On the consistency of \tpdf\lcdm with CMB measurements in light of the latest Planck, ACT and SPT data

We start by considering the \lcdm’s best fit to the latest CamSpec data [70] as the mean function in our analysis. In Fig. 3, we show the two-dimensional likelihood profiles for the CamSpec residuals with respect to \lcdm’s best-fit $\mathcal{D}_{\ell}$’s, as a function of $(\sigma_{f},\ell_{f})$. The color bar shows the goodness of fit, where $\Delta\chi^{2}=-2(\ln\mathcal{L}^{\rm GP}-\ln\mathcal{L}^{\Lambda\rm CDM})$ and $\ln{\mathcal{L}^{\rm GP}}$ is the log-marginal likelihood (LML), defined in Eq. (2.3). Negative values of $\Delta\chi^{2}$ (in blue) reflect regions in parameter space yielding an improvement in the fit with respect to \lcdm. Conversely, red-colored regions correspond to deviations from the mean function leading to a degraded fit to the data ($\Delta\chi^{2}>0$), whereas white-shaded regions represent no improvement at all. The black dot represents the set of hyperparameters $(\sigma_{f},\ell_{f})$ yielding the highest likelihood and we report the corresponding improvement in fit⁶⁶6Strictly speaking, this quantity is analogous to a $\chi^{2}$ difference, but the exact $\Delta\chi^{2}$ values should not be interpreted in the usual sense, as the LML receives additional (possibly large) contributions from the second and third terms in Eq.(2.3) which depend on the values of the kernel hyperparameters $(\sigma_{f},\ell_{f})$. ($\Delta\chi^{2}$) in Table 3. As mentioned before, if the mean function is a good description of the data, the LML in (2.3) should peak at $\sigma_{f}\to 0$ and/or $\ell_{f}\to\infty$. In other words, no smooth deviations away from the best-fit \lcdm are needed to explain the data. However, if the LML peaks at finite (possibly large) values of $(\sigma_{f},\ell_{f})$, it might point towards the need for a different mean function or indicate the existence of hidden structures/systematics in the data.

In this case, as can be seen from Fig. 3, the \lcdm model provides a relatively good fit to TT, TE, and EE data. The LML seems to prefer small deviations from the best fit \lcdm spectra: $\sigma_{f}\lesssim 1$ for TT and TE and $\sigma_{f}\lesssim 0.1$ for the case of EE. Any larger deviation from the mean function is highly penalized by the data, as can be seen by the color bar on the right ($\Delta\chi^{2}>0$ meaning a degraded fit), and the improvement in fit by the GP is negligible in both TT and TE; see Table 3. Meanwhile, there is a noticeable improvement in fit in EE for $\ell_{f}\lesssim 1$. Such a GP realisation is essentially a white noise where the values at each $\ell$ are uncorrelated with each other. Preferences toward the inclusion of white noise indicate that the fluctuations in the data around the mean are more than what is expected from the data covariance matrix. In other words, the covariance matrix may have been slightly underestimated for this EE data. This may alter the weights and hence the optimality of the cosmological parameter estimation, but is less likely to have a significant effect on the estimated parameter values.

In fact, the LML is expected to have a local minimum at some $\sigma_{f}$ with $\ell_{f}\ll 1$ about half of the time. Taking the limit $\ell_{f}\rightarrow 0$ in (2.3), we found that the necessary and sufficient condition for such a minimum is given by ${\|\Sigma^{-1}\bf{r}\|^{2}>\rm{tr}\left(\Sigma^{-1}\right)}$, where $\bf{r}$ is the residual vector. Assuming that the mean and the covariance matrix are exact, the expected value of the left-hand side is equal to the right-hand side. When the residual vector is large enough either by statistical fluctuations and/or underestimation of the errors, then we expect to see a local minimum at some $\sigma_{f}$ satisfying ${\|(\sigma_{f}^{2}I+\Sigma)^{-1}{\bf r}\|^{2}={\rm tr}\left[(\sigma_{f}^{2}I+\Sigma)^{-1}\right]}.$

Despite this fact, it is still intriguing that we find $\Delta\chi^{2}$ as low as $-10.21$ in EE for CamSpec PR4. As we discuss in Appendix B.1, most of these improvements in fit come from low-$\ell$ data, and notably is not present in the \planck Official Release (PR3). Our non-parametric approach using GP indicates that the updated analysis pipeline of CamSpec PR4 has caused the residuals in the EE coadded spectrum to vary more than the expected amount given by the covariance matrix; the errors seem to have been slightly underestimated. Indeed, we confirm that the usual chi-squared statistic for the EE data in the range $30\leq\ell\leq 650$ is $\chi^{2}=709$, larger than the expected value of $620$ by $2.51\sigma$.⁷⁷7For the full data range of $30\leq\ell\leq 2000$, $\chi^{2}=2023$, which is $0.83\sigma$ above the expected value of 1971. Note that here we look at the TT+TE+EE best-fit values. This excess in residual error has been investigated in [70, see e.g. Table 1] and is reported to be statistically consistent with random fluctuations. Moving one step further than these simple chi-squared checks, we investigate the scales where these excess residuals are correlated through GP analyses. The fact that the marginal likelihood is maximised at small $\ell_{f}$ implies that these excess variances appear to be uncorrelated random fluctuations rather than smooth deformations in the mean, unlike the cases of TT and TE. Nonetheless, the deviations from zero are $\mathcal{O}(10^{-2})$, with relatively minor improvements in fit, suggesting \lcdm is a good description of the \planck data.

