Cosmic Microwave Background Temperature Power Spectrum Research Articles

Probing the physics of reionization using kinematic Sunyaev–Zeldovich power spectrum from current and upcoming cosmic microwave background surveys

ABSTRACT The patchiness in the reionization process alters the statistics of cosmic microwave background (CMB), with the kinematic Sunyaev–Zeldovich (kSZ) effect in the CMB temperature power spectrum being a notable consequence. In this work, we aim to explore the potential of future kSZ power spectrum measurements in inferring the details of the reionization process. In this pursuit, we capitalize on the recent developments in foreground mitigation techniques using the Cross-Internal Linear Combination (Cross-ILC) technique, which enables robust detection of the kSZ power spectrum with signal-to-noise ratio roughly 20σ–30σ in this decade by South pole Telescope (SPT-3G) and Simons Observatory (SO); and ≥80σ by CMB-S4 – substantially improving on the recent evidence for kSZ binned at ℓ = 3000 using SPT-SZ + SPTpol surveys. We use a fiducial kSZ power spectrum along with realistic error bars expected from the above technique for SPT-3G, SO, and CMB-S4 to constrain the parameter space for a physical model of reionization. We find that with the improved error bars it will be possible to place stringent constraints on reionization using solely the Cross-ILC recovered SPT-3G kSZ without imposing any prior on τ in the Bayesian inference. Notably, high-fidelity kSZ measurements from CMB-S4 coupled with τ measurements through LiteBIRD will enable unprecedented constraint on the midpoint of reionization with an error bar of ∼0.25 and the duration of reionization with an error bar at ∼0.21 exclusively using CMB data. This study highlights the need to capture kSZ power spectrum on a broad range of multipoles to gain insights into the inhomogeneous reionization era.

COSMOGLOBE DR1 results

We present COSMOGLOBEData Release 1, which implements the first joint analysis of WMAP andPlanckLFI time-ordered data, processed within a single Bayesian end-to-end framework. This framework directly builds on a similar analysis of the LFI measurements by the BEYONDPLANCKcollaboration, and approaches the cosmic microwave background (CMB) analysis challenge through Gibbs sampling of a global posterior distribution, simultaneously accounting for calibration, mapmaking, and component separation. The computational cost of producing one complete WMAP+LFI Gibbs sample is 812 CPU-h, of which 603 CPU-h are spent on WMAP low-level processing; this demonstrates that end-to-end Bayesian analysis of the WMAP data is computationally feasible. We find that our WMAP posterior mean temperature sky maps and CMB temperature power spectrum are largely consistent with the official WMAP9 results. Perhaps the most notable difference is that our CMB dipole amplitude is 3366.2 ± 1.4 μK, which is 11 μK higher than the WMAP9 estimate and 2.5σhigher than BEYONDPLANCK; however, it is in perfect agreement with the HFI-dominatedPlanckPR4 result. In contrast, our WMAP polarization maps differ more notably from the WMAP9 results, and in general exhibit significantly lower large-scale residuals. We attribute this to a better constrained gain and transmission imbalance model. It is particularly noteworthy that theW-band polarization sky map, which was excluded from the official WMAP cosmological analysis, for the first time appears visually consistent with theV-band sky map. Similarly, the long standing discrepancy between the WMAPK-band and LFI 30 GHz maps is finally resolved, and the difference between the two maps appears consistent with instrumental noise at high Galactic latitudes. Relatedly, these updated maps allowed us for the first time to combine WMAP and LFI polarization data into a single coherent model of large-scale polarized synchrotron emission. Still, we identified a few issues that require additional work, including (1) low-level noise modeling; (2) large-scale temperature residuals at the 1–2 μK level; and (3) a strong degeneracy between the absoluteK-band calibration and the dipole of the anomalous microwave emission component. We conclude that leveraging the complementary strengths of WMAP and LFI has allowed the mitigation of both experiments’ weaknesses, and resulted in new state-of-the-art WMAP sky maps. All maps and the associated code are made publicly available through the COSMOGLOBEweb page.

