arXiv subjects

P. Natoli

Publications and source records attributed to P. Natoli.

At least 19 recordsLinked to original sources

arXiv2026

Galactic Science with the LiteBIRD satellite: Spectral characterization of diffuse Galactic polarized emission at the angular power spectrum level

Detection of primordial $B$-mode polarization in the cosmic microwave background (CMB) from tensor perturbations generated during inflation is a major scientific goal of future CMB missions. Its success will strongly depend on the characterization of polarized foregrounds, a challenge that the LiteBIRD satellite aims to tackle with its 15 frequency bands ranging from 40 to 402 GHz. In this work, we forecast the ability of LiteBIRD to characterize polarized dust and synchrotron emission in the diffuse interstellar medium (ISM), at the angular power spectrum level. From simulated LiteBIRD intensity and polarization maps with different foreground complexities, we compute cross-frequency angular power spectra and fit them to dust and synchrotron spectral energy distributions, which are modeled by a modified black body and a power law, respectively. We find that LiteBIRD will be able to measure the dust temperature, dust and synchrotron spectral indices and spatial correlation with dispersions as low as $\sigma(T_{\rm d})\sim0.2$ K, $\sigma(\beta_{\rm d})\sim0.006$, $\sigma(\beta_{\rm s})\sim0.04$ and $\sigma(\rho)\sim10^{-2}$, as well as to detect and quantify deviations from the proposed parametric model due to variations of the emission properties in the three dimensions of our Galaxy. Additionally, LiteBIRD is likely to rule out the power-law model of polarized foreground angular power spectra suggested by Planck data. It will also be able to detect differences in the values of $\beta_{\rm d}$, $T_{\rm d}$, and $\beta_{\rm s}$ between $E$ modes, $B$ modes, and intensity in the diffuse ISM for the first time, highlighting the joint variations of the physical conditions and the magnetic field structure across the Galaxy. We conclude that in addition to detailed studies of CMB polarization, LiteBIRD will open a new window onto the physical conditions governing the ISM of the Milky Way.

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arXiv2026

A cross-calibration approach for polarisation-sensitive detectors in CMB experiments: application to LiteBIRD's polarisation angle calibration

One of the current challenges in observational cosmology is obtaining high-precision polarisation maps of the CMB to measure primordial $B$-modes and constrain the tensor-to-scalar ratio ($r$). The weakness of this signal compared to foregrounds and $E$-to-$B$ leakage makes this task particularly challenging, requiring large detector arrays operating at multiple frequencies and extremely precise calibration. We present a cross-calibration algorithm to determine relative calibration of detectors within the same frequency band of a CMB experiment. The method iteratively compares single-detector maps with band-averaged maps and can be applied to any calibration parameter that can be observed and corrected at the map level, relaxing pre-flight calibration requirements and enabling post-processing validation. We validate the pipeline by calibrating the polarisation angle of simulated LiteBIRD observations, including both random detector miscalibration and wafer-level rotations. The algorithm converges to correct values with arcminute precision. Finally, we propagate residual calibration uncertainties through component separation and tensor-to-scalar ratio estimation pipelines using both parametric (FgBuster) and blind (HILC) methods. The induced bias on $r$ remains well below the LiteBIRD systematics budget of $\delta r < 6.5\times10^{-6}$, demonstrating that the method is suitable for next-generation CMB experiments.

astro-ph.CO↗

arXiv2026

Mitigating half-wave plate systematics at the map-making level: calibration requirements for LiteBIRD

Although half-wave plates (HWPs) are becoming a popular choice of polarization modulators for cosmic microwave background (CMB) experiments, their non-idealities can introduce systematic effects that should be carefully characterized and mitigated. One possible mitigation strategy is to incorporate information about the non-idealities at the map-making level, which helps to reduce the HWP-induced distortions of the reconstructed CMB. Nevertheless, the non-idealities can only be known with finite precision. In this paper we investigate the consequences of discrepancies between their true frequency profiles and those assumed by the map-maker. We present an end-to-end framework, including a blind component-separation step, and use it to translate these discrepancies into a bias on the tensor-to-scalar ratio, $r$, for the LiteBIRD satellite mission. We subsequently derive realistic and conservative measurement requirements for accurately characterizing the HWP non-idealities to ensure they do not compromise LiteBIRD's ambitious scientific goals. We find that the obtained results are robust against sky models with varying complexity.