We then use the set of $(\sigma_{f},\ell_{f})$ maximizing the likelihood in Eq. (2.3) (shown as a black dot in Fig. 3) to obtain the mean (in orange), $68\%$ and $95\%$ C.L. (gray-shaded bands) shown in Fig. 4, using Eqs. (2.1) and (2.2), respectively. Again, we see that the reconstructions are perfectly consistent with zero across the entire multipole range covered by the data.

These results suggest that despite the aforementioned issues in EE, the best-fit \lcdm model is overall consistent with the CamSpec data. This should not come as a surprise, since we have chosen the best-fit predictions to CamSpec data as the mean function in our analysis. However, as explained before, this serves as a consistency test for the different CMB measurements. In the presence of systematics, or physics beyond \lcdm, some inconsistencies might appear when using a (possibly incorrect) \lcdm mean function.

Similarly, we take the \lcdm’s best-fit predictions to CamSpec data and confront it to the official \planck likelihood (Plik) as a way of testing the internal consistency of the \planck measurements and the differences between the CamSpec and Plik likelihoods. In Fig. 5, we show the 2D LML profiles for TT, TE, and EE measurements from Plik (which we simply refer to as Planck). As discussed before, finding a significantly low $\Delta\chi^{2}$ by our GP means that a smooth deformation away from the mean function is preferred, which would hint towards inconsistencies between CamSpec and Plik in this case. While we find TE and EE to be consistent, we find notable inconsistencies in TT for the two likelihoods. There is a large improvement in fit from GP for $\sigma_{f}\simeq 5$ and $\ell_{f}\simeq 10^{3}$; corresponding to the blue regions in Fig. 5.

As seen in the upper panel of Fig. 6, the corresponding GP reconstruction also shows large deviations ($>3\sigma$) from zero. The corresponding improvements in fit, $\Delta\chi^{2}$, are reported in the first column of Table 3. See also Table 3 in [70] and discussions therein for a more detailed comparison between the official Planck release (PR3) and CamSpec (PR4) within the \lcdm model.

Interestingly, our GP analysis shows that these smooth deformations away from the CamSpec mean function not only fit the Plik likelihood significantly better but also have a similar correlation length $\ell_{f}$ to those that improve the fit to the ACT data (Figs. 7 and 8), albeit with a smaller amplitude ($\sigma_{f}$); see Section 3.2.1 below. These results imply that the best-fit predictions from CamSpec and Plik differ mainly in the TT power spectrum.

The fact that the GP captures differences in temperature between CamSpec and Plik is not so surprising. The temperature measurements from the official \planck data release are known to prefer deviations away from the standard model in some simple (one-parameter) extensions to \lcdm. Two well-known examples are the tendencies of the TT spectra to favour an excess of lensing amplitude $A_{L}>1$ – which should be equal to unity in \lcdm – and the preference for a closed Universe, with positive spatial curvature $\Omega_{k,0}<0$. As already noted in [27], these tendencies in TT to favour higher lensing amplitudes have similar effects in the $\mathcal{C}_{\ell}$’s as other parameters, such as curvature or the dark energy equation of state in extended models. Interestingly, these “anomalies” disappear when going from Plik to CamSpec (i.e. $A_{L}\to 1$ and $\Omega_{k,0}\to 0$) and are attributed to statistical fluctuations in the multipole range $800<\ell<1600$ [29]. Our GP reconstruction for the TT power spectrum also shows significant deviations from zero in the same multipole range; see the upper panel of Fig. 6. However, we note that our GP analysis prefers larger correlation scales ($\ell_{f}\simeq 10^{3}$) than what one would expect from random fluctuations, even considering the off-diagonal correlations from lensing.

Finally, we should stress that in this comparison, we use the best-fit power spectra to the CamSpec likelihood. An alternative analysis is to study deviations away from the TTTEEE best-fit to the official \planck data release, shown in green in Fig. 1. We refer the reader to Appendix A.1, where we use the best-fit $\mathcal{D}_{\ell}$’s to Plik instead, and test the consistency with both CamSpec measurements and other ground-based experiments.

In Fig. 7 we show the posteriors for $(\sigma_{f},\ell_{f})$ when using ACT DR4 data and \lcdm’s best-fit to CamSpec as mean function. In this case, an interesting feature appears in the TT data. The LML peaks at $(\sigma_{f},\ell_{f})\simeq(9,2\times 10^{3})$ where the GP finds an improvement in fit with respect to the mean function (\lcdm), corresponding to a $\Delta\chi^{2}=-15.11$; see Table 3. Interestingly, the TE and EE posteriors show similar (bimodal) distributions, with a preference for non-vanishing values of $(\sigma_{f},\ell_{f})$, although the statistical significance of these deviations from \lcdm is milder than in the TT case. The improvements in fit are reported in the third column of Table 3.

