Cosmological measurements from the CMB and BAO are insensitive to the tail probability in the assumed likelihood

When fitting cosmological models to data, a Bayesian framework is commonly used, requiring assumptions on the form of the likelihood and model prior. In light of current tensions between different data, it is interesting to investigate the robustness of cosmological measurements to statistical assumptions about the likelihood distribution from which the data was drawn. We consider the impact of changes to the likelihood caused by uncertainties due to the finite number of mock catalogs used to estimate the covariance matrix, leading to the replacement of the standard Gaussian likelihood with a multivariate t-distribution. These changes to the likelihood have a negligible impact on recent cosmic microwave background (CMB) lensing and baryon acoustic oscillation (BAO) measurements, for which covariance matrices were measured from mock catalogs. We then extend our analysis to perform a sensitivity test on the Gaussian likelihoods typically adopted, considering how increasing the size of the tails of the likelihood (again using a t-distribution) affects cosmological inferences. For an open ΛCDM model constrained by BAO alone, we find that increasing the weight in the tails shifts and broadens the resulting posterior on the parameters, with a ∼0.2–0.4σ effect on ΩΛ and Ωk. In contrast, the CMB temperature and polarization constraints in ΛCDM showed less than 0.03σ changes in the parameters, except for {τ, ln(1010 A s), σ 8, S 8, σ 8Ω0.25 m, z re, 109 A s e -2τ } which shifted by around 0.1–0.2σ. If we use solely ℓ < 30 data, the amplitude A s e -2τ varies in the posterior mean by 0.7σ and the error bars increase by 6%. We conclude, at least for current-generation CMB and BAO measurements, that uncertainties in the shape and tails of the likelihood do not contribute to current tensions.


Introduction
The standard cosmological model with a cosmological constant (Λ) and cold dark matter (CDM) model agrees well with an extensive range of observational results.Experiments that support this ΛCDM model include spectroscopic galaxy surveys such as the extended Baryon Oscillation Spectroscopic Survey (BOSS and eBOSS; [1,2]), photometric weak lensing surveys such as the Dark Energy Survey (DES; [3]), cosmic microwave background (CMB) measurements like Planck [4], and the Pantheon and Union supernova compilations [5,6].However, there are tensions in the derived parameters of the ΛCDM model from different datasets.In particular, the Hubble constant (H 0 ) measurements from local distance ladder measurements [7] disagree with those from the CMB [4] or from Big Bang nucleosynthesis (BBN) analysed together with Baryon Acoustic Oscillation (BAO) data [8].In addition, different measurements of the clustering amplitude of matter (S 8 ) disagree, particularly those from weak lensing [9,10] when compared to the predictions from Planck. 1iven these tensions and future improvements in data precision [12,13], there has been a renewed focus on parameter uncertainties stemming from systematic errors.For example, the reionization optical depth, τ , as measured using the CMB is very sensitive to changes in the analysis pipeline.The value has changed significantly between analyses: the WMAP experiment measured τ = 0.17 ± 0.04 [14] from the year one data, and τ = 0.089 ± 0.014 after nine years of data [15].In contrast, Planck published τ = 0.097 ± 0.038 in its first set of results [16], and τ = 0.058 ± 0.006 using the final data and an updated pipeline [17].Research collaborations independent from Planck, such as BeyondPlanck [18], continue to improve the analysis technique to achieve a higher control over the uncertainty propagation and provide a more robust estimate.
In the standard Bayesian analysis pipelines used to make inferences from cosmological data (e.g.[19]), credible intervals for cosmological model parameters are determined by exploring the posterior surface. 2 The results depend on the prior assumed for each parameter and any assumptions made when determining the likelihood.The likelihood is typically approximated to be Gaussian, but this is generally not exact.For example, the likelihood of CMB multipoles C ℓ is a Wishart distribution, which deviates from Gaussianity particularly at low ℓ where the central limit theorem does not hold [21,22].A further complication results from errors in the covariance matrix [23,24], which also modify the simple Gaussian likelihood.
In this paper, we consider how assumptions about the form of the likelihood affect recovered credible intervals for fits to the Sloan Digital Sky Survey (SDSS) [8] and Planck [4] data.We first consider the case where the covariance on an intermediate statistic itself has an error.This is the situation when covariance matrices for correlation functions or power spectra are determined from sets of mocks, as for recent analyses of eBOSS data [8] and CMB lensing [25] data from Planck, for example.In this case, to complete a Bayesian analysis we need a prior on the covariance matrix, and a number of suggestions have previously been made for this, including an Independence-Jeffreys prior [26], or a prior matching frequentist confidence intervals with Bayesian credible intervals [20].We consider the effect of this choice of prior, comparing against the situation where we ignore the errors in the covariance matrix.
We then extend this work to test the sensitivity to the Gaussian assumptions for the likelihood when fitting BAO and CMB data.This can be considered a sensitivity analysis for the cosmological inferences made.Within the fields of biostatistics and epidemiology, sensitivity analyses play a crucial role in assessing the robustness of conclusions drawn from observational data (eg.[27][28][29]).They are a critical way of assessing the impact, effect, or influence of key modelling choices like definitions of outcomes, protocol deviations, missing data assumptions, outliers, or prior specifications.Many experiments make a Gaussian approximation for the likelihood even if the model does not intrinsically support this, or the central limit theorem does not hold.Additionally, noise is not necessarily Gaussian distributed, and large-scale structure analyses often use quasi-linear modes where gravitational non-Gaussianity is nonnegligible.Hence, considering heavy-tailed likelihoods in cosmology is well-motivated and this has been done in past work [30][31][32].We focus on the tails of the distribution assumed for the likelihood, using the multivariate t-distribution as a replacement for the Gaussian distribution to vary the fraction of probability within the tails and determine how sensitive the final parameter constraints are to this choice.
Section (2) provides background information on statistical techniques and introduces the procedure we implemented to stress test components of the data analysis.Then, Section (3) outlines the datasets that we studied.Following this, Section (4) considers analyses where the covariance matrix itself has errors, while Section (5) details our sensitivity analysis of more general BAO and CMB measurements.Finally, we conclude in Section (6).

