The Atacama Cosmology Telescope: Multi-probe cosmology with unWISE galaxies and ACT DR6 CMB lensing
Gerrit S. Farren Alex Krolewski Frank J. Qu Simone Ferraro Erminia Calabrese Jo Dunkley Carmen Embil Villagra J. Colin Hill Joshua Kim Mathew S. Madhavacheril Kavilan Moodley Lyman A. Page Bruce Partridge Neelima Sehgal Blake D. Sherwin Cristóbal Sifón Suzanne T. Staggs Alexander Van Engelen Edward J. Wollack











    figures/rcases.}unWISE x ACT DR6 3×2pt cosmologyFarren, Krolewski, Qu, Ferraro et al.0000-0001-5704-1127]Gerrit S. FarrenDAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UKKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge CB3 0HA, UK0000-0003-2183-7021]Alex KrolewskiPerimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON NL2 2Y5, Canada0000-0001-7805-1068]Frank J. QuDAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UKKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge CB3 0HA, UK0000-0003-4992-7854]Simone FerraroPhysics Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USABerkeley Center for Cosmological Physics, University of California, Berkeley, CA 947200000-0003-0837-0068]Erminia CalabreseSchool of Physics and Astronomy, Cardiff University, The Parade, Cardiff, Wales CF24 3AA, UK0000-0002-7450-2586]Jo DunkleyJoseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ USA 085440009-0001-3987-7104]Carmen Embil VillagraDAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UK0000-0002-9539-0835]J. Colin HillDepartment of Physics, Columbia University, 538 West 120th Street, New York, NY, USA 100270000-0002-0935-3270]Joshua KimDepartment of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA0000-0001-6740-5350]Mathew S. MadhavacherilDepartment of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA, USA 191040000-0001-6606-7142]Kavilan MoodleyAstrophysics Research Centre, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4001, South Africa0000-0002-9828-3525]Lyman A. PageJoseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 085440000-0001-6541-9265]Bruce PartridgeDepartment of Physics and Astronomy, Haverford College, Haverford PA, USA 190410000-0002-9674-4527]Neelima SehgalPhysics and Astronomy Department, Stony Brook University, Stony Brook, NY USA 117940000-0002-4495-1356]Blake D. SherwinDAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UKKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge CB3 0HA, UK0000-0002-8149-1352]Cristóbal SifónInstituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile0000-0002-7020-7301]Suzanne T. StaggsJoseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 085440000-0002-3495-158X]Alexander Van EngelenSchool of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287, USA0000-0002-7567-4451]Edward J. WollackNASA/Goddard Space Flight Center, Greenbelt, MD, USA 20771Gerrit S. Farrengsf29@cam.ac.uk§ ABSTRACT  
    We present a joint analysis of the CMB lensing power spectra measured from the Data Release 6 of the Atacama Cosmology Telescope and Planck PR4, cross-correlations between the ACT and Planck lensing reconstruction and galaxy clustering from unWISE, and the unWISE clustering auto-spectrum. We obtain 1.5% constraints on the matter density fluctuations at late times parametrised by the best constrained parameter combination S_8^ 3x2pt≡σ_8 (Ω_m/0.3)^0.4 = 0.815 ± 0.012. The commonly used S_8 ≡σ_8 (Ω_m/0.3)^0.5 parameter is constrained to S_8 = 0.816±0.015. In combination with baryon acoustic oscillation (BAO) measurements we find σ_8=0.815± 0.012. We also present sound-horizon-independent estimates of the present day Hubble rate of H_0=66.4^+3.2_-3.7 from our large scale structure data alone and H_0=64.3^+2.1_-2.4 in combination with uncalibrated supernovae from . Using parametric estimates of the evolution of matter density fluctuations, we place constraints on cosmic structure in a range of high redshifts typically inaccessible with cross-correlation analyses. Combining lensing cross- and auto-correlations, we derive a 3.3% constraint on the integrated matter density fluctuations above z=2.4, one of the tightest constraints in this redshift range and fully consistent with a ΛCDM model fit to the primary CMB from Planck. Finally, combining with primary CMB observations and using the extended low redshift coverage of these combined data sets we derive constraints on a variety of extensions to the ΛCDM model including massive neutrinos, spatial curvature, and dark energy. We find in flat ΛCDM ∑ m_ν<0.12 eV at 95% confidence using the LSS data, BAO measurements from SDSS and primary CMB observations.
    
    
    § INTRODUCTION
    
    
    Measurements of the matter density fluctuations at low redshifts inform our understanding of the formation of cosmic structure, probe the nature of dark matter and dark energy, and constrain the masses of neutrinos. They also provide an important test of the predictions of general relativity. Gravitational lensing observations that are sensitive to the total matter distribution, including the invisible dark matter, have become an indispensable tool for studying cosmic structure. Several lensing related techniques have been developed to study both the weak gravitational lensing of galaxies as well as of the cosmic microwave background (CMB).
    
    Over the past two decades a standard model of cosmology has emerged primarily based on high precision observations of the CMB. Measurements by WMAP first established the now prevailing six parameter ΛCDM model <cit.>. It posits that the Universe is dominated by phenomenological cold dark matter (CDM), is spatially flat, and its expansion is driven by a cosmological constant Λ. These results were sharpened by measurements made by the Planck satellite <cit.>. The model also makes predictions for other cosmological observables, which in recent years have reached increasing precision enabling new tests of this model. Despite the ΛCDM model's overall success, some discrepancies have been observed and several extensions have been put forward. Here we weigh in on some of these discrepancies and constrain extensions beyond the standard model.
    
    One such discrepancy and a primary focus of this work is the amplitude of matter density fluctuations typically parametrised in terms of σ_8, the RMS (root-mean-square) of the linear matter density contrast smoothed on scales of 8, which provides the normalisation of the matter power spectrum. The shape and redshift evolution of the matter power spectrum are predicted from the ΛCDM model.
    
    In recent work <cit.> and <cit.> reconstructed the gravitational lensing field over 9400 deg^2 from new high resolution CMB observations by the Atacama Cosmology Telescope (ACT). They showed percent level constraints on the integrated matter density fluctuations over a wide range of redshifts (z≲ 5) from the CMB lensing power spectrum. The σ_8 constraints are in excellent agreement with model extrapolations from a ΛCDM model fit to observations of the primary fluctuations in the CMB as observed by Planck<cit.>. Their results are also in excellent agreement with CMB lensing measurements from Planck<cit.> and the combination of both measurements yields improved constraints on cosmic structure.
    
    <cit.> used the ACT DR6 and Planck CMB lensing reconstruction together with galaxies detected in imaging data from the Wide-Field Infrared Survey Explorer <cit.> to focus on a lower redshift range, approximately 0.2≲ z ≲ 1.8. Using the correlation between the galaxy distribution in two redshift bins, which acts as a biased tracer of the underlying matter density, and the CMB lensing reconstruction, <cit.> similarly found good agreement with the ΛCDM prediction for σ_8 and CMB lensing auto-correlation results.
    
    Meanwhile, several galaxy weak lensing surveys like the Dark Energy Survey <cit.>, the Kilo-Degree Survey <cit.>, and the Hyper Suprime-Cam <cit.>, among others, have found a 2-3σ lower amplitude of matter density fluctuations, compared to the prediction from Planck primary CMB assuming a standard ΛCDM model. Such surveys typically probe lower redshifts, z ≲ 1, than the CMB lensing work discussed above. Similar results have also been obtained by some other studies of CMB lensing cross-correlations with galaxy surveys, albeit at varying levels of significance <cit.>. While the modelling assumptions and the range of scales probed in these works vary, they mostly sensitive to lower redshifts than the analysis presented in <cit.>.
    
    This motivates a further investigation of the formation of structure at low redshifts. We note that the  σ_8 constraints from <cit.> and <cit.> differ from the galaxy weak lensing result not only in terms of the redshift range, but also in the scales probed. As pointed out for example by <cit.> and <cit.> galaxy weak lensing draws significant information from highly non-linear scales which the CMB lensing auto- and cross-correlations are insensitive to. The observed discrepancy (∼2-3σ) with the ΛCDM prediction for σ_8 from Planck may therefore also be explained by a suppression of the matter power spectrum on small scales that exceeds expectations of the baryon feedback induced suppression based on the most recent hydrodynamical simulations <cit.>.
    
    Motivated in part by further investigation of the formation of structure at low redshifts and on linear to mildly non-linear scales we combine the lensing auto-spectrum analysis from <cit.> and <cit.> with the cross-correlation analysis from <cit.>. Throughout the paper we will often refer to this combination as `3x2pt' given that it contains three two-point correlation functions (or rather their harmonic space equivalent, the power spectrum). These are the auto-spectrum of the CMB lensing convergence, C_ℓ^κκ, the cross-correlation between galaxies and CMB lensing, C_ℓ^κ  g, and the galaxy auto-correlation, C_ℓ^gg. By contrast we will often refer to the cross-correlation analysis as `2x2pt' (C_ℓ^κ  g&C_ℓ^gg). In Sec. <ref> we show constraints on cosmic structure formation from our `3x2pt' data.
    
    Another discrepancy, which has reached ∼5σ with recent data, is between the present day expansion rate, parameterised by the Hubble constant H_0, inferred within the ΛCDM model from the CMB <cit.> and a local measurement based on Cepheid-calibrated supernovae from <cit.>. Meanwhile results from baryon acoustic oscillation (BAO) observations have generally found results consistent with the CMB-derived values <cit.>. CMB and BAO constraints are predominantly informed by the angular size of the sound horizon scale[Here, we do not make a careful distinction between the sound horizon scale relevant for BAO (r_ d) and CMB (r_ s) observations, although to be precise these are defined at the baryon drag epoch and at photon decoupling, respectively.]. This fact has motivated theoretical work to explain the tension by invoking new physics that decreases the physical size of the sound horizon at recombination by approximately 10% (e.g., ). It also motivates new measurements of the Hubble constant that are derived from a different physical scale present in the large-scale structure, namely, the matter-radiation equality scale (with comoving wave-number k_ eq) which sets the turn-over in the matter power spectrum. As pointed out in <cit.> such measurements can be obtained from the CMB lensing power spectrum and other large scale structure (LSS) tracers and we investigate the implications of our data for the H_0-tension in Sec. <ref>.
    
