Consistency tests of $\mathrm{\Lambda }\mathrm{CDM}$ from the early integrated Sachs-Wolfe effect: Implications for early-time new physics and the Hubble tension

New physics increasing the expansion rate just prior to recombination is among the least unlikely solutions to the Hubble tension and would be expected to leave an important signature in the early integrated Sachs-Wolfe (eISW) effect, a source of cosmic microwave background (CMB) anisotropies arising from the time variation of gravitational potentials when the Universe was not completely matter dominated. Why, then, is there no clear evidence for new physics from the CMB alone, and why does the $\mathrm{\Lambda }$ cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$) model fit CMB data so well? These questions and the vastness of the Hubble tension theory model space provide the motivation for general consistency tests of $\mathrm{\Lambda }\mathrm{CDM}$. I perform an eISW-based consistency test of $\mathrm{\Lambda }\mathrm{CDM}$ introducing the parameter $A_{\mathrm{eISW}}$, which rescales the eISW contribution to the CMB power spectra. A fit to Planck CMB data yields $A_{\mathrm{eISW}}=0.988\pm 0.027$, in perfect agreement with the $\mathrm{\Lambda }\mathrm{CDM}$ expectation $A_{\mathrm{eISW}}=1$ and posing an important challenge for early-time new physics, which I illustrate in a case study focused on early dark energy (EDE). I explicitly show that the increase in ${\omega }_c$ needed for EDE to preserve the fit to the CMB, which has been argued to worsen the fit to weak lensing and galaxy clustering measurements, is specifically required to lower the amplitude of the eISW effect, which would otherwise exceed $\mathrm{\Lambda }\mathrm{CDM}$’s prediction by $\approx 20\%$: this is a generic problem beyond EDE that likely applies to most models enhancing the expansion rate around recombination. Early-time new physics models invoked to address the Hubble tension are therefore faced with the significant challenge of making a similar prediction to $\mathrm{\Lambda }\mathrm{CDM}$ for the eISW effect while not degrading the fit to other measurements in doing so.

Figure 1

Impact of varying $A_{\mathrm{eISW}}$ on the CMB temperature anisotropy power spectrum. Upper panel: CMB temperature anisotropy power spectrum for different values of $A_{\mathrm{eISW}}$, specified by the color coding, with the black curve corresponding to the standard case $A_{\mathrm{eISW}}=1$. Notice that, as standard in the field, plotted on the $y$ axis is $T_{\mathrm{CMB}}^2{D}_{\ell }^{TT}\equiv T_{\mathrm{CMB}}^2\ell (\ell +1)C_{\ell }^{TT}$, with $T_{\mathrm{CMB}}\approx 2.725\mathrm{K}$ the CMB temperature today. Bottom panel: relative differences between the power spectra shown in the upper panel, relative to the baseline $A_{\mathrm{eISW}}=1$ power spectrum, with the color coding used in the upper panel. It is clear that varying $A_{\mathrm{eISW}}$ mostly affects scales around the first acoustic peak, with $A_{\mathrm{eISW}}>0$ ($A_{\mathrm{eISW}}<0$) enhancing (suppressing) power.

Figure 2

Triangular plot showing 2D joint and 1D marginalized posterior probability distributions for the eISW amplitude $A_{\mathrm{eISW}}$, the physical baryon density ${\omega }_b$, and the scalar spectral index $n_s$, obtained from a fit to the Planck dataset. The blue contours are obtained within the seven-parameter $\mathrm{\Lambda }\mathrm{CDM}+A_{\mathrm{eISW}}$ model, while the red contours are obtained within the six-parameter $\mathrm{\Lambda }\mathrm{CDM}$ model, where $A_{\mathrm{eISW}}$ is fixed to $A_{\mathrm{eISW}}=1$, as indicated by the vertical red line.

Figure 3

One-dimensional marginalized normalized posterior distribution for $A_{\mathrm{eISW}}$, obtained within the baseline seven-parameter $\mathrm{\Lambda }\mathrm{CDM}+A_{\mathrm{eISW}}$ model, from four different datasets/dataset combinations: the baseline Planck data using the high-$\ell$ Plik likelihood (blue curve); a combination of Planck and a BBN prior on ${\omega }_b$ (black curve); a combination of Planck with late-time BAO and Pantheon Hubble flow SNeIa measurements (red curve); and the baseline Planck data, where the high-$\ell$ Plik likelihood is replaced by the CamSpec one (green curve). It is clear that the inferred value of $A_{\mathrm{eISW}}$ is very stable overall against these choices of datasets/dataset combinations.

Figure 4

One-dimensional marginalized normalized posterior distribution for $A_{\mathrm{eISW}}$, obtained from the Planck data, within the baseline seven-parameter $\mathrm{\Lambda }\mathrm{CDM}+A_{\mathrm{eISW}}$ model (blue curve), as well as extensions where the following parameters are allowed to vary as well: the effective number of relativistic species $N_{\mathrm{eff}}$ (red curve), the helium fraction $Y_P$ (green curve), the lensing amplitude $A_{\mathrm{lens}}$ (black curve), the running of the scalar spectral index ${\alpha }_s$ (magenta curve), and both the running ${\alpha }_s$ and the running of the scalar spectral index ${\beta }_s$ (cyan curve). It is clear that the inferred value of $A_{\mathrm{eISW}}$ is very stable overall against these parameter space extensions.

Figure 5

CMB temperature anisotropy power spectra for $\mathrm{\Lambda }\mathrm{CDM}$ and early dark energy (EDE). Top left panel: CMB temperature anisotropy power spectrum for a $\mathrm{\Lambda }\mathrm{CDM}$ model (black curve) with parameters given by the left column of Table 4, and for an EDE model (red curve) with parameters given by the middle column of Table 4, where, in particular, the physical cold DM density is fixed to the low value ${\omega }_c=0.1177$. Bottom left panel: relative differences between the EDE and $\mathrm{\Lambda }\mathrm{CDM}$ power spectra shown in the upper panel, with the former clearly predicting an excess of power compared to the latter, particularly around the scale of the first peak. Right panel: as in the left panel, but with the EDE parameters given by the right column of Table 4, where, in particular, the physical cold DM density is fixed to the high value ${\omega }_c=0.1320$. As the lower panel shows, this increase in ${\omega }_c$ makes the two power spectra nearly indistinguishable.

Figure 6

Like Fig. 5 but focusing on the early ISW (eISW) contribution to the CMB temperature anisotropy power spectrum. It is clear that the increase in ${\omega }_c$ significantly decreases the excess eISW contribution on most scales of interest. Note that the $x$ axis covers multipoles only in the range $50<\ell <300$, where the eISW effect is most prominent, as shown in Fig. 1.