Statistical Techniques
We will focus on Bayesian parameter inference (e.g.[33]), where one derives credible regions for parameters based on the posterior, inferred using Bayes' theorem: where H is the hypothesis to be tested and D is the data.The posterior P(H|D) is determined from the likelihood P(D|H) and prior P(H).So, to determine the posterior distribution of the model parameters, the prior must be chosen to reflect the knowledge of the parameters before the new data is considered, and a likelihood function specified.The multivariate Gaussian (Normal) distribution is often adopted for the likelihood, if the distribution from which the data are drawn is not known: [34] Here, Σ is the covariance matrix, the data is x d , the data model is x(p), the parameters are p, and In order to test how robust modern cosmology is to the choices made for the likelihood, we consider two different cases as detailed in the next section.

Covariance Matrix Estimated From Simulations
An unbiased estimate of the true covariance matrix Σ from a set of simulated data is given by where n s is the number of mock simulations, x m i are the mock data, and xm is the mean of x m i .Inverting this estimate (S −1 ), however, gives a biased estimate of the inverse covariance Σ −1 [35].This systematically biases the credible regions placed on cosmological parameters.Simply correcting the skewness in the inverse covariance matrix as advocated by [35] does not correct the credible regions because it does not take into account the way in which the posterior surface is explored, which can be considered a constrained inversion of the inverse covariance back to a covariance of new parameters [20].An approximate correction for this effect is to preserve the Gaussian likelihood function and simply scale the covariance [20,36], such that the matrix to invert is S ′ where Here n d is the number of data points and n p is the number of parameters.
Ref. [26] instead considered a fully Bayesian approach where an Independence-Jeffreys prior is placed on the covariance matrix, and it is considered a random variable in the analysis.This results in a multivariate t-distribution posterior [26], where x 0 are the data with dimension n d .For the Independence-Jeffreys prior we have m = n s .Ref. [20] considered instead a prior on the covariance matrix designed to match Bayesian credible intervals with frequentist confidence intervals.Again, this results in a multivariate t-distribution posterior, but this time with We note that for the multivariate t-distribution, S is referred to as the scale matrix, as the covariance is Using either the Independence-Jeffreys or frequentist matching prior for the true covariance matrix results in a posterior with larger tails than a Gaussian, increasing the likelihood of parameter values further away from the central region.This raises the question of how much this changes the cosmological constraints for previously published datasets.We will investigate this for both choices of priors, hereafter calling the posteriors the Independence-Jeffreys t-distribution and the matching prior t-distribution, comparing against a Gaussian posterior where the error in the covariance matrix is ignored.

