Challenges of the Standard Cosmological Model

Abstract

Measurements of the temperature and polarization anisotropy of the cosmic microwave background (CMB) provided strong confirmation of the vanilla flat $\Lambda$CDM model of structure formation. Even if this model fits incredibly well, the cosmological and astrophysical observations in a wide range of scales and epochs, some interesting tensions between the cosmological probes, and anomalies in the CMB data, have emerged. These discrepancies have different statistical significance, and although some parts may be due to systematic errors, their persistence strongly indicates possible cracks in the standard $\Lambda$CDM cosmological scenario.

1. Introduction

Among a number of different cosmological models introduced in the literature, the lambda cold dark matter ($\Lambda$CDM) scenario is the mathematically simplest model, and has now been practically selected as the “standard” cosmological scenario, because it provides a terrific description of a wide range of astrophysical and cosmological probes in a wide range of scales and epochs. However, despite its fantastic fit to the available observations, $\Lambda$CDM contains vast areas of phenomenology and ignorance. For example, it is not yet able to explain key concepts in our understanding of the structure and evolution of the Universe, currently based on three completely unknown quantities that are not expected by theoretical first principles and are not supported by laboratory experiments, although a big effort has been devoted to this research in the recent decades. In actuality, the physical evidence for these unknown pillars comes only from cosmological and astrophysical observations, and the $\Lambda$CDM model assumes the simplest assumptions for all of them. We can list these three quantities and their specific solutions for the standard model as follows. First, the inflation [1,2,3], introduced to solve crucial puzzles of the early evolution of the Universe, in the standard model is given by a single, minimally coupled, slow-rolling scalar field. Second, the dark matter (DM) [4,5] is expected to interact only gravitationally to facilitate the formation of the structures, and in the standard model it is assumed to be pressure-less and cold, but there is no direct evidence of DM particles. Third, the dark energy (DE) [6,7] component, introduced to explain the current stage of accelerated expansion, in the standard model is the cosmological constant term $\Lambda$, without any strong physical basis.

The standard $\Lambda$CDM cosmological model, while hugely successful in explaining the observations, is therefore mainly favoured by its simplicity. Actually, with just six free parameters, it manages to fit well all the current cosmological observations [8,9,10,11,12,13,14,15,16,17]. This is a spectacular success, but it must not be forgotten that the model has many weaknesses. For example, it remains a theoretical challenge to understand the magnitude of the cosmological constant $\Lambda$ from first principles. Deviations from a power law in the primordial spectrum of perturbations are expected in practically all the inflationary scenarios, but are set to zero in the $\Lambda$CDM model. The DM can be made of different particles, it could interact, and it can happen to have a non-negligible momentum in certain epochs. The $\Lambda$CDM scenario is, therefore, in some ways similar to a phenomenological effective theory, i.e., an approximation to the underlying real model theoretically motivated which has yet to be discovered, and with the improvement in the number and accuracy of the observations, the deviations from $\Lambda$CDM are not unexpected. Thus, discrepancies between key cosmological parameters of the models that have emerged with different statistical significance in recent years could reveal a more complete description of our Universe, bringing new understanding to several areas of physics. Although some of these discrepancies may have a systematic origin, their persistence across probes should require multiple and unrelated errors, strongly suggesting the need for new physics beyond the canonical $\Lambda$CDM model.

Current cosmological tensions and anomalies are the subject of the Cosmology Intertwined review paper I submitted for the SNOWMASS 1 call [18], which includes contributions from over 200 people who have participated in brainstorming sessions and provided feedback through regular Zoom seminars and meetings. This White Paper is based on the four letters of interest Ipresented in August 2020 [19,20,21,22], which deal with the $H_0$ disagreement between the cosmic microwave background (CMB) and the local distance ladder measurements [22], the growth tension quantified by the parameter $S_8$ with the cosmic shear and cluster counts data [19], and a non-zero curvature of the Universe ${\Omega }_k$ [20] related to the $A_{\mathrm{lens}}$ problem, which is an internal anomaly of the Planck data. However, one can also find a section where there is room for discussion of many less-well-known less-standard existing signals in cosmological and astrophysical data that seem to be in tension ($2\sigma$ or more) with the standard $\Lambda$CDM model as defined by the Planck 2018 parameter values [12]. In many cases, the signals are controversial and a debate is currently underway in the literature on the possible systematic origin of some of these signals. The reader is invited to take a look at the paper [18] if interested in knowing more.

In this brief short overview paper, I will mainly focus instead on the Hubble constant discussion, and I will conclude with some hints at other curious tensions and anomalies of the current CMB data.

