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We introduce the Parity-Odd Power (POP) spectra, a novel set of observables for probing parity violation in cosmological $N$-point statistics. POP spectra are derived from composite fields obtained by applying nonlinear transformations, involving also gradients, curls, and filtering functions, to a scalar field. This compresses the parity-odd trispectrum into a power spectrum. These new statistics offer several advantages: they are computationally fast to construct, estimating their covariance is less demanding compared to estimating that of the full parity-odd trispectrum, and they are simple to model theoretically. We measure the POP spectra on simulations of a scalar field with a specific parity-odd trispectrum shape. We compare these measurements to semi-analytic theoretical calculations and find agreement. We also explore extensions and generalizations of these parity-odd observables., Comment: 18 pages, 5 figures

The Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) is designed to detect and measure the redshifts of more than one million Ly$\alpha$ emitting galaxies (LAEs) between $1.88 < z < 3.52$. In addition to its cosmological measurements, these data enable studies of Ly$\alpha$ spectral profiles and the underlying radiative transfer. Using the roughly half a million LAEs in the HETDEX Data Release 3, we stack various subsets to obtain the typical Ly$\alpha$ profile for the $z \sim 2-3$ epoch and to understand their physical properties. We find clear absorption wings around Ly$\alpha$ emission, which extend $\sim 2000$ km $\mathrm{s}^{-1}$ both redward and blueward of the central line. Using far-UV spectra of nearby ($0.002 < z < 0.182$) LAEs in the CLASSY treasury and optical/near-IR spectra of $2.8 < z < 6.7$ LAEs in the MUSE-Wide survey, we observe absorption profiles in both redshift regimes. Dividing the sample by volume density shows that the troughs increase in higher density regions. This trend suggests that the depth of the absorption is dependent on the local density of objects near the LAE, a geometry that is similar to damped Lyman-$\alpha$ systems. Simple simulations of Ly$\alpha$ radiative transfer can produce similar troughs due to absorption of light from background sources by HI gas surrounding the LAEs., Comment: 15 pages, 10 figures, accepted for publication in The Astrophysical Journal

Polarization of the cosmic microwave background (CMB) can help probe the fundamental physics behind cosmic inflation via the measurement of primordial $B$ modes. As this requires exquisite control over instrumental systematics, some next-generation CMB experiments plan to use a rotating half-wave plate (HWP) as polarization modulator. However, the HWP non-idealities, if not properly treated in the analysis, can result in additional systematics. In this paper, we present a simple, semi-analytical end-to-end model to propagate the HWP non-idealities through the macro-steps that make up any CMB experiment (observation of multi-frequency maps, foreground cleaning, and power spectra estimation) and compute the HWP-induced bias on the estimated tensor-to-scalar ratio, $r$. We find that the effective polarization efficiency of the HWP suppresses the polarization signal, leading to an underestimation of $r$. Laboratory measurements of the properties of the HWP can be used to calibrate this effect, but we show how gain calibration of the CMB temperature can also be used to partially mitigate it. On the basis of our findings, we present a set of recommendations for the HWP design that can help maximize the benefits of gain calibration., Comment: 17 pages + appendices and bibliography, 7 figures, 1 table

The statistics of thermal gas pressure are a new and promising probe of cosmology and astrophysics. The large-scale cross-correlation between galaxies and the thermal Sunyaev-Zeldovich effect gives the bias-weighted mean electron pressure, $\langle b_\mathrm{h}P_e\rangle$. In this paper, we show that $\langle b_\mathrm{h}P_e\rangle$ is sensitive to the amplitude of fluctuations in matter density, for example $\langle b_\mathrm{h}P_e\rangle\propto \left(\sigma_8\Omega_\mathrm{m}^{0.81}h^{0.67}\right)^{3.14}$ at redshift $z=0$. We find that at $z<0.5$ the observed $\langle b_\mathrm{h}P_e\rangle$ is smaller than that predicted by the state-of-the-art hydrodynamical simulations of galaxy formation, MillenniumTNG, by a factor of $0.93$. This can be explained by a lower value of $\sigma_8$ and $\Omega_\mathrm{m}$, similar to the so-called "$S_8$ tension'' seen in the gravitational lensing effect, although the influence of astrophysics cannot be completely excluded. The difference between Magneticum and MillenniumTNG at $z<2$ is small, indicating that the difference in the galaxy formation models used by these simulations has little impact on $\langle b_\mathrm{h}P_e\rangle$ at this redshift range. At higher $z$, we find that both simulations are in a modest tension with the existing upper bounds on $\langle b_\mathrm{h}P_e\rangle$. We also find a significant difference between these simulations there, which we attribute to a larger sensitivity to the galaxy formation models in the high redshift regime. Therefore, more precise measurements of $\langle b_\mathrm{h}P_e\rangle$ at all redshifts will provide a new test of our understanding of cosmology and galaxy formation., Comment: 16 pages, 10 figures

We present the Simple Intensity Map Producer for Line Emission (SIMPLE), a public code for quickly simulating mock line-intensity maps, and an analytical framework for modeling intensity maps including observational effects. SIMPLE can be applied to any spectral line sourced by galaxies. The SIMPLE code is based on lognormal mock catalogs of galaxies including positions and velocities and assigns luminosities following the luminosity function. After applying a selection function to distinguish between detected and undetected galaxies, the code generates an intensity map, which can be modified with anisotropic smoothing, noise, a mask, and sky subtraction, and calculates the power spectrum multipoles. We show that the intensity autopower spectrum and the galaxy-intensity cross-power spectrum agree well with the analytical estimates in real space. We derive and show that the sky subtraction suppresses the intensity autopower spectrum and the cross-power spectrum on scales larger than the size of an individual observation. As an example application, we make forecasts for the sensitivity of an intensity mapping experiment similar to the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) to the cross-power spectrum of Ly$\alpha$-emitting galaxies and the Ly$\alpha$ intensity. We predict that HETDEX will measure the galaxy-intensity cross-power spectrum with a high signal-to-noise ratio on scales of $0.04\, h\,\mathrm{Mpc}^{-1} < k < 1\, h\,\mathrm{Mpc}^{-1}$., Comment: 15 pages, 5 figures, published in ApJ, added discussion, updated Fig. 1 and 4