astro-ph.CO↗

arXiv2025

Stairway to Axions: the cross-correlation of birefringence and galaxies from NPIPE and Quaia

We present the first measurement of the cross-correlation between anisotropic birefringence and galaxy number counts, utilizing polarization data from Planck NPIPE and the Quaia quasar catalog. By employing a QML/pseudo-$C_\ell$ combined estimator, we compute the angular power spectrum up to $\ell=191$ from birefringence and clustering maps at $N_{\rm side}=64$. Our analysis indicates that the observed spectrum is well consistent with the null-hypothesis, with a probability to exceed of 37% and an estimated scale-invariant amplitude of $A^{\mathcal{D}_\ell}=(2.22\pm2.09)\times10^{-4}\,\text{deg}$, at the 68% confidence level. Finally, we derive constraints on the axion-parameters within an early dark energy model of birefringence. Our findings reveal an unprecedented upper bound on the axion-photon coupling down to $g_{\phi\gamma}=10^{-15}\,\text{GeV}^{-1}$ for masses around $10^{-32}\,\text{eV}$ and high initial misalignment angles. This result opens a previously unexplored window in parameter space, providing the first constraint in this ultra-light mass regime.

astro-ph.CO↗

arXiv2025

Planck constraints on the scale dependence of isotropic cosmic birefringence

The rotation of the linear polarisation plane of photons during propagation, also known as cosmic birefringence, is a powerful probe of parity-violating extensions of standard electromagnetism. Using Planck legacy data, we confirm previous estimates of the isotropic birefringence angle, finding $\beta \simeq 0.30 \pm 0.05$ [deg] at 68% CL, not including the systematic error from the instrumental polarisation angle. If this is a genuine signal, it could be explained by theories of Chern--Simons-type coupled to electromagnetism, which could lead to a harmonic scale-dependent birefringence signal, if the hypothesis of an ultra-light (pseudo) scalar field does not hold. To investigate these models, we pursue two complementary approaches: first, we fit the birefringence angle estimated at different multipoles, $\beta_{\ell}$, with a power-law model and second, we perform a non-parametric Bayesian reconstruction of it. Both methods yield results consistent with a non-vanishing constant birefringence angle. The first method shows no significant dependence on the harmonic scale (up to $1.8\sigma$ CL), while the second method demonstrates that a constant model is favored by Bayesian evidence. This conclusion is robust across all four published Planck CMB solutions. Finally, we forecast that upcoming CMB observations by Simons Observatory, LiteBIRD and a wishful CMB-Stage 4 experiment could reduce current uncertainties by a factor of approximately 7.

astro-ph.CO↗

arXiv2025

LiteBIRD Science Goals and Forecasts: constraining isotropic cosmic birefringence

Cosmic birefringence (CB) is the rotation of the photons' linear polarisation plane during propagation. Such an effect is a tracer of parity-violating extensions of standard electromagnetism and would probe the existence of a new cosmological field acting as dark matter or dark energy. It has become customary to employ cosmic microwave background (CMB) polarised data to probe such a phenomenon. Recent analyses on Planck and WMAP data provide a hint of detection of the isotropic CB angle with an amplitude of around $0.3^\circ$ at the level of $2.4$ to $3.6\sigma$. In this work, we explore the LiteBIRD capabilities in constraining such an effect, accounting for the impact of the more relevant systematic effects, namely foreground emission and instrumental polarisation angles. We build five semi-independent pipelines and test these against four different simulation sets with increasing complexity in terms of non-idealities. All the pipelines are shown to be robust and capable of returning the expected values of the CB angle within statistical fluctuations for all the cases considered. We find that the uncertainties in the CB estimates increase with more complex simulations. However, the trend is less pronounced for pipelines that account for the instrumental polarisation angles. For the most complex case analysed, we find that LiteBIRD will be able to detect a CB angle of $0.3^\circ$ with a statistical significance ranging from $5$ to $13 \, \sigma$, depending on the pipeline employed, where the latter uncertainty corresponds to a total error budget of the order of $0.02^\circ$.

astro-ph.CO↗

arXiv2024

Requirements on the gain calibration for LiteBIRD polarisation data with blind component separation