Subtracting the kinetic Sunyaev-Zeldovich effect from the cosmic microwave background with surveys of large-scale structure

The kinetic Sunyaev-Zeldovich (kSZ) effect will be an important source of cosmological and astrophysical information in upcoming surveys of the cosmic microwave background (CMB). However, the kSZ effect will also act as the dominant source of noise for several other measurements that use small angular scales in CMB temperature maps, since its blackbody nature implies that standard component separation techniques cannot be used to remove it from observed maps. In this paper, we explore the idea of ``de-kSZing'': constructing a template for the late-time kSZ effect using external surveys of large-scale structure, and then subtracting this template from CMB temperature maps in order to remove some portion of the kSZ signal. After building intuition for general aspects of the de-kSZing procedure, we perform forecasts for the de-kSZing efficiency of several large-scale structure surveys, including BOSS, DESI, Roman, MegaMapper, and PUMA. We also highlight potential applications of de-kSZing to cosmological constraints from the CMB temperature power spectrum, CMB lensing reconstruction, and the moving-lens effect. While our forecasts predict achievable de-kSZing efficiencies of 10%--20% at best, these results are specific to the de-kSZing formalism adopted in this work, and we expect that higher efficiencies are possible using improved versions of this formalism.

On Low Hubble Expansion Rate from Planck Data Anomalies

From the linear perturbations of Nash–Greene fluctuations of a background metric, we obtain profiles of Hubble function evolution H(z) and fσ8(z) measurements as compared with the ΛCDM results at intermediate redshifts 0.1<z<1. For parameter estimation, we use joint data from Planck Cosmic Microwave Background (CMB) likelihoods of CMB temperature and polarization angular power spectra, Barionic Acoustic Oscillations (BAO) and local measurements of Hubble constant H0 from the Hubble Space Telescope (HST). We analyze the stability of the effective Newtonian constant Geff and its agreement with Big Bang Nucleosynthesis (BBN) constraints. We show that our results are highly compatible with the ΛCDM paradigm, rather extending the perspective for further studies on redshift-space galaxy clustering data. Moreover, we obtain the CMB TT angular spectra with the Integrated Sachs–Wolfe (ISW) effect, which is weakened on low-l scales. The resulting linear matter power spectrum P(k) profile is also compatible with ΛCDM results but somewhat degenerate with an early dark energy (DE) contribution. Finally, posing a dilemma to the solution of Hubble tension, our results indicate a low Hubble expansion rate suggesting possible anomalies in Planck data in consonance with the recent South Pole Telescope (SPT-3G) data.

Forecast analysis on interacting dark energy models from future generation PICO and DESI missions

ABSTRACT The next-generation cosmic microwave background (CMB) satellite missions are expected to provide robust constraints on a wide range of cosmological parameters with unprecedented precision. But these constraints on the parameters could weaken if we do not attribute dark energy to a cosmological constant. The cosmological models involving interaction between dark energy and dark matter can give rise to comparable energy densities at the present epoch, thereby alleviating the so-called cosmic coincidence problem. In the present paper, we perform a forecast analysis to test the ability of the future generation high-sensitive CMB, and baryon acoustic oscillation (BAO) experiments to constrain phenomenological interacting dark energy models. We consider cosmic variance limited future CMB experiment Probe of Inflation and Cosmic Origins (PICO) along with BAO information from the Dark Energy Spectroscopic Instrument (DESI), to constrain the parameters of the interacting dark sector. Based on the stability of the cosmological perturbations, we consider two possibilities for the interaction scenario. We investigate the impact of both coupling constant and equation of state parameter of dark energy on CMB temperature power spectrum, matter power spectrum, and fσ8. We have used simulated temperature and polarization data from PICO within the multipole ranges (ℓ = 2–4000), and as expected, we do see PICO alone produces better constraints than Planck on the ΛCDM parameters. With the integration of PICO and DESI missions, we observe a significant improvement in the constraints on several cosmological parameters, especially the equation of state parameter of dark energy. However, we note that additional data are required to constrain a small positive coupling constant.