We describe the ensemble properties of the $1.9 < z < 3.5$ Lyman Alpha Emitters (LAEs) found in the HETDEX survey's first public data release, HETDEX Public Source Catalog 1 (Mentuch Cooper et al. 2023). Stacking the low-resolution ($R \sim$ 800) spectra greatly increases the signal-to-noise ratio, revealing spectral features otherwise hidden by noise, and we show that the stacked spectrum is representative of an average member of the set. The flux limited, Ly$\alpha$ signal-to-noise ratio restricted stack of 50K HETDEX LAEs shows the ensemble biweight ``average" $z \sim 2.6$ LAE to be a blue (UV continuum slope $\sim -2.4$ and E(B-V) $< 0.1$), moderately bright (M$_{\text{UV}} \sim -19.7$) star forming galaxy with strong Ly$\alpha$ emission (log $L_{Ly\alpha}$ $\sim$ 42.8 and $W_{\lambda}$(Ly$\alpha$) $\sim$ 114\AA), and potentially significant leakage of ionizing radiation. The restframe UV light is dominated by a young, metal poor stellar population with an average age 5-15 Myr and metallicity of 0.2-0.3 Z$_{\odot}$., Comment: 17 pages, 11 figures, 2 data files (ApJ Accepted)

Polarization of the cosmic microwave background (CMB) is sensitive to new physics violating parity symmetry, such as the presence of a pseudoscalar "axionlike" field. Such a field may be responsible for early dark energy (EDE), which is active prior to recombination and provides a solution to the so-called Hubble tension. The EDE field coupled to photons in a parity-violating manner would rotate the plane of linear polarization of the CMB and produce a cross-correlation power spectrum of $E$- and $B$-mode polarization fields with opposite parities. In this paper, we fit the $EB$ power spectrum predicted by the photon-axion coupling of the EDE model with a potential $V(\phi)\propto [1-\cos(\phi/f)]^3$ to polarization data from Planck. We find that the unique shape of the predicted $EB$ power spectrum is not favored by the data and obtain a first constraint on the photon-axion coupling constant, $g=(0.04\pm 0.16)M_{\text{Pl}}^{-1}$ (68% CL), for the EDE model that best fits the CMB and galaxy clustering data. This constraint is independent of the miscalibration of polarization angles of the instrument or the polarized Galactic foreground emission. Our limit on $g$ may have important implications for embedding EDE in fundamental physics, such as string theory., Comment: 7 pages, 3 figures, 1 table. Accepted for publication in Physical Review Letters. The stacked EB power spectrum is publicly available at https://github.com/LilleJohs/Observed-EB-Power-Spectrum

We present the first publicly released catalog of sources obtained from the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX). HETDEX is an integral field spectroscopic survey designed to measure the Hubble expansion parameter and angular diameter distance at 1.88

The Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) is an untargeted spectroscopic survey that aims to measure the expansion rate of the Universe at $z \sim 2.4$ to 1% precision for both $H(z)$ and $D_A(z)$. HETDEX is in the process of mapping in excess of one million Lyman Alpha emitting (LAE) galaxies and a similar number of lower-z galaxies as a tracer of the large-scale structure. The success of the measurement is predicated on the post-observation separation of galaxies with Ly$\alpha$ emission from the lower-$z$ interloping galaxies, primarily [OII], with low contamination and high recovery rates. The Emission Line eXplorer (ELiXer) is the principal classification tool for HETDEX, providing a tunable balance between contamination and completeness as dictated by science needs. By combining multiple selection criteria, ELiXer improves upon the 20 Angstrom rest-frame equivalent width cut commonly used to distinguish LAEs from lower-$z$ [OII] emitting galaxies. Despite a spectral resolving power, R $\sim800$, that cannot resolve the [OII] doublet, we demonstrate the ability to distinguish LAEs from foreground galaxies with 98.1% accuracy. We estimate a contamination rate of Ly$\alpha$ by [OII] of 1.2% and a Ly$\alpha$ recovery rate of 99.1% using the default ELiXer configuration. These rates meet the HETDEX science requirements., Comment: 38 pages, 11 figures

Polarization of the cosmic microwave background (CMB) can probe new parity-violating physics such as cosmic birefringence (CB), which requires exquisite control over instrumental systematics. The non-idealities of the half-wave plate (HWP) represent a source of systematics when used as a polarization modulator. We study their impact on the CMB angular power spectra, which is partially degenerate with CB and miscalibration of the polarization angle. We use full-sky beam convolution simulations including HWP to generate mock noiseless time-ordered data, process them through a bin averaging map-maker, and calculate the power spectra including $TB$ and $EB$ correlations. We also derive analytical formulae which accurately model the observed spectra. For our choice of HWP parameters, the HWP-induced angle amounts to a few degrees, which could be misinterpreted as CB. Accurate knowledge of the HWP is required to mitigate this. Our simulation and analytical formulae will be useful for deriving requirements for the accuracy of HWP calibration., Comment: 12 pages + appendices and bibliography, 5 figures, 1 table; accepted for publication in JCAP

A tantalizing hint of isotropic cosmic birefringence has been found in the $E B$ cross-power spectrum of the cosmic microwave background (CMB) polarization data with a statistical significance of $3\sigma$. A pseudoscalar field coupled to the CMB photons via the Chern-Simons term can explain this observation. The same field may also be responsible for early dark energy (EDE), which alleviates the so-called Hubble tension. Since the EDE field evolves significantly during the recombination epoch, the conventional formula that relates $E B$ to the difference between the $E$- and $B$-mode auto-power spectra is no longer valid. Solving the Boltzmann equation for polarized photons and the dynamics of the EDE field consistently, we find that currently favored parameter space of the EDE model yields a variety of shapes of the $EB$ spectrum, which can be tested by CMB experiments., Comment: 7 pages, 4 figures, 1 table. (v2) Expanded discussion on forecasts for SO and CMB-S4. Accepted for publication in Phys. Rev. D

We present extended Lyman-{\alpha} (Ly{\alpha}) emission out to 800 kpc of 1034 [O III]-selected galaxies at redshifts 1.940 kpc from the galaxy centers. The surface brightness in the inner parts (r<10 kpc) around the [O III]-selected galaxies, however, is ten times fainter than that of the LAEs. Our results are consistent with the notion that photons dominating the outer regions of the Ly{\alpha} halos are not produced in the central galaxies but originate outside of them., Comment: 7 pages, 3 figures, accepted by Astrophysical Journal Letters

