$\bar T$: A New Cosmological Parameter?

TL;DR

This paper discusses the fundamental cosmological parameter $ar T$, highlighting that its unknown monopole contribution can bias parameter estimation in future CMB experiments, despite current precise measurements.

Contribution

It introduces the issue of fixing $ar T$ to the observed temperature and shows how this causes systematic errors that are irreducible and can bias cosmological inferences.

Findings

Systematic errors from fixing $ar T$ are smaller than Planck errors.

Future surveys may misinterpret measurements due to irreducible biases.

Fixing $ar T$ biases parameter estimates proportionally to $ heta_0$.

Abstract

The background photon temperature $\bar T$ is one of the fundamental cosmological parameters. Despite its significance, $\bar T$ has never been allowed to vary in the data analysis, owing to the precise measurement of the comic microwave background (CMB) temperature by COBE FIRAS. However, even in future CMB experiments, $\bar T$ will remain unknown due to the unknown monopole contribution $\Theta_0$ at our position to the observed (angle-averaged) temperature $\langle T\rangle^{\rm obs}$. By fixing $\bar T\equiv\langle T\rangle^{\rm obs}$, the standard analysis underestimates the error bars on cosmological parameters, and the best-fit parameters obtained in the analysis are biased in proportion to the unknown amplitude of $\Theta_0$. Using the Fisher formalism, we find that these systematic errors are smaller than the error bars from the $Planck$ satellite. However, with $\bar T\equiv\langle T\rangle^{\rm obs}$, these systematic errors will always be present and irreducible, and future cosmological surveys might misinterpret the measurements.

Equations

$${\Theta }_0:=\int \frac{d^2\hat{n}}{4\pi }\Theta (x_o,\hat{n}),$$

$$C_l^{\mathrm{biased}}:=⟨\frac{|a_{lm}{|}^2}{(1+{\Theta }_0{)}^2}⟩=C_l\left(1+\frac{3}{4\pi }{C}_0+\cdots \right),$$

$$F_{00}$$

$$F_{i0}$$

$$F_{ij}$$

$${\sigma }_p^2=diag.{\left(F_{ij}-\frac{F_{i0}{F}_{0j}}{F_{00}}\right)}^{-1}$$

$$0$$

$$0=\sum _{l=2}^{\infty }(2l+1){\tilde{C}}_l^{-1}\frac{\partial }{\partial p_i}{\tilde{C}}_l\left[1-\frac{1}{2l+1}\sum _m\frac{{\bar{T}}^2|a_{lm}^{\mathrm{obs}}{|}^2}{({⟨T⟩}^{\mathrm{obs}}{)}^2{\tilde{C}}_l}\right],$$

$$C_l^{\mathrm{biased}}(p_{\mu }^t)\simeq {\tilde{C}}_l\left(1+\frac{3}{4\pi }{\tilde{C}}_0-\frac{\partial \ln {\tilde{C}}_l}{\partial \ln \bar{T}}{\Theta }_0-\frac{\partial \ln {\tilde{C}}_l}{\partial p_i}\delta p_i\right),$$

$$\delta p_i=-{\left(F_{\mathrm{std}}^{-1}\right)}_{ij}{F}_{j0}{\Theta }_0,$$

Full text

Introduction.— Cosmology has seen enormous development in recent decades (see, e.g., Weinberg et al. (2013) for a review). In particular, the cosmic microwave background (CMB) experiments have greatly improved in recent years with the Wilkinson Microwave Anisotropy Probe (WMAP) and the Planck satellites Spergel et al. (2003); Ade et al. (2014). The primary cosmological parameters are now constrained at the sub-percent level Planck Collaboration et al. (2018a, b), and the angular scale of the acoustic peak is even better constrained by an order-of-magnitude. This level of precision in cosmological parameter estimation demands a matching accuracy in our theoretical predictions.