    Beyond modifications to the flat ΛCDM model motivated by these observed discrepancies other extensions are motivated by physical considerations. Given the phenomenological nature of dark energy it is natural to consider departures from the cosmological constant model which allow an equation of state w≠ -1 or evolution in the dark energy equation of state. We consider such models in Sec. <ref>; our data unfortunately only marginally improves on existing constraints on such models derived from the primary CMB, BAO and supernovae. Furthermore, observations of neutrino oscillations require neutrinos to be massive, and while the mass splitting between the neutrino mass eigenstates is well determined the absolute mass scale is poorly constrained <cit.>. In Sec. <ref> we derive constraints on the neutrino mass from the characteristic suppression of the formation of structure on scales smaller than the neutrino free streaming scale which can be probed with our `3x2pt' data. It has been pointed out that those neutrino mass constraints are model dependent, as the effect of massive neutrinos can be mimicked also by some beyond-ΛCDM models <cit.>. Thus we consider massive neutrinos in the context of extended dark energy models in Sec. <ref>. In addition to dark energy and neutrinos, we also revisit the assumption of spatial flatness in Sec. <ref> where our data provides competitive cross-checks on constraints from the primary CMB and BAO.
    
    Before presenting these results in Sec. <ref> we discuss the data used in this analysis in Sec. <ref> including the external data sets we employ in our analysis (Sec. <ref>). In Sec. <ref> we briefly describe how we obtain the covariance for the likelihood analysis described in Sec. <ref>. Finally, in Sec. <ref> we summarise our findings and conclude with an outlook to future work. For the convenience of the reader we also provide an overview of the key results from this work in the following section (Sec. <ref>).
    
    
    
    
    § KEY RESULTS
    
    
    The key results from this work are constraints on the amplitude of matter density fluctuations at low redshift parameterised by σ_8. As in CMB lensing autocorrelation analyses <cit.>, CMB lensing cross-correlations <cit.>, or galaxy weak lensing <cit.>, when using the projected large scale structure tracers only, σ_8 is significantly degenerate with the matter density, Ω_m (as can be seen in the left panel of Fig. <ref>). In our case the parameter combination best constrained by the combination of CMB lensing auto-spectrum, galaxy-galaxy clustering auto-power spectra, and the cross-correlation between CMB lensing and galaxies is approximately σ_8 Ω_m^0.4. In analogy with the commonly used parameter S_8 ≡σ_8 (Ω_m/0.3)^0.5, corresponding to the parameter combination best constrained by galaxy weak lensing surveys, we therefore define the parameter S_8^ 3x2pt≡σ_8 (Ω_m/0.3)^0.4. 
    
    
    
    Using the lensing auto-spectra from ACT and Planck, the respective cross-correlations with the unWISE galaxies, and the galaxy auto-correlation (3x2pt) we constrain this parameter combination to ∼1.5%,
     S_8^ 3x2pt≡   σ_8 (Ω_m/0.3)^0.4 = 0.815± 0.012 ( 3x2pt).
    Within the ΛCDM model this parameter is similarly well constrained by primary CMB data (S_8^ 3x2pt = 0.826± 0.012 from the primary CMB data sets discussed in Sec. <ref>) and the two constraints are in good agreement as can be seen in the left panel of Fig. <ref>. Even though S_8 differs slightly from the best constrained combination of σ_8 and Ω_m in our work we nevertheless also obtain highly competitive constraints on this parameter combination which can be compared, for example, to results from DES, KiDS and HSC <cit.>. Using again both ACT and Planck CMB lensing auto-spectra, the cross-correlations with unWISE, and the unWISE auto-spectrum we find  S_8=0.816± 0.015  ( 3x2pt). While our value of S_8 falls between the typical values preferred by galaxy weak lensing analyses and those predicted from ΛCDM fits to the CMB, it is not in statistically significant tension with either of those data sets (∼1σ and ∼1.5-2.5σ relative to primary CMB and various galaxy weak lensing measurements respectively).
    
    BAO data provides constraints primarily in the Ω_m-H_0 parameter space and can therefore be used to break the degeneracy between σ_8 and Ω_m. By adding BAO data, primarily from the Baryon Oscillation Spectroscopic Survey (BOSS), to our analysis we obtain
    σ_8=0.815 ± 0.012  ( 3x2pt +  BAO), also a ∼1.5% measurement. This value is again in good agreement with model predictions derived from ΛCDM fits to primary CMB data (σ_8 = 0.8107± 0.0064; the posteriors are shown in the σ_8-Ω_m plane in the right panel of Fig. <ref>).
    
    Following the approach suggested in <cit.> we use our CMB lensing and cross-correlation data to place constraints on the Hubble constant which arise exclusively from the measurement of the angular size of the matter-radiation equality scale imprinted in the turnover of the matter power spectrum. These constraints are independent of the sound horizon scale, knowledge of which is crucial for inferring the Hubble constant from the baryon oscillation feature either in the primary CMB or through BAO observations. We find  H_0=66.5^+3.2_-3.7 ( 3x2pt) from CMB lensing, cross-correlation, and galaxy auto-correlation data alone (using both ACT and Planck). When additionally including uncalibrated supernovae from the  data set <cit.> to further break the degeneracy between H_0 and Ω_m we obtain  H_0=64.3^+2.1_-2.4 ( 3x2pt +  SN). This represents a ∼20% improvement over the constraint presented in <cit.> from the lensing auto-spectrum alone. Our results are consistent with the value of H_0 preferred by BAO and primary CMB data (at the ∼1σ level), but in tension (∼3.6σ) with local measurements of H_0 from <cit.>. This is consistent with the results found using similar methods on three-dimensional galaxy clustering data <cit.>.
    
    Posteriors for the key ΛCDM parameters (H_0, Ω_m, σ_8, and S_8^ 3x2pt) probed in our analysis are shown in Fig. <ref>. We show posteriors for the 3x2pt analysis jointly using ACT and Planck lensing data, as well as each lensing data set separately.
    
    Furthermore, we explore a model-agnostic parametrisation of the growth of perturbations. Using the combination of lensing auto- and cross-correlations together with BAO we find tight (≲ 4%) constraints on σ_8(z) in three redshift bins z=0-1.15, z=1.15-2.4, and z>2.4. Our reconstruction of σ_8(z) is in good agreement with predictions within a ΛCDM model fit to the primary CMB from Planck.
    
    In going beyond the standard ΛCDM model we explore various extensions, including non-minimum mass neutrinos, evolving dark energy and curvature. We find constraints on the neutrino mass sum from the combination of our 3x2pt data with BAO and primary CMB data from Planck consistent with results from analyses using only the lensing auto-spectrum. When including both ACT and Planck lensing auto- and cross-spectra we constrain the sum of the neutrinos to 
    
        ∑ m_ν <    0.124 eV at 95% confidence
           ( 3x2pt +  CMB +  BAO).
     
    These constraints are degraded when also considering a time varying dark energy equation of state but by additionally including uncalibrated supernovae from  we still obtain a constraint of ∑ m_ν < 0.231  eV (95% C.I.).
    
    We constrain the curvature of the Universe to 
    
        -0.011 < Ω_k < 0.004    at 95% confidence
           ( 3x2pt +  CMB)
     
    from the primary CMB and our data alone (without BAO), about 20% tighter than previous results from the primary CMB and the CMB lensing auto-spectrum only <cit.>. 
    
    We make our data and likelihood publicly available enabling the community to perform further investigation into models not explored in this work (see Appendix <ref> for details).
    
    § DATA
    
    
    In this section we briefly describe the data sets we use in our analysis as well as several external likelihoods we include to break degeneracies and probe both the ΛCDM model as well as potential extensions with all available data.
    
    
    
    
     §.§ CMB lensing power spectrum
    
    Throughout we adopt the CMB lensing power spectrum measurements from ACT DR6 <cit.> and Planck PR4 <cit.>.
    
    The ACT DR6 lensing reconstruction covers 9400 deg^2 of the sky and is signal-dominated on lensing scales of L<150. This reconstruction is based on CMB measurements made between 2017 and 2021 (relying only on the night-time data) at ∼90 and ∼150 GHz and uses CMB scales 600<ℓ<3000. The use of cross-correlation based estimators reduces the sensitivity to the modelling of the instrumental noise <cit.>, and profile hardening is employed to mitigate against extragalactic foregrounds <cit.>. Since the CMB lensing signal is reconstructed using quadratic estimators the power spectrum of the reconstruction is a four-point function containing several biases which need to be subtracted using simulations. The largest of these biases is the Gaussian disconnected bias, which depends on the two-point power spectrum of the observed CMB maps and is thus non-zero even in the absence of lensing. The debiasing is discussed in detail in <cit.>. Since some of the bias corrections as well as the normalisation of the lensing estimator depend weakly on cosmology we implement corrections capturing the dependency of the lensing normalisation and bias subtraction on cosmological parameters (see Appendix <ref> for details). The CMB lensing power spectrum from <cit.> is determined at 2.3% precision, corresponding to a measurement signal-to-noise ratio of 43σ.
    
    Potential sources of systematic biases have been investigated in detail in <cit.> and <cit.> and were found to be comfortably subdominant to statistical uncertainties.
    
    The Planck PR4 lensing analysis <cit.> reconstructs lensing with CMB angular scales from 100≤ℓ≤2048 using the quadratic estimator. This analysis is based on the reprocessed PR4  maps that incorporated around 8% more data compared to the 2018 Planck PR3 release. It also includes pipeline improvements such as optimal (anisotropic) filtering of the input CMB fields resulting in an increase of the overall signal-to-noise ratio by around 20% compared to  Planck PR3 <cit.> and a detection of the lensing power spectrum  at 42σ.
    
