Testing Einstein's Equivalence Principle and Its Cosmological Evolution from Quasar Gravitational Redshifts

Einstein's equivalence principle (EEP) is the central premise of gravitation. It meshes gravity with all the laws of physics and, in particular, predicts the gravitational redshift of photons. EEP is very well tested via gravitational redshift experiments in the solar neighborhood (Pound & Snider 1964; Vessot et al. 1980; Roca Cortés & Pallé 2014; Gravity Collaboration et al. 2018; Joyce et al. 2018; González Hernández et al. 2020), but very scarcely on cosmic scales. Departures from EEP are not usually allowed in cosmological tests of gravitation (Bonvin & Fleury 2018), and when taken into consideration (Rapetti et al. 2010; Reyes et al. 2010), the results are not independent of the assumption of the ΛCDM model (Wojtak et al. 2011).

A notable exception to the lack of cosmological tests of EEP based on astrophysical phenomena (albeit limited to a very narrow range of cosmological redshift, z_cosm ≃ 0.22) is the study of the gravitational redshift effect in clusters of galaxies (Wojtak et al. 2011; see also Domínguez Romero et al. 2012; Kaiser 2013; Jimeno et al. 2015; Sadeh et al. 2015; Alam et al. 2017). The basis of this test is to average the emission line shifts of a huge number of cluster member galaxies moving in the gravitational potentials of their clusters to detect very small gravitational redshift differences (Δ ∼ −10 km s⁻¹) between the outskirts and the centers of the clusters. Wojtak et al. (2011) obtained an integrated value of ${z}_{g}^{m}/{z}_{g}^{p}\sim 0.9\pm 0.3$ for the ratio between the measured and predicted gravitational redshifts in agreement with EEP. Subsequent studies have shown, however, that this measurement is affected by transverse Doppler (Zhao et al. 2013), an effect that can have the same order of magnitude as the gravitational redshift for systems in virial equilibrium, and by light cone and surface brightness effects (Kaiser 2013; Jimeno et al. 2015; Sadeh et al. 2015). Taking into account these effects, the initial prediction of Wojtak et al. (2011) modifies to ${z}_{g}^{m}/{z}_{g}^{p}\sim 0.6\pm 0.3$ (Kaiser 2013). Sadeh et al. (2015) follow up the analysis by Wojtak et al. (2011) with a larger data set finding an average redshift of ${\rm{\Delta }}=-{11}_{-5}^{+7}\,\mathrm{km}\,{{\rm{s}}}^{-1}$, consistent within uncertainties with Wojtak et al. (2011) and in good agreement with the prediction (Δ ∼ −12 km s⁻¹) by Kaiser (2013), but with larger uncertainties (${z}_{g}^{m}/{z}_{g}^{p}\sim {0.9}_{-0.4}^{+0.6}$). Jimeno et al. (2015) examine three major cluster samples from the Sloan Digital Sky Survey (SDSS) finding statistical agreement with the model predictions for two of them (${z}_{g}^{m}/{z}_{g}^{p}\sim 0.95\pm 0.3$ for GMBCG and ${z}_{g}^{m}/{z}_{g}^{p}\sim 0.7\pm 0.2$ for redMaPPer) but anomalous measurements (Δ ∼ +2.5 km s⁻¹ measured versus Δ ∼ −20 km s⁻¹ predicted) for WHL12.

We propose here a different astrophysical scenario where the gravitational redshift effect is stronger by up to three orders of magnitude: the accretion disks formed by ionized gas rotating around the supermassive black holes (SMBH) located at the centers of active galactic nuclei (AGNs) and quasars. For typical distances from the SMBH of ∼10 lt-day and masses of the SMBH of ∼10⁹ M_⊙ we expect gravitational redshifts z_g ∼ GM/c² R ∼ 0.006 equivalent to ∼1700 km s⁻¹. Recently, we have discovered that a spectroscopic UV feature of iron, the Fe III λ λ 2039-2113 emission line blend, appears redshifted by typical amounts of ∼1000 km s⁻¹ in many AGNs and quasars (Mediavilla et al. 2018, 2019, 2020). Much evidence (see Appendix A) from microlensing (Guerras et al. 2013; Fian et al. 2018), photoionization calculations (Temple et al. 2020), modeling of the Fe III λ λ 2039-2113 emission line profile (Mediavilla et al. 2018, 2019), and reverberation mapping (RM; Mediavilla et al. 2018) shows that this blend originates close to the SMBH. This UV iron blend allows to perform single object measurements of the gravitational redshift with very high significance: a mean value of 6.5σ and a maximum of 15σ for individual SDSS quasar spectra with a signal-to-noise ratio (S/N) ≳ 20. ⁵

