[a,b,c]Maria Dainotti

# The Hubble constant tension: current status and future perspectives through
new cosmological probes

Biagio De Simone  Giovanni Montani  Tiziano Schiavone  Gaetano Lambiase

###### Abstract

The Hubble constant ($H_{0}$) tension is one of the major open problems in
modern cosmology. This tension is the discrepancy, ranging from 4 to 6
$\sigma$, between the $H_{0}$ value estimated locally with the combination of
Supernovae Ia (SNe Ia) + Cepheids and the cosmological $H_{0}$ obtained
through the study of the Cosmic Microwave Background (CMB) radiation. The
approaches adopted in [1] and [2] are introduced. Through a binning division
of the Pantheon sample of SNe Ia [3], the value of $H_{0}$ has been estimated
in each of the redshift-ordered bins and fitted with a function lowering with
the redshift. The results show a decreasing trend of $H_{0}$ with redshift. If
this is not due to astrophysical biases or residual redshift evolution of the
SNe Ia parameters, it can be explained in light of modified gravity theories,
e.g., the $f(R)$ scenarios. We also briefly describe the possible impact of
high-$z$ probes on the Hubble constant tension, such as Gamma-ray bursts
(GRBs) and Quasars (QSOs), reported in [2] and [4], respectively.

##  1 Introduction

One of the major goals of modern science is to understand the structure of the
universe, relying on the most accredited paradigm for describing its current
state and future evolution. To date, the strongest candidate for such a
purpose is the so-called $\Lambda$CDM model for cosmology, based on the
standard FLRW metric [5]. The basic hypothesis under this formulation are: the
presence of the cold - namely, non-relativistic dark matter (CDM) - whose
density, in combination with the baryonic matter, is described by the
$\Omega_{0m}$ parameter; the acceleration of the universe driven by the dark
energy which, in turn, is represented by the $\Omega_{0\Lambda}$ parameter.
One of the greatest achievements by the $\Lambda$CDM is the capability of
predicting the accelerated expansion phase of the universe, as proven by the
works of Riess [6] and Perlmutter [7] through their analysis of SNe Ia, which
are among the best standard candles so far discovered.
Despite all the great success that the $\Lambda$CDM achieved in the scientific
community, there are still many unsolved problems related to this model. First
of all, the nature of dark matter and dark energy is not clear. Second, the
void energy measurement through quantum mechanics and the void energy
estimated with the cosmological observations differ by 120 orders of magnitude
(the so-called fine-tuning problem). Third, in principle, there is no clear
reason why the universe’s dark matter and dark energy contributions must have
the same magnitude order today - the coincidence problem [8, 9]. It is worth
mentioning also the lithium problem [10], namely the discrepancy between the
predicted abundance of primordial lithium as inferred from the CMB
observations and the local observations in the halo stars of our Galaxy.
Finally, one of the most intriguing problems that affect the $\Lambda$CDM
model is the so-called Hubble constant ($H_{0}$) tension: this is the
discrepancy, in more than 4 $\sigma$, between the value of the Hubble constant
estimated through the local probes [11] and the cosmological value inferred
with the power spectrum of the CMB [12]. The solution to this tension is not
an easy task, and many suggestions have been provided by the scientific
community. Considering the theoretical scenarios, many researchers have
proposed a modification of gravity [13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27], in particular the $f(R)$ theories [28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42], the $f(T)$ theories [43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57], or the $f(Q)$ theories [58, 59].
It is also important to mention that many alternative cosmological models have
been suggested by the scientific community [60, 61, 62, 63, 64, 65, 66, 67,
68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86,
87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104,
105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117].
The modifications can also be theorized on the fundamental constants, or
physical observables [118, 119, 120, 121, 122, 123, 123, 124, 125, 126, 127],
in particular, the gravitational constant [128, 129, 130, 131, 132, 133, 134,
135, 136, 137].
Of particular interest are the theoretical frameworks that consider the
contribution of dark energy [138, 139, 140, 141, 142, 143, 144, 145, 146, 147,
148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162,
163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177,
178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192,
193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203], the phantom dark
energy [204, 205, 206, 207], the dark matter [208, 209, 210, 211, 212, 213,
214, 215, 216, 217, 218, 219, 220, 221, 222], the dark radiation [223, 224,
225], the dark energy gravity models [226], the dark fluid models [227, 228],
and the models the dark energy and dark matter interaction between themselves