The observed pattern of linear polarization of the cosmic microwave background (CMB) photons is a sensitive probe of physics violating parity symmetry under inversion of spatial coordinates. A new parity-violating interaction might have rotated the plane of linear polarization by an angle $\beta$ as the CMB photons have been traveling for more than 13 billion years. This effect is known as "cosmic birefringence." In this paper, we present new measurements of cosmic birefringence from a joint analysis of polarization data from two space missions, Planck and WMAP. This dataset covers a wide range of frequencies from 23 to 353 GHz. We measure $\beta = 0.342^{\circ\,+0.094^\circ}_{\phantom{\circ\,}-0.091^\circ}$ (68% C.L.) for nearly full-sky data, which excludes $\beta=0$ at 99.987% C.L. This corresponds to the statistical significance of $3.6\sigma$. There is no evidence for frequency dependence of $\beta$. We find a similar result, albeit with a larger uncertainty, when removing the Galactic plane from the analysis., Comment: 9 pages, 5 figures, 1 table. The code to reproduce the results of this paper is available at https://github.com/LilleJohs/Cosmic_Birefringence. Accepted for publication in Physical Review D

We derive a new upper bound on the tensor-to-scalar ratio parameter $r$ using the frequentist profile likelihood method. We vary all the relevant cosmological parameters of the $\Lambda$CDM model, as well as the nuisance parameters. Unlike the Bayesian analysis using Markov Chain Monte Carlo (MCMC), our analysis is independent of the choice of priors. Using $Planck$ Public Release 4, BICEP/Keck Array 2018, $Planck$ CMB lensing, and BAO data, we find an upper limit of $r<0.037$ at 95% C.L., similar to the Bayesian MCMC result of $r<0.038$ for a flat prior on $r$ and a conditioned $Planck$ lowlEB covariance matrix., Comment: 7 pages, 2 Figures, 2 Tables. Submitted to MNRAS. Updated references

We present the first nonlinear lattice simulation of an axion field coupled to a U(1) gauge field during inflation. We use it to fully characterize the statistics of the primordial curvature perturbation {\zeta}. We find high-order statistics to be essential in describing non-Gaussianity of {\zeta} in the linear regime of the theory. On the contrary, non-Gaussianity is suppressed when the dynamics becomes nonlinear. This relaxes bounds from overproduction of primordial black holes, allowing for an observable gravitational waves signal at pulsar timing array and interferometers scales. Our work establishes lattice simulations as a crucial tool to study the inflationary epoch and its predictions., Comment: 6 pages, 5 figures; Minor changes from the first versions, matching the version published in PRD

A pseudoscalar "axionlike" field, $\phi$, may explain the $3\sigma$ hint of cosmic birefringence observed in the $EB$ power spectrum of the cosmic microwave background (CMB) polarization data. Is $\phi$ dark energy or dark matter? A tomographic approach can answer this question. The effective mass of dark energy field responsible for the accelerated expansion of the Universe today must be smaller than $m_\phi\simeq 10^{-33}$ eV. If $m_\phi \gtrsim 10^{-32}$ eV, $\phi$ starts evolving before the epoch of reionization and we should observe different amounts of birefringence from the $EB$ power spectrum at low ($l\lesssim 10$) and high multipoles. Such an observation, which requires a full-sky satellite mission, would rule out $\phi$ being dark energy. If $m_\phi \gtrsim 10^{-28}$ eV, $\phi$ starts oscillating during the epoch of recombination, leaving a distinct signature in the $EB$ power spectrum at high multipoles, which can be measured precisely by ground-based CMB observations. Our tomographic approach relies on the shape of the $EB$ power spectrum and is less sensitive to miscalibration of polarization angles., Comment: 11 pages, 12 figures

In this paper we will list a few important goals that need to be addressed in the next decade, also taking into account the current discordances between the different cosmological probes, such as the disagreement in the value of the Hubble constant $H_0$, the $\sigma_8$--$S_8$ tension, and other less statistically significant anomalies. While these discordances can still be in part the result of systematic errors, their persistence after several years of accurate analysis strongly hints at cracks in the standard cosmological scenario and the necessity for new physics or generalisations beyond the standard model. In this paper, we focus on the $5.0\,\sigma$ tension between the {\it Planck} CMB estimate of the Hubble constant $H_0$ and the SH0ES collaboration measurements. After showing the $H_0$ evaluations made from different teams using different methods and geometric calibrations, we list a few interesting new physics models that could alleviate this tension and discuss how the next decade's experiments will be crucial. Moreover, we focus on the tension of the {\it Planck} CMB data with weak lensing measurements and redshift surveys, about the value of the matter energy density $\Omega_m$, and the amplitude or rate of the growth of structure ($\sigma_8,f\sigma_8$). We list a few interesting models proposed for alleviating this tension, and we discuss the importance of trying to fit a full array of data with a single model and not just one parameter at a time. Additionally, we present a wide range of other less discussed anomalies at a statistical significance level lower than the $H_0$--$S_8$ tensions which may also constitute hints towards new physics, and we discuss possible generic theoretical approaches that can collectively explain the non-standard nature of these signals.[Abridged], Comment: Contribution to Snowmass 2021. 224 pages, 27 figures. Accepted for publication in JHEAp

We present the median-stacked Lyman-$\alpha$ surface brightness profile of 968 spectroscopically selected Lyman-$\alpha$ emitting galaxies (LAEs) at redshifts $1.980\,\mathrm{kpc}$, our profile is flatter than the MUSE model. The measured profile agrees at most radii with that of galaxies in the Byrohl et al. (2021) cosmological radiative transfer simulation at $z=3$. This suggests that the surface brightness of a Lyman-$\alpha$ halo at $r\lesssim 100$ kpc is dominated by resonant scattering of Lyman-$\alpha$ photons from star-forming regions in the central galaxy, whereas at $r > 100$ kpc it is dominated by photons from galaxies in surrounding dark matter halos., Comment: 15 pages, 7 figures, accepted by The Astrophysical Journal

