# Questioning the H0 tension via the look-back time

Salvatore Capozziello, Giuseppe Sarracino, Alessandro D. A. M. Spallicci

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Salvatore Capozziello, Giuseppe Sarracino, Alessandro D. A. M. Spallicci. Questioning the H0 tension via the look-back time. Physics of the Dark Universe, 2023, 40, 10.1016/j.dark.2023.101201. insu-04043842

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Questioning the $H_0$ tension via the look-back time

Salvatore Capozziello,¹,²,³ Giuseppe Sarracino,¹,² and Alessandro D.A.M. Spallicci⁴,⁵,⁶

¹Dipartimento di Fisica "E. Pancini", Università degli Studi di Napoli "Federico II",
Complesso Universitario Monte S. Angelo, Via Cinthia 9 Edificio G, 80126 Napoli, Italy,

²Istituto Nazionale di Fisica Nucleare, Sezione di Napoli,
Complesso Universitario Monte S. Angelo, Via Cinthia 9 Edificio G, 80126 Napoli, Italy,

³Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, Italy.

⁴Laboratoire de Physique et Chimie de l'Environnement et de l'Espace, UMR 7328,
Centre National de la Recherche Scientifique, Université d'Orléans, Centre National d'Études Spatiales,
3A Avenue de la Recherche Scientifique, 45071 Orléans, France,

⁵UFR Sciences et Techniques, Université d'Orléans, Rue de Chartres, 45100 Orléans, France,

⁶Observatoire des Sciences de l'Univers en région Centre, UMS 3116,
Université d'Orléans, Centre National de la Recherche Scientifique, Observatoire de Paris, Université Paris Sciences & Lettres,
1A rue de la Férollerie, 45071 Orléans, France.

(Dated: March 3, 2023)

The Hubble tension is investigated taking into account the cosmological look-back time. Specifically, considering a single equation, widely used in standard cosmology, it is possible to recover both values of the Hubble constant $H_0$ reported by the SH0ES and Planck collaborations: the former is obtained through cosmological ladder methods (e.g. Cepheids, Supernovae Type IA) and the latter through measurements of the Cosmic Microwave Background. Also, other values obtained in the literature are achieved with the same approach. We conclude that the Hubble tension can be removed if the look-back time is correctly referred to the redshift where the measurement is performed.

Keywords: Observational cosmology; Hubble-Lemaître constant; look-back time.

# I. INTRODUCTION

The so-called $H_0$ tension, related to the reported discrepancy on the measured values of Hubble-Lemaître (henceforth Hubble) constant, $H_0$, is claimed to be one of the most critical issues for the Λ Cold Dark Matter (ΛCDM) model, considered the standard picture of the observed Universe. This staggering discrepancy arises when comparing estimates on $H_0$ provided by standard candles with those inferred by the Cosmic Microwave Background Radiation (CMBR) [1]. In fact, the results from the SH0ES collaboration, obtained by considering the cosmological ladder and probes like classical Cepheids and Supernovae Type Ia (SNe Ia), in a region with redshift interval $0.02333 < z < 0.15$, tell us that $H_0 = 73.04 \pm 1.04$ km/(s Mpc) at 68% confidence level (CL) [2], while the Planck collaboration, reporting observations of the CMBR, shows, instead, that $H_0 = 67.4 \pm 0.5$ km/(s Mpc) at 68% CL [3]. Despite the accuracy of both approaches, a $5.0\sigma$ tension emerges and it seems to be irreconcilable under the standard of the ΛCDM model.

The problem has to be addressed both from experimental and theoretical points of view. Considering the increasing precision achieved by recent observations, an experimental reason for the tension, related to some systematic effects, is becoming less and less probable [1, 4]. Thus this tension (and other ones revealed in cosmological parameters [1]) is seemingly hinting at some undefined new physics, going beyond the ΛCDM model. The last conclusion can be inferred also from other critical issues of the standard cosmological model, like the nature of dark matter and dark energy, which should account for approximately 25% and 70%, respectively, of the energy-density budget of the observed Universe [4–13]. However, the nature of the dark side of the Universe is still unknown at a fundamental level, even if very precise tests are performed by several running experiments [14–17].

Different approaches are considered in the literature to fix the above problems. A possibility is taking into account extensions and modifications of General Relativity (GR) where the ΛCDM model evolves along the cosmic history. For example, extensions like $f(R)$ gravity [18–22]), have been considered in cosmological applications [23–27] to address different issues related to the ΛCDM model, like the late-time dark energy [12], and the inflationary behaviour of the early Universe [28].

Many possible extensions and modifications of GR have been proposed (see e.g. [29]), but, no final theory, capable of addressing the whole cosmic history and the large-scale structure has been formulated up to date. As in the case of cosmological inflation, the paradigm seems consistent but a final model, comprehensive of the whole phenomenology, if any, is still missing. In any case, modified gravity can be adopted to investigate the $H_0$ tension, as recently proposed in [30].

