varphi }^2$ with $m=1$ and was calculated using oscode [33]. For the approximations the free parameters were taken to be best fit values of $k_t=9.34$ and ${\nu }_s=1.52$. The Contaldi method correctly predicts high $k$ oscillations but fails to account for the decay in peak height, while the slow-roll method appears to fit very well at high $k$. Both approximations at low $k$ correctly predict a cutoff but underestimate the value at which it occurs.

Figure 6

Primordial power spectrum for BD initial conditions using the slow-roll approximation and power-law times Contaldi approximation. In both ${\nu }_s$ was taken to be 1.5187. The two different approximations are in good agreement at high $k$. The error only exceeds 1% below the low $k$ cutoff.

Figure 7

CMB temperature spectrum for BD, HD, RHM, and RSET initial conditions along with the best fit $\mathrm{\Lambda }\mathrm{CDM}$ model and experimental data from Planck 2018. The free parameter $k_t$ is taken to be $5\times {10}^{-4}{\mathrm{Mpc}}^{-1}$, which is chosen so that the cutoffs in the primordial power spectra were in the same region as the best fit cutoffs found in [19]. At high $\ell$ there is no noticeable difference between any of the initial conditions and $\mathrm{\Lambda }\mathrm{CDM}$. However at low $\ell$ there is some variation between $\mathrm{\Lambda }\mathrm{CDM}$, RSET, and the other three. All four initial conditions lead to a suppression of the low multipoles compared to $\mathrm{\Lambda }\mathrm{CDM}$ which may make them favored over the baseline $\mathrm{\Lambda }\mathrm{CDM}$ model. The top axis gives an approximate correspondence between $k$ and $\ell$ using the Limber approximation [39].

Figure 8

CMB TE and EE spectra for BD, HD, RHM, and RSET initial conditions along with experimental data from Planck 2018. Spectra presented as differences from the Planck 2018 best fit $\mathrm{\Lambda }\mathrm{CDM}$ model. Note that unlike Fig. 7 the right-hand portion of the graphs have different scales to their left-hand portion as there is such a large magnitude difference. As above, the free parameter $k_t$ is taken to be $5\times {10}^{-4}{\mathrm{Mpc}}^{-1}$. Similarly to the TT spectra, for all vacuum initial conditions both the TE and EE spectra are indistinguishable from $\mathrm{\Lambda }\mathrm{CDM}$ at high $\ell$ and have a relative suppression of power at low multipoles, with noticeable differences between RSET and the other initial conditions spectra at low $\ell$.

Figure 9

Unlensed CMB temperature spectrum produced by frozen initial conditions for various values of the $\mathrm{\Delta }k$ parameter along with the best fit $\mathrm{\Lambda }\mathrm{CDM}$ model and experimental data from Planck 2018. For larger $\mathrm{\Delta }k$ the FIC spectra show noticeable oscillations about the $\mathrm{\Lambda }\mathrm{CDM}$ spectrum. However for small $\mathrm{\Delta }k$ the oscillations seem to average out producing a spectra very similar, or even identical, to $\mathrm{\Lambda }\mathrm{CDM}$. The oscillations for some intermediate values of $\mathrm{\Delta }k$, for example $5.0\times {10}^{-4}{\mathrm{Mpc}}^{-1}$, may provide a potential explanation for the drop in power seen at around $\ell =20$.

Figure 10

Unlensed CMB TE and EE spectra for frozen initial conditions with various values of the $\mathrm{\Delta }k$ parameter along with experimental data from Planck 2018. Spectra presented as differences from the Planck 2018 best fit $\mathrm{\Lambda }\mathrm{CDM}$ model. The right and left hand portion of the graphs have different scales due to there being a large magnitude difference. As was seen in the TT spectra the FIC spectra vary strongly with the value of $\mathrm{\Delta }k$, with higher values displaying large oscillations and low values being nearly indistinguishable from $\mathrm{\Lambda }\mathrm{CDM}$.

Figure 11

Posterior distributions of the parameters of $\mathrm{\Lambda }\mathrm{CDM}$, BD, and RSET models. RHM and HD posterior distributions were found to be near identical to the BD posterior shown here. For the six shared parameters the posteriors are all very similar and in good agreement with the Planck 2018 results for $\mathrm{\Lambda }\mathrm{CDM}$. The marginalized $k_t$ posterior of BD shows a noticeable peak around $2.0\times {10}^{-4}{\mathrm{Mpc}}^{-1}$ whereas for RSET the $k_t$ posterior is maximum at 0 and generally decays as $k_t$ increases. Given that the predictions of BD, RSET, and $\mathrm{\Lambda }\mathrm{CDM}$ are equivalent when $k_t=0$ this suggests the BD model can provide a better fit to the Planck data than $\mathrm{\Lambda }\mathrm{CDM}$. In addition it would seem $\mathrm{\Lambda }\mathrm{CDM}$ is a better model then RSET on grounds of being more parsimonious. Plot produced using GetDist [46].

Figure 12

Profile likelihood of BD and RSET models over the parameter $k_t$, with the ${\chi }^2$ for the best fit $\mathrm{\Lambda }\mathrm{CDM}$ model shown for comparison. At each fixed value of $k_t$ a Nelder-Mead optimization is performed over the remaining parameters to find the best fit BD/RSET model with that value of $k_t$. From this model we then calculate ${\chi }^2=-2\ln (\mathrm{P}(\text{Data}|\text{Model}))$. We find BD shows a significant range of $k_t$ values where it can provide a better fit to the Planck 2018 data than $\mathrm{\Lambda }\mathrm{CDM}$. Conversely RSET only improves on the best fit $\mathrm{\Lambda }\mathrm{CDM}$ in a very narrow region of parameter space and only by a small amount.

Figure 13

Log Bayesian evidences of the vacuum initial condition and $\mathrm{\Lambda }\mathrm{CDM}$ models. For each model the central black line is the calculated log evidence and the coloured bar about it gives the $2\sigma$ confidence interval in that value. The dashed lines shows the relative Occam penalty incurred by the vacuum initial condition models, estimated from difference in KL divergence [47] between the model and $\mathrm{\Lambda }\mathrm{CDM}$, calculated using anesthetic [48]. The results show that $\mathrm{\Lambda }\mathrm{CDM}$ is significantly better supported by the observations than the vacuum initial condition models, primarily due to the Occam penalty.