The background CMB temperature $\bar{T}$ is one of the fundamental cosmological parameters that characterize the evolution of the Universe. In particular, it is tantamount to the photon energy density $\omega_{\gamma}$, and it sets the total radiation density $\omega_{r}$ (hence the epoch $z_{\rm eq}$ of the matter-radiation equality), once the other cosmological parameters such as the matter density $\omega_{m}$ and the neutrino masses $m_{\nu}$ are provided. Despite its significant role in cosmology, the background CMB temperature $\bar{T}$ has rarely been treated as a free cosmological parameter in literature, because of the pioneering work Mather et al. (1994); Fixsen et al. (1996); Fixsen (2009) by the COBE Far Infrared Absolute Spectrometer (FIRAS) in 1990, which provided the precise measurements of the observed CMB temperature $\left\langle T\right\rangle^{\rm obs}$ at our position by averaging the CMB temperature measurements over the sky.

The final release Fixsen et al. (1996) of the COBE FIRAS measurements is $\left\langle T\right\rangle^{\rm obs}=2.728\pm 0.004$ K, and the measurements were later further calibrated in Ref. Fixsen (2009) by using the WMAP differential temperature measurements Hinshaw et al. (2009): $\left\langle T\right\rangle^{\rm obs}=2.7255\pm 5.7\cdot 10^{-4}$ K. This measurement of the CMB temperature with exquisite precision underpins the standard practice, in which the background CMB temperature $\bar{T}$ is set equal to the observed CMB temperature $\left\langle T\right\rangle^{\rm obs}$ without any error associated to this number. Ref. Hamann and Wong (2008) investigated the impact of the measurement error of $\left\langle T\right\rangle^{\rm obs}$ on the other cosmological parameters and found a negligible inflation of their error bars.

In this Letter, we show that this practice is formally incorrect, because it neglects the uncertainty related to cosmic variance Mitsou et al. (2019), i.e. the fact that we can only observe a single light-cone. Instead, $\bar{T}$ should in principle be considered as an extra free cosmological parameter to be varied in the Bayesian analysis. With $\bar{T}\equiv\left\langle T\right\rangle^{\rm obs}$, the standard practice leads to underestimation of the error bars on the cosmological parameters (consistent with the results in Hamann and Wong (2008)), and systematic biases in the cosmological parameter estimation (an effect absent in Hamann and Wong (2008)), even in the era of future CMB experiments with virtually no measurement errors in $\left\langle T\right\rangle^{\rm obs}$. Although the overall impact on parameter estimation is negligible today, it might become relevant in the future.

The cosmological parameter $\bar{T}$.— The background CMB temperature $\bar{T}$ is really one of the other cosmological parameters such as the background matter density $\omega_{m}$ or the (background) Hubble parameter $H_{0}$ that are defined in a homogeneous and isotropic universe and control the evolution of the perturbations in an inhomogeneous universe. The observed CMB temperature $\left\langle T\right\rangle^{\rm obs}$ from the COBE FIRAS is, on the other hand, obtained by averaging the CMB temperature measurements on the sky, and it differs from the background CMB temperature $\bar{T}$ due to the monopole perturbation $\Theta_{0}$. As any other physical quantities, the CMB temperature at a given position $x$ and direction $\hat{n}$ in general includes not only the background $\bar{T}$, but also the perturbation $\Theta(x,\hat{n})$, and the separation of the background and perturbation is made for our theoretical convenience. Therefore, when averaged over the sky at our position $x_{o}$, the observed CMB temperature can be expressed as $\left\langle T\right\rangle^{\rm obs}=\bar{T}(1+\Theta_{0})$, where the monopole perturbation is

[TABLE]

and we suppressed the dependence of $\Theta_{0}$ on the observer position $x_{o}$.

