<figure><img src="image_22.png" /><figcaption>Fig. 17. Period-Luminosity Relation for short-period, presumed O-rich Miras in the SNIa host galaxy M101 from Ref. [20].</figcaption></figure>

intervention. ML can further identify patterns and correlations in large datasets, improving the efficiency of SBF measurements and reducing measurement time. Sufficient training datasets will be available after the first years of Euclid or LSST public data releases.

In conclusion, the SBF method currently stands as a robust and versatile tool in extragalactic distance measurements. To date, the method's range of usefulness allows it to be adopted for tests on the $H_0$ tension, providing results that agree with the late-time/direct Hubble constant estimates. With advancements in telescope technology, data processing, and collaborative efforts, the SBF method is well-positioned to make significant contributions to resolving cosmic tensions, possibly even beyond the $H_0$ parameter, and enhancing our understanding of the Universe. By implementing the proposed future developments, the astronomical community can further refine the SBF technique and extend its applicability to new frontiers in extragalactic astronomy.

## 2.1.6. Mira variables

**Coordinator:** Caroline Huang

**Contributors:** Antonio Capodagli, Lucas Macri, and Massimo Marengo

Mira variables are fundamental-mode, thermally-pulsing AGB stars with periods ranging from ~100–1000 days or longer. They fall into a broader category of variables known as Long-Period Variables (LPVs), which includes semi-regular variables, and OGLE small-amplitude RGB stars, which are overtone or irregular pulsators. The pulsation of Miras is likely driven by a $\kappa$-mechanism similar to that found in Cepheids. As highly-evolved stars, they contribute to the chemical enrichment of the ISM through stellar mass loss, have low effective temperatures ($T_{\text{eff}} < 3500$ K) with spectral intensity peaking between 1–2 $\mu$m, and are often among the brightest stars in an intermediate-to-old population ($L \sim 10^4 L_{\odot}$). They are also ubiquitous as nearly all stars ($0.8 M_{\odot} < M < 8 M_{\odot}$) will experience a Mira phase in evolution.

Like other AGB stars, they may be classified into two main spectral types based on their photospheric carbon-to-oxygen ratio (C/O ratio). The exact boundaries are somewhat fluid, but typical C/O values are: C/O ratio $\gtrsim 1$ are classified as C-rich (C-type), C/O ratio $\lesssim 0.5$ are O-rich (M-type), and C/O ratio $\sim 0.5–1.0$ are intermediate, or S-type. While the identification of carbonaceous or silicate molecular features in stellar spectra is the gold standard for classification, in practice, spectra are difficult or expensive to obtain, and colors, or color-color diagrams, are typically used as a proxy for spectral type.

The first Period-Luminosity Relations (PLRs) derived for Miras used bolometric magnitudes [196,197]. Over the past several decades, Miras have been observed in bands ranging from optical to far-infrared have been used to measure distances to many Local Group galaxies (see reviews by Refs. [198–200] and references therein) and map the structure

of the MW [201]. Studies of nearby dwarf galaxies have suggested that metallicity does not appear to have an observable effect on the pulsational properties of these stars [202,203], although possible metallicity effects have been suggested by theoretical models. Recently, nonlinear stellar pulsational models have shown significant improvements in matching theoretical predictions of long-period variable PLRs with observations [204] which may shed light on this apparent discrepancy in the future.

### 2.1.6.1. Application.
For the purposes of this review, we will focus on the O-rich Miras, which are more commonly used as distance indicators because they follow PLRs with lower scatter in near-infrared wavelengths (~0.12 mag) compared to their C-rich counterparts. In order to create the PLRs, time-series observations — often with either an uneven, power-law spacing or monthly sampling — are also required in order to determine periods.

The first rung of Mira distance ladder uses Hubble Space Telescope observations of Miras in the water megamaser host galaxy NGC 4258 and ground-based near-infrared observations of Miras in the LMC [205,206] as “anchor” galaxies to obtain an absolute magnitude calibration [19]. In order to tie the ground-based observations to the HST photometric system, Ref. [207] used a photometric transformation derived from O-rich Mira spectra. Unlike the Cepheid distance ladder, there is currently no precise, parallax-based absolute magnitude calibration. This is due to the fact that Miras (as well as AGB stars in general, and other evolved stars such as red supergiants) exist at the intersection of many difficulties for Gaia — they are highly luminous, red, have angular sizes larger than their parallaxes, and are known to have photocenter variations, all of which can contribute to underestimated uncertainties and bias in the inversion of their parallaxes [208–211].

The second rung of the distance ladder is currently built on observations of Miras in two nearby Type Ia Supernova host galaxies — NGC 1559 and M101 (hosts of SN 2005df and SN 2011fe respectively, with PL relation for M101 shown in Fig. 17) [19,20]. Thus far, these are the only two SNIa host galaxies with Mira-based distances. The statistical uncertainty in the SNIa calibration is also the dominant source of error in the Mira-$H_0$ measurement (even after standardization, the uncertainty in SNIa peak magnitude is ~0.1 mag). Thus the greatest reduction in uncertainty will be obtained from obtaining more observations of Miras in SNIa host galaxies.

### 2.1.6.2. Implications for hubble constant and hubble constant tension.
The current most precise Mira-based measurement yields $H_0 = 72.37 \pm 2.97 \text{ km s}^{-1}\text{Mpc}^{-1}$ (~4% total uncertainty, including both statistical and systematic components) [20]. While it is not yet as precise as more established distance indicators, as an independent measurement, this
result does support the hypothesis that the local measurement of $H_0$ is greater than the early-Universe measurement with ~95% confidence. In addition, it provides an important cross-check to distances made with other more established distance indicators such as Cepheids and TRGB.

### 2.1.6.3. Prospects.
The next decade should be particularly exciting for using Miras and other LPVs as distance indicators. In the NIR/MIR bands accessible with HST, JWST and Roman, Miras have PLRs with smaller scatter than in the optical, nearly-sinusoidal light curves, and are relatively easy to identify as some of the brightest stars in resolved stellar populations. The Vera C. Rubin Observatory's LSST is expected to have sufficient depth and number of epochs to enable the detection of Miras at optical wavelengths (griz) [212] in galaxies out to D ~ 10 Mpc [213].

### 2.1.7. Type Ia supernovae

**Coordinator:** Maria Vincenzi

**Contributors:** Jenny Wagner, Lluís Galbany, Luca Izzo, M. Pilar Ruiz Lapuente, and Sanjay Mandal

SNIa are thermonuclear explosions of carbon oxygen white dwarfs in close binary systems. Their peak brightness after various empirical corrections is highly homogeneous (r.m.s. <0.15 mag in B-band), hence their usage in cosmology as standardizable candles. SNIa have played and still play a crucial role in measurements of the Hubble constant, and they are typically used in two out of three rungs in the typical distance ladder. In the second rung, their luminosities are calibrated with stellar distance indicators, e.g., Cepheids or TRGB or JAGB, in the same galaxy. In the third rung, their brightness values calibrate the Hubble-Lemaître relation into what is now deemed the ‘Hubble flow’ (~100 to 600 Mpc, or z < 0.15), where the cosmological expansion dominates over the peculiar motions of galaxies. Ideally, one could remove the intermediate step (second rung), and go straight from geometric calibration to SN. Unfortunately, the rate of SN in the local Universe is not nearly frequent enough to provide multiple SN in the few galaxies used for geometric anchors (the MW, the LMC, or the mega-maser NGC 4258). Even with Cepheids/TRGB/JAGB as a go-between, the low rate of SN in the nearby Universe (roughly one per galaxy per 100 years) is the limiting component of the precision of $H_0$ measurements, for instance Ref. [34] utilizes every SNIa that pass cosmological quality requirements and Cepheid suitability within 40 Mpc. The most comprehensive three-rung ladder $H_0$ measurements using Cepheids are shown in Fig. 8.

#### 2.1.7.1. SN Ia standardization and derivation of $H_0$. We review here the formalism for deriving the Hubble constant with SNIa in the local distance ladder, as presented by Ref. [34]. Using a set of Cepheids or TRGB distance moduli ($\mu_0$) calibrated with geometric measurements such as parallax or megamasers (first rung in the distance ladder) and comparing these distances to brightnesses $m_X$ of SNIa exploding in the same galaxies, we can estimate the single offset $M_B$, which describes the absolute magnitude of an SNIa (second rung). Following the SALT modeling framework [214,215],<fnref n="4" /> SNIa standardized brightness, $m_X$,<fnref n="5" /> is generally measured with the Tripp formula [218] such that:

$$m_X = m_B + \alpha x_1 - \beta c - \delta_{\text{Bias}} + \delta_{\text{Host}}, \quad (2.1)$$

where $m_B$, $x_1$ and $c$ are all independent properties of each light curve derived using the SALT light-curve model, $\alpha$ and $\beta$ are correlation coefficients that help standardize the brightness, $\delta_{\text{Bias}}$ is a correction due to selection effects and other biases as predicted by simulations,

and $\delta_{\text{Host}}$ is a final correction due to residual correlations with host galaxy properties.

For a SNIa in the $i$th Cepheid host,

$$m_{X,i} = \mu_{0,i} + M_B, \quad (2.2)$$

where $M_B$ is the fiducial SNIa absolute magnitude (assumed to be the same across the whole sample), and $\mu_{0,i}$ is the distance modulus derived from Cepheid measurements for the $i$th galaxy. The ladder is completed with a set of SNIa that measure the expansion rate quantified as the intercept, $a_B$, of the distance (or magnitude)-redshift relation. For an arbitrary expansion history and for $z > 0$ as

$$a_B = \log cz \left\{ 1 + \frac{1}{2} [1 - q_0] z - \frac{1}{6} [1 - q_0 - 3q_0^2 + j_0] z^2 + \mathcal{O}(z^3) \right\} - 0.2m_X^{\text{HF}}, \quad (2.3)$$

measured from a set of SNIa ($z, m_X^{\text{HF}}$) in the Hubble flow, where $z$ is the redshift due to expansion, $q_0$ is the deceleration parameter, and $j_0$ is the jerk parameter. Typically, for $\Lambda$CDM, $j_0$ is set to 1. The determination of $H_0$ follows from

$$\log H_0 = 0.2M_B + a_B + 5. \quad (2.4)$$

Covariances between rungs are taken into account following the approach in Refs. [219,220]. Finally, $q_0$ (and $j_0$) can only be constrained from SNIa data, without the requirement of any additional information.

#### 2.1.7.2. Systematics on the path to $H_0$ with SN Ia. $H_0$ measurements using SNIa are affected by various sources of systematic uncertainties.

Ref. [33] gives a comprehensive overview of many of these systematics and how they may affect measurements of $H_0$. In Fig. 18, adapted from Ref. [33], we show the impact on $H_0$ of applying $1\sigma$ shift of each systematic. We group the systematics into various categories: Redshifts, SN Physics, Selection, Calibration, and MW Dust. Before delving into each category of systematics in detail, it is crucial to understand which systematics $H_0$ is most sensitive to, and conversely, which have minimal impact due to being mitigated. The distance ladder is structured so that the same probe is used in two of its three rungs (e.g., Cepheids in the first and second rungs, and SNIa in the second and third rungs). If a systematic error disproportionately affects SNIa in the second and third rungs, or introduces significant differences in the SNIa populations between these rungs, it will have a substantial impact on $H_0$. Conversely, if a systematic introduces a consistent offset that affects all SNIa uniformly, the impact of that systematic will be negligible due to the cancellation effect inherent in this formalism. In other words, consistent offsets will cancel out, mitigating the effect of the systematic error.

**Calibration:** Since many of the same surveys measure SN in both the second and third rungs of the distance ladder, the effects of calibration systematics on $H_0$ is mitigated by the effect discussed above. As illustrated in Ref. [223] in the context of the latest SHOES analysis, per-survey “gray” calibration offsets affect the uncertainty by $0.2\,\text{km s}^{-1}\text{Mpc}^{-1}$ at most. However, when different surveys are used for the second and third rungs, as shown in Ref. [15,225], this cancellation does not occur and the effects of calibration can be larger.

**Redshifts:** $H_0$ measurements are more sensitive to systematics related to redshift since redshift information is used only in the third rung of the distance ladder, with no systematic mitigation. Ref. [222] explore various bulk flow models in the nearby Universe, finding that changes in $H_0$ could be up to $0.2\,\text{km s}^{-1}\text{Mpc}^{-1}$. They also note that including or excluding peculiar velocity corrections could lead to $\Delta H_0 \sim 0.5$. Additionally, Refs. [230,231] find redshift measurement biases causing uncertainties around $0.1\,\text{km s}^{-1}\text{Mpc}^{-1}$.

**SN Physics and Selection:** SN intrinsic astrophysics and the role of SN dust remain one of the most poorly understood aspects of SNIa cosmology and can affect $H_0$ measurements [221,232]. Most analyses have found that these effects have a small impact on $H_0$ because the differences between SN sub-populations selected in the second and

<footnote n="4">⁴ The SALT2 and SALT3 frameworks have been used in all most recent cosmological analyses. Other frameworks are also available, i.e., SnooPy [216], BayeSN [217].</footnote>

<footnote n="5">⁵ We call the standardized brightness $m_X$ instead of $m_B$ as in Ref. [34] to be clear that the brightness is standardized.</footnote>
<figure><img src="image_23.png" /><figcaption>**Fig. 18.** Ref. [33], the impact on recovery of $H_0$ of the various systematic uncertainties tabulated. The units of these measurements are $\text{km s}^{-1}\text{Mpc}^{-1}$. The dashed lines are given at $\Delta H_0$ of 0.7, which is the entire contribution of the uncertainty in Ref. [34] from SN measurements. We labeled the different categories of systematic uncertainties.</figcaption></figure>

*Source:* Adapted from

<caption>**Table 1**</caption>

A summary of the various cross-checks and systematics on the supernova component of the distance ladder. If two uncertainties are given, the first one is the statistical uncertainty and the second one is the systematic uncertainty.

<table><thead><tr><th>Reference</th><th>Notes on the specific systematics check implemented</th><th>Result (km s<sup>-1</sup>Mpc<sup>-1</sup>)</th></tr></thead><tbody><tr><td colspan="3"><strong>Optical SNIa:</strong></td></tr><tr><td>[221]</td><td>Uses spectral feature twinning process to improve standardization; checks dust modeling, intrinsic scatter modeling;</td><td>73.01 ± 0.92</td></tr><tr><td>[222]</td><td>Checks different models of peculiar velocities/bulk flows</td><td>σ<sub>H<sub>0</sub></sub> &lt; 0.2</td></tr><tr><td>[223]</td><td>Checks SNIa Calibration by allowing individual SN survey offsets</td><td>σ<sub>H<sub>0</sub></sub> &lt; 0.2</td></tr><tr><td>[224]</td><td>Checks impact of mass step, global vs. local correlations</td><td>σ<sub>H<sub>0</sub></sub> &lt; 0.15</td></tr><tr><td>[225]</td><td>Checks light-curve fitting method; also does NIR fits</td><td>73 ± 2</td></tr><tr><td>[194]</td><td>Uses 4-rung distance ladder, checks SNIa host demographic systematic</td><td>74.6 ± 0.9 ± 2.7</td></tr><tr><td>[226]</td><td>Uses ZTF data alone, check on SNIa calibration</td><td>76.94 ± 6.4</td></tr><tr><td colspan="3"><strong>NIR SNIa:</strong></td></tr><tr><td>[227]</td><td>Uses literature NIR SN (restframe J) and peak fitting</td><td>72.8 ± 2.8</td></tr><tr><td>[36]</td><td>Uses literature NIR SN (restframe J and H band)</td><td>72.3 ± 1.4 ± 1.4</td></tr><tr><td>[228]</td><td>Uses RAISIN\+literature NIR SN (restframe Y band) and SNooPy fitting, check on dust</td><td>75.9 ± 2.2</td></tr><tr><td>[229]</td><td>Uses literature NIR SN and BayesN fitting</td><td>74.82 ± 0.97 ± 0.84</td></tr></tbody></table>

third rung are not expected to be significantly different. For example, Ref. [233] showed evidence for a correlation between standardized brightness and the age of the host galaxy (quantified estimating the specific star-formation at the SN location). In earlier measurements like Ref. [234], the third rung of SN had no galaxy-based selection applied. Only the second rung had this selection, tied to Cepheid discovery, which favored star-forming host galaxies. This would potentially lead to a bias in the recovery of $H_0$. The size of the bias would depend on the relative differential fraction of host-galaxy demographics between the second and third rung multiplied by the size of the correlation. Subsequent analyses [224] showed that this effect would likely be insufficient to explain the Hubble tension. Still, in the most recent SHOES analysis [34], the selection of SN in the third rung of the distance ladder was done to be as similar as possible as the second rung. Only SN found in star-forming galaxies are selected, which thereby removed the sensitivity to this systematic. Yet, the impact of this change was less than the statistical uncertainty from the supernova component of the distance ladder. Similarly, significant differences in dust extinction and/or color-related effects between SN in the second and third rung could potentially bias $H_0$ measurements [235]. SN dust extinction and color-dependent corrections are encapsulated in the nuisance parameter $\beta$ (see Eq. (2.1)). As a cross-check, the $\beta$ parameter was fitted separately in SN used in the second and third rung, and it was found to be consistent [33]. This test, together with various NIR SN $H_0$ measurements, see Table 1 [36,227–229], suggests that dust or color-dependent effects are not expected to significantly bias $H_0$, or to be a significantly underestimated systematic in current $H_0$ measurements.

**Other systematics:** Additional systematic tests, such as changing the light-curve fitter or adding spectroscopic information, have shown consistent results, within $\approx 0.3 \text{ km s}^{-1}\text{Mpc}^{-1}$, e.g., see Ref. [221,225]. Even when isolating data to a single survey, as in Ref. [229], the results

remain consistent, though with larger uncertainties due to the smaller sample size.

2.1.7.3. *Variants on the path to $H_0$ with Supernovae Ia.* Some of the main cross-checks on the SNIa used for these analyses is varying the wavelength regime in which light curves are measured (i.e., optical to NIR) or the dataset used (i.e., the survey used to measure the light curves). As $H_0$ constraints are limited by the number of SN found within 40 Mpc, there is considerable overlap in the data between these various studies. The most popular path to check and improve the distance ladder with SNIa is by measuring NIR light curves. We list these papers in Table 1. Overall, even though the rest-frame band in which light curves are measured varies between these analyses, and the fitting method varies between these methods, there is generally very good agreement in the recovered values of $H_0$. One challenge multiple of these studies have found (e.g., [227,228]) is larger calibration offsets between samples than those found for optical studies. A benefit of this type of study is the possibility of improved precision of distance measurements from NIR data, but the quality of older light curves has not typically been good enough to evaluate this possibility.

An additional path is creating a “4-rung distance ladder”, as done in Ref. [194]. SNIa used in the SHOES distance ladder are those found in late-type galaxies. To avoid this specific sub-sample, one can add another rung in the distance ladder from SBF between TRGB/Cepheids and SN. The analysis of Ref. [194] improves on that of Ref. [193] because the latter follows a similar method, but uses an inhomogeneous set of SBF measurements. The inhomogeneous set significantly increases the scatter of the tie between SBF and SN measurements and appears to bias $H_0$ to lower values. Ref. [194] find a value of $H_0 = 74.6 \pm 2.8 \text{ km s}^{-1}\text{Mpc}^{-1}$, in good agreement with the SHOES value.
2.1.7.4. *Inverse distance ladder to H₀ with supernova.* The same set of SNIa used in the SHOES distance ladder can also serve as uncalibrated relative distance indicators to constrain H₀ when combined with other probes like BAO. BAO constrain the expansion history H(z) and extrapolate H₀. But this approach assumes the sound horizon from CMB constraints and is model-dependent, relying on ΛCDM to infer H(z = 0) from BAO data at z ~ 0.5. However, SNIa can address the latter issue. Instead of calibrating SNIa to the distance ladder, they can be calibrated to the BAO distance scale at typical BAO redshifts (z ~ 0.5), allowing SNIa to constrain the expansion history at later times (z < 0.1) without assuming ΛCDM. Studies such as Refs. [236–238] using this technique found H₀ = 68.57±0.9 km s⁻¹Mpc⁻¹, consistent with CMB under ΛCDM. Since BAOs obtain the physical scale from the sound speed in the early Universe, these low H₀ values have prompted discussions about the impact of the sound horizon value.

2.1.7.5. *Improving measurements of H₀ in the future.* Improving constraints on H₀ using SNIa and the distance ladder is challenging. The limiting factor in the past decade has been the fact that HST can discover Cepheids/TRGB within a radius of 40 Mpc, and there is one to three SNIa per year exploding within this volume. However, new telescopes like the JWST (or future instruments like the Nancy Grace Roman Space Telescope and the ELT) are already showing impressive improvements (both in quality and depth) in Cepheids, TRGB and JAGB measurements<fnref n="6" /> as shown by Refs. [26,174], with exciting hints of shifts to lower H₀ values presented by Ref. [26], even though the statistics is still too low to draw firm conclusions.

The other path toward improving the constraint from SNIa is to improve the precision of the measurements. This is the path followed by papers like Refs. [221,239] which tried using spectral features to improve the standardization, or the large number of papers that measure SNIa in the NIR as better standard candles. The main challenge is that these new standardization approaches depend on the amount of data available for its application in past literature measurements. New types of measurements can be made for nearby SN in the future, but we cannot re-measure past SNIa.