Cosmology requires new physics beyond the Standard Model of elementary particles and fields. What is the fundamental physics behind dark matter and dark energy? What generated the initial fluctuations in the early Universe? Polarised light of the cosmic microwave background (CMB) may hold the key to answers. In this article, we discuss two new developments in this research area. First, if the physics behind dark matter and dark energy violates parity symmetry, their coupling to photons rotates the plane of linear polarisation as the CMB photons travel more than 13 billion years. This effect is known as `cosmic birefringence': space filled with dark matter and dark energy behaves as if it were a birefringent material, like a crystal. A tantalising hint for such a signal has been found with the statistical significance of $3\sigma$. Next, the period of accelerated expansion in the very early Universe, called `cosmic inflation', produced a stochastic background of primordial gravitational waves (GW). What generated GW? The leading idea is vacuum fluctuations in spacetime, but matter fields could also produce a significant amplitude of primordial GW. Finding its origin using CMB polarisation opens a new window into the physics behind inflation. These new scientific targets may influence how data from future CMB experiments are collected, calibrated, and analysed., Comment: 16 pages, 5 figures + Supplementary information (2 pages, 3 figures). Pre-edited version, to appear in Nature Reviews Physics. (v2) References updated. (v3) Minor typos corrected. (v4) Minor typos in Eq.(17), (21) and (22) have been corrected

A dark energy-like component in the early universe, known as early dark energy (EDE), is a proposed solution to the Hubble tension. Currently, there is no consensus in the literature as to whether EDE can simultaneously solve the Hubble tension and provide an adequate fit to the data from the cosmic microwave background (CMB) and large-scale structure of the universe. In this work, we deconstruct the current constraints from the Planck CMB and the full-shape clustering data of the Baryon Oscillation Spectroscopic Survey (BOSS) to understand the origin of different conclusions in the literature. We use two different analyses, a grid sampling and a profile likelihood, to investigate whether the current constraints suffer from volume effects upon marginalization and are biased towards some values of the EDE fraction, $f_\mathrm{EDE}$. We find that $f_\mathrm{EDE}$ allowed by the data strongly depends on the particular choice of the other parameters of the model and that several choices of these parameters prefer larger values of $f_\mathrm{EDE}$ than in the Markov Chain Monte Carlo analysis. This suggests that volume effects are the reason behind the disagreement in the literature. Motivated by this, we use a profile likelihood to analyze the EDE model and compute a confidence interval for $f_\mathrm{EDE}$, finding $f_\mathrm{EDE} = 0.072\pm 0.036$ ($68\%$ C.L.). This confidence interval is not subject to volume effects; thus, our approach yields more robust constraints on EDE and provides a powerful tool to understand whether EDE is a possible solution to the Hubble tension., Comment: 6 pages, 5 figures; v2: small changes and additional references

Phenomenological success of inflation models with axion and SU(2) gauge fields relies crucially on control of backreaction from particle production. Most of the previous study only demanded the backreaction terms in equations of motion for axion and gauge fields be small on the basis of order-of-magnitude estimation. In this paper, we solve the equations of motion with backreaction for a wide range of parameters of the spectator axion-SU(2) model. First, we find a new slow-roll solution of the axion-SU(2) system in the absence of backreaction. Next, we obtain accurate conditions for stable slow-roll solutions in the presence of backreaction. Finally, we show that the amplitude of primordial gravitational waves sourced by the gauge fields can exceed that of quantum vacuum fluctuations in spacetime by a large factor, without backreaction spoiling slow-roll dynamics. Imposing additional constraints on the power spectra of scalar and tensor modes measured at CMB scales, we find that the sourced contribution can be more than ten times the vacuum one. Imposing further a constraint of scalar modes non-linearly sourced by tensor modes, the two contributions can still be comparable., Comment: 30 pages, 20 figures, minor corrections, accepted by JCAP

We present a demonstration of the in-flight polarization angle calibration for the JAXA/ISAS second strategic large class mission, LiteBIRD, and estimate its impact on the measurement of the tensor-to-scalar ratio parameter, r, using simulated data. We generate a set of simulated sky maps with CMB and polarized foreground emission, and inject instrumental noise and polarization angle offsets to the 22 (partially overlapping) LiteBIRD frequency channels. Our in-flight angle calibration relies on nulling the EB cross correlation of the polarized signal in each channel. This calibration step has been carried out by two independent groups with a blind analysis, allowing an accuracy of the order of a few arc-minutes to be reached on the estimate of the angle offsets. Both the corrected and uncorrected multi-frequency maps are propagated through the foreground cleaning step, with the goal of computing clean CMB maps. We employ two component separation algorithms, the Bayesian-Separation of Components and Residuals Estimate Tool (B-SeCRET), and the Needlet Internal Linear Combination (NILC). We find that the recovered CMB maps obtained with algorithms that do not make any assumptions about the foreground properties, such as NILC, are only mildly affected by the angle miscalibration. However, polarization angle offsets strongly bias results obtained with the parametric fitting method. Once the miscalibration angles are corrected by EB nulling prior to the component separation, both component separation algorithms result in an unbiased estimation of the r parameter. While this work is motivated by the conceptual design study for LiteBIRD, its framework can be broadly applied to any CMB polarization experiment. In particular, the combination of simulation plus blind analysis provides a robust forecast by taking into account not only detector sensitivity but also systematic effects., Comment: Submitted to JCAP

We use a lattice simulation to study a model of axion inflation where the inflaton is coupled to a U(1) gauge field through Chern-Simons interaction. These kinds of models have already been studied with a lattice simulation in the context of reheating. In this work, we focus on the deep inflationary phase and discuss the new aspects that need to be considered in order to simulate gauge fields in this regime. Our main result is reproducing with precision the growth of the gauge field on the lattice induced by the rolling of the axion on its potential, thus recovering the results of linear perturbation theory for this model. In order to do so, we study in detail how the spatial discretization, through the choice of the spatial derivatives on the lattice, influences the dynamics of the gauge field. We find that the evolution of the gauge field is highly sensitive to the choice of the spatial discretization scheme. Nevertheless, we are able to identify a discretization scheme for which the growth of the gauge field on the lattice reproduces the one of continuous space with good precision., Comment: 13 pages, 8 figures; typos corrected, added references, slight modifications to the text; various changes matching the version published in PRD

We describe the survey design, calibration, commissioning, and emission-line detection algorithms for the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX). The goal of HETDEX is to measure the redshifts of over a million Ly$\alpha$ emitting galaxies between 1.88