    
    
     §.§ CMB lensing-galaxy cross-correlations and galaxy-galaxy auto-correlation
    
    We include measurements of the cross-correlation between CMB lensing (from ACT DR6 and Planck PR4) and galaxies from the unWISE catalog, along with the auto-correlation of the unWISE galaxies <cit.>. The unWISE galaxy catalogue is constructed from three band imaging by the Wide-Field Infrared Survey Explorer (WISE) survey <cit.>, including four years of the post-hibernation NEOWISE phase <cit.>. We employ two colour selected galaxy samples which we refer to as Blue and Green, with broad redshift distributions centred at z ∼ 0.6 and 1.1. These samples are extensively described in <cit.> and <cit.>. To obtain redshift distributions for these samples of galaxies we employ the method of clustering redshifts estimated from cross-correlations with spectroscopic samples from the Sloan Digital Sky Survey (SDSS) (see  and  for further details).
    
    As with the lensing power spectrum measurements, <cit.> undertook an extensive evaluation of potential systematic biases using different versions of the lensing reconstructions and analysis masks, as well as realistic simulations of the contamination due to extragalactic foregrounds. They did not find any evidence for statistically significant biases.
    
    
    
     §.§ External Likelihoods
    
    We also combine our results with external data sets from observations of the primary CMB temperature and polarisation anisotropies, Baryon Acoustic Oscillations, and uncalibrated supernovae.
    
    
    
      §.§.§ Primary CMB
    
    We jointly analyse our data with measurements of the temperature and polarisation power spectra of the primary CMB as observed by the Planck satellite. In keeping with <cit.> we adopt the analysis of the Planck PR4 maps based on the  likelihood for the small scale (ℓ>30) temperature and polarisation power spectra <cit.>. Additionally, we include the Planck PR3 likelihood for the large scale temperature power spectrum <cit.>. Finally, to include information from Planck's large-scale polarisation data that constrains the optical depth to reionisation we use the likelihood estimated in <cit.> from the  maps.
    
    
    
    
    
      §.§.§ Baryon Acoustic Oscillations
    
    Furthermore, we add observations of the BAO feature. We use a combination of BAO measurements based on the clustering of galaxies with samples spanning redshifts up to z≃ 1, including 6dFGS <cit.>, SDSS DR7 Main Galaxy Sample <cit.>, BOSS DR12 LRGs <cit.>, and eBOSS DR16 LRGs <cit.>. In contrast to earlier work on the ACT DR6 CMB lensing auto-spectrum <cit.>, we additionally include the higher-redshift ELGs <cit.>, Lyman-α forest <cit.>, and quasar samples <cit.> from eBOSS DR16.
    
    As this work was nearing completion the Year-1 BAO results from the Dark Energy Spectroscopic Instrument <cit.> were released <cit.>. These provide improved BAO measurements tightening constraints on the cosmic expansion history. We do not reanalyse all our results with the DESI BAO, but because <cit.> showed that this data favours tight limits on the neutrino mass, we address our neutrino mass constraints also with the recent DESI BAO likelihood (see Sec. <ref>). Given the consistency and similar constraining power on Ω_m from DESI Y1 and earlier BAO data sets in the context of flat ΛCDM we do not expect the choice of BAO likelihoods to impact our constraints on structure formation.
    
      §.§.§ Uncalibrated Supernovae
    
    
    Finally, we present some constraints which additionally employ `uncalibrated' measurements of the relationship between the apparent brightness of Type IA supernovae and their redshifts from the  data set <cit.>. Here, `uncalibrated' refers to the fact that the absolute magnitudes of the supernovae have not been calibrated, e.g., with Cepheid variables or the tip-of-the-red-giant-branch (TRGB) technique, such that only information from the relative (not absolute) distance-redshift relation is included. This data set provides matter density information independently of the primary CMB and BAO, which enables estimates of the Hubble constant independent of the sound horizon scale as proposed in <cit.>. Uncalibrated supernovae also constrain potential dark energy evolution and break degeneracies between evolving dark energy and neutrino mass when considering multiple extensions to the baseline ΛCDM model jointly.
    
    
    
    
    § 3×2PT COVARIANCE
    
    As in <cit.>, we use a simulation derived covariance for the CMB lensing and lensing cross-correlation measurements. <cit.> described how to obtain Gaussian realisations of the galaxy field that exhibit the correct correlations with lensing reconstruction simulations from <cit.> for ACT and <cit.> for Planck. They also detailed the analytic estimates for the cross-covariance between different galaxy samples and the simulations used to estimate the covariance between ACT and Planck cross-correlations, which are obtained by running the ACT lensing reconstruction pipeline on Planck simulations. For this work we additionally adopt the bias subtracted CMB lensing auto-power spectra measured on the same simulations to estimate the cross-covariance between the cross-correlation and auto-correlation measurements.
    
    We find significant correlations of up to 50-60% between the ACT lensing auto-spectrum and the cross-correlations with the ACT lensing map and slightly smaller (≲40%) between the respective Planck spectra. Correlations between C_ℓ^gg and lensing auto-spectra and between cross- and auto-spectra from ACT and Planck respectively (or vice versa) are small (≲20%).
    
    Furthermore, when analysing the low redshift data only (i.e., not in combination with primary CMB data) we propagate uncertainties in the lensing estimator normalisation due to the uncertainty in the CMB two-point functions into additional contributions to the covariance as described in <cit.> and <cit.> for the cross- and auto-spectra respectively. When combining with primary CMB data, we explicitly correct for errors in the normalisation and reconstruction bias subtraction as described in Appendix <ref>.
    
    
    
    § COSMOLOGICAL ANALYSIS
    
    
    We obtain cosmological constraints by constructing a Gaussian likelihood
    
    -2 lnℒ∝∑_bb'[  ΔĈ_b^gg; ΔĈ_b^κ g;  ΔĈ_b^κκ ]ℂ^-1_b b'[  ΔĈ_b'^gg; ΔĈ_b'^κ g;   ΔĈ_b^κκ ]
    where the ΔĈ_b^gg, ΔĈ_b^κ g, and ΔĈ_b^κκ are the residuals between our observed galaxy-galaxy, galaxy-CMB lensing, and CMB lensing-CMB lensing spectra, Ĉ_b^gg, Ĉ_b^κ g, and Ĉ_b^κκ, and the respective binned and band window convolved theory spectra, C_b^gg, C_b^κ g, and C_b^κκ. The covariance ℂ has the form
    ℂ_b b' = [      ℂ_b b'^gg-gg     ℂ_b b'^gg-κ g      ℂ_b b'^gg-κκ; (ℂ_b b'^gg-κ g)^T    ℂ_b b'^κ g-κ g     ℂ_b b'^κ g-κκ;  (ℂ_b b'^gg-κκ)^T (ℂ_b b'^κ g-κκ)^T      ℂ_b b'^κκ-κκ ]
    where ℂ_b b'^gg-gg, ℂ_b b'^κ g-κ g, and ℂ_b b'^gg-κ g are the galaxy auto-spectrum covariance, the galaxy-CMB lensing cross-spectrum covariance, and the cross-covariance between them. ℂ_b b'^gg-κκ and ℂ_b b'^κ g-κκ are the cross-covariance between the galaxy-galaxy and galaxy-CMB lensing spectra on one hand and the lensing auto spectrum on the other hand. Finally, ℂ_b b'^κκ-κκ is the CMB lensing power spectrum covariance. These are estimated from simulations as described above in Sec. <ref>. When combining the lensing power spectrum and cross-spectrum likelihood with that for the CMB anisotropy power spectra, we ignore the covariance between the measured lensing and anisotropy spectra, as these are negligible for DR6 noise sensitivities <cit.>. 
    
    We infer cosmological parameters via Markov Chain Monte Carlo (MCMC) methods performing Metropolis-Hastings sampling using the [<https://github.com/CobayaSampler/cobaya>] code <cit.>. We consider MCMC chains to be converged if the Gelman-Rubin statistic <cit.> satisfies R-1 ≤ 0.01 for the cosmological parameters of interest.
    
    We use a hybrid perturbation theory expansion to second order to model the galaxy auto and cross-spectra up to k∼0.3 (see  for details). We impose conservative priors on lensing magnification, shot noise, and the parameters of the bias expansion (second order and shear bias) based on simulations. <cit.> showed that these results are insensitive to these prior choices, higher order corrections contribute at most at the few percent level to the signal, and even neglecting all higher order terms leads to only minor shifts in inferred parameters (≪1σ). We also marginalise over uncertainties in the redshift distribution of unWISE galaxies (see  for details). The CMB lensing power spectrum is modeled using the non-linear matter power spectrum from  (see  for details).
    
     §.§ Priors
    
    
    In our baseline analysis we consider a spatially flat, ΛCDM universe with massive neutrinos of the minimum mass allowed in the normal hierarchy (∑ m_ν = 0.06eV). When analysing only low redshift data (i.e., when not including observations of the primary CMB) our analysis is insensitive to the optical depth to reionisation, τ, and we thus fix it to the mean value obtained in <cit.>. For the remaining five cosmological parameters we adopt the priors from <cit.> sampling the logarithm of the scalar perturbation amplitude, log(10^10 A_s), the primordial spectral tilt, n_s, the physical density in baryons and cold dark matter, Ω_b h^2 and Ω_c h^2, and the angular size of the sound horizon at recombination, θ_ MC. We place flat priors on all parameters except the baryon density and the primordial spectral tilt, which are only weakly constrained by our observations. We choose a prior motivated by Big Bang Nucleosynthesis (BBN) measurements of deuterium abundance from <cit.> for Ω_b h^2 (see Table <ref>). As pointed out in <cit.> the primordial spectral tilt and the amplitude of fluctuations are somewhat degenerate given only a measurement of the projected lensing auto- or cross-spectra. We conservatively adopt a prior centred on but also about five times broader than the n_s-constraint obtained from Planck measurements of the CMB anisotropy power spectra in the ΛCDM model <cit.>, and two times broader than constraints obtained there from various extensions of ΛCDM. To avoid exploring unphysical parts of parameter space we furthermore limit the range over which H_0 may vary to between 40 and 100. This limitation is relevant for some of the CMB lensing auto-spectrum-only runs we perform for comparison; all other analyses do not allow for such a wide range of H_0 values.
    
