CMB lensing in a modified $\mathrm{\Lambda }\mathrm{CDM}$ model in light of the $H_0$ tension

The observed discrepancy of the Hubble parameter measurements in the local universe with the cosmic microwave background (CMB) data may indicate a new physics. It is vital to test the alternative models that reconcile the Hubble tension with other cosmological observations in this direction. The CMB lensing is a crucial observation that relates the early universe perturbations to the matter’s late-time distribution. In this work, we study the prediction of the $ü\mathrm{\Lambda }\mathrm{CDM}$ as a solution for $H_0$ tension for CMB lensing and the low- and high-$\ell$’s temperature (TT) power spectrum internal inconsistency. We show that this model relaxes the low- and high-$\ell$’s TT mild inconsistency and the CMB lensing tensions simultaneously. Accordingly, $ü\mathrm{\Lambda }\mathrm{CDM}$ having the same number of free parameters as $\mathrm{\Lambda }\mathrm{CDM}$ with lensing amplitude $A_L$ added, has a better fit with $\mathrm{\Delta }{\chi }^2=-3.3$.

Figure 1

The contour plot üof $ü\mathrm{\Lambda }\mathrm{CDM}$ parameters is represented for high-$\ell$’s with green dash-dotted lines and low-$\ell$’s with red dashed lines. The blue contour plots show the $1\sigma$ and $2\sigma$ constraints with full Planck TT data.

Figure 2

The angular power spectrum of the CMB lensing is plotted for $ü\mathrm{\Lambda }\mathrm{CDM}$, $\mathrm{\Lambda }\mathrm{CDM}$ with and without free $A_L$. It is obvious that $ü\mathrm{\Lambda }\mathrm{CDM}$ mimics $\mathrm{\Lambda }\mathrm{CDM}$ with $A_L=1$ without any inconsistency with the data points. Note that the temperature power spectrum prefers $\mathrm{\Lambda }\mathrm{CDM}$ with $A_L=1.2$ which is not consistent with CMB lensing power spectrum.

Figure 3

The relative likelihood of the $A_L$ parameter is plotted for the standard $\mathrm{\Lambda }\mathrm{CDM}$ and $ü\mathrm{\Lambda }\mathrm{CDM}$ models in left and right, respectively. Both models have $\sim 2–3\sigma$ tension when we checked them against only the Planck dataset (green curves). When we add the lensing data (blue curves), both models are consistent with $A_L=1$ in their $1\sigma$ regions. Adding R19 (red curves) restores the $2-3\sigma$ tension for the $\mathrm{\Lambda }\mathrm{CDM}$. CMB lensing and R19 datasets are not compatible in the $\mathrm{\Lambda }\mathrm{CDM}$. However $ü\mathrm{\Lambda }\mathrm{CDM}$ shows no inconsistency between lensing and R19 datasets.

Figure 4

We have plotted two-dimensional likelihood for $A_L$ vs $H_0$ for $\mathrm{\Lambda }\mathrm{CDM}$ and $ü\mathrm{\Lambda }\mathrm{CDM}$ in the left and right, respectively. All the datasets’ combinations of the $\mathrm{\Lambda }\mathrm{CDM}$ model show a positive correlation between $A_L$ and $H_0$. That is why it is difficult for $\mathrm{\Lambda }\mathrm{CDM}$ to be consistent for both $A_L=1$ and R19 simultaneously. However, this correlation seems broken for the case of $ü\mathrm{\Lambda }\mathrm{CDM}$.

Figure 5

The standard model’s two-dimensional contour plots investigating the lensing amplitude versus matter density parameters of total matter, baryons, cold dark matter, and ${\sigma }_8$ are plotted.

Figure 6

Two-dimensional contour plots of $ü\mathrm{\Lambda }\mathrm{CDM}$, investigating the lensing amplitude versus matter density parameters of total matter, baryons, cold dark matter, and ${\sigma }_8$, are plotted.

Figure 7

The ISW-galaxy angular cross power spectrum is plotted for $\mathrm{\Lambda }\mathrm{CDM}$ (black solid line) and $ü\mathrm{\Lambda }\mathrm{CDM}$ (blue dashed line). The observational data are from the NVSS galaxy sample (hexagon points) and WISE (circle dots).