The Hobby-Eberly Telescope (HET) Dark Energy Experiment (HETDEX) is undertaking a blind wide-field low-resolution spectroscopic survey of 540 square degrees of sky to identify and derive redshifts for a million Lyman-alpha emitting galaxies (LAEs) in the redshift range 1.9 < z < 3.5. The ultimate goal is to measure the expansion rate of the Universe at this epoch, to sharply constrain cosmological parameters and thus the nature of dark energy. A major multi-year wide field upgrade (WFU) of the HET was completed in 2016 that substantially increased the field of view to 22 arcminutes diameter and the pupil to 10 meters, by replacing the optical corrector, tracker, and prime focus instrument package and by developing a new telescope control system. The new, wide-field HET now feeds the Visible Integral-field Replicable Unit Spectrograph (VIRUS), a new low-resolution integral field spectrograph (LRS2), and the Habitable Zone Planet Finder (HPF), a precision near-infrared radial velocity spectrograph. VIRUS consists of 156 identical spectrographs fed by almost 35,000 fibers in 78 integral field units arrayed at the focus of the upgraded HET. VIRUS operates in a bandpass of 3500-5500 Angstroms with resolving power R~800. VIRUS is the first example of large scale replication applied to instrumentation in optical astronomy to achieve spectroscopic surveys of very large areas of sky. This paper presents technical details of the HET WFU and VIRUS, as flowed-down from the HETDEX science requirements, along with experience from commissioning this major telescope upgrade and the innovative instrumentation suite for HETDEX., Comment: 65 pages, 25 figures, published in the Astronomical Journal; replaced with final published version

The thermal Sunyaev-Zeldovich effect contains information about the thermal history of the universe, observable in maps of the Compton $y$ parameter; however, it does not contain information about the redshift of the sources. Recent papers have utilized a tomographic approach, cross-correlating the Compton $y$ map with the locations of galaxies with known redshift, in order to deproject the signal along the line of sight. In this paper, we test the validity and accuracy of this tomographic approach to probe the thermal history of the universe. We use the state-of-the-art cosmological hydrodynamical simulation, Magneticum, for which the thermal history of the universe is a known quantity. The key ingredient is the Compton-$y$-weighted halo bias, $b_y$, computed from the halo model. We find that, at redshifts currently available, the method reproduces the correct mean thermal pressure (or the density-weighted mean temperature) to high accuracy, validating and confirming the results of previous papers. At higher redshifts ($z\gtrsim 2.5$), there is significant disagreement between $b_y$ from the halo model and the simulation., Comment: 5 pages, 4 figures

SU(2) gauge fields coupled to an axion field can acquire an isotropic background solution during inflation. We study homogeneous but anisotropic inflationary solutions in the presence of such (massless) gauge fields. A gauge field in the cosmological background may pose a threat to spatial isotropy. We show, however, that such models $\textit{generally}$ isotropize in Bianchi type-I geometry, and the isotropic solution is the attractor. Restricting the setup by adding an axial symmetry, we revisited the numerical analysis presented in Wolfson et.al (2020). We find that the reported numerical breakdown in the previous analysis is an artifact of parametrization singularity. We use a new parametrization that is well-defined all over the phase space. We show that the system respects the cosmic no-hair conjecture and the anisotropies always dilute away within a few e-folds., Comment: 23 pages, 14 figures. Published 23 September 2021 Updated to published version

The construction of catalogues of a particular type of galaxy can be complicated by interlopers contaminating the sample. In spectroscopic galaxy surveys this can be due to the misclassification of an emission line; for example in the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) low redshift [OII] emitters may make up a few percent of the observed Ly${\alpha}$ emitter (LAE) sample. The presence of contaminants affects the measured correlation functions and power spectra. Previous attempts to deal with this using the cross-correlation function have assumed sources at a fixed redshift, or not modelled evolution within the adopted redshift bins. However, in spectroscopic surveys like HETDEX, where the contamination fraction is likely to be redshift dependent, the observed clustering of misclassified sources will appear to evolve strongly due to projection effects, even if their true clustering does not. We present a practical method for accounting for the presence of contaminants with redshift-dependent contamination fractions and projected clustering. We show using mock catalogues that our method, unlike existing approaches, yields unbiased clustering measurements from the upcoming HETDEX survey in scenarios with redshift-dependent contamination fractions within the redshift bins used. We show our method returns auto-correlation functions with systematic biases much smaller than the statistical noise for samples with at least as high as 7 per cent contamination. We also present and test a method for fitting for the redshift-dependent interloper fraction using the LAE-[OII] galaxy cross-correlation function, which gives less biased results than assuming a single interloper fraction for the whole sample., Comment: This is a pre-copy edited, author-produced PDF of an article accepted for publication in Monthly Notices of the Royal Astronomical Society following peer review. The version of record Farrow et al, MNRAS, 2021 is available online at: https://academic.oup.com/mnras/advance-article-abstract/doi/10.1093/mnras/stab1986/6322852

The scalar field theory of cosmological inflation constitutes nowadays one of the preferred scenarios for the physics of the early universe. In this paper we aim at studying the inflationary universe making use of a numerical lattice simulation. Various lattice codes have been written in the last decades and have been extensively used for understating the reheating phase of the universe, but they have never been used to study the inflationary phase itself far from the end of inflation (i.e. about 50 e-folds before the end of inflation). In this paper we use a lattice simulation to reproduce the well-known results of some simple models of single-field inflation, particularly for the scalar field perturbation. The main model that we consider is the standard slow-roll inflation with an harmonic potential for the inflaton field. We explore the technical aspects that need to be accounted for in order to reproduce with precision the nearly scale invariant power spectrum of inflaton perturbations. We also consider the case of a step potential, and show that the simulation is able to correctly reproduce the oscillatory features in the power spectrum of this model. Even if a lattice simulation is not needed in these cases, that are well within the regime of validity of linear perturbation theory, this sets the basis to future work on using lattice simulations to study more complicated models of inflation., Comment: 18 pages, 10 figures. Various changes from the previous version, matching the version published in JCAP. Added a new appendix and two figures

We search for evidence of parity-violating physics in the Planck 2018 polarization data, and report on a new measurement of the cosmic birefringence angle, $\beta$. The previous measurements are limited by the systematic uncertainty in the absolute polarization angles of the Planck detectors. We mitigate this systematic uncertainty completely by simultaneously determining $\beta$ and the angle miscalibration using the observed cross-correlation of the $E$- and $B$-mode polarization of the cosmic microwave background and the Galactic foreground emission. We show that the systematic errors are effectively mitigated and achieve a factor-of-$2$ smaller uncertainty than the previous measurement, finding $\beta=0.35 \pm 0.14\,\deg$ (68% C.L.), which excludes $\beta = 0$ at $99.2$% C.L. This corresponds to the statistical significance of $2.4\sigma$., Comment: 6 pages, 2 figures, 1 table. Published in Physical Review Letters