    We note that these prior choices differ somewhat from those adopted in <cit.>, where the baryon density and the spectral tilt were fixed to the Planck best-fit values. Furthermore, this previous work fixed the parameter combination Ω_m h^3, which within ΛCDM is closely related to θ_ MC<cit.>, to the Planck best-fit value.
    
    When combining our low redshift data sets with primary CMB observations discussed in Sec. <ref> we remove the informative priors on Ω_b h^2 and n_s and additionally also sample the optical depth to reionsiation with a uniform prior. When exploring beyond-ΛCDM extensions we adopt the same priors as in <cit.>. All priors are summarised in Table <ref>.
    
    
    
    § CONSTRAINTS ON COSMOLOGY
    
    
    In the following section we present our constraints on cosmological parameters obtained from jointly analysing the ACT and Planck lensing auto- and cross-spectra together with the auto-correlation of the unWISE galaxies and in some cases external data as discussed in Sec. <ref>. We begin by outlining our findings in the context of a flat ΛCDM model before discussing potential extensions beyond this model including non-minimum mass neutrinos, dark energy with equation of state w ≠ -1, and spatial curvature.
    
    
    
     §.§ Constraints on flat Λ CDM
    
    
    In the context of a flat ΛCDM cosmology the weak lensing and galaxy clustering is primarily sensitive to the amplitude of matter density fluctuations in the late universe. Different probes derive their information from different scales and redshifts, providing complementary information on the evolution of matter density perturbations. As discussed in <cit.> the lensing cross-correlations primarily derive their information from linear and moderately non-linear scales 0.05 ≲ k ≲ 0.3 and redshifts z ≃ 0.2 - 1.6. By contrast the lensing auto-spectrum is sensitive to a large range of redshifts between approximately 0.5 and 5 and is dominated by linear scales <cit.>. With the combination of these two data sets we can thus probe a wide range of redshifts from z ≃ 0.2 to 5 while focusing mostly on linear and mildly non-linear scales.
    
    These results can be compared for example to recent results from galaxy weak lensing surveys <cit.>. These are largely sensitive to smaller scales, but partially overlap with the lower end of the redshift range probed here <cit.>. As described above, such surveys have consistently found a somewhat smaller amplitude of matter density fluctuations than expected within the flat ΛCDM model fit to the primary CMB data from Planck.
    
    In addition to constraints on matter density fluctuations, our data are also sensitive to the Universe's expansion which sets the distances to the observed structures. Features in the distribution of matter of known physical size can be used to constrain the distance to the observed structures independently of redshift estimates and therefore provide the opportunity to constrain the present day expansion rate of the Universe, H_0. Two such scales are imprinted onto the matter density field in the early universe: The sound horizon scale, the distance a sound wave may travel in the early universe prior to recombination (up to z ≃ 1100), and the matter-radiation equality scale, related to the size of the horizon when the matter and radiation densities become equal (z ≃ 3500). 
    
    Our data are not directly sensitive to the former, as it manifests through oscillations in the power spectrum which are largely smoothed out due to the projection over a wide range of redshifts. However, the expansion rate can be estimated from the sound horizon feature using BAO observation from spectroscopic galaxy surveys (see e.g., those discussed in Sec. <ref>). Such measurements suffer from significant degeneracies between H_0 and the global matter density Ω_m. Our data, in which Ω_m and H_0 have a different degeneracy direction, can therefore serve to improve BAO derived H_0 estimates. 
    
    On the other hand, our data is directly sensitive to the latter scale which is related to the turnover of the matter power spectrum. It was pointed out in <cit.> that CMB lensing data could therefore be used to obtain constraints on the expansion rate independent of the sound horizon scale. This scale has been the target of several modifications to the ΛCDM model which aim to resolve the tension between CMB derived H_0 estimates (which also depend on the sound horizon) and local measurements using the distance ladder <cit.>. This measurement equally suffers from the H_0-Ω_m degeneracy, but the direction of the degeneracy varies with redshift and with the scales probed. Combining lensing auto- and cross-spectrum information thus partially alleviates this degeneracy, allowing us to derive a constraint on H_0 from our CMB lensing and galaxy clustering data alone. However, the degeneracy is more effectively broken by the addition of uncalibrated supernovae data.
    
    
    
      §.§.§ Constraints on structure growth
    
    
    Within the ΛCDM model the parameter combination best constrained by the combination of CMB lensing auto-spectrum data and our cross-correlation measurements is what we have defined as S_8^ 3x2pt≡σ_8 (Ω_m/0.3)^0.4. With lensing data from ACT alone (ACT DR6 C_ℓ^κκ + ACT DR6 × unWISE C_ℓ^κ g + unWISE C_ℓ^gg) we constrain this parameter combination to ∼1.8%,[Here and for the remainder of the section, we will label the 3x2pt analyses using the ACT DR6 or Planck PR4 CMB lensing data sets (or their combination) as ACT+ unWISE or Planck+ unWISE (ACT+Planck+ unWISE) respectively. In the subsequent sections where we only consider the joint 3x2pt data set using ACT and Planck CMB lensing data we will simply refer to these as `3x2pt'. Meanwhile, we label the primary CMB from Planck PR4 simply as `CMB'.]
        S_8^ 3x2pt = 0.819 ± 0.015 ( ACT+ unWISE).
    
    This can be compared to results using Planck PR4 lensing data only 
    
        S_8^ 3x2pt = 0.803 ± 0.015 (Planck+ unWISE),
    
    a 1.9% constraint. The combination of ACT and Planck data tightens these constraints by a factor of ∼1.25 to
    
        S_8^ 3x2pt = 0.815 ± 0.012 ( ACT+Planck+ unWISE),
    
    
    For comparison the best constrained parameters in the lensing-only analysis is S_8^ CMBL≡σ_8 (Ω_m/0.3)^0.25 while the cross-correlation analysis with unWISE using C_ℓ^κ g and C_ℓ^gg best constrains S_8^×≡σ_8 (Ω_m/0.3)^0.45. These parameters are constrained to 2.2% and 1.7% from those analyses respectively (using ACT and Planck data jointly in both cases) and improvements on these parameters by moving to the 3x2pt analysis are small.
    
    For better comparability with galaxy weak lensing surveys, which commonly report the parameter combination S_8≡σ_8 (Ω_m/0.3)^0.5, we also present constraints on this parameter. We find the following constraints
    
        S_8    =    0.820 ± 0.021 ( ACT+ unWISE),
        
            S_8    =    0.806 ± 0.018 (Planck+ unWISE), and
        
            S_8    =    0.816 ± 0.015 ( ACT+Planck+ unWISE).
    
    We show the constraints from our 3x2pt analysis using ACT, Planck or ACT+Planck comparison with primary CMB constraints from Planck in Fig. <ref>.
    
    
    
    To directly constrain σ_8 we need to break the degeneracy with Ω_m. We have two ways of doing this: either by adding BAO data which provide constraints primarily in the Ω_m-H_0 parameter space or by adding uncalibrated supernovae which constrain Ω_m. With BAO we find
    
        σ_8    =    0.815 ± 0.013
              ( ACT+ unWISE+  BAO),
        σ_8    =    0.806 ± 0.013
              (Planck+ unWISE +  BAO), and
        σ_8    =    0.815 ± 0.012
              ( ACT+Planck+ unWISE +  BAO)
    
    compared to 
    
        σ_8    =    0.796 ± 0.020
              ( ACT + unWISE +  SN),
        σ_8    =    0.780 ± 0.019
              (Planck + unWISE +  SN),  and
        σ_8    =    0.794 ± 0.016
              ( ACT+Planck + unWISE +  SN)
    
    with supernovae data.
    
    Our constraints in combination with BAO are primarily limited by lack of knowledge of n_s. As noted above we adopted a conservative prior of σ(n_s)=0.02 which is about five times wider than the constraint from Planck<cit.>. We investigate instead adopting a more aggressive prior of σ(n_s)=0.01 corresponding approximately to the largest 1σ-posterior found for various beyond-ΛCDM extensions examined by Planck<cit.>. This tightens the constraints on σ_8
        σ_8    =    0.815 ± 0.012
              ( ACT + unWISE +  BAO),
        σ_8    =    0.805 ± 0.012
              (Planck + unWISE +  BAO), and
        σ_8    =    0.814 ± 0.010
              ( ACT+Planck + unWISE +  BAO).
    
    
    
    As we can see the constraints from ACT and Planck data alone, while affected by the n_s prior, are less sensitive to this choice than the combination of both, which is increasingly limited by our conservative prior and improved by ∼20% when tightening the prior on n_s.
    
    
    Given the agreement with primary CMB predictions reported in <cit.> and <cit.> we do not expect strong disagreement with Planck constraints from the primary CMB. Indeed, our lensing constraints are  consistent with the primary CMB data sets discussed in Sec. <ref>. From the primary CMB we have S_8^ 3x2pt = 0.826 ± 0.012 (S_8=0.830 ± 0.014, σ_8=0.8107 ± 0.0064) about 0.6σ (0.7σ, 0.3σ or 1σ comparing to the analyses with BAO or SN respectively) away from the joint result reported above. 
    
    We also combine our data with the primary CMB to further break degeneracies. We find even tighter constraints on σ_8 of
    
        σ_8    =    0.8124 ± 0.0048
              ( ACT + unWISE +  CMB),
        σ_8    =    0.8105 ± 0.0047
              (Planck + unWISE +  CMB), and
        σ_8    =    0.8127 ± 0.0044
              ( ACT+Planck + unWISE +  CMB).
    This constraint represents a ∼30% improvement over the constraint from Planck primary CMB data alone, demonstrating that with precise knowledge of the scalar spectral index, the matter density and other cosmological parameters our data provides a powerful probe of matter density fluctuations.We also note, however, that this result only represents a marginal, ∼2.5% improvement over the combination of Planck primary CMB data with the ACT and Planck lensing auto-spectra. This is not entirely unexpected given that within the ΛCDM model low redshift structure formation is constrained tightly by the lensing auto-spectrum alone.
    