The test we propose and perform here is the iconic gravitational redshift experiment ⁶ corrected to take into account the transverse Doppler effect ⁷ arising from the orbital motion of the ionized gas around the SMBH. Let us write the measured gravitational plus transverse Doppler redshift as

$$\begin{eqnarray}&&{z}_{g}^{m}={\left(\displaystyle \frac{{\rm{\Delta }}\lambda }{\lambda }\right)}_{\mathrm{Fe}{\rm\small{III}}}.\end{eqnarray} \tag{ 1 }$$

From EEP ⁸ the predicted gravitational redshift is (Mediavilla & Insertis 1989)

$$\begin{eqnarray}&&{z}_{g}^{p}=\displaystyle \frac{3}{2}\displaystyle \frac{{{GM}}_{\mathrm{BH}}}{{c}^{2}{R}_{\mathrm{Fe}{\rm\small{III}}}}.\end{eqnarray} \tag{ 2 }$$

Thus, to compare the observed and predicted gravitational redshifts we need independent estimates of the SMBH mass, M_BH, and of the size of the region emitting the Fe III blend, R_{Fe III} . There are several options to estimate SMBH masses (Mejía-Restrepo et al. 2018; Campitiello et al. 2020): to apply the virial theorem from the Doppler broadening of the emission lines, to use the M_BH − σ_* or M_BH − M_bulge relationships, or to model the quasar emission, for instance. Here, we consider the first option (virial mass estimate). In this case, for the emission line of species χ, the BH mass is given by

$$\begin{eqnarray}&&{M}_{\mathrm{BH}}={f}_{\chi }\displaystyle \frac{{\mathrm{FWHM}}_{\chi }^{2}{R}_{\chi }}{G},\end{eqnarray} \tag{ 3 }$$

where FWHM_χ and R_χ are the full width at half-maximum of the spectral line used to estimate the mass, and the radial distance from the SMBH of the emitting region, respectively. f_χ is the virial factor, which accounts for the geometry and kinematics of the emitting region. Thus, ⁹

$$\begin{eqnarray}&&\displaystyle \frac{{z}_{g}^{m}}{{z}_{g}^{p}}=\displaystyle \frac{2}{3}{f}_{\chi }^{-1}\displaystyle \frac{{R}_{\mathrm{Fe}{\rm\small{III}}}}{{R}_{\chi }}\displaystyle \frac{{\left(\tfrac{{\rm{\Delta }}\lambda }{\lambda }\right)}_{\mathrm{Fe}{\rm\small{III}}}}{{({\mathrm{FWHM}}_{\chi }/c)}^{2}}\end{eqnarray} \tag{ 4 }$$

is the predicted ratio to be experimentally tested.

The paper is organized as follows. In Section 2 we determine ${z}_{g}^{m}/{z}_{g}^{p}$ (Equation (4)) using quasar spectra from X-shooter@VLT, SDSS, and BOSS databases and UV spectra from two AGNs: NGC 5548 and NGC 7469. In Section 3 we compare our estimates with previous results at low cosmological redshift and discuss the cosmic evolution of EEP. Finally, in Section 4 we summarize the main conclusions.

According to Equation (4), we can estimate the ratio between the observed and predicted gravitational redshifts using widths corresponding to the Fe III λ λ 2038-2113 blend or to other emission lines. In the latter case, both the value of the virial factor, f_χ , and the ratio between the emitting regions, R_{Fe III} /R_χ , are needed. If we use, instead, Fe III λ λ 2038-2113 data to estimate the widths, we only need to know the virial factor, f_{Fe III} . However, we lack independent measurements of f_{Fe III} and we need a procedure to derive it.

