Primordial standard clock models and CMB residual anomalies

The residuals of the power spectra of Wilkinson Microwave Anisotropy Probe and Planck’s cosmic microwave background (CMB) anisotropies data are known to exhibit a few interesting anomalies at different scales with marginal statistical significance. Combining bottom-up and top-down model-building approaches and using a pipeline that efficiently compares model predictions with data, we construct a model of primordial standard clock that is able to link and address the anomalies at both the large and small scales. This model, and its variant, provide some of the best fits to the feature anomalies in CMB. According to Bayes evidences, these models are currently statistically indistinguishable from the Standard Model. We show that the difference between them will soon become statistically significant with various higher quality data on the CMB polarization. We demonstrate that such a model-building and data-analyses process may be used to uncover a portion of detailed evolutionary history of our universe during its primordial epoch.

Figure 1

A bird’s eye view of the model configuration and the background trajectory, plotted over equipotential surfaces (the redder color corresponds to lower potential). In terms of the Cartesian coordinates $x$ and $y$ shown here, for $x<0$, the $\mathrm{\Theta }$ and $\sigma$ are Cartesian with $x=\mathrm{\Theta }-({\mathrm{\Theta }}_T+{\mathrm{\Theta }}_0)$ and $y=\sigma +{\xi }^{-1}$, so the straight solid line at the top of the plateau represents $\sigma =0$. For $x>0$, the $\mathrm{\Theta }$ and $\sigma$ become radial coordinates with $r=\sigma +{\xi }^{-1}$ and $\theta =\pi /2-\xi (\mathrm{\Theta }-{\mathrm{\Theta }}_0-{\mathrm{\Theta }}_T)$. Together with the noncanonical coupling (2.5), they describe a curved path at the bottom of the potential valley, as in [15, 16]. The insets show how the inflaton overshoots the bottom of the valley (yellow line) and climbs onto a side of the cliff (top-right), and the clock oscillation (bottom-left).

Figure 2

Example of the evolution of the clock field [top] and the time derivative of the inflaton $\dot{\mathrm{\Theta }}$ (bottom). In the top panel, we also plot the effective minimum of $\sigma$ given by Eq. (2.5) in white dashed line.

Figure 3

Effects of varying the effective parameters of the model on the primordial scalar power spectrum. We fix the parameters of the baseline model to $C_{\mathrm{\Theta }}\times {10}^2=1.85$, ${\log }_{10}{P}_{\zeta *}=-8.64$, $C_{\sigma }/{\mathrm{\Theta }}_f^2=0.15$, $\mathrm{\Delta }{P}_{\mathrm{dip}}/P=0.24$, $\mathrm{\Delta }{P}_{\text{clock}}/P=0$, $m_{\sigma }/H=30$, $N_0=14.3$, $N_T=0$ and vary one parameter at a time. Specifically, we vary $C_{\sigma }/{\mathrm{\Theta }}_f^2\in \{0.07,0.15,0.45\}$ in the top-left panel, $\mathrm{\Delta }{P}_{\mathrm{dip}}/P\in \{0.07,0.24,0.33\}$ in the top-right panel, $\mathrm{\Delta }{P}_{\text{clock}}/P\in \{0,0.046,0.077\}$ in the center-left panel, $m_{\sigma }/H\in \{18,30,50\}$ in the center-right panel, $N_0\in \{13.9,14.3,14.7\}$ in the bottom-left panel and $N_T\in \{0,0.3,0.6\}$ in the bottom-right panel. In the last three panel, we fix $\mathrm{\Delta }{P}_{\text{clock}}/P=0.046$. In each plot, the convention is that purple, dark orange and light blue lines represent the smallest, the central and the largest value of the parameter which is being varied respectively.

Figure 4

Triangle plot of the full model for PlikHM bin1 (P18, purple) and Clean camspec v12.5hmcln (EG20, light blue) datasets. We also plot better likelihood samples in the top-right corner of the plot ($\mathrm{\Delta }{\chi }^2$ in the color bar is defined as $\mathrm{\Delta }{\chi }^2\equiv \mathrm{\Delta }{\chi }_{\text{featureless}}^2-\mathrm{\Delta }{\chi }_{\mathrm{CPSC}}^2$).

Figure 5

PPS of the low-frequency best-fit candidates (LFCI, LFC IIa, and LFC IIb), and high-frequency best-fit candidate (HFC).

Figure 6

Residual plots of the full model for the low-frequency best-fit candidates to P18 (top) and EG20 (bottom) likelihood. Orange, purple and light blue lines correspond to LFC I, LFC IIa, and LFC IIb respectively.