Future cosmic microwave background (CMB) experiments are primarily targeting a detection of the primordial $B$-mode polarisation. The faintness of this signal requires exquisite control of systematic effects which may bias the measurements. In this work, we derive requirements on the relative calibration accuracy of the overall polarisation gain ($\Delta g_\nu$) for LiteBIRD experiment, through the application of the blind Needlet Internal Linear Combination (NILC) foreground-cleaning method. We find that minimum variance techniques, as NILC, are less affected by gain calibration uncertainties than a parametric approach, which requires a proper modelling of these instrumental effects. The tightest constraints are obtained for frequency channels where the CMB signal is relatively brighter (166 GHz channel, $\Delta {g}_\nu \approx 0.16 \%$), while, with a parametric approach, the strictest requirements were on foreground-dominated channels. We then propagate gain calibration uncertainties, corresponding to the derived requirements, into all frequency channels simultaneously. We find that the overall impact on the estimated $r$ is lower than the required budget for LiteBIRD by almost a factor $5$. The adopted procedure to derive requirements assumes a simple Galactic model. We therefore assess the robustness of obtained results against more realistic scenarios by injecting the gain calibration uncertainties, according to the requirements, into LiteBIRD simulated maps and assuming intermediate- and high-complexity sky models. In this case, we employ the so-called Multi-Clustering NILC (MC-NILC) foreground-cleaning pipeline and obtain that the impact of gain calibration uncertainties on $r$ is lower than the LiteBIRD gain systematics budget for the intermediate-complexity sky model. For the high-complexity case, instead, it would be necessary to tighten the requirements by a factor $1.8$.

astro-ph.CO↗

arXiv2024

Unveiling V Modes: Enhancing CMB Sensitivity to BSM Physics with a Non-Ideal Half-Wave Plate

V-mode polarization of the cosmic microwave background is expected to be vanishingly small in the $\Lambda$CDM model and, hence, usually ignored. Nonetheless, several astrophysical effects, as well as beyond standard model physics could produce it at a detectable level. A realistic half-wave plate - an optical element commonly used in CMB experiments to modulate the polarized signal - can provide sensitivity to V modes without significantly spoiling that to linear polarization. We assess this sensitivity for some new-generation CMB experiments, such as the LiteBIRD satellite, the ground-based Simons Observatory and a CMB-S4-like experiment. We forecast the efficiency of these experiments to constrain the phenomenology of certain classes of BSM models inducing mixing of linear polarization states and generation of V modes in the CMB. We find that new-generation experiments can improve current limits by 1-to-3 orders of magnitude, depending on the data combination. The inclusion of V-mode information dramatically boosts the sensitivity to these BSM models.

astro-ph.CO↗

arXiv2024

Conversations in the dark: cross-correlating birefringence and LSS to constrain axions

Unveiling the dark sector of the Universe is one of the leading efforts in theoretical physics. Among the many models proposed, axions and axion-like particles stand out due to their solid theoretical foundation, capacity to contribute significantly to both dark matter and dark energy, and potential to address the small-scale crisis of $\Lambda$CDM. Moreover, these pseudo-scalar fields couple to the electromagnetic sector through a Chern-Simons parity-violating term, leading to a rotation of the plane of linearly polarized waves, namely cosmic birefringence. We explore the impact of the axion-parameters on anisotropic birefringence and study, for the first time, its cross-correlation with the spatial distribution of galaxies, focusing on ultralight axions with masses $10^{-33}\,{\rm eV}\le m_\phi\le10^{-28}\,{\rm eV}$. Through this novel approach, we investigate the axion-parameter space in the mass $m_\phi$ and initial misalignment angle $\theta_i$, within the framework of early dark energy models, and constrain the axion-photon coupling $g_{\phi\gamma}$ required to achieve unity in the signal-to-noise ratio of the underlying cross-correlation, computed with the instrument specifications of Euclid and forthcoming CMB-polarization data. Our findings reveal that for masses below $10^{-32}\,{\rm eV}$ and initial misalignment angles greater in absolute value than $\pi/4$, the signal-to-noise ratio not only exceeds unity but also surpasses that achievable from the auto-correlation of birefringence alone (up to a factor 7), highlighting the informative potential of this new probe. Additionally, given the late-time evolution of these low-mass axions, the signal stems from the epoch of reionization, providing an excellent tool to single out the birefringence generated during this period.