      §.§.§ Constraints on Hubble expansion
    
    
    As discussed above our data can be used in two ways to shed light on the expansion rate of the Universe, H_0. Firstly, when combining our data with BAO to break the degeneracy between H_0 and Ω_m we find
    
        H_0    =    67.53± 0.80 
              ( ACT + unWISE +  BAO +  BBN),
        
            H_0    =    67.24± 0.78 
              (Planck + unWISE+  BAO +  BBN), and
        
            H_0    =    67.35± 0.78 
              ( ACT+Planck + unWISE +  BAO +  BBN).
    We note that these constraints depend sensitively on the BBN prior on the baryon density discussed in Sec. <ref> which is crucial in determining the sound horizon size. These results are consistent with results from the primary CMB (H_0=67.32 ± 0.51  from the Planck primary CMB), approximately 20% tighter than results from BOSS BAO alone (H_0=67.33±0.98 when including the BBN prior used in this work as well; ), and comparable to the constraint from the newer DESI BAO measurements (H_0=68.53±0.80 when calibrating the sound horizon ruler with BBN information; ).Secondly, we can use the fact that, as discussed above, our data are not directly sensitive to the sound horizon to place independent constraints on the Hubble rate through measurements of the matter-radiation equality scale. 
     From the 3x2pt data alone we find
    
        H_0    =    68.4^+5.2_-6.5
              ( ACT + unWISE),
        
            H_0    =    65.4^+3.3_-4.0
              (Planck + unWISE), and
        
            H_0    =    66.5^+3.2_-3.7
              ( ACT+Planck + unWISE).
    
    When combining our data with uncalibrated supernovae to further break the H_0-Ω_m degeneracy we obtain
    
        H_0    =    64.8^+2.6_-3.0
              ( ACT+ unWISE +  SN),
        
            H_0    =    63.5^+2.2_-2.4
              (Planck+ unWISE +  SN), and
        
            H_0    =    64.3^+2.1_-2.4
              ( ACT+Planck+ unWISE +  SN).
    The posteriors on H_0 and degeneracies with Ω_m are shown in Fig. <ref>.
    
      §.§.§ Comparison of ΛCDM constraints with external analyses
    
    
    As discussed in Sec. <ref> our constraints on the amplitude of matter density fluctuations are in excellent agreement with model predictions from a flat ΛCDM model (with minimum neutrino mass, ∑ m_ν = 0.06eV) fit to the primary CMB from Planck (see Sec. <ref> for a detailed description of the data sets used). In Figs. <ref> and <ref> we show these comparisons graphically and additionally show a comparison with independent primary CMB observations from the combination of WMAP and ACT <cit.> (shown in magenta in Figs. <ref> and <ref>). 
    
    We also provide comparisons to a wider set of other large scale structure probes. We include results from CMB lensing auto-spectrum analyses, galaxy weak lensing surveys (DES, KiDS and HSC), other cross-correlations with CMB lensing from ACT, SPT, and Planck, and from the three dimensional clustering of galaxies from BOSS and eBOSS. We subsequently briefly introduce the datasets we employ:
    
      * CMBL: These are constraints from the analysis of the auto-spectrum of CMB lensing reconstructions (shown in green in Figs. <ref> and <ref>). These results are mostly sensitive to linear scales at z=1-2but with a broad tail to higher redshifts. CMB lensing primarily constrains the parameter combination σ_8 Ω_m^0.25. In addition to the two CMB lensing auto-spectra which also enter our analysis, Planck PR4 <cit.>, ACT DR6 <cit.>, we also compare to the independent analysis from SPT-3G<cit.>. To make fair comparisons we combine the CMB lensing measurements with BAO information which breaks the degeneracy between σ_8 and Ω_m.
    
        
      * CMBLX: We also compare our results with cross-correlations between CMB lensing and galaxy surveys (using only the galaxy-CMB lensing cross-correlation and the galaxy-galaxy auto-correlation; 2x2pt). These are shown in red in Figs. <ref> and <ref>. We use results from <cit.> and <cit.> which analysed the cross-correlation between DESI LRG targets and the ACT DR6 lensing reconstruction. <cit.> also present a reanalysis of the DESI LRGs' cross-correlation with Planck PR4 updating the results from <cit.>. 
        
        We also include work cross-correlating DES-Y3 galaxy shear (γ) and galaxy densities (δ_g) with a joint SPT and Planck lensing reconstruction based on a much smaller survey footprint than considered here<cit.>. The joint lensing reconstruction from SPT-SZ and Planck is presented in <cit.>. A cross-correlation between DES-Y3 clustering (δ_g) and ACT DR4 lensing <cit.> based on the lensing reconstruction from <cit.> is also included. While all aforementioned analyses use photometric galaxy samples in their cross-correlations, the final cross-correlation study included in our comparisons, <cit.>, jointly models the three dimensional clustering of the spectroscopic BOSS galaxies and their cross-correlation with Planck.
    
        
      * GL: From galaxy weak lensing surveys we include constraints from `3x2pt'-analyses, combining measurements of galaxy clustering, galaxy shear, and their cross-correlation (shown in blue in Figs. <ref> and <ref>). For DES we adopt the results obtained in <cit.> when fixing the neutrino mass. For KiDS we show results presented in <cit.>. We note that in contrast to the DES analysis the KiDS results are obtained from the combination of projected galaxy shear and galaxy-galaxy lensing measurements with a measurement of the three dimensional clustering of galaxies in the spectroscopic BOSS and 2dFLenS surveys. Therefore, the KiDS analysis already contains the BAO information. For HSC we adopt a set of results based on measurements of the galaxy shear, galaxy-galaxy lensing, and galaxy clustering from <cit.>. We show results for an analysis using a linear bias model <cit.> and from an analysis using a halo model based emulator which includes non-linear scales <cit.>.
    
        
      * GC: Finally, we compare our results with constraints from the three dimensional clustering of galaxies as measured by BOSS and eBOSS (shown in gold in Figs. <ref> and <ref>). We include the analysis of BAO and Redshift Space Distortions (RSD) from <cit.>. Furthermore, we include an independent analysis based on the effective theory of large scale structure (EFTofLSS) that fits the `full shape' of the power spectrum and bispectrum measured in redshift-space <cit.>. Similar, previous analyses found compatible results <cit.>.
    
    
    Comparisons for the S_8 and σ_8 parameters are shown in Figs. <ref> and <ref>. We also show comparisons in the σ_8-Ω_m plane for a few selected external data sets in Fig. <ref>. Generally we find good agreement with the CMB lensing results, although we note only the estimate from SPT-3G is not already included in our 3x2pt analysis. When comparing to the four different galaxy weak lensing 3x2pt analyses we find that these generally favour a lower value of S_8 at moderate significance between ∼1σ and 2.3σ. The same is true for the other CMB lensing cross-correlations we compare our results to. From the DESI LRG targets we find a ∼1.6σ lower value of S_8 using CMB lensing data from ACT DR6 and Planck PR4 (0.9σ and 1.9σ for ACT DR6 and Planck PR4 alone respectively). 
    
    Using DES galaxies and cosmic shear together with the SPT+Planck lensing reconstruction also yields a ∼2.4σ lower value for S_8, while the discrepancy with the cross-correlation between DES and ACT DR4 is slightly less significant (1.4σ). The results from the cross-correlation of spectroscopic galaxies from BOSS with Planck PR3 lensing also yields a value of S_8 that is 2.7σ lower than our inference. Meanwhile the disagreement with galaxy clustering is less significant (0.6-1.0σ).
    
    Where available we also consider results that directly constrain σ_8 either by combining with BAO or because they include the BAO information as part of a three dimensional clustering analysis and compare these to our analysis with BAO. We broadly find similar levels of agreement/discrepancy as for S_8 except in the case of the `full-shape' galaxy clustering analysis which finds a value of σ_8 about 2.3 σ lower than our results.
    
    We conclude that while our results are in good agreement with primary CMB and CMB lensing results, there are some moderate discrepancies with other large scale structure tracers from galaxy weak lensing surveys and (mostly lower redshift) CMB lensing cross-correlations. However, as we can see in Fig. <ref>, the posteriors for several of these data sets have significant overlap with our results in the σ_8-Ω_m plane. The projection onto S_8 slightly exaggerates the level of disagreement. At current levels of precision the various data sets are in broad agreement and we are unable to conclusively rule out statistical fluctuations as the source of the observed discrepancies.
    
    Furthermore, we also compare our constraints on the Hubble constant to a range of external measurements. This includes model dependent measurements based on the apparent size of the sound horizon such as those from the primary CMB <cit.> and BAO <cit.>, as well as measurements based on the matter-radiation equality scale from <cit.> and <cit.>. Finally, we compare to several local measurements of H_0, including the TDCOSMO strong-lensing time-delay measurement with marginalisation over lens profiles <cit.>, an alternative TDCOSMO measurement with different lens-mass assumptions <cit.>, the Cepheid-calibrated  supernovae measurement <cit.>, the TRGB-calibrated supernovae measurement <cit.>[We note that a more recent TRGB-calibrated measurement using data from the James Web Space Telescope finds similar values of H_0 but with slightly larger uncertainties due to a smaller sample size <cit.>.], and recent results employing calibration based on observation of asymptotic giant branch stars in the J-band <cit.>.
    
    We find good agreement with other equality scale based measurements. The constraint from the 3x2pt data alone is in excellent agreement with sound horizon based measurements while the measurements including the supernovae data set yield values about 1.3σ lower than the value inferred from the CMB, but in no statistically significant tension. At the same time the combination of 3x2pt data and supernovae is in ∼3.6σ tension with local measurements of H_0 from the Cepheid-calibrated supernovae (). We find no significant tension with TRGB- or JAGB-calibrated supernovae measurements (∼1.7σ and ∼0.8σ respectively); the change is driven in part by larger uncertainties but also due to a lower central value of H_0 inferred with these methods.
    