2. The ${\mathit{H}}_{\mathbf{0}}$ Tension

The Hubble constant $H_0$ describes the expansion rate of the Universe today. This can be achieved mainly in two ways: (i) by measuring the luminosity distance and the recessional velocity (or redshift) due to the cosmic expansion of known standard candles, and by computing the proportionality factor (Hubble–Lemaître law), this approach is model-independent and it is based on geometrical measurements; (ii) by considering early Universe probes and, assuming a model for the expansion history of the Universe, estimating the expansion rate today. For example, we have CMB measurements and assume the standard model of cosmology, i.e., the $\Lambda$CDM scenario, where the expansion history of the Universe is obtained with the Friedmann’s equations.

The $H_0$ tension is the name given to the $5\sigma$ disagreement between the CMB Planck 2018 estimate of the Hubble constant obtained by assuming a standard $\Lambda$CDM cosmological model, i.e., $H_0=67.27\pm 0.60$ km/s/Mpc at 68% CL [12], and the local value measured by the SH0ES (Supernova, $H_0$, for the Equation of State of Dark Energy) collaboration [23], i.e., $H_0=73.04\pm 1.04$ km/s/Mpc at 68% CL, calibrating the Type Ia Supernovae (SNeIa) with the Cepheids.

The latest local measurement obtained by the SH0ES collaboration is based on the distance ladder approach shown in Figure 12 of Ref. [23]. In this plot, one can see that $H_0$ is measured with a simultaneous fit, performed by optimizing a ${\chi }^2$ statistic to determine the most probable values of the parameters of a three-rung distance ladder: (i) geometric distance measurements of the anchors used to standardize Cepheid variables (lower left panel), (ii) standardized Cepheids used to calibrate SNeIa in nearby galaxies (middle panel), and (iii) calibrated SNeIa used to obtain the Hubble flow (top right panel).

On the other side, we have, instead, a CMB measurement, i.e., the black body radiation emitted at recombination characterized by extremely small anisotropies (of the order of ${10}^{-5}$). We can expand in Legendre polynomials the two-point correlation function between the difference of temperature in two directions in the sky, averaged over the total sky, and thus obtain the temperature angular power spectrum. At the same time, we can choose a set of parameters describing our theoretical model to compute our theoretical angular power spectrum, usually with the publicly available Boltzmann solver codes CAMB [24] or CLASS [25]. Due to the correlations present between the parameters, the variation of different quantities can produce similar effects on the CMB. We can finally compare the angular power spectrum obtained from the data with the one computed by varying the parameters of the model and, using a Bayesian analysis, we obtain the best combination of cosmological parameter values that fits the data. Among the various statistics that can be used to achieve this, the simplest form of the Markov chain Monte Carlo method is widely used, and some of the codes most commonly used by the scientific community are CosmoMC [26], MontePyhton [27], and CosmoSIS [28]. In this way, we can obtain the constraints of the parameters. We can extract four independent angular spectra from a CMB experiment: the autocorrelation of the temperature, type E polarization (density fluctuations), and type B polarization (gravitational waves), and the cross-correlation between the temperature and the type E polarization. If we compare the constraints on the parameters obtained separately for a $\Lambda$CDM scenario from Planck EE, TE, and TT high-ℓ power spectra combined with low-ℓ polarization and temperature data, we see a perfect agreement [12], which shows internal consistency between the different independent probes. We can therefore conclude that the 2018 Planck results are a wonderful confirmation of the flat standard $\Lambda$CDM cosmological model. However, these results have two important caveats: (1) the cosmological constraints are obtained by assuming a cosmological model, and if we change the assumptions, completely different results can be obtained; (2) these results are affected by the degeneracy between the parameters that induce similar effects on the observables, so that they can be biased towards a correlation direction.

2.1. The $H_0$ Measurements

Supporting a lower value for the Hubble constant $H_0$ are the other CMB experiments that assume a $\Lambda$CDM cosmological model, such as the ground-based CMB telescope South Pole Telescope (SPT-3G) [29], that finds $H_0=68.8\pm 1.5$ km/s/Mpc at 68% CL, or the Atacama Cosmology Telescope (ACT-DR4) + Wilkinson Microwave Anisotropy Probe (WMAP) [13], which gives $H_0=67.6\pm 1.1$ km/s/Mpc at 68% CL. However, a similar value is also obtained from the baryon acoustic oscillations (BAO) measurements from the BOSS and eBOSS estimates [15] plus the Big Bang nucleosynthesis (BBN) data, which assume a $\Lambda$CDM scenario and are CMB-independent. In general, we find that all of the early Universe measurements, assuming a vanilla $\Lambda$CDM cosmological scenario for the expansion history, are completely in agreement with a lower value for the Hubble constant today such as that estimated by Planck. An extensive list of all the Hubble constant measurements in the early and late Universe made by different astronomical missions and groups over the years are summarized in Figure 2 of Ref. [18], where the orange vertical band corresponds to the $H_0$ value measured by the SH0ES team and the light pink vertical band corresponds to the $H_0$ value as reported by the Planck 2018 team within a $\Lambda$CDM scenario 2. This figure is really busy, with many measurements with error bars so large that do not help discriminate the true value for the Hubble constant, so we filtered it to obtain Figure 1, where we only keep the high-precision measurements of $H_0$, corresponding to the early Universe estimates with error bars less than $1.5$ km/s/Mpc, and the measurements in the late Universe with error bars less than $3.0$ km/s/Mpc. In this way we can fully appreciate the high accuracy and consistency of the data at both sides of this plot, and the need for an exact solution (new physics or unrelated errors) that can explain all these multiple observations.

