Magnitude-redshift Relation Research Articles

An updated measurement of the Hubble constant from near-infrared observations of Type Ia supernovae

We present a measurement of the Hubble constant (H0) using type Ia supernovae (SNe Ia) in the near-infrared (NIR) from the recently updated sample of SNe Ia in nearby galaxies with distances measured via Cepheid period-luminosity relations by the SH0ES project. We collected public near-infrared photometry of up to 19 calibrator SNe Ia and 57 SNe Ia in the Hubble flow (z > 0.01), and directly measured their peak magnitudes in the J- and H-band by Gaussian processes and spline interpolation. Calibrator peak magnitudes together with Cepheid-based distances were used to estimate the average absolute magnitude in each band, while Hubble-flow SNe were used to constrain the zero-point intercept of the magnitude–redshift relation. Our baseline result of H0 is 72.3 ± 1.4 (stat) ±1.4 (syst) km s−1 Mpc−1 in the J-band and 72.3 ± 1.3 (stat) ±1.4 (syst) km s−1 Mpc−1 in the H-band, where the systematic uncertainties include the standard deviation of up to 21 variations of the analysis, the 0.7% distance scale systematic from SH0ES Cepheid anchors, a photometric zero-point systematic, and a cosmic variance systematic. Our final measurement represents a measurement with a precision of 2.8% in both bands. Among all the analysis variants, the largest change in H0 comes from limiting the sample to those SNe from the CSP and CfA programs; they are noteworthy because they are the best calibrated, yielding H0 ∼ 75 km s−1 Mpc−1 in both bands. We explore applying stretch and reddening corrections to standardize SN Ia NIR peak magnitudes, and we demonstrate that they are still useful to reduce the absolute magnitude scatter and, which improves its standardization, at least up to the H-band. Based on our results, in order to improve the precision of the H0 measurement with SNe Ia in the NIR in the future, we would need to increase the number of calibrator SNe Ia, to be able to extend the Hubble–Lemaître diagram to higher redshift, and to include standardization procedures to help reduce the NIR intrinsic scatter.

The quadrupole in the local Hubble parameter: first constraints using Type Ia supernova data and forecasts for future surveys

ABSTRACT The cosmological principle asserts that the Universe looks spatially homogeneous and isotropic on sufficiently large scales. Given its fundamental implications, it is important to empirically test its validity. In this paper, we use the Type Ia supernova (SN Ia) magnitude–redshift relation, from both the Pantheon and joint light-curve analysis compilations, to constrain theoretically motivated anisotropies in the Hubble flow. In particular, we constrain the quadrupole in the effective Hubble parameter and the dipole in the effective deceleration parameter. We find no significant quadrupole term regardless of the redshift frame used. Our results are consistent with the theoretical expectation of a quadrupole moment of a few percent at scales of ∼100 h−1 Mpc. We place an upper limit of an $\sim 10{{\%}}$ quadrupole amplitude relative to the monopole, H0, at these scales. We find that we can detect an $\sim 7{{\%}}$ quadrupole at the 5σ level, for a forecast low-z sample of 1055 SNe Ia. We find the signficance of an exponentially decaying dipole of the deceleration parameter depends on the redshift frame used. In the heliocentric frame, as expected, it is detected at ∼3σ significance. In the cosmic microwave background (CMB) rest frame, we find a marginal ∼2σ dipole, however, after applying peculiar velocity (PV) corrections, the dipole is insignificant. Finally, we find the best-fitting frame of rest relative to the supernovae to differ from that of the CMB at ∼2σ for both compilations, which reduces to <1σ when including PV covariance.

A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km s−1 Mpc−1 Uncertainty from the Hubble Space Telescope and the SH0ES Team