     §.§ Reconstructing the growth of perturbations
    In addition to measuring the growth of structure within the ΛCDM model through constraints on a single parameter, S_8^ 3x2pt or σ_8, we are able to derive information on the evolution of structures over time by leveraging the different redshift sensitivity of the cross-correlations with the two unWISE samples and the CMB lensing auto-correlation. In Fig. <ref> we show the constraints from the two cross-correlations and the CMB lensing auto-correlation, each analysed jointly with BAO. The redshift kernels shown at the bottom of the top panel give an indication of the redshift sensitivity of the samples given by the fractional contribution to the signal-to-noise, log SNR /  z. The computation of log SNR /  z includes an approximate marginalisation over galaxy nuisance parameters, achieved by linearising the model for C_ℓ^gg and C_ℓ^κ g in small fluctuations around the best-fit linear bias and shot noise and propagating the uncertainty in these parameters to the covariance matrix.We adopt the median of the redshift sensitivity kernel to represent the effective redshift of each of the three measurements and compute σ_8(z) as σ_8(z=0)D(z), where D(z) is the linear growth function which is primarily dependent on Ω_m. These results are also summarised in Table <ref>. We find excellent agreement with the growth of structures predicted by the ΛCDM model fit to the primary CMB from Planck (grey band in Fig. <ref>).However, as can be easily seen from Fig. <ref>, the three samples have significant redshift overlap. In particular, while the median redshift of the measurement from the lensing auto-spectrum is z_ Med≃ 3.5 it receives significant contributions from lower redshifts where we also have information from the cross-correlation measurements. To optimally combine the available information we explore a reconstruction of the growth of (linear) perturbations with redshift through a parametric form of σ_8(z) which we constrain jointly with all three samples taking into account their overlap. With this method we are able to use the cross-correlation measurements to constrain the low redshift contribution to the lensing auto-spectrum and extract information on the integrated growth of structure at high redshifts, above the two galaxy samples (z≳ 2.4).
    
    Due to the broad redshift kernels of our data sets we cannot constrain the growth of perturbations with arbitrary resolution. Instead we adopt the following simple parametrisation similar to <cit.>: We rescale the linear power spectrum as follows
    
        P_ lin^ new(k, z) =    P_ lin^ input(k, z) A(z)
         =    P_ lin^ input(k, z) 
                A_0     0≤ z < z_1
        
                A_1     z_1≤ z < z_2
        
                A_2     z_2≤ z
            .
    
    Where P_ lin.^ input(k, z) is the linear matter power spectrum computed by  at a given cosmology and A_0, A_1, and A_2 are free parameters. The redshift bins, z_1=1.15 and z_2=2.4, are motivated by the redshift origin of the signal for the two cross-correlation measurements and are chosen to separate the two samples as optimally as possible. A_0 is primarily constrained by the Blue sample of unWISE galaxies, while the Green sample is primarily sensitive to A_1. The lensing auto-correlation receives contributions from a wide range of redshifts, but in combination with the two cross-correlation measurements allows us to constrain the amplitude at high redshift, A_2.
    
    We marginalise over A_i with uniform priors in the range 0 to 2. The parameters of interest are then σ_8 √(A_i). Since our model depends on the non-linear  matter power spectrum (see <cit.> and <cit.> for detailed descriptions of the models used to fit the lensing auto- and cross-correlations respectively) we take 
    
        P_ non-lin.^ new(k, z) =    P_ lin.^ new(k, z) + P_ non-lin.^ input(k, z) - P_ lin.^ input(k, z)
         =    P_ lin.^ input(k, z) [A(z) - 1] + P_ non-lin.^ input(k, z).
    
    
    The non-linear contributions to the  matter power spectrum have a non-trivial dependence on the amplitude of linear fluctuations. In our parametrisation σ_8 on its own (in contrast to its product with √(A_i)) is only constrained by the size of the non-linear terms. This approach effectively allows us to marginalise over the size of these non-linear contributions.
    
    
    
    Fig. <ref> shows the results of this analysis in the form of the posteriors on σ_8 √(A_i) for the 3x2pt, 2x2pt and lensing-only data sets. In each case we combine with BAO data (described in Sec. <ref>) to break the degeneracy between σ_8 and Ω_m. We find good consistency with the amplitude of fluctuations predicted by a ΛCDM fit to the primary CMB in each of the three bins. 
    
    We can also use these results to reconstruct σ_8(z) which we show in Fig. <ref>. We show the 1σ confidence intervals from our inference chains on the combination √(A_i)σ_8(z=0) D(z) within the relevant bins. We do not separately constraint the shape of D(z) within each bin, but rather assume a single D(z) across all bins with its shape determined primarily by Ω_m. Within each bin we compute the median redshift of the joint signal as we did above for the individual measurements. The parameter constraints are summarised in Table <ref>.The representative constraints at the median signal redshift can be compared to results from the literature for example from the cross-correlation of CMB lensing with DESI LRGs <cit.> or the Quaia quasar catalog <cit.>. Our results are broadly consistent with these external reconstructions across the entire redshift range. At low redshifts (z≲1) we achieve similar precision (∼4%) to constraints presented in <cit.> but cannot match the redshift resolution of the DESI samples. In the redshift range 1 ≲ z  ≲ 2.5 our results represent some of tightest constraint on the amplitude of matter density fluctuations; tighter than results in a similar redshift range from <cit.> by a factor of about 4. The main power of our method, however, is the ability to place constraints on structure formation at higher redshifts, typically inaccessible with cross-correlation analysis. The median signal redshift within our highest redshift bin is z_ Med = 5.6, significantly higher than the highest redshift constraint available from cross-correlation with Quaia (z=2.7). In this high redshift range we obtain a ∼4% constraint on the amplitude of matter density fluctuations, consistent with predictions based on a ΛCDM fit to the primary CMB from Planck.  
    
    As one can see in Fig. <ref> some residual correlations remain between the redshift bins. We find that bin 1 and 2 are about 37% correlated while bin 2 and 3 are 45% correlated. The correlation between bin 1 and 3 is about 24%.
    
    When using only the lensing auto- and cross-correlations as well as the unWISE auto-correlation, but without BAO, σ_8 √(A_i) is degenerate with Ω_m, as expected. In each of the three bins we therefore determine the best constrained S_8-equivalent combination for our 3x2pt data alone. Using all lensing data from ACT and Planck we find
    
        σ_8 √(A_0) (Ω_m/0.3)^0.66   =    0.784± 0.035,
        σ_8 √(A_1) (Ω_m/0.3)^0.49   =    0.839± 0.042, and
        σ_8 √(A_2) (Ω_m/0.3)^0.39   =    0.783± 0.033.
    
    We can see that, as expected, the lower redshift results are more degenerate with Ω_m.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
     §.§ Constraints on beyond Λ CDM cosmologies
    
    
    We explore several extensions beyond the standard flat ΛCDM cosmology with minimum mass neutrinos. Generally we combine our data for this purpose with primary CMB observations. We also add BAO and supernovae data in some cases. Unless otherwise noted the constraints reported in the following sections consider the 3x2pt data set using both ACT and Planck lensing data. We summarise the constraints on the beyond ΛCDM parameters in Table <ref>.
    
    
    
      §.§.§ νΛ CDM
    
    
    Observations of neutrino flavour oscillations require that neutrinos have non-vanishing mass. Since any mechanism to give neutrinos mass requires physics beyond the standard model, this has significant implications for our understanding of fundamental physics. However, oscillation experiments only constrain the difference between the squared masses of the three mass eigenstates, Δ_12 m^2 ≡ m_1^2 - m_2^2 and |Δ_32 m^2 | where |Δ_32 m^2 |≫Δ_12 m^2, but not the absolute mass scale. Current constraints dictate that the sum of the neutrino masses ∑ m_ν is at least 0.058eV in what is known as the normal hierarchy (Δ_32 m^2>0; two of the masses are significantly smaller than the third) or at least about 0.1eV in the inverted hierarchy (Δ_32 m^2<0; two of the masses are significantly larger than the third).
    
    Direct detection experiments like <cit.> have placed upper limits on the effective electron anti-neutrino mass, m_ν_e<0.8eV at 90% confidence, from tritium beta decay observations. However, the most stringent limits on the sum of the neutrino masses are derived from cosmological observations. After neutrinos become non-relativistic they cluster similarly to CDM on scales above the neutrino free streaming length, but on smaller scales they suppress the growth of perturbations due to their large velocity dispersion. Meanwhile the neutrino energy density also contributes to the Universe's background expansion. The scale dependence of the suppression effect is only poorly constrained by current generation data like ours, but we are sensitive to the overall suppression of the formation of structure.
    
    For this purpose we combine our lensing auto- and cross-spectrum information which probes the matter power spectrum at late times, after neutrinos have become non-relativistic, with primary CMB data which provides information on the early time matter power spectrum before suppression due to massive neutrinos can manifest. We furthermore also add BAO information to probe the background expansion and break degeneracies, for example with the total matter density. It should be noted that the amplitude of the primordial power spectrum extracted from the small scale CMB power spectrum is completely degenerate with the optical depth to reionisation, τ. It is thus important to include the large scale CMB polarisation data from the  analysis discussed in Sec. <ref>.
    
    We extend the flat ΛCDM model by a single free parameter, ∑ m_ν, corresponding to the sum of the neutrino masses[As in <cit.> we follow <cit.> and <cit.> and consider a degenerate combination of three equally massive neutrinos.]. Using ACT and Planck lensing and cross-correlation data we find an upper limit on the sum of the neutrino masses of
    
        ∑ m_ν <    0.124 eV at 95% confidence
           ( 3x2pt +  CMB +  BAO).
    This represents a small improvement over the constraint reported in <cit.> (∑ m_ν < 0.13 eV; 95% C.I.) based on the lensing power spectrum only. However, the difference is driven by the additional, higher redshift, BAO likelihoods used in this work which favour a lower value of Ω_m. When reanalysing the lensing auto-spectrum  with these likelihoods we find ∑ m_ν < 0.123 eV; 95% C.I. (see also Fig. <ref>). The lack of improvement over auto-spectrum only constraints is not unexpected given that we find a slightly smaller amplitude of matter density fluctuations in the combined analysis of lensing auto- and cross-correlation data than from the auto-correlation alone, compatible with slightly more suppression due to massive neutrinos and therefore a larger neutrino mass. On idealised mock observations we show that, assuming minimum mass normal hierarchy neutrinos, the 3x2pt analysis leads to a marginally (∼5%) tighter upper limit on the neutrino mass sum than the C_ℓ^κκ-only analysis.
    
