11institutetext: David Fernández-Arenas (✉) 22institutetext:
Canada–France–Hawaii Telescope, 65-1238 Mamalahoa Hwy Kamuela, Hawaii 96743,
USA
22email<EMAIL_ADDRESS>33institutetext: Ricardo Chávez (✉)
44institutetext: Instituto de Radioastronomía y Astrofísica, UNAM, Campus
Morelia, C.P. 58089, Morelia, Mexico
Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582, 03940
Mexico City, Mexico 44email<EMAIL_ADDRESS>
# Determination of the local Hubble constant using Giant extragalactic H ii
regions and H ii  galaxies

David Fernández-Arenas and Ricardo Chávez

###### Abstract

The relationship between the integrated H$\beta$ line luminosity and the
velocity dispersion of the ionized gas of H ii galaxies and giant H ii regions
represents an exciting standard candle that presently can be used up to
redshifts $z\sim~{}4$. Locally it is used to obtain precise measurements of
the Hubble constant by combining the slope of the relation obtained from
nearby ($z<0.2$) H ii galaxies with the zero point determined from giant H ii
regions belonging to an ‘anchor sample’ of galaxies for which accurate
redshift-independent distance moduli are available (e.g Cepheids, TRGBs).
Through this chapter, we present a general description of the method and
results obtained so far in the determination of the local value of the Hubble
constant and cosmological constraints. We account for the main systematic
effects associated with the method and the possibility of improvement in the
future. The discussion presented here is the product of an independent
approach to the standard methods used in the literature to measure distances
and, although at present less precise than the latest SNIa results, it is
amenable to substantial improvement.

##  0.1 Introduction

The search for standard candles or rulers to measure cosmic distances is one
of the fundamental efforts in the last decades in the interface of
observational astronomy and cosmology. However, it has been a difficult task
for a long time. In this chapter, we present a historical review of the use of
H ii galaxies (HIIGs) and giant H ii regions (HIIRs) to carry out this task of
measuring distances and its application in the determination of the Hubble
constant, from its beginnings to recent years. We discuss its applicability,
its potential physical origin and the associated errors which can be improved
in the future.

“Peculiar” objects from Zwicky list of compact galaxies were identified by
Sargent1970a ; Sargent1970b  and further studied by Sargent1970 . Those that
have spectral characteristics indistinguishable from giant extragalactic H ii
regions (like 30 Dor in the LMC) were designated by them as “Isolated
Extragalactic H ii Regions”. Later on, these compact, dwarf, star-forming
galaxies, were dubbed H ii Galaxies cam86 ; Terlevich1981 ; Maza1991 .

In the early 1960s, Sersic discovered that the diameters of the largest HIIRs
in spiral galaxies increase with the galaxy luminosity. The use of the HIIRs
for the determination of distances goes back to the mid-70s, with the studies
of Sandage and Tammann sandage_and_tamman1974 , mainly with the “classical”
application, an empirical correlations between the diameters of the HIIRs and
the parent galaxies luminosity. However, the correlation was not useful beyond
20 Mpc where the angular diameters become comparable with the seeing even for
the largest HIIRs.

Later, Melnick in 1977 Melnick1977  studying the structures of HIIRs, noticed
that their linear diameters also correlated well with the width of their
integrated H$\alpha$ line. This correlation between the velocity dispersion
(line-width) in HIIRs and its size is analyzed and established in members of
the Local Group and M81 and NGC 2403 group allowing a new determination of the
distance to M101. In 1978, Melnick Melnick1978  explored the close correlation
between the linear diameters of HIIRs in late-type galaxies and the width of
their global H$\alpha$ emission and used it to establish a close correlation
between this width and the absolute magnitude of the parent galaxies; this
correlation was calibrated in members of the local and NGC 2403 groups, and
then used again to estimate the distance to NGC 300 and M101. The values
obtained were consistent with those reported in the literature at that epoch.

In 1979, the relation between the absolute H$\beta$ fluxes from HIIRs and the
corresponding linear diameters and turbulent velocities (width of their
integrated H$\alpha$ line) is examined and the results presented in
Melnick1979 . In this study, a strong correlation was found between the linear
diameters and the absolute H$\beta$ luminosities, no longer of the galaxies
but of the HIIRs themselves. However, the large scatter precludes its
application to distance determination with the properties of the dust being
largely responsible for the observed scatter in the relation.

In 1981, Terlevich and Melnick Terlevich1981  analyzed the correlations
between H$\beta$ luminosities, linear diameters and the widths of the global
emission line profiles of 19 giant extragalactic HIIRs in 6 galaxies and 4 HII
galaxies. They found that the relations: luminosity$\propto$(linewidth)4 and
size$\propto$(linewidth)2 which are valid for pressure-supported stellar
systems (elliptical galaxies, bulges of spiral galaxies and globular clusters)
are also valid for giant HIIRs.