Time-delay cosmography with gravitationally lensed quasars plays an important role in anchoring the absolute distance scale and hence measuring the Hubble constant, $H_{0}$, independent of traditional distance ladder methodology. A current potential limitation of time delay distance measurements is the "mass-sheet transformation" (MST) which leaves the lensed imaging unchanged but changes the distance measurements and the derived value of $H_0$. In this work we show that the standard method of addressing the MST in time delay cosmography, through a combination of high-resolution imaging and the measurement of the stellar velocity dispersion of the lensing galaxy, depends on the assumption that the ratio, $D_{\rm s}/D_{\rm ds}$, of angular diameter distances to the background quasar and between the lensing galaxy and the quasar can be constrained. This is typically achieved through the assumption of a particular cosmological model. Previous work (TDCOSMO IV) addressed the mass-sheet degeneracy and derived $H_{0}$ under the assumption of $\Lambda$CDM model. In this paper we show that the mass sheet degeneracy can be broken without relying on a specific cosmological model by combining lensing with relative distance indicators such as supernovae type Ia and baryon acoustic oscillations, which constrain the shape of the expansion history and hence $D_{\rm s}/D_{\rm ds}$. With this approach, we demonstrate that the mass-sheet degeneracy can be constrained in a cosmological-model-independent way, and hence model-independent distance measurements in time-delay cosmography under mass-sheet transformations can be obtained., Comment: 11 pages, 8 figures

We present an algorithm to self-consistently generate mock weak gravitational lensing convergence fields and galaxy distributions in redshift space. We generate three-dimensional cosmic density fields that follow a log-normal distribution, and ray-trace them to produce convergence maps. As we generate the galaxy distribution from the same density fields in a manner consistent with ray-tracing, the galaxy-convergence cross-power spectrum measured from the mock agrees with the theoretical expectation with high precision. We use this simulation to forecast the quality of galaxy-shear cross-correlation measurements from the Subaru Hyper Suprime-Cam (HSC) and Prime Focus Spectrograph (PFS) surveys. We find that the nominal HSC and PFS surveys would detect the cross power spectra with signal-to-noise ratios of 20 and 5 at the lowest ($z = 0.7$) and highest ($z = 2.2$) redshift bins, respectively., Comment: 22 pages, 10 figures, accepted to JCAP. The simulation code is available at https://wwwmpa.mpa-garching.mpg.de/~komatsu/codes.html

We investigate the possibility of measuring the primordial gravitational wave (GW) signal across 21 decades in frequencies, using the cosmic microwave background (CMB), pulsar timing arrays (PTA), and laser and atomic interferometers. For the CMB and PTA experiments we consider the LiteBIRD mission and the Square Kilometer Array (SKA), respectively. For the interferometers we consider space mission proposals including the Laser Interferometer Space Antenna (LISA), the Big Bang Observer (BBO), the Deci-hertz Interferometer Gravitational wave Observatory (DECIGO), the $\mu$Ares experiment, the Decihertz Observatory (DO), and the Atomic Experiment for Dark Matter and Gravity Exploration in Space (AEDGE), as well as the ground-based Einstein Telescope (ET) proposal. We implement the mathematics needed to compute sensitivities for both CMB and interferometers, and derive the response functions for the latter from the first principles. We also evaluate the effect of the astrophysical foreground contamination in each experiment. We present binned sensitivity curves and error bars on the energy density parameter, $\Omega_{GW}h^2$, as a function of frequency for two representative classes of models for the stochastic background of primordial GW: the quantum vacuum fluctuation in the metric from single-field slow-roll inflation, and the source-induced tensor perturbation from the spectator axion-SU(2) inflation models. We find excellent prospects for joint measurements of the GW spectrum by CMB and space-borne interferometers mission proposals., Comment: Accepted in JCAP. Improved treatment of foregrounds for interferometers and more realistic analytical formula for $\Omega_{GW}$. Python code to reproduce figures is available at the link https://github.com/pcampeti/SGWBProbe

As cosmic structures form, matter density fluctuations collapse gravitationally and baryonic matter is shock-heated and thermalized. We therefore expect a connection between the mean gravitational potential energy density of collapsed halos, $\Omega_{W}^{\rm halo}$, and the mean thermal energy density of baryons, $\Omega_{\rm th}$. These quantities can be obtained using two fundamentally different estimates: we compute $\Omega_{W}^{\rm halo}$ using the theoretical framework of the halo model which is driven by dark matter statistics, and measure $\Omega_{\rm th}$ using the Sunyaev-Zeldovich (SZ) effect which probes the mean thermal pressure of baryons. First, we derive that, at the present time, about 90% of $\Omega_{W}^{\rm halo}$ originates from massive halos with $M>10^{13}\,M_\odot$. Then, using our measurements of the SZ background, we find that $\Omega_{\rm th}$ accounts for about 80% of the kinetic energy of the baryons available for pressure in halos at $z\lesssim 0.5$. This constrains the amount of non-thermal pressure, e.g., due to bulk and turbulent gas motion sourced by mass accretion, to be about $\Omega_{\rm non-th}\simeq 0.4\times 10^{-8}$ at $z=0$., Comment: 11 pages + references, 4 figures, 2 tables. (v2) Expanded discussion on the modelling uncertainty. (v3) Fixed a minor typo in Eq.(22). Accepted for publication in the Astrophysical Journal. Julia codes to reproduce the figures and tables are available in https://github.com/komatsu5147/OmegaGrav.jl

We develop a strategy to determine the cosmic birefringence and miscalibrated polarization angles simultaneously using the observed $EB$ polarization power spectra of the cosmic microwave background and the Galactic foreground emission. We extend the methodology of Y. Minami et al. (Prog. Theor. Exp. Phys. $\mathbf{2019}$, 083E02, 2019), which was developed for auto frequency power spectra, by including cross frequency spectra. By fitting one global birefringence angle and independent miscalibration angles at different frequency bands, we determine both angles with significantly smaller uncertainties (by more than a factor of two) compared to the auto spectra., Comment: 14 pages, 2 figures, 3 tables