or with the other sectors of the universe [229, 230, 231, 232, 233, 234, 235,
236, 237]. The running or interacting vacuum models are also a proposal for
relieving the Hubble tension [238, 239, 240, 241, 242, 243, 244, 245, 246,
247, 248, 238, 249].
Other ideas stem from the early time physics or inflation scenario
modifications [250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261,
262, 263, 264, 265, 266, 267, 268], late time modifications [269, 270, 271,
272, 273, 274, 275, 276, 277, 278], fluctuations and perturbation theories
[279, 280, 281, 282, 283, 284, 285], or even from the possibility of
inhomogeneities in the universe, local underdensities or voids [286, 287, 288,
289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303,
304].
Other discussions have been provided taking into account a modification to the
physics of standard model particles, like the neutrino, [305, 306, 307, 308,
309, 310, 311, 312, 313, 314, 315, 316, 317, 318], the gravitino and gravitone
conjectures [319, 320] or the most exotic scenarios that involve strings
[321], axions [322, 323, 324, 325, 326, 327, 328], B-meson decays [329],
lepton flavor universality violation [330], and the Planck mass [331].
Not only the theoretical scenarios have been called into question, but of
course, the statistical analysis on the $H_{0}$ measurements constitutes a
significant fraction of the literature regarding the Hubble tension [332, 333,
334, 335, 336, 337, 338, 339, 340, 341, 342].
The most tricky discussion concerning the Hubble tension must be done
considering the probes used for its measurement: Cepheids [343, 344, 345, 346,
11, 347, 348, 349, 350, 351, 352, 353, 354], SNe Ia [355, 356, 357, 358, 359,
360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374,
375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389,
390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404,
405, 406], lensed objects [407, 408, 409, 410, 411, 412, 413, 414, 415, 416,
417], Cosmic Chronometers (CCs) [418, 419, 420, 421], CMB [422, 423, 424, 12,
425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437], Cosmic Dawn
observations [438], galaxies clusters and sky-surveys [439, 440, 441, 442,
443, 444, 445, 446, 447, 448, 449, 450], large scale structures [451, 452,
453], galaxies parallax [454], Baryon Acoustic Oscillations (BAOs) [455, 456,
457, 458, 459, 460, 461, 462], Big Bang Nucleosynthesis (BBN) [463, 464], Tip
of the Red Giant Branch method (TRGB) [465, 466, 467], Mira variables [468],
Tully-Fisher data [469], Megamaser [470], Old Astrophysical Objects (OAO)
[471, 472, 473], SNe Type II [474], gravitational signals and dark sirens
[475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489,
490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501]. Many papers
consider the combination of different astrophysical probes for testing the
cosmological models and investigating the Hubble tension [502, 503, 504, 505,
506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520,
521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535,
536, 537, 538, 539, 540, 541, 542, 543]. Together with the assessed
cosmological probes, new promising standardizable candles like the GRBs [544,
545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 509, 557, 558,
559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573,
574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588,
589, 590, 591, 592, 593, 594, 595, 596, 597], the QSOs [598, 599, 600, 601,
602, 603, 604, 605, 606, 607], the active galactic nuclei (AGNs) [608, 609,
610], the planetary nebulae luminosity function (PNLF) [611], the Fast Radio
Bursts (FRBs) [612, 613, 614], J-Branch Asymptotic Giant Branch stars (JAGB)
[615], and the extragalactic radio jets [616] are promising tools for casting
more light on the open cosmological problems.
Other suggestions in literature are given by the model-independent approaches
[617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631,
632, 633, 634, 635], cosmography [636, 637, 638, 639, 640, 641, 642, 643, 644,
645, 646, 647, 648], cosmic triangles schemes [649], machine learning
techniques [609, 650], the ABC algorithm [651], the alternative resolution of
Friedmann equations [652, 653] and the discussion of distance measurement in
cosmology [654].
Finally, it is essential to discuss also the presence of biases in the
collected data samples [655, 656, 657, 658, 659, 660, 661] since, if this is
the case, the biased variables may lead to unreliable cosmological results.
A more complete description of the Hubble constant tension approaches and the
basics of cosmological models can be found in the review papers and books
[662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676,
677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687].
In the current work, we will describe the analysis performed through the use
of SNe Ia [1] and SNe Ia combined with Baryon Acoustic Oscillations (BAOs) [2]
to investigate the Hubble constant tension at local redshifts ($z$), given
that the redshift of SNe Ia is $z_{SNeIa}\leq 2.26$. We will then show how the
GRBs and QSOs can contribute to the estimation of cosmological parameters and
discuss their application in the $H_{0}$ tension problem.