astro-ph.CO↗

arXiv2024

LiteBIRD Science Goals and Forecasts: Primordial Magnetic Fields

We present detailed forecasts for the constraints on primordial magnetic fields (PMFs) that will be obtained with the LiteBIRD satellite. The constraints are driven by the effects of PMFs on the CMB anisotropies: the gravitational effects of magnetically-induced perturbations; the effects on the thermal and ionization history of the Universe; the Faraday rotation imprint on the CMB polarization; and the non-Gaussianities induced in polarization anisotropies. LiteBIRD represents a sensitive probe for PMFs and by exploiting all the physical effects, it will be able to improve the current limit coming from Planck. In particular, thanks to its accurate $B$-mode polarization measurement, LiteBIRD will improve the constraints on infrared configurations for the gravitational effect, giving $B_{\rm 1\,Mpc}^{n_{\rm B} =-2.9} < 0.8$ nG at 95% C.L., potentially opening the possibility to detect nanogauss fields with high significance. We also observe a significant improvement in the limits when marginalized over the spectral index, $B_{1\,{\rm Mpc}}^{\rm marg}< 2.2$ nG at 95% C.L. From the thermal history effect, which relies mainly on $E$-mode polarization data, we obtain a significant improvement for all PMF configurations, with the marginalized case, $\sqrt{\langle B^2\rangle}^{\rm marg}<0.50$ nG at 95% C.L. Faraday rotation constraints will take advantage of the wide frequency coverage of LiteBIRD and the high sensitivity in $B$ modes, improving the limits by orders of magnitude with respect to current results, $B_{1\,{\rm Mpc}}^{n_{\rm B} =-2.9} < 3.2$ nG at 95% C.L. Finally, non-Gaussianities of the $B$-mode polarization can probe PMFs at the level of 1 nG, again significantly improving the current bounds from Planck. Altogether our forecasts represent a broad collection of complementary probes, providing conservative limits on PMF characteristics that will be achieved with LiteBIRD.

astro-ph.CO↗

arXiv2024

Planck constraints on Cosmic Birefringence and its cross-correlation with the CMB

Cosmic birefringence is the in-vacuo, frequency independent rotation of the polarization plane of linearly polarized radiation, induced by a parity-violating term in the electromagnetic Lagrangian. We implement an harmonic estimator for the birefringence field that only relies on the CMB E to B mode cross-correlation, thus suppressing the effect of cosmic variance from the temperature field. We derive constraints from Planck public releases 3 and 4, revealing a cosmic birefringence power spectrum consistent with zero at about $2\sigma$ up to multipole $L=1500$. Moreover, we find that the cross-correlations of cosmic birefringence with the CMB T-, E- and B-fields are also well compatible with null. The latter two cross-correlations are provided here for the first time up to $L=1500$.

astro-ph.CO↗

arXiv2023

Impact of beam far side-lobe knowledge in the presence of foregrounds for LiteBIRD

We present a study of the impact of an uncertainty in the beam far side-lobe knowledge on the measurement of the Cosmic Microwave Background $B$-mode signal at large scale. It is expected to be one of the main source of systematic effects in future CMB observations. Because it is crucial for all-sky survey missions to take into account the interplays between beam systematic effects and all the data analysis steps, the primary goal of this paper is to provide the methodology to carry out the end-to-end study of their effect for a space-borne CMB polarization experiment, up to the cosmological results in the form of a bias $\delta r$ on the tensor-to-scalar ratio $r$. LiteBIRD is dedicated to target the measurement of CMB primordial $B$ modes by reaching a sensitivity of $\sigma \left( r \right) \leq 10^{-3}$ assuming $r=0$. As a demonstration of our framework, we derive the relationship between the knowledge of the beam far side-lobes and the tentatively allocated error budget under given assumptions on design, simulation and component separation method. We assume no mitigation of the far side-lobes effect at any stage of the analysis pipeline. We show that $\delta r$ is mostly due to the integrated fractional power difference between the estimated beams and the true beams in the far side-lobes region, with little dependence on the actual shape of the beams, for low enough $\delta r$. Under our set of assumptions, in particular considering the specific foreground cleaning method we used, we find that the integrated fractional power in the far side-lobes should be known at a level as tight as $\sim 10^{-4}$, to achieve the required limit on the bias $\delta r < 1.9 \times 10^{-5}$. The framework and tools developed for this study can be easily adapted to provide requirements under different design, data analysis frameworks and for other future space-borne experiments beyond LiteBIRD.