## 2.1.8. J-regions of the asymptotic giant branch methods

**Coordinator:** Siyang Li

**Contributors:** Adam Riess, Bartek Zgirski, Caroline Huang, Dan Scolnic, Gagandeep S. Anand, Greg Sloan, Louise Breuval, Richard I. Anderson, and Stefano Casertano

The JAGB refers to a group of stars in a NIR color magnitude diagram (see Fig. 19) that contains thermally pulsating carbon-rich AGB stars with photospheric C/O greater than 1. The potential for using carbon stars (CS) as distance indicators was first realized in the 1980s [240–244], and the method was later revived in the 2000s [129, 177,245–248]. These pioneering studies, among others, have proposed that the mean, median, mode, or model fit of a near-infrared J-region LF can be used as a standard candle to measure extragalactic distances. The basis for doing so originates from the expectation that only oxygen-rich AGB stars that have masses falling within a relatively narrow range can evolve into carbon-rich AGB stars, constrained by hot bottom burning and the efficiency of the 3rd dredge-up events.

2.1.8.1. *Application.* The JAGB method relies on a measurement of the CS infrared magnitude distribution in a color range between weakly pulsating CS that produce little dust and strongly pulsating CS (i.e., Mira variables) that produce significant amounts of amorphous carbon dust. The JAGB is typically measured in the “outer disk” of a galaxy where the stellar population is young enough to contain a substantial

<figure><img src="image_24.png" /><figcaption>Fig. 19. J vs. J - K color-magnitude diagram of stars in the LMC using the sample from Ref. [250].</figcaption></figure>

amount of carbon AGB stars and far enough from the center to minimize the effects of crowding and internal reddening [249]. The region is also typically chosen to be close enough to avoid sparser fields which increases the statistical uncertainty. In addition, the red background of galaxies can be similar to that of CS and contaminate the CS sample [248]. Consistent selection of fields is also important in maintaining consistent stellar populations.

Several methods have been proposed to measure the JAGB reference magnitude in the outer disk; these are typically designed to measure some variation of the peak of the J-region LF. For instance, Refs. [129, 247] use the mean and median, Ref. [177] smooth a binned J-region LF and take the mode, while Refs. [246,251] introduce maximum likelihood and binned versions of medians and calibrate their JAGB measurements using the skew parameter in a Lorentzian fit. Ref. [248] fit a Gaussian+Quadratic model to the LF. There is an important need for standardization of the technique to ensure accuracy.

2.1.8.2. *Prospects.* The JAGB method has the potential to independently support or refute local measurements of the Hubble constant from the second rung of the distance ladder. Recent studies have found JAGB-based Hubble constants of 74.7 ± 2.1 (stat) ± 2.3 (sys) km s⁻¹Mpc⁻¹, with a full range spanning 71 to 78 km s⁻¹Mpc⁻¹ depend- ing on the measurement method used [18], and 67.96 ± 1.85 (stat) ± 1.90 (sys) km s⁻¹Mpc⁻¹ [174,252], noting that the lower value here originates from galaxy subsample selection, see Ref. [26]. However, it is important to be aware that this method is still much less mature than other standard candles, such as Cepheids. For this method to be robust, several aspects will need to be better understood, such as the empirical effects of metallicity, molecular atmospheric diversity effects on photometry, and the shift of CS from weak to strong pulsations in the context of population diversity.

In addition, past literature has found evidence of non-uniform asymmetry in the J-region LF [79,246,251,253]. This can produce methodological variations (i.e., the mean will be different from the median, mode, and model fit). Non-uniformity of the asymmetry can also result in a mismatch in the degree of methodological variations across the distance ladder, thus increasing uncertainties of the method. It will be important to standardize this effect.

The JAGB method introduces observational flexibility for measuring H₀. JAGB only requires a single epoch while Cepheids need multiple epochs to measure their periods and mean magnitudes. CS are ubiquitous in most galaxies, unlike Cepheids which are generally observed in face-on spiral galaxies. In the near-infrared, JAGB stars are as bright as long-period Cepheids, making them competitive for reaching large distances. On the other hand, the metallicity dependence of Cepheids is very well calibrated [58], which is not yet the case for CS. The JAGB feature is also brighter than TRGB. However, JAGB stars are an intermediate-age (300 Myr - 1 Gyr [129]) population, and are

<footnote n="6">⁶ Even a 25% increase in the Cepheid distance would allow a doubling in the number of usable SNIa in the second rung of the distance ladder (the volume of discovered SN will increase with distance cubed).</footnote>
therefore not as ubiquitous as TRGB stars, which populate essentially every galaxy.

The JAGB is primarily measured in the NIR, typically in the J- and H-bands (or space-based equivalents: HST F110W [249], JWST, F115W, [17] & F150W [18]). It is important to understand whether either of these bands necessarily offers more consistency in color dependence over the other and whether either is sloped as a function of color. Multiple studies have found evidence of asymmetric LF and non-flat color dependence in both the J- and H- bands [79,246,251,253].

Increasing the SNIa sample size decreases fluctuations in H0 arising from cosmic variance and facilitates closer reversion to the mean (see Ref. [26]). Recently, Ref. [254] augmented the JAGB sample used to measure H0 by combining all available JAGB distances to SNIa host galaxies in the literature, as well as measuring new distances to galaxies from JWST Cycles 1 & 2 for a total of 15 galaxies hosting 18 SNIa. As in Ref. [18], they find methodical variations consequent of non-uniform asymmetry in the J-region LF; taking the middle measurement variant (as described in Ref. [18]) yields $H_0 = 73.3 \pm 1.4(\text{stat}) \pm 2.0(\text{sys}) \text{ km s}^{-1}\text{Mpc}^{-1}$.

A powerful way to characterize, and potentially better standardize, variations in the JAGB method is to conduct a field-to-field comparison of the JAGB. This approach has been implemented for the TRGB by the Comparative Analysis for TRGBs (CATs) team [122,125,255]. They compared multiple fields in a given galaxy and developed a standardization procedure via a contrast ratio. Future missions, such as Roman, and more observations with JWST can obtain a larger field coverage that can be used to measure the JAGB and make a similar analysis possible.

## 2.1.9. The Hubble constant from Type II supernovae

**Coordinator:** Lluís Galbany

**Contributors:** Anil Kumar Yadav, Anto Idicherian Lonappan, David Benisty, Géza Csörnyei, Ismailov Nariman Zeynalabdi, and Vladas Vansavičius

With the currently ongoing tension between local and distant measurements of the Hubble constant, it is crucial to test and employ independent methods that adhere to a separate set of systematic uncertainties. SNII offer such independent routes with sufficient accuracy. SNII are the explosions of red supergiant stars, with multiple direct progenitor detections to date (e.g., see Refs. [256,257]). Given the well constrained progenitor type along with their relatively simple composition (with the red supergiant retaining its hydrogen envelope and being made up mostly of H and He before the explosion) and the recent advancements in the understanding of the explosion mechanism, multiple theoretically well-founded distance estimation techniques exist for SNII. The spectra of SNII are characterized by the presence of broad P-Cygni profiles, which allow constraining the photospheric properties. To date, three techniques have been used to measure the Hubble constant: the expanding photosphere method (EPM, [258]), the spectral modeling based techniques (either the spectral expanding atmosphere method SEAM [259] or the tailored EPM [260]) and the standardizable candle method (SCM, [261]). Of the three, SCM is an empirical technique, which employs the relation present between the luminosity and the expansion velocity of the supernova. On the other hand, EPM and SEAM both aim to constrain the physical parameters and the luminosity based on the spectra directly, providing a single step measurement tool. Both Refs. [23,262] demonstrated that SCM and EPM distance measurements up to redshifts of $z \approx 0.34$ are feasible, providing important alternative paths for cosmology, as depicted in Fig. 20.

### 2.1.9.1. Methods and approaches to measure $H_0$ using SNIa
The use of EPM for SNII, which essentially consists of estimating the size of the photosphere, then comparing it to the observed flux by assuming a blackbody spectrum, was first demonstrated by Ref. [258]. This initial

exploration yielded $H_0 = 65 \pm 15 \text{ km s}^{-1}\text{Mpc}^{-1}$. However, it was highlighted later by Ref. [263], SNII radiate with a much smaller surface flux than a blackbody of the same color temperature (the photosphere flux appears diluted), owing to the scattering dominated atmosphere of SNII and the fact that the thermalization layer from which the blackbody radiation is generated is deeper than the photosphere. To take this into account, the EPM was refined by incorporating a multiplicative $\xi$ dilution factor, which relates the position of the thermalization layer to the photosphere, which was shown to depend largely only on the color temperature (see Refs. [264–266]). Systematic discrepancies between the different sets of dilution factors can explain differences on the scale of 20% in the inferred distance [267,268]. The EPM is to date one of the most frequently used distance estimation technique for SNII, with the latest $H_0$ estimate being $72.9_{-4.3}^{+5.7} \text{ km s}^{-1}\text{Mpc}^{-1}$ based on 12 SNII following Ref. [269], using the dilution factors from Ref. [265]. To rule out the possible systematic effects introduced by dilution factors, one has to carry out radiative transfer based spectral modeling of observations, which yields self-consistent results for the physical parameters.

This is achieved in both the customized EPM and in the SEAM, which incorporate radiative transfer modeling of spectra into the distance estimation process. In both techniques, the modeling of the complete spectral time series is carried out, which yields the relevant physical parameters at each epoch in a self-consistent way, avoiding the use of dilution factors. However, estimating a single radiative transfer model takes hours to days, hence estimating a distance this way is a time-consuming process. Due to this, until recent years, this method was only used for a handful of SN [259,260,270]. Recent advances in ML allowed for the faster estimation of radiative transfer models, hence significantly speeding up this modeling process, as described in Ref. [271]. This method was shown to provide internally consistent results [24] at a competitive precision [146] and yielded a Hubble constant of $H_0 = 74.9_{-1.9}^{+1.9} \text{ km s}^{-1}\text{Mpc}^{-1}$ based on the analysis of literature [272]. However, this method is currently largely limited to local redshifts: to date only a few SNII were observed far enough out into the Hubble flow in sufficient detail for the application of either tailored EPM or SEAM.

To date, the most widely used method for estimating $H_0$ using SNII is the SCM [23,261,273]. The method is based on an empirically found correlation between the luminosity and the photospheric expansion velocity of the objects [274,275], and in terms of philosophy is similar to the method employed for SNIa (e.g., see Ref. [34]). In contrast to the Ia's, however, to standardize SNII, spectroscopic information is required in the form of line velocity measurements. Over the years, the standardization has been further extended with color term accounting for the varying level of extinction present for different SN [276]. Recently, Ref. [23] applied this method with nine calibrator SN to estimate the Hubble constant by using SNII as the final rung of the distance ladder, obtaining $H_0 = 75.4_{-3.7}^{+3.8} \text{ km s}^{-1}\text{Mpc}^{-1}$. The method is still under further development, with the aim of increasing the currently available set of calibrators with high precision cases, and also with attempts to investigate the need for additional standardization terms based on spectroscopic information.

These results show that SNII indeed provides important and independent distance measures that can serve as the tools for a sanity check for multiple rungs of the distance ladder. With the steady increase in the number of objects with detailed observations, the coming years will see multiple SNII-based Hubble constant estimates with competitive precision, which will allow for an alternative look at the Hubble tension.

### 2.1.10. HII galaxy distance indicators

**Coordinator:** Ricardo Chávez

**Contributors:** Ana Luisa González Morán, David Fernández-Arenas, David Valls-Gabaud, Elena Terlevich, Fabio Bresolin, Iryna Vavilova,
<figure><img src="image_25.png" /><figcaption>**Fig. 20.** SNII for Hubble parameter estimation. **Left:** The distance ladder formalism applied on SNII, as presented by Ref. [23]. The bottom panels with the blue dots depict the intermediate rung of the distance ladder, where the SNII brightnesses are calibrated through Cepheids and TRGB, along with the residuals after the standardization. The top panels with the red dots show the final rung of the distance ladder, where this standardization is applied on SNII in the Hubble flow which allow for a Hubble constant estimation, while the residuals are shown on the panel below. **Right:** SNII Hubble diagram using EPM distances, as updated from Ref. [277]. EPM distances do not require external calibration, hence they provide a single step measurement of $H_0$. The empty circles denote the EPM distances that employ dilution factors, while the filled squares mark the tailored EPM estimates that make use of radiative transfer modeling. The references for the distance estimates are as follows: B14 – [278], E96 – [264], G18 – [262], J09 – [267], D24 – [269], V24 – [272] and C23 – [24].</figcaption></figure>

Ismailov Nariman Zeynalabdi, Manolis Plionis, Roberto Terlevich, Rodrigo Sandoval-Orozco, Spyros Basilakos, and Vladas Vansevičius

HII galaxies (HIIGs) are characterized by intense, compact episodes of star formation predominantly occurring within dwarf irregular galaxies, significantly enhancing their luminosity. These galaxies are spectroscopically identified based on the prominent equivalent width of their Balmer emission lines, specifically $EW(H\beta) > 50$ Å, which indicates their young age (less than 5 Myr). Similarly, Giant Extragalactic HII Regions (GEHRs) exhibit vigorous star formation but are typically found in the outer disks of late-type galaxies. The rest-frame optical spectra of both HIIGs and GEHRs display pronounced emission lines, indicative of gas ionization by massive Young Stellar Clusters (YSC) or Super Star Clusters (SSC), leading to similar spectral features [279–283].

Numerous studies have confirmed a consistent correlation, found by Ref. [281], between the luminosity of Balmer lines, such as $L(H\beta)$, and the ionized gas velocity dispersion, $\sigma$, measured through these emission lines, in both HIIGs and GEHRs. This relationship, known as the $L-\sigma$ relation [283–285], is recognized as a potent cosmological distance indicator [286–289], where GEHRs and nearby HIIGs are used as the “anchor” sample because their distances can be independently estimated from Cepheid variables or TRGB measurements. As a result, the $L-\sigma$ relation provides an exceptional method for utilizing this distance metric to investigate the Hubble flow over an extensive range of redshifts (z).

The characteristic strong emission lines within the rest-frame optical spectra of GEHRs and HIIGs render them effective tools for exploring nascent star formation at high redshifts. Using instruments such as NIRSpec [290] on board the JWST [291], it is possible to study these regions up to $z \sim 6.5$ through the H$\alpha$ emission line, or even up to $z \sim 9$ using the H$\beta$ and [OIII]$\lambda\lambda4959, 5007$ Å emission lines. This capability enables the observation of luminous HIIGs that date back to the Epoch of Reionization (EoR) as shown in Fig. 21.

Using a dataset of 231 GEHRs and HIIGs, some observed with NIRSpec on the JWST up to $z \sim 7.5$, a study by Ref. [289] employs the MultiNest Bayesian inference algorithm [292–294] to derive constraints for various cosmological models. They used uniform, non-informative priors across all parameters for unbiased estimations. The derived constraints are detailed in Table 2, showing marginalized best-fit values and $1\sigma$ uncertainties for each parameter, with some parameters held constant during the analysis. The study explores a generalized parameter space $\theta = \{\alpha, \beta, h, \Omega_m, w_0, w_a\}$, where $\theta_n = \{\alpha, \beta\}$ represents nuisance parameters of the $L-\sigma$ relation for GEHRs and HIIGs with $\alpha$ as the intercept and $\beta$ as the slope. For the flat $\Lambda$CDM model, $\theta_c = \{h, \Omega_m, -1, 0\}$ sets $h$ as the reduced Hubble constant and $\Omega_m$ as the total matter density, fixing the first two DE equation of state (DE EoS) parameters at $w_0 = -1$ and $w_a = 0$, corresponding to a cosmological constant ($\Lambda$). Adjusting $w_0$ allows for models with evolving DE EoS, akin to quintessence [295,296], while including $w_a$ aligns with the CPL model [297–299].

The constraints derived on cosmological parameters, specifically $\{h, \Omega_m, w_0\} = \{0.731 \pm 0.039, 0.302_{-0.069}^{+0.12}, -1.01_{-0.29}^{+0.52}\}$ (stat) from GHIIR and HIIG data, align closely with recent Pantheon+ results from 1550 SNIa, which produces $\{h, \Omega_m, w_0\} = \{0.735 \pm 0.011, 0.334 \pm 0.018, -0.90 \pm 0.14\}$ [33]. This agreement emphasizes the validity and importance of HIIGs as distance indicators within the context of current cosmological studies.

The enduring validity of the $L-\sigma$ relation at high redshifts ($z > 3$), reaching into the EoR, suggests remarkable uniformity in HIIGs properties over vast cosmic timescales. This not only confirms the reliability of the $L-\sigma$ relation as a cosmological tool but also illuminates the fundamental processes underlying the formation and evolution of early Universe galaxies.

In our efforts to improve the accuracy of cosmological parameters using HIIGs, the addition of data from JWST, extending to $z \sim 9$
<caption>**Table 2**</caption>

Marginalized best-fit parameter values and associated 1σ uncertainties for the HIIGs and anchor samples. Values enclosed in parentheses indicate parameters that were held constant during the analysis.

<table><thead><tr><th>Data set</th><th>α</th><th>β</th><th>h</th><th>Ω<sub>m</sub></th><th>w<sub>0</sub></th><th>w<sub>a</sub></th><th>N</th></tr></thead><tbody><tr><td>HIIG</td><td>–</td><td>(5.022 ± 0.058)</td><td>–</td><td>0.282<sup>+0.037</sup><sub>−0.045</sub></td><td>(−1.0)</td><td>(0.0)</td><td>195</td></tr><tr><td>HIIG</td><td>–</td><td>(5.022 ± 0.058)</td><td>–</td><td>0.278<sup>+0.092</sup><sub>−0.051</sub></td><td>−1.21<sup>+0.45</sup><sub>−0.40</sub></td><td>(0.0)</td><td>195</td></tr><tr><td>HIIG</td><td>(33.268 ± 0.083)</td><td>(5.022 ± 0.058)</td><td>0.715 ± 0.018</td><td>0.267<sup>+0.038</sup><sub>−0.048</sub></td><td>(−1.0)</td><td>(0.0)</td><td>195</td></tr><tr><td>HIIG</td><td>(33.268 ± 0.083)</td><td>(5.022 ± 0.058)</td><td>0.718 ± 0.020</td><td>0.278<sup>+0.091</sup><sub>−0.050</sub></td><td>−1.22<sup>+0.46</sup><sub>−0.40</sub></td><td>(0.0)</td><td>195</td></tr><tr><td>Anchor+HIIG</td><td>33.28 ± 0.11</td><td>4.997 ± 0.089</td><td>0.730 ± 0.038</td><td>(0.3)</td><td>(−1.0)</td><td>(0.0)</td><td>231</td></tr><tr><td>Anchor+HIIG</td><td>33.28 ± 0.14</td><td>5.00 ± 0.11</td><td>0.730 ± 0.040</td><td>0.335<sup>+0.044</sup><sub>−0.055</sub></td><td>(−1.0)</td><td>(0.0)</td><td>231</td></tr><tr><td>Anchor+HIIG</td><td>33.29 ± 0.14</td><td>4.99 ± 0.11</td><td>0.731 ± 0.039</td><td>0.302<sup>+0.12</sup><sub>−0.069</sub></td><td>−1.01<sup>+0.52</sup><sub>−0.29</sub></td><td>(0.0)</td><td>231</td></tr><tr><td>Anchor+HIIG</td><td>33.29 ± 0.14</td><td>4.99 ± 0.11</td><td>0.730 ± 0.039</td><td>0.321<sup>+0.10</sup><sub>−0.063</sub></td><td>−0.91<sup>+0.55</sup><sub>−0.33</sub></td><td>−0.71<sup>+0.65</sup><sub>−1.2</sub></td><td>231</td></tr></tbody></table>

<figure><img src="image_26.png" /><figcaption>**Fig. 21.** Hubble diagram for GEHRs and HIIGs, here *z* is the redshift and *μ* is the distance modulus. In cyan we present the “anchor” sample of 36 GEHRs which have been analyzed in Ref. [300], in black we present the full sample of 181 HIIGs which have been analyzed in Ref. [289], while in blue we present the 9 new HIIGs from Ref. [301] and in green 5 new HIIGs studied with JWST by Ref. [302]. The black line is the cosmological model that best fits the data with the red shaded area representing the 1σ uncertainties to the model, while the gray dashed line is a flat cosmological model without DE. The inset at the left shows a close-up of the Hubble diagram for *z* ≤ 0.15. The inset at the right presents the pulls PDF of the entire sample of GEHRs and HIIGs and the red line shows the best Gaussian fit to the PDF.</figcaption></figure>

*Source:* Adapted from Ref. [289].

and beyond, is proving crucial. The JWST's exceptional sensitivity and resolution enable detailed observations of HIIGs at these higher redshifts, offering a unique view of the early Universe. This data range is essential for analyzing the dynamics of the Universe's expansion across various cosmological scenarios.

### 2.1.11. The baryonic Tully-Fisher relation approach

**Coordinator:** Khaled Said

**Contributors:** Benoit Famaey, Cláudio Gomes, Duško Borka, Esha Bhata, Jenny G. Sorce, Maurice H.P.M. van Putten, Milan Milošević, Paolo Salucci, Paula Boubel, Predrag Jovanović, and Vesna Borka Jovanović

The BTFR [303–312] extends the classic Tully-Fisher (TF; [25,313–335]) relation by incorporating both the stellar and gas masses of galaxies, correlating a galaxy's baryonic mass ($M_b$) with its rotational velocity ($V_{rot}$). This relation provides critical insights into galaxy formation and dynamics and is crucial in testing and constraining models of galaxy evolution within both the ΛCDM framework and alternative theories [336–349].

The BTFR is defined as: $M_b \propto V_{rot}^x$, where $M_b$ is the sum of stellar and gas masses, $M_b = M_* + M_{gas}$ and $x$ is typically around 4, suggesting

a deep connection between the visible matter and the dynamics of galaxies [303,350–352].