In Fig. 8, we show the mean and $2\sigma$ reconstructions from the GP when using the set of hyperparameters maximizing the LML in Eq. (2.3), shown as black dots in Fig. 7. Note that the ACT reconstructions of the TT spectra seem to prefer lower amplitudes at $\ell\lesssim 3000$ (at more than $2\sigma$) and a slightly larger amplitude at $\ell\gtrsim 3000$ with respect to what is predicted by \lcdm’s best-fit to the CamSpec data. This is yet another (non-parametric) indication that the ACT data seem to favor a scale-invariant, Harrison-Zel’dovich ($n_{s}\simeq 1$) spectrum of fluctuations [24, 37, 35, 36]. At the same time, a larger value for $n_{s}$ might also imply an increased value of $H_{0}$⁸⁸8However, we should note that such a shift in the cosmological parameters, $H_{0}\to 73\;\rm km/s/Mpc$ and $n_{s}\to 1$, would typically lead to larger values of $S_{8}$, worsening the fit to low-redshift (weak-lensing) measurements of the clustering amplitude., through a reduction of the size of the sound horizon [34, 38].

While the LML improvements at $\ell_{f}\sim 2000$ relate to the overall scaling through $n_{\mathrm{s}}$, the other mode in LML found at $\ell_{f}\sim 90$ in TE and EE spectra may be closely related to the cosmological parameters affecting the width and height of the acoustic peaks. Roughly speaking, a realisation of a GP with $\ell_{f}\sim 90$ has a typical full width at half maximum (FWHM) of $\sim 210$ and is likely to have oscillations that mimic the acoustic peaks in the CMB ($\Delta\ell\sim 300$). Indeed, the GP reconstruction of Fig. 8 for EE spectra has oscillation scales similar to those of the differences in the best-fit predictions from ACT+WMAP and CamSpec data (red line in Fig. 1). The features present in our non-parametric reconstructions can therefore be a manifestation of the mild discordance in the ($\omega_{b},\omega_{c}$)-plane between Planck and ACT (seen for instance in Fig. 7 in [6]), or vice versa. An interesting feature is also captured in the EE reconstructions around $\ell\sim 400-700$, with milder oscillations extending up to $\ell\sim 3000$. These might be a slight hint of new features, a manifestation of the mild disagreement in the estimated cosmological parameters, or some unaccounted systematics affecting the low/high-$\ell$ part of the ACT and/or \planck data. Together with the issue in TT, it might explain why recent analyses reach slightly different conclusions when considering ACT data alone, or performing cuts at a given $\ell_{\rm max}$ in the \planck data [e.g. 81, 51, 82]. Our results seem to support previous findings in the context of \lcdm and simple extensions [e.g. 24, 55, 35, 83], suggesting that these mild discrepancies are mainly driven by the ACT data and in particular by the TT measurements. Whether such discrepancies arise from physical, systematic, or statistical origin, however, remains to be determined by upcoming (more precise) CMB observations. In particular, the ACT collaboration is soon expected to update its results with the ACT DR6 data release.

Similarly, we inspect for structures in the residuals of SPT-3G data, when subtracting \lcdm’s best fit to CamSpecPR4 data. The results are shown in Fig. 9. The posteriors of the temperature auto-correlation (TT) and temperature/polarisation cross-correlation (TE) are in good agreement with the \lcdm predictions, and the GP finds negligible improvements with respect to \lcdm; see also Fig. 10. However, this could also be explained by the larger uncertainties in the temperature measurements with respect to \planck (see Fig. 1). The situation is slightly different for EE, suggesting again a bimodal distribution in the $(\sigma_{f},\ell_{f})$-plane, with typical deviations from a zero mean-function of the order $\sigma_{f}\lesssim 1$ and with a preferred correlation length of $\ell_{f}\simeq 30$, indicating that the improvement in fit is likely due to subtle oscillations around the \lcdm best-fit predictions. The improvements in fit with respect to \lcdm are again reported in the last column of Table 3, with a maximum $\Delta\chi^{2}=-8.196$ for EE, which is of the same order of magnitude as the improvement in fit as for the case of ACT data. Curiously, the SPT reconstructions also show a prominent feature at intermediate scales ($\ell\sim 500-700$). As discussed before, these oscillations might be linked to the mild differences in the cosmological parameters (such as $\omega_{b}$ or $\omega_{c}$) affecting the width, height and position of the acoustic peaks with respect to the ones inferred by \planck. We should mention that the SPT-3G results are overall consistent with those of \planck at the parameter level; see Table IV and Fig. 7 in [6]. However, our results suggest that the very mild differences between the two are mostly driven by the EE measurements; in agreement with the conclusions from Fig. 3 in [6].