If we now focus on the late Universe measurements since 2020 of Figure 2 of Ref. [18], we see that they are instead preferring higher values of $H_0$. We have, in orange, the SNeIa calibrated with Cepheids [23,30,31,32], as performed by the SH0ES collaboration. Then, we find, in dark green, the SNeIa calibrated with the tip of the red-giant branch (TRGB) [33,34,35,36,37,38] where, focusing on high-precision measurements, there is the well-discussed Freedman 2021 [33] measurement which is exactly in between the Planck and SH0ES values of $H_0$, i.e., in agreement with both sides within $1.5\sigma$, and the new independent reanalysis of the targets presented by the Carnegie-Chicago Hubble Program (CCHP) performed in [36] that is shifted more in agreement with the SH0ES value. Then, we see in fuchsia the surface brightness fluctuation (SBF) measurements [39,40], where Ref. [40] uses the SBF to calibrate the SNeIa, while Ref. [39] uses the SBF as a replacement distance ladder for long-range indicator, calibrated by both Cepheids and TRGB that separately give the same result. Then, we have, in brown, the Type II Supernovae [41,42] used as standardisable candles and calibrated by both Cepheids and TRGB, in purple, the Megamaser Cosmology Project [43], which measures $H_0$ avoiding any distance ladder and providing the geometric distance directly in the Hubble flow, and in red, the measurements based on the Tully–Fisher relation [44,45] calibrated with both Cepheids and TRGB. Finally, we find, in blue, the strong lensing measurements of the time delays of multiple images of quasar systems caused by the strong gravitational lensing from a foreground galaxy [46,47,48,49,50]. While those based on the H0LiCOW data were more in agreement with SH0ES, the latest TDCOSMO results relax the assumptions on the lens mass density profile, obtaining much weaker bounds, and once combined with SLACS, they shift $H_0$ towards a lower value. These types of measurements have relaxed bounds on the Hubble constant, which is in agreement with both SH0ES and Planck within $1.5\sigma$, and are highly dependent on the astrophysical model.

In the absence of a full covariance matrix between the late Universe measurements of the Hubble constant discussed before, we can quantify the robustness of the disagreement between the measurements in the early and late Universe using the same conservative approach adopted in Ref. [51], i.e., by computing an arithmetic average of all the late Universe measurements in a colour group, and associating with this average the smaller error bar in the group. After that, we can compute a weighted average of all the results of the different colour groups. If we combine all the values in the references above using this method, we have an optimistic value, which is $H_0=72.97\pm 0.63$ km/s/Mpc at 68% CL, i.e., at about $6.55\sigma$ tension with Planck. If we now exclude one colour group of data and take the average with the largest error bars, which is obtained when we exclude the most precise Hubble constant measurement we have, which is based on SNeIa calibrated with Cepheids, we find a conservative estimate that is equal to $H_0=72.73\pm 0.80$ km/s/Mpc at 68% CL, i.e., at about $5.5\sigma$ tension with Planck. Finally, if we repeat the process ruling out two colour groups of measurements and choosing the average with the largest error bar, we obtain $H_0=73.3\pm 1.1$ km/s/Mpc at 68% CL in $4.8\sigma$ tension with Planck. We call this an ultraconservative estimate obtained by excluding measurements based on SNeIa calibrated with Cepheids and the time-delay lensing data. Therefore, we can conclude that the disagreement is robust and statistically significant, even by removing one or two groups of measurements, and ranges between ∼4–6$\sigma$ (see also [52,53]). In this regard, it was extensively discussed Figure A1 in Ref. [33], where a convergence of the Hubble constant measurements in the last ∼40 years is shown. However, what is missing in that plot is an indication of what kind of measurement was obtained with which experiment, because in the past the tension was within the same types of measurements and at the same redshifts and thus indicating directly the presence of systematic errors, while now the puzzling aspect is that there are no late Universe measurements below the early Universe estimates and vice versa. Given the robustness of the tension removing some groups of data and the distribution of the values, it is difficult to find a single type of systematic error that can resolve the Hubble constant tension, and at the same time, it is less probable to have multiple, unrelated systematic errors that bias the Hubble constant values in the same direction, instead of scattering them around the true value.