Compared to the other multipole moments $\Theta_{l}$ ($l\geq 1$) in CMB, the monopole is not an observable, as it is absorbed into the observed CMB temperature $\left\langle T\right\rangle^{\rm obs}$ together with the background temperature $\bar{T}$. Despite this peculiarity, the monopole perturbation $\Theta_{0}$ at our position is very unlikely to be zero. The Ergodic theorem states that once the fluctuations are averaged over a sufficiently large volume, the resulting average is equivalent to the ensemble average or the average over many realizations of our Universe. While the ensemble average of the monopole is zero, it is shown in Ref. Mitsou et al. (2019) that the angle average is not quite the ensemble average, as it is obtained only at our own position. This implies that if we were to perform the angle average of the CMB temperature at the Andromeda galaxy, we would obtain $\left\langle T\right\rangle^{\rm obs}$ different from the COBE FIRAS result, due to the fluctuation of the monopole from place to place. Only if we could average the CMB temperature $\left\langle T\right\rangle^{\rm obs}(x)$ over all the possible observer positions, we would be able to replace the average with the ensemble average and obtain the background CMB temperature $\bar{T}$. As this procedure is impossible, the background CMB temperature $\bar{T}$ can never be measured and needs to be treated as a free cosmological parameter as the other cosmological parameters.

As an extra cosmological parameter in the Bayesian analysis, the prior distribution of $\bar{T}$ should have a mean of $\left\langle T\right\rangle^{\rm obs}$ and a standard deviation $\sigma_{\ln\bar{T}}\simeq(\sigma_{\Theta_{0}}^{2}+\sigma_{m}^{2})^{1/2}$, where $\sigma_{m}\sim 2\cdot 10^{-4}$ is the current measurement uncertainty and $\sigma_{\Theta_{0}}\sim 10^{-5}$ is the cosmic variance contribution of the monopole. Since currently $\sigma_{m}\sim 20\sigma_{\Theta_{0}}$, the effect of cosmic variance will be negligible as well. However, the fact that $\sigma_{m}$ is already close to $\sigma_{\Theta_{0}}$ implies that future CMB measurements might cross the threshold. Note that the Planck team did allow $\bar{T}$ to vary in their analysis Planck Collaboration et al. (2016), but by ignoring the COBE FIRAS input at the prior level. The aim of this exercise was to establish how well $\bar{T}$ can be constrained by the anisotropy and galaxy clustering data alone and whether the result would be consistent with the COBE FIRAS measurement of $\left\langle T\right\rangle^{\rm obs}$, under the assumption $\bar{T}\equiv\left\langle T\right\rangle^{\rm obs}$.

CMB observations and theoretical predictions.— In observations, the CMB temperature map as well as the polarization map obtained in the CMB experiments is decomposed with spherical harmonics $Y_{lm}$ as $T^{\rm obs}(\hat{n}):=\sum_{lm}T^{\rm obs}_{lm}Y_{lm}(\hat{n})$, and the angular multipoles $T_{lm}$ are used to construct the observed CMB power spectra $D^{\rm obs}_{l}:=\sum_{m}|T^{\rm obs}_{lm}|^{2}/(2l+1)$ for $l\geq 1$. The angle average of the CMB temperature is equivalent to the monopole $\left\langle T\right\rangle^{\rm obs}\equiv T^{\rm obs}_{00}/\sqrt{4\pi}$. The theoretical predictions are, however, based on the separation of the background and the perturbation around it, so that the CMB temperature is modeled as $T^{\rm obs}(\hat{n}):=\bar{T}(1+\Theta)$ and the angular decomposition of the temperature anisotropies $\Theta(\hat{n}):=\sum_{lm}a_{lm}Y_{lm}(\hat{n})$ yields the angular multipole $a_{lm}$ and their power spectra $C_{l}:=\left\langle|a_{lm}|^{2}\right\rangle$, where the angular multipoles and the power spectra are both dimensionless, as opposed to the dimensionful quantities $T^{\rm obs}_{lm}$ and $D_{l}^{\rm obs}$ in observation.