#### 2.1.11.1. Historical context and development.
The original TF relation, proposed by Ref. [313], established a link between the intrinsic brightness of spiral galaxies and their maximum rotational velocities. This relation has been pivotal in determining distances to galaxies and measuring the Hubble constant [353]. For example, Ref. [313] applied their TF relation to derive distances to the Virgo and Ursa Major clusters, estimating a Hubble constant of approximately $H_0 \approx 84 \text{ km s}^{-1} \text{Mpc}^{-1}$ for Virgo and $H_0 \approx 75 \text{ km s}^{-1} \text{Mpc}^{-1}$ for Ursa Major. The extension to the BTFR was motivated by the need to include galaxies with significant gaseous components, particularly dwarf galaxies [303]. Over the years, numerous studies have validated and expanded the BTFR, demonstrating its robustness and utility in various cosmological and astrophysical applications [312,347,354]. A crucial step toward the understanding of the physics of the TF and BTFR relationships has been the finding, in disk systems, of a family of local Tully-Fisher-like relationships [355–357], holding at various specific galactocentric radii and thus dubbed as the Radial Tully Fisher (RTF: for a review see Ref. [358]).

#### 2.1.11.2. Theoretical models and their implications.
<figure><img src="image_27.png" /><figcaption>**Fig. 22.** Comparison between the BTFR obtained from error-weighted fits of galaxies with accurate distances from Ref. [359] to the BTFR predicted by ΛCDM cosmology. The observed BTFR (blue solid line) includes a light blue band representing the intrinsic scatter of 0.1 dex. The ΛCDM prediction is shown as a black dashed line. Both relations are plotted using equations from Ref. [359]: Eq. (6) for the observed BTFR and Eq. (6) for the ΛCDM prediction. [Reprinted under CC BY 4.0, from: V. Borka Jovanović, D. Borka, and P. Jovanović, Contrib. Astron. Obs. Skalnate Pleso 55/2, 24 - 33 (2025)].</figcaption></figure>

**The standard Λcdm model** In the ΛCDM, the TF and BTFR are seen as consequences of the gravitational dynamics dictated by DM halos and baryonic matter. The rotational velocities of galaxies are thought to be influenced by both visible and DM, with the BTFR providing a means to probe these interactions [347]. However, as illustrated in Fig. 22, the observed slope and intrinsic scatter of the BTFR often differ from ΛCDM predictions, which typically forecast a lower slope and higher scatter [359]. This discrepancy poses challenges to the ΛCDM model and prompts further investigation into galaxy dynamics and mass distribution.

**Modified gravity models** Some studies propose that the BTFR can be explained without DM through modified theories like $f(R)$ theories of gravity [360–363]. Alternative gravitational theories, such as Modified Newtonian dynamics (MOND) and non-minimal matter-curvature coupling models, offer different perspectives on the BTFR. MOND, proposed by [364], adjusts Newtonian dynamics at low accelerations, predicting a BTFR with a slope exactly equal to 4, consistent with many observations [347,354]. MOND has made several successful predictions regarding the detailed shapes of rotation curves, galaxy dynamics, and galaxy lensing [365–367].

Another approach, non-minimal matter-curvature coupling model, generalizes the pure gravity sector by introducing a generic function of the curvature scalar, $f_1(R)$, and a non-minimal coupling between the matter Lagrangian and another generic function of the curvature scalar, $f_2(R)$ [368],

$$S = \int d^4x \sqrt{-g} (f_1(R) + f_2(R)L). \qquad (2.5)$$

This model leads to an extra force term in the geodesics for a perfect fluid. In three dimensions and in the Newtonian limit, assuming that the total acceleration $\vec{a}$ is collinear with the 3-force, $f$, and in the limit of very small gravitational accelerations, the Newtonian acceleration, $\vec{a}_N$, is $\vec{a}_N \approx \frac{a}{a_E}\vec{a}$, where $a_E^{-1} = (2f)^{-1}(1 - \frac{f^2}{a^2})$, which is remarkably similar to MOND's result. Furthermore, $a \approx \sqrt{a_EGM/r} = v^2/r$, hence a Tully-Fisher relation appears with a luminosity of the form $L \sim v^4$ with $v^4 = a_EGM$. Thus $a_E = 10^{-10} \text{ms}^{-2}$ as in MOND or $a_E =$

(8.5 ± 1.3) × 10⁻¹⁰ ms⁻² [368]. This model has been shown to be consistent with several observations ranging from the cosmic version of the virial theorem at Abell 586 cluster [369], to Jeans instability in Bok globules [370,371], or to SN distance data and the BAO data allowing for an attempt to solve the Hubble tension [372].

A finite sensitivity of weak gravitation in galaxy dynamics to background cosmology may be detectable below the de Sitter acceleration scale [373] using SPARC [374] and large galaxy surveys such as MaNGA [375]. Redshift dependence would point to a potentially unified picture, linking BTFR to JWST observations of ultra-high redshift galaxies at cosmic dawn [376,377]. This approach may also place novel observational constraints on cosmological parameters, notably $H_0$ and the deceleration parameter $q_0$ [375].

### 2.1.11.3. Observations and cosmic tensions.
Various studies have extensively explored the BTFR using data from large galaxy surveys. The Spitzer Photometry and Accurate Rotation Curves (SPARC), for example, includes measurements of 175 rotationally supported galaxies in the near-IR, minimizing the effect of star-halo degeneracy and providing precise baryonic mass and rotational velocity data [378,379]. Such robust datasets enable accurate fits to the BTFR and facilitate comparisons between observed galaxy properties and theoretical predictions. The BTFR for the SPARC sample aligns with a scale of $x = 4$. Research by Ref. [380] using the BTFR with a sample of 95 independent galaxies also challenges the Hubble constant value $H_0 < 70 \text{ km s}^{-1}\text{Mpc}^{-1}$ with 95% confidence level. A More recent study by Ref. [25] demonstrated that using the BTFR allowed them to determine a Hubble constant of $H_0 = 75.5 \pm 2.5 \text{ km s}^{-1}\text{Mpc}^{-1}$, which again challenges the standard cosmological model that suggests lower $H_0$ values [192]. Additionally, studies have shown variations in BTFR parameters across different distance bins, which could contribute to the understanding of the Hubble tension [381–386].

The RTF relationship, applied to a large sample of 843 local galaxies extending out to $z = 0.03$, has recently been used as a distance indicator to determine the maximum allowed variance of the $H(z)/H_0$ parameter [356]. They found that the maximum allowed local ‘radial’ variation is 1%, which is not enough to resolve the $H_0$ tension. In the near future, with the PROBES sample [387] (3000 objects) we will be able to use the RTF to test the hypothesis of a Giant Local Void at greater distances and investigate the anisotropy of the expansion of the Universe (see Ref. [356] for the present situation). Finally, the RTF can be applied at $z \sim 1-2$ to investigate a major part of the expansion history of the Universe [388].

The original TF relation is also useful for tracing the distance-redshift relation, provided that the relation is carefully calibrated. Unlike the BTFR, the TF relation is not universally linear, so studies typically introduce more complex models to account for the curvature and varying intrinsic scatter [333,389]. The Cosmicflows-4 (CF4) Tully-Fisher catalog [25] of ~10,000 spiral galaxies is currently the largest TF dataset, combining HI line widths with photometric magnitudes. There have been several published $H_0$ measurements using this dataset. For the full CF4 TF catalog, Ref. [25] derived $H_0 = 75.5 \pm 2.5 \text{ km s}^{-1}\text{Mpc}^{-1}$ with an estimated systematic error of $\pm 3 \text{ km s}^{-1}\text{Mpc}^{-1}$. Using an improved measurement methodology [389] and updated primary distance calibration [390], this was re-measured as $H_0 = 76.3 \pm 2.1 \text{ (stat)} \pm 1.5 \text{ (sys)} \text{ km s}^{-1}\text{Mpc}^{-1}$, where the statistical error reflects the relatively small number of primary calibrators in the sample (~50 objects with CPLR and/or TRGB distances). These are in agreement with other recent $H_0$ measurements using the TF relation as the final rung of the distance ladder, which consistently return $H_0 > 70 \text{ km s}^{-1}\text{Mpc}^{-1}$ (see Ref. [353] for a review). The next generation of Tully-Fisher datasets resulting from large surveys such as WALLABY [391], DESI [392], and FAST [393] will increase the precision of TF-derived measurements of the Hubble constant.
### 2.1.12. The Hubble tension in our Backyard: DESI and the nearness of the Coma cluster

**Coordinator:** Yukei Murakami

**Contributors:** Daniel Scolnic

The recent work by DESI [394] analyzed their first samples of early-type galaxies in the DESI peculiar velocity survey to study the fundamental plane (FP). FP is an empirical three-dimensional relation that links elliptical galaxies' radii, surface brightness, and the velocity dispersions [395,396]$^7$. This tight correlation serves as a standardization to measure distances to the galaxies over a large redshift range once the absolute distance scale is anchored to an independent, well-measured distance. In the DESI work [394], the Hubble Constant $H_0$ is measured with FP using an SBF measurement to the Coma cluster to calibrate the distance scale (i.e., distance ladder). Alternatively, recent study [397] showed that one can calibrate the FP in the early universe using Planck+ΛCDM inference of $H_0$ to measure the distance to the Coma cluster (inverse distance ladder). The inverse distance ladder with Planck+ΛCDM-calibration of FP places Coma at ~10% further than the direct measurement of distance by SBF. In addition, the new SNIa distance, as well as other independent distance measurements to Coma collected over the past few decades, are all consistent with the SBF, creating a tension against Planck+ΛCDM-prediction. This discrepancy provides a unique opportunity to shed light on the implications of the Hubble Tension in our local universe.

#### 2.1.12.1. DESI FP and inverse distance ladder.
DESI has provided an extensive FP measurement and a Hubble diagram over the redshift range of 0.01 < z < 0.1 using a sample of 4191 early-type galaxies in the Hubble flow and 226 galaxies in Coma [394]. The FP relation, which correlates a galaxy's velocity dispersion, surface brightness, and physical size, serves as a secondary distance indicator when calibrated with independent distance anchors. When the FP relation is calibrated using a near-infrared SBF distance of $D_{Coma} = 99.1 \pm 5.8$ Mpc [124], DESI derives a value of $H_0 = 76.05 \pm 1.3$ km s⁻¹ Mpc⁻¹. This result relates the Hubble constant and the Coma distance as $H_0 = (76.05 \pm 1.3) \times (99.1 \text{ Mpc} / D_{Coma})$ km s⁻¹ Mpc⁻¹. This relation can be used to obtain the distance to Coma predicted by Planck+ΛCDM, using the Planck-inferred value of the Hubble constant $H_0 = 67.4$ km s⁻¹ Mpc⁻¹. The implied distance to Coma shifts to $D_{Coma} = 111.8 \pm 1.8$ Mpc, a distance larger than any of the historical measurements that we discuss later.

#### 2.1.12.2. Latest Coma distance with ATLAS SNe Ia.
A recent analysis [397] compiled a sample of 10 spectroscopically confirmed SNIa within the Coma Cluster observed by the ATLAS survey [398] and YSE [399], as well as two additional SN previously included in the Pantheon+ dataset [32]. Applying the standardization and calibration methods established in the Pantheon+ analysis, the study derived a mean standardized peak brightness of $m_B^h = 15.712 \pm 0.041$ mag for SN in Coma cluster. Using the absolute magnitude calibration of SNIa from the HST Cepheids [34], this results in a distance modulus of $\mu = 34.97 \pm 0.05$ mag, corresponding to $D_{Coma} = 98.5 \pm 2.2$ Mpc. This measurement is in excellent agreement with the SBF distance used to calibrate DESI, with a significantly reduced uncertainty. The measured distance is at $4.6\sigma$ tension from the Planck-ΛCDM prediction.

#### 2.1.12.3. New & historical measurements in tension with Λcdm.
Historical measurements, as well as additional distance indicators and calibration methods further support a distance to Coma around 100 Mpc (see Fig. 23). A compilation of historical measurements summarized in [400], including I-band Tully-Fisher [319], K-band SBF [401], I-band SBF [402], $D_n - \sigma$ [403], FP [404] and Globular Cluster LF (GCLF) [405], produce an average of $D_{Coma} = 95.1 \pm 3.1$ Mpc. The HST

Key Project (KP) calibrated the FP relation using Cepheid variable stars, and its results, accounted for the recent updates, measure $D_{Coma} = 85 \pm 8$ Mpc [406]. A more recent JWST measurement of TRGB and SBF, allows to re-calibrate the FP relation by HST-KP, which places Coma at 90 ± 9 Mpc. When combined with the SBF distance used in the DESI work and the new SNIa distance, these independent approaches yield a consensus distance of 98.0 ± 2.0 Mpc.

The tension between local and Planck-calibrated distances to Coma reflects a broader challenge in reconciling the local and early-universe distance scales. If Coma were truly at $D_{Coma} > 110$ Mpc as ΛCDM-Planck calibration of DESI FP predicts, it would be inconsistent with decades of direct distance measurements based on a wide range of methodologies. The upcoming year 1 data release for DESI FP is expected to be significantly larger than the current study, and the targeted JWST observation of Coma galaxies (PI Jensen, GO 5989) is underway. When combined, these data will significantly reduce the statistical uncertainties in the FP-based $H_0$ or distance measurements, and the increased sky coverage in DESI data will allow FP to be calibrated with more than one galaxy clusters. Additionally, a few SNIa are being discovered in Coma every year, and the SNIa-based distance measurements are expected to be more robust in the next few years. Coma distance serves as an alternative or additional perspective on the issues surrounding the cosmological distance scales, and it showcases a new type of distance ladder analysis that is expected to develop rapidly in the coming years.

### 2.1.13. Cosmic chronometers

**Coordinator:** Michele Moresco

**Contributors:** Adrià Gómez-Valent, Anto Idicherian Lonappan, Arianna Favale, David Benisty, David Valls-Gabaud, Dinko Milakovic, Elena Tomasetti, Marek Biesiada, Rodrigo Sandoval-Orozco, Ruth Lazkoz, Sanjay Mandal, Swayamtrupta Panda, and Vavilova Iryna

CC have emerged in the last few years [182] as one of the most promising cosmological probes that can provide direct measurements of the expansion history of the Universe without the need to rely on any assumed cosmological model. The method [408], which is only based on the validity of the Cosmological Principle and on the assumption that General Relativity and standard physics hold in the environment of the galactic stars, proposes to derive the Hubble parameter $H(z)$ directly from the differential age evolution of the Universe ($dt$) in a redshift interval ($dz$), as:

$$H(z) = -\frac{1}{(1+z)} \frac{dz}{dt} \qquad (2.6)$$

To map the age evolution of the Universe, the best tracers that have been found are very massive ($\log_{10}(M/M_\odot) > 11$) and passively evolving galaxies since they have been proven to be extremely homogeneous systems, with synchronized formation, representing at each redshift the oldest population of galaxies [182]. Several authors have applied this method with different approaches to derive the differential age $dt$ [409–417]. It currently counts more than 30 $H(z)$ measurements in the range $0 < z < 2.1$, with accuracy from ~10%–15% at the higher redshifts down to ~5% at the lower redshifts (see Fig. 24). See Refs. [182,418] for extensive reviews on the topic.

Since it provides a measurement of the Hubble parameter without assuming any cosmological model, recently, CCs have been widely employed in discussing the Hubble tension. To fully take advantage of the potential of these data, several different cosmology-independent techniques have been explored, from GP [419–431], to non-parametric smoothing [432,433], Weighted Polynomial Regression method [422], Padé approximations [427,434], and ANN architectures [431]; all these analysis provide a reconstruction of $H(z)$ independent of any cosmological model that can, in turn, be extrapolated to $z = 0$ for a determination of the Hubble constant $H_0$. The other alternative is, instead, to assume a cosmological model and derive cosmological constraints from

<footnote>$^7$ See Section 2.1.11 for the Tully-Fisher relation, a late-type galaxy counterpart</footnote>
<figure><img src="image_28.png" /><figcaption>Fig. 23. Historical (1990 onward) distance modulus measurements of the Coma cluster (as reviewed in [406]). Only distance measurements that do not depend on redshift and $H_0$ are included. Figure taken from [397], and the rightmost point represents the SNIa distance measured in their work.</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>Fig. 24. Left plot. $H(z)$ measurements obtained with the CC method from the literature (white points) and from future surveys (green points, from Refs. [182,407]). Right plot. Cosmological constraints in the $\Omega_m H_0$ plane; the different colors show the 68% CL results (also reported in the left panel) obtained by fitting current data with a flat $\Lambda$CDM (in blue), open $\Lambda$CDM (in yellow), and flat $w$CDM (in red) cosmological model, and future CC data with a flat $\Lambda$CDM model (in gray). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

the parametric fit of the data [182,435,436]. Both these approaches, considering all the sources of statistical and systematic errors, give comparable results, yielding a constraint on the Hubble constant of $H_0 = 70.7 \pm 6.7 \text{ km s}^{-1}\text{Mpc}^{-1}$ [429]. Future measurements, however, leveraging on much larger statistics (like from the ESA mission Euclid [437] or from future spectrographs like WST [407]) and better modeling of the data, promise to significantly improve these constraints, improving the accuracy from ~8% to ~2%. An example is given in Fig. 24, where current and future CC data are fitted with different cosmological models (flat $\Lambda$CDM, open $\Lambda$CDM, and $w$CDM) to provide constraints in the $\Omega_M H_0$ plane (for more details, see Refs. [182,407]).

A difference in $H_0$ can be reinterpreted also as a tension in the calibrators of the cosmic distance ladders, i.e., the absolute magnitude $M$ of standard candles such as SNIa and the standard ruler represented by the comoving sound horizon at the baryon-drag epoch, $r_d$. Assuming CC as reliable CCs, it is also possible to use them to measure these distance calibrators independently from the CMB and the first rungs of the direct distance ladder. Thus, it is clear why CC plays an important role in the discussion. So far, the uncertainties for $M$ and $r_d$ derived by applying these calibrations are still too large to arbitrate the tension [424,429,438], but also this is bound to improve in the near future with the advent of upcoming surveys and data. Euclid [437], for instance, is expected to increase by 2 orders of magnitude the currently

available statistics [182]. CCs have been used to calibrate baryon acoustic oscillations and SNIa [422–424,429,439–441] and also other standardizable objects like ultra-compact radio QSOs [420], GRBs or QSOs [430,438,442,443] which can be exploited to extend the ladder beyond the SNIa redshifts, $z > 2$.

Beyond the $H_0$ tension, CCs are also the ideal data to test and compare with different cosmological models, from the simplest to the more exotic ones. Among the litmus tests of the $\Lambda$CDM model, the $Om(z)$ diagnostic and its two-point version $Om(z_1, z_2)$ have a distinguishing feature [444] that depends only upon the expansion rate $H(z)$. Hence, CCs have been used [445,446] to perform tests of scenarios assuming either a cosmological constant or an evolving DE equation of state.

### 2.1.14. Strong lensing and time delay measurements

**Coordinator:** Tommaso Treu

**Contributors:** Alba Domi, Alexander Bonilla Rivera, Ali Övgün, Anowar J. Shajib, Clecio Roque De bom, David Valls-Gabaud, Dominique Sluse, Eoin O Colgain, Frédéric Courbin, Jenny Wagner, LINDITA HAMOLLI, Predrag Jovanović, Sherry H. Suyu, Simon Birrer, Tanja Petrushevska, and Veronica Motta

Time variable sources that are multiply imaged — i.e., strongly lensed — provide an opportunity to measure absolute distances, and
therefore cosmological parameters including the Hubble constant $H_0$. The main advantages of this so-called time delay cosmography (TDC) with respect to other cosmological probes described in this white paper are: (i) it is a direct measurement of distances well into the Hubble flow; (ii) it is independent of all other types of distance measurements; (iii) it relies on well understood fundamental physics such as general relativity. In this section, we provide a brief summary of TDC, the current state of the art, and future prospects. The reader is referred to dedicated reviews for more extensive discussion, e.g., see Ref. [447-451].

The method was originally proposed for lensed SNIa by Refsdal in Ref. [452], well before the discovery of any actual strong gravitational lens. Since the mid-eighties it has been applied to lensed QSOs [447], and only recently — with the discovery of multiply imaged SN [453, 454] — it has been possible to fulfill Refsdal’s dream [455,456].

TDC measures angular diameter distances, typically between $z = 0.5-3$, where the majority of deflector galaxies and lensed sources lie. More precisely, it measures the so-called time delay distance $D_{\Delta t} = (1+z_d) \frac{D_d D_s}{D_{ds}}$, where D indicates angular diameter distance, and the subscripts s and d stand for source and deflector, respectively. In combination with stellar kinematics, it also measures $D_d$ [457].

TDC is thus primarily sensitive to $H_0$. It is, however, also sensitive to other cosmological parameters, such as curvature or the DE equation of state, in a complementary way to probes such as the CMB and SNIa, breaking some of the degeneracies of those probes [458-461].

Several representative measurements using lensed QSOs prior to 2022 are summarized in Fig. 25. Already in 2020, just 7 lenses yielded $H_0$ at 2% precision [462,463], under standard assumptions about the mass density profile of the deflector. That measurement is in agreement with, and completely independent, of late Universe probes. If those assumptions are relaxed, and a parameterization of the mass density profile that is maximally degenerate with the Hubble constant via the mass-sheet degeneracy [464,465] is considered, the uncertainty increases to 8% for the same 7 lenses [466]. In 2023, the first lensed supernova [455] yielded an $H_0$ measurement with ~7% precision. Whereas this precision is very encouraging for a single system, especially given the complexity of modeling a cluster of galaxies, see also Ref. [456], it is not sufficient to distinguish between the early and late-time measurements and resolve the “Hubble tension”. As more lensed SN are being discovered and analyzed, e.g., see Ref. [467-474], the precision is expected to improve rapidly, especially once the Roman and Rubin telescopes come on line [475].

