# Cosmological tests with strong gravitational lenses using Gaussian processes

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## Abstract

Strong gravitational lenses provide source/lens distance ratios \({\mathcal {D}}_{\mathrm{obs}}\) useful in cosmological tests. Previously, a catalog of 69 such systems was used in a one-on-one comparison between the standard model, \(\varLambda \)CDM, and the \(R_{\mathrm{h}}=ct\) universe, which has thus far been favored by the application of model selection tools to many other kinds of data. But in that work, the use of model parametric fits to the observations could not easily distinguish between these two cosmologies, in part due to the limited measurement precision. Here, we instead use recently developed methods based on Gaussian Processes (GP), in which \({\mathcal {D}}_{\mathrm{obs}}\) may be reconstructed directly from the data without assuming any parametric form. This approach not only smooths out the reconstructed function representing the data, but also reduces the size of the \(1\sigma \) confidence regions, thereby providing greater power to discern between different models. With the current sample size, we show that analyzing strong lenses with a GP approach can definitely improve the model comparisons, producing probability differences in the range \(\sim \) 10–30%. These results are still marginal, however, given the relatively small sample. Nonetheless, we conclude that the probability of \(R_{\mathrm{h}}=ct\) being the correct cosmology is somewhat higher than that of \(\varLambda \)CDM, with a degree of significance that grows with the number of sources in the subsamples we consider. Future surveys will significantly grow the catalog of strong lenses and will therefore benefit considerably from the GP method we describe here. In addition, we point out that if the \(R_{\mathrm{h}}=ct\) universe is eventually shown to be the correct cosmology, the lack of free parameters in the study of strong lenses should provide a remarkably powerful tool for uncovering the mass structure in lensing galaxies.

## 1 Introduction

The degree to which light from high-redshift quasars is deflected by intervening galaxies can be calculated precisely if one has enough information concerning the distribution of mass within the gravitational lens [1, 2]. Depending on the mass of the galaxy, and the alignment between source, lens, and observer, gravitational lenses may be classified either as macro (with sub-classes of strong and weak lensing) or micro lensing systems. Strong lensing occurs when the source, lens, and observer are sufficiently well aligned that the deflection of light forms an Einstein ring. Using the angle of deflection, one may derive the radius of this ring, from which one may then also compute the angular diameter distance to the lens. This distance, however, is model dependent. Hence, together with the measured redshift of the source, this angular diameter distance may be used to discriminate between various cosmological models (see, e.g., Ref. [3, 4, 5, 6, 7]).

In this paper, we use a recent compilation of 118 [8] plus 40 [9] strong lensing systems, with good spectroscopic measurements of the central velocity dispersion based on the Sloan Lens ACS (*SLACS*) Survey [4, 10, 11], and the Lenses Structure and Dynamics (*LSD *) Survey (see, e.g., refs. [12, 13]), to conduct a comparative study between \(\varLambda \)CDM [14, 15] and another Friedmann–Robertson–Walker (FRW) cosmology known as the \(R_{\mathrm{h}}=ct\) universe [16, 17]. Over the past decade, such comparative tests between this alternative model and \(\varLambda \)CDM have been carried out using a wide assortment of data, most of them favouring the former over the latter (for a summary of these tests, see Table 1 in ref. [18]). These studies have included high *z*-quasars [19], gamma-ray bursts [15], Type Ia SNe [21, 22], and cosmic chronometers [23]. The \(R_{\mathrm{h}}=ct\) model is characterized by a total equation of state \(p=-\rho /3\), in terms of the total pressure *p* and density \(\rho \) in the cosmic fluid.