Primordial non-gaussianities of inflationary step-like models

We use Minkowski Functionals (MF) to explore the possible presence of non-Gaussian signatures in the recently released cosmic microwave background (CMB) Planck maps. In particular, we investigate MF outcomes for two sets of simulated CMB maps: one seeded by the ΛCDM concordance model, another seeded by the ‘best-fit’ CMB angular power spectrum. By ‘best-fit’ we mean a simulated CMB spectrum seeded by an inflationary model that well reproduce the observed low-ℓ features of the CMB temperature power spectrum. In this way, we obtain the spurious signal introduced by the power spectra with different shapes, signal to be taken into account in the corresponding MF analyses. Interestingly, we investigate the response of the four MF (i.e., Area, Perimeter, Genus, Nclusters ) for several sets of simulated maps. Our analyses confirm that, as already reported in the literature, the Perimeter exhibits the best performance. Moreover, our results show that the Perimeter obtained from the Planck maps (2015) is better fitted by the simulated CMB maps seeded by the angular power spectrum with low-ℓ features with respect to the curves obtained from the CMB maps seeded by the ΛCDM model.

Dark energy with oscillatory tracking potential: observational constraints and perturbative effects

ABSTRACT The cosmological models exhibiting tracker properties have great significance in the context of dark energy as they can reach the present value of dark energy density from a wide range of initial conditions, thereby alleviating both the fine-tuning and the cosmic coincidence problem. The α-attractors, which are originally discussed in the context of inflation, can exhibit the properties of dark energy as they can behave like cosmological trackers at early times and show the late-time behaviour of a cosmological constant. In this paper, we study the oscillatory tracker model (OTM), which belongs to the family of α-attractor dark energy models. Using the current observational data sets like cosmic microwave background (CMB), baryon acoustic oscillation, and type 1a supernova data (Pantheon compilation), we constrain the parameters of the model and estimate both the mean and best-fitting values. Although the OTM contains a larger set of parameters than the usual lambda cold dark matter (LCDM) model, the common set of parameters of both agree within $1\, \sigma$ error limits. Our observations using both high redshift and low redshift data support Hubble parameter value H0 = 67.4 Km s−1 Mpc−1. We study the effect of the OTM on the CMB temperature and polarization power spectra, matter power spectrum and fσ8. Our analysis of the CMB power spectrum and matter power spectrum suggests that the oscillatory tracker dark energy model has noticeable differences from usual LCDM predictions. Yet, in most cases, the agreement is very close.

Observational Constraints in Delta-gravity: CMB and Supernovae

Abstract Delta-gravity (DG) is a gravitational model based on an extension of general relativity given by a new symmetry called . In this model, new matter fields are added to the original matter fields, motivated by the additional symmetry. We call them matter fields. This model predicts an accelerating universe without the need to introduce a cosmological constant. In this work, we study the scalar cosmic microwave background (CMB) temperature (TT) power spectrum predicted by DG using an analytical hydrodynamic approach. To fit the Planck satellite’s data with the DG model, we used a Markov Chain Monte Carlo analysis. We also include a study about the compatibility between Type Ia supernovae (SNe Ia) and CMB observations in the DG context. Finally, we obtain the scalar CMB TT power spectrum and the fitted parameters needed to explain both SN Ia data and CMB measurements. The results are in reasonable agreement with both observations considering the analytical approximation. We also discuss whether the Hubble constant and the accelerating universe are in concordance with the observational evidence in the DG context.

Generation of inflationary perturbations in the continuous spontaneous localization model: The second order power spectrum

Cosmic inflation, which describes an accelerated expansion of the early Universe, yields the most successful predictions regarding temperature anisotropies in the cosmic microwave background (CMB). Nevertheless, the precise origin of the primordial perturbations and their quantum-to-classical transition is still an open issue. The continuous spontaneous localization model (CSL), in the cosmological context, might be used to provide a solution to the mentioned puzzles by considering an objective reduction of the inflaton wave function. In this work, we calculate the primordial power spectrum at the next leading order in the Hubble flow functions that results from applying the CSL model to slow roll inflation within the semiclassical gravity framework. We employ the method known as uniform approximation along with a second order expansion in the Hubble flow functions. We analyze some features in the CMB temperature and primordial power spectra that could help to distinguish between the standard prediction and our approach.