    When including supernovae observations in addition to BAO and primary CMB, the data favours a slightly larger matter density. Since the quantity best determined by our data is approximately S_8^ 3x2pt∝σ_8 Ω_m^0.4 this results in a lower inference for σ_8 (see also results in Sec. <ref>) and compatibility with larger neutrino induced power suppression. Hence the posterior on ∑ m_ν is shifted to larger values which leads to a small degradation in the one sided constraint to
    
        ∑ m_ν <    0.137 eV at 95% c.l.
           ( 3x2pt +  CMB +  BAO +  SN).
    
    
    When replacing our baseline BAO likelihoods from 6dFGS, SDSS, BOSS, and eBOSS with the new DESI BAO likelihood we find significantly tighter constraints on the neutrino mass. From the 3x2pt data with primary CMB and DESI BAO from the first year of data we obtain
    
        ∑ m_ν <    0.082 eV at 95% confidence
           ( 3x2pt +  CMB +  DESI BAO)
    
    This represents only a very marginal improvement over constraints from the lensing auto-spectrum only (∑ m_ν < 0.083 eV at 95% confidence from C_ℓ^κκ + CMB + DESI BAO using the ACT DR6 and Planck PR4 lensing power spectra[Note that this is a slightly weaker constraint than presented in <cit.> for two reasons. Firstly, we use the Planck PR4 likelihood as our default high-ℓ CMB likelihood, and secondly, the older version of the ACT lensing likelihood used in that work effectively deactivated the lensing norm corrections discussed in Appendix <ref>.]). The 3x2pt results formally disfavour the minimum mass allowed in the inverted hierarchy (min_ inv.[∑ m_ν] = 0.1eV) at about 98% confidence. We note that the significantly tighter constraint is in part explained by the DESI preference for a slightly lower value of Ω_m, leading to a larger value of σ_8 at fixed lensing amplitude, requiring less suppression due to neutrinos.
    
      §.§.§ Non-flat Λ CDM
    
    
    Within the standard model of cosmology, the Universe is predicted to be spatially flat as a consequence of inflation. However, primary CMB data from Planck has at times shown a preference for a closed universe with the curvature parameter, Ω_k<0<cit.>. Since the primary CMB anisotropies alone do not constrain curvature due to a “geometric degeneracy”<cit.> this preference is driven entirely by the lensing-induced peak smearing in Planck measurements of the CMB anisotropies which has been shown to be somewhat larger than expected from the measured lensing power spectrum <cit.>. This preference is not present in independent analyses of CMB anisotropy measurements from ACT+WMAP<cit.> and also disappears in combination with BAO. 
    
    
    
    When combining CMB lensing and unWISE cross-correlation data with primary CMB anisotropies we also no longer observe a preference for a closed universe. We find
    
        -0.011 < Ω_k < 0.004    at 95% confidence
           ( 3x2pt +  CMB)
    
    using both ACT and Planck lensing data. We show these constraints in the Ω_k-Ω_Λ plane compared to an analysis of primary CMB anisotropies only as well as the combination of primary CMB anisotropies and BAO in Fig. <ref>. While this constraint represents a significant improvement over the constraint from CMB lensing alone <cit.> the combination of primary CMB anisotropies and BAO provides a tighter constraint (-0.003 < Ω_k < 0.003 at 95% confidence). Despite the fact that the combination of BAO, lensing auto- and cross-spectrum data, and primary CMB does not improve on the constraints from BAO and primary CMB only, our result still represents a valuable cross-check on the BAO derived constraints.
    
    
    
      §.§.§ Extended dark energy models
    
    
    In this work we consider two extended dark energy scenarios. In the standard ΛCDM framework dark energy is assumed to be a cosmological constant, Λ, equivalent to a cosmological fluid with equation of state w=-1. First, we consider allowing w to take on values different from -1 but remain constant in time. Our data alone is only very weakly sensitive to w and the effect is largely degenerate with other parameters. However, when combining the 3x2pt data set with the primary CMB from Planck we find
    
        w = -1.53^+0.20_-0.31 ( 3x2pt+  CMB)
    
    with large uncertainties but consistent with w=-1 at ∼2σ. This is not competitive with the constraint obtained using all external data sets considered in this work (CMB + BAO + SN) which yields w = -0.979 ± 0.026. Adding the 3x2pt data improves constraints only very marginally by ∼4% to
    
        w = -0.982± 0.024 ( 3x2pt+  CMB+  BAO+  SN).
    
    Without supernovae data we find
    
        w = -1.027^+0.050_-0.043 ( 3x2pt+  CMB+  BAO)
    
    compared to w = -1.022^+0.053_-0.048 from CMB and BAO without our 3x2pt data (a ∼7% improvement). These constraints are shown in Fig. <ref>.
    
    
    
    We also consider a phenomenological parameterisation of an evolving dark energy equation of state. As in <cit.> we adopt the CPL model <cit.>:
    
        w(a) = w_0 + (1-a) w_a
    
    where a=1/(1+z) is the scale factor and both w_0 and w_a are free parameters. This commonly considered parametrisation has been shown to provide a good fit to several physically motivated dark energy models <cit.>.
    
    Our constraints on w_0 and w_a are shown in Fig. <ref>. When using only 3x2pt data and the primary CMB w_0 and w_a are so significantly degenerate that we are unable to provide meaningful constraints. However, in combination with BAO and SN we find 
    
        w_0 =    -0.881 ± 0.060
        
                w_a =    -0.43^+0.25_-0.22( 3x2pt + CMB + BAO + SN)
    
    only a very minor ∼5% and ∼9% improvement on w_0 and w_a respectively over constraints from CMB, BAO and SN alone. Removing the supernovae data yields 
    
        w_0 =    -0.56 ± 0.24
        
                w_a =    -1.27 ± 0.66
            ( 3x2pt + CMB + BAO)
    
    not improving significantly on constraints from the external data alone (by about 4% and 3% respectively). 
    
    To alleviate some of the significant degeneracy between w_0 and w_a evident from the left panel in Fig. <ref> we also determine the `pivot' redshifts, z_p, at which we obtain the best constraints on w(a) for various data combinations. Following <cit.> we estimate z_p by considering the marginalised parameter covariance, defining
    
        a_p = 1 + C_w_0 w_a/C_w_a w_a,
     effectively obtaining the best constrained linear combination of w_0 and w_a which guarantees that w_a is uncorrelated with w_p ≡ w(a_p). The model can then be reparametrised as w(a) = w_p + (a_p - a)w_a. We find z_p=0.44, 0.55, and 0.29 when combining the 3x2pt data with CMB, CMB + BAO, and CMB + BAO + SN respectively. For comparison the pivot redshift for the external data sets alone (CMB + BAO + SN) is z_p=0.28 (or z_p=0.54 for CMB + BAO). The equation of state at the pivot redshift is constrained to
    
        w_p    =    -1.56^+0.28_-0.37( 3x2pt +  CMB; z_p=0.44),
        
            w_p    =    -1.005^+0.062_-0.055(+  BAO; z_p=0.55),  and
        
            w_p    =    -0.976 ± 0.028(+  SN; z_p=0.29).
    
    These represent small improvements over constraints from the external data sets alone of about 11%, 8% and 7% over CMB, CMB + BAO, and CMB + BAO + SN respectively, in line with the improvements we saw for the wCDM model.
    
    We show the two dimensional posterior in the w_p-w_a plane in the right panel of Fig. <ref>. We can see that all our constraints are consistent with (w_p, w_a)=(-1, 0) within ∼2σ. However, at present the constraining power is insufficient to further shed light on recent hints at deviations from the cosmological constant model with our data similarly consistent with the values preferred by the combination of DESI BAO data and supernovae <cit.>.
    
    
    
      §.§.§ Massive neutrinos in the context of dynamical dark energy
    
    
    As has been pointed out several times, the characteristic effect of massive neutrinos on the Universe's background evolution and the suppression of structure formation due to neutrino free streaming can be mimicked by beyond-ΛCDM extensions <cit.>. In part because the scale dependence of the neutrino effect is only poorly constrained with current data, the effect is difficult to distinguish from other non-standard physics. Here we consider neutrino mass constraints in the presence of a modified dark energy equation of state[For constraints on massive neutrinos in a wider range of extended models from the CMB lensing power spectrum alone see <cit.>.]. We derive constraints on ∑ m_ν, both in the two-parameter extension, wCDM+∑ m_ν, as well as in the case of evolving dark energy, w_0 w_aCDM+∑ m_ν. Our constraints are summarised in Table <ref>. Additionally, we show the 1D marginalised posteriors on the sum of the neutrino masses in Fig. <ref>. 
    
    As expected, we generally see a degradation of constraints in extended dark energy models. One exception occurs when allowing w to vary while including supernovae in the analysis. The supernovae data set provides tight constraints on w with a slight preference for w>-1. When w>-1 structures grow more slowly and our data are consequently compatible with less suppression due to neutrinos. Within the wCDM+∑ m_ν cosmology our neutrino mass constraint is thus tightened to
    
        ∑ m_ν <    0.123 eV at 95% c.l.
           ( 3x2pt +  CMB +  BAO +  SN)
    
    compared to ∑ m_ν < 0.137eV from the same data in a ΛCDM+∑ m_ν model. In the presence of evolving dark energy we find
    
        ∑ m_ν <    0.231 eV at 95% c.l.
           ( 3x2pt +  CMB +  BAO +  SN).
    
    
    § CONCLUSION
    
    
    We have presented cosmological constraints from the joint analysis of the CMB lensing auto-correlation, the cross-correlation between CMB lensing and galaxy data, and the galaxy auto-correlation using CMB lensing data from ACT DR6 and Planck PR4 together with galaxy data from unWISE.
    