Sensitivity Analysis
In the previous section, we considered well motivated changes within a Bayesian context to the likelihood following consideration of how the covariance matrix was estimated.Changes to the likelihood can also form part of a sensitivity analysis, where we consider how robust our inferences are to the analysis method assumed.The approximation of a particular form for the likelihood may not be easy to determine, but we can easily test how important that approximation is for particular parameters.For this, following on from Section 2.1, we consider the general form of the multivariate t-distribution: . (2.9) Here, x is the data and µ is the model, which both have dimension n d × 1. Σ is the n d × n d scale matrix and ν is the number of degrees of freedom.In the limit ν → ∞, the distribution tends towards a Gaussian form.Using the multivariate t-distribution we can set the number of degrees of freedom based on how much more probability is in the tails of the chosen tdistribution compared to a Gaussian.In order to quantify the extent to which we are adjusting the likelihoods, we consider the multivariate t-distribution for a single data point, marginalized over the other data points, and calculate the 2σ interval for different values of ν.We choose the two ν values which correspond to 2σ intervals that are 1% or 10% larger than those of a marginalized multivariate Gaussian with matching variance.To do this, we use the marginalized multivariate t-distribution, which is the Student t-distribution, and calculate the cumulative distribution function and evaluate the 2.5 th and 97.5 th percentiles.These are then compared to the percentiles of the Gaussian distribution so the value of ν that increased the region by 1% and 10% could be determined.For the 1% scaling, ν = 104, and for 10% scaling, ν = 13.An example of how the distribution changes is shown in Fig. (1) for the 10% case.In the Monte Carlo code, the multivariate Gaussian distribution was replaced with Eq. (2.9) using the derived values of ν, along with the corresponding value of n d for the dataset considered.

SDSS BAO
Over 20 years, the Sloan Digitial Sky Survey (SDSS; [37]) has undertaken a series of galaxy redshift surveys, from which the BAO scale can be measured.At low redshift, 0.07 < z < 0.2, there is the Main Galaxy Sample (MGS; [38,39]) from Data Release 7 (DR7; [40]); at 0.2 < z < 0.5, the Baryon Oscillation Spectroscopic Survey (BOSS; [41]) from DR12 ( [42]); and at z > 0.6, extended BOSS (eBOSS) [43] from DR16 ( [44]).In our work, we focus on the analyses where galaxies are used as discrete tracers of the density field, and the BAO were measured and fitted from these samples using both the correlation function and power spectrum.For the 2-point functions, covariance matrices are calculated using mock catalogues.We primarily focus on fitting the DR12 BOSS LRG correlation function [45] for redshift bins z = 0.38 and z = 0.61 (since incorporating bin z = 0.51 adds negligible extra information), as these data represent the best large-scale structure data currently available.We directly fit the cosmological parameters to the two-point correlation function using the post-reconstruction damped BAO model of [46].In Appendix (A), we instead work with all BAO samples described above (which are summarized in Table (8), including the number of mocks, size of the data vector, and number of parameters in the model), but use their publicly released compressed parameters (α ∥ , α ⊥ ) rather than re-fitting the correlation function measurements.The SDSS team conducted these fits in two phases, first compressing the correlation function into (model-independent) Alcock-Paczysnki (AP) parameters, α ∥ and α ⊥ , which parameterise the measured dilation of the BAO peak along and across the line of sight with respect to a fiducial cosmology.They then fit cosmological parameters to the compressed dataset.Our likelihood directly fits the cosmological parameters to the correlation functions by converting them into α ∥ and α ⊥ and then shifting the post-reconstruction BAO model of [45] by these AP parameters.The likelihood of [45] includes a prior on the bias B 0 ; we first split this prior from the likelihood, and then add it back after scaling χ 2 or changing from a Gaussian to a t-distribution.
We use the publicly available post-reconstruction correlation functions and covariance matrices of [45], 3 and removed the Hartlap factor before switching to either a frequentistmatching prior or Independence-Jeffreys t-distribution.We considered the open ΛCDM (oΛCDM) model that allows for non-zero curvature, sampling over cosmological parameters Ω m , and Ω k and fixing Ω b = 0.0468 and H 0 = 70.We use Cobaya [19] to obtain the cosmological parameter constraints and considered the chains converged for Gelman-Rubin R − 1 ≤ 0.02 [47] along with a further exploration of the tails beyond the 95% confidence interval, with a permitted quantile chain variance of 0.02 standard deviations in the 95% confidence interval.