2.1. Results for X-shooter@VLT Quasar Spectra (Capellupo et al. 2015, 2016) Using the Mg II, Hβ, and Hα Emission Lines to Estimate the FWHM

It is convenient to start applying Equation (4) to the available data set with the widest wavelength coverage and with a high S/N: a sample of 10 quasars at z ∼ 1.55 observed with X-shooter@VLT (Capellupo et al. 2015, 2016) for which the redshifts of the Fe III λ λ 2039-2113 blend have been measured (Mediavilla et al. 2019). Mg II, Hβ, and Hα widths are also available (Mejía-Restrepo et al. 2016). To estimate the R_{Fe III} /R_Hβ ratio we adopt the commonly used radius–luminosity relationship (Bentz et al. 2013), which scales Hβ sizes with luminosities at λ5100,

$$\begin{eqnarray}&&{R}_{{\rm{H}}\beta }={391.74}_{-26.99}^{+28.99}{\left(\displaystyle \frac{\lambda {L}_{\lambda 5100}}{{10}^{46}{\rm{erg}}{{\rm{s}}}^{-1}}\right)}^{0.533}{\rm{light}}\mbox{--}{\rm{days}}.\end{eqnarray} \tag{ 5 }$$

For the Fe III λ λ 2039-2113, we adopt a similar scaling relationship anchored to the probability density function of the microlensing based size estimates obtained for a sample of lensed quasars by Fian et al. (2018),

$$\begin{eqnarray}&&{R}_{{\rm{Fe}}{\rm\small{III}}}={17.35}_{-6.73}^{+8.39}{\left(\displaystyle \frac{\lambda {L}_{\lambda 1350}}{{10}^{45.79}{\rm{erg}}{{\rm{s}}}^{-1}}\right)}^{0.533}{\rm{light}}\mbox{--}{\rm{days}},\end{eqnarray} \tag{ 6 }$$

where 17.35 lt-day and 10^45.79 erg s⁻¹ are, respectively, the average of the Fe III sizes and λ L_λ1350 luminosities of the quasars considered by Fian et al. (2018). The use in the Fe III case of the same exponent as for Hβ is supported by previous studies (Mediavilla et al. 2018), which find a best-fit value of 0.57 ± 0.08. Using the average values for L_λ5100 and L_λ1350 of the sample of 10 quasars observed with X-shooter, we obtain an average value of the ratio between sizes, 〈R_{Fe III} /R_Hβ 〉 = 0.106, remarkably coincident with the photoionization model prediction, which indicates that the Fe III emission originates 1 dex closer to the SMBH than the C IV one ¹⁰ (Temple et al. 2020). To obtain the 〈R_{Fe III} /R_χ 〉 ratio for the other emission lines we take R_{H α} = R_{C IV} = R_{Mg II} /2 = R_Hβ (from RM estimates; Shen et al. 2019). ¹¹ Finally, we need an estimate of the virial factor, for which we take 〈f〉 = 0.93 ± 0.09 (see Appendix B), obtained averaging several values from the literature (Collin et al. 2006; Graham et al. 2011; Ho & Kim 2014; Woo et al. 2015; Mejía-Restrepo et al. 2018; Williams et al. 2018; Wang et al. 2019; Yu et al. 2020). The resulting average ratio between the observed and predicted gravitational redshift for the X-shooter quasars sample, $\langle {z}_{g}^{m}/{z}_{g}^{p}\rangle \,=0.87\pm 0.12$, at z = 1.55 is shown (light red open circle) in Figure 1 (see also Table 1).