The results of these comparative tests are not yet universally accepted, however, and several counterclaims have been made in recent years. One may loosely group these into four general categories: (1) that the gravitational radius (and therefore also the Hubble radius) \(R_{\mathrm{h}}\) is not really physically meaningful [24, 25, 26]; (2) that the zero active mass condition \(\rho +3p=0\) at the basis of the \(R_{\mathrm{h}}=ct\) cosmology is inconsistent with the actual constitutents in the cosmic fluid [27]; (3) that the *H*(*z*) data favour \(\varLambda \)CDM over \(R_{\mathrm{h}}=ct\) [25, 28]; and (4) that Type Ia SNe also favour the concordance model over \(R_{\mathrm{h}}=ct\) [25, 28, 29]. These works, and papers published in response to them [17, 23, 30, 31, 32, 33], have generated an important discussion concerning the viability of \(R_{\mathrm{h}}=ct\) that we aim to continue here. In Sect. 7 below, we will discuss at greater length the need to use truly model-independent data in these tests, basing their analysis on sound statistical practices. Such due diligence is of utmost importance in any serious attempt to compare different cosmologies in an unbiased fashion.

The test most directly relevant to the work reported here was carried out using strong lenses by Ref. [7], who based their comparison on parametric fits from the models themselves, and concluded that both cosmologies account for the data rather well. The precision of the measurements used in that application, however, was not good enough to favour either model over the other. In this paper, we revisit that sample of strong lensing systems and use an entirely different approach for the comparison, based on Gaussian Processes (GP) to reconstruct the function representing the data non-parametrically. In so doing, the angular diameter distance to the lensing galaxies is determined without pre-assuming any model, providing a better comparison of the competing cosmologies using a functional area minimization statistic described in Sect. 5. An obvious benefit of this approach is that a reconstructed function representing the data may be found regardless of whether or not any of the models being tested is actually the correct cosmology.

In Sect. 2 of this paper, we describe the lensing equation used in cosmological tests, and we then describe the data used with this application in Sect. 3. The Gaussian processes and the cosmological models being tested here are summarized in Sect. 4. The area minimization statistic is introduced in Sect. 5, and we explain how this is used to obtain the model probabilities. We end with our conclusions in Sect. 6.

## 2 Theory of lensing

## 3 Data

Strong gravitational lensing systems with \(0.45<z_s<0.475\)

Galaxy | \(z_l\) | \(\theta _{\mathrm{E}}\) (arc s) | \(\sigma _0\) (\(\text {km s}^{-1}\)) | \({\mathcal {D}}_{\mathrm{obs}}\,f_{SIS}=1.010\) | \(\sigma _{\mathcal {D}}\) | \({\mathcal {D}}_{R_{\mathrm{h}}=ct}\) | \({\mathcal {D}}_{\varLambda \mathrm{CDM}}\) | Refs.\(^{\text {a}}\) |
---|---|---|---|---|---|---|---|---|

SDSS J1134 + 6027 | 0.1528 | 1.10 | \(239\pm 12\) | 0.6689 | 0.735 | 0.634 | 0.652 | 10 |

SDSS J1403 + 0006 | 0.1888 | 0.83 | \(213\pm 17\) | 0.635 | 0.069 | 0.553 | 0.573 | 10 |

SDSS J2300 + 0022 | 0.2285 | 1.25 | \(305\pm 19\) | 0.446 | 0.0512 | 0.460 | 0.479 | 1–9 |

SDSS J0956 + 5100 | 0.2405 | 1.32 | \(318\pm 17\) | 0.4536 | 0.0498 | 0.441 | 0.459 | 1–9 |

SDSS J0935 − 0003 | 0.3475 | 0.87 | \(396\pm 35\) | 0.192 | 0.0211 | 0.222 | 0.234 | 10,11 |

Strong gravitational lensing systems with \(0.46<z_s<0.485\)

Galaxy | \(z_l\) | \(\theta _{\mathrm{E}}\) (arc s) | \(\sigma _0 (\text {km s}^{-1})\) | \({\mathcal {D}}_{\mathrm{obs}}\,f_{SIS}=1.010\) | \(\sigma _{\mathcal {D}}\) | \({\mathcal {D}}_{R_{\mathrm{h}}=ct}\) | \({\mathcal {D}}_{\varLambda \mathrm{CDM}}\) | Refs.\(^{\text {a}}\) |
---|---|---|---|---|---|---|---|---|

SDSS J1134 + 6027 | 0.1528 | 1.10 | \(239\pm 12\) | 0.6689 | 0.735 | 0.634 | 0.652 | 10 |

SDSS J1403 + 0006 | 0.1888 | 0.83 | \(213\pm 17\) | 0.635 | 0.069 | 0.553 | 0.573 | 10 |