A novel CMB component separation method: hierarchical generalized morphological component analysis

ABSTRACT We present a novel technique for cosmic microwave background (CMB) foreground subtraction based on the framework of blind source separation. Inspired by previous work incorporating local variation to generalized morphological component analysis (GMCA), we introduce hierarchical GMCA (HGMCA), a Bayesian hierarchical graphical model for source separation. We test our method on Nside = 256 simulated sky maps that include dust, synchrotron, free–free, and anomalous microwave emission, and show that HGMCA reduces foreground contamination by $25{{\ \rm per\ cent}}$ over GMCA in both the regions included and excluded by the Planck UT78 mask, decreases the error in the measurement of the CMB temperature power spectrum to the 0.02–0.03 per cent level at ℓ > 200 (and $\lt 0.26{{\ \rm per\ cent}}$ for all ℓ), and reduces correlation to all the foregrounds. We find equivalent or improved performance when compared to state-of-the-art internal linear combination type algorithms on these simulations, suggesting that HGMCA may be a competitive alternative to foreground separation techniques previously applied to observed CMB data. Additionally, we show that our performance does not suffer when we perturb model parameters or alter the CMB realization, which suggests that our algorithm generalizes well beyond our simplified simulations. Our results open a new avenue for constructing CMB maps through Bayesian hierarchical analysis.

Footprints of Doppler and aberration effects in cosmic microwave background experiments: statistical and cosmological implications

ABSTRACT In the frame of the Solar system, the Doppler and aberration effects cause distortions in the form of mode couplings in the cosmic microwave background (CMB) temperature and polarization power spectra and, hence, impose biases on the statistics derived by the moving observer. We explore several aspects of such biases and pay close attention to their effects on CMB polarization, which, previously, have not been examined in detail. A potentially important bias that we introduce here is boost variance—an additional term in cosmic variance, induced by the observer’s motion. Although this additional term is negligible for whole-sky experiments, in partial-sky experiments it can reach 10 per cent (temperature) to 20 per cent (polarization) of the standard cosmic variance (σ). Furthermore, we investigate the significance of motion-induced power and parity asymmetries in TT, EE, and TE as well as potential biases induced in cosmological parameter estimation performed with whole-sky TTTEEE. Using Planck-like simulations, we find that our local motion induces $\sim 1\!-\!2 {{\ \rm per\ cent}}$ hemispherical asymmetry in a wide range of angular scales in the CMB temperature and polarization power spectra; however, it does not imply any significant amount of parity asymmetry or shift in cosmological parameters. Finally, we examine the prospects of measuring the velocity of the Solar system w.r.t. the CMB with future experiments via the mode coupling induced by the Doppler and aberration effects. Using the CMB TT, EE, and TE power spectra up to ℓ = 4000, the Simons Observatory and CMB-S4 can make a dipole-independent measurement of our local velocity, respectively, at 8.5σ and 20σ.

Do we have any hope of detecting scattering between dark energy and baryons through cosmology?

ABSTRACT We consider the possibility that dark energy and baryons might scatter off each other. The type of interaction we consider leads to a pure momentum exchange, and does not affect the background evolution of the expansion history. We parametrize this interaction in an effective way at the level of Boltzmann equations. We compute the effect of dark energy-baryon scattering on cosmological observables, focusing on the cosmic microwave background (CMB) temperature anisotropy power spectrum and the matter power spectrum. Surprisingly, we find that even huge dark energy-baryon cross-sections $\sigma _{xb} \sim {\cal O}({\rm b})$, which are generically excluded by non-cosmological probes such as collider searches or precision gravity tests, only leave an insignificant imprint on the observables considered. In the case of the CMB temperature power spectrum, the only imprint consists in a sub-per cent enhancement or depletion of power (depending whether or not the dark energy equation of state lies above or below −1) at very low multipoles, which is thus swamped by cosmic variance. These effects are explained in terms of differences in how gravitational potentials decay in the presence of a dark energy-baryon scattering, which ultimately lead to an increase or decrease in the late-time integrated Sachs–Wolfe power. Even smaller related effects are imprinted on the matter power spectrum. The imprints on the CMB are not expected to be degenerate with the effects due to altering the dark energy sound speed. We conclude that, while strongly appealing, the prospects for a direct detection of dark energy through cosmology do not seem feasible when considering realistic dark energy-baryon cross-sections. As a caveat, our results hold to linear order in perturbation theory.

Gravitational Waves, CMB Polarization, and the Hubble Tension.

The discrepancy between the Hubble parameter inferred from local measurements and that from the cosmic microwave background (CMB) has motivated careful scrutiny of the assumptions that enter both analyses. Here we point out that the location of the recombination peak in the CMB B-mode power spectrum is determined by the light horizon at the surface of last scatter and thus provides an alternative early-Universe standard ruler. It can thus be used as a cross-check for the standard ruler inferred from the acoustic peaks in the CMB temperature power spectrum and to test various explanations for the Hubble tension. The measurement can potentially be carried out with a precision of ≲2% with stage-IV B-mode experiments. The measurement can also be used to measure the propagation speed of gravitational waves in the early Universe.