In order to increase the precision on $H_0$ to the desired level of 1% under these weaker assumptions, three complementary approaches can be pursued [476]. The first is to increase the sample size. The Euclid, Roman, and Rubin Telescopes are expected to discover thousands of strong lenses, including lensed SN [477-480]. There is little doubt that large numbers of lenses will be discovered. The challenge will be to transform them into cosmological probes by obtaining the necessary additional data and computing the necessary cosmography grade models. Additional data include high-resolution images, time delays, deflector and source redshift, deflector kinematics, and supernova classification if relevant. Some of these data will be provided by the surveys themselves, e.g., high-resolution images by Roman and Euclid [475,481] and some time delays by Rubin [482], but others will require concerted follow-up effort. A coordinated follow-up for measuring precise time delays is critical. QSO time delays require high precision lightcurves minimally affected by microlensing [483,484], while SN require dedicated multi-color or spectroscopic follow-up [485,486]. The effort will be substantial, and it will require, e.g., repurposing a 4 m class telescope for time delays or building a dedicated satellite. The advent of ELTs will simplify matters, especially for high-resolution imaging and kinematics.

Modeling hundreds of lenses with precision sufficient for cosmography hundreds of lenses would be prohibitive at the moment, both in terms of human power and computing power. Fortunately, efforts are

<figure><img src="image_30.png" /><figcaption>**Fig. 25.** The measurements are categorized according to (i) assumptions on the mass distribution of the main deflector, “assertive” and “conservative” for simply parameterized models or “free form” for pixilated models; (ii) to the amount of information used per lens, “low info” utilizes QSO positions and time delays, “high info” adds the extended surface brightness distribution of the multiple images of the QSO host galaxy (or “Einstein Ring”), stellar kinematics of the main deflector, and number counts or WL to estimate the line-of-sight convergence. For each measurement, the reference and the number of time delay lenses are given. The measurements shown in red were performed with blinding to prevent experimenter bias.</figcaption></figure>

*Source:* The figure is taken from Ref. [449], see therein for details.

underway to develop automated pipelines [487-490], in some cases with the help of ML algorithms [491-493], to speed things up. The first results are promising, in the sense that models from independent automated pipelines are found to agree in their predicted time delays within the uncertainties, when the data contain sufficient information models that agree to within the uncertainties in predicted time delay can be found [494].

The second approach consists of increasing the precision per lens. In order to break the mass sheet degeneracy, non-lensing information is needed, for example, stellar kinematics of the deflector [461,495-497], the absolute luminosity of a lensed supernova Ia [498], or WL [499]. If the quality of the kinematics is sufficient, in principle, each lens can deliver distances to 2%–4% precision, limited by systematic errors in stellar velocity dispersion measurements [500].

The third approach consists of utilizing the much more abundant non-time delay lenses (a.k.a. “external samples”) to learn about the internal structure of time delay lenses. This method has been shown to increase the precision of TDC significantly [466]. The key to using external samples without introducing biases is to understand the selection function of both the time delay and non-time delay lenses sufficiently well. At the moment, this is not a limiting factor [501] - statistical errors are larger - but the selection function will have to be properly modeled and understood in order to achieve 1% precision and accuracy on $H_0$, sufficient to solve the Hubble tension.

As more lensed SN are being discovered and analyzed, the precision is expected to improve rapidly, especially once the Roman and Rubin telescopes come online [502].
## 2.1.15. Extended QSO cosmologies

**Coordinator:** Maria Giovanna Dainotti

**Contributors:** Aleksander Łukasz Lenart, Catarina Marques, Celia Escamilla-Rivera, Giada Bargiacchi, Giovanni Montani, Rodrigo Sandoval-Orozco, and Swayamtrupta Panda

QSOs are one of the most luminous non-transient energy sources, whose use at cosmological scales allows us to study the Universe at higher redshifts, e.g., up to $z \sim 7$ [503,504]. At this redshift range, it is possible to reveal some interesting aspects and deviations from different cosmological models that can be indistinguishable at low $z$. In this path, some examples of the QSO cosmological use are: (i) the reverberation mapping technique [505,506], and (ii) the relation between variability in X-ray amplitude and black hole mass scatter [507]. However, there is a lack of a clear definition of QSOs to order the diversity of AGN objects [508], although recent studies have begun to streamline the various QSO categories [509,510] with particular emphasis on sources accreting at or above the Eddington limit [511]. There is a need to homogenize the modeling of broadband QSO spectra and examine and assess the evolution of QSO properties (Black Hole mass, Eddington ratio, viewing angle to the source) over a broad range of redshifts [512, 513]. Under such techniques, several QSO catalogs have been used to find cosmological constraints under certain theoretical assumptions, e.g., consistencies or inconsistencies between time delays from reverberation mapping of prominent emission lines extending beyond the cosmic noon [514] and impact of the inherent heterogeneity in the data set on cosmological parameter constraints [515], constraints using GRBs with QSO cosmology analysis for future landscapes [516].

Furthermore, studies including QSO catalogs combined with GRB observations achieve small uncertainties on cosmological parameters [517,518]. Moreover, combining the Risaliti-Lusso relation for QSO with SNIa can allow us to obtain corrections on the redshift evolution as a function of cosmology [519]. Recent studies have been going through the Risaliti-Lusso relation, which has been validated in Ref. [513], where $\Omega_{m,0}$ value seems larger when only it is considered QSO baselines [519–523], and some systematics in the QSOs measurements should be expected in future new cosmological probes [524]. Despite the systematics, this relation is a promising cosmological probe since its observed dispersion on average is ~0.20–0.25 dex, and becomes significantly smaller when only the sources with the highest quality X-ray observations are considered [525], suggesting that intrinsically the X-ray to UV relation has a scatter lower than 0.1 dex, pointing to a tight, universal physical process regulating the energy flow between the accretion disk and the X-ray emitting region in QSOs. In this scheme, several studies related to extended theories of gravity have been performed and cosmologically constrained using QSO baselines, e.g., $f(T)$ cosmologies have been constrained through cosmography by considering non-flat and flat geometries [526] with GRB observables, using QSO objects [527] detected through high-quality UV and X-ray fluxes up to $z \sim 5.1$ [528]. In view of possible discrepancies among measurements of the matter critical parameter via sources with different $z$-ranges, it has been proposed a promising model in which $\Omega_{0,m}$ becomes a dynamical quantity. The proposed scenario relies on a metric $f(R)$-gravity in the so-called Jordan frame and dominant linear contribution is retained in the potential term of the non-minimally coupled scalar field (a small deviation is included too). It is such a dominant contribution that is responsible for a non-standard critical matter parameter [529]. From the cosmographic point of view, other analyses have highlighted a significant tension between the standard cosmological model and the best-fit cosmographic model, when using QSOs, SNIa, and GRBs [528], QSOs and SNIa [530], and QSOs combined with GRBs, BAO, and SNIa [531]. Also, some studies use QSOs as standard rulers [527] by their angular size–luminosity using very-long-baseline interferometry [532]. In addition, the joint

analysis of spectroastrometry and reverberation mapping can measure AGN distances and provide a new way to measure $H_0$ [533]. Furthermore, recent results have considered QSO physics from UV, X-ray, and optical plane techniques behind the local observations as cosmological probes to study the $H_0$ tension [534]. Another application of QSOs in cosmology is the Sandage test of the cosmological redshift drift, i.e., a small dynamic change in the redshift of objects following the Hubble flow [535], which requires not only high instrumental resolution but also very bright targets. One of the main objectives of the QUBRICS (QUasars as BRight beacons for Cosmology in the Southern Hemisphere) survey [536] is to provide a sample of bright targets for the Sandage test. The two brightest QSOs in this Golden Sample have started to be analyzed in the pilot program “An ESPRESSO Redshift Drift Experiment”, which will be a complete end-to-end proof of concept for this experiment at ANDES in ELT, where the redshift drift signal is expected to be detected [537]. Although the Sandage test is a direct and real-time model-independent mapping of the expansion rate of the Universe, recent theoretical studies such as Refs. [538,539] show synergies with other cosmological probes, in particular regarding the characterization of the physical properties of DE.

Regarding the efforts to reach a higher precision in the determination of cosmological parameters, also a purely statistical approach has been employed. This is based on the fact that the normalized residuals of QSO logarithmic luminosities of the Risaliti-Lusso relation are not normally distributed. Hence, the Gaussian is not the most appropriate distribution to be used as cosmological likelihood. Indeed, Ref. [517] prove that the QSO have the logistic distribution as a best-fit. This is shown in the right upper panel of Fig. 26 along with the same investigation for GRBs (left upper panel) and SNIa (lower panels). Remarkably, the employment of this proper likelihood for QSOs, compared to the standard Gaussian one, reduces the uncertainty on $H_0$ up to 35%, on $\Omega_{m,0}$ up to 27%, on $\Omega_k$ up to 32%, and on $w$ up to 31%, when non-flat and $w$CDM models are investigated by combining QSOs with GRBs, BAOs, and SNIa, where every single probe is fitted with its own best-fit cosmological likelihood [518]. Another approach aiming at inferring cosmological parameters with lower uncertainties consists of identifying a QSO sub-sample composed only of sources that better follow the Risaliti-Lusso relation, i.e., are closer to the best-fit relation. Indeed, this method allows us to reduce the intrinsic dispersion of the correlation, thus yielding a better precision of the cosmological parameters. Different techniques have been applied to select such a sub-sample [529,540,541] which have promoted QSOs as standalone cosmological probes, even with the same precision as SNIa.

## 2.1.16. GRBs as cosmological standard candles

**Coordinator:** Maria Giovanna Dainotti

**Contributors:** Aleksander Łukasz Lenart, Arianna Favale, Denitsa Staicova, Giada Bargiacchi, Hassan Abdalla, and Radosław Wojtak

Long-duration GRBs (LGRBs) could be used as standard candles, extending the capabilities of the Hubble diagram to measure distances further than currently possible with SNIa, thereby helping to constrain cosmological parameters. Phenomenological relations derived from spectral modeling, e.g., the Amati relation [542–544] can be used for this purpose. This relation connects the source-frame energy ($E_{i,p}$), where the gamma-ray spectral energy peaks, with the isotropic-equivalent bolometric energy ($E_{iso}$) released during the prompt phase [545]. Another significant empirical correlation in the prompt emission is the Yonetoku relation [546] between $E_{peak}$ with the intrinsic peak luminosity $L_{iso}$. This correlation has been validated for both LGRBs and short-duration GRBs (SGRBs) [546,547]. Given the large variability in the feature of the prompt emission, a more promising approach is obtained with the use of correlations involving the plateau emission [548], a relatively flat segment in the light curve (LC). The Dainotti relation relates the rest-frame time at the end of the plateau ($T_a^*$) and its corresponding luminosity ($L_a$). This relation succeeds in
<figure><img src="image_31.png" /><figcaption>**Fig. 26.** Normalized residuals $\Delta\mu$ histogram for GRBs (left panel), QSOs (middle panel), and SNIa from Pantheon + samples (right panel). The red curve is the best-fit distribution, while the orange one is the Gaussian distribution. In the case of GRBs, the two coincide since the Gaussian is the best-fit distribution.</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>**Fig. 27.** The distance luminosity vs redshift when a calibration with CC is considered. Source: The figure has been taken by Ref. [430].</figcaption></figure>

standardizing GRBs by utilizing the shape of their LCs. Indeed, around 500 GRBs observed by Swift exhibit a distinct, plateau phase in X-rays. The great advantage of this relation, compared to the ones in the prompt emission, is that the plateau phase can be attributed more straightforwardly to various theoretical models, including magnetar spin-down [549–552] and forward/reverse shock mechanisms [553–555].

This correlation was later updated in Refs. [556–558], and subsequently extended to a 3D correlation by incorporating the peak luminosity ($L_{\text{peak}}$) [556,559–561], the so-called 3D Dainotti relation. [558, 559] demonstrated that this correlation is devoid of biases and it can be corrected for the redshift evolution.

Further studies by Refs. [560,562] confirmed the robustness of this correlation for high-quality LGRBs, which show well-defined plateau properties that obey the fundamental plane relation and constitute the so-called Gold sample. An improvement of this sample has been classified as the Platinum sample [563]. As a result, this correlation has been successfully applied as a cosmological tool [518,564–567], providing insights into the high-redshift Universe. GRBs, being at high-z, could be one of the important tools to differentiate between competing cosmological models. Indeed, Ref. [517] pinpointed traces of rising tension between flat $\Lambda$CDM and the equivalent model with $\Omega_k \neq 0$. There is a trend that reveals $\Omega_k < 0$, marking the importance of further studies in this direction at high-z.

The future cosmological role of GRBs has also been investigated in Ref. [566] where it has been determined how many GRBs are needed as stand-alone probes to achieve a comparable precision on $\Omega_{m,0}$ to the one obtained by SNIa only. They obtained the same error measurements derived using SNIa in 2011 and 2014 with 142 and 284 simulated optical GRBs, respectively, considering the error bars on the variables halved. These error limits will be reached in 2038 and in 2047, respectively. Using a doubled sample, obtained by the current ML approaches [568,569] allowing a LC reconstruction [570] and the estimates of GRB redshifts, with error bars halved, the same precision as SNIa in 2011 and 2014, is reached now and in 2026, respectively.

If we need to reach the current precision of SNIa, it would require 18 years from now, but this estimate is very conservative since it does not consider the new redshifts from Euclid and JWST.

Indeed, in light of the existing cosmological tensions, such as the one on the Hubble constant, $H_0$, it has become fundamental to obtain unbiased cosmological distances via intermediate redshift probes. Thus, the calibration procedure of the GRB correlations must not be subject to strong model-dependent assumptions. Works have shown that this can be done by employing low-z probes, such as SNIa from the Pantheon Plus sample, a collection of 1701 SNIa, discussed already in the previous section, in combination with advanced statistical techniques such as GP [571] or cosmographical analyses [572,573]. On the other hand, data on the $H(z)$ obtained from CC can play the role of calibrating the ladders [429] (see Fig. 27) since these data only rely on General Relativity and the validity of the standard physics in the environment of the galaxy stars. Ref. [430] found that current data on CC can identify a subset of GRBs in the Platinum sample in $0.553 \le z \le 1.96$ that reveals a tight fundamental plane relation, with one of the lowest intrinsic scatter observed to date, $\sigma_{\text{int}} = 0.20_{-0.05}^{+0.03}$. This result is obtained considering a relation corrected for evolutionary effects and with parameters compatible with the magnetar model [550–552,574]. The GRB sub-sample pinpointed by this model-independent calibration can represent a valuable set of standardizable candles that is used to extend the cosmic distance ladder by providing unbiased luminosity distances up to $z = 5$. For QSO, calibrating with SNIa, varying only $H_0$ in the cases without and with correction for evolution, $H_0 = 73.76 \pm 2.18 \text{ km s}^{-1}\text{Mpc}^{-1}$ and $H_0 = 69.82 \pm 2.27 \text{ km s}^{-1}\text{Mpc}^{-1}$, respectively [519]. For GRBs, using the Gaussian priors, varying only $H_0$ in the cases without and with correction for evolution, $H_0 = 73.23 \pm 3.31 \text{ km s}^{-1}\text{Mpc}^{-1}$ and $H_0 = 72.87 \pm 2.92 \text{ km s}^{-1}\text{Mpc}^{-1}$, respectively [567].

Additionally, the 3D X-ray Dainotti relation has also been recently employed in cosmographic analyses to investigate the reliability of the standard cosmological model in a cosmology-independent way. In this concern, as mentioned in the previous section, GRBs combined with SNIa, QSOs, and BAO point toward a statistically significant discrepancy between the prediction of the concordance model and the observational data Ref. [531]. From another point of view, [575] constrained the transition epoch between a matter-dominated and a DE-dominated universe by using two cosmographic approaches and the combination of GRBs, SNIa, and BAO obtaining results compatible with the concordance model. Concerning instead the statistical analyses already mentioned for QSOs, the residuals of the distance moduli of GRBs remarkably prove to be effectively Gaussian, as reported in the upper left panel of Fig. 26, differently from the cases of QSOs and SNIa.

Given the importance of the 3D Dainotti relation, it is crucial to explore the existence of the same correlations but with different associated classes (not only the Platinum sample). The LGRBs associated with Supernovae Type Ib/c (GRB-SN) are interesting since they have been discussed in the literature as possible standard candles [563,576]. Ref. [577] investigate the existence of probable correlations among SN parameters and GRB prompt and afterglow features considering the largest compilation of GRB-v possible associations observed from 1997
up to 2021 and find a possible correlation between the GRB optical luminosity at the end of the plateau (log₁₀ Lₐ,ₐ₉₄) and the log₁₀ of SN rest-frame peak time (log₁₀ t*ₚ). The correlation can be expressed as log₁₀ Lₐ,ₐ₉₄ = (9.43 ± 1.47) log₁₀ t*ₚ - (13.60 ± 11.89). The uncertainties on the fitting parameters (16%-87%) are too high to allow using this correlation for standardizing GRBs, thus new observations are needed to validate it. If confirmed, then it will represent a crucial support for cosmological analysis.

Another interesting application of GRB physics in cosmology comes from the search for quantum gravity models. Some such models predict an energy-dependent speed of light, which can be observed as simultaneously emitted high and low-energy photons arriving at different times. Such a tiny quantum effect is expected to be amplified by very high energies of the photons and cosmological distances, making GRBs a promising probe. In Refs. [578,579], the authors explored the use of different GRB time-delay datasets in combination with the Pantheon/Pantheon+ SNIa datasets, BAOs, and the CMB distance priors under different approximations for the intrinsic lag to constrain cosmological parameters and investigate the impact on the H₀ tension. The analysis revealed that the inclusion of such time-delay datasets still leads to a deviation in the parameter c/H₀r₉, where c is the speed of light and r₉ is the comoving sound horizon at the drag epoch. Such deviation can be interpreted as an artifact of the H₀ tension on the remaining quantities, demonstrating again that the tension is spread to the whole H₀ - r₉ = Ωₘ,₀ plane. Such a dataset is not directly affected by the Eₚ - Eᵢₐₛₚ correlation, but relies critically on the assumption of Lorentz Invariance Violation (LIV), which so far has not been detected. In the case where LIV is not zero, GRB time delays would be a new independent cosmological probe.

In addition, the constraints on the cosmological parameters can be obtained with the interaction of the gamma rays with their surrounding medium or other radiations. Gamma rays with energies above around 10 GeV are attenuated via interactions with the extragalactic background light (EBL) photons, resulting in electron-positron pair production. The attenuation effect was systematically measured in spectra of blazars and GRBs, using a wide range of instruments and techniques, e.g., see Ref. [580,581]. Since the attenuation scales with the comoving distance, these measurements can be used to constrain cosmological parameters, in particular H₀, given a model of the EBL based on integrating light from galaxies in deep cosmological surveys [582]. Recent joint modeling of the EBL and γ-ray attenuation yields H₀ = 65.1±6.0 km s⁻¹Mpc⁻¹, and H₀ = 66.5±2.2 km s⁻¹Mpc⁻¹ when combined with the BAO observations and the BBN prior [583]. The H₀ measurements are independent of any external distance calibration, and the best-fit values are in close agreement with the CMB-based Planck value [192].

### 2.1.17. Gravitational wave constraints

**Coordinator:** Antonella Palmese

**Contributors:** Bangalore S. Sathyaprakash, Ivan de Martino, Matteo Tagliazucchi, Nicola Borghi, and Nicola Tamanini

The field of GW astronomy has recently facilitated novel measurements of the Universe's expansion rate by exploiting unique properties of compact binary mergers. By providing a distinct perspective on the Universe, in contrast to traditional astronomical and cosmological probes (such as electromagnetic radiation, cosmic rays, and neutrons), GWs offer a complementary approach to addressing cosmological tensions.

Since 2015, a ground-based GW detector network, including the Laser Interferometer Gravitational Wave Observatory (LIGO) [584] and later Virgo [585] and KAGRA [586], has been detecting coalescences of compact object binaries, which so far include binary neutron star (BNS) [587], neutron star-black hole (NSBH) [588], and binary black hole (BBH) [589] mergers. It is anticipated that LIGO-Virgo-KAGRA (LVK) will complete their fourth and fifth observing runs (O4 and

O5) by the early 2030s [590], after which a significantly upgraded detector network, called A# [591,592] is expected to become operational. This will be followed by the deployment of next-generation (XG) observatories, such as the Cosmic Explorer [593–595] and Einstein Telescope (ET) [596]. Additionally, the Laser Interferometer Space Antenna (LISA) [597] is scheduled for launch in the next decade and will be sensitive to GWs from the inspiral and merger of binaries within the 0.1 mHz to 1 Hz frequency range, including massive BBHs and extreme mass ratio inspirals (EMRIs). LISA and XG observatories will observe mergers throughout the cosmos from an epoch before the first stars formed.

GW events can be used as “standard sirens” (StS; [598,599]), as they can act as absolute distance indicators. The GW signal of a compact binary merger is directly sensitive to the luminosity distance of the source and can therefore be combined with redshift information to act as a probe of the expansion of the Universe. Depending on where the redshift information is derived from, StSs are typically divided into different classes, with bright and dark StSs being the two main classes. In what follows, we briefly describe each method and the state of the field, including the latest measurements and systematics studies, and end with prospects for future measurements, focusing on H₀ constraints as a pathway to shed light on the Hubble tension.