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Table 1. Measured versus Predicted Gravitational Redshift Ratios for 10 z_cosm Bins

〈zcosm〉	$\langle {z}_{g}^{m}/{z}_{g}^{p}\ ({\rm{C}}\,{\rm\small{IV}},\mathrm{Mg}\,{\rm\small{II}},{\rm{H}}\beta ,{\rm{H}}\alpha )\rangle$	〈zcosm〉	$\langle {z}_{g}^{m}/{z}_{g}^{p}\ (\mathrm{Fe}{\rm\small{}}\mathrm{III})\rangle$	Data Source
1.27	0.80 ± 0.09	1.29	0.95 ± 0.21	SDSS
1.50	0.75 ± 0.09	1.52	0.95 ± 0.16	SDSS
1.55	0.87 ± 0.12	⋯	⋯	X-shooter
1.73	1.19 ± 0.12	1.75	1.38 ± 0.22	SDSS
1.91	0.94 ± 0.10	1.93	0.96 ± 0.11	SDSS
2.10	1.32 ± 0.21	2.12	1.52 ± 0.23	SDSS
2.25	0.89 ± 0.07	2.27	0.72 ± 0.06	BOSS
2.27	1.17 ± 0.21	2.29	1.07 ± 0.28	SDSS
2.46	0.97 ± 0.17	2.48	0.87 ± 0.20	BOSS
2.85	1.34 ± 0.18	2.88	1.16 ± 0.24	BOSS

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2.2. Results for SDSS Quasar Spectra Using the Mg II Emission Lines to Estimate the FWHM

2.3. Results for BOSS Composite Quasar Spectra Using the C IV and Mg II Emission Lines to Estimate the FWHM

2.4. Results for NGC 5548 and NGC 7469 Using Hβ Emission Lines to Estimate the FWHM

2.5. Results for SDSS and BOSS Quasar Spectra Using the Fe III λ λ 2038-2113 Line Emission Blend to Estimate the FWHM

Our experimental results to test EEP along with others from the literature are summarized in Figure 1, in which two remarkable results can be readily seen: (i) the wide coverage in cosmological redshift of our data compared with the tiny region around z_cosm ∼ 0.22 previously studied and (ii) the comparatively good statistical uncertainties of our measurements (see also Table 1), which range from ∼9% to ∼20% (∼35% if we include the NGC 5548 and NGC 7469 data at z_cosm ∼ 0.017), similar or better than the errors of other measurements outside the solar system.

Regarding EEP, we do not observe any statistically significant trend with cosmological redshift. Averaging the data corresponding to C IV, Mg II, Hβ, and Hα in the z_cosm ∼ 1 to 3 redshift range, we obtain a mean value $\langle {z}_{g}^{m}/{z}_{g}^{p}\rangle =1.02\pm 0.07$ with a scatter of the data of 0.21 ± 0.05. We obtain similar results averaging the Fe III λ λ 2039-2113 data, $\langle {z}_{g}^{m}/{z}_{g}^{p}\rangle =1.06\pm 0.08$ with a scatter of 0.25 ± 0.06. The average value corresponding to Fe III λ λ 2039-2113 is tied to Mg II through the derivation of f_{Fe III} , but the comparison between the ratios at different cosmological redshifts inferred from the Fe III λ λ 2039-2113 widths is an independent confirmation of the constancy of EEP along the studied cosmological redshift range.

So far we have considered the data grouped in 10 bins in z_cosm respecting the data source (see Figure 1 and Table 1). It is convenient to increase the S/N by regrouping the data, irrespective of their origin, in four z_cosm bins, shown as red points in Figure 1 (see also Table 2). As expected, the mean does not change significantly with the regrouping, $\langle {z}_{g}^{m}/{z}_{g}^{p}\rangle =1.05\pm 0.06$, while the scatter improves by almost a factor of 2 (0.13 ± 0.05). Thus, we find that the gravitational redshift derived from EEP holds with less than a 13% of deviation in this cosmological redshift range, and averaging the whole redshift range (blue data point in Figure 1), we validate EEP with a statistical uncertainty of 6%.