2.2. Looking for a Solution beyond Systematic Errors

On the other hand, since the early Universe constraints are model-dependent, we can try to modify the phenomenological $\Lambda$CDM standard cosmological model used to analyse the data and see if a new model can work to solve the Hubble constant tension. Many alternative models have been proposed as a solution to the $H_0$ tension, and many of them are listed in the following review papers: Refs. [18,22,52,54,55,56,57,58,59]. Usually, the solutions are classified in early solutions, corresponding to modifications of the standard cosmological model before recombination, such as the variation of the effective number of relativistic degrees of freedom $N_{\mathrm{eff}}$ [60] or the introduction of an early dark energy (EDE) [61], or late time solutions, corresponding to modifications of the standard cosmological model after recombination, such as exotic dark energy models with an equation of state free to vary [62] or interacting dark energy scenarios [63]. I will comment on the actual validity of the early vs. late time solutions to solve the Hubble tension in the next subsections, but it can be anticipated that there is currently no consensus on the validity of the models proposed as a solution for the Hubble constant problem.

2.2.1. The $H_0-r_d$ Plane

If we assume a different model for the expansion history of the Universe and use this model to fit the Planck power spectra, then we can classify the formally successful models in solving the $H_0$ tension only by comparing the estimated $H_0$ parameter in the new model with the SH0ES result, as was performed in Table B1 of Ref. [58]. In this case, a model comparison is not performed and the goodness of the fit is not taken into account, but the tension between the $H_0$ value estimated by Planck assuming a non-standard cosmological scenario ($H_{0,new}\pm {\sigma }_{H_{0,new}}$) and SH0ES ($H_{0,SH0ES}\pm {\sigma }_{H_{0,SH0ES}}$) is computed using the formula $T=(H_{0,SH0ES}-H_{0,new})/\sqrt{{\left({\sigma }_{H_{0,SH0ES}}\right)}^2+{\left({\sigma }_{H_{0,new}}\right)}^2}$. We see that most of the successful models with tension T less than $1\sigma$ are late time solutions. However, for these models, the inclusion of BAO and Pantheon SNeIa measurements significantly worsens the fit of the data because these scenarios fail to recover the shape of $H\left(z\right)$ at low redshifts. In actuality, a combination of BAO and Pantheon data constrains the product of $H_0$ and the sound horizon $r_s$, as one can see in Figure 1 of Ref. [54]. This implies that, to have a higher $H_0$ value from BAO + Pantheon in agreement with SH0ES, we need $r_s\sim 137$ Mpc, while to agree with Planck, assuming $\Lambda$CDM, we need $r_s\sim 147$ Mpc. For this reason, the solutions that can increase $H_0$ and at the same time decrease $r_s$ are most promising. In Figure 3 of Ref. [64], one can see that late time solutions, such as wCDM, increase $H_0$ because they reduce the expansion history at intermediate redshift, but do not affect the sound horizon $r_s$. However, early time solutions, such as $N_{\mathrm{eff}}$ or EDE, move in the right direction of correlation for both parameters, but cannot fully solve the Hubble constant tension between Planck and SH0ES without including a prior on $H_0$. In this case, in fact, the best fit obtained by these models is always the $\Lambda$CDM one and the reduction of the tension is mainly due to a volume effect due to the increase of the error bars of the parameters. If we now assume a different model for the expansion history of the Universe and use this model to fit a combination of different data, not just Planck, we can now classify the model that formally succeeded in solving the $H_0$ tension, as was performed in Table B2 of Ref. [58], without considering a model comparison but only computing the tension T. In this case, many of the successful models are early time solutions.

At this point it is worth noting that we should be careful to rule out all the late time solutions due to their disagreement with the BAO data. In actuality, BAO is formed in the early Universe, when baryons are strongly coupled to photons, and the gravitational collapse due to the cold dark matter is counterbalanced by the radiation pressure, but these fluctuations have evolved and we can observe BAO at low redshifts in the distribution of galaxies. Since the data reduction process leading to these measurements requires assumptions about the fiducial cosmology, BAO is model-dependent (see Ref. [65] for a discussion about the BAO dependence on $\Lambda$CDM assumptions). In other words, the tension between Planck + BAO, assuming an exotic dark energy model, and SH0ES could be due to a statistical fluctuation in this case. The BAO reliability in extended models was evaluated for early time modifications of the expansion history and DE late time modifications that can be parametrized by $w_0{w}_a$ (see Refs. [15,66,67]). Indeed, the whole procedure leading to the BAO dataset performed by the different collaborations may not necessarily be valid in extended DE models, where perturbations in the mildly nonlinear regime are important. For this reason, the BAO data may need to be non-trivially revised on a case-by-case basis when applied to constrain exotic DE cosmologies 3.