From this and other considerations, they proposed that giant extragalactic HII
regions are self-gravitating systems in which the observed emission-line
profile widths represent the velocity dispersion of discrete gas clouds in the
gravitational potential of these massive gas—young star complexes.

The analysis also showed that the scatter observed in the H$\beta$ luminosity-
linewidth relation for their sample is largely due to a metallicity effect and
after correction for this effect, the scatter in the relation is reduced with
strong implications as a new method for distance calibrations and although
their results are based on a very small sample, they highlight the need for
more and better observations of HIIRs. They proposed that these correlations
can be calibrated using HIIRs in nearby galaxies for which the distances are
known to obtain distances to field spirals and irregular galaxies beyond the
Virgo cluster.

From that moment on, the empirical correlations among integrated parameters of
HIIRs are considered to have fundamental importance for the understanding of
the nature and origin of these objects and represent an independent indicator
of distance.

In 1987, with a new homogeneous sample of data for 22 HIIRs in 10 galaxies
with known distances, Melnick et. al. melnick1987  studied the different
correlations among integrated parameters of HIIRs. They confirmed the
existence of significant correlations between core radius, emission line
width, H$\beta$luminosity, and oxygen abundance, providing a powerful distance
indicator for galaxies with HIIRs. Since, the distance indicator of HIIRs
relies on the correlations between H$\beta$ luminosity, the emission line-
profile widths and the oxygen abundance of giant HIIRs, they proposed a
possible physical scenario where the stellar winds and gravity are the most
promising mechanisms for explaining the observations, although not explaining
completely the presence of supersonic motions in the ionized gas.

In 1987, Melnick et. al. Melnick1987RMxAA  proposed that the most useful
application of HIIRs to the determination of distances, is the correlation
find by Terlevich1981 , which, we will call the $L(\mathrm{H}\beta)-\sigma$
relation. Using this extragalactic distance indicator tied to galactic scale
via cepheid variables, 41 HIIGs from the group’s spectrophotometric catalogue
tel91  and the zero point calibrated with the giant HIIRs in nearby spiral
galaxies, for the first time a value of H0=$95\pm 10$
$\rm{km~{}s^{-1}}~{}Mpc^{-1}$was obtained for the Hubble parameter, with the
distance calibrator given by:

$\log\rm L(\rm H\beta)=\log\frac{\sigma^{5}}{\rm(O/H)}+41.32$ (1)

A year later, Melnick et. al. melnick1988  calibrated the correlation between
luminosity, line width and oxygen abundance. They reported a value for H0 of
$89.0\pm 10$ $\rm{km~{}s^{-1}}~{}Mpc^{-1}$, after correcting the flows for
Malmquist bias and the radial velocities for the movement of the Local Group.
In this study, they discuss the initial potential problems related to the
application of the method: the zero point errors and the curvature of the
$L(\mathrm{H}\beta)-\sigma$ relation. An underestimate of the zero point may
arise if many HIIGs are composed of more than one HIIR superimposed along the
line of sight. On the other hand, the curvature related to the broadening of
the emission lines may be affected by the gravity of the overall galaxy or
systems with rotation. If present, this effect will saturate the
$L(\mathrm{H}\beta)-\sigma$ relation at high luminosities where the line
broadening will be dominated by the gravity of the old stellar component.
Excluding systems with a linewidth $>60$ km s-1 the correlation between
H$\beta$ luminosity, the width of the line and oxygen abundance, HIIGs show
the same functional form and the same scatter as that for local HIIRs,
suggesting that any broadening of the lines by a source not related to the
young stellar population must be small. Additionally, by imposing the
condition that the equivalent width of H$\beta$$>30$Å, effectively the number
of galaxies where the light of the young component is severely affected by the
underlying older stellar population is reduced.

In the last two decades, with the advance of technology in new detectors, the
new generation of big telescopes and given the potential use of HIIRs and
HIIGs as distance indicators, a strategy to measure the Hubble constant and to
constrain cosmological parameters was devised. It is a long-term program using
HIIGs, as standard candles that can, in principle, be applied out to $z\sim
3.5$ Melnick2000 ; Plionis2011 ; terlevich2015  with present-day ground-based
instrumentation and up to $z\sim 9$ with the Near Infrared Spectrograph
(NIRSpec) component of the James Webb Space Telescope (JWST) (deGraaff2023 ;
Bunker2023  and Chavez et al in preparation). Let’s remember that the
potential of HIIGs as distance indicators stems from the existence of a
correlation between the luminosity of a hydrogen recombination line, e.g.
L(H$\beta$) (proportional to the number of ionising photons), and the line
width of the emission line, $\sigma$.

In what follows, we show the latest advances in the description of the
$L(\mathrm{H}\beta)-\sigma$ relation, its calibrations with new data sets and
the application to the determination of the local value of the Hubble
constant.