Planck CMB Lensing
We also consider the Planck 2018 CMB lensing measurements [25].Gravitational lensing creates distinctive, non-Gaussian structure in CMB temperature and polarization maps, which can be extracted using quadratic estimators [48].The Planck team applied these estimators to the Planck temperature (T ) and E-mode polarization (E) maps to produce a map of the CMB lensing potential φ across 60% of the sky.Estimating the lensing potential power spectrum requires (i) a mean field normalization correction, (ii) a subtraction of noise biases (arising from the disconnected four-point function of the Gaussian CMB, non-primary couplings of the connected four-point function, and point source biases), and (iii) the application of a simulation-determined Monte Carlo calculation.Complications such as the Galactic mask and sky varying noise couple previously independent Fourier modes of ϕ.The covariance matrix used to include such effects in [25] was estimated by applying lensing reconstruction to 240 realistic FFP10 CMB simulations. 4Any dependence on the theoretical model of the CMB power spectra is removed by marginalization over the primary CMB power spectrum, which adds a term to the covariance matrix (Eq.( 34) in [25]).It is this covariance matrix containing the additive correction that we include in our analysis.
We test the sensitivity of the measurements from CMB lensing for the ΛCDM model, sampling over the cosmological parameters Ω b h 2 , Ω c h 2 , 100θ MC , ln(10 10 A s ) and n s .We used a modified version of the code CosmoMC [49] to obtain the parameter constraints with a convergence criterion of R − 1 ≤ 0.01 and further sample the tails beyond the 99% confidence level, with a limit on the quantile chain variance of 0.2 standard deviations.To calculate the covariance matrix, the number of simulations used was n s = 240, the data vector had n d = 9 (corresponding to the conservative multipole range 8 ≤ ℓ ≤ 400), and the number of parameters was n p = 5 [25].We also checked that using the aggressive multipole range 8 ≤ ℓ ≤ 2048 gave results similar to the conservative range, so for this test the data vector was n d = 14. 5 Unlike SDSS, only the Hartlap correction factor [35] was considered, so we removed this and then changed from the Gaussian to the Independence-Jeffreys and matching prior tdistribution posteriors.We used the lensing convergence power spectrum from the minimum variance (MV) combination of T and E maps. 6.