#### 2.1.17.1. Bright standard sirens.
For bright StSs, an electromagnetic (EM) counterpart is identified, and the redshift is derived from its host galaxy, assuming that both are unique. So far, there exists only one EM counterpart that is confidently associated with a GW event, GW170817 [600]. This association has been used to derive the first StS measurement [601], finding $H_0 = 70_{-8}^{+12} \text{ km s}^{-1} \text{Mpc}^{-1}$, $a \sim 14\%$ precision measurement. Subsequent analyses have taken a more in-depth approach to estimate the peculiar velocity of the host galaxy [602–604]. Moreover, various works attempted to take advantage of the EM observations to constrain the viewing angle of the binary and break the distance–inclination angle degeneracy which exists when estimating these parameters from the GW data [605–608]. The latest of these measurements reached a $\sim 7\%$ precision and they are all consistent with both early and late-time measurements of $H_0$.

It is worth noting some of the major systematics that may affect future bright StS measurements. A StS analysis only using the GW data and the host galaxy spectroscopic redshift of a high-confidence EM counterpart provides an $H_0$ measurement which is not expected to first order to be affected by major systematic uncertainties, assuming that the peculiar velocities measurements are well constrained (also considering that their uncertainties will become less important as we move toward larger GW detector distance horizons [609]) and detector calibration errors are kept under control ($\lesssim 2\%$ [610]). Although GW selection effects are well understood and can be more easily modeled, selection effects from EM searches also need to be taken into account and may be coupled to binary parameters such as source luminosity distance and viewing angle [611]. Beyond selection effects, if including EM estimates of the viewing angle, $H_0$ measurements may also be affected by modeling uncertainties of the jet structure [612], kilonova geometry [613], or even a possible jet misalignment with the binary angular momentum [614].

Future bright StSs are most likely expected to arise from binary neutron star (BNS) and neutron star black hole (NSBH; which would prove valuable StSs [615]) mergers, but candidate EM counterparts to BBH mergers also exist [616–618]. Although the BBH EM counterparts association is currently uncertain [619,620], they may in the future be used to infer the Hubble constant along with other cosmological parameters [621–625].

#### 2.1.17.2. Dark standard sirens.
Most mergers detected by the LVK (currently in the order of $\sim 200$ when including significant detection candidates from the ongoing observing run) do not have an associated EM counterpart and are mostly comprised of BBH mergers. In that case,
the redshift information necessary for an StS analysis may be taken
from galaxy catalogs (e.g., see Refs. [626,627]), from the redshifted
mass measured with the GW data [628,629], or a combination of the
two [630–632].

In the first case, the redshift distribution of galaxies along the line of sight of a GW event's localization can be used to infer cosmological parameters, including $H_0$, from the distance–redshift relation. Various works have produced such measurements so far [633–641], in general finding a precision on $H_0$ down to ~ 20% following the third LVK observing run, again in agreement with both Planck and SH0ES constraints. Even when combined with the bright StS available, the precision reaches ~ 10% (when ignoring EM viewing angle constraints) and does not yet allow us to distinguish between the two leading $H_0$ measurements.

Golden dark sirens are a subclass of dark sirens when the cata-
log contains, or follow-up observations reveal, a single galaxy in the
3D localization volume. Although rare, they may provide exquisite
few percent-level measurements of $H_0$ [642]. It is also possible to
cross-correlate GW events with galaxy catalogs to infer cosmological
parameters (e.g., see Ref. [643]). In all cases, the GW localization
precision and the availability of extended spectroscopic galaxy cata-
logs [632] will be crucial to establish dark StS as a cosmological probe
competitive with bright StSs.

In the second case, the source redshift is reconstructed at the
statistical level by taking advantage of the presence of features in
the distributions of the BBH population properties, such as the mass
gap that may be explained by the theory of pair-instability super-
nova [644,645], or the overdensity peaks in the mass distribution
observed in LVK data [646,647]. This “spectral sirens” approach has
allowed constraints on $H_0$ at the ~ 60% level [636] and on modified GW
propagation [648,649]. These constraints are expected to significantly
improve with XG detectors, as all BBH events can be used as spectral
sirens. However, this poses a computational challenge, since the time
required for a full cosmological and population analysis scales linearly
with the number of events.

The major sources of systematics for dark sirens methods are ex-
pected from incorrect assumptions on the mass function [636]; while
inferring cosmological parameters in conjunction may mitigate this
effect [630], it is still crucial to correctly model the mass function in-
cluding its potential evolution [629]. Other systematics may arise when
using photometric redshifts [650,651], and host galaxy weighting or
galaxy assumptions that do not match the true underlying distribution
of merger hosts [652,653].

2.1.17.3. *Prospects for ground-based detectors.* Fig. 28 in Ref. [654] summarizes the prospect of terrestrial GW observatories in resolving the Hubble tension. The x-axis lists GW detector networks that are expected to become operational or newly built over the next decade or more. For each network, the y-axis shows the precision with which the Hubble constant would be measured, for four classes of binary merger events: BBH, NSBH and BNS as golden dark sirens, as well as BNS mergers with associated EM counterpart. For each network the figure also shows the number of events expected to be detected in each source class. A target precision of 2% could be achieved by the HLV network with dark sirens but is not guaranteed since the number of events expected is $O(1)$. If the current median BNS merger rate holds, then the HLV network might observe ~ 130 bright sirens and accomplish the 2% target. Assuming one ET detector and its synergy with Transient High Energy Sources and Early Universe Surveyor (THE-SEUS) [655] for bright StSs, one may achieve accuracy on $H_0$ of only 0.40 km s⁻¹Mpc⁻¹ in the case of a non-flat ΛCDM model after five years of observations (which corresponds to ~ 166 bright StSs detections with redshift below $z \sim 4.3$) [656]. ET alone, or a network of only two CE detectors, will not be able to accomplish the 2% target with dark sirens, although a network of two CE detectors could do so with bright sirens. A network consisting of an ET and at least one CE will determine $H_0$ to

sub-percent precision with dark sirens of all source classes, as well as
with hundreds of bright sirens. In the case of DE models, the accuracy
on the Hubble constant may achieve 1% after 10 years of observations
due to the correlations between $H_0$ and the DE parameters of the
specific model under consideration [657]. It is also worth noting that
bright sirens with XG will reach a percent level precision in distance in
the local Universe, thus enabling precision measurements of $\sigma_8$ [658]
which will inform our understanding of the $S_8$ tension, should it persist.
With XG observatories we will enter the era of precision cosmology
with GW observations.

2.1.17.4. **LISA perspective.** The LISA [659] will push GW cosmology at high redshift ($z \ge 3$), mapping the expansion of the Universe in a still poorly charted cosmic epoch [660]. LISA will observe both bright sirens, in the form of massive black hole binary mergers with an identified EM counterpart [661,662], and dark sirens, in the form of extreme mass ratio inspirals (EMRIs) [663]. Independently these two populations of standard sirens are expected to deliver percent constraints on the Hubble constant [664–669]. Combined together, however, they should provide a one-percent, or even better, constraint on $H_0$ [670], making LISA a rightful competitor to solve the Hubble tension. The large redshift of LISA standard sirens will furthermore provide original tests of ΛCDM and of general relativity, delivering new potential insights on the nature of DE [665,671–675].

2.1.18. *The cosmic microwave background radiation*

Coordinator: Margherita Lembo and Martina Gerbino

Contributors: Anto Idicherian Lonappan, Chandra Shekhar Saraf, En-
rico Specogna, Luca Caloni, Özgür Akarsu, Nils A. Nilsson, Pawel-
Bielewicz, Simone Paradiso, and Venus Keus

2.1.18.1. *Review of CMB basics and recent observations.* The CMB plays a crucial role in shaping our understanding of modern physics and remains a powerful tool for advancing knowledge in cosmology and particle physics. The CMB dates back to roughly 380000 years after the Big Bang, at a redshift of $z_* \simeq 1100$, when CMB photons decoupled from electrons, close to the moment in the thermal history of the Universe when electrons and protons combined into neutral hydrogen atoms. Since then, CMB photons free stream with a nearby blackbody distribution with a temperature today of approximately $T \simeq 2.7$ K [676]. The CMB spectrum peaks around 100 GHz, within the microwave frequency range of the electromagnetic spectrum. Primordial density perturbations at the last scattering surface gave rise to temperature anisotropies in the CMB field of the order of $\Delta T/T \sim 10^{-5}$ [677]. The CMB radiation, because of its quadrupolar anisotropy, is indeed also partially linearly polarized due to Thompson scattering, with an amplitude of ~ 10% of temperature anisotropies.

The polarization pattern is usually decomposed into so-called (curl-
free) E-modes and (divergence-free) B-modes. On small angular scales,
both the temperature and the polarization fields undergo a tiny dis-
tortion when they pass close to large distributions of matter. The
weak gravitational lensing effect (CMB lensing) is one of the most
important mechanisms that can generate secondary anisotropies in the
CMB. In particular, it smears out anisotropies at small angular scales in
temperature and E-mode polarization, and it gives rise to an additional
B-mode contribution sourced by the lensed E-mode pattern.

The first full sky mapping of the CMB anisotropies was made by
the COBE satellite [678], refined by WMAP [679] and finally by
Planck [192], that delivered the state-of-the-art for full-sky CMB mea-
surements both in temperature and polarization. At smaller (arcmin)
angular scales in temperature and E-mode polarization, observations
have been dominated in sensitivity by the ground-based experiments
ACT and SPT; the search for B modes has been led so far by ground-
based telescopes observing degree angular scales, i.e., BICEP/KECK,
POLARBEAR/Simons Array.
<figure><img src="image_33.png" /><figcaption>**Fig. 28.** Precision with which $H_0$ would be constrained by standard sirens with the HLV network upgraded to A# sensitivity and networks containing one or more XG observatories (see Section 2.1.17.3 text for details).</figcaption></figure>

Source: Figure taken from Ref. [654].

Next-generation CMB experiments, both space-borne (e.g., LITE-BIRD [680]) and ground-based (e.g., Simons Observatory [681] and CMB-S4 [682]), aim to achieve precise measurements of CMB polarization. A clear science goal is to detect the imprint on CMB pattern of primordial GWs, which are the smoking gun of an inflationary phase in the early Universe [680,683,684]. Moreover, improved measurements of CMB polarization would provide insights on potential new physics beyond the standard model [685–693]. CMB lensing is one of the most relevant observable of near-future CMB experiments, as it provides an invaluable tool to reconstruct the integrated distribution of matter in the Universe and to place stringent constraints on physical properties and effects mostly related to the late-time phases of the Universe, such as the growth of structures and neutrino masses (see e.g., Planck [694], ACT [695], SPT [696]).

### 2.1.18.2. CMB-driven constraints in the context of cosmological tensions.
CMB observations allowed us to test the predictions of the standard cosmological model and to shape our knowledge of the Universe, its history, and composition. Despite its phenomenological success, the ΛCDM model is still incomplete in that, for example, fails to address the fundamental nature of the most abundant dark components. In addition, it has been facing challenges due to statistical tensions between results obtained with recent cosmological and astrophysical surveys of increased accuracy. Most notably, there is a severe (> 5σ) discrepancy in the value of the Hubble constant $H_0$ estimated with CMB observations with respect to the result obtained with local distance ladder measurements [34,192]. Furthermore, CMB and LSS measurements show a less severe albeit still intriguing disagreement on the amplitude of matter perturbations quantified via the $\sigma_8$ parameter [192, 697–699]. These discrepancies and their implications will be explored in more detail in the following subsections from the perspective of CMB observations.

### 2.1.18.3. Constraints on $H_0$ from CMB measurements.
The Hubble tension is one of the major unresolved issues in modern cosmology. This tension refers to the discrepancy between local measurements of the Hubble constant, $H_0$, and the value inferred from early Universe observations assuming the ΛCDM model. Early Universe estimates of $H_0$ are driven by CMB observations made by the Planck satellite, which yields the value $H_0 = 67.27 \pm 0.60 \text{ km s}^{-1}\text{Mpc}^{-1}$ at 68% CL, using a

combination of TT,TE,EE+ lowE data [192] (see also Refs. [700,701] for re-analyses of Planck data, with no significant deviations of $H_0$ from the Planck result). On the other hand, the most sensitive local measurement of $H_0$ is obtained by the SH0ES collaboration, which finds $H_0 = 73.04 \pm 1.04 \text{ km s}^{-1}\text{Mpc}^{-1}$ exploiting type SNIa calibrated with Cepheids [34]. These two values are in 5σ tension.

Alternative CMB datasets, such as WMAP [679], ACT [702,703], and SPT [704], or CMB-independent probes, such as combinations of BBN with BAO measurements [705], yield $H_0$ values that align with those obtained from Planck. The analysis of ACT-DR6 data [703] yields a 68% CL value of $H_0 = 66.11 \pm 0.79 \text{ km s}^{-1}\text{Mpc}^{-1}$ (ACT TT-TEEE+Planck+Sroll2 EE at large scales to constrain tau) consistent with measurements obtained by Planck satellite and lower than local measurements. Similarly, the combination of ACT-DR6 and WMAP data results in a value of $H_0 = 66.78 \pm 0.68 \text{ km s}^{-1}\text{Mpc}^{-1}$ at 68% CL [703,706]. Additionally, the combination of ACT and WMAP data results in a value of $H_0 = 67.6 \pm 1.1 \text{ km s}^{-1}\text{Mpc}^{-1}$ at 68% CL [702]. The same collaboration also reported an estimate of $H_0 = 68.3 \pm 1.1 \text{ km s}^{-1}\text{Mpc}^{-1}$ from measurements of CMB lensing in combination with BAO data (6dF and SDSS) [695]. The SPT collaboration finds a compatible result, with a value of $H_0 = 68.3 \pm 1.5 \text{ km s}^{-1}\text{Mpc}^{-1}$ at 68% CL [704], improved to $66.81 \pm 0.81 \text{ km s}^{-1}\text{Mpc}^{-1}$ with the use of the latest unlensed EE and CMB lensing data in combination with a prior on the optical depth $\tau$ [707]. It is relevant to emphasize that all the results above are independent of Planck data. On the other hand, late-time measurements different from SH0ES, including additional methods for calibrating SNIa at large distances (see Section 2.1.7) or observations independent of SNIa (see e.g., Section 2.1.9, Section 2.1.13, Section 2.1.10, Section 2.1.17), are all suggesting higher values of the Hubble parameter.

Depending on the combination of measurements, the tension ranges roughly from 4σ to 6σ. Fig. 29 shows a comprehensive list of CMB measurements of the Hubble parameter. It is important to emphasize that these results were derived under the assumption of the ΛCDM model. For example, as shown in Ref. [708], the uncertainty on $H_0$ increases when considering extensions to the ΛCDM framework.

Assuming a different cosmological model may lead to different estimates of $H_0$, either in the mean value or in the width (i.e., in the errorbar) of the PDF (or both) [3,709,710].
<figure><img src="image_34.png" /><figcaption>Fig. 29. Whisker plot with 68% CL constraints on the Hubble constant $H_0$ through different indirect measurements performed over the years. The constraints shown in this plot have been derived assuming $\Lambda$CDM model. If not specified, “BAO” refers to a combination of BAO data prior to DESI (see the corresponding paper for more details). The green vertical band corresponds to the $H_0$ value from SH0ES team [34], while the red one corresponds to the value from Planck 2018 [192].
Source: This figure has been adapted from Fig. 1 of Ref. [709].</figcaption></figure>

Since the value of $H_0$ inferred from CMB observations depends on the underlying cosmological model, this discrepancy with the local measurements of the Hubble parameter might be an indication of physics beyond the $\Lambda$CDM model. An alternative possibility, which will be discussed in Section 2.1.18.5, is that this tension is due to unknown systematics that affect either early-time or late-time observations (or both). However, given the vast array of independent probes of $H_0$, it seems quite unlikely that systematic effects may be the only responsible for this discrepancy. This further motivates the search for an explanation in terms of new physics. The landscape of theoretical scenarios to address the $H_0$ tension is broadly discussed in Section 4. Here, we just recall that the easiest extensions to $\Lambda$CDM are unable to solve the tension. More exotic scenarios, like those involving new BSM interactions between the components of the stress–energy tensor, the existence of new components or variation of fundamental constants, usually do not properly alleviate the tension, but rather yield estimates of $H_0$ with a larger uncertainty than the $\Lambda$CDM-based inference. Some models that gained visibility in the past, like those with strongly self-interacting neutrinos, require new physics that is at odds with complementary constraints from laboratory probes [711]. The search for new physics is still ongoing.

In order to understand the possible physical origin of this tension, we need first to discuss how CMB observations constrain $H_0$ (see Refs. [712,713]). The key quantity from which the Hubble constant is inferred is the angular scale of the sound horizon at recombination, which we assume to happen instantaneously at redshift $z_* \simeq 1100$. This is defined as

$$ \theta_s \equiv \frac{r_s(z_*)}{D_A(z_*)}, \qquad (2.7) $$

where $r_s(z_*)$ and $D_A(z_*)$ are the comoving sound horizon at recombination and the comoving angular diameter distance to recombination, respectively:

$$ r_s(z_*) = S_k \left( \int_{z_*}^{\infty} \frac{c_s(z)}{H(z)} dz \right), \quad D_A(z_*) = S_k \left( \int_{0}^{z_*} \frac{1}{H(z)} dz \right), \qquad (2.8) $$

with

$$ S_k(x) = \begin{cases} \frac{1}{\sqrt{|k|}} \sinh(\sqrt{|k|}x), & \text{for } k < 0 \\ x, & \text{for } k = 0. \\ \frac{1}{\sqrt{|k|}} \sinh(\sqrt{|k|}x), & \text{for } k > 0 \end{cases} \qquad (2.9) $$

Here, $c_s(z) = 1/\sqrt{3[1+R(z)]}$ is the sound speed of the acoustic waves in the baryon-photon plasma, where $R(z) = 3\rho_b(z)/(4\rho_\gamma(z)) = 3\omega_b/(4\omega_\gamma)(1+z)^{-1}$ denotes the ratio between the baryon and photon energy densities, and $k$ is the spatial curvature of the Universe. Because of the so called “geometric degeneracy” [714], CMB anisotropy measurements alone do not constrain the curvature of the Universe. To a large extent, the degeneracy can be broken using CMB lensing effect. Tighter constraints can be imposed by analyzing CMB data in combination with other cosmological probes (see Section 2.1.18.4). The Hubble parameter, $H(z)$, is given in terms of the physical densities of DE, matter, photons, neutrinos and curvature as

$$ H(z) = (100 \text{ km s}^{-1} \text{ Mpc}^{-1}) \sqrt{\omega_{\text{DE}}(z) + \omega_{\text{m},0}(1+z)^3 + \omega_{\gamma}(1+z)^4 + \omega_{\nu}(z) + \omega_k(1+z)^2}. \qquad (2.10) $$

Within the $\Lambda$CDM model, $\omega_{\text{DE}}(z) = \omega_\Lambda$ and $\omega_k = 0$. The only unknown quantities in Eqs. (2.8)–(2.10) are $\omega_b$, $\omega_{\text{m},0}$ and $\omega_\Lambda$. The radiation density $\omega_\gamma$ is fixed by the CMB temperature obtained from measurements of the CMB blackbody spectrum [676] and the neutrino density can be obtained at any time by properly integrating the neutrino distribution function<fnref n="8" />.

<footnote n="8">⁸ The parameter of the neutrino equation of state is not constant. Therefore, it is not possible to generally parameterize $\omega_\nu$ via a simple scaling of the redshift, unless one considers the limiting cases of ultra-relativistic and non-relativistic neutrinos.</footnote>
Then, $H_0$ is inferred from CMB data as follows (see Ref. [713] for more details). First, we determine the baryon and matter densities from their impact on CMB power spectra [715]. The baryon density affects the relative heights of the acoustic peaks and the diffusion damping in the CMB power spectrum, while the matter density is mostly determined via the “potential envelope” effect, i.e., the increase in power for modes that re-enter the horizon before matter–radiation equality. The characteristic scale of this power boost corresponds to the comoving size of the horizon at matter–radiation equality projected on the last scattering surface, which depends on the ratio between the matter and radiation densities. Since $\omega_A$ is negligible at redshifts $z \ge z_*$, we have everything we need to determine $r_s(z_*)$ via Eq. (2.8).

Then, we obtain the acoustic scale $\theta_s$ from the spacing between the acoustic peaks, and, using Eq. (2.7) we derive $D_A(z_*)$. As a last step, we adjust $\omega_A$ in order to match the value of $D_A(z_*)$ calculated via Eq. (2.8) with the one inferred as described above. Having determined the physical density of each component, we can finally reconstruct $H(z)$ via Eq. (2.10). For $z=0$, this provides us with the value of $H_0$.

Note that $\theta_s$ is very tightly constrained by observations. In particular, it is measured to 0.03% accuracy by Planck data, which gives $100\theta_s = 1.04109 \pm 0.00030$ using a combination of TT,TE,EE+ lowE data [192]. Other CMB probes measure $\theta_s$ to be consistent with Planck, such as $10^4\theta_s = 104.056 \pm 0.031$ from ACT-DR6 (TTTEEE+Planck-Sroll2 EE [703]) and $100\theta_s = 1.04016 \pm 0.00067$ (SPT+WMAP [704]).