astro-ph.CO↗

arXiv2023

LiteBIRD Science Goals and Forecasts: A full-sky measurement of gravitational lensing of the CMB

We explore the capability of measuring lensing signals in $LiteBIRD$ full-sky polarization maps. With a $30$ arcmin beam width and an impressively low polarization noise of $2.16\,\mu$K-arcmin, $LiteBIRD$ will be able to measure the full-sky polarization of the cosmic microwave background (CMB) very precisely. This unique sensitivity also enables the reconstruction of a nearly full-sky lensing map using only polarization data, even considering its limited capability to capture small-scale CMB anisotropies. In this paper, we investigate the ability to construct a full-sky lensing measurement in the presence of Galactic foregrounds, finding that several possible biases from Galactic foregrounds should be negligible after component separation by harmonic-space internal linear combination. We find that the signal-to-noise ratio of the lensing is approximately $40$ using only polarization data measured over $90\%$ of the sky. This achievement is comparable to $Planck$'s recent lensing measurement with both temperature and polarization and represents a four-fold improvement over $Planck$'s polarization-only lensing measurement. The $LiteBIRD$ lensing map will complement the $Planck$ lensing map and provide several opportunities for cross-correlation science, especially in the northern hemisphere.

astro-ph.CO↗

arXiv2023

LiteBIRD Science Goals and Forecasts: Improving Sensitivity to Inflationary Gravitational Waves with Multitracer Delensing

We estimate the efficiency of mitigating the lensing $B$-mode polarization, the so-called delensing, for the $LiteBIRD$ experiment with multiple external data sets of lensing-mass tracers. The current best bound on the tensor-to-scalar ratio, $r$, is limited by lensing rather than Galactic foregrounds. Delensing will be a critical step to improve sensitivity to $r$ as measurements of $r$ become more and more limited by lensing. In this paper, we extend the analysis of the recent $LiteBIRD$ forecast paper to include multiple mass tracers, i.e., the CMB lensing maps from $LiteBIRD$ and CMB-S4-like experiment, cosmic infrared background, and galaxy number density from $Euclid$- and LSST-like survey. We find that multi-tracer delensing will further improve the constraint on $r$ by about $20\%$. In $LiteBIRD$, the residual Galactic foregrounds also significantly contribute to uncertainties of the $B$-modes, and delensing becomes more important if the residual foregrounds are further reduced by an improved component separation method.

astro-ph.CO↗

arXiv2023

LiteBIRD Science Goals and Forecasts. A Case Study of the Origin of Primordial Gravitational Waves using Large-Scale CMB Polarization

We study the possibility of using the $LiteBIRD$ satellite $B$-mode survey to constrain models of inflation producing specific features in CMB angular power spectra. We explore a particular model example, i.e. spectator axion-SU(2) gauge field inflation. This model can source parity-violating gravitational waves from the amplification of gauge field fluctuations driven by a pseudoscalar "axionlike" field, rolling for a few e-folds during inflation. The sourced gravitational waves can exceed the vacuum contribution at reionization bump scales by about an order of magnitude and can be comparable to the vacuum contribution at recombination bump scales. We argue that a satellite mission with full sky coverage and access to the reionization bump scales is necessary to understand the origin of the primordial gravitational wave signal and distinguish among two production mechanisms: quantum vacuum fluctuations of spacetime and matter sources during inflation. We present the expected constraints on model parameters from $LiteBIRD$ satellite simulations, which complement and expand previous studies in the literature. We find that $LiteBIRD$ will be able to exclude with high significance standard single-field slow-roll models, such as the Starobinsky model, if the true model is the axion-SU(2) model with a feature at CMB scales. We further investigate the possibility of using the parity-violating signature of the model, such as the $TB$ and $EB$ angular power spectra, to disentangle it from the standard single-field slow-roll scenario. We find that most of the discriminating power of $LiteBIRD$ will reside in $BB$ angular power spectra rather than in $TB$ and $EB$ correlations.