We report observations from the Hubble Space Telescope (HST) of Cepheid variables in the host galaxies of 42 Type Ia supernovae (SNe Ia) used to calibrate the Hubble constant (H 0). These include the complete sample of all suitable SNe Ia discovered in the last four decades at redshift z ≤ 0.01, collected and calibrated from ≥1000 HST orbits, more than doubling the sample whose size limits the precision of the direct determination of H 0. The Cepheids are calibrated geometrically from Gaia EDR3 parallaxes, masers in NGC 4258 (here tripling that sample of Cepheids), and detached eclipsing binaries in the Large Magellanic Cloud. All Cepheids in these anchors and SN Ia hosts were measured with the same instrument (WFC3) and filters (F555W, F814W, F160W) to negate zero-point errors. We present multiple verifications of Cepheid photometry and six tests of background determinations that show Cepheid measurements are accurate in the presence of crowded backgrounds. The SNe Ia in these hosts calibrate the magnitude–redshift relation from the revised Pantheon+ compilation, accounting here for covariance between all SN data and with host properties and SN surveys matched throughout to negate systematics. We decrease the uncertainty in the local determination of H 0 to 1 km s−1 Mpc−1 including systematics. We present results for a comprehensive set of nearly 70 analysis variants to explore the sensitivity of H 0 to selections of anchors, SN surveys, redshift ranges, the treatment of Cepheid dust, metallicity, form of the period–luminosity relation, SN color, peculiar-velocity corrections, sample bifurcations, and simultaneous measurement of the expansion history. Our baseline result from the Cepheid–SN Ia sample is H 0 = 73.04 ± 1.04 km s−1 Mpc−1, which includes systematic uncertainties and lies near the median of all analysis variants. We demonstrate consistency with measures from HST of the TRGB between SN Ia hosts and NGC 4258, and include them simultaneously to yield 72.53 ± 0.99 km s−1 Mpc−1. The inclusion of high-redshift SNe Ia yields H 0 = 73.30 ± 1.04 km s−1 Mpc−1 and q 0 = −0.51 ± 0.024. We find a 5σ difference with the prediction of H 0 from Planck cosmic microwave background observations under ΛCDM, with no indication that the discrepancy arises from measurement uncertainties or analysis variations considered to date. The source of this now long-standing discrepancy between direct and cosmological routes to determining H 0 remains unknown.

Intervening or associated? Machine learning classification of redshifted H i 21-cm absorption

ABSTRACT In a previous paper, we presented the results of applying machine learning to classify whether an H i 21-cm absorption spectrum arises in a source intervening the sightline to a more distant radio source or within the host of the radio source itself. This is usually determined from an optical spectrum giving the source redshift. However, not only will this be impractical for the large number of sources expected to be detected with the Square Kilometre Array, but bright optical sources are the most ultraviolet luminous at high redshift and so bias against the detection of cool, neutral gas. Adding another 44, mostly newly detected absorbers, to the previous sample of 92, we test four different machine learning algorithms, again using the line properties (width, depth, and number of Gaussian fits) as features. Of these algorithms, three gave some improvement over the previous sample, with a logistic regression model giving the best results. This suggests that the inclusion of further training data, as new absorbers are detected, will further increase the prediction accuracy above the current ≈80 per cent. We use the logistic regression model to classify the zabs = 0.42 absorption towards PKS 1657−298 and find this to be associated, which is consistent with a previous study that determined zem ≈ 0.42 from the K-band magnitude–redshift relation.

Constraining neutrino mass and dark energy with peculiar velocities and lensing dispersions of Type Ia supernovae

We show that peculiar velocities of Type Ia supernovae can be used to derive constraints on the sum of neutrino masses, $\Sigma m_{\nu}$, and dark energy equation of state, $w = w_0+w_a(1-a)$, from measurements of the magnitude-redshift relation, complementary to galaxy redshift and weak lensing surveys. Light from a supernova propagates through a perturbed Universe so the luminosity distance is modified from its homogeneous prediction. This modification is proportional to the matter density fluctuation and its time derivative due to gravitational lensing and peculiar velocity respectively. At low redshifts, the peculiar velocity signal dominates while at high redshifts lensing does. We show that using lensing and peculiar velocity of supernovae from the upcoming surveys WFIRST and ZTF, without other observations, we can constrain $\Sigma m_{\nu} \lesssim 0.31$ eV, $\sigma(w_0) \lesssim 0.02$, and ${\sigma(w_a)} \lesssim 0.18$ ($1-\sigma$ CL) in the $\Sigma m_{\nu}$-$w_0$-$w_a$ parameter space, where all the other cosmological parameters are fixed. We find that adding peculiar velocity information from low redshifts shrinks the volume of the parameter ellipsoid in this space by $\sim 33$%. We also allow $\Omega_{\text{CDM}}$ to vary as well as $\Sigma m_{\nu}$, $w_0$ and $w_a$, and demonstrate how these constraints degrade as a consequence.