Modified gravity is not the only approach to fix tensions in cosmology. New physics can be pursued also by taking into account other paradigms like extensions of Electromagnetism and the Standard Model of Particles [31–33], or revising the concept of cosmological measurements according to the principles of Quantum Mechanics [34, 35], as well as many different other approaches reported in the literature [36–64]. Furthermore, requiring a phenomenological evolution of $H_0$ with the redshift could be a straightforward way out for the problem [65–78] even without investigating some fundamental counterpart.
A more conservative approach is assuming, in any case, the validity of the ΛCDM model and the correctness of the reported measurements. In other words, we can investigate the possibility that both ΛCDM and observations can be somehow reconciled by searching for suitable observables. In other words, the issue of $H_0$ and other tensions could be nothing else but a problem of misinterpretation of measurements at high and low redshift which have to be related in the right way.

In this paper, we wish to show that the $H_0$ tension can be addressed by the correct interpretation of the look-back time at various redshifts. It is defined as

$$T_{lt}(z) = \frac{1}{H_0} \int_{0}^{z} \frac{dz'}{(1+z')E(z')}, \qquad (1)$$

which expresses the time a photon takes to reach us from a given redshift $z$ in the expanding Universe. It is directly related to the light-travel distance

$$d_{lt}(z) = cT_{lt}(z) \qquad (2)$$

where $c$ is the speed of light. The key ingredient is the function $E(z)$, defined as,

$$E(z) = \frac{H(z)}{H_0} = \sqrt{\Omega_r(1+z)^4 + \Omega_M(1+z)^3 + \Omega_k(1+z)^2 + \Omega_\Lambda}, \qquad (3)$$

the entries of which are the different contributions to the density of the Universe. $\Omega_r$ is the radiation density, $\Omega_M$ the matter density (baryonic and dark), $\Omega_k$ is the density associated with the spatial curvature, and $\Omega_\Lambda$ is the density associated to the cosmological constant. Clearly, $\Omega_\Lambda$ can be given by the standard cosmological constant or by some evolving contribution of the Universe vacuum energy. It is worth stressing that Eq. (3) represents a class of models including ΛCDM and its extensions. The role of Precision Cosmology is to fix consistently the values of $\Omega_i$ parameters as correctly as possible.

Previous works [79–84] have already used the look-back time and the age of the Universe for investigations on the Hubble tension.

The layout of the paper is the following. Sec. 2 is a general discussion devoted on how to infer $H_0$ from Eq. (1) starting from the age of the Universe, $T_0$, via the same equation valid at any redshift $z$. In Sec. 3, we show how, inserting the values of $T_0, \Omega_r, \Omega_M, \Omega_k$, and $\Omega_\Lambda$, provided by the Planck collaboration, it is possible to derive values of $H_0$ consistent with both SH0ES and Planck. Also, $H_0$ measurements, reported by other collaborations, are consistent with the presented approach. Conclusions are drawn in Sec. 4.

## II. LOOK-BACK TIME AND THE HUBBLE TENSION

The Hubble constant can be inferred by the look-back time at a given redshift by inverting Eq.(1)

$$H_0 = \frac{1}{T_{lt}(z)} \int_{0}^{z} \frac{dz'}{(1+z')E(z')}. \qquad (4)$$

On the other hand, $T_{lt}$ can be linked to the age of the Universe considering the difference between this quantity, evaluated today, and that computed at a given redshift $T(z)$. It is

$$T_{lt} = T_0 - T(z) = T_0 - \frac{T_0}{(1+z)} = T_0 - a(t)T_0, \qquad (5)$$

where we used the definition of the scale factor of the Universe $a(t)$ as

$$\frac{a_0}{a(t)} = 1 + z. \qquad (6)$$

Here $a_0 = 1$ is the scale factor corresponding to our epoch<fnref n="1" />. In doing so, we can infer the value of the Hubble parameter $H_0$ at any redshift starting from the age of the Universe and the function $E(z)$ from Eq.(4). It is

$$H_0^{(z)} = \frac{(1+z)}{T_0 z} \int_0^z \frac{dz'}{(1+z')E(z')}. \qquad (7)$$

<footnote n="1">¹ The parametrization $T(z) = a(t)T_0$ is assumed as a label for the age of the Universe at a given redshift. Thus $a(t)$ is a projection factor and we do not need an integration over the cosmic time.</footnote>
We notice that, for $z \to +\infty$, it is $\lim_{z \to +\infty} \frac{z+1}{z} = 1$, and thus

$$H_0^{(\infty)} = \frac{1}{T_0} \int_0^\infty \frac{dz'}{(1+z')E(z')} \quad (8)$$

Eqs. (7) and (8) tell us that it is possible to find different values for $H_0$ at different values of the redshift, and that the $H_0$ tension could be addressed by reconciling the value of $H_0$, derived at a given $z$, with the one derived at $z \to +\infty$. In this perspective, we may conclude that the $H_0$ tension is not due to neither systematic errors, nor to some new physics beyond the ΛCDM model, but simply to the fact that, in measuring the same quantity at different redshift, different results arise thanks to the look-back time evaluated at different epochs.