SDSS J1402 + 6321 | 0.2046 | 1.39 | \(290\pm 16\) | 0.5743 | 0.063 | 0.526 | 0.546 | 1–9 |

SDSS J1205 + 4910 | 0.2150 | 1.22 | \(281\pm 14\) | 0.5368 | 0.059 | 0.504 | 0.524 | 10 |

SDSS J2300 + 0022 | 0.2285 | 1.25 | \(305\pm 19\) | 0.446 | 0.0512 | 0.460 | 0.479 | 1–9 |

SDSS J0956 + 5100 | 0.2405 | 1.32 | \(318\pm 17\) | 0.4536 | 0.0498 | 0.441 | 0.459 | 1–9 |

SDSS J0935 − 0003 | 0.3475 | 0.87 | \(396\pm 35\) | 0.192 | 0.0211 | 0.222 | 0.234 | 10,11 |

Strong gravitational lensing systems with \(0.5<z_s<0.525\)

Galaxy | \(z_l\) | \(\theta _{\mathrm{E}}\) (arc s) | \(\sigma _0\,\text {(km s}^{-1})\) | \({\mathcal {D}}_{\mathrm{obs}}\,f_{SIS}=1.010\) | \(\sigma _{\mathcal {D}}\) | \({\mathcal {D}}_{R_{\mathrm{h}}=ct}\) | \({\mathcal {D}}_{\varLambda \mathrm{CDM}}\) | Refs.\(^{\text {a}}\) |
---|---|---|---|---|---|---|---|---|

SDSS J1451 − 0239 | 0.1254 | 1.04 | \(223\pm 14\) | 0.726 | 0.079 | 0.718 | 0.735 | 10,11 |

SDSS J2303 + 1422 | 0.1553 | 1.64 | \(271\pm 16\) | 0.775 | 0.0852 | 0.654 | 0.673 | 1–9 |

SDSS J1627 − 0053 | 0.2076 | 1.21 | \(295\pm 13\) | 0.482 | 0.052 | 0.552 | 0.573 | 1–9 |

SDSS J1142 + 1001 | 0.2218 | 0.98 | \(221\pm 22\) | 0.697 | 0.0766 | 0.509 | 0.529 | 10,11 |

SDSS J0109 + 1500 | 0.2939 | 0.69 | \(251\pm 19\) | 0.3807 | 0.0418 | 0.389 | 0.407 | 10 |

SDSS J0216 − 0813 | 0.3317 | 1.15 | \(349\pm 24\) | 0.3287 | 0.0361 | 0.320 | 0.336 | 1–9 |

## 4 Gaussian processes and model comparisons

*f*(

*x*) at

*x*using GP creates a Gaussian random variable with mean \(\mu (x)\) and variance \(\sigma (x)\). The function reconstructed at

*x*using GP, however, is not independent of that reconstructed at \(\tilde{x}=(x+dx)\), these being related by a covariance function \(k(x,\tilde{x})\). Although one can use many possible forms of

*k*, we use one that depends on the distance between

*x*and \(\tilde{x}\), i.e., the squared exponential covariance function defined as

*y*-direction and \(\varDelta \) represents a distance over which a significant change in the

*x*-direction occurs. Overall, these two hyperparameters characterize the smoothness of the function

*k*, and are trained on the data using a maximum likelihood procedure, which leads to the reconstructed \({\mathcal {D}}_{\mathrm{obs}}(z_l,\langle z_s\rangle )\) function for each source redshift shell centered on \(z_s\). For this paper, we have found that these hyperparameter values are 0.144 and 0.661, respectively.