### 2.1.18.4. CMB in combination with other cosmological probes.
The same acoustic oscillations that we observe in the CMB are also left imprinted in the galaxy power spectra in the form of BAO. The characteristic scale of BAO is the sound horizon at the drag epoch, $z_d \simeq 1059$ [192], i.e., the time when baryons were released from the drag of CMB photons. This characteristic scale provides us with a cosmological standard ruler that serves as an independent way to measure the expansion rate of the Universe and improve our bounds on the cosmological parameters. Notably, the inclusion of BAO data allows us to better constrain the spatial curvature of the Universe and break the aforementioned geometric degeneracy [714,716]. We refer to Section 2.1.19 for a more detailed discussion of BAO measurements of $H_0$. In what follows, we report an incomplete list of the most recent estimates of $H_0$ obtained with the combination of CMB and LSS data, to give a sample of the complementarity and constraining power of these combinations.

A recent analysis of DESI BAO data in combination with Planck PR4 primary anisotropies (using small, commander and CamSpec likelihoods), Planck and ACT-DR6 CMB lensing, leads to $H_0 = 68.17 \pm 0.28 \text{ km s}^{-1}\text{Mpc}^{-1}$ [717]. A combined analysis of Planck CMB lensing, BOSS DR12 galaxy power spectra, and the PANTHEON+supernova constraints yields $H_0 = 64.8_{-2.5}^{+2.2} \text{ km s}^{-1}\text{Mpc}^{-1}$ [718], by imposing a BBN prior on physical baryon density and assuming $\Lambda$CDM cosmology. A multi-tracer full-shape analysis of luminous red galaxy (LRG) and emission line galaxy (ELG) samples from the eBOSS DR16 measures $H_0 = 70.0 \pm 2.3 \text{ km s}^{-1}\text{Mpc}^{-1}$ in the $\Lambda$CDM framework when combining the BBN prior and fixing the spectral tilt $n_s$ to Planck value [719]. Another analysis of the BOSS DR12 data by fixing $n_s$ and the baryon-to-DM ratio $\Omega_b/\Omega_{\text{DM}}$ to Planck value gives $H_0 = 68.5 \pm 2.2 \text{ km s}^{-1}\text{Mpc}^{-1}$ [720]. Adopting a forward modeling approach for the BOSS DR12 bispectrum monopole, with BBN prior on $\Omega_b h^2$, [721] found $H_0 = 67.6_{-1.8}^{+2.2} \text{ km s}^{-1}\text{Mpc}^{-1}$. Another study of the BOSS galaxy power spectrum with bispectrum monopole and quadrupole estimated $H_0 = 69.2 \pm 1.1 \text{ km s}^{-1}\text{Mpc}^{-1}$ [722] within the $\Lambda$CDM model, using BBN and fixing spectral tilt to Planck value.

It is crucial to highlight that these results assume a $\Lambda$CDM cosmology. A combination of Planck TT,TE,EE+lowE, Planck lensing, BAO measurements from BOSS and eBOSS, and Pantheon+ SNIa data, assuming a non-flat $\Lambda$CDM model yielded $H_0 = 68.24 \pm 0.54 \text{ km s}^{-1}\text{Mpc}^{-1}$ with $\Omega_k = 0.0004 \pm 0.0017$ [723]. A similar constraint of $H_0 = 68.53 \pm 0.56 \text{ km s}^{-1}\text{Mpc}^{-1}$ was found when treating $A_{\text{lens}}$ as a free parameter

(for more details, see the discussion on the $A_{\text{lens}}$ anomaly in Section 2.1.18.5). Extending the analysis to non-flat $\omega$CDM model resulted in $H_0 = 67.95 \pm 0.66 \text{ km s}^{-1}\text{Mpc}^{-1}$ and $\Omega_k = 0.0016 \pm 0.0019$ (see Ref. [723] and references therein). To conclude this list of constraints, the take-home message is that the combination of CMB and LSS measurements provide estimates of $H_0$ which are in agreement with the (lower) value inferred from Planck alone. This is true also in case of combinations that do not include Planck anisotropies — or do not include Planck data at all — in their fit.

### 2.1.18.5. The role of potential systematic effects in CMB-based estimates of $H_0$.
Unresolved systematic effects (hereafter systematics) in CMB measurements could potentially bias the cosmological constraints. This possibility has motivated a collective effort to examine potential sources of systematics within the CMB dataset.

Features in the Planck spectra (especially temperature and at small scales) have been also interpreted as hints to unsolved systematics. In this context, the $A_{\text{lens}}$ anomaly in Planck data is noteworthy. First introduced in Ref. [724], $A_{\text{lens}}$ is a phenomenological (somehow “unphysical”) parameter used to rescale the effects of gravitational lensing on the CMB angular power spectra. Therefore, the expected value is $A_{\text{lens}} = 1$. Interestingly, Planck data (primary anisotropies) show a preference for $A_{\text{lens}} > 1$ at more than $2\sigma$, increasing to more than $3\sigma$ when combined with BAO data [192]. Theoretical explanations for such values are challenging, as it would require either a closed universe, posing conflicts with other datasets and simple inflationary models, or more exotic solutions such as modifications to General Relativity. Furthermore, the lensing anomaly does not appear in Planck trispectrum data (i.e., in the lensing power spectrum), which offers an independent and complementary measurement of CMB lensing. If not indicative of new physics or not interpreted as a statistical fluke, the $A_{\text{lens}}$ anomaly might arise from an undetected systematics error in Planck data and this systematic could potentially bias the measurement of the $H_0$ parameter. However, when the effect of $A_{\text{lens}}$ is marginalized over in the analyses, the Planck and Planck + BAO constraints on $H_0$ shift only slightly toward higher values: $H_0 = 68.28 \pm 0.72 \text{ km s}^{-1}\text{Mpc}^{-1}$ and $H_0 = 68.23 \pm 0.49 \text{ km s}^{-1}\text{Mpc}^{-1}$ at 68% CL, respectively [192], not enough to provide a satisfying solution to the $H_0$ tension. If the $A_{\text{lens}}$ anomaly is due to systematics which may impact the final $H_0$ constraints, it raises questions whether the same systematics can be fully captured by $A_{\text{lens}}$ alone or if further modeling is needed. Moreover, recent analyses by ACT and SPT, which also find a lower value of $H_0$ than local measurements, find no deviation from the standard lensing effect predicted by the $\Lambda$CDM model [702,703,724], supporting the idea that the $A_{\text{lens}}$ anomaly might not be the right solution to the tension.

The consistency of cosmological parameters estimated from different multipole ranges of Planck data has also been investigated extensively in the past years [726–728], sometimes claiming evidence for internal inconsistencies which may impact the $H_0$ estimates. However, it has been showed [715] that the shifts in cosmological parameters recovered from different ranges — mostly due to the combined effects in temperature of non-lensing-related residuals at high multipoles and power deficit at large scales ($\ell < 30$) — are not very significant and consistent with sample variance.

In the context of potential systematics in CMB data, it is also worth mentioning that the mild discrepancy seen between ACT-DR4 and both WMAP and Planck has been solved with the final data release of ACT. Indeed, ACT-DR6 finds very good agreement with both Planck Legacy data (PR3) and Planck PR4 (NPIPE) already at the power spectrum level [703]. In terms of cosmological parameters, the difference between the best fit values in $\Lambda$CDM obtained with ACT-DR6 and Planck is estimated to be within $1.6\sigma$ (for Planck Legacy) and $2.5\sigma$ (for Planck PR4), see [703,706] for details.

As far as SPT results are concerned, good agreement has been found [704] between SPT and Planck, both at the power spectrum level
(in TT over angular scales that are signal-dominated in both experiments) and at the cosmological parameter level (assuming ΛCDM). As noted by the SPT collaboration, this excellent agreement between two effectively independent experiments (given the negligible overlap between observed sky fractions and different sensitivity to angular scales) is a strong argument in favor of the robustness of the results and of the consistency of ΛCDM across different scales and spectra. A good agreement has been found also between SPT and WMAP. The agreement between SPT and ACT is acceptable [704], with the largest shift in recovered ΛCDM parameters being on θ (2σ larger in ACT than in SPT). However, a multi-dimensional statistical test results in no significant deviations to be noted and agree with an explanation in terms of statistical fluctuations.

Another crucial aspect of the CMB data analysis to be explored in the context of $H_0$ estimation is the possible impact of instrumental systematic effects. In CMB data analysis, particular care is devoted to the quantification of residual systematics, which are corrected for through the pipeline and/or propagated in the likelihood analysis either by incorporating it in the noise model or modeling specific residuals in the data vector. In this context, the use of end-to-end simulations is key. Among the different sources of systematics, beam characterization and instrument calibration (especially in polarization) are particularly worrisome. As an example, the determination of polarization efficiencies is considered one of the main limitations of the Planck data products [729]. Nevertheless, the use of a large suite of null tests and cross-checks on several data splits, and the comparison with realistic end-to-end simulations allow us to show, for all the data products released by the main CMB experiments, that possible biases of cosmological parameters — included $H_0$ — induced by residual systematics are small-to-negligible [702,704,707,729–732]. Similar analyses on dedicated simulations for next generation CMB experiments have recently shown that, even in the presence of unaccounted-for systematics (i.e., effects which have not been corrected for in the analysis pipeline) due to, e.g., beam chromaticity, bandpass mismatch, polarization angle miscalibration and incorrect calibration in polarization, the bias on $H_0$ is at the level of a small fraction of $\sigma$ [733,734]. Therefore, we conclude that instrumental systematic effects cannot be the (only) source of discrepancy between the CMB-based estimate of $H_0$ and local measurements.

Finally, the interplay between instrumental systematics and foreground contamination can have non-linear effects on the final results, potentially biasing cosmological parameter estimates, including the Hubble constant. The presence of galactic and extragalactic foregrounds, such as dust emission and radio sources, complicates the separation of the CMB signal from other astrophysical contributions. This complexity underscores the need for precise modeling and mitigation strategies to ensure accurate parameter estimation, but it is unlikely the source of the $H_0$ tension.

### 2.1.18.6. $S_8$ tension.
The amplitude of the CMB power spectrum, especially CMB lensing, places stringent constraints on the matter density and hence on $\sigma_8$. Here, we briefly address the tension related to the amplitude of matter clustering in the late Universe, described by the parameter $S_8 \equiv \sigma_8(\Omega_{m,0}/0.3)^{0.5}$. For a more detailed discussion, refer to Section 2.2.

CMB-based estimates of $S_8$ are consistently larger than those derived from galaxy-based measurements [192,695,700–702,704,707]. Observations of weak gravitational lensing at low redshifts ($z \lesssim 0.5–1$) indeed suggest weaker matter clustering than predicted by the ΛCDM model with parameters inferred from CMB data. Simply put, the distribution of galaxies and matter in the late Universe appears smoother than expected based on the evolution of the fluctuations seen in the CMB. The CMB estimate of $S_8$ is model-dependent and can shift with other parameters that affect the growth and amplitude of matter fluctuations. Notably, the optical depth to reionization ($\tau$), which carries the largest uncertainty among the ΛCDM parameters, and

the sum of neutrino masses ($\sum m_\nu$), which is a derived parameter and is usually fixed. Uncertainties in these parameters affect the derived $S_8$ constraints. Additionally, Planck's excess lensing anomaly, as discussed in Ref. [735], can mimic a larger $S_8$. However, the CMB-LSS tension persists even without Planck's lensing excess, as shown by the latest ACT+WMAP analysis [703,706], which reports a high value of $S_8 = 0.857 \pm 0.020$ without an anomalous lensing amplitude, and SPT-3G analysis [707], which is independent of temperature data and reports $S_8 = 0.850 \pm 0.017$.

On the theoretical side, efforts to resolve the $S_8$ tension involve modifying the matter or gravity sectors of the ΛCDM model, leading to a range of alternative cosmological scenarios (see e.g., Sections 4.1 and 4.2 for more details). Although some models alleviate the $S_8$ tension, most fail to provide consistent solutions when all cosmological probes are considered. In particular, due to the correlation between $H_0$ and $S_8$, models that resolve the $S_8$ tension often worsen the $H_0$ tension, and vice versa [736–738]. For example, late-time dark sector transitions, which prefer a higher $H_0$ value, often result in lower $\Omega_{m,0}$ to preserve the measured value of $\Omega_{m,0}h^2$, leading to changes in structure growth and CMB anisotropies and typically yielding a higher $\sigma_8$ than ΛCDM due to extended matter domination. Similarly, EDE models that address the $H_0$ tension tend to increase $\sigma_8$ because they require a higher initial curvature perturbation amplitude to counterbalance the unclustered component. It is therefore crucial to perform joint analyses fitting a full array of multiple datasets, without fixing any of the parameters of the model. Simultaneously any solution to the $S_8$ tension must be consistent with the growth history (typically studied through the parameter $f\sigma_8(z)$), probed by e.g., BAO, galaxy power spectrum, and void measurements.

### 2.1.19. Baryonic acoustic oscillations

**Coordinator:** Florian Beutler

**Contributors:** Armando Bernui, David Benisty, Denitsa Staicova, Felipe Avila, Ignacio Sevilla-Noarbe, Özgür Akarsu, Maret Einasto, Mustapha Ishak, Nicola Deiosso, Rafael C. Nunes, Ruchika, Samuel Brieden, and Sveva Castello

BAO represent a characteristic scale in the distribution of galaxies, or more generally of any tracer of the matter density field [739,740]. This scale, known as the sound horizon $r_d$, originated from sound waves traveling through the early ($z \gtrsim 1060$) Universe before baryons and photons decoupled (see Refs. [741–743] for reviews of the subject). We can employ this scale as a standard ruler, which allows us to measure the expansion history of the Universe. In practice, the BAO features can be extracted from the two-point correlation function of galaxies, where it appears as a peak [744,745], or from the Fourier-space equivalent, the power spectrum, where it is manifested as a series of oscillations [746–748]. In the following, we will review the different methods to extract the BAO scale from the galaxy distribution. We will then discuss the most recent measurements from DESI.

### 2.1.19.1. Methods.
The 3D position of a galaxy can be determined by measuring its location on the sky (in right ascension and declination) and its redshift. For this reason, measuring the BAO scale in the perpendicular direction to the line-of-sight and along the line-of-sight provides constraints on slightly different quantities. The angular component constrains the comoving angular diameter distance $D_M/r_d = 1/\Delta\theta$, where $\Delta\theta$ is the angular separation of the pair of galaxies, while the line-of-sight component constrains the comoving Hubble distance $D_H/r_d = 1/\Delta z$, where $\Delta z$ is the redshift separation of the galaxy pair. In the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, the comoving angular diameter distance $D_M$, the related $D_A$ and the Hubble distance $D_H$ are given by

$$D_M = \frac{c}{H_0 \sqrt{|\Omega_K|}} \operatorname{sinn} \left[ |\Omega_K|^{1/2} \int_0^z \frac{dz'}{E(z')} \right], \quad D_A = \frac{D_M}{1+z}, \quad D_H = \frac{c}{H_0 E(z)}, \quad (2.11)$$
where $\sinn(x) \equiv \sin(x)$, $x$, $\sinh(x)$ for $\Omega_K < 0$, $\Omega_K = 0$, $\Omega_K > 0$ respectively, and $E(z)$ is the normalized Hubble function. Rather than separating $D_M$ and $D_H$, often measurements are reported as a combination of the two given by $D_V \equiv [zD_M(z)^2D_H(z)]^{1/3}$. BAO constraints from photometric galaxy surveys usually have significant redshift uncertainties that only allow for constraints on $D_M$. One important point to note from the equations above is that BAO measurements always constrain distances relative to the sound horizon scale. This introduces degeneracies between cosmological parameters and the sound horizon scale $r_d$. In $\Lambda$CDM for example, BAO can only constrain $H_0 r_d$, and breaking this degeneracy is crucial to obtain measurements of $H_0$ and provide key information in light of the Hubble tension. We will discuss this further in Section 2.1.19.3.

The density field sources a velocity field that will displace galaxies away from their initial position, effectively leading to a redshift-dependent smoothing of the localized BAO feature. This effect can reduce the signal-to-noise by up to a factor of two at low redshift (e.g., see Ref. [749]) and also results in a sub-percent level systematic biasing of the BAO scale [750,751]. Given that the velocity field is sourced by the density, one can estimate the displacement of galaxies assuming standard gravity and correct for this effect [751,752]. Such techniques are known as density field reconstruction and are routinely applied to most spectroscopic BAO measurements.

The standard BAO analysis relies on converting the angular position and the redshift into Cartesian coordinates, from which the 3D clustering statistics can be calculated. This step requires adopting a fiducial cosmology, which typically is taken to be a flat $\Lambda$CDM model with parameters based on the CMB. The BAO analysis accounts for these assumptions by including geometric scaling parameters and so far studies with alternative models (e.g., EDE or MG) have not found that the fiducial cosmology assumptions inflict any significant bias in the standard BAO analysis [753–756]. Through density field reconstruction, the standard analysis also makes assumptions about the connection between the measured galaxy density field, the matter density field, and the large-scale velocity field. However, just like with the fiducial cosmology assumptions, the impact on the measurements has been shown to be negligible (e.g., see Refs. [757,758]). Assumptions about the number of relativistic particles, $N_{\text{eff}}$ at high redshift have been shown to impact the BAO phase and could bias BAO measurements [759,760]. The standard analysis usually fixes this parameter to the standard model value. While a Planck prior on $N_{\text{eff}}$ (within $\Lambda$CDM) reduces any potential bias to well below the uncertainties of current measurements, one still should be aware of these prior assumptions.

Given the priors implicit in the standard BAO analysis, alternative analysis methods have been developed that can (at least partly) avoid the assumption of a fiducial cosmology [761]. This technique often runs under the name transverse BAO [762,763], which should not be confused with the many angular BAO measurements in the literature which follow the standard analysis technique described above (see e.g., DES Y6 in Fig. 30). The basic idea of such methods is to bin the data into redshift bins and measure the angular BAO signature in those redshift bins through two-point angular-clustering statistics, avoiding the need to convert redshifts into distances. However, to account for the redshift evolution within the redshift bin one still needs to use a fiducial cosmology, and density field reconstruction is generally not applied in such cases. For this reason, such methods usually have much larger measurement uncertainties (see e.g., see Refs. [764–770]).

### 2.1.19.2. BAO measurements.
The BAO feature has been measured both with photometric and spectroscopic surveys. Photometric surveys allow for a direct identification of a huge number of galaxies, but generally only provide poor photometric redshift information. Even for measurements of the angular BAO scale, the redshift information is necessary to split the sample into redshift bins, to avoid the smearing of the BAO signal due to its redshift evolution. Although the photometric redshift quality can be improved with an adequate subselection, this

usually also reduces the number of galaxies in the sample [771]. To date, the best photometric BAO measurement comes from DES year-6 analysis using about 16 000 000 galaxies and yielding a 2.1% constraint on $D_M/r_d$ [772]. Observational systematics for such measurements are dominated by uncertainties in the redshift distribution, which are nevertheless still below the statistical error budget.

Compared to photometric surveys, spectroscopic surveys have the advantage of providing precise redshift measurements and detailed spectral information. This is achieved through the identification of specific spectral lines or features, such as the $H\alpha$, $H\beta$, OII lines or the 4000 Å break for galaxies, and broad emission lines for QSOs. The Lyman-$\alpha$ (Ly$\alpha$) forest represents a series of absorption features in the spectra of distant QSOs that can be used to map the distribution of intergalactic hydrogen gas and provides additional BAO measurements at higher redshift [773].

The first convincing detections of the BAO signal in the distribution of galaxies came from the 2-degree Field Galaxy Redshift Survey (2dFGRS) and the Sloan Digital Sky Survey (SDSS) in the early and mid-2000s [744,746,774], while the first detection in the Ly$\alpha$ forest was made by the BOSS [775]. There have been many more subsequent detections in other galaxy surveys [748,749,776–782]. The best constraints to date come from DESI, which measured the BAO signal in 8 independent redshift bins with a detection significance ranging from 3.3 to 9.1$\sigma$ [745,783]. A comparison between these DESI measurements and previous BAO measurements is shown in Fig. 30.

Observational and instrumental systematics, such as fiber collisions [784], distortions due to peculiar velocities (Redshift-Space Distortions (RSD)), non-linear matter and galaxy clustering, atmospheric dispersion, and spectrograph calibration are critical factors in any analysis based on spectroscopic surveys datasets. Since the BAO signal is located on very large scales ($r_d \sim 150$ Mpc) and represents a distinguishable feature unlikely to be mimicked by any instrumental or physical process, BAO measurements have proven to be very robust (see e.g., see Ref. [758]). The galaxy clustering analysis of DESI identified the galaxy-halo connection (HOD) as the dominant systematic for their BAO measurement. However, all combined systematics still only added 5% to the final statistical error budget even for the highest precision BAO measurement (see table 13 and 15 in Ref. [783]).

The BAO signal is most commonly measured as a mixture of radial and transversal modes. Alternatively, the BAO signature can be extracted in a thin redshift bin using the two-point angular-clustering statistics (where the angular separation of pairs instead of the comoving distances are computed, avoiding converting redshifts into distances using a fiducial cosmology), providing a measurement of the transverse BAO, $\Delta\theta_{\text{BAO}}(z)$. The finer the redshift bin, the purer this transverse contribution. If $r_d$ is known, $\Delta\theta_{\text{BAO}}(z)$ determines $D_M(z)$, as explained in Section 2.1.19.1. Fig. 30 shows such transverse BAO measurements from SDSS data (labeled SDSS-tr) derived in Refs. [762,763,785–788]. However, for the remainder of this section, we will focus on the most recent DESI analysis.