2.2.2. The $M_B$ Tension

Furthermore, it was pointed out in Ref. [73] that we should be careful in quantifying the goodness of a model in solving the Hubble tension just by looking at the $H_0$ estimate that such a model can provide. In actuality, SH0ES measures the SNeIa absolute magnitude $M_B$, and to obtain the $H_0$ value it uses the Hubble flow of the SNeIa in the redshift range $0.023\le z\le 0.15$ and a cosmography with $q_0=-0.55$ and $j_0=1$, which is information beyond the local Universe. This implies that a model with a fast transition at very low redshift ($z<0.1$), such as a hockey-stick dark energy scenario, can completely solve the Hubble tension, but may disagree with the measured $M_B$ at more than $5\sigma$ (see Ref. [32]). Therefore, instead of explaining the $H_0$ tension, we should deal with the $M_B$ tension. The advantage of using an $M_B$ prior is that the all the information about the shape of the SNeIa magnitude–redshift relation is not lost, and that it is possible to combine, at the same time, the entire SNeIa sample Pantheon without the risk of double-counting the SNeIa used in the Hubble flow. We can now classify successful models in solving the $M_B$ tension, and this was carried out in Table 1 of Ref. [74], where the best solution appears to be a closed Universe with a varying electron mass [75]. Finally, it has been argued that the $M_B$ tension is another reason to rule out modifications of the expansion history at late times: while they can provide a higher $H_0$ value, these could be strongly disfavoured by the Pantheon magnitude–redshift relation (see Ref. [76]). However, this is true for all the transitions at very low redshift ($z<0.1$), such as a hockey-stick dark energy scenario, but should be tested on a case-by-case basis for smooth scenarios, where using a $H_0$ prior or an $M_B$ prior gives the same results, i.e., for those models where the Hubble–Lemaître law is still valid.

2.2.3. The $H_0-r_d-{\Omega }_m$ Plane

Finally, we have a further complication: the early time solutions proposed to alleviate the $H_0$ tension increase the $S_8$ tension (see the next section). In actuality, a model that simply changes the value of the sound horizon $r_d$ would not completely resolve the tension, as it will affect the inferred value of the matter density ${\Omega }_m$ and transfer the tension to it. Figure 1 of Ref. [56] shows that achieving full agreement between CMB, BAO, and SH0ES through a reduction of $r_d$ requires a higher value of ${\Omega }_m{h}^2$ as measured by the CMB, significantly exacerbating the tension with the weak lensing experiments, as shown in Figure 2 of Ref. [56]. Therefore, the tension between the different probes should be considered not only in the plane $H_0-r_d$, but it should be extended to the triplet of parameters $H_0-r_d-{\Omega }_m$. A plot showing the density of the cosmological models proposed to alleviate the Hubble tension can be found in Figure B1 of Ref. [58]. A model in agreement with all available datasets should have a data point that crosses the $H_0$ value measured by SH0ES (light blue band), the matter density ${\Omega }_m{h}^2$ measured by the CMB (light yellow band), and the sound horizon $r_{d}h$ measured by BAO (light green band), and at the moment, we can safely conclude that there are no specific proposals that prove to be highly probable or far better than all the others.

3. Additional Anomalies and Tensions

3.1. The $S_8$ Tension

A tension on the $S_8$ parameter, defined as a combination of the matter density ${\Omega }_m$ and the clustering amplitude of the matter power spectrum ${\sigma }_8$, i.e., $S_8\equiv {\sigma }_8\sqrt{(}{\Omega }_m/0.3)$, is present between the Planck data and the cosmic shear data, assuming the $\Lambda$CDM scenario. This tension is becoming statistically significant and is now at the level of $3.1\sigma$ with KiDS-100 [77], that gives $S_8={0.766}_{-0.014}^{+0.020}$, and $2.5\sigma$ with DES-Y3 [78], that gives $S_8={0.759}_{-0.025}^{+0.025}$, while Planck prefers a higher value for $S_8={0.834}_{-0.016}^{+0.016}$. Preferring a lower $S_8$ value, we have all of the late Universe measurements, in particular all the data of weak lensing, galaxy cluster, cluster counts, and redshift space distortions. A plot showing the distribution of the $S_8$ estimates, with a list of all the references, can be found in Figure 4 of Ref. [18]. Again, similar to the $H_0$ tension, the puzzling aspect is that there are no late Universe measurements above the CMB estimates and vice versa. A summary of the possible candidates proposed to solve the $S_8$ tension is present in the review papers [18,19]. However, a systematic error in the Planck data could explain the $S_8$ tension, as clarified in the next subsection.

3.2. The $A_{\mathrm{lens}}$ Anomaly

The path of the CMB photons emitted at recombination is weakly deflected by the gravitational lensing effect along the line of sight due to the presence of massive structures. We can define the lensing amplitude $A_{\mathrm{lens}}$ [79] as the rescaling of the lensing potential in the CMB power spectra. If we leave this parameter free to vary, its effect on the power spectrum will be the smoothing of the acoustic peaks when increasing $A_{\mathrm{lens}}$. An interesting consistency check, therefore, consists of verifying whether the amplitude of the smoothing effect in the CMB power spectra corresponds to the theoretical expectation $A_{\mathrm{lens}}=1$, and if the amplitude of the smoothing is consistent with that measured by the lensing reconstruction, i.e., the four-point correlation function. In other words, if $A_{\mathrm{lens}}=1$, then the theory is correct; otherwise, we have new physics or systematic errors affecting the results.