Fractal universe and cosmic acceleration in a Lemaître–Tolman–Bondi scenario

In this paper we attempt to answer the question: can cosmic acceleration of the universe have a fractal solution? We give an exact solution of a Lemaitre–Tolman–Bondi (LTB) universe based on the assumption that such a smooth metric is able to describe, on average, a fractal distribution of matter. While the LTB model has a center, we speculate that, when the fractal dimension is not very different from the space dimension, this metric applies to any point of the fractal structure when chosen as center so that, on average, there is not any special point or direction. We examine the observed magnitude-redshift relation of type Ia supernovae (SNe Ia), showing that the apparent acceleration of the cosmic expansion can be explained as a consequence of the fractal distribution of matter when the corresponding space-time metric is modeled as a smooth LTB one and if the fractal dimension on scales of a few hundreds Mpc is .

The effect of inhomogeneities on dark energy constraints

Constraints on models of the late time acceleration of the universe assume the cosmological principle of homogeneity and isotropy on large scales. However, small scale inhomogeneities can alter observational and dynamical relations, affecting the inferred cosmological parameters. For precision constraints on the properties of dark energy, it is important to assess the potential systematic effects arising from these inhomogeneities. In this study, we use the Type Ia supernova magnitude-redshift relation to constrain the inhomogeneities as described by the Dyer-Roeder distance relation and the effect they have on the dark energy equation of state (w), together with priors derived from the most recent results of the measurements of the power spectrum of the Cosmic Microwave Background and Baryon Acoustic Oscillations. We find that the parameter describing the inhomogeneities (η) is weakly correlated with w. The best fit values w = −0.933 ± 0.065 and η = 0.61 ± 0.37 are consistent with homogeneity at < 2 σ level. Assuming homogeneity (η =1), we find w = −0.961 ± 0.055, indicating only a small change in w. For a time-dependent dark energy equation of state, w0 = −0.951 ± 0.112 and wa = 0.059 ± 0.418, to be compared with w0 = −0.983 ± 0.127 and wa = 0.07 ± 0.432 in the homogeneous case, which is also a very small change. We do not obtain constraints on the fraction of dark matter in compact objects, fp, at the 95% C.L. with conservative corrections to the distance formalism. Future supernova surveys will improve the constraints on η, and hence, fp, by a factor of ∼10.

Is it no Longer Necessary to Test Cosmologies with Type Ia Supernovae?

We look at the current practice of analyzing the magnitude–redshift relation from the data on Type Ia supernovae. We show that, if the main aim of such analysis were to check the validity of a cosmological model, then the recently advanced arguments do not serve the purpose. Rather, the procedure followed tells us only about the statistical significance of the internal parameters used in the model, whereas the model itself is tacitly assumed to give a good fit to the data. A statistical assessment of the procedure is given and it is argued that given the growing data, the validity of the cosmological model should be checked first rather than the spread of any internal parameters. In passing we also discuss some aspects of the Milne model in the light of the present test.

Narrowing down the possible explanations of cosmic acceleration with geometric probes

Recent re-calibration of the Type Ia supernova (SNe Ia) magnitude-redshift relation combined with cosmic microwave background (CMB) and baryon acoustic oscillation (BAO) data have provided excellent constraints on the standard cosmological model. Here, we examine particular classes of alternative cosmologies, motivated by various physical mechanisms, e.g. scalar fields, modified gravity and phase transitions to test their consistency with observations of SNe Ia and the ratio of the angular diameter distances from the CMB and BAO. Using a model selection criterion for a relative comparison of the models (the Bayes Factor), we find moderate to strong evidence that the data prefer flat ΛCDM over models invoking a thawing behaviour of the quintessence scalar field. However, some exotic models like the growing neutrino mass cosmology and vacuum metamorphosis still present acceptable evidence values. The bimetric gravity model with only the linear interaction term as well as a simplified Galileon model can be ruled out by the combination of SNe Ia and CMB/BAO datasets whereas the model with linear and quadratic interaction terms has a comparable evidence value to standard ΛCDM. Thawing models are found to have significantly poorer evidence compared to flat ΛCDM cosmology under the assumption that the CMB compressed likelihood provides an adequate description for these non-standard cosmologies. We also present estimates for constraints from future data and find that geometric probes from oncoming surveys can put severe limits on non-standard cosmological models.