The conversion between these quantities is trivial in theory: $T_{lm}\equiv\bar{T}a_{lm}$ and $D_{l}\equiv\bar{T}^{2}C_{l}$ for $l\geq 1$, but it is impossible in observation, as the background CMB temperature $\bar{T}$ is unknown. However, this poses no problem, as we can include an additional cosmological parameter $\bar{T}$ in our data analysis and obtain the best-fit value for $\bar{T}$ as the other (unknown) cosmological parameters in a given model. The problems arise because the data analysis is performed by fixing $\bar{T}\equiv\left\langle T\right\rangle^{\rm obs}$ by hand. This procedure results in two problems: 1) the background evolution in our theoretical predictions never matches the correct background in our Universe, unless the monopole at our position happens to be zero, and 2) by using $\left\langle T\right\rangle^{\rm obs}$ instead of $\bar{T}$, the observed temperature and the CMB power spectra are in practice compared to $T^{\rm obs}_{lm}/\left\langle T\right\rangle^{\rm obs}=a_{lm}/(1+\Theta_{0})$ and

[TABLE]

where the monopole of the power spectrum is $C_{0}\simeq 1.7\cdot 10^{-9}$ in our fiducial $\Lambda$CDM model. Though negligible in the Planck data analysis, the point 1) causes systematic errors in the standard data analysis larger than the point 2).

Underestimation of the error bars.— One immediate consequence of the standard practice with $\bar{T}\equiv\left\langle T\right\rangle^{\rm obs}$ is the underestimation of the error bars on the cosmological parameters in a given model, as there exists one less degree of freedom in the parameter estimation than in reality. The true error bars on the cosmological parameters can be estimated by considering the full model with extra cosmological parameter $p_{0}:=\ln\bar{T}$, in addition to the standard model parameters $p_{i}$ ($i=1,\cdots,N$) and by marginalizing over the nuisance parameter $p_{0}$. To estimate the inflation of the error bars, we adopt the Fisher information matrix formalism. For the Gaussian fluctuations on the sky, the Fisher matrix takes the standard form with one critical difference: the observables contain both the background and the perturbation. For CMB, the observables are $T^{\rm obs}_{lm}$ and $D^{\rm obs}_{l}$, and the Fisher matrix is then obtained in Ref. Yoo et al. (2019) as

[TABLE]

where the standard Fisher analysis corresponds to the sub-matrix of the full Fisher matrix ($F^{\rm std}_{ij}\equiv F_{ij}$). The true error bars on the cosmological parameters after marginalizing over $p_{0}$ can be obtained as the diagonal elements of the $N$-$N$ sub-matrix

[TABLE]

of the inverse of the full Fisher information matrix.

For the proof of concept, we apply the Fisher formalism to a CMB experiment like the Planck satellite, where we used the temperature $C_{l}^{\rm TT}$ at $l=2\sim 2500$, the polarization $C_{l}^{\rm EE}$ at $l=2\sim 2000$, and the cross $C_{l}^{\rm TE}$ power spectra at $l=30\sim 2000$ as our CMB observables. The Fisher matrix is computed by accounting for the covariance among the temperature and the polarization observables Zaldarriaga and Seljak (1997); Zaldarriaga et al. (1997). We adopt that the sky coverage is $f_{\rm sky}=0.86$, the detector pixel noise is $\Delta^{2}_{T}=(0.55\mu\rm K~{}deg)^{2}$, and the beam size is $\sigma_{b}=7.22$ arcmin in FWHM for 143 GHz channel. These specifications are taken into consideration in the Fisher matrix by modifying the factor $(2l+1)/2C_{l}^{2}$. Finally, for our fiducial cosmological parameters, we adopted the best-fit $\Lambda$CDM model parameters reported in Table 7 of the Planck 2018 results Planck Collaboration et al. (2018b) (Planck alone). The CMB power spectra are computed by using the CLASS Boltzmann code Blas et al. (2011).