Strong gravitational lensing systems with \(0.51<z_s<0.535\)

Galaxy | \(z_l\) | \(\theta _{\mathrm{E}}\) (arc s) | \(\sigma _0\, (\text {km s}^{-1})\) | \({\mathcal {D}}_{\mathrm{obs}}\,f_{SIS}=1.010\) | \(\sigma _{\mathcal {D}}\) | \({\mathcal {D}}_{R_{\mathrm{h}}=ct}\) | \({\mathcal {D}}_{\varLambda \mathrm{CDM}}\) | Refs.\(^{\text {a}}\) |
---|---|---|---|---|---|---|---|---|

SDSS J2321 − 0939 | 0.0819 | 1.57 | \(245\pm 70\) | 0.9082 | 0.0999 | 0.816 | 0.829 | 1–9 |

SDSS J1451 − 0239 | 0.1254 | 1.04 | \(223\pm 14\) | 0.726 | 0.079 | 0.718 | 0.735 | 10,11 |

SDSS J0959 + 0410 | 0.1260 | 1.00 | \(229\pm 13\) | 0.6616 | 0.07277 | 0.723 | 0.740 | 1–9 |

SDSS J1538 − 5817 | 0.1428 | 1.00 | \(189\pm 12\) | 0.9717 | 0.106 | 0.687 | 0.705 | 10,11 |

SDSS J2303 + 1422 | 0.1553 | 1.64 | \(271\pm 16\) | 0.775 | 0.0852 | 0.654 | 0.673 | 1–9 |

SDSS J1627 − 0053 | 0.2076 | 1.21 | \(295\pm 13\) | 0.482 | 0.052 | 0.552 | 0.573 | 1–9 |

SDSS J0959 + 4416 | 0.2369 | 0.96 | \(244\pm 19\) | 0.5597 | 0.0615 | 0.501 | 0.521 | 10 |

SDSS J0109 + 1500 | 0.2939 | 0.69 | \(251\pm 19\) | 0.3807 | 0.0418 | 0.389 | 0.407 | 10 |

SDSS J0216 − 0813 | 0.3317 | 1.15 | \(349\pm 24\) | 0.3287 | 0.0361 | 0.320 | 0.336 | 1–9 |

Strong gravitational lensing systems with \(0.52<z_s<0.545\)

Galaxy | \(z_l\) | \(\theta _{\mathrm{E}}\) (arc s) | \(\sigma _0 \,(\text {km s}^{-1})\) | \({\mathcal {D}}_{\mathrm{obs}}\,f_{SIS}=1.010\) | \(\sigma _{\mathcal {D}}\) | \({\mathcal {D}}_{R_{\mathrm{h}}=ct}\) | \({\mathcal {D}}_{\varLambda \mathrm{CDM}}\) | Refs.\(^{\text {a}}\) |
---|---|---|---|---|---|---|---|---|

SDSS J1420 + 6019 | 0.0629 | 1.04 | \(206\pm 5\) | 0.851 | 0.0936 | 0.858 | 0.869 | 1–9 |

SDSS J2321 − 0939 | 0.0819 | 1.57 | \(245\pm 70\) | 0.9082 | 0.0999 | 0.816 | 0.829 | 1–9 |

SDSS J1451 − 0239 | 0.1254 | 1.04 | \(223\pm 14\) | 0.726 | 0.079 | 0.718 | 0.735 | 10, 11 |

SDSS J0959 + 0410 | 0.1260 | 1.00 | \(229\pm 13\) | 0.6616 | 0.07277 | 0.723 | 0.740 | 1–9 |

SDSS J1538 − 5817 | 0.1428 | 1.00 | \(189\pm 12\) | 0.9717 | 0.106 | 0.687 | 0.705 | 10, 11 |

SDSS J1627 − 0053 | 0.2076 | 1.21 | \(295\pm 13\) | 0.482 | 0.052 | 0.552 | 0.573 | 1–9 |

SDSS J0959 + 4416 | 0.2369 | 0.96 | \(244\pm 19\) | 0.5597 | 0.0615 | 0.501 | 0.521 | 10 |

SDSS J0109 + 1500 | 0.2939 | 0.69 | \(251\pm 19\) | 0.3807 | 0.0418 | 0.389 | 0.407 | 10 |

SDSS J0216 − 0813 | 0.3317 | 1.15 | \(349\pm 24\) | 0.3287 | 0.0361 | 0.320 | 0.336 | 1–9 |

*K*is the covariance matrix for the original dataset. Note that the dispersion at point \(x_i\) will be less than \(\sigma _{{\mathcal {D}}i}\) when \(K_*K^{-1}K_*^T>\sigma _f\), i.e., when for that point of estimation there is a large correlation between the data. From Eq. (9) it is clear that the correlation between any two points

*x*and \(\tilde{x}\) will be large only when \(x-\tilde{x}<\sqrt{2}\varDelta \). This condition, however, is satisfied most frequently for the strong lenses used in our study, which results in GP estimated \(1\sigma \) confidence regions that are smaller than the errors in the original data. We refer the reader to Ref. [51] for further details.