Table 2. Measured versus Predicted Gravitational Redshift Ratios for Four z_cosm Bins

〈zcosm〉	$\langle {z}_{g}^{m}/{z}_{g}^{p}\ ({\rm{C}}{\rm\small{}}\mathrm{IV},\mathrm{Fe}{\rm\small{}}\mathrm{III},\mathrm{Mg}\,{\rm\small{}}\mathrm{II},{\rm{H}}\beta ,{\rm{H}}\alpha )\rangle$
1.42	0.86 ± 0.03
1.83	1.12 ± 0.09
2.22	1.12 ± 0.11
2.67	1.09 ± 0.09

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Although until now we have only taken into account statistical (random) errors, there are also systematic uncertainties in our estimates arising from the factor R_{Fe III} /(f_χ R_χ ) in Equation (4). This factor does not affect at all the study of the cosmic evolution as we can remove it by normalizing the ${z}_{g}^{m}/{z}_{g}^{p}$ ratios to the mean ¹² : $({z}_{g}^{m}/{z}_{g}^{p})/\langle {z}_{g}^{m}/{z}_{g}^{p}\rangle$ . With respect to the averaged values, $\langle {z}_{g}^{m}/{z}_{g}^{p}\rangle$, we could anchor them to z ∼ 0 where we know from other redshift experiments that EEP holds. ¹³ In any event, the evaluation and mitigation of systematics errors is interesting if one wants to consider each measurement separately from the others. The budget of systematic uncertainties is largely dominated by the error in R_{Fe III} , roughly inferred from microlensing measurements (from Equation (6) we estimate an uncertainty of ∼44%) but that can be greatly reduced (also the uncertainty in R_χ and f_χ ) by precise RM measurements or complementary observations. Moreover, the factor R_χ f_χ can be eliminated by using a different method to estimate BH masses in Equation (2) (e.g., using M_BH–σ_* or M_BH–M_bulge relationships, or quasar emission modeling ¹⁴ ).

On the other hand, the statistical uncertainties can also be greatly reduced with the help of new observations (notice that we are just considering about 10 spectra per redshift bin!). Most of the used data come from the SDSS and, hence, have been obtained with a modest size telescope, but observation of ∼1000 sources with a 4–10 m class telescope could bring statistical uncertainties down to the 1% level.

We propose a new method to test EEP in a large range in cosmological redshift. We explore an uncharted cosmic territory covering ∼4 × 10⁹ yr of the history of the universe (in principle, the method can be extended to other epochs using quasar spectra with an adequate wavelength coverage). We show the feasibility of the method, obtaining errors comparable or better than other classical methods applied outside the SS. The main results are:

1. We measure the gravitational redshift directly in individual objects (85 SDSS quasars among them) in the early universe (as far as 10 Gyr ago). These single object measurements have a high statistical significance (≫3σ).

2. At low redshift, we estimate the ratio between the observed gravitational redshifts and the theoretical predictions in the AGNs NGC 5548 and NGC 7469 (${z}_{g}^{m}/{z}_{g}^{p}=0.88\pm 0.35$ and ${z}_{g}^{m}/{z}_{g}^{p}=0.98\pm 0.30$, respectively). These measurements, on their own, represent a qualitative leap to test EEP.

3. For the first time we measure the ratio between the observed gravitational redshifts, and the theoretical predictions, ${z}_{g}^{m}/{z}_{g}^{p}$, in 10 independent cosmological redshift bins covering the z_cosm ∼ 1–3 range.

4. We find that EEP predictions for the gravitational redshift hold in this cosmological redshift range (1 ≲ z_cosm ≲ 3) with a statistical uncertainty of 6%, without evidence for cosmic evolution above 13%.

The present results can be extended to fill the gap at cosmological redshifts below 1 with UV observations. On the other edge, the gravitational redshift of massive quasars detected at high cosmological redshift (up to z_cosm ∼ 7) could be probed with IR observations. In the future, application of our method to larger data sets and better-quality spectra will open extraordinary possibilities for precise tests of EEP over most of the history of the universe

We thank the referee for the thorough and constructive review of the paper. We thank SDSS and BOSS surveys for providing data. We tank Temple et al. for providing the data of their narrow and broad composites. This research was supported by the Spanish MINECO with the grants AYA2016-79104-C3-1-P and AYA2017-84897-P, by the Fondo Europeo de Desarrollo Regional (FEDER), and by project FQM-108 financed by Junta de Andalucía.