Figure 1 illustrates the underestimation of the true error bars on the cosmological parameters in the standard practice. We consider three cases, in which the observed CMB temperature $\left\langle T\right\rangle^{\rm obs}$ is constrained with different precision: no measurement uncertainty ($\sigma_{m}\equiv 0$; solid), COBE FIRAS measurement uncertainty calibrated with the WMAP measurements (dotted), and original COBE FIRAS measurement uncertainty (dashed). In none of these three cases, we have the precise information about the background CMB temperature $\bar{T}$. However, given the monopole power $C_{0}\simeq 1.7\cdot 10^{-9}$, the 1-$\sigma$ rms fluctuation of the monopole is $\Theta_{0}\equiv a_{00}/\sqrt{4\pi}\sim 1.2\cdot 10^{-5}$, so the background CMB temperature $\bar{T}$ is likely to be within the current measurement uncertainty $6\cdot 10^{-4}$ K from $\left\langle T\right\rangle^{\rm obs}=2.7255$ K.

Under the assumption that the monopole happens to vanish at our position, the standard data analysis underestimates the error bars on the cosmological parameters, for instance, by two percent for the baryon density $\omega_{b}$, when the measurement of $\left\langle T\right\rangle^{\rm obs}$ from COBE FIRAS is calibrated with the WMAP measurements and by tens of percents when the original COBE FIRAS measurement is used. Note that the inflation of error bars in Figure 1 is relative to the error bar in the standard practice. The amplitude $A_{s}$ of the curvature perturbation is equally affected, while the angular size $\theta$ or the spectral index $n_{s}$ are less sensitive. The inflation of the error bars is largely determined by two factors: the uncertainty in $\bar{T}$ (or $C_{0}$ in $F_{00}$) and the correlation $F_{i0}$ of the parameter $p_{i}$ and the temperature $\bar{T}$ variations. $F_{i0}$ is stronger for $\omega_{b}$ and $\omega_{c}$, and this trend is amplified by the correlation $F^{-1}_{\rm std}$ among the model parameters. The error bars in $A_{s}$ is enhanced largely by the parameter correlation. With an order-of-magnitude reduction of the uncertainty in $\left\langle T\right\rangle^{\rm obs}$ in Ref. Fixsen (2009), the inflation of the error bars (dotted) is less than a few percents for the $\Lambda$CDM cosmological parameters. Propagating the errors on $\omega_{b}$, $\omega_{c}$, and $100\theta$, we obtain the inflation of the error on the Hubble parameter $h$: 2%, 0.04%, $10^{-4}\%$ for the three cases. What is important is to note that the error bars are always underestimated (solid lines) in the standard data analysis, even with no measurement uncertainty in $\left\langle T\right\rangle^{\rm obs}$ from future CMB experiments.

Cosmological parameter bias.— By fixing $\bar{T}\equiv\left\langle T\right\rangle^{\rm obs}$, the standard data analysis contains systematic errors in terms of biases in the cosmological parameter estimation. Assuming that the systematic errors are small, the best-fit cosmological parameters $p^{b}_{\mu}$ are characterized by the parameter biases $\delta p_{\mu}$ from the true parameter set $p^{t}_{\mu}$ as $p^{b}_{\mu}:=p^{t}_{\mu}+\delta p_{\mu}$ ($\mu=0,1,\cdots,N$), where in the standard practice $p^{b}_{0}\equiv\ln\left\langle T\right\rangle^{\rm obs}=\ln[\bar{T}(1+\Theta_{0})]\simeq\ln\bar{T}+\Theta_{0}$, so that the parameter bias for $p_{0}=\ln\bar{T}$ is the unknown monopole at our position: $\delta p_{0}\equiv\Theta_{0}$.

The relation between two parameter sets can be obtained by considering that the likelihood ${\cal L}(p_{\mu})$ of the CMB observables is maximized at the best-fit parameters $p^{b}_{\mu}$:

[TABLE]

where the commas represent derivative of the covariance matrix ${\bf C}$ with respect to the parameter $p_{i}$ and the observed data set $d^{\rm obs}$ includes the observed temperature and polarization anisotropies. The covariance matrix ${\bf C}(p_{\mu})$ and the mean $\mu(p_{\mu})$ are the theoretical predictions in a given model, where $\mu=\bar{T}$ for temperature anisotropies and $\mu=0$ for polarization anisotropies. However, due to the assumption $\bar{T}\equiv\left\langle T\right\rangle^{\rm obs}$ in the standard practice, the theoretical predictions for ${\bf C}$ and $\mu$ depend only on the model parameters $p_{i}$, but not on $\bar{T}$, and we used tilde to represent that the theoretical predictions are evaluated at $p^{b}_{\mu}$, not at $p^{t}_{\mu}$.