Planck CMB Temperature and Polarization
Next, we consider the Planck CMB measurements of temperature and polarization (T &P ) [50].Planck considers different combinations of auto-and cross-correlations of the T and E spectra, as well as different ℓ ranges.These are contained in separate codes exploring high-ℓ and low-ℓ multipoles separately [51].In detail, Plik is the joint T T , EE and T E likelihood in the multipole range 30-2508 for T T , and 30 − 1996 for T E and EE.It is based on the binned cross-spectra from 100, 145 and 217 GHz channels, and represents the likelihood as a correlated Gaussian: where Ĉ is the vector with observed spectra, C(θ) are the predicted spectra for the cosmological parameter set θ, and Σ is the covariance matrix as computed for a fiducial realisation.Even though the Gaussian shape is an approximation to the true Wishart distribution, it has been demonstrated to perform reasonably well even for ℓ ∼ 30, as discussed in [22].In the [52] analysis, large angular scales (ℓ < 30) use the SimAll (EE) and commander (T T ) likelihoods.The latter is based on a Gaussianised Blackwell-Rao estimator of the T T power spectrum from foreground cleaned CMB samples [53], whereas the former consists of a brute-force inversion of the likelihood for polarisation power spectra estimates from foreground-cleaned maps [51].Hence commander is a Gaussian likelihood, and SimAll is not.Our procedure consists of changing Gaussian likelihoods to multivariate t-distributions, and should not be applied to likelihood forms that are already non-Gaussian.
We therefore use the LFI-based likelihood bflike for large angular scales [51], replacing commander and SimAll.bflike is a map-based Gaussian likelihood, and unlike commander, it includes polarization as well as temperature.The likelihood for bflike is where m is the CMB-plus-Noise map, and S(θ) + N is the Signal-plus-Noise covariance.The signal covariance here is computed for every cosmological parameter set θ in order to explore the full joint posterior distribution for T T , T E, EE and BB power spectra in the multipole range 2 − 30.In order for the covariance to be non-singular, it is required that the number of pixels in the Stokes I, Q, and U maps is larger than ∼ (2ℓ max + 1) 2 .We considered the value of our data vector in the t-distribution to be n d = 2289 for Plik and n d = 6467 for bflike, where the latter value comes from the number of unmasked pixels in I, Q, and U CMB maps at HEALPix N side = 16, and the former from the number of bins used in TT, TE, and EE in different frequency channels (given in Table (20) of [51]).For the MCMC runs, the base cosmological parameters were Ω b h 2 , Ω c h 2 , 100θ MC , τ , ln(10 10 A s ) and n s when we considered both Plik (T T T EEE) and bflike (lowT EB) 7 .For runs with only bflike, we sampled over cosmological parameters τ and ln(10 10 A s ).We again used CosmoMC with the same convergence settings as given above for CMB lensing.The left panel shows constraints for the original BAO results from using SDSS's method (red), with the correction factors removed (purple), the matching prior t-distribution (orange), and the Independence-Jeffreys t-distribution (blue).On the right is again the original Gaussian set-up (red) along with the sensitivity test with 1% larger tails (purple) and 10% larger tails (blue).
We begin by producing cosmological constraints for the oΛCDM model using SDSS BOSS's original likelihood setup with the Gaussian distribution, before moving on to consider results using instead the t-distribution with Independence-Jeffreys or frequentist-matching priors on the covariance matrix.We also compare against the scenario where all of the corrections are removed, equivalent to assuming no error in the covariance matrix.The cosmological constraints are illustrated in Fig. (2), which shows a high level of agreement between the contours.Additionally, the constraints for Ω Λ and Ω k are listed in in Table (1).The correction factors used by SDSS increase the size of the confidence intervals by 1%, compared to not using the correction factors and using a Gaussian likelihood.
For the two t-distribution likelihoods, the parameters only have up to a 0.04σ difference in comparison to the original Gaussian results, and the confidence intervals are similar in size to the Gaussian likelihood without the correction factors.The primary take-home message from Table (1) is that none of the choices results in a significant change in constraints.This was also seen when instead of fitting to the correlation function data, we fit to the compressed data (AP parameters) in Appendix (A).Therefore, while these methods of marginalizing over BOSS DR12 LRG BAO, z = 0. Table 2: Parameter constraints for the ΛCDM model using CMB lensing measurements.The uncertainties are given by the 68% credible intervals.We considered using either the original Gaussian method, removing the correction factors, or a t-distribution with an Independence-Jeffreys or frequentist-matching prior.
Next, we consider the Planck 2018 CMB lensing measurements within the ΛCDM model.We first do this for both the original Gaussian setup and then including no correction factors, which are compared in Table (2) and Fig. (3) against methods that consider a posterior that allows for the covariance matrix error.The marginalised parameter credible regions are effectively the same with less than a 0.05σ disagreement.We conclude that the error on the covariance matrix used for the CMB lensing analysis was sufficiently small that the corrections for it are negligible.

Figure 3:
The contour plot for the ΛCDM model, illustrating the Ω m -σ 8 constraints at 68% and 95% credible intervals for Planck CMB lensing.Shown on the plot is the case of using Planck 's original Gaussian method (red), correction factors removed (purple), the Independence-Jeffreys t-distribution (orange), and the matching prior t-distribution (blue).

Results from the Sensitivity Analysis
In the previous section we evaluated the importance of the correction to the likelihood when considering uncertainty in a covariance matrix constructed using mocks.In this section we take this one step further and consider the sensitivity of the analysis to generic changes to the heaviness of the tails, without a specific motivation from the details of the analysis.This is a form of sensitivity analysis that is common in other research fields, and is designed to test the robustness of the inferences made to the statistical form assumed for the data (e.g. in the presence of "unknown unknowns" that may affect the tails of the distribution).For this test, we consider the BAO and CMB lensing data as well as the Planck temperature and lensing data.To perform our sensitivity analysis, we have changed the likelihood to a t-distribution with degrees of freedom ν = 104 (1% extra tail probability) or ν = 13 (10% extra tail probability) as described in Section (2.2).