Geodesics and the magnitude-redshift relation on cosmologically symmetric Finsler spacetimes

We discuss the geodesic motion of both massive test particles, following timelike geodesics, and light, following null geodesics, on Finsler spacetimes with cosmological symmetry. Using adapted coordinates on the tangent bundle of the spacetime manifold, we derive the general form of the geodesic equation. Further, we derive a complete set of constants of motion. As an application of these findings, we derive the magnitude-redshift relation for light propagating on a cosmologically symmetric Finsler background, both for a general Finsler spacetime and for particular examples, such as spacetimes equipped with Bogoslovsky and Randers length measures. Our results allow a confrontation of these geometries with observations of the magnitude and redshift of supernovae.

A 2.4% DETERMINATION OF THE LOCAL VALUE OF THE HUBBLE CONSTANT*

ABSTRACT We use the Wide Field Camera 3 (WFC3) on the Hubble Space Telescope (HST) to reduce the uncertainty in the local value of the Hubble constant from 3.3% to 2.4%. The bulk of this improvement comes from new near-infrared (NIR) observations of Cepheid variables in 11 host galaxies of recent type Ia supernovae (SNe Ia), more than doubling the sample of reliable SNe Ia having a Cepheid-calibrated distance to a total of 19; these in turn leverage the magnitude-redshift relation based on ∼300 SNe Ia at z &lt; 0.15. All 19 hosts as well as the megamaser system NGC 4258 have been observed with WFC3 in the optical and NIR, thus nullifying cross-instrument zeropoint errors in the relative distance estimates from Cepheids. Other noteworthy improvements include a 33% reduction in the systematic uncertainty in the maser distance to NGC 4258, a larger sample of Cepheids in the Large Magellanic Cloud (LMC), a more robust distance to the LMC based on late-type detached eclipsing binaries (DEBs), HST observations of Cepheids in M31, and new HST-based trigonometric parallaxes for Milky Way (MW) Cepheids. We consider four geometric distance calibrations of Cepheids: (i) megamasers in NGC 4258, (ii) 8 DEBs in the LMC, (iii) 15 MW Cepheids with parallaxes measured with HST/FGS, HST/WFC3 spatial scanning and/or Hipparcos, and (iv) 2 DEBs in M31. The Hubble constant from each is 72.25 ± 2.51, 72.04 ± 2.67, 76.18 ± 2.37, and 74.50 ± 3.27 km s−1 Mpc−1, respectively. Our best estimate of H 0 = 73.24 ± 1.74 km s−1 Mpc−1 combines the anchors NGC 4258, MW, and LMC, yielding a 2.4% determination (all quoted uncertainties include fully propagated statistical and systematic components). This value is 3.4σ higher than 66.93 ± 0.62 km s−1 Mpc−1 predicted by ΛCDM with 3 neutrino flavors having a mass of 0.06 eV and the new Planck data, but the discrepancy reduces to 2.1σ relative to the prediction of 69.3 ± 0.7 km s−1 Mpc−1 based on the comparably precise combination of WMAP+ACT+SPT+BAO observations, suggesting that systematic uncertainties in CMB radiation measurements may play a role in the tension. If we take the conflict between Planck high-redshift measurements and our local determination of H 0 at face value, one plausible explanation could involve an additional source of dark radiation in the early universe in the range of ΔN eff ≈ 0.4–1. We anticipate further significant improvements in H 0 from upcoming parallax measurements of long-period MW Cepheids.

Supernova constraints on higher-dimensional cosmology with a phantom field

We use observational data on the magnitude-redshift relation for Type Ia supernovae (SNeIa) together with constraints on the ages of the oldest stars to rule out a higher-dimensional extension of General Relativity with a negative kinetic energy scalar field. This theory is of considerable physical interest because it produces accelerated expansion at both early and late times with a single new field, as in quintessential inflation scenarios. It is also of mathematical interest because it is characterized by an analytic expression for the macroscopic scale factor [Formula: see text]. We show that cosmological solutions of this theory can be usefully parametrized by a single quantity, the lookback time [Formula: see text] corresponding to the transition from deceleration to acceleration. Supernovae data from the recently released Supernova Cosmology Project (SCP) Union 2.1 compilation single out a narrow range of values for [Formula: see text]. In the context of the theory, however, these same values of [Formula: see text] imply that the universe is much older than the oldest observed stars.