*H*(

*z*) from Eq. (11), one gets

*p*are the total energy density and pressure, respectively. With this additional constraint, the \(R_\mathrm{h}=ct\) universe always expands at a constant rate, which depends on only one parameter – the Hubble constant \(H_0\). Using the Friedmann equation with zero active mass, we find that

## 5 The area minimization statistic

*r*is a Gaussian random variable with zero mean and a variance of 1. Next, these \({\mathcal {D}}_i(z_l,\langle z_s\rangle )\) are used together with the errors \(\sigma _{{\mathcal {D}}_i}\) to reconstruct the function \({\mathcal {D}}_{\mathrm{mock}}(z_l,\langle z_s\rangle )\) corresponding to each mock sample, and finally we calculate the weighted absolute area difference between \({\mathcal {D}}_{\mathrm{mock}}(z_l,\langle z_s\rangle )\) and the GP reconstructed function of the actual data according to

*DA*, and from it construct the cumulative probability distribution. In Fig. 1 we show these quantities for the illustrative source shell \(0.50<z_s<0.525\) (the frequency is shown in the top panel, and the cumulative probability distribution is on the bottom). This procedure generates a 1-to-1 mapping between the value of

*DA*and the frequency with which it arises. With the additional assumption that curves with a smaller

*DA*are a better match to \({\mathcal {D}}_{\mathrm{obs}}\), one can then use the cumulative distribution to estimate the probability that the difference between a model’s prediction and the reconstructed curve is merely due to Gaussian randomness. When comparing a model’s prediction to the data, we therefore calculate its

*DA*and use our 1-to-1 mapping to determine the probability that its inconsistency with the data is just due to variance, rather than the model being wrong. These are the probabilities we then compare to determine which model is more likely to be correct. This basic concept is common to many kinds of statistical approaches, though none of the existing ones can be used when comparing two continuous curves, as we have here.

The reconstructed curves for our five subsamples are shown in the left-hand panels of Fig. 2. These correspond to the five source redshift shells in Tables 1, 2, 3, 4 and 5. The corresponding cumulative probability distributions are plotted in the right-hand panels, which also locate the *DA* values for \(R_{\mathrm{h}}=ct\) (yellow) and \(\varLambda \)CDM (red). The probabilities associated with these differential areas are summarized in Table 6. Along with the reconstructed functions, the left-hand panels also show the corresponding \(1\sigma \) (dark) and \(2\sigma \) (light) confidence regions provided by the GP, and the theoretical predictions in \(\varLambda \)CDM (dashed) and \(R_{\mathrm{h}}=ct\) (dotted). As we highlighted earlier, the functions \({\mathcal {D}}_{\mathrm{obs}}(z_l,\langle z_s\rangle )\) have been reconstructed without pre-assuming any parametric form, so in principle they represent the actual variation of \({\mathcal {D}}\) with redshift, regardless of whether or not either of the two models being tested here is the correct cosmology.