##  2 Data analysis

###  2.1 The Hubble tension in the Pantheon sample: part I

In this subsection, we summarize the results obtained in [1].
To investigate the $H_{0}$ tension through SNe Ia, a binning approach was
adopted. The Pantheon sample [3], a collection of 1048 SNe Ia with a redshift
range in $0<z_{SNeIa}\leq 2.26$, was divided into 3 and 4 equally populated
SNe bins ordered in redshift. The division into 3 bins is the best option for
giving each bin a reliable statistical weight, while the 4 bins have been
extracted to compare our results with the ones in [342]. The value of SNe Ia
fiducial absolute magnitude $M$ was assumed such that the local $H_{0}$ in the
first bin of each division (first out of 3 and first out of 4) is calibrated
on the value of $H_{0}=73.5\,km\,s^{-1}\,Mpc^{-1}$.
For each bin, a 1-dimensional Monte Carlo Markov Chain (MCMC) analysis has
been performed using the Cobaya
package111https://cobaya.readthedocs.io/en/latest/index.html [688], allowing
only $H_{0}$ to vary in both the $\Lambda$CDM and $w_{0}w_{a}$CDM [63] models.
Subsequently, the $H_{0}$ values obtained have been fitted with the following
parametric form:

$f(z)=\frac{\tilde{H_{0}}}{(1+z)^{\alpha}},$ (1)

where $\alpha$ is the evolutionary parameter for $H_{0}$ with redshift and
$\tilde{H_{0}}$ is the local value of the Hubble constant obtained with the
fitting, namely $f(z=0)=H_{0}(z=0)$.
The results show that the $H_{0}$ has a mild decreasing trend with redshift,
with an $\alpha$ parameter in the order of $10^{-2}$. More specifically, from
the test of the $\Lambda$CDM cosmological model, we found out that
$\alpha=0.009\pm 0.004$ in the 3 bins division, thus $\alpha$ is compatible
with zero only in 2.0 $\sigma$. In 4 bins, the parameter $\alpha=0.008\pm
0.006$ is compatible with zero in 1.5 $\sigma$. The results of $w_{0}w_{a}$CDM
model are compatible in 1 $\sigma$ with the ones of $\Lambda$CDM model and
will not be reported. The fitting for 3 and 4 bins in the $\Lambda$CDM model
are visible in the upper and lower panels of Figure 1, respectively.

Figure 1: Upper panel. The decreasing trend of $H_{0}$ in the 3 bins division
of the Pantheon sample. Lower panel. The decreasing trend of $H_{0}$ in the 4
bins division of the Pantheon sample. Both the plots are taken from [1].