Figure 7

Residual plots of the full model for the high-frequency best-fit candidate to P18 (top) and EG20 (bottom) likelihood.

Figure 8

$\mathrm{\Delta }{\chi }^2$ histogram for our model from fitting 1000 simulations of Planck noise. The dashed line is the probability distribution function estimated from the data points, that are shown with yellow ticks on the x-axis. Details about the simulation are provided in the main text.

Figure 9

Triangle plot for the restricted model for PlikHM bin1 (P18, purple) and Clean camspec v12.5hmcln (EG20, light blue) datasets.

Figure 10

Comparison of the amplitude and frequency 2-dimensional posterior contours of the restricted CPSC (purple) and SiRe (blue) model. We show contours for P18 and EG20 in the upper and lower panels respectively.

Figure 11

Best-fit candidates for the restricted CPSC model and the SiRe model. The candidates for P18 and EG20 are shown in the upper and lower panel respectively. For a comparison, we also plot a best-fit candidate found in [35] for another CPSC model [8, 9].

Figure 12

Residual plots for the best-fit candidates to P18 (top) and EG20 (bottom) for the restricted CPSC model (purple) and for the SiRe model (blue).

Figure 13

$\mathrm{\Delta }{\chi }^2$ histogram for the restricted model from fitting 1000 simulations of Planck noise. The dashed line is the probability distribution function estimated from the data points, that are shown with yellow ticks on the x-axis.

Figure 14

The noise power spectra for E-modes from four observations—Planck, LiteBIRD, SO, CMB S4, and PICO are plotted in dashed lines. The $\mathrm{\Delta }{\mathcal{C}}_{\ell }$ due to cosmic variance are also plotted as contours around the ${\mathcal{C}}_{\ell }$. The cosmic variance differs only by the $f_{\mathrm{sky}}$. Here we only plot it for $f_{\mathrm{sky}}=0.7$.

Figure 15

Residual plot of the two best-fit candidates showing P18 error bars together with the error bars expected by future experiments such as LiteBIRD (green), the Simons Observatory (blue), CMB S4 (gold), and PICO (pink). In the EE panel, where data points are outside the y-axis range, we have moved the errors for SO and PICO near the upper or lower frame edge in order to demonstrate their sizes.

Figure 16

Projected constraints on model parameters with future experiments using LFC-IIa in Fig. 5 as fiducial model. In the top-right corner, we plot constraints on the frequency and amplitude of the clock signal obtained using only $\ell >600$ information from PICO. In order to show the accuracy to which the frequency of the signal can be determined, we plot derived constraints on $m_{\sigma }/H$ rather than on ${\log }_{10}{m}_{\sigma }/H$, which is the actual parameter varied in the analysis.

Figure 17

Projected constraints on model parameters with future observations using the restricted model candidate in Fig. 11 as fiducial model. In the top-right corner, we plot constraints on the frequency and amplitude of the clock signal obtained using only $\ell >600$ information from PICO. In order to show the accuracy to which the frequency of the signal can be determined, we plot derived constraints on $m_{\sigma }/H$ rather than on ${\log }_{10}{m}_{\sigma }/H$, which is the actual parameter varied in the analysis.

Figure 18

Linear matter power spectrum at $z=0$ for our best-fit candidates.

Figure 19

Effects of varying the $\epsilon$ parameter. We plot the background slow-roll parameter $\epsilon$ and the scalar and tensor power spectra in the top, center, and bottom panel respectively. The parameters for the case with $\epsilon ={10}^{-7}$ are the same used in Fig. 3. For other values of $\epsilon$, we perform the rescaling mentioned in the text.

Figure 20

Scalar power spectra for very small values of $m_{\sigma }/H$. The parameters are the same as used in Fig. 3, except for $N_T=1$ and for $m_{\sigma }/H$ which is varied according to the legend. We also plot the power spectra for ${\mathrm{\Theta }}_f\to \infty$ in dashed lines.

Figure 21

Top: evolution of the clock field and (bottom) predictions on the PPS for a set of different values of $C\equiv {\sigma }_i/{\sigma }_f$.

Figure 22

Triangle plot for the restricted model for Atacama Cosmology Telescope (ACT, pink), Clean camspec v12.5hmcln (EG20, light blue) and their combination (olive). In the inset, we also show the best-fit for the combined dataset and compare it to the one for EG20.

Figure 23

Same as Fig. 4 but with a different choice of priors and parametrization described in Appendix pp3.

Figure 24

Same as Fig. 9 but with a different choice of priors and parametrization described in Appendix pp3.

Figure 25

Same as Fig. 10 but with a different choice of priors and parametrization described in Appendix pp3.