Although the Planck lensing reconstruction power spectrum is consistent with the expected amplitude for the $\Lambda$CDM model that fits the CMB temperature and polarization power spectra, that is, the Planck lensing measurement is compatible with $A_{\mathrm{lens}}=1$ [80], the distribution of $A_{\mathrm{lens}}$ inferred from the CMB power spectra indicates a preference for $A_{\mathrm{lens}}>1$ [12]. The joint TTTEEE likelihood, including the polarization data, shifts the preferred value by TT data down towards $A_{\mathrm{lens}}=1$, but the error also is reduced, increasing the significance of $A_{\mathrm{lens}}>1$ to $2.8\sigma$ 4. The preference for a high $A_{\mathrm{lens}}\ne 1$ is not just a volume effect in the full parameter space, because the best fit improves by $\Delta {\chi }^2\sim 9$ for the temperature power spectrum, when adding a single degree of freedom $A_{\mathrm{lens}}$. In other words, $A_{\mathrm{lens}}>1$ is a failed consistency check of Planck 2018 data.

Furthermore, it was shown in Ref. [82], and confirmed in Ref. [12], that the constraints on some cosmological parameters obtained by assuming $\Lambda$CDM from two separate fits of the low-ℓ and the high-ℓ multipoles of the Planck temperature power spectrum data are in tension at more than $2\sigma$ level, and that fixing the parameter $A_{\mathrm{lens}}=1.2$, similar to its best fit obtained by Planck, could be a way to resolve this internal disagreement of the two multipole regions, directly indicating a systematic error present in the data.

In addition, if we leave this parameter free to vary, there is a strong correlation between the $S_8$ parameter and $A_{\mathrm{lens}}$ [83], so we can lower $S_8$ to agree with local measurements once we marginalize over $A_{\mathrm{lens}}$ (see also Figure 2 of Ref. [84]). This anomaly $A_{\mathrm{lens}}>1$ is very robust, showing a significance at more than $99\%$ CL even when BAO or SNeIa data are included in the analysis.

In a few words, $A_{\mathrm{lens}}\ne 1$ is a failed consistency check of the Planck data, a way to resolve an internal tension of the temperature power spectrum, and a way to alleviate the external disagreement of Planck with the $S_8$ late Universe measurements. It is worth mentioning that the official position of the Planck collaboration is that this $A_{\mathrm{lens}}$ anomaly is only a statistical fluctuation [12,85], so it is not a physical problem. However, the fact that Planck data are failing a consistency check poses some problems on the reliability of the results that are lensing-related (such as the curvature of the Universe or the total neutrino mass), suggesting that they could be influenced by an unsolved systematic in the data, and this presents serious limitations to the precision cosmology.

3.3. The ${\Omega }_k$ Tension

Now we want to understand if, assuming general relativity, there is a physical explanation for the $A_{\mathrm{lens}}$ anomaly. In fact, a similar effect seems to be captured by another important parameter, which is the curvature of the Universe. Specifically, Planck 2018 found an indication for a closed Universe [12] at approximately $3.4\sigma$, with a $99\%$ probability region of $-0.095\le {\Omega }_k\le -0.007$ 5. The first question that we want to answer is whether Planck provides an unbiased and reliable estimate of the curvature of the Universe. This may not be the case, as there is a “geometrical degeneracy” with ${\Omega }_m$ that could skew the results. However, since gravitational lensing depends on the matter density, its detection in the CMB power spectra breaks the geometrical degeneracy and, using simulated Planck measurements, it was shown in Ref. [86] that such an experiment could constrain the curvature of the Universe to be flat with an uncertainty of $2\%$, with no significant bias towards closed models. Planck therefore prefers a closed Universe (${\Omega }_k<0$) with a probability of $99.985\%$, and this is not entirely a volume effect, as the best-fit $\Delta {\chi }^2$ changes by $-11$ compared to the base $\Lambda$CDM when adding one additional degree of freedom as the curvature parameter. The improvement is due also to the fact that the observed low CMB anisotropy quadrupole could be in agreement with a large-scale cut-off in the primordial density fluctuations, as predicted by closed models. Therefore, a model with ${\Omega }_k<0$ is slightly preferred over a flat model with $A_{\mathrm{lens}}>1$, because closed models better fit not only the smoothing of the peaks in the damping tail, but also the low-multipole data. A lower quadrupole than that predicted by the $\Lambda$CDM scenario was already present in the WMAP data, and a closed Universe to explain this effect was already proposed in Ref. [87].