BAO and CMB Lensing
First, we applied this test to BAO data and we fitted the oΛCDM model, and focused on Ω Λ and Ω k as the parameters of interest.We have deliberately chosen these data, as we know that the data only weakly constrains these parameters.As shown in Table (3) and Fig.
(2), the parameter medians increase respectively by (Ω Λ , Ω k )=(0.2σ,0.2σ) and (Ω Λ , Ω k )=(0.4σ,0.3σ) for 1% and 10% more power in the tails of the likelihood.Moreover, the error bars increase by 7% (25%) for Ω Λ and are nearly unchanged for Ω k .It is important to note that we find this sensitivity to the heaviness of the tails when constraining an extended model with BAO only.
Hence, the increased sensitivity to the likelihood assumptions is likely linked to the data only weakly constraining the model degeneracy we are moving along, which explains why this form of robustness test may be even more relevant to perform on these types of poorly constrained models.We find that fitting directly to the correlation function (rather than the intermediate compressed statistics α ∥ and α ⊥ ) is critical.In Appendix (A), we fit to α ∥ and α ⊥ instead of Table 3: The parameter values for the oΛCDM model using the BOSS DR12 LRG correlation function data for redshift bins centered at z eff = 0.38 and 0.61.We compare the results found using the Gaussian likelihood that SDSS had implemented versus a t-distribution likelihood of 1% or 10% extra probability in the tails.The uncertainties are given by the 68% credible intervals.
the correlation function and find that the constraints do not change significantly (∼ 0.02σ) with the form of the likelihood.
Planck CMB Lensing Table 4: These are the 68% credible intervals for the ΛCDM model using CMB lensing measurements.The table compares the results for a Gaussian likelihood to t-distribution likelihoods derived from a 1% and 10% increase of a Gaussian's 2σ region.
We also applied this sensitivity analysis to CMB lensing data for the ΛCDM model, which resulted in the parameter constraints outlined in Table (4).The CMB lensing results exhibited almost no variation in the parameter constraints with regards to the form of the likelihood, with only up to 0.02σ differences.We further confirmed that CMB lensing with the aggressive multipole range (with larger correction factors due to larger n d ) had results consistent with that found for the conservative range.

CMB T &P
Fitting to the CMB temperature and polarization data, we again replaced the Gaussian likelihood with a multivariate t-distribution for different choices of degrees of freedom, ν.The results for T T T EEE and bflike lowT EB are shown in Tables (5) and (6), then for only lowT EB in Table (7).We found for T T T EEE + lowT EB that parameters related to optical depth and the overall power spectrum amplitude {τ, ln(10 10 A s ), σ 8 , S 8 , σ 8 Ω 0.25 m , z re , 10 9 A s e −2τ } exhibited similar behaviour, ranging between a 0.1-0.2σdifference from the t-distributions to the Gaussian.This corresponds to a lower value of σ 8 and S Table 5: Difference between the ΛCDM model parameter constraints when using Planck CMB Plik high-l TT,TE,EE and bflike low-l TT,TE,EE,BB measurements for a Gaussian likelihood versus a t-distribution likelihood with 1% larger tails (ν = 104).Included are the 68%, 95%, and 99.7% credible intervals.We used the units of km s −1 Mpc −1 for H 0 and defined S 8 ≡ σ 8 (Ω m /0.3) 0.5 .In the first grouping of rows are the base parameters used in our MCMC analysis and the bottom group are the derived parameters.
tension.The other parameters did not differ significantly from the original Gaussian likelihood, with a maximum discrepancy of 0.03σ.For the lowT EB t-distribution results, we found that τ changed by 0.15σ whereas ln(10 10 A s ) and 10 9 A s e −2τ had a 0.7σ difference from the Gaussian case.By increasing power in the tails of the distribution, we obtain a lower estimate of τ and A s .This prompted us to investigate another choice of degrees of freedom, ν = 213, to confirm this trend in parameter constraints.This corresponds to a 0.3% increase of the Gaussian's 2σ region, so it is a mid-point between the Gaussian case and the 1% increase.).The uncertainties are the 68%, 95%, and 99.7% intervals, the units of H 0 are km s −1 Mpc −1 , and S 8 ≡ σ 8 (Ω m /0.3) 0.5 .In the first section of rows are the base parameters sampled over in MCMC, and the bottom section are the derived parameters.
that the change in constraints may be a discontinuous one caused by changing the likelihood from Gaussian to a t-distribution.We should note that the polarization of the CMB at large angular scales is dominated by noise and, because there is a strong degeneracy between τ and A s , these parameters are sensitive to this noisy signal.Since we use the variance of the map to infer cosmological parameters, any residual or additional noise not captured by the noise covariance matrix will be interpreted as additional signal, thus causing a systematic preference for larger values of τ and A s .Therefore, our results show that more weight in the tails of the distribution means more weight is given to the noise, lowering 10 There is a well-known tendency for τ to shift between different CMB analyses, due to the difficulty of modelling the extremely large angular scales that are sensitive to τ .For instance, the official Planck 2018 low-l LFI (bflike) likelihood value is τ = 0.063 ± 0.020 [51] and the combined Commander and SimAll result is τ = 0.0506 ± 0.0086.Natale et al. [54] reported τ = 0.069 +0.011 −0.012 for their WMAP + LFI likelihood, which roughly corresponds to a 0.9σ difference from the t-distributions.Moreover, the BeyondPlanck LFI results are τ = 0.065 ± 0.012, which corresponds to a 0.5σ offset [18].These shifts in τ are comparable to or somewhat larger than the shifts we find of ∆τ = 0.003, when switching from a Gaussian to a t-distribution.This also shows that τ is very sensitive to choices made in the likelihood, possibly explaining discrepancies seen in the literature.However, interestingly, the overall amplitude A s e −2τ and the matter fluctuation amplitude A s are more sensitive to the form of the likelihood than τ for low-l LFI data.When high-ℓ and HFI data are included this sensitivity decreases significantly.