PROBING THE ISOTROPY OF COSMIC ACCELERATION TRACED BY TYPE Ia SUPERNOVAE

We present a method to test the isotropy of the magnitude-redshift relation of Type Ia Supernovae (SNe Ia) and single out the most discrepant direction (in terms of the signal-to-noise ratio) with respect to the all-sky data. Our technique accounts for possible directional variations of the corrections for SNe Ia and yields all-sky maps of the best-fit cosmological parameters with arbitrary angular resolution. To show its potential, we apply our method to the recent Union2.1 compilation, building maps with three different angular resolutions. We use a Monte Carlo method to estimate the statistical significance with which we could reject the null hypothesis that the magnitude-redshift relation is isotropic based on the properties of the observed most discrepant directions. We find that, based on pure signal-to-noise arguments, the null hypothesis cannot be rejected at any meaningful confidence level. However, if we also consider that the strongest deviations in the Union2.1 sample closely align with the dipole temperature anisotropy of the cosmic microwave background, we find that the null hypothesis should be rejected at the $95-99$ per cent confidence level, slightly depending on the angular resolution of the study. If this result is not due to a statistical fluke, it might either indicate that the SN data have not been cleaned from all possible systematics or even point towards new physics. We finally discuss future perspectives in the field for achieving larger and more uniform data sets that will vastly improve the quality of the results and optimally exploit our method.

Milestones of general relativity: Hubble’s law (1929) and the expansion of the universe

Hubble’s announcement of the magnitude–redshift relation (Hubble 1929 Proc. Natl. Acad. Sci. USA 15 168–73) brought about a major change in our understanding of the Universe. After tracing the pre-history of Hubble’s work, and the hiatus in our understanding which his underestimate of distances led to, this review focuses on the development and success of our understanding of the expanding Universe up to the present day, and the part that general relativity plays in that success.

Sound speed of scalar field dark energy: Weak effects and large uncertainties

The possibility of reconstruction of Lagrangian for the scalar field dark energy with constant effective sound speed $c_s$ is analyzed. It is found that such reconstruction can be made with accuracy up to an arbitrary constant. The value of $c_s$ is estimated together with other dark energy parameters ($\Omega_{de}$, $w_0$, $c_a^2$) and main cosmological ones on the basis of data including Planck-2013 results on CMB anisotropy, BAO distance ratios from recent galaxy surveys, galaxy power spectrum from WiggleZ, magnitude-redshift relations for distant SNe Ia from SNLS3 and Union2.1 compilations, the HST determination of the Hubble constant. It is shown that no value of $c_s$ from the range [0,1] is preferred by the used data because of very weak influence of dark energy perturbations on the large scale structure formation and CMB temperature fluctuations.

The magnitude-redshift relation in a realistic inhomogeneous universe

The light rays from a source are subject to a local inhomogeneous geometry generated by inhomogeneous matter distribution as well as the existence of collapsed objects. In this paper we investigate the effect of inhomogeneities and the existence of collapsed objects on the propagation of light rays and evaluate changes in the magnitude-redshift relation from the standard relationship found in a homogeneous FRW universe. We give the expression of the correlation function and the variance for the perturbation of apparent magnitude, and calculate it numerically by using the non-linear matter power spectrum. We use the lognormal probability distribution function for the density contrast and spherical collapse model to truncate the power spectrum in order to estimate the blocking effect by collapsed objects. We find that the uncertainties in Ωm is ∼ 0.02, and that of w is ∼ 0.04 . We also discuss a possible method to extract these effects from real data which contains intrinsic ambiguities associated with the absolute magnitude.

Constraining the dynamical dark energy parameters: Planck-2013 vs WMAP9

We determine the best-fit values and confidence limits for dynamical dark energy parameters together with other cosmological parameters on the basis of different datasets which include WMAP9 or Planck-2013 results on CMB anisotropy, BAO distance ratios from recent galaxy surveys, magnitude-redshift relations for distant SNe Ia from SNLS3 and Union2.1 samples and the HST determination of the Hubble constant. We use a Markov Chain Monte Carlo routine to map out the likelihood in the multi-dimensional parameter space. We show that the most precise determination of cosmological parameters with the narrowest confidence limits is obtained for the Planck+HST+BAO+SNLS3 dataset. The best-fit values and 2σ confidence limits for cosmological parameters in this case are Ωde = 0.718±0.022, w0 = −1.15+0.14−0.16, ca2 = −1.15+0.02−0.46, Ωbh2 = 0.0220±0.0005, Ωcdmh2 = 0.121±0.004, h = 0.713±0.027, ns = 0.958+0.014−0.010, As = (2.215+0.093−0.101)⋅10−9, τrei = 0.093+0.022−0.028. For this dataset, the ΛCDM model is just outside the 2σ confidence region, while for the dataset WMAP9+HST+BAO+SNLS3 the ΛCDM model is only 1σ away from the best fit. The tension in the determination of some cosmological parameters on the basis of two CMB datasets WMAP9 and Planck-2013 is highlighted.

