arXiv subjects

Graeme E. Addison

Publications and source records attributed to Graeme E. Addison.

At least 19 recordsLinked to original sources

arXiv2025

Nearly Full-Sky Low-Multipole Cosmic Microwave Temperature Anisotropy: I. Foreground Cleaned Maps

Studies of cosmic microwave background (CMB) are often limited by foreground contamination. Foreground cleaning is performed either in harmonic or pixel space after data cuts have excluded sky areas of strong contamination. We present a nearly full-sky CMB temperature map with only 1% of pixels masked. To derive this map, we make use of six full-sky template maps at foreground-dominated frequencies from different experiments smoothed to $1^\circ$ and rely on the combination of these weighted maps to trace the morphology of foreground contamination. We do not impose any spectral index constraints, but only fit for template amplitudes at each target frequency. We clean WMAP and Planck maps at a set of target frequencies and conduct quality tests at the level of the maps, pixel histograms and power spectra to select four CMB maps that are cleaned with negligible foreground contamination and only 1% masked pixels and no inpainting. We recommend use of these cleaned CMB maps for low multipole ($\ell < 30$) studies.

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arXiv2025

Nearly Full-Sky Low-Multipole Cosmic Microwave Background Temperature Anisotropy: III. CMB Temperature Anomalies

Unexpected features have been observed in the cosmic microwave background (CMB) temperature on large scales. We revisit these CMB anomalies using new foreground-cleaned CMB temperature maps derived in a companion paper from WMAP and Planck data, which are tailored to low-resolution analysis and require only minimal masking of $1\%$ of the sky. These maps allow us to assess the impact of foreground-cleaning methods and the choice of sky cut on the significance of five commonly studied CMB anomalies. We find a notable impact of the choice of galactic mask on the significance of two anomalies: the significance of the low real-space correlation function and of the local-variance asymmetry reduces from $\sim 3\sigma$ for the Planck common mask with $26\%$ masked fraction to $\sim 2\sigma$ for the $1\%$ mask. We find good agreement between the two sky cuts for the low northern variance, $\sim 3\sigma$, and the parity asymmetry, $\sim 2\sigma$. For the quadrupole-octopole alignment, we find good agreement between the 1\%-mask result and the full-sky results in the literature, $\sim 3\sigma$. Thus using a larger fraction of the sky enabled by improved foreground cleaning reduces the significance of two commonly studied CMB anomalies. Overall, for an alternative physical model to be convincingly favored over $\Lambda$CDM with statistically isotropic Gaussian fluctuations, it would need to explain multiple CMB anomalies, or better describe some other type of measurement in addition to a CMB anomaly.

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arXiv2025

Nearly Full-Sky Low-Multipole Cosmic Microwave Background Temperature Anisotropy: II. Angular Power Spectra and Likelihood

We present a CMB temperature power spectrum measurement at large angular scales from WMAP and Planck maps that were cleaned of foregrounds using a template-based approach described in the first paper of this series. We recover essentially the full-sky CMB information at multipoles $\ell<30$ with only 1% of pixels near the Galactic plane masked and no inpainting. Notable features continue to appear: (a) a low quadrupole power compared to the best-fit Planck 2018 $\Lambda$CDM spectrum at $2.2\sigma$, (b) a dip in the range $20 \le \ell \le 27$, and (c) an overall $\ell<30$ power level low of the $\Lambda$CDM prediction derived from higher multipole moments. Given the different methodology from previous analyses and the nearly full-sky solution presented here, these features do not plausibly arise from foreground contamination, systematic errors, masking, or mode-mixing. Our overall $\ell<30$ amplitude constraint is consistent with published WMAP (77% sky fraction) and Planck (86%) results at $1.2\sigma$ and $0.6\sigma$, respectively, accounting for the improvement in statistical precision. We present a new $\ell<30$ likelihood for use with the Cobaya package. Parameter constraints from joint fits with the higher-multipole Planck data are consistent with the published Planck results, for example we find $H_0=67.35\pm0.54$ km s$^{-1}$ Mpc$^{-1}$ in a joint $\Lambda$CDM fit.