## 6 Conclusions

Model comparison using strong gravitational lenses with Gaussian processes

This is reflected in the probabilities quoted in Table 6 for the two models we have examined here. Unlike previous model comparisons based on the use of parametric fits to the strong-lensing data, we now find that \(R_{\mathrm{h}}=ct\) is favoured over \(\varLambda \)CDM with consistently higher likelihoods in all five source redshift shells we have assembled for this work. Though these statistics are still quite limited, it is nonetheless telling that the differentiation between models improves as the number of sources within each shell increases. Also, at least for \(R_{\mathrm{h}}=ct\), the probability of its predictions matching the GP reconstructed functions generally increases as the size of the lens sample grows. The outcome of this work underscores the importance of using unbiased data and sound statistical methods when comparing different cosmological models. As a counterexample, consider the use of *H*(*z*) measurements based on BAO observations instead of cosmic chronometers [28], constituting an unwitting use of model-dependent measurements to test competing models. Such an approach ignores the significant limitations in all but the three most recent BAO measurements [58, 59] for this type of work. Previous applications of the galaxy two-point correlation function to measure the BAO scale were contaminated with redshift distortions associated with internal gravitational effects [59]. To illustrate the significance of these limitations, and the impact of the biased BAO measurements of *H*(*z*), note how the model favoured by the data switches from \(\varLambda \)CDM to \(R_{\mathrm{h}}=ct\) when only the unbiased measurements are used [61].

A second counterexample is provided by the merger of disparate sub-samples of Type Ia SNe to improve the statistical analysis. We have already published an in-depth explanation of the perils associated with the blending of data with different systematics for the purpose of model selection [62], but let us nonetheless consider a brief synopsis here. The Union2.1 catalog [63, 64] includes \(\approx 580\) SN detections, though each sub-sample has its own systematic and intrinsic uncertainties. The conventional approach minimizes an overall \(\chi ^2\), while each sub-sample is assigned an intrinsic dispersion to ensure that \(\chi ^2_{\mathrm{dof}}=1\) [28, 29]. Instead, the statistically correct approach would estimate the unknown intrinsic dispersions simultaneously with all other parameters [62, 65]. Quite tellingly, the outcome of the model selection is reversed when one switches from the improper statistical approach to the correct one. To emphasize how critical this reversal is in the case of \(\varLambda \)CDM, one simply needs to compare the outcome of using a merged super-sample with that produced with a large, single homogeneous sample, such as the Supernova Legacy Survey Sample [66].

Within this context, we highlight the fact that the features of the GP reconstruction approach in the study of strong lenses are promising because, in spite of the fact that the use of these systems to measure cosmological parameters has been with us for over a decade (see, e.g., Refs. [4, 5, 6, 67]), the results of this effort have thus far been less precise than those of other kinds of observations. For the large part, these earlier studies were based on the use of parametric fits to the data, but it is quite evident (e.g., from Figs. 1 and 2 in Ref. [7]) that the scatter in \({\mathcal {D}}_{\mathrm{obs}}\) about the theoretical curves generally increases significantly as \(D_A(z_l,z_s) \rightarrow D_A(0,z_s)\). That is, measuring \({\mathcal {D}}\) incurs a progressively greater error as the distance to the gravitational lens becomes a smaller fraction of the distance to the quasar source. This has to do with the fact that \(\theta _{\mathrm{E}}\) changes less for large values of \(z_s/z_l\) so, for a fixed error in the Einstein angle, the measurement of \({\mathcal {D}}_{\mathrm{obs}}\) becomes less precise. As we have demonstrated in this paper, the analysis of strong lensing systems based on a GP reconstruction of \({\mathcal {D}}_{\mathrm{obs}}\) improves our ability to distinguish between different models, albeit by a modest amount given the current sample.

Upcoming survey projects, such as the Dark Energy Survey (DES; [68]), the Large Synoptic Survey Telescope (LSST; [69]), the Joint Dark Energy Mission (JDEM; [70]), and the Square Kilometer Array (SKA; e.g., Ref. [71]), are expected to greatly grow the size of the lens sample. The ability of GP reconstruction methods to differentiate between models will increase in tandem with this growth. Several sources of uncertainty still remain, however, including the actual mass distribution within the lens. And since such errors appear to be more restricting for lens systems with large values of \(z_s/z_l\), a priority for future work should be the identification of strong lenses with small angular diameter distances between the source and lens relative to the distance between the lens and observer.

## Notes

### Acknowledgements

FM is grateful to the Instituto de Astrofísica de Canarias in Tenerife and to Purple Mountain Observatory in Nanjing, China for their hospitality while part of this research was carried out. FM is also grateful for partial support to the Chinese Academy of Sciences Visiting Professorships for Senior International Scientists under Grant 2012T1J0011, and to the Chinese State Administration of Foreign Experts Affairs under Grant GDJ20120491013.

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