Testing the ABS Method with the Simulated Planck Temperature Maps

Abstract In this study, we apply the Analytical method of Blind Separation (ABS) of the cosmic microwave background (CMB) from foregrounds to estimate the CMB temperature power spectrum from multi-frequency microwave maps. We test the robustness of the ABS estimator and assess the accuracy of the power spectrum recovery using realistic simulations based on the seven-frequency Planck data, including various frequency-dependent and spatially varying foreground components (synchrotron, free–free, thermal dust, and anomalous microwave emission), as well as an uncorrelated Gaussian-distributed instrumental noise. Considering no prior information about the foregrounds, the ABS estimator can analytically recover the CMB power spectrum over almost all scales with less than 0.5% error for maps where the Galactic plane region ( ) is masked out. To further test the flexibility and effectiveness of the ABS approach in a variety of situations, we apply the ABS to the simulated Planck maps in three cases: (1) without any mask, (2) imposing a two-times-stronger synchrotron emission, and (3) including only the Galactic plane region ( ) in the analysis. In such extreme cases, the ABS approach can still provide an unbiased estimate of band powers at the level of 1 μK2 on average over all ℓ ranges, and the recovered powers are consistent with the input values within 1σ for most ℓ bins.

Can early dark energy explain EDGES?

The Experiment to Detect the Global Epoch of Reionization Signature (EDGES) collaboration has reported the detection of an absorption feature in the sky-averaged spectrum at ≈ 78 MHz. This signal has been interpreted as the absorption of cosmic microwave background (CMB) photons at redshifts 15 ≲ z ≲ 20 by the 21cm hyperfine transition of neutral hydrogen, whose temperature is expected to be coupled to the gas temperature by the Wouthuysen-Field effect during this epoch. Because the gas is colder than the CMB, the 21cm signal is seen in absorption. However, the absorption depth reported by EDGES is more than twice the maximal value expected in the standard cosmological model, at ≈ 3.8σ significance. Here, we propose an explanation for this depth based on “early dark energy” (EDE), a scenario in which an additional component with equation of state w=−1 contributes to the cosmological energy density at early times, before decaying rapidly at a critical redshift, zc. For 20 ≲ zc ≲ 1000, the accelerated expansion due to the EDE can produce an earlier decoupling of the gas temperature from the radiation temperature than that in the standard model, giving the gas additional time to cool adiabatically before the first luminous sources form. We show that the EDE scenario can successfully explain the large amplitude of the EDGES signal. However, such models are strongly ruled out by observations of the CMB temperature power spectrum. Moreover, the EDE models needed to explain the EDGES signal exacerbate the current tension in low- and high-redshift measurements of the Hubble constant. We conclude that non-finely-tuned modifications of the background cosmology are unlikely to explain the EDGES signal while remaining consistent with other cosmological observations.

Structure formation in f(T) gravity and a solution for H0 tension

We investigate the evolution of scalar perturbations in f(T) teleparallel gravity and its effects on the cosmic microwave background (CMB) anisotropy. The f(T) gravity generalizes the teleparallel gravity which is formulated on the Weitzenböck spacetime, characterized by the vanishing curvature tensor (absolute parallelism) and the non-vanishing torsion tensor. For the first time, we derive the observational constraints on the modified teleparallel gravity using the CMB temperature power spectrum from Planck's estimation, in addition to data from baryonic acoustic oscillations (BAO) and local Hubble constant measurements. We find that a small deviation of the f(T) gravity model from the ΛCDM cosmology is slightly favored. Besides that, the f(T) gravity model does not show tension on the Hubble constant that prevails in the ΛCDM cosmology. It is clear that f(T) gravity is also consistent with the CMB observations, and undoubtedly it can serve as a viable candidate amongst other modified gravity theories.