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arXiv2025

The Atacama Cosmology Telescope: DR6 Maps

We present Atacama Cosmology Telescope (ACT) Data Release 6 (DR6) maps of the Cosmic Microwave Background temperature and polarization anisotropy at arcminute resolution over three frequency bands centered on 98, 150 and 220 GHz. The maps are based on data collected with the AdvancedACT camera over the period 2017--2022 and cover 19,000 square degrees with a median combined depth of 10 uK arcmin. We describe the instrument, mapmaking and map properties and illustrate them with a number of figures and tables. The ACT DR6 maps and derived products are available on LAMBDA at https://lambda.gsfc.nasa.gov/product/act/actadv_prod_table.html. We also provide an interactive web atlas at https://phy-act1.princeton.edu/public/snaess/actpol/dr6/atlas and HiPS data sets in Aladin (e.g. https://alasky.cds.unistra.fr/ACT/DR4DR6/color_CMB).

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arXiv2025

The Atacama Cosmology Telescope: DR6 Power Spectra, Likelihoods and $\Lambda$CDM Parameters

We present power spectra of the cosmic microwave background (CMB) anisotropy in temperature and polarization, measured from the Data Release 6 maps made from Atacama Cosmology Telescope (ACT) data. These cover 19,000 deg$^2$ of sky in bands centered at 98, 150 and 220 GHz, with white noise levels three times lower than Planck in polarization. We find that the ACT angular power spectra estimated over 10,000 deg$^2$, and measured to arcminute scales in TT, TE and EE, are well fit by the sum of CMB and foregrounds, where the CMB spectra are described by the $\Lambda$CDM model. Combining ACT with larger-scale Planck data, the joint P-ACT dataset provides tight limits on the ingredients, expansion rate, and initial conditions of the universe. We find similar constraining power, and consistent results, from either the Planck power spectra or from ACT combined with WMAP data, as well as from either temperature or polarization in the joint P-ACT dataset. When combined with CMB lensing from ACT and Planck, and baryon acoustic oscillation data from DESI DR1, we measure a baryon density of $\Omega_b h^2=0.0226\pm0.0001$, a cold dark matter density of $\Omega_c h^2=0.118\pm0.001$, a Hubble constant of $H_0=68.22\pm0.36$ km/s/Mpc, a spectral index of $n_s=0.974\pm0.003$, and an amplitude of density fluctuations of $\sigma_8=0.813\pm0.005$. Including the DESI DR2 data tightens the Hubble constant to $H_0=68.43\pm0.27$ km/s/Mpc; $\Lambda$CDM parameters agree between the P-ACT and DESI DR2 data at the $1.6\sigma$ level. We find no evidence for excess lensing in the power spectrum, and no departure from spatial flatness. The contribution from Sunyaev-Zel'dovich (SZ) anisotropy is detected at high significance; we find evidence for a tilt with suppressed small-scale power compared to our baseline SZ template spectrum, consistent with hydrodynamical simulations with feedback.

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arXiv2025

The Atacama Cosmology Telescope: DR6 Constraints on Extended Cosmological Models

We use new cosmic microwave background (CMB) primary temperature and polarization anisotropy measurements from the Atacama Cosmology Telescope (ACT) Data Release 6 (DR6) to test foundational assumptions of the standard cosmological model and set constraints on extensions to it. We derive constraints from the ACT DR6 power spectra alone, as well as in combination with legacy data from Planck. To break geometric degeneracies, we include ACT and Planck CMB lensing data and baryon acoustic oscillation data from DESI Year-1, and further add supernovae measurements from Pantheon+ for models that affect the late-time expansion history. We verify the near-scale-invariance (running of the spectral index $d n_s/d\ln k = 0.0062 \pm 0.0052$) and adiabaticity of the primordial perturbations. Neutrino properties are consistent with Standard Model predictions: we find no evidence for new light, relativistic species that are free-streaming ($N_{\rm eff} = 2.86 \pm 0.13$, which combined with external BBN data becomes $N_{\rm eff} = 2.89 \pm 0.11$), for non-zero neutrino masses ($\sum m_\nu < 0.082$ eV at 95% CL), or for neutrino self-interactions. We also find no evidence for self-interacting dark radiation ($N_{\rm idr} < 0.134$), early-universe variation of fundamental constants, early dark energy, primordial magnetic fields, or modified recombination. Our data are consistent with standard BBN, the FIRAS-inferred CMB temperature, a dark matter component that is collisionless and with only a small fraction allowed as axion-like particles, a cosmological constant, and the late-time growth rate predicted by general relativity. We find no statistically significant preference for a departure from the baseline $\Lambda$CDM model. In general, models introduced to increase the Hubble constant or to decrease the amplitude of density fluctuations inferred from the primary CMB are not favored by our data.