    Building on previous separate analyses of these data sets we provide tight constraints on the amplitude of matter density fluctuations in the redshift range z≃ 0.2-5. Within the ΛCDM model we constrain the relevant parameter S_8^ 3x2pt≡σ_8 (Ω_m/0.3)^0.4 to 1.5% and in combination with BAO we obtain similarly tight constraints on σ_8. Our constraints are in excellent agreement with model predictions from ΛCDM fits to the primary CMB. At the same time our results are not in any statistically significant tension with other LSS probes despite favouring a slightly larger amplitude of matter density fluctuations. Nevertheless, our findings suggest that if the `S_8 tension' is indicative of new or so far neglected physics rather than systematic effects, the effect has to be confined either to very low redshifts (probably below z≃ 0.5) where our data are less sensitive or small scales (k ≳ 0.3h/Mpc) which our data equally does not probe well.
    
    In addition, we constrained the Hubble constant independently of the sound horizon to 3.5%. Our constraints are in excellent agreement with constraints from observations of the primary CMB and BAO. While our current uncertainties are too large to provide conclusive evidence, these results raise further questions about the viability of modifications to the sound horizon to resolve the Hubble tension.
    
    We derive constraints on the amplitude of matter density fluctuations as a function of redshift. When the amplitude of low redshift structure is constrained by the cross-correlation data, the lensing auto-spectrum provides constraints on high redshift structures. We find one of the tightest constraints to date (∼3.3%) on the integrated matter density fluctuations above z≃2.4. The reconstructed σ_8(z) is in excellent agreement with predictions from the primary CMB within the standard ΛCDM model.
    
    Using the 3x2pt data we also revisited constraints on the neutrino mass sum previously presented only from the CMB lensing auto-spectrum. Despite the larger information content we do not tighten existing upper limits. This is because we obtain a slightly smaller amplitude of matter density fluctuations in the joint analysis than in the lensing auto-spectrum analysis, consistent with larger neutrino induced structure suppression and therefore larger neutrino masses. This analysis nevertheless represents a valuable step towards detection of massive neutrinos with future cross- and auto-correlation analyses.
    
    We also investigate, for the first time with this data, a series of beyond-ΛCDM models including spatial curvature and extended dark energy models. We show that our data marginally tightens some of the existing constraints and provides competitive cross-checks on others. We also make our data and likelihood publicly available enabling the community to perform further investigation into models not explored in this work (see Appendix <ref> for details).
    
    This work represents the first 3x2pt analysis with the new ACT DR6 lensing reconstruction. Such analyses, combining all possible auto- and cross-correlations between a galaxy sample and the lensing reconstruction, have become the standard in the galaxy weak lensing field. We showed the excellent constraining power of this analysis approach also for CMB lensing data. Similar analyses using future CMB lensing data for example from Simons Observatory <cit.> and CMB-S4 <cit.>, and galaxy samples from future surveys such as Euclid <cit.> and eventually the Vera C. Rubin Observatory's Legacy Survey of Space and Time <cit.> will provide further improved constraints on cosmological parameters and contribute to improving our understanding of the cosmos.
    
    
    
    § ACKNOWLEDGEMENTS
    
    
    The authors wish to thank Hironao Miyatake and the HSC team for making HSC chains available to us slightly before their public release. We thank Noah Sailer for useful discussions.
    
    Support for ACT was through the U.S. National Science Foundation through awards AST-0408698, AST-0965625, and AST-1440226 for the ACT project, as well as awards PHY-0355328, PHY-0855887 and PHY-1214379. Funding was also provided by Princeton University, the University of Pennsylvania, and a Canada Foundation for Innovation (CFI) award to UBC. The development of multichroic detectors and lenses was supported by NASA grants NNX13AE56G and NNX14AB58G. Detector research at NIST was supported by the NIST Innovations in Measurement Science program. 
    ACT operated in the Parque Astronómico Atacama in northern Chile under the auspices of the Agencia Nacional de Investigación y Desarrollo (ANID). We thank the Republic of Chile for hosting ACT in the northern Atacama, and the local indigenous Licanantay communities whom we follow in observing and learning from the night sky.
    
    Computing was performed using the Princeton Research Computing resources at Princeton University, the Niagara supercomputer at the SciNet HPC Consortium, and the Symmetry cluster at the Perimeter Institute. This research also used resources provided through the STFC DiRAC Cosmos Consortium and hosted at the Cambridge Service for Data Driven Discovery (CSD3). SciNet is funded by the CFI under the auspices of Compute Canada, the Government of Ontario, the Ontario Research Fund–Research Excellence, and the University of Toronto. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Industry Canada and by the Province of Ontario through the Ministry of Colleges and Universities. This research also used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231 using NERSC award HEP-ERCAPmp107.
    
    GSF acknowledges support through the Isaac Newton Studentship and the Helen Stone Scholarship at the University of Cambridge. 
    GSF, FJQ, CEV, and BDS acknowledge support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 851274). BDS further acknowledges support from an STFC Ernest Rutherford Fellowship. SF is supported by Lawrence Berkeley National Laboratory and the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. 
    EC acknowledges support from the European Research Council
    (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 849169). CEV received the support from “la Caixa” Foundation (ID 100010434, fellowship code LCF/BQ/EU22/11930099). JK acknowledges support from NSF grants AST-2307727 and AST-2153201. KM acknowledges support from the National Research Foundation of South Africa. MM acknowledges support from NSF grants AST-2307727 and  AST-2153201 and NASA grant 21-ATP21-0145.  CS acknowledges support from the Agencia Nacional de Investigación y Desarrollo (ANID) through Basal project FB210003.
    
    
    
     §.§ Software
    
    
    
    
    We acknowledge use of the <cit.> package and the Python Image Library for producing plots in this paper. Furthermore, we acknowledge use of the <cit.> and <cit.> packages. We use the Boltzmann code <cit.> for calculating theory spectra, and use <cit.> and <cit.> for likelihood analysis.
    
    
    
    § DATA AVAILABILITY
    Pre-release versions of the Markov Chain Monte Carlo runs from this paper and <cit.> are available through the NERSC (National Energy Research Scientific Computing Center) Science Gateway, https://portal.nersc.gov/project/act/act_x_unWISE_xcorr+3x2pthere[<https://portal.nersc.gov/project/act/act_x_unWISE_xcorr+3x2pt>].Alongside this publication we also distribute various data products and software relevant to this work. The likelihood, bandpowers, covariances and various auxiliary data products required to perform the analysis presented here will be made available upon publication. The likelihood can then be found https://github.com/ACTCollaboration/unWISExLens_lklhhere[<https://github.com/ACTCollaboration/unWISExLens_lklh>] and all required data products will be available through the NERSC Science Gateway above and . We caution that, because of significant correlations, this likelihood should not be combined with any other CMB lensing or CMB lensing cross-correlation likelihoods, like those from <cit.> or <cit.>. Instead the likelihood provided here can be used to include the auto-correlation measurements in a self consistent manner.
    
    § LIKELIHOOD CORRECTIONS
    
    
    The CMB lensing reconstruction is obtained using quadratic estimators which depend on two powers of the observed CMB fields. The normalisation of the estimator and the bias corrections which are required for the lensing power spectrum depend in principle on the underlying CMB power spectra and the lensing convergence power spectrum. In practice we compute the normalisation and bias corrections given a fiducial choice of spectra[The ACT lensing reconstruction adopts fiducial spectra from a ΛCDM model fit to Planck 2015 TTTEEE data with an updated τ prior as in <cit.>.]. While the CMB power spectrum is well constrained by Planck, some residual uncertainty remains that must be propagated to our likelihood analysis.
    
    Let us denote a set of cosmological parameters as θ and the assumed fiducial cosmology as θ_0. As discussed in more detail in <cit.> the unnormalised lensing reconstruction is sensitive to the product of the lensing convergence field, κ_L m(θ), and the lensing response function, ℛ_L(θ). The latter is computed within the fiducial cosmology. When comparing theory to observations we have to account for this fact in the lensing auto- and cross-spectra as
    
     C_L^κ g,  obs(θ)    =   ℛ^-1_L(θ_0)/ℛ^-1_L(θ) C_L^κ g,  th(θ), and
    
     C_L^κκ,  obs(θ)    =   [ℛ^-1_L(θ_0)]^2/[ℛ^-1_L(θ)]^2C^κκ_L(θ)-N^1_L(θ_0)+N^1_L(θ).
    
    Here we have also included the N^1-bias in the lensing power spectrum which we compute from simulations and which depends on the fiducial CMB power spectra as well as the lensing power spectrum present in the simulations (see  for more details on the N^1-bias).
    
    Fully calculating the above for each point in the sampled parameter space is unfeasible, and hence we follow the approach of <cit.> and <cit.> and forward model the linearised corrections to the theory spectrum due to the parameter deviations from the fiducial cosmology. Given the excellent constraints on the CMB power spectra any deviations from the fiducial spectrum, C^CMB_ℓ(θ_0), are expected to be small justifying this expansion. The spectra to be compared to the observed data spectra are thus given by
    
        C_L^κ g,  obs(θ)    ≈    C_L^κ g,  th(θ)[1 - M_L^ℓ[C^ CMB_ℓ(θ)-C^ CMB_ℓ(θ_0)]], and
    
        C_L^κκ,  obs(θ)    ≈   
        C^κκ_L(θ)-2M_L^ℓ[C^ CMB_ℓ(θ)-C^ CMB_ℓ(θ_0)]C^κκ_L(θ_0) 
          +dN^1_L/dC^ CMB_ℓ[C^ CMB_ℓ(θ)-C^ CMB_ℓ(θ_0)]+dN^1_L/dC^κκ_L^'[C^κκ_L^'(θ)-C^κκ_L^'(θ_0)] 
    where M_L^ℓ = ∂lnℛ^-1_L / ∂ C_ℓ^CMB|_θ_0 is the linearised normalisation-correction matrix.
    
    For joint constraints with CMB anisotropy spectra, we correct the normalisation and N^1-bias subtraction at each point in parameter space according to Eqs. <ref> and <ref>. For cosmology runs that do not include information from the primary CMB, we propagate the uncertainty in the lensing normalisation due to possible fluctuations in the CMB power spectrum into an additional contribution to the covariance. Specifically, we obtain 1000 posterior samples from the ACT DR4 + Planck primary CMB chains and propagate these to the covariance matrix as described in Appendix B of <cit.> and Sec. 6.1 of <cit.>. This step is done consistently to both the ACT and Planck parts of the covariance matrix.