Optical galaxy clusters in the Deep Lens Survey

We present the first sample of 882 optically selected galaxy clusters in the Deep Lens Survey (DLS), selected with the Bayesian Cluster Finder. We create mock DLS data to assess completeness and purity rates, and find that both are at least $70\%$ within 0.1$\le z \le$ 1.2 for clusters with $M_{200}\ge 1.2\times 10^{14}M_{\odot}$. We verified the integrity of the sample by performing several comparisons with other optical, weak lensing, X-ray and spectroscopic surveys which overlap the DLS footprint: the estimated redshifts are consistent with the spectroscopic redshifts of known clusters (for $z>0.25$ where saturation in the DLS is not an issue); our richness estimates in combination with a previously calibrated richness-mass relation yields individual cluster mass estimates consistent with available SHeLS dynamical mass estimates; synthetic mass maps made from the optical mass estimates are correlated ($>3\sigma$ significance) with the weak lensing mass maps; and the mass function thus derived is consistent with theoretical predictions for the CDM scenario. With the verified sample we investigated correlations between the brightest cluster galaxies (BCG) properties and the host cluster properties within a broader range in redshift (0.25 $\le z \le$ 0.8) and mass ($\ge2.4\times 10^{14}M_{\odot}$) than in previous work. We find that the slope of the BCG magnitude-redshift relation throughout this redshift range is consistent with that found at lower redshifts. This result supports an extrapolation to higher redshift of passive evolution of the BCG within the hierarchical scenario.

A possible resolution of tension between Planck and Type Ia supernova observations

There is an apparent tension between cosmological parameters obtained from Planck cosmic microwave background radiation observations and that derived from the observed magnitude-redshift relation for the type Ia supernova (SNe Ia). Here, we show that the tension can be alleviated, if we first calibrate, with the help of the distance-duality relation, the light-curve fitting parameters in the distance estimation in SNe Ia observations with the angular diameter distance data of the galaxy clusters and then re-estimate the distances for the SNe Ia with the corrected fitting parameters. This was used to explore their cosmological implications in the context of the spatially flat cosmology. We find a higher value for the matter density parameter, Ωm, as compared to that from the original SNLS3, which is in agreement with Planck observations at 68.3% confidence. Therefore, the tension between Planck measurements and SNe Ia observations regarding Ωm can be effectively alleviated without invoking new physics or resorting to extensions for the standard concordance model. Moreover, with the absolute magnitude of a fiducial SNe Ia, M, determined first, we obtained a constraint on the Hubble constant with SNLS3 alone, which is also consistent with Planck.

Combined constraints on intergalactic dust from quasar colours and the soft X-ray background

Unless properly corrected for, the existence of intergalactic dust will introduce a redshift-dependent magnitude offset to standard candle sources. This would lead to overestimated luminosity distances compared to a dust-free universe and bias the cosmological parameter estimation as derived from, e.g., Type Ia supernova observations. In this paper, we model the optical extinction and X-ray scattering properties of intergalactic dust grains to constrain the intergalactic opacity using a combined analysis of observed quasar colours and the soft X-ray background. Quasar colours effectively constrain the amount of intergalactic dust grains smaller than m, to the point where we expect the corresponding systematic error in the Type Ia supernova magnitude–redshift relation to be sub-dominant. Soft X-ray background observations are helpful in improving the constraints on very large dust grains for which the amount of optical reddening is very small and therefore is more difficult to correct for. Our current upper limit corresponds to ∼0.25 mag dimming at optical wavelengths for a source at redshift z = 1, which is too small to alleviate the need for dark energy but large in terms of relative error. However, we expect it to be possible to lower this bound considerably with an improved understanding of the possible sources of the X-ray background, in combination with observations of compact X-ray sources such as active galactic nuclei.