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arXiv2024

Using Two-Frequency Dust Spectral Matching to Separate Galactic Synchrotron and Free-Free Temperature Foregrounds from the CMB

We introduce a method for removing CMB and anomalous microwave emission (AME, or spinning dust) intensity signals at high to intermediate Galactic latitudes in temperature sky maps at frequencies roughly between 5 and 40 GHz. The method relies on the assumption of a spatially uniform combined dust (AME and thermal) rms spectral energy distribution for these regions, but is otherwise model independent. A difference map is produced from input maps at two different frequencies in thermodynamic temperature: the two frequencies are chosen such that the rms AME signal in the lower frequency (~5 - 40 GHz) map is equivalent to the thermal dust emission rms in the higher frequency (~95 - 230 GHz) map. Given the high spatial correlation between AME and thermal dust, the resulting difference map is dominated by synchrotron and free-free foreground components, and can thus provide useful insight into the morphology and possible spectral variations of these components at high latitudes. We show examples of these difference maps obtained with currently available WMAP and Planck data and demonstrate the efficacy of CMB and dust mitigation using this method. We also use these maps, in conjunction with Haslam 408 MHz and WHAM H-alpha observations, to form an estimate of the diffuse synchrotron spectral index in temperature on degree scales. The hybrid analysis approach we describe is advantageous in situations where frequency coverage is insufficient to break spectral degeneracies between AME and synchrotron.

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arXiv2024

JWST Validates HST Distance Measurements: Selection of Supernova Subsample Explains Differences in JWST Estimates of Local H0

JWST provides new opportunities to cross-check the HST Cepheid/SNeIa distance ladder, which yields the most precise local measure of H0. We analyze early JWST subsamples (~1/4 of the HST sample) from the SH0ES and CCHP groups, calibrated by a single anchor (N4258). We find HST Cepheid distances agree well (~1 sigma) with all 8 combinations of methods, samples, and telescopes: JWST Cepheids, TRGB, and JAGB by either group, plus HST TRGB and Miras. The comparisons explicitly include the measurement uncertainty of each method in N4258, an oft-neglected but dominant term. Mean differences are ~0.03 mag, far smaller than the 0.18 mag "Hubble tension." Combining all measures produces the strongest constraint yet on the linearity of HST Cepheid distances, 0.994+-0.010, ruling out distance-dependent bias or offset as the source of the tension at ~7 sigma. Yet, measurements of H0 from current JWST subsamples produce large sampling differences whose size and direction we can directly estimate from the full HST set. We show that Delta(H0)~2.5 km/s/Mpc between the CCHP JWST program and the full HST sample is entirely consistent with differences in sample selection. Combining all JWST samples produces a new, distance-limited set of 16 SNeIa at D<25 Mpc and more closely resembles the full sample thanks to "reversion to the mean" of larger samples. Using JWST Cepheids, JAGB, and TRGB, we find 73.4+-2.1, 72.2+-2.2, and 72.1+-2.2 km/s/Mpc, respectively. Explicitly accounting for SNe in common, the combined-sample three-method result from JWST is H0=72.6+-2.0, similar to H0=72.8 expected from HST Cepheids in the same galaxies. The small JWST sample trivially lowers the Hubble tension significance due to small-sample statistics and is not yet competitive with the HST set (42 SNeIa and 4 anchors), which yields 73.2+-0.9. Still, the joint JWST sample provides important crosschecks which the HST data passes.

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arXiv2024

Does the Correlation between 2MRS Galaxies and the CMB Indicate an Unmodeled CMB Foreground?

We revisit the claimed detection of a new cosmic microwave background (CMB) foreground based on the correlation between low-redshift 2MASS Redshift Survey (2MRS) galaxies and CMB temperature maps from the Planck and WMAP missions. We reproduce the reported measurements but argue that the original analysis significantly underestimated the uncertainties. We cross-correlate the 2MRS galaxy positions with simulated CMB maps and show that the correlation measured with the real data for late-type spiral galaxies at angular scales $θ\geq0.1^{\circ}$ and redshift $cz<4500$ km s$^{-1}$ is consistent with zero at the $1.7σ$ level or less, depending on the exact CMB map and simulation construction. This was the sample that formed the basis for the original detection claim. For smaller angular separations the results are not robust to galaxy type or CMB cleaning method, and we are unable to draw firm conclusions. The original analysis did not propose a specific, falsifiable physical correlation mechanism, and it is impossible to rule out any contribution from an underlying physical effect. However, given our calculations, the lack of signal from expanding the redshift range, and the lack of corroboration from other galaxy surveys, we do not find the evidence for a new CMB foreground signal compelling.