A joint constraint Planck + lensing + BAO is instead very consistent with a flat Universe, finding ${\Omega }_k=0.0007\pm 0.0019$ at 68% CL [12] (${\Omega }_k=0.0004\pm 0.0019$ at 68% CL using the alternative likelihood [88]). Given the significant change in the conclusions from Planck alone, it is mandatory to investigate whether the datasets combined together are indeed consistent. In fact, a fundamental prerequisite for combining complementary probes is that they derive from the same cosmological model and therefore must be consistent. While in a $\Lambda$CDM model the BAO data agree very well with the Planck measurements (see Figure 1 of Ref. [12]), when we let the curvature vary there is a strong $>3\sigma$ disagreement between the Planck constraints and the BAO measurements (see Figure 4 of Ref. [86]). The same disagreement is evident in the independent analysis shown in Figure 1 of Ref. [89], as well as in Figure 1 of Ref. [90], which reports a similar tension with the full-shape (FS) galaxy power spectrum measurements from the BOSS DR12 CMASS sample.

The closed models have substantially higher lensing amplitudes than in the $\Lambda$CDM model, because a closed Universe means more matter and therefore more lensing. This is why varying ${\Omega }_k$ or $A_{\mathrm{lens}}$ gives a similar best fit of the lensing reconstruction data (see Figure 5 of Ref. [86]). There is clearly a degeneracy between the curvature and the $A_{\mathrm{lens}}$ parameters, as shown in Figure 2 of Ref. [86], and we can see how fixing the Universe to be flat leads to this anomalous value for $A_{\mathrm{lens}}>1$, while $A_{\mathrm{lens}}$ can be in agreement with its expected value if we are open to the possibility of a closed Universe. However, we should be extremely careful in comparing these parameters, because while the $A_{\mathrm{lens}}$ parameter is just a phenomenological consistency check that is failing, without any theoretical motivation, the curvature of the Universe is not new physics beyond the standard model [91], but it is predicted by general relativity, and depends on the energy content of the Universe. Furthermore, similarly to what happens with $A_{\mathrm{lens}}$, in a closed Universe with ${\Omega }_k=-0.045$, which is the preferred value for the best fit of the Planck data, the cosmological parameters obtained from the fit of the low-ℓ and the high-ℓ multipoles of the Planck data are now fully compatible. The problem is that by varying ${\Omega }_k$, both the well-known tensions on $H_0$ and $S_8$ are exacerbated: in a $\Lambda$CDM + ${\Omega }_k$ model, Planck gives $H_0={54.4}_{-4.0}^{+3.3}$ km/s/Mpc at 68% CL, increasing the tension with SH0ES at $5.5\sigma$, and $S_8$ in disagreement at about $3.8\sigma$ with KiDS-1000 [77], and more than $3.5\sigma$ with DES-Y3 [78].

The compatibility of non-CMB data combinations with a closed Universe was also investigated in Ref. [86], finding that BAO + SNeIa + BBN gives $H_0=79.6\pm 6.8$ km/s/Mpc at 68% CL, perfectly consistent with SH0ES, but at $3.4\sigma$ tension with Planck, and that BAO + SNeIa + BBN + SH0ES gives ${\Omega }_k=-0.091\pm 0.037$ at 68% CL, i.e., slightly preferring a closed Universe. Furthermore, Ref. [65] uses effective field theories of large-scale structure (EFTofLSS) to constrain ${\Omega }_k$ independently of some flatness assumptions that usually go into different large-scale structure analyses, and finds ${\Omega }_k=-{0.089}_{-0.046}^{+0.049}$ at 68% CL, but still agrees with a flat Universe when CMB data are included.

In order to test the robustness of different dataset combinations in preferring a flat Universe, we can extend the parameter space to include variation in the dark energy equation of state. In Ref. [92], it was shown that Planck data are in perfect agreement with the SNeIa and the local Hubble constant measurements, even by strongly preferring a closed Universe, while Planck is still in strong tension with the BAO measurements, so their combination should be considered with some caution. Therefore, the conclusions of a closed Universe incompatible with the luminosity distance measurements are a result of the assumption of a cosmological constant. In practice, Planck plus the luminosity distance measurements disagree with both a cosmological constant and a flat Universe at more than $99\%$ CL, preferring a phantom closed Universe with the same statistical significance.

This whole debate over the curvature of the Universe has grown to question whether and how the choice of data type affects the parameter constraints, and has stimulated the investigation of how data should be safely chosen and combined in the data analysis to obtain physically meaningful results (see, for example, Ref. [93]).