Probing features in inflaton potential and reionization history with future CMB space observations

We consider the prospects of probing features in the primordial power spectrum with future Cosmic Microwave Background (CMB) polarization measurements. In the scope of the inflationary scenario, such features in the spectrum can be produced by local non-smooth pieces in an inflaton potential (smooth and quasi-flat in general) which in turn may originate from fast phase transitions during inflation in other quantum fields interacting with the inflaton. They can fit some outliers in the CMB temperature power spectrum which are unaddressed within the standard inflationary ΛCDM model. We consider Wiggly Whipped Inflation (WWI) as a theoretical framework leading to improvements in the fit to the Planck 2015 temperature and polarization data in comparison with the standard inflationary models, although not at a statistically significant level. We show that some type of features in the potential within the WWI models, leading to oscillations in the primordial power spectrum that extend to intermediate and small scales can be constrained with high confidence (at 3σ or higher confidence level) by an instrument as the Cosmic ORigins Explorer (CORE). In order to investigate the possible confusion between inflationary features and footprints from the reionization era, we consider an extended reionization history with monotonic increase of free electrons with decrease in redshift. We discuss the present constraints on this model of extended reionization and future predictions with CORE. We also project, to what extent, this extended reionization can create confusion in identifying inflationary features in the data.

Interacting neutrinos in cosmology: exact description and constraints

We consider the impact of neutrino self-interactions described by an effective four-fermion coupling on cosmological observations. Implementing the exact Boltzmann hierarchy for interacting neutrinos first derived in [1] into the Boltzmann solver CLASS, we perform a detailed numerical analysis of the effects of the interaction on the cosmic microwave background (CMB) anisotropies, and compare our results with known approximations in the literature. While we find good agreement between our exact approach and the relaxation time approximation used in some recent studies, the popular (ceff2,cvis2)-parameterisation fails to reproduce the correct scale dependence of the CMB temperature power spectrum. We then proceed to derive constraints on the effective coupling constant Geff using currently available cosmological data via an MCMC analysis. Interestingly, our results reveal a bimodal posterior distribution, where one mode represents the standard ΛCDM limit with Geff ≲ 108 GF, and the other a scenario in which neutrinos self-interact with an effective coupling constant Geff ≃ 3 × 109 GF.

On the regularity of the covariance matrix of a discretized scalar field on the sphere

We present a comprehensive study of the regularity of the covariance matrix of a discretized field on the sphere. In a particular situation, the rank of the matrix depends on the number of pixels, the number of spherical harmonics, the symmetries of the pixelization scheme and the presence of a mask. Taking into account the above mentioned components, we provide analytical expressions that constrain the rank of the matrix. They are obtained by expanding the determinant of the covariance matrix as a sum of determinants of matrices made up of spherical harmonics. We investigate these constraints for five different pixelizations that have been used in the context of Cosmic Microwave Background (CMB) data analysis: Cube, Icosahedron, Igloo, GLESP and HEALPix, finding that, at least in the considered cases, the HEALPix pixelization tends to provide a covariance matrix with a rank closer to the maximum expected theoretical value than the other pixelizations. The effect of the propagation of numerical errors in the regularity of the covariance matrix is also studied for different computational precisions, as well as the effect of adding a certain level of noise in order to regularize the matrix. In addition, we investigate the application of the previous results to a particular example that requires the inversion of the covariance matrix: the estimation of the CMB temperature power spectrum through the Quadratic Maximum Likelihood algorithm. Finally, some general considerations in order to achieve a regular covariance matrix are also presented.

Optimal scan strategies for future CMB satellite experiments

The B-mode polarisation power spectrum in the Cosmic Microwave Background (CMB) is about four orders of magnitude fainter than the CMB temperature power spectrum. Any instrumental imperfections that couple temperature fluctuations to B-mode polarisation must therefore be carefully controlled and/or removed. We investigate the role that a scan strategy can have in mitigating certain common systematics by averaging systematic errors down with many crossing angles. We present approximate analytic forms for the error on the recovered B-mode power spectrum that would result from differential gain, differential pointing and differential ellipticity for the case where two detector pairs are used in a polarisation experiment. We use these analytic predictions to search the parameter space of common satellite scan strategies in order to identify those features of a scan strategy that have most impact in mitigating systematic effects. As an example we go on to identify a scan strategy suitable for the CMB satellite proposed for the ESA M5 call. considering the practical considerations of fuel requirement, data rate and the relative orientation of the telescope to the earth. Having chosen a scan strategy we then go on to investigate the suitability of the scan strategy.