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arXiv2023

Revisiting the $A_L$ Lensing Anomaly in Planck 2018 Temperature Data

We revisit the lensing anomaly in the Planck 2018 temperature (TT) data and examine its robustness to frequency selection and additional sky masking. Our main findings are: (1) The phenomenological lensing amplitude parameter, $A_L$, varies with ecliptic latitude, with a $2.9\sigma$ preference for $A_L>1$ near the ecliptic, and $1.0\sigma$ preference near the ecliptic poles, compared to $2.5\sigma$ on the original masks. This behavior is largely or solely from 217 GHz and suggestive of some non-random effect given the Planck scan strategy. (2) The 217 GHz TT data also show a stronger preference for $A_L>1$ than the lower frequencies. The shifts in $A_L$ from 217 GHz with additional Galactic dust masking are too large to be explained solely by statistical fluctuations, indicating some connection with the foreground treatment. Overall, the Planck $A_L$ anomaly does not have a single simple cause. Removing the 217 GHz TT data leaves a $1.8\sigma$ preference for $A_L>1$. The low-multipole ($\ell<30$) TT data contribute to the preference for $A_L>1$ through correlations with $\Lambda$CDM parameters. The 100 and 143 GHz data at $\ell\geq30$ prefer $A_L>1$ at $1.3\sigma$, and this appears robust to the masking tests we performed. The lensing anomaly may impact fits to alternative cosmological models. Marginalizing over $A_L$, optionally applied only to Planck TT spectra, can check this. Models proposed to address cosmological tensions should be robust to removal of the Planck 217 GHz TT data.

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arXiv2022

An exploration of an early gravity transition in light of cosmological tensions

We study a step-like transition in the value of the effective Planck mass (or effective gravitational constant) on cosmological scales prior to recombination. We employ CMB, BAO, and SNIa data and find they are sufficient to strongly constrain our implementation of the Effective Field Theory of Dark Energy and Modified Gravity, used to model the transition, to a limited parameter space. The data prefer a $\sim 5\%$ shift in the value of the effective Planck mass ($<10 \%$ at $2 σ$) prior to recombination. This Transitional Planck Mass (TPM) model is free to undergo its transition at any point over multiple decades of scale factor prior to recombination, $\log_{10}(a) = -5.32^{+0.96}_{-0.72}$ (68\% CL). This lowers the sound horizon at last scattering, which increases the Hubble constant to $71.09 \pm 0.75$ km $\textrm{s}^{-1}\textrm{Mpc}^{-1}$ with a combination of local measurements as prior and to $69.22^{+0.67}_{-0.86}$ km $\textrm{s}^{-1}\textrm{Mpc}^{-1}$ when the prior is excluded. The TPM model improves $χ^2$ with respect to $Λ$CDM by $Δχ^2 = -23.72$ with the $H_0$ prior and $Δχ^2 = -4.8$ without the prior. The model allows for both $H_0 > 70$ km$\textrm{s}^{-1}\textrm{Mpc}^{-1}$ and $S_8 < 0.80$ simultaneously with lower values of $S_8$ due to a reduction in the matter density $Ω_m$ to offset the increase in $H_0$ relative to $Λ$CDM. While this is a particular modified gravity model, studying other variants of modified gravity may be a productive path for potentially resolving cosmological tensions, while avoiding the need for a cosmological constant.

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arXiv2022

Polarized Synchrotron Foreground Assessment for CMB Experiments

Polarized Galactic synchrotron emission is an undesirable foreground for cosmic microwave background (CMB) experiments observing at frequencies $< 150$ GHz. We perform a combined analysis of observational data at 1.4, 2.3, 23, 30 and 33 GHz to quantify the spatial variation of the polarized synchrotron spectral index, $β^{pol}$, on $\sim3.5^\circ$ scales. We compare results from different data combinations to address limitations and inconsistencies present in these public data, and form a composite map of $β^{pol}$. Data quality masking leaves 44% sky coverage (73% for $|b|> 45^\circ$). Generally $-3.2 < β^{pol} \lesssim -3$ in the inner Galactic plane and spurs, but the Fan Region in the outer Galaxy has a flatter index. We find a clear spectral index steepening with increasing latitude south of the Galactic plane with $Δβ^{pol}=0.4$, and a smaller steepening of $0.25$ in the north. Near the south Galactic pole the polarized synchrotron spectral index is $β^{pol} \approx -3.4$. Longitudinal spectral index variations of $Δβ^{pol} \sim 0.1$ about the latitudinal mean are also detected. Within the BICEP2/Keck survey footprint, we find consistency with a constant value, $β^{pol} = -3.25 \pm 0.04$ (statistical) $\pm 0.02$ (systematic). We compute a map of the frequency at which synchrotron and thermal dust emission contribute equally to the total polarized foreground. The limitations and inconsistencies among datasets encountered in this work make clear the value of additional independent surveys at multiple frequencies, especially between $10-20$ GHz, provided these surveys have sufficient sensitivity and control of instrumental systematic errors.