### 2.1.19.3. BAO and the $H_0$ tension.
One of the challenges in the use of BAO measurements in cosmology is the degeneracy between the sound horizon $r_d$ and Hubble constant $H_0$. As mentioned in Section 2.1.19.1, this arises because all measurable quantities in the BAO data depend on the product $c/H_0r_d$. The only way to decouple them is by using additional information beyond the BAO scale [713,718,789–796] or imposing an external prior on $r_d$ from the CMB or BBN [699,713,797]. The 2024 DESI BAO results combined with a BBN prior from Ref. [798] yield a constraint of $H_0 = 68.53 \pm 0.80 \text{ km s}^{-1} \text{Mpc}^{-1}$ [783]. This measurement is independent of the CMB and solely based on DESI BAO + BBN. While this value of the Hubble constant is slightly higher than the one preferred by Planck [192] as well as the one reported in previous data from SDSS [797], it is still in 3.4$\sigma$ tension with the SH0ES result of Ref. [78]. Fig. 31 shows the values for $H_0$ and $\Omega_{m,0}$ inferred from the DESI data, along with the SH0ES results on $H_0$ as a shaded band. One
<figure><img src="image_35.png" /><figcaption>**Fig. 30.** DESI BAO measurements (filled circles) compared to prior BAO measurements (empty symbols) from spectroscopic (6dFGS, WiggleZ, SDSS) and photometric (DES Y6) surveys, where SDSS includes the MGS, BOSS DR12 and eBOSS DR16 galaxy and Lyman-α Samples. Additionally, we show the transverse BAO measured from SDSS catalogs (SDSS-tr) in thin redshift shells, neglecting the longitudinal modes. Note that these measurements are not official SDSS products, but were instead obtained independently in Refs. [763,785–788] (small gray squares) and Ref. [762] (small pink squares). Horizontal arrows and colored regions indicate the redshift range spanned by each DESI tracer type. Solid lines show BAO distances (scaled by $\sqrt{z}$ for improved visibility) as a function of redshift $z$ for the standard flat $\Lambda$CDM DESI best-fit, and dashed lines indicate the Planck best-fit with shaded region denoting the $\pm 1\sigma$ regime. **Upper panel:** Comparison between transverse (pink), radial (blue), and isotropic (orange) BAO measurements, where the latter is only displayed for data that does not exhibit an anisotropic measurement due to low signal-to-noise. **Lower panel:** Isotropic BAO distance residuals where the anisotropic BAO measurements from the upper panel (apart from DES-Y6 and SDSS-tr, which do not measure radial BAO) have been combined into an isotropic measurement (not shown in the upper panel to avoid redundancy). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_36.png" /><figcaption>**Fig. 31.** 68% credible-interval constraints on the Hubble constant $H_0$ (left panel) and the matter density $\Omega_{m,0}$ (right panel), assuming different cosmological models and different combinations of datasets (on the plot, DESI, Planck 2018, Pantheon Plus, DESY5, Union3) and the BBN and $\theta_*$ priors. The gray band indicates the SHOES measurement for $H_0$.</figcaption></figure>

sees that only the $w_0$CDM model comes close to resolving the Hubble tension for the DESI+CBM dataset to the price of much lower matter density. A further look at the DESI results is presented in Fig. 32, where we show the $1\sigma$ posteriors for some of the models presented in Fig. 31. In this figure, the degeneracy between $H_0$ and $\Omega_{m,0}$ manifests as the covariance of the inferred data, which is particularly visible on the right panel of Fig. 32 zooming in on the DESI+BBN models. This leads to

the known issue that, across different models, increasing $H_0$ generally decreases $\Omega_{m,0}$, and vice versa. On the plots, $r_d$ is missing since it is not a free parameter for all tested models.

BAO measurements rely on a model-dependent extrapolation to redshift zero when constraining the Hubble constant. However, combining BAO with SNIa datasets and the CMB does constrain the redshift
evolution and makes it difficult to find extensions to $\Lambda$CDM that could resolve the Hubble tension [238,789,799–802].

Ref. [803] has found that marginalizing over $r_d$ can lead to a degeneracy between $H_0$ and the primordial power spectrum and to a model-dependent $H_0$ measurement. A similar observation has been published in Ref. [804] showing that models that only reduce $r_d$ cannot resolve the Hubble tension while at the same time remaining consistent with other cosmological datasets. A possible solution to this consists of working with the sound horizon as a free parameter (thus not making any assumption concerning the recombination physics) and using other quantities to break the degeneracy [805–807]. In these works, it was shown that while this approach can produce interesting results when examining $\Lambda$CDM alternatives, it does not solve the Hubble tension. Another approach consists in marginalizing over both $r_d$ and $H_0$ [808], which integrates the quantity $c/H_0r_d$ out of the likelihood ($\chi^2$) and thus removes entirely the dependence on these parameters from the theoretical predictions. The results obtained in this way show a slight preference for Dynamical Dark Energy (DDE) models but they do not solve the Hubble tension.

While early BOSS and eBOSS Ly-$\alpha$ BAO results showed a 1.5 to 2.5$\sigma$ tension with Planck-$\Lambda$CDM [789,797], the most recent Ly-$\alpha$ analysis in DESI [745] shows no tension anymore. However, DE density reconstructions using DESI BAO still suggest the possibility of zero or negative DE densities for $z \gtrsim 1.5–2$ in some dataset combinations [809]. Additionally, a shift of the Ly-$\alpha$ BAO peak at the 2.2$\sigma$ and 3.5$\sigma$ levels in real and redshift space, respectively, was recently suggested in Ref. [810]. The reasons behind these findings are yet to be understood, warranting further investigation. Since the comoving angular diameter distance to last scattering, $D_M(z_a)$, is strictly constrained by CMB data almost model-independently, a lower $H(z)$ for $z \gtrsim 1.5–2$ due to vanishing or negative DE density should be compensated by a higher $H(z)$ for $z \lesssim 1.5–2$, provided that the pre-recombination Universe is not modified [801,802,811–813]. This results in a higher $H_0$ and correspondingly smaller $\Omega_m$, which can reduce the $S_8$ value and address the most discussed tensions in the $\Lambda$CDM model [801,802,811–813].

The most recent DESI data release, DESI DR2, improves on the DR1 findings, analyzing over 14 million galaxies and QSOs from three years of observations (surpassing SDSS in the effective volume of all tracers). Compared to DR1, the precision of BAO measurements has increased by approximately a factor of two, with uncertainties reduced to about 0.24% for galaxy/QSO measurements and 0.65% at high redshift from the Lyman-$\alpha$ forest (which is now twice the DR1 sample). The DESI DR2 combined with a BBN prior yield $H_0 = 68.51 \pm 0.58 \text{ km s}^{-1}\text{Mpc}^{-1}$, 28% more precise than DR1. The cosmological analysis shows stronger evidence for evolving DE, with statistical significance amounting to 2.8$\sigma$, 3.8$\sigma$ and 4.2$\sigma$ depending on which datasets are combined with DESI (Pantheon+, Union3 or DES Y5) and 3.1$\sigma$ for DESI+CMB. The results show a preference for a DE equation of state with $w_0 > -1$ and $w_a < 0$ and a possible phantom crossing in the past. While the degeneracy direction for the constraints in the $w_0 - w_a$ plane approximately points toward the $\Lambda$CDM solution, the statistical evidence consistently challenges the standard $\Lambda$CDM across multiple analysis approaches and dataset combinations [717].

2.1.19.4. **Outlook.** The DESI experiment is expected to publish its Y3 BAO analysis in 2025, while the final Y5 analysis should be made public in 2026/27 [814]. The final dataset will be three times larger than the current Y1 catalog, and proposals for a DESI extension until 2029 and a second phase (DESI-II) well into the 2030s are under consideration. At the same time, the Euclid mission has also started, with the first cosmology analysis expected in 2025/26 [164], and the Vera Rubin Observatory will see first light in 2025.

On a longer timescale, there have been several proposals for stage-V spectroscopic experiments with a start in the early to mid-2030s. These experiments increase the telescope aperture and the spectroscopic multiplexing capabilities with the aim of collecting hundreds of millions of

galaxy redshifts (e.g., see Refs. [407,815–817]). After the DESI mission is completed, the BAO signal in the low redshift Universe ($z \lesssim 1$) will have been measured up to the sample variance limit over almost the entire sky available for cosmological measurements ($\sim 15\,000$ to $18\,000 \text{ deg}^2$), see figure 16 in Ref. [818]. Future experiments therefore focus on the high redshift regime, where significant gains are still possible.

## 2.2. $S_8$ tension: measurements and systematics

### 2.2.1. Weak lensing

**Coordinator:** Marika Asgari, Daniela Grandón, Cora Uhlemann

**Contributors:** David Sanchez Cid, Ignacio Sevilla, Judit Prat, Konrad Kuijken, Maciej Bilicki, Markus Michael Rau, Masahiro Takada, Nikolina Šarčević, and Shun-Sheng Li

Cosmic shear is one of the key probes of the LSS. A cosmic shear analysis measures the weak gravitational lensing signal [819–824] imprinted in the observed galaxy shapes by LSS (see Ref. [825–827] for a review and references therein). The strength of this signal depends directly on the total matter distribution between the source galaxies and the observer. Therefore, cosmic shear provides a precise mapping of the projected matter density field, capturing its statistical properties and enabling constraints on cosmological parameters.

To perform a cosmic shear analysis, we need a galaxy catalog that comprises two measurements: galaxy shapes (Section 2.2.1.1) and the redshift information (Section 2.2.1.2). The redshifts are used to divide source galaxies into tomographic bins. The redshift distribution in each bin determines the strength of the lensing effect and serves as an input for the theoretical predictions of the cosmic shear signal, which require prescriptions for baryonic effects (Section 2.2.1.3) and the intrinsic alignment of galaxies (Section 2.2.1.4). In addition, the tomographic analysis allows us to extract information about the evolution of structures and to disentangle the astrophysical systematics from the cosmological information.

The most commonly used estimator for the cosmic shear signal is the two-point statistic. In real space, the two-point shear correlation functions are measured as the average correlation between the shapes of galaxy pairs in parallel and orthogonal directions as a function of their angular separation [828]. Both of those correlations can be related to one angular convergence power spectrum, which is a line-of-sight projection of the matter power spectrum, and thus sensitive to non-linear effects on small scales and low redshifts. However, because the lensing field is a non-Gaussian random field, cosmic shear surveys have also implemented estimators that capture information beyond the traditional two-point statistics [829]. These higher-order or non-Gaussian statistics, such as the three-point shear correlation function [830–833], the bispectrum [834,835], the lensing convergence PDF [836–839] or its moments [840], and peak statistics [841–844], among others [845–848], have proven effective in enhancing the constraining power and promise to be valuable in Stage IV analysis [829,849].

Cosmic shear analysis is most sensitive to the $S_8$ parameter and the strength of the intrinsic alignment signal of galaxies. In fact, with the available data, $S_8$ is the only cosmological parameter that cosmic shear analysis can robustly constrain from two-point statistics. A joint analysis of two-point function and higher-order statistics however can improve constraints in $\Omega_{m,0}$, as well as self-calibrate systematics and break degeneracies between cosmological and nuisance parameters. These surveys consistently infer $S_8$ values that are $1–3\sigma$ lower than what is expected from the CMB anisotropies.

The current generation of Stage III cosmic shear surveys — such as the KiDS [850,851], DES [852,853], the Subaru HSC Survey [854–856] and the Dark Energy Camera All Data Everywhere (DECADE; [857]) — have conducted multiple cosmic shear analyses from at least two separate data releases. These surveys consistently infer $S_8$ values that are $2–3\sigma$ lower than what is expected from the CMB anisotropies and
<figure><img src="image_37.png" /><figcaption>**Fig. 32.** Illustrations of the degeneracy between the Hubble constant and the matter density parameter ($\Omega_{m,0}$) constrained by different data combinations (indicated by line style) for different models (indicated by color) at $1\sigma$. “CMB” refers to Planck + ACT lensing data and “SN” to the Pantheon+ SNIa sample. The parameter region constrained by CMB alone is shown as a prior on $\omega_m$ (black dotted contours) that is independent of the late-time model extensions considered here. The most constraining data combination (DESI+CMB+SN) is shown on the right as a zoom into the black rectangle on the left for better visibility. For conciseness, the SDSS BAO + BBN constraint (dash-dot-dotted) is shown for $\Lambda$CDM (orange) only. Note that the legends shown in the left and right panels apply to both simultaneously.</figcaption></figure>

CMB lensing measurements by Planck and ACT [855,856,858–864]. A summary of these results is presented in Fig. 33. This disagreement in the $S_8$ value between early- and late-time probes, often referred to as the $S_8$ tension, was first observed between the tomographic analysis of the Canada France Hawaii Telescope Lensing Survey (CFHTLenS) [865] and the first Planck cosmology results [866], at the level of $2.3\sigma$. More recently, the DES and KiDS teams collaborated on a unified analysis pipeline which resulted in a joint constraint with a slightly higher value of $S_8$, translating to a $1.7\sigma$ difference with Planck [867]. The most recent KiDS results (KiDS-Legacy) shows consistent results with Planck at the level of $1\sigma$. The most recent KiDS results (KiDS-Legacy) show consistent results with Planck at the level of $1\sigma$ [868].

The “3 × 2pt” analysis has emerged in the past decade as a mature cosmological probe that extends cosmic shear by incorporating two additional two-point correlations: galaxy-galaxy lensing (the cross-correlation of lens galaxy positions and source galaxy shapes) and photometric galaxy clustering (the autocorrelation of lens galaxy positions). The primary advantage of this combination is its ability to break the degeneracy between $\Omega_{m,0}$ and $\sigma_8$, enabling robust constraints on $\Omega_{m,0}$ in addition to $S_8$, as well as improving the self-calibration of the nuisance parameters, especially of the intrinsic alignment and redshift parameters. The three major current-generation surveys have conducted this kind of analysis, with their results presented in Fig. 33. Generally, the $S_8$ values measured by 3 × 2pt have been found to be ~1–2$\sigma$ below the Planck value predicted assuming $\Lambda$CDM cosmology, thus with a slightly smaller tension with respect to the cosmic shear results.

A summary of the $S_8$ constraints from low-redshift probes, including lensing surveys, is shown in Fig. 4. In particular, results from peak count analyses — the most studied higher-order statistic — are presented for KiDS-1000 [841], DES Y3 [877], and HSC Y1 [842]. The most precise constraints on $S_8$ come from the DES Y3 peak counts analysis, yielding $S_8 = 0.797^{+0.015}_{-0.013}$, which is in $1.5\sigma$ tension with Planck 2018 [192]. The recently published KiDS-Legacy COSEBIs results report $S_8 = 0.815^{+0.016}_{-0.021}$, showing less than $1\sigma$ disagreement with Planck [868]. On the other hand, a shear two-point correlation function analysis using only blue (star-forming) galaxies in DES Y1 yields $S_8 = 0.822^{+0.019}_{-0.020}$ [878], indicating no tension with Planck CMB results. These findings highlight the importance of mitigating astrophysical systematics — such as intrinsic alignments — to better understand the nature of the $s_8$ discrepancy.

<figure><img src="image_38.png" /><figcaption>**Fig. 33.** The $S_8$ constraints derived from the Stage III WL surveys DES Y3 [861,862,869,870], HSC Y3 [864,871–873], KiDS-1000 [860,874,875] and KiDS-Legacy [868] from two-point shear statistics in harmonic space using $C_l$ or in angular space using $\hat{r}_{\perp}$, COSEBIS and 3 × 2-point statistics including galaxy clustering. Also shown is the joint constraint from DES Y3+KiDS-1000 [867]. We include the CMB measurements from the Planck satellite [694], ACT [695], and SPT [876] on top for comparison. The error bars denote $1\sigma$ uncertainties.</figcaption></figure>

The constraints on $S_8$ and other cosmological parameters can also be affected by the adopted statistical inference pipeline. Most cosmic shear analyses use a Gaussian likelihood, which is defined using the data, the theoretical predictions, and the data covariance matrix (for exceptions employing simulation-based inference (SBI), e.g., see Ref. [879–881]).
Studies have shown that an accurate estimation of the covariance matrix is crucial to avoid biases in cosmological constraints [882, 883]. Additionally, properly accounting for likelihood asymmetry in two-point functions is essential for robust parameter inference [884].

In this section, we focus on weak gravitational lensing and the associated systematics that influence cosmic shear cosmological constraints. These systematics can be broadly categorized as observational or theoretical. Observational systematics affect either the accuracy of shape measurements or the redshift distributions. On the theoretical side, achieving percent-level precision in predicting the matter power spectrum is required across a wide range of scales [885], determined by the survey depth and analysis choices. Since these scales extend into the nonlinear regime, an accurate nonlinear power spectrum prescription is necessary. Additionally, two key astrophysical systematics — intrinsic galaxy alignments and baryonic feedback — remain among the least understood aspects of theoretical modeling for this signal. Various modeling approaches have been developed to address these issues, which can generally be categorized into emulators based on N-body simulations [886,887], and the more widely used halo model-based approaches, see Ref. [888] for a recent review, such as *halofit* [889, 890] and *HMCode* [891,892]. While emulators may offer greater accuracy within the specific parameter range they are trained on, they lack the flexibility of halo model approaches, which are capable of extrapolating beyond this range. As a result, most cosmic shear analyses to date, including all of the flagship publications by current surveys have relied on the latter. To limit the contributions from small scales, suitable linear combinations of the signal can achieve a nulling of the effective WL kernel [893]. This can be used to perform scale cuts in physical rather than angular space and render predictions more robust to small-scale physics, e.g., see Ref. [894,895]. While this discussion focuses on current cosmic shear surveys, the next generation of cosmic shear observations (Stage IV surveys) is approaching [437,896,897]. As constraints tighten, more precise modeling and calibration will be required. Consequently, the interplay of these systematic effects will become increasingly important in future analyses.

### 2.2.1.1. Observational effects: Shape measurement and calibration.
Extracting WL signals from observed galaxy images is a non-trivial task. The cosmic shear signal is typically orders of magnitude smaller than the intrinsic galaxy ellipticities, necessitating robust statistical measurements of a large number of galaxies. Furthermore, systematic biases introduced by various instrumental and measurement effects further complicate the process. While the statistical limitations are readily overcome with the ever-growing galaxy samples in ongoing and upcoming WL surveys; addressing the systematic biases remains a challenging task. This challenge is further exacerbated by increasingly stringent requirements driven by the growing statistical power of these surveys. For upcoming Stage IV surveys, the overall residual shear biases must be reduced to the order of $10^{-3}$, compared to the current percent-level requirements of Stage III surveys, e.g., see Ref. [898]. Achieving this stringent requirement demands continuous development of shear measurement and calibration techniques, with particular attention to subtle effects arising from object detection, blending, selection, and redshift estimation.

**Shape measurement and data-based calibration** Given the critical importance of accurate shear measurement, the WL community has dedicated significant early efforts to developing robust shear measurement algorithms. A number of methods have now achieved an overall bias at the percent level. Broadly, these shear measurement methods can be categorized into two main classes: moment-based methods, e.g., see Ref. [899] and model-fitting methods, e.g., see Ref. [900, 901]. Moment-based methods estimate galaxy ellipticities from the second moments of observed galaxy images after correcting for the point-spread function (PSF) caused by instrumental and observational effects. Conversely, model-fitting methods employ forward modeling with PSF-convolved parametric galaxy profiles. These techniques are

susceptible to specific biases. Moment-based methods are affected by missing pixels, and contamination by light from detected and undetected neighboring galaxies, whereas model-fitting approaches suffer from “model bias”. Additionally, both are affected by common biases such as “noise bias” and selection effects, see Ref. [902] for a recent review. These biases need to be calibrated and corrected for shear measurement methods to meet the requirement of modern WL surveys.

A recent significant advancement in shear measurement algorithms is the introduction of self-calibration or meta-calibration procedures that serve as a first-order correction to the initial shear measurement, e.g., see Ref. [903,904]. These techniques, which can rely on model profiles, observed galaxy images, or priors from deep observations, have demonstrated the ability to control the residual shear bias to a percent or sub-percent level, nearly meeting the requirements of current Stage III surveys, e.g., see Ref. [905–907]. However, these data-based calibrations cannot correct biases introduced at the detection stage as required for the accuracy of Stage IV surveys, e.g., see Ref. [908].

**Detection biases and simulation-based calibration** Detection biases arise primarily from blending, where the light of neighboring galaxies overlaps in the image plane. Pixel noise and PSF convolution further exacerbate these effects, introducing shear-dependent detection biases, e.g., see Ref. [909–912]. Because these biases arise before shear measurement and are already imprinted in the data, they are difficult to calibrate using observations alone. Meta-calibration methods attempt to correct detection biases by introducing an additional detection step, known as metadetection, but this becomes challenging when accounting for redshift estimation [913]. Furthermore, the shear of galaxies varies with their environment and redshift, resulting in redshift-shear interplay effects that cannot be calibrated with observational data where the true shear is unknown, e.g., see Ref. [905,907]. Thus, simulation-based calibration is essential for shear measurement methods to meet the stringent requirements of Stage IV surveys. This approach originated from a series of community-wide blind challenges, where mock images with realistic galaxy properties and real data features were used to assess shear measurement algorithms [914–918]. With simulated images, where the ground truth is known, shear biases can be directly measured for any algorithm.

If the simulated images accurately replicate a survey's properties (e.g., resolution, PSF, signal-to-noise SNR, size and ellipticity distributions, source density, and clustering), the measured shear biases can be used to calibrate residual biases in real observations. This approach has been adopted by all Stage III surveys, e.g., see Ref. [905–907] and will remain vital for the Stage IV surveys.

**Impact on $S_8$ tension and future direction** Decades of dedicated work on shear measurement and calibration have significantly improved the accuracy of shear measurement. The latest KiDS cosmic shear analysis demonstrates that residual shear biases, primarily stemming from uncertainties in galaxy profiles, introduce only sub-percent uncertainties in $S_8$ estimates [919]. Shear measurement uncertainties are thus unlikely to be the main driver of the current $S_8$ tension.