astro-ph.CO↗

arXiv2023

Tensor-to-scalar ratio forecasts for extended LiteBIRD frequency configurations

LiteBIRD is a planned JAXA-led CMB B-mode satellite experiment aiming for launch in the late 2020s, with a primary goal of detecting the imprint of primordial inflationary gravitational waves. Its current baseline focal-plane configuration includes 15 frequency bands between 40 and 402 GHz, fulfilling the mission requirements to detect the amplitude of gravitational waves with the total uncertainty on the tensor-to-scalar ratio, $\delta r$, down to $\delta r<0.001$. A key aspect of this performance is accurate astrophysical component separation, and the ability to remove polarized thermal dust emission is particularly important. In this paper we note that the CMB frequency spectrum falls off nearly exponentially above 300 GHz relative to the thermal dust SED, and a relatively minor high frequency extension can therefore result in even lower uncertainties and better model reconstructions. Specifically, we compare the baseline design with five extended configurations, while varying the underlying dust modeling, in each of which the HFT (High-Frequency Telescope) frequency range is shifted logarithmically towards higher frequencies, with an upper cutoff ranging between 400 and 600 GHz. In each case, we measure the tensor-to-scalar ratio $r$ uncertainty and bias using both parametric and minimum-variance component-separation algorithms. When the thermal dust sky model includes a spatially varying spectral index and temperature, we find that the statistical uncertainty on $r$ after foreground cleaning may be reduced by as much as 30--50 % by extending the upper limit of the frequency range from 400 to 600 GHz, with most of the improvement already gained at 500 GHz. We also note that a broader frequency range leads to better ability to discriminate between models through higher $\chi^2$ sensitivity. (abridged)

astro-ph.CO↗

arXiv2022

Polarization angle requirements for CMB B-mode experiments. Application to the LiteBIRD satellite

A methodology to provide the polarization angle requirements for different sets of detectors, at a given frequency of a CMB polarization experiment, is presented. The uncertainties in the polarization angle of each detector set are related to a given bias on the tensor-to-scalar ratio $r$ parameter. The approach is grounded in using a linear combination of the detector sets to obtain the CMB polarization signal. In addition, assuming that the uncertainties on the polarization angle are in the small angle limit (lower than a few degrees), it is possible to derive analytic expressions to establish the requirements. The methodology also accounts for possible correlations among detectors, that may originate from the optics, wafers, etc. The approach is applied to the LiteBIRD space mission. We show that, for the most restrictive case (i.e., full correlation of the polarization angle systematics among detector sets), the requirements on the polarization angle uncertainties are of around 1 arcmin at the most sensitive frequency bands (i.e., $\approx 150$ GHz) and of few tens of arcmin at the lowest (i.e., $\approx 40$ GHz) and highest (i.e., $\approx 400$ GHz) observational bands. Conversely, for the least restrictive case (i.e., no correlation of the polarization angle systematics among detector sets), the requirements are $\approx 5$ times less restrictive than for the previous scenario. At the global and the telescope levels, polarization angle knowledge of a few arcmins is sufficient for correlated global systematic errors and can be relaxed by a factor of two for fully uncorrelated errors in detector polarization angle. The reported uncertainty levels are needed in order to have the bias on $r$ due to systematics below the limit established by the LiteBIRD collaboration.

astro-ph.CO↗

arXiv2021

$Euclid$ preparation: XV. Forecasting cosmological constraints for the $Euclid$ and CMB joint analysis

The combination and cross-correlation of the upcoming $Euclid$ data with cosmic microwave background (CMB) measurements is a source of great expectation since it will provide the largest lever arm of epochs, ranging from recombination to structure formation across the entire past light cone. In this work, we present forecasts for the joint analysis of $Euclid$ and CMB data on the cosmological parameters of the standard cosmological model and some of its extensions. This work expands and complements the recently published forecasts based on $Euclid$-specific probes, namely galaxy clustering, weak lensing, and their cross-correlation. With some assumptions on the specifications of current and future CMB experiments, the predicted constraints are obtained from both a standard Fisher formalism and a posterior-fitting approach based on actual CMB data. Compared to a $Euclid$-only analysis, the addition of CMB data leads to a substantial impact on constraints for all cosmological parameters of the standard $\Lambda$-cold-dark-matter model, with improvements reaching up to a factor of ten. For the parameters of extended models, which include a redshift-dependent dark energy equation of state, non-zero curvature, and a phenomenological modification of gravity, improvements can be of the order of two to three, reaching higher than ten in some cases. The results highlight the crucial importance for cosmological constraints of the combination and cross-correlation of $Euclid$ probes with CMB data.

astro-ph.CO↗