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arXiv2022

Snowmass 2021 CMB-S4 White Paper

This Snowmass 2021 White Paper describes the Cosmic Microwave Background Stage 4 project CMB-S4, which is designed to cross critical thresholds in our understanding of the origin and evolution of the Universe, from the highest energies at the dawn of time through the growth of structure to the present day. We provide an overview of the science case, the technical design, and project plan.

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arXiv2021

Limits on Polarized Dust Spectral Index Variations for CMB Foreground Analysis

Using Planck polarization data, we search for and constrain spatial variations of the polarized dust foreground for cosmic microwave background (CMB) observations, specifically in its spectral index, $β_d$. Failure to account for such variations will cause errors in the foreground cleaning that propagate into errors on cosmological parameter recovery from the cleaned CMB map. It is unclear how robust prior studies of the Planck data which constrained $β_d$ variations are due to challenges with noise modeling, residual systematics, and priors. To clarify constraints on $β_d$ and its variation, we employ two pixel space analyses of the polarized dust foreground at $>3.7^{\circ}$ scales on $\approx 60\%$ of the sky at high Galactic latitudes. A template fitting method, which measures $β_d$ over three regions of $\approx 20\%$ of the sky, does not find significant deviations from an uniform $β_d = 1.55$, consistent with prior Planck determinations. An additional analysis in these regions, based on multifrequency fits to a dust and CMB model per pixel, puts limits on $σ_{β_d}$, the Gaussian spatial variation in $β_d$. At the highest latitudes, the data support $σ_{β_d}$ up to $0.45$, $0.30$ at mid-latitudes, and $0.15$ at low-latitudes. We also demonstrate that care must be taken when interpreting the current Planck constraints, $β_d$ maps, and noise simulations. Due to residual systematics and low dust signal to noise at high latitudes, forecasts for ongoing and future missions should include the possibility of large values of $σ_{β_d}$ as estimated in this paper, based on current polarization data.

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arXiv2021

High $H_0$ Values from CMB E-mode Data: A Clue for Resolving the Hubble Tension?

The E-mode (EE) CMB power spectra measured by Planck, ACTPol, and SPTpol constrain the Hubble constant to be $70.0\pm2.7$, $72.4^{+3.9}_{-4.8}$, and $73.1^{+3.3}_{-3.9}$ km s$^{-1}$ Mpc$^{-1}$ within the standard $Λ$CDM model (posterior mean and central 68% interval bounds). These values are higher than the constraints from the Planck temperature (TT) power spectrum, and consistent with the Cepheid-supernova distance ladder measurement $H_0=73.2\pm1.3$ km s$^{-1}$ Mpc$^{-1}$. If this preference for a higher value was strengthened in a joint analysis it could provide an intriguing hint at the resolution of the Hubble disagreement. We show, however, that combining the Planck, ACTPol, and SPTpol EE likelihoods yields $H_0=68.7\pm1.3$ km s$^{-1}$ Mpc$^{-1}$, $2.4σ$ lower than the distance ladder measurement. This is due to different degeneracy directions across the full parameter space, particularly involving the baryon density, $Ω_bh^2$, and scalar tilt, $n_s$, arising from sensitivity to different multipole ranges. We show that the E-mode $Λ$CDM constraints are consistent across the different experiments within $1.4σ$, and with the Planck TT results at $0.8σ$. Combining the Planck, ACTPol, and SPTpol EE data constrains the phenomenological lensing amplitude, $A_L=0.89\pm0.10$, consistent with the expected value of unity.