There is a lot of evidence that the Fe III λλ2039-2113 blend originates close to the SMBH: (i) it shows very strong gravitational microlensing (Guerras et al. 2013; Fian et al. 2018) with magnifications comparable to those of the continuum generated by the accretion disk; (ii) according to photoionization calculations (Temple et al. 2020), the UV emission of the Fe III ion traces especially high density gas confined in a region very close to the central SMBH and moving in quasi-ordered flows in the equatorial plane of the AGN (likely in the accretion disk; Temple et al. 2020); (iii) the profile of the Fe III λ λ 2039-2113 blend is very well fitted with a single kinematic component of the standard template of the blend (Vestergaard & Wilkes 2001), supporting the confinement of the gas and explaining the clean measurement of the gravitational redshift (see below the fit to the SDSS narrow and broad composite spectra constructed by Temple et al. 2020); (iv) the observed redshifts match not only the expected magnitude of the gravitational redshift but also the theoretical correlation between the gravitational redshift and the squared widths of the emission lines ¹⁵ (Mediavilla et al. 2018, 2019; i.e., correlate with M_BH/R); and (v) the only reverberation mapping estimate available (Mediavilla et al. 2018) of the size of the region emitting the Fe III λ λ 2039-2113 blend, confirms that this region has a size comparable to that of the continuum.

The narrow and broad high S/N composites created by Temple et al. (2020) by averaging SDSS spectra with FWHM_{Mg II} in the range 2000–4000 km s⁻¹ (narrow) and with FWHM_{Mg II} in the range 9000–15,000 km s⁻¹ (broad) are perfectly suited to illustrate several results of the study of the Fe III λ λ 2039-2113 blend: the good matching of the line profile with a single kinematic component and the existence of a gravitational redshift (larger for wider lines). We model the blend in both composite spectra (kindly provided by Temple et al. 2020) using the standard template (Vestergaard & Wilkes 2001) and following the same steps as in Mediavilla et al. (2018). The resulting fits (notice the remarkable good matching, especially of the characteristic features of the narrow composite shape) are presented in Figure 2. Both composites are redshifted with respect to the systemic velocity of the quasar: the narrow by +3.95 Å and the broad by a larger, as expected, quantity, +7.54 Å. We measure broadenings of 2163 and 4632 km s⁻¹.

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In Figure 3 we compare the original broad composite blend (blue points) with the narrow blend convolved with a Gaussian to match the broadening of the broad composite according to the widths estimated from the fits (orange dotted line). The redshift of the blue points with respect to the orange dotted line is evident. The green dotted line is the orange dotted line shifted by the difference in redshifts estimated from the fits. It matches the blue points fairly well.

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Finally, we would like to remark on the striking good fits based on the template of Vestergaard & Wilkes (2001) to a variety of objects with quite different properties (masses, emission line broadenings, degree of activity, etc.). The Vestergaard & Wilkes (2001) template is inferred from IZw1, a narrow line Seyfert 1 galaxy, which may be not representative of objects with very broad emission lines. However, we have applied this template to many AGNs and quasars (cf. Section 2; see also Mediavilla et al. 2018, 2019, 2020) obtaining surprisingly good fits if we take into account the scatter in properties among these objects. Moreover, in Mediavilla et al. (2018) we obtain consistent fits to other UV Fe III features like the Fe III λ λ1970-2039 blend and the Fe III λ2419 emission line. These results seem to indicate that the UV Fe III emission arises from a homogeneous region with quite regular physical conditions.

The used value of the virial factor, f, has been calculated as the weighted mean of the estimates given in the following references (see Figure 4): Mejía-Restrepo et al. (2018), Collin et al. (2006), Graham et al. (2011), Ho & Kim (2014), Woo et al. (2015), Williams et al. (2018), Wang et al. (2019), and Yu et al. (2020). (Notice that we use two values from Collin et al. 2006, one of them based on the dependence with the FWHM.) Each estimate is weighted according to $1/{\sigma }_{f}^{2}$, where σ_f is the error estimate given in the original data source. (For Mejía-Restrepo et al. 2018 and Collin et al 2006 we use the f versus FWHM relationships given by these authors to estimate f and σ_f using the average FWHM derived from our X-shooter sample.) The resulting weighted mean is 〈f〉 = 0.93 ± 0.09 with scatter 0.25 ± 0.06. Using a nonweighted average instead produces a similar value, 〈f〉 = 0.99 ± 0.08.

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