###  2.2 The Hubble tension in the Pantheon sample: part II

In this second subsection, the approach adopted in [2] is reported.
To investigate if this trend is still visible when leaving more cosmological
parameters free to vary and adding another cosmological probe, the BAOs, we
performed the following modifications to the approach shown in [1]:

  * •

the $H_{0}$ and $\Omega_{0m}$ were both allowed to vary in the $\Lambda$CDM
model;

  * •

the same consideration has been done for $H_{0}$ and the $w_{a}$ parameters in
the $w_{0}w_{a}$CDM model, being $w(z)=w_{0}+w_{a}*z/(1+z)$ the equation of
state in the CPL parametrization;

  * •

A further cosmological probe has been added; namely, the BAOs from [457];

  * •

The Pantheon sample combined with BAOs was divided into 3 bins to avoid the
statistical fluctuations dominating the analysis;

  * •

The fiducial value of SNe Ia absolute magnitude is $M=-19.25$, like in the
original release of Pantheon [3] where the value of
$H_{0}=70\,km\,s^{-1}\,Mpc^{-1}$.

Following the aforementioned approach, the results still show a decreasing
trend for $H_{0}$ with redshift and an $\alpha$ in the order of $10^{-2}$.
According to the $\Lambda$CDM model, the value of $\alpha=0.008\pm 0.006$ is
compatible with zero in 1.2 $\sigma$, while in the $w_{0}w_{a}$CDM model
$\alpha=0.033\pm 0.005$ is compatible with zero in only 5.8 $\sigma$. The
fitting for the $\Lambda$CDM and the $w_{0}w_{a}$CDM models are reported in
the upper and lower panel of Figure 2, respectively.

Figure 2: Upper panel. The decreasing trend of $H_{0}$ in the 3 bins division
of the Pantheon sample, adding the BAOs and also allowing $\Omega_{0m}$ to
vary. In red is the case where only SNe Ia are present in the bins, while in
blue is the case of SNe Ia + BAOs present. Lower panel. The decreasing trend
of $H_{0}$ in the 4 bins division of the Pantheon sample, with BAOs and the
variation of $H_{0}$ and $w_{a}$. The same color coding of the upper panel is
adopted here. The plots are extracted from [2].

##  3 Towards a physical interpretation of the Hubble constant trend

We now discuss the possibility of a physical interpretation of the observed
profile of $H_{0}\left(z\right)$. In other words, we investigate new physics
that could account for a redshift evolution of the $\Lambda$CDM model,
resulting in a rescaling of the luminosity distance, which reproduces an
effective evolution of the Hubble constant, i.e.,
$H_{0}(z)\sim(1+z)^{-\alpha}$.

In [1], it was argued that a natural framework to obtain the desired effect is
the $f(R)$ modified gravity in the Jordan frame [689, 690, 691, 692, 693,
694], simply because it provides a rescaling of the Einstein constant,
precisely what it would need for an additional redshift dependence for the
$\Lambda$CDM cosmological dynamics. The quantity that could be responsible for
this rescaling of the Einstein constant is the non-minimally coupled scalar
field $\phi$ to standard gravity, as it emerges in the Jordan frame. The
dynamics of $\phi$ are influenced by a potential term $V\left(\phi\right)$,
which is entirely fixed by the particular form of the $f(R)$ function we are
considering. However, this perspective was investigated in detail in [2],
where some discrepancies for the observed $H_{0}(z)$. On the one hand, the use
of the Hu-Sawicki model [695, 696], namely one of the most reliable $f(R)$
proposal for the description of a dark energy dynamics, was unable to
reproduce the amended luminosity distance for removing the dependence
$H_{0}(z)$ from data, see the plot in Figure 3; on the other hand, the
possibility for slow-rolling dynamics of the scalar field encountered some
difficulties. Only in the dark energy-dominated era a viable model has been
derived.

Here, we briefly elucidate the theoretical paradigm that in [697] led to
propose a revised $\Lambda$CDM dynamics from $f(R)$ gravity, which reproduces
the desired $H_{0}(z)$ behavior. The starting point is the modified Friedmann
equation for an isotropic late Universe, as it emerges in the Jordan frame,
i.e.:

$H^{2}=\frac{\chi\rho_{m}}{3\phi}-H\frac{\dot{\phi}}{\phi}+\frac{V}{6\phi}\,,$
(2)

where $\rho_{m}$ denotes the energy density of the matter component in the
late Universe, setting $c=1$.