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arXiv2021

Quantifying the CMB Degeneracy Between the Matter Density and Hubble Constant in Current Experiments

We revisit the degeneracy between the Hubble constant, $H_0$, and matter density, $Ω_m$, for current cosmic microwave background (CMB) observations within the standard $ΛCDM$ model. We show that Planck, Wilkinson Microwave Anisotropy Probe (WMAP), South Pole Telescope (SPT), and Atacama Cosmology Telescope Polarimeter (ACTPol) temperature power spectra produce different values of the exponent $x$ from minimizing the variance of the product $Ω_mH_0^x$. The distribution of $x$ from the different data sets does not follow the Markov Chain Monte Carlo (MCMC) best-fit values for $H_0$ or $Ω_m$. Particularly striking is the difference between Planck multipoles $\ell\leq800$ ($x=2.81$), and WMAP ($x = 2.94$), despite very similar best-fit cosmologies. We use a Fisher matrix analysis to show that, in fact, this range in exponent values is exactly as expected in $ΛCDM$ given the multipole coverage and power spectrum uncertainties for each experiment. We show that the difference in $x$ from the Planck $\ell \leq 800$ and WMAP data is explained by a turning point in the relationship between $x$ and the maximum effective multipole, at around $\ell=700$. The value of $x$ is determined by several physical effects, and we highlight the significant impact of gravitational lensing for the high-multipole measurements. Despite the spread of $H_0$ values from different CMB experiments, the experiments are consistent with their sampling of the $Ω_m-H_0$ degeneracy and do not show evidence for the need for new physics or for the presence of significant underestimated systematics according to these tests. The Fisher calculations can be used to predict the $Ω_m-H_0$ degeneracy of future experiments.

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arXiv2021

Analytic Calculation of Covariance between Cosmological Parameters from Correlated Data Sets, with an Application to SPTpol

Consistency checks of cosmological data sets are an important tool because they may suggest systematic errors or the type of modifications to $Λ$CDM necessary to resolve current tensions. In this work, we derive an analytic method for calculating the level of correlations between model parameters from two correlated cosmological data sets, which complements more computationally expensive simulations. This method is an extension of the Fisher analysis that assumes a Gaussian likelihood and a known data covariance matrix. We apply this method to the SPTpol temperature and polarization CMB spectra (TE and EE). We find weak correlations between $Λ$CDM parameters with a 9$\%$ correlation between the TE-only and EE-only constraints on $H_0$ and a 25$\%$ and 32$\%$ correlation for log($A_s$) and $n_s$ respectively. Despite the negative correlations between the TE and EE power spectra, the correlations in the parameters are positive. The TE-EE parameter differences are consistent with zero, with a PTE of 0.53, in contrast to the PTE of 0.017 reported by SPTpol for the consistency of the TE and EE power spectra with $Λ$CDM. Using simulations we find that the results of these two tests are independent and that this difference can arise simply from statistical fluctuations. Ignoring correlations in the TT-TE and TE-EE comparisons biases the $χ^2$ low, artificially making parameters look more consistent. Therefore, we conclude that these correlations need to be accounted for when performing internal consistency checks of the TT vs TE vs EE power spectra for future CMB analyses.

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arXiv2021

Deconstructing the Planck TT Power Spectrum to Constrain Deviations from $Λ$CDM

Consistency checks of $Λ$CDM predictions with current cosmological data sets may illuminate the types of changes needed to resolve cosmological tensions. To this end, we modify the CLASS Boltzmann code to create phenomenological amplitudes, similar to the lensing amplitude parameter $A_L$, for the Sachs-Wolfe, Doppler, early Integrated Sachs-Wolfe (eISW), and Polarization contributions to the CMB temperature anisotropy, and then we include these additional amplitudes in fits to the Planck TT power spectrum. We find that allowing one of these amplitudes to vary at a time results in little improvement over $Λ$CDM alone suggesting that each of these physical effects are being correctly accounted for given the current level of precision. Further, we find that the only pair of phenomenological amplitudes that results in a significant improvement to the fit to Planck temperature data results from varying the amplitudes of the Sachs-Wolfe and Doppler effects simultaneously. However, we show that this model is really just refinding the $Λ$CDM + $A_L$ solution. We test adding our phenomenological amplitudes as well as $N_{\textrm{eff}}$, $Y_{\textrm{He}}$, and $n_{\textrm{run}}$ to $Λ$CDM + $A_L$ and find that none of these model extensions provide significant improvement over $Λ$CDM + $A_L$ when fitting Planck temperature data. Finally, we quantify the contributions of both the eISW effect and lensing on the constraint of the physical matter density from Planck temperature data by allowing the phenomenological amplitude from each effect to vary. We find that these effects play a relatively small role (the uncertainty increases by $3.5\%$ and $16\%$ respectively) suggesting that the overall photon envelope has the greatest constraining power.

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