### III. RECOVERING $H_0$ FROM PLANCK AND SH0ES RESULTS

Let us now test the above considerations by adopting observational results provided by the Planck and SH0ES collaborations. The claim is that Eq. (7) gives $H_0$ reported in [2] and Eq. (8) gives the one reported in [3]. According to the latter reference, the values that fix the ΛCDM model are

$$T_0 = 13.797\text{Gyr}, \quad \Omega_r = 9.252 \times 10^{-5}, \quad \Omega_M = 0.3153, \quad \Omega_\Lambda = 0.6847. \quad (9)$$

Taking into account these values, and fixing $z = 0.15$ (the upper limit of the redshift interval of the observations performed by the SH0ES collaboration), we find, from Eq. (7)

$$H_0^S = 73.29 \text{ km/(s Mpc)}, \quad (10)$$

which is remarkably consistent with the value inferred in Ref. [2]. It is worth noticing that this result is achieved by using values provided by the Planck collaboration, further proving the concordance between the two results. Furthermore, from Eq. (8), we find

$$H_0^P = 67.40 \text{ km/(s Mpc)}, \quad (11)$$

which is consistent with the $H_0$ value reported by the Planck collaboration. In both cases, ΛCDM has been adopted but the look-back time has been considered at redshifts consistent with observational data.

The approach can be generalised to any redshift, finding how $H_0$ changes as a function of the redshift. In Fig.1, we plot Eq.(7) confronting the values of $H_0$ reported by various collaborations. We choose the upper limit of the $z$ interval reported in the related literature, or the redshift adopted to estimate $H_0$ corresponding to a given physical phenomenon. Besides the $H_0$ results, provided by [2] and [3], we report on the plot the following measurements, all at 68% CL:

* $H_0 = 69.9 \pm 1.9 \text{ km/(s Mpc)}$, obtained by using the tip of the red giants branch as an anchor for SNe Ia, at $z = 0.08$ [85];

* $H_0 = 75.8 \pm 5.0 \text{ km/(s Mpc)}$, derived from SNe Type II, at $z = 0.45$ [86];

* $H_0 = 73.3 \pm 4.0 \text{ km/(s Mpc)}$, derived from the Mira Variables and SNe Ia, at $z \sim 0.15$ [87];

* $H_0 = 76.0 \pm 2.6 \text{ km/(s Mpc)}$, derived from the Tully-Fisher relation, at $z = 0.5$ [88];

* $H_0 = 73.3 \pm 2.5 \text{ km/(s Mpc)}$, derived from the surface brightness fluctuations, at $z = 0.33$ [89];

* $H_0 = 69.5 \pm 3.3 \text{ km/(s Mpc)}$, inferred from the Large Scale Structure $t_{eq}$ standard ruler, and thus confronted to our computations at the redshift of equivalence $z_{eq} \sim 3300$ [90];

* $H_0 = 72.0 \pm 1.9 \text{ km/(s Mpc)}$, inferred from the masers+SNe Ia and compared at $z \sim 0.15$ [91];

* $H_0 = 73.3 \pm 1.8 \text{ km/(s Mpc)}$, derived from gravitational lensed quasars, confronted with our plot at $z = 0.745$;

* $H_0 = 67.9 \pm 1.5 \text{ km/(s Mpc)}$, which is a measurement obtained by the CMBR independently from the Planck collaboration, and thus corresponding, in our plot, at the reionization epoch $z \sim 1100$ [92];

* $H_0 = 69.6 \pm 2.1 \text{ km/(s Mpc)}$, linked to the 21 cm absorption line and corresponding at the beginning of the so-called Cosmic Dawn, the epoch when the first stars formed ($z \sim 17.2$), in combination with CMBR data and considering a Chaplygin gas model for the dark sector [93];

* Finally, $H_0 = 73.4 \pm 8.8 \text{ km/(s Mpc)}$, deduced by gravitational waves, at $z = 0.438$ [94].
<figure><img src="image_2.png" /><figcaption>FIG. 1: The value of $H_0$ derived from Eq. (7) vs the redshift, confronted with observational data. The values for the redshifts representing each measurement have been chosen either at the upper limit of the redshift interval sample, or at the redshift corresponding to the given phenomenon considered for the estimate (Cosmic Dawn, Recombination, Equivalence). The x-axis is in logarithmic scale.</figcaption></figure>