The cosmic thermal history, quantified by the evolution of the mean thermal energy density in the universe, is driven by the growth of structures as baryons get shock heated in collapsing dark matter halos. This process can be probed by redshift-dependent amplitudes of the thermal Sunyaev-Zeldovich (SZ) effect background. To do so, we cross-correlate eight sky intensity maps in the $\it{Planck}$ and Infrared Astronomical Satellite missions with two million spectroscopic redshift references in the Sloan Digital Sky Surveys. This delivers snapshot spectra for the far-infrared to microwave background light as a function of redshift up to $z\sim3$. We decompose them into the SZ and thermal dust components. Our SZ measurements directly constrain $\langle bP_{\rm e} \rangle$, the halo bias-weighted mean electron pressure, up to $z\sim 1$. This is the highest redshift achieved to date, with uncorrelated redshift bins thanks to the spectroscopic references. We detect a threefold increase in the density-weighted mean electron temperature $\bar{T}_{\rm{e}}$ from $7\times 10^5~{\rm K}$ at $z=1$ to $2\times 10^6~{\rm K}$ today. Over $z=1$-$0$, we witness the build-up of nearly $70\%$ of the present-day mean thermal energy density $\rho_{\rm{th}}$, with the corresponding density parameter $\Omega_{\rm th}$ reaching $1.5 \times10^{-8}$. We find the mass bias parameter of $\it{Planck}$'s universal pressure profile of $B=1.27$ (or $1-b=1/B=0.79$), consistent with the magnitude of non-thermal pressure in gas motion and turbulence from mass assembly. We estimate the redshift-integrated mean Compton parameter $y\sim1.2\times10^{-6}$, which will be tested by future spectral distortion experiments. More than half of which originates from the large-scale structure at $z<1$, which we detect directly., Comment: 22 pages, 11 figures. Accepted for publication in ApJ

Building on Weinberg's approach to effective field theory for inflation, we construct an effective Lagrangian for a pseudo scalar (axion) inflaton field with shift symmetry. In this Lagrangian we allow the axion field to couple to non-Abelian gauge fields via a Chern-Simons term. We then analyze a class of inflation models driven by kinetic terms. We find that the observational constraints on the amplitudes of curvature perturbations and non-Gaussianity yield a lower bound for the tensor-to-scalar ratio of $r\gtrsim 5\times 10^{-3}$ from the vacuum fluctuation. The sourced gravitational wave from SU(2) gauge fields further increases the tensor-to-scalar ratio and makes the total gravitational wave partially chiral and non-Gaussian, which can be probed by polarization of the cosmic microwave background and direct detection experiments. We discuss constraints on parameter space due to backreaction of spin-2 particles produced by the gauge field., Comment: 18 pages, 3 figures

In this paper, we examine the viability of inflation models with a spectator axion field coupled to both gravitational and SU(2) gauge fields via Chern-Simons couplings. Requiring phenomenological success of the axion-SU(2) sector constrains the coupling strength of the gravitational Chern-Simons term. We find that the impact of this term on the production and propagation of gravitational waves can be as large as fifty percent enhancement for the helicity that is not sourced by the gauge field, if the cut-off scale is as low as $\Lambda$ = 20H. The effect becomes smaller for a larger value of $\Lambda$, while the impact on the helicity sourced by the gauge field is negligible regardless of $\Lambda$., Comment: 15 pages, 4 figures (v2) References added and typos corrected. Submitted to JCAP

The key to the phenomenological success of inflation models with axion and SU(2) gauge fields is the isotropic background of the SU(2) field. Previous studies showed that this isotropic background is an attractor solution during inflation starting from anisotropic (Bianchi Type I) spacetime; however, not all possible initial anisotropic parameter space was explored. In this paper, we explore more generic initial conditions without assuming the initial slow-roll dynamics. We find some initial anisotropic parameter space which does not lead to the isotropic background, but to violation of slow-roll conditions, terminating inflation prematurely. The basin of attraction increases when we introduce another scalar field acting as inflaton and make the axion-SU(2) system a spectator sector. Therefore, the spectator axion-SU(2) model is phenomenologically more attractive., Comment: 22 pages, 9 figures. Updated to version accepted by JCAP

We examine the impact of fiber assignment on clustering measurements from fiber-fed spectroscopic galaxy surveys. We identify new effects which were absent in previous, relatively shallow galaxy surveys such as Baryon Oscillation Spectroscopic Survey . Specifically, we consider deep surveys covering a wide redshift range from z=0.6 to z=2.4, as in the Subaru Prime Focus Spectrograph survey. Such surveys will have more target galaxies than we can place fibers on. This leads to two effects. First, it eliminates fluctuations with wavelengths longer than the size of the field of view, as the number of observed galaxies per field is nearly fixed to the number of available fibers. We find that we can recover the long-wavelength fluctuation by weighting galaxies in each field by the number of target galaxies. Second, it makes the preferential selection of galaxies in under-dense regions. We mitigate this effect by weighting galaxies using the so-called individual inverse probability. Correcting these two effects, we recover the underlying correlation function at better than 1 percent accuracy on scales greater than 10 Mpc/h., Comment: 17 pages, 11 figures

Redshifts of an astronomical body measured at multiple epochs (e.g., separated by 10 years) are different due to the cosmic expansion. This so-called Sandage-Loeb test offers a direct measurement of the expansion rate of the Universe. However, acceleration in the motion of Solar System with respect to the cosmic microwave background also changes redshifts measured at multiple epochs. If not accounted for, it yields a biased cosmological inference. To address this, we calculate the acceleration of Solar System with respect to the Local Group of galaxies to quantify the change in the measured redshift due to local motion. Our study is motivated by the recent determination of the mass of Large Magellanic Cloud (LMC), which indicates a significant fraction of the Milky Way mass. We find that the acceleration towards the Galactic Center dominates, which gives a redshift change of 7 cm/s in 10 years, while the accelerations due to LMC and M31 cannot be ignored depending on lines of sight. We create all-sky maps of the expected change in redshift and the corresponding uncertainty, which can be used to correct for this effect., Comment: 6 pages, 3 figures. Accepted for publication in PASJ