Using the spherical harmonics decomposition, the condition for the best-fit parameter set is expressed as

[TABLE]

where the power spectra $\tilde{C}_{l}$ account for the covariance among the temperature, the polarization, and their cross power spectra together with the detector noise and beam smoothing Zaldarriaga and Seljak (1997); Zaldarriaga et al. (1997). To make further progress, we take the ensemble average to replace the ratio of $a^{\rm obs}_{lm}$ and $\left\langle T\right\rangle^{\rm obs}$ with $C_{l}^{\rm biased}$ and expand the power spectra around $p^{b}_{\mu}$ as

[TABLE]

where the first correction arises from $C_{l}^{\rm biased}$ and the remaining corrections arise due to the difference between $p^{b}_{\mu}$ and $p^{t}_{\mu}$. Ignoring the small correction due to the first term, the cosmological parameter bias can be neatly expressed as

[TABLE]

and it is in proportion to the amplitude of the unknown monopole at our position, while it is independent of the measurement uncertainty in $\left\langle T\right\rangle^{\rm obs}$, given our assumption $p^{t}_{\mu}\simeq p^{b}_{\mu}$.

Figure 2 shows the bias $\delta p_{i}$ in units of the parameter’s standard deviation $\sigma_{p_{i}}$ in the best-fit cosmological parameters with $\Theta_{0}$ assumed to be at 1-$\sigma$ fluctuation. If the monopole happens to vanish at our position, there would be no bias in the cosmological parameters by using the standard practice. However, if the monopole at our position is non-zero, the standard analysis yields the biases in the best-fit cosmological parameters in proportion to the unknown amplitude of the monopole. For instance, the baryon density parameter $\omega_{b}$ is off by 0.01$\sigma_{\omega_{b}}$ at 1-$\sigma$ fluctuation of $\Theta_{0}$, and this level of bias is readily tolerable today. While the biases in $\omega_{c}$ and $\ln(10^{10}A_{s})$ are of similar magnitude, their error bars are larger, hence the impacts are slightly smaller. The impacts for $100\theta$, $\tau$, and $n_{s}$ are negligible.

Conclusions.— We showed that in principle the background CMB temperature $\bar{T}$ has to be considered as an unknown cosmological parameter, because the observed (angle-average) CMB temperature $\left\langle T\right\rangle^{\rm obs}$ includes the unknown monopole contribution at our position. We investigated the impact of this “new” cosmological parameter $\bar{T}$ on the CMB data analysis. With the current uncertainty in $\left\langle T\right\rangle^{\rm obs}$, the standard data analysis underestimates the error bars on the cosmological parameters by a relative amount of up to 2%, and if the monopole is non-vanishing at our position, the best-fit cosmological parameters in the standard analysis are biased by about 1% of their current standard deviation or 1-$\sigma$ error bar.

We conclude that these systematic errors are negligible in the Planck data analysis, providing a further justification to the standard practice. However, these systematic errors are always present and irreducible in the standard data analysis, so that cosmological measurements might be misinterpreted in future experiments with better precision than the Planck satellite. Of course, these systematic errors can be readily avoided by including one extra cosmological parameter $\bar{T}$.

We thank Antony Lewis, Pavel Motloch, Douglas Scott, David Spergel, Matias Zaldarriaga, and James Zibin for useful discussions. We acknowledge support by the Swiss National Science Foundation. J.Y., E.M., Y.D. are further supported by a Consolidator Grant of the European Research Council (ERC-2015-CoG grant 680886).