Since shear measurement is closely tied to redshift estimation in real observations, developing a consistent, joint calibration of shear and redshift estimates using multi-band image simulations remains essential and will be a key focus for future simulation-based calibration, e.g., see Ref. [905,907]. Furthermore, due to the inherent limitations of image simulations — especially in capturing galaxy profile details — advancing shear measurement algorithms to reduce sensitivity to these intricacies remains a crucial goal.

### 2.2.1.2. Observational effects: Redshift measurement and calibration.
WL measurements rely on two-dimensional (projected) quantities, meaning individual source redshifts are not strictly required. However, redshift estimates are essential for cosmic shear tomography, which enhances the signal and probes the time evolution of the growth of structures. Historically, the requirements for photometric redshifts of individual lensing sources have been relatively lenient, but upcoming surveys such
as Euclid [164] and LSST [897] demand higher precision, e.g., see Ref. [920] for a review.

**Photometric redshift estimation** Photometric redshift estimation is crucial for extracting cosmological information in optical surveys like LSST and Euclid. In contrast to spectroscopic redshift measurement, photometric redshift inference uses image, or photometric information. Broad band optical surveys like LSST and Euclid take images in 100–200 nanometer wide optical filter bands, which does not allow the recovery of atomic lines. This limitation means that redshift estimation must rely on the overall shape and characteristic features of a galaxy’s spectral energy distribution (SED), for a review see Ref. [920,921]. Two primary approaches exist: ML and template fitting. ML methods [922–928] construct a direct mapping from observed photometry of individual galaxies to their redshift based on training data. While this training data does not need to spatially overlap, it has to be representative and complete in the color-redshift mapping being learned. Template fitting methods, e.g., see Ref. [929–935] match observed photometry to pre-existing SED models. It relies on accurate SED modeling and requires validation using similar calibration datasets. Although complementary, both techniques face challenges such as epistemic uncertainty in the SED and selection function models and incompleteness of accurate reference data at the faint end of the color-magnitude space, e.g., see Ref. [936–940].

**Redshift distribution calibration for cosmic shear** Beyond individual redshift estimates for cosmic shear sources, accurately calibrating their redshift distributions is crucial for interpreting observed signals with underlying cosmological models. For a given set of sources — usually selected in tomographic redshift bins — it is essential to recover their underlying true redshift distribution as closely as possible [941]. This is necessary to relate the observed signal to theoretical predictions, as the redshift distribution of sources determines the observed shape correlations via the matter power spectrum. Calibrating the redshift distribution in current WL surveys is challenging due to the faint nature of most cosmic shear sources and the limited availability of spectroscopic measurements. Consequently, a range of calibration methods have been developed, and efforts are ongoing to extend spectroscopic samples to better match the needs of cosmic shear surveys [937,942].

One of the earliest approaches for redshift distribution calibration, employed by e.g., the Deep Lens Survey (DLS) [943] or CFHTLenS [944], involved stacking photo-z posterior PDFs, derived with Bayesian Photometric Redshifts (BPZ) [945], to estimate the population distribution $dN/dz$. As Stage III surveys increased in statistical power, more reliable calibration methods were developed. These methods generally rely on either mapping the relation between colors and redshifts, or on the cross-correlation (clustering) approach. The former employs spectroscopic or narrow-filter multiband photometric calibration data to re-weight $dN/dz_{phot}$, either directly in magnitude space [858,859,946], by incorporating additional properties, e.g., see Ref. [947], or via self-organizing maps (SOM) [948]. Two key ingredients are required for these calibration methods: spectroscopic data that closely match the cosmic shear sample (in magnitude, color, and redshift range) and a sufficient number of multi-band filters to break degeneracies in the color-redshift mapping. This is most readily achievable in KiDS, which covers nine bands and includes multiple deep calibration fields [949, 950]. The DES redshift inference and calibration utilizes a mapping from deep multi-band to wide four-band data that is combined with spatial cross-correlations [951–953]. This choice is motivated by studies [940] that found that the photometric redshift biases from direct calibration strategies induced by the incompleteness in spectroscopic calibration samples are unacceptably high for DES. Similar to DES, the HSC calibration [954] relies on a combination of spectroscopic and narrow filter multiband calibration data as well as spatial cross-correlation measurements described in the following section. We note that the choice of photometric redshift calibration method and the

incorporation of systematic effects is dictated by the survey specification and cannot be generalized across survey designs. However it can be concluded that the treatment of selection functions of calibration sources poses a major challenge for DES, HSC, and KiDS.

The cross-correlation method, also known as clustering redshifts, leverages the spatial correlation of galaxy positions to estimate $dN/dz$ for a photometric sample by measuring its angular cross-correlation with a reference sample that has known redshifts, e.g., see Ref. [955, 956]. Either spectroscopic or high-fidelity photometric redshifts can serve as reference samples in this method. A major limitation of the clustering redshift approach is the availability of wide-angle calibration samples that extend to sufficiently high redshifts. As a result, its application in current cosmic shear surveys has been restricted primarily to the calibration of low-redshift bins [949,951,954].

**Impact on $S_8$ tension and future direction** Ref. [957] showed that an improved redshift calibration of the full sample or calibrations using different subsets change the $S_8$ obtained from the fiducial KiDS-1000 analysis at most at the level of $0.5\sigma$. For upcoming surveys, photometric redshift estimation based on ML methods could benefit from training sample augmentation with simulated galaxies possessing otherwise unrepresented features [958]. The complex selection function of spectroscopic surveys presents a major challenge in the calibration of photometric redshifts, especially at the faint end of the color-magnitude space [940].

Computationally efficient approaches for the marginalization over photometric redshift uncertainties are crucial to tackling the challenge of running inference in high-dimensional parameter spaces. Strategies can involve simplified $n(z)$ parametrizations, such as in terms of shifts of a mean redshift $\Delta z$ per bin, or a resampling approach using multiple MCMC chains at a fixed $n(z)$ sampled from the uncertainties. Ref. [959] showed that for the forecasted precision for HSC Y3, these methods recover statistically consistent error bars. However, when the constraining power of the full HSC survey (or other surveys of comparable or greater statistical power) is considered, the choice of the marginalization method may modify the $1\sigma$ uncertainties on $\Omega_{m,0} - S_8$ constraints by a few percent.

### 2.2.1.3. Astrophysical effects: Baryonic feedback.
With the increasing precision of modern cosmological surveys, small-scale astrophysical processes — commonly referred to as baryonic feedback — have become increasingly relevant in WL analyses. Baryonic feedback, driven by galaxy formation, supernova explosions, and AGN, suppresses the total matter power spectrum on scales of $k \sim 1-10\hbar\,\text{Mpc}^{-1}$ by re-distributing gas within and beyond halos and influencing through back-reaction [960]. Unlike gravity, which acts as a long-range force, baryonic effects follow an inside-out pattern: halos, stars, galaxies, and central supermassive black holes form on small scales through hierarchical structure formation, while supernova and AGN feedback modify the baryon distribution by, for example, expelling intrahalo gas into the IGM [961–965]. In particular, hydrodynamical simulations [966–970] indicate that AGN feedback is one of the most significant effects in suppressing the matter power spectrum at small scales. However, the spatial extent of the AGN ejected gas remains an open question — whether it affects a few Mpc or extends to tens of Mpc [969,971]. Below this maximum scale, the total matter distribution remains unchanged due to the conservation of mass and momentum. On smaller scales of $k > 10\hbar\,\text{Mpc}^{-1}$, baryonic effects such as gas cooling and star formation become efficient, leading to an upturn in the matter power spectrum [885]. Since these effects alter the total matter distribution, cosmic shear summary statistics — such as two-point correlation functions — are significantly affected compared to a purely collisionless scenario.

**Mitigation strategies** Despite the importance of baryonic feedback, modeling it from first principles remains challenging due to the complex and nonlinear nature of these processes. This introduces significant theoretical uncertainties in WL analyses. A common mitigation
<figure><img src="image_39.png" /><figcaption>**Fig. 34.** Suppression of the total matter power spectrum on small scales due to baryonic effects for various baryon models [887,891,960,972,973]. All spectra were computed for a flat ΛCDM cosmology with Ω_c = 0.25, Ω_b = 0.05, Ω_k = 0.0, σ_8 = 0.81, n_s = 0.96, and h = 0.67, assuming no massive neutrinos. The y-axis shows the ratio F(k) of each model including modeling of baryons, to the nonlinear matter power spectrum without baryonic effects. The nonlinear power spectrum was computed using pyccl.nonlin_matter_power method, while baryonic suppression was modeled with the baryonic modules in pyCCL [974], all evaluated at a = 1 (z = 0) over the same k-range.</figcaption></figure>

Source: Adapted from Ref. [975].

strategy is to apply scale cuts to the summary statistics, excluding small scale data where hydrodynamical simulations suggest feedback effects dominate [860–862,864,871]. While this method preserves the robustness of cosmological constraints, it comes at the cost of discarding high SNR scales that are rich in cosmological information, thereby limiting the constraining power of WL surveys. To fully exploit the potential of upcoming Stage IV surveys, it is crucial to incorporate baryonic effects into cosmological analyses rather than simply removing affected scales.

Several modeling strategies have been developed to address this issue by incorporating baryonic feedback into the total matter power spectrum. One widely used approach is the baryon correction model (BCM), which modifies the position of particles in N-body simulations using parametric prescriptions that approximate the impact of feedback on the DM halo profiles [887,962,972,976].

Another method integrates baryonic effects within the halo model framework by modifying density profiles and gas expulsion mechanisms [891,892,977]. Some of these models and their impact on the matter power spectrum are summarized in Fig. 34. Principal Component Analysis (PCA) and alternative basis function approaches provide a more data-driven way to capture a broad range of feedback scenarios [978]. The most detailed recipe comes from cosmological hydrodynamical simulations, which explicitly simulate baryonic physics using sub-grid prescriptions for unresolved processes such as supernova and AGN feedback [885]. However, discrepancies arise due to differences in sub-grid implementations, calibration strategies, and simulation methods. These variations lead to disagreements in both the amplitude and scale dependence of the suppression of the total matter power spectrum [971]. Consequently, an effective model for WL must be flexible enough to encompass the full range of plausible baryonic effects. Moreover, other nonlinear effects — including intrinsic alignments, nonlinear clustering, and reduced shear corrections — must be accounted for to ensure robust cosmological constraints. The aforementioned strategies provide a way to marginalize over baryonic feedback uncertainties while preserving small-scale cosmological information. However, degeneracies between baryonic feedback and cosmological parameters may degrade the constraints on $S_8$, highlighting the need of estimators beyond the two-point functions that can disentangle these

effects while maximizing the constraining power of next-generation surveys.

Higher-order statistics of the WL field, such as bispectrum, peak counts [849,979] and the one-point lensing PDF [836], are more sensitive to non-Gaussian and smaller-scale structures than two-point correlations [847,980–982]. Thus, combining the two-point correlations with the higher-order statistics would allow us to obtain tighter constraints on cosmological parameters [842,846], while self-calibrating the baryonic effects [983].

Finally, multi-probe approaches — combining WL with X-ray, measurements of the Sunyaev–Zeldovich (SZ) effect (thermal tSZ and kinetic kSZ), and FRB observations — offer complementary constraints on the distribution of baryons in large-scale structures [984,985,985–988]. These techniques collectively help preserve cosmological information while minimizing biases from astrophysical processes.

**Impact on cosmic tension and $S_8$** To explore the impact of baryons on the inferred $S_8$, Stage III surveys have conducted parameter inference with two-point functions down to small scales. They found that moderate to small baryonic effects are present in the data [976, 989–991]. In particular, Ref. [989] performed a joint cosmic shear analysis of the Stage III surveys DES-Y3, KiDS-1000 and HSC-DR1 using small scale data in harmonic space. This reanalysis implements a single pipeline that extends the cosmic shear angular power spectrum to scales up to $\ell < 4500$, using BACC0emu to model the non-linear regime and baryonic effects in the total matter power spectrum. Their resulting $S_8$ constraint is $1.8\sigma$ lower than Planck, however, [989] found a $\Omega_{m,0}$ tension between both data sets of $3.5\sigma$. When analyzing the parameter space in terms of variables without an implicit dependence on the Hubble constant $H_0$, ($S_{12}, w_m$), the authors find no tension with Planck. On the other hand, Ref. [990] explore signatures of baryonic effects in the HSC Y3 two-point correlation functions $\xi_{\pm}$ down to small scales of $0.28$ arcminutes (up to $k \sim 20h \text{ Mpc}^{-1}$). The theoretical modeling is implemented by means of a non-linear DM-only matter power spectrum emulator (similar to DARKEMULATOR [992]). The authors found no significant shift in the inferred $S_8$ when including the small scales. Moreover, the DM-only theory fits the observations within the statistical error of the survey, meaning baryons are not playing an important role in these measurements. Similar results were obtained for HSC Y1 data using multiple higher-order statistics at small scales [847]. These studies of cosmic shear data, together with other analyses from the Sunyaev-Zel'dovich observations, indicate that strong feedback scenarios are needed to reconcile the $S_8$ inferred from cosmic shear surveys with the value derived from Planck ΛCDM [993]. This translates into a large suppression of the matter power spectrum, of the order of ~25% at $k \sim 1h \text{ Mpc}^{-1}$. The latter, however, is larger than the suppression found from hydrodynamical simulations, and contradicts the constraints imposed by X-ray observations of the baryon mass fraction for cluster-scale halos [990,991]. Therefore, it is possible that baryons in combination with other non-linear effects and unmodeled systematics may explain the discrepancy between cosmic shear results and Planck ΛCDM cosmology, or extensions of ΛCDM and new physics are required.

2.2.1.4. Astrophysical effects: Intrinsic alignments of galaxies. Intrinsic alignments (IA) of galaxies represent a significant astrophysical effect that can systematically affect the measurement of cosmic shear and, consequently, the inference of cosmological parameters (see Ref. [994] for a comprehensive review). These alignments arise due to gravitational interactions and the tidal forces exerted by the large-scale structure of the Universe, causing the shapes of nearby galaxies to correlate with each other and with the surrounding matter density field.

For cosmic shear two-point correlation functions, intrinsic alignments can be categorized primarily into two types: intrinsic–intrinsic (II) alignments and gravitational–intrinsic (GI) and intrinsic–gravitational (IG) alignments. II alignments occur when the shapes of
physically close galaxies are directly influenced by the same gravita-
tional tidal field, leading them to align with each other. GI and IG
alignments describe the correlation between intrinsic galaxy shapes and
the shear induced by gravitational lensing. GI refers to the correlation
between the intrinsic shape of a foreground galaxy and the shear
of a background galaxy, while IG represents the reverse case [995].
These terms are related through symmetry properties of the shear field,
with IG often being treated as the transpose of GI in theoretical and
numerical treatments.

Galaxy-galaxy lensing (GGL) probe, or the correlation between galaxy shapes and galaxy positions, is also sensitive to intrinsic alignments when the shape and the position catalog overlap in redshift. GGL is sensitive only to GI types of alignments and through a different kernel than for cosmic shear (see equations in e.g., Ref. [996]). Because of this, 3 × 2pt analyses significantly help to constrain the intrinsic alignment parameters. Recently, GGL shear-ratios have been used to fold in prior IA information to the cosmic shear analysis [861,862,997]. Recently, inverse galaxy-galaxy lensing has also been proposed as an additional probe to constrain IA parameters [998].

**Modeling intrinsic alignments** Intrinsic alignment (IA) models vary in complexity and scale-dependent effectiveness. The Nonlinear Alignment Model (NLA) [999] has been the standard approach to model red (elliptical) galaxies. This model assumes a linear coupling between the nonlinear tidal field and galaxy shapes, typically parameterized by an amplitude parameter alongside an optional redshift evolution parameter. In this framework, blue (spiral) galaxies are assumed to have negligible intrinsic alignments. However, the linear tidal shear assumption breaks down on small scales, necessitating more sophisticated models.

The Tidal Alignment and Tidal Torquing (TATT) model [1000] represents an important extension, introducing additional terms to capture the alignments of spiral galaxies where angular momentum plays a critical role. This model incorporates a separate amplitude parameter for the tidal torquing effect, along with its own redshift scaling parameter. In total, the basic TATT model typically employs four parameters: $A_1$ (tidal alignment amplitude), $A_2$ (tidal torquing amplitude), $\eta_1$ (alignment redshift scaling), and $\eta_2$ (torquing redshift scaling). A fifth parameter, $b_{TA}$ (the linear bias of source galaxies), is frequently included to account for the clustered distribution of source galaxies in observational data. Many studies such as Refs. [1001–1003] demonstrate that while NLA fits measurements well above 6–8 Mpc/h, TATT extends this range down to 1–2 Mpc/h. These findings emerge from both simulation-based tests (N-body with semi-analytic IA components or hydrodynamic simulations) and direct fits to observational data.

For modeling at even smaller scales, the halo model describes
alignments using one-halo and two-halo terms, based on galaxy cor-
relations within and between DM halos [994,1004] while the Effective
Field Theory (EFT) framework treats alignments as a tensor field and
incorporates small-scale physics through free parameters. Ref. [1005]
shows that EFT describes DM halo intrinsic alignments up to $k_{max} =$
0.30 h/Mpc at $z = 0$, significantly outperforming both NLA and TATT
which they find to only reach $k_{max} = 0.05$ h/Mpc.

Despite these advances in model complexity, practical applications raise important questions about necessary sophistication. The DES Y3 cosmic shear analysis [861] found that NLA was sufficient to fit their data, even though TATT had been chosen for fiducial results in a blinded fashion. Similarly, Ref. [1001] found no significant evidence for non-zero tidal torquing amplitude ($A_2$) in Illustris TNG hydrodynamic simulations, challenging the need for TATT's full 5-parameter framework in typical applications. Moreover, it is important to note that increasing model complexity does not necessarily reduce biases. The interplay between IA and other uncertainties (e.g., from photometric redshift errors [1006]) can introduce additional complications. Recent work [1007] shows that jointly modeling IA and source redshift

distributions via a shared LF can help mitigate such biases while main-
taining consistency in WL analyses. Careful calibration and validation
remain crucial to ensure these models accurately represent physical
processes without introducing unintended distortions in cosmologi-
cal analyses. To navigate these modeling challenges, researchers have
developed strategic approaches for IA model selection. Ref. [1008]
recently introduced a data-driven methodology to objectively deter-
mine which IA model best suits a particular analysis, providing a more
systematic framework for model choice. In parallel, many cosmological
studies have adopted a multi-model approach, presenting results across
several IA formulations to demonstrate robustness and identify poten-
tial model-dependent biases. Below, we examine how these modeling
decisions have shaped recent cosmological parameter constraints and
what they reveal about the relative importance of IA modeling in
contemporary WL analyses.

**2.2.1.5. Modeling IA for higher order statistics.** While IA modeling has primarily focused on two-point statistics, recent efforts have extended these frameworks beyond 2-point statistics. Analytical advances have provided foundations for modeling IA in bispectra [1009,1010] and the lensing PDF [1011], while Ref. [1012] demonstrated that high signal-to-noise peaks in aperture mass maps can experience deviations up to 30% for Euclid-like surveys due to IA effects. Many recent analyses employ simulation-based inference, which relies on map-level IA modeling and presents unique challenges. For example, Ref. [1013] showed that source clustering impacts non-Gaussian statistics significantly more than two-point functions [1014]. Current approaches address this either by implementing additional clustering terms in analytical predictions [1011] or by directly incorporating clustering effects in simulations [879]. The latter approach extends beyond traditional NLA implementation by using simulation-based clustering rather than tree-level perturbation theory, showing that modeling IA at the map level can also offer advantages for various summary statistics.

**Impact on cosmic tension and $S_8$ Mise stimations in intrinsic**
alignments can lead to significant biases in the structure growth pa-
rameter $S_8$, thereby affecting our understanding of cosmic tensions.
Recent cosmic shear analyses, as illustrated in Fig. 35, demonstrate
that among various analysis choices, the selection of IA model often
produces shifts in the $S_8$ constraints, which will become substantial for
Stage IV surveys [1015]. The TATT model (pink points) tends to yield
somewhat lower $S_8$ values compared to the NLA model (blue points)
across different analysis combinations. This difference, while modest,
may be relevant when considering potential tensions between early and
late-Universe probes, as indicated by the comparison with the reference
line in the figure. The largest change comes from the joint analysis of
data from the DES and KiDS, which revealed that transitioning from the
NLA model to the more complex TATT model can lead to heightened
tension in $S_8$ measurements. Quantitatively, the difference between
the redshift-dependent NLA and TATT analysis results was 0.9$\sigma$ in
the $S_8$ marginalized posterior, but reached 1.3$\sigma$ when comparing the
maximum a posteriori (MAP) values in the full parameter space. This
distinction is particularly significant as it indicates that the discrepancy
arises from genuine differences in model physics rather than merely
from differences in prior volumes or projection effects. These results
highlight the importance of carefully considering IA modeling choices
when interpreting cosmological constraints from WL surveys. 3 × 2pt
analyses in DES Y3 were less susceptible to the IA model compared to
statistics based on cosmic shear alone.

**Future directions** As WL analyses continue to advance, refin-
ing IA models and developing more realistic cosmological simulations
will be essential for providing robust validation tests. These improve-
ments will work synergistically with upcoming surveys such as LSST
by the Vera C. Rubin Observatory, which will deliver high-quality
imaging across unprecedented sky areas, enabling studies of IA with
greater statistical power and tightening constraints on cosmic shear
measurements.