The local expansion rate of the Universe is parametrized by the Hubble constant, $H_0$, the ratio between recession velocity and distance. Different techniques lead to inconsistent estimates of $H_0$. Observations of Type Ia supernovae (SNe) can be used to measure $H_0$, but this requires an external calibrator to convert relative distances to absolute ones. We use the angular diameter distance to strong gravitational lenses as a suitable calibrator, which is only weakly sensitive to cosmological assumptions. We determine the angular diameter distances to two gravitational lenses, $810^{+160}_{-130}$ and $1230^{+180}_{-150}$~Mpc, at redshifts of $z=0.295$ and $0.6304$. Using these absolute distances to calibrate 740 previously-measured relative distances to SNe, we measure the Hubble constant to be $H_0=82.4^{+8.4}_{-8.3} ~{\rm km\,s^{-1}\,Mpc^{-1}}$., Comment: This paper presents the measurements of angular diameter distances to two time-delay lenses, and the Hubble constant derived only from these two distances and the JLA supernova sample. One of the distance measurements is further used for the cosmological inference in the H0LiCOW XIII paper (arxiv:1907.04869). Published in Science

The Cerro Chajnantor Atacama Telescope-prime (CCAT-prime) is a new 6-m, off-axis, low-emissivity, large field-of-view submillimeter telescope scheduled for first light in the last quarter of 2021. In summary, (a) CCAT-prime uniquely combines a large field-of-view (up to 8-deg), low emissivity telescope (< 2%) and excellent atmospheric transmission (5600-m site) to achieve unprecedented survey capability in the submillimeter. (b) Over five years, CCAT-prime first generation science will address the physics of star formation, galaxy evolution, and galaxy cluster formation; probe the re-ionization of the Universe; improve constraints on new particle species; and provide for improved removal of dust foregrounds to aid the search for primordial gravitational waves. (c) The Observatory is being built with non-federal funds (~ \$40M in private and international investments). Public funding is needed for instrumentation (~ \$8M) and operations (\$1-2M/yr). In return, the community will be able to participate in survey planning and gain access to curated data sets. (d) For second generation science, CCAT-prime will be uniquely positioned to contribute high-frequency capabilities to the next generation of CMB surveys in partnership with the CMB-S4 and/or the Simons Observatory projects or revolutionize wide-field, sub-millimetter line intensity mapping surveys., Comment: Astro2020 APC White Paper

Thermal Sunyaev-Zeldovich (tSZ) power spectrum is a powerful probe of the present-day amplitude of matter density fluctuations, and has been measured up to $\ell\approx 10^3$ from the {\it Planck} data. The largest systematic uncertainty in the interpretation of this data is the so-called "mass bias" parameter $B$, which relates the true halo mass to the mass proxy used by the {\it Planck} team as $M_{\rm 500c}^{Planck}=M_{\rm 500c}^{\rm true}/B$. Since the power spectrum of the cosmic weak lensing shear is also sensitive to the amplitude of matter density fluctuations via $S_8\equiv \sigma_8 \Omega_m^{\alpha}$ with $\alpha\sim 0.5$, we can break the degeneracy between the mass bias and the cosmological parameters by combining the tSZ and cosmic shear power spectra. In this paper, we perform a joint likelihood analysis of the tSZ power spectrum from {\it Planck} and the cosmic shear power spectrum from Subaru Hyper Suprime-Cam. Our analysis does not use the primordial cosmic microwave background (CMB) information. We obtain a new constraint on the mass bias as $B = 1.37 ^{+0.15}_{-0.23}$ or $(1-b) = B^{-1}=0.73^{+0.08}_{-0.13}$ (68\%~C.L.), for $\sigma_8 < 0.9$. This value of $B$ is lower than that needed to reconcile the tSZ data with the primordial CMB and CMB lensing data, i.e., $B = 1.64 \pm 0.19$, but is consistent with the mass bias expected from hydrodynamical simulations, $B = 1.28 \pm 0.20$. Our results thus indicate that the mass bias is consistent with the non-thermal pressure support from mass accretion of galaxy clusters via the cosmic structure formation, and that the cosmologies inferred from the tSZ and the cosmic shear are consistent with each other., Comment: 8 pages, 3 figures, PASJ accepted

We present a measurement of the Hubble constant ($H_{0}$) and other cosmological parameters from a joint analysis of six gravitationally lensed quasars with measured time delays. All lenses except the first are analyzed blindly with respect to the cosmological parameters. In a flat $\Lambda$CDM cosmology, we find $H_{0} = 73.3_{-1.8}^{+1.7}$, a 2.4% precision measurement, in agreement with local measurements of $H_{0}$ from type Ia supernovae calibrated by the distance ladder, but in $3.1\sigma$ tension with $Planck$ observations of the cosmic microwave background (CMB). This method is completely independent of both the supernovae and CMB analyses. A combination of time-delay cosmography and the distance ladder results is in $5.3\sigma$ tension with $Planck$ CMB determinations of $H_{0}$ in flat $\Lambda$CDM. We compute Bayes factors to verify that all lenses give statistically consistent results, showing that we are not underestimating our uncertainties and are able to control our systematics. We explore extensions to flat $\Lambda$CDM using constraints from time-delay cosmography alone, as well as combinations with other cosmological probes, including CMB observations from $Planck$, baryon acoustic oscillations, and type Ia supernovae. Time-delay cosmography improves the precision of the other probes, demonstrating the strong complementarity. Allowing for spatial curvature does not resolve the tension with $Planck$. Using the distance constraints from time-delay cosmography to anchor the type Ia supernova distance scale, we reduce the sensitivity of our $H_0$ inference to cosmological model assumptions. For six different cosmological models, our combined inference on $H_{0}$ ranges from $\sim73$-$78~\mathrm{km~s^{-1}~Mpc^{-1}}$, which is consistent with the local distance ladder constraints., Comment: Accepted for publication in MNRAS; 23 pages, 13 figures, 8 tables

We show that the cosmic birefringence and miscalibrated polarisation angles can be determined simultaneously by cosmic microwave background (CMB) experiments using the cross-correlation between $E$- and $B$-mode polarisation data. This is possible because polarisation angles of the CMB are rotated by both the cosmic birefringence and miscalibration effects,whereas those of the Galactic foreground emission only by the latter. Our method does not require prior knowledge of the $E$- and $B$-mode power spectra of the foreground emission, but uses only the knowledge of the CMB polarisation spectra. Specifically, we relate the observed $EB$ correlation to the difference between the $\mathit{observed}$ $E$- and $B$-mode spectra in the sky, and use different multipole dependence of the CMB (given by theory) and foreground spectra (with no assumption) to derive the likelihood for the miscalibration angle $\alpha$ and the birefringence angle $\beta$. We show that a future satellite mission similar to LiteBIRD can determine $\beta$ with a precision of ten arcminutes., Comment: 13 pages, 4 figures, 2 tables