3.4. The CMB Ground-Based Experiments

On the other hand, if we consider the alternative CMB experiments, we see that ACT-DR4 + WMAP gives at $68\%$ CL ${\Omega }_k=-0.001\pm 0.012$ [13] (see also the left panel of Figure 2), while SPT-3G finds ${\Omega }_k={0.001}_{-0.019}^{+0.018}$ [14]. This is a confirmation that it is possible to obtain precise measurements of ${\Omega }_k$ from a CMB experiment, as suggested by the simulations in Ref. [86], because the gravitational lensing that can be measured by the power spectra depends on the matter density, and its detection breaks the geometrical degeneracy. For these datasets, there is perfect agreement between the CMB and BAO data when the curvature is free to vary, as shown in the right panel of Figure 2, so they can be safely combined together.

The significant difference in the estimates of the curvature between Planck and the ground-based CMB telescopes is due to their mutual tension. It is possible to compute the global tensions between these CMB datasets using the suspiciousness statistics [94], finding that between Planck and ACT there is a tension at $2.6\sigma$ level, assuming a $\Lambda$CDM model. While some parameters can have a larger disagreement over $3\sigma$, this global $2.6\sigma$ tension is intriguing because they are looking at the same observable and are assuming the same standard $\Lambda$CDM scenario to analyse the data.

If they have such a large difference assuming the standard cosmological model, the tension between Planck and ACT-DR4 can increase even more when considering an extended scenario. In particular, they disagree significantly when a running ${\alpha }_{\mathrm{s}}=(d{n}_s/dlogk)$ and a running of the running ${\beta }_s=(d{\alpha }_{\mathrm{s}}/dlogk)$ of the scalar spectral index $n_s$ are free to vary: ACT-DR4 + WMAP prefer both these parameters to be different from zero at the level of $2.9\sigma$ and $2.7\sigma$, respectively [95], while for Planck they are consistent with zero.

Furthermore, Planck and ACT-DR4 disagree on the results for the early dark energy scenario [96], which is mostly known to be a promising candidate in resolving the Hubble tension: considering ACT-only data or combined with Planck TT up to multipoles 650, there is evidence for EDE $>3\sigma$, which completely solves the Hubble tension [96,97]. Moreover, the evidence for EDE $>3\sigma$ persists with the inclusion of Planck lensing and BAO data (shifting $H_0$ towards a lower value, but still able to alleviate the tension), but it disappears completely once the full Planck multipoles data are considered, with $H_0$ in tension again with SH0ES. In a few words, the Planck damping tail is at odds with EDE different from zero.

Finally, Planck and ACT-DR4 (and SPT-3G) are in tension for the neutrino sector, both for the neutrino effective number $N_{\mathrm{eff}}$ and the total neutrino mass $\Sigma m_{\nu }$ constraints. In particular, in Ref. [98], it was shown that ACT-DR4 + WMAP slightly suggests a neutrino mass with $\Sigma m_{\nu }=0.68\pm 0.31$ eV at 68% CL, and SPT-3G + WMAP prefers $\Sigma m_{\nu }={0.46}_{-0.36}^{+0.14}$ eV at 68% CL, and both agree with the Planck + CMB lensing constraint of $\Sigma m_{\nu }={0.41}_{-0.25}^{+0.17}$ eV at 68% CL obtained by marginalizing over $A_{\mathrm{lens}}$, which is the possible systematic effect in the Planck data. Furthermore, by extending the parameter space to include variations in the DE and inflationary sector, Ref. [98] finds that both ACT-DR4 and SPT-3G can accommodate larger neutrino masses, and that the anticorrelation present between $\Sigma m_{\nu }$ and $H_0$ in a $\Lambda$CDM framework disappears (see Figure 3 of Ref. [98]). This extended parameter space with a non-zero total neutrino mass is in agreement with the Hubble constant measured locally and can also relieve the $S_8$ tension.

4. Conclusions

Most of the anomalies and tensions involve the CMB data, and we currently have (i) the $5\sigma$ Hubble constant disagreement between the CMB data and the local measurements, (ii) the $2-3\sigma$$S_8$ tension between the CMB data and the late Universe measurements, (iii) an anomalous $2-3\sigma$ indication for an excess of lensing $A_{\mathrm{lens}}$ in the Planck data, corresponding to a $3\sigma$ indication for a closed Universe ${\Omega }_k<0$, (iv) a $2-3\sigma$ indication for a running ${\alpha }_s$ and a running of the running ${\beta }_s$ of the scalar spectral index $n_s$ from ACT-DR4 and SPT-3G data, (v) an indication for a total neutrino mass different from zero from ACT-DR4 and SPT-3G, and, finally, (vi) an indication of a $>3\sigma$ for an EDE for ACT-DR4. These anomalies and tensions present a serious limitation to “precision cosmology” and question whether the CMB results are a confirmation of the flat standard $\Lambda$CDM cosmological model. At this point, given the quality of all the analyses, these discrepancies could indicate a problem with the underlying cosmology and our understanding of the Universe, requiring new observations and stimulating the investigation of alternative theoretical models and solutions.