From Fig. 1, it is clear that most of the $H_0$ measurements are consistent, within $1\sigma$, with respect to the curve obtained by Eq. (7) for any $z$. The two outliers, corresponding to the measurements reported in [85, 95], are still consistent within $2\sigma$. It is interesting to see that the trend of $H_0$ values matches Eq. (7) also if different methods and cosmological probes have been adopted. This means that any $H_0$ measurement can be somehow related to the look-back time. We comment that, according to Fig. 1, $H_0$ measurements in the range $z \sim 2 \div 3 (0.30 \div 0.48 \text{ in log})$ are expected to show even higher values for this quantity, and thus a stronger tension with respect to the Planck results. Cosmological observations and computations in this intermediate redshift range exist but they strictly depend on calibrations at lower redshift [96–100]. Thus, new observations are expected by novel probes, possibly providing a standalone estimate of cosmological parameters in this redshift region, like quasars, novel gravitational wave standard sirens, galaxy clusters, Lyman-$\alpha$ lines, or gamma-ray bursts [101–104].

## IV. DISCUSSION AND CONCLUSIONS

The problem of the $H_0$ tension can be addressed in the framework of the ΛCDM model in terms of look-back time considered at the redshift where measurements are performed. In this derivation, the values of the age of the Universe and the cosmic densities, provided by the Planck collaboration, are adopted. The look-back time curve, matching the observational values of $H_0$, is shown in Fig. 1. According to this approach, we are able to recover most of the reported $H_0$ measurements and, more remarkably, the two most representatives for the $H_0$ tension, namely the ones provided by the Planck [3] and SH0ES [2] collaborations. The result is interesting because the concordance of $H_0$ measurements is achieved independently from the type of performed measurement. Indeed, we have considered estimates obtained by different standard candles (SNe Ia, TRGB, Cepheids, etc), phenomena concerning early stages of the Universe (CMBR, Cosmic Dawn, Equivalence Epoch), and also novel probes (gravitational waves). According to this picture, we may conclude that no systematic errors nor new physics are necessary to explain the tension, which, instead, can be accounted for with a single formula in the framework of the ΛCDM model. In other words, by adopting the look-back time, we can consistently account for late-type and early-type measurements, confirming the underlying consistency of the ΛCDM model. We also point out that this evolution of $H_0$ constant in the framework of the
ΛCDM model does not change its role as a formal integration constant derived from the Friedman equations, but its operative
value depends on the redshift where it is inferred. This conclusion is important for the definition of the cosmological distances,
which are all depending on $H_0$.

It is worth stressing that the value of the age of the Universe, $T_0$, has been derived by the Planck collaboration starting from
six independent parameters (the minimum number of parameters necessary to describe the ΛCDM model), among which $H_0$
is not included. In fact, also $H_0$ is inferred from the same minimum number of independent parameters. This means that any
circularity problem is avoided. Furthermore, we can note that the plot in Fig. 1 is in agreement with the general trend of $H_0$
late-type measurements. See, for example, Ref. [105] where the estimate $H_0 = 72.94 \pm 0.75\text{km/(s Mpc)}$ at 68% CL is reported.
Furthermore, our results are consistent with constraints on $H_0$ obtained from the age of the Universe: for instance, in [52], an
upper limit at $H_0 = 73.2 \text{ km/(s Mpc)}$ with 95% CL has been inferred by considering the ages of old astrophysical objects up to
$z \sim 8$. This value is consistent with Fig. 1, because, according to our approach, we find $H_0 = 72.3 \text{ km/(s Mpc)}$ for that particular
value of the redshift.

Finally, computations reported here are consistent with the interpretation proposed in [34, 35], according to which a funda-
mental uncertainty on $H_0$ exists *a priori* in the framework of the ΛCDM scenario. Furthermore, by the light-travel distance,
a cosmological Compton length can be always defined. This distance is strictly connected to the look-back time [34] without
contradiction with present results.

As a final remark, it is useful noticing that we do not exclude the idea of new physics beyond the ΛCDM model, because other
issues, like dark energy and dark matter, at late time, or inflation, at early time, could require to go beyond GR and standard
cosmology. However, from the above discussion, it is possible to state that new physics is not necessary for the particular issue
of the $H_0$ tension, which can be completely accounted for in the framework of the ΛCDM model.

Acknowledgements

This paper is based upon work from the COST Action CA21136 Addressing observational tensions in cosmology with sys-
tematics and fundamental physics (CosmoVerse) supported by COST (European Cooperation in Science and Technology). SC
and GS are grateful to the LPC2E laboratory, Université d'Orléans, for the hospitality while writing this work. They also
acknowledge the support of Istituto Nazionale di Fisica Nucleare, Sez. di Napoli, Iniziative Specifiche MOONLIGHT-2 and
QGSKY.

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