Now, without loss of generality, we can express the scalar field as
$\phi=\phi(z)$, and we can assume that for a range $0<z\ll 1$, the potential
term remains sufficiently flat to be approximated with a constant value
$V\simeq 2\chi\rho_{\Lambda}\simeq\textrm{const}$ with $\rho_{\Lambda}$ being
the value of the late universe vacuum energy. Hence, it is immediate to
recognize that the Friedmann equation (2) can be rewritten as:

$H^{2}=\frac{1}{\phi-(1+z)\frac{d\phi}{dz}}\frac{\chi}{3}\left(\rho_{m}+\rho_{\Lambda}\right)\,.$
(3)

We see that the desired rescaling of the Einstein constant is ensured by the
factor $1/[\phi-(1+z)d\phi/dz]$, which must equate the observed profile
$1/(1+z)^{2\alpha}$. This request fixes $\phi(z)$. Furthermore, the necessary
consistency of the dynamics equation governing $\phi$ is checked by evaluating
the derivative of the scalar field potential $V\left(\phi\right)$ for low
redshifts.

In [697], it was shown that the proposed scenario consistently works,
reproducing both an effective evolution of the Hubble constant and a current
cosmic accelerated phase. Furthermore, the same picture was implemented on a
purely numerical level without imposing the existence of a flat region of the
potential, which naturally emerges for $z<0.3$, i.e., in the dark energy-
dominated Universe.

Figure 3: $H_{0}$ versus $z$ plot in the case of $|F_{R0}|\leq 10^{-7}$,
according to four different values of $\Omega_{0m}=0.298,0.301,0.303,0.305$
and combining SNe Ia+BAOs (plot taken from [2]).

##  4 GRBs and QSOs: promising cosmological probes

GRBs are the most energetic phenomena observed in the universe after the Big
Bang. Their origin is cosmological, and they arise from the collapse of
massive stars in the case of Long GRBs (with intrinsic duration $>2\,s$),
while in the case of Short GRBs ($<2\,s$) the progenitor is believed to be the
merging of two compact objects - such as two neutron stars or a neutron star
with a black hole. GRBs are observed up to high redshifts; the distance record
is currently held by GRB 090429B with a redshift value $z=9.4$ [578]. It is
necessary to invoke reliable correlations between the luminosity or energy of
the GRBs and other physical parameters independent of the cosmological model
to use GRBs as standardizable candles.
In the past years, many contributions in this field have been provided
focusing on the prompt properties of GRBs [585, 583, 584]. More recently, the
idea of considering the afterglow properties for the search of tight and
unbiased correlations has led to the discovery of the 2-dimensional Dainotti
relations [557, 563, 564, 565, 566, 567, 568, 574, 698] between the end of the
GRB plateau emission $T^{*}_{a}$ and the luminosity at the end of the plateau
$L_{a}$ in the X-rays, optical, and radio wavelengths. Adding the peak prompt
luminosity, $L_{p}$, the so-called fundamental plane relation or (Dainotti
relation) is unveiled [545, 509, 570, 571, 572, 548, 577, 575, 573, 699, 700,
701, 702, 703, 560, 561, 704, 592, 596, 597]. This correlation can be
expressed through the following equation:

$log_{10}L_{a}=a*log_{10}T^{*}_{a}+b*log_{10}L_{p}+c,$ (4)

being $a,b$ the parameters of the plane and $c$ the renormalization constant.
It is important to stress that the physical observables $T^{*}_{a}$, $L_{a}$,
and $L_{p}$ have been corrected for selection biases and redshift evolution
effects through the Efron and Petrosian method [656]. First, for the
fundamental plane reference, the platinum sample of X-ray GRBs is chosen
[573]. Then, in [705] the simulations of 1000 GRBs and 1200 GRBs according to
the platinum sample properties have been performed. For each of the two
simulated samples, a MCMC analysis with the D’Agostini method [706] has been
done varying the parameters $a,b,c,\sigma,\Omega_{0m}$, where $\sigma$ is the
intrinsic scatter of the fundamental plane relation in Equation 4. The results
show how in the case of 1000 GRBs, a value of $\Omega_{0m}=0.280\pm 0.111$ is
obtained, while in the 1200 GRBs simulated sample, the same parameter is
$\Omega_{0m}=0.270\pm 0.092$. The results are reported in Figure 4. According
to the rate of GRBs with plateau observed by the Swift satellite, it is
expected that by 2030 the observations of GRBs with properties similar to the
ones from the platinum sample will lead to a standalone estimation of the
$\Omega_{0m}$ parameters comparable with the one reported in [379].

Figure 4: Upper panel. The simulation of 1000 GRBs according to the
fundamental plane properties, with 5600 steps elapsed. Lower panel. The
simulation of 1200 GRBs, with 4500 steps elapsed. The plots are taken from
[705].

QSOs are very bright active galactic nuclei observed up to redshift $z=7.642$
[602]. Similarly to GRBs, to use QSOs as distance measurement tools a
statistically reliable correlation between their luminosity and other
parameters is needed. To this end, the use of Risaliti & Lusso (RL) relation
[707] between the X-ray and ultraviolet fluxes of quasars represents a
reliable candidate. The RL relation can be expressed with the following:

$log_{10}L_{X}=g*log_{10}L_{UV}+b,$ (5)

denoting with $sv$ the intrinsic scatter of the correlation. The unbiased
nature of this correlation for QSO has been proved in [708, 4]. The RL
relation has been applied to QSOs [4]: assuming a flat $\Lambda$CDM model,
combining QSOs and SNe Ia, the results yield different values of $H_{0}$ which
are in middle way between the results of SNe Ia and Planck data.

This outcome shows how the use of QSOs may significantly impact the study of
the Hubble tension.

##  5 Conclusions

The $H_{0}$ tension is a long-standing problem that still challenges our
current knowledge of astrophysics and cosmology. In particular, the approach
adopted for the analysis of the SNe Ia Pantheon sample shows a mild decreasing
trend of $H_{0}$ with redshift. This trend is present in the data, regardless
of the number of bins in which the Pantheon sample has been divided or the
number of parameters allowed to vary or, even more, the presence of a further
cosmological probe (BAO) even binned.
These outcomes could be due to the presence of hidden astrophysical biases or
redshift evolution effects in the SNe Ia data. One of the possible causes can
be the evolution of the SNe Ia stretch parameter with redshift which has
already been discussed in [709].
In case results are not due to biases or astrophysical effects, they can be
explained through alternative theoretical scenarios, e.g., the modified
gravity theories like the $f(R)$ theories of gravity. Since the Hu-Sawicki
model has not fully accounted for the observed $H_{0}$ trend, it is likely
that this effect could be still due to modified gravity dynamics, but with
different features from the Hu-Sawicki model.
Finally, we have also shown that GRBs and QSOs are promising tools for the
measurement of cosmological distances and the estimation of cosmological
parameters. In the case of GRBs, the analysis of 1200 events simulated from
the fundamental plane relation gives a value $\Omega_{0m}=0.270\pm 0.092$,
confirming the possibility for the GRBs to reach, by 2030, the same precision
on this parameter as the estimation in [379].
For the QSOs, when these are combined with SNe Ia, mixed outcomes about the
Hubble constant are found.
These phenomena extend the Hubble diagram up to $z=7.642$ and $z=9.4$ in the
cases of QSOs and GRBs, respectively: these values are far beyond the current
observational limit of SNe Ia, being the furthest SNe Ia observed at $z=2.26$.
This important feature of GRBs and QSOs suggests how these probes can be used
to cast more light on open cosmological problems, in particular, the Hubble
tension.

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