Conclusions
We have examined how cosmological constraints from BAO and CMB measurements rely on the form of the likelihood used in the parameter inferences.Initially we considered how, when the covariance matrix is estimated through simulations, we can marginalize over the true covariance matrix instead of assuming a Gaussian likelihood with a scaled covariance.Then, we tested the effect of different choices of priors on the covariance matrix, including the Independence-Jeffreys prior and the frequentist matching prior.We found minimal differences between the different approaches when applied to current BAO and CMB lensing data from SDSS and Planck, respectively (less than 0.05σ differences).We further confirmed that the insensitivity to the form of the likelihood held regardless of whether we fit to the correlation function data or the compressed AP parameters.This demonstrates that the previously published constraints are robust to the change in the likelihood due to uncertainties in the covariance matrix, but future data analyses should consider using the t-distribution likelihood when the covariance is estimated from simulations, as it is more statistically robust.
Second, we have performed a sensitivity test on the assumption of Gaussianity in the likelihood by examining how parameter constraints are affected by increasing the probability in the tails for a wider range of data.When constraining extensions to flat ΛCDM with low by the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference #DIS-2022-568580.

Figure 1 :
Figure 1: A Gaussian distribution (orange) versus the Student t-distribution with its 2σ interval 10% larger than the Gaussian's (blue).

Figure 2 :
Figure2: Comparison of the Ω k -Ω Λ constraints at 68% and 95% credible intervals for the oΛCDM model using BOSS DR12 LRG correlation function data for redshift bins centered at z eff = 0.38 and 0.61.The left panel shows constraints for the original BAO results from using SDSS's method (red), with the correction factors removed (purple), the matching prior t-distribution (orange), and the Independence-Jeffreys t-distribution (blue).On the right is again the original Gaussian set-up (red) along with the sensitivity test with 1% larger tails (purple) and 10% larger tails (blue).

Table 1 :
Parameter constraints for the oΛCDM model, using the BOSS DR12 LRG correlation function data for redshift bins centered at z eff = 0.38 and 0.61.The uncertainties are the 68% credible intervals.Results were found using the original SDSS method, no correction factors, then the Independence-Jeffreys prior t-distribution and the matching prior t-distribution.theunknown covariance are statistically more rigorous, in practice it does not mean that results from previously published studies need to be reanalyzed.
8, which slightly reduces the S 8 Indeed, it does show the same behaviour of decreased mean values of τ and A s , suggesting

Table 6 :
Comparison of parameter constraints for the ΛCDM model using Planck CMB temperature and polarization measurements for a Gaussian likelihood and a t-distribution likelihood with 10% more probability in the tails (ν = 13

Table 7 :
9A s e −2τ .For Planck CMB low-l T T ,T E,EE,BB measurements, the ΛCDM model parameter constraints are compared for a Gaussian likelihood and t-distribution likelihoods that have 0.3%, 1% and 10% larger 2σ regions than the Gaussian.These are the 68% credible intervals.