##  0.2 The L-$\sigma$ relation

Two of the more recent works related to the $L(\mathrm{H}\beta)-\sigma$
relation for HIIGs have been presented in the works of Bordallo & Telles
Bordallo2011  and Chávez et. al.Chavez2014  in 2011 and 2014, respectively.
They presented new observational datasets of high- and low- spectral
resolution. After multi-parametric fits, they found that the H$\beta$
equivalent width is more important for determining the scatter in the
$L(\mathrm{H}\beta)-\sigma$ relation than the dependence on metallicity,
contrary to the suggestion by Terlevich et. al. Terlevich1981 . Assuming that
the EW(H$\beta$) is an indicator of the burst’s age, the interpretation is
that the age of the underlying older stellar population could be contaminating
the luminosity of a recent dominant young burst more than its velocity
dispersion. Chávez et. al. Chavez2014  show that the majority of the scatter
in the $L(\mathrm{H}\beta)-\sigma$ relation is reduced using the size of the
burst as a second parameter, which allows us to think of a fundamental plane
defined by the luminosity, velocity dispersion and size in HIIGs and HIIRs.

Both studies conclude that, in addition, part of the scatter in the
$L(\mathrm{H}\beta)-\sigma$ relation can be reduced by using the objects with
the most Gaussian profiles in their emission lines. However, the cost of
reducing the samples is of the order of 50% using this criterion, which can be
parameterized in different ways, (see section 4.2 in Bordallo2011  and sec 6.1
in Chavez2014 ). The main parametric relations presented in the above papers
are reproduced in table 0.2. According to different settings, constraints are
placed on the slope between 3-5, which could be consistent with a fundamental
plane similar to that followed by elliptical galaxies and old star systems
supported by gravity. A study on the evolution of these young systems and
their scale relationships compared to old systems can be found in Terlevich
et. al. Terlevich2018 . However, the existence of the $L-\sigma$ relation for
HIIRs and HIIGs poses an intriguing question about the origin of the
supersonic velocity widths of the emission line profiles. A simple explanation
would be that metal rich galaxies produce more massive starbursts, and this
should be verified by observations. Another hint about the origin of velocity
widths found in HIIGs is provided by the systematic difference found between
the width of Balmer lines and [O iii] lines, the most recent studies relating
to “The $L-\sigma$ relation for HIIGs in green” can be found in Melnick et.
al. 2017 Melnick2017 . A possible explanation would be the existence of a well
behaved ionization structure, in which more excited ions are concentrated
closer to the ionizing sources and densest regions.

Table 1: Fits for the $L-\sigma$ relation in local HIIGs

$\log\rm L=\alpha+\beta\times\log\sigma$
---
|  |  |  |
Linear Regression |  Intercept ($\alpha$) |  Slope ($\alpha$) |  rms |  N
|  |  |  |
(BT11, Here the luminosity of H$\alpha$ has been used instead of H$\beta$
luminosity)
OLS(Y$\mid$X) |  36.21 $\pm$ 0.32 |  3.01 $\pm$ 0.23 |  0.37 |  81a
OLS(X$\mid$Y) |  34.52 $\pm$ 0.38 |  4.18 $\pm$ 0.27 |  |
OLS Bisector |  35.49 $\pm$ 0.32 |  3.51 $\pm$ 0.23 |  |
OLS(Y$\mid$X) |  35.29 $\pm$ 0.42 |  3.72 $\pm$ 0.31 |  0.31 |  53b
OLS(X$\mid$Y) |  33.73 $\pm$ 0.47 |  4.85 $\pm$ 0.34 |  |
OLS Bisector |  34.61 $\pm$ 0.41 |  4.22 $\pm$ 0.30 |  |
OLS(Y$\mid$X) |  34.80 $\pm$ 0.41 |  4.14 $\pm$ 0.29 |  0.29 |  37c
OLS(X$\mid$Y) |  33.45 $\pm$ 0.53 |  5.13 $\pm$ 0.38 |  |
OLS Bisector |  34.19 $\pm$ 0.43 |  4.58 $\pm$ 0.30 |  |
(C14)
Error in both axes |  33.71 ± 0.21 |  4.65 ± 0.14 |  0.332 |  107d
Error in both axes |  33.22 ± 0.27 |  4.97 ± 0.17 |  0.332 |  69e

a All galaxies with homogeneous spectrophotometry in. bThe sample showing
regular Gaussian profiles. cThe sample showing regular Gaussian profiles and
homogeneous data. dAll galaxies with homogeneous data in. eThe sample showing
regular Gaussian profiles.

##  0.3 The data

The use of the $L-\sigma$ relation as a distance indicator and as a tool to
derive the Hubble constant requires accurate determination of both the
luminosity and the FWHM or velocity dispersion of the emission lines in giant
HIIRs and HIIGs. Observationally HIIGs and HIIRs represent the youngest Super
Star Clusters that can be observed in any detail. The measurements that are
required for the calibration of the $L(\mathrm{H}\beta)-\sigma$ relation are
the following:

  1. 1.

Narrow-band imaging or low-resolution-wide slit spectroscopy to obtain the
integrated H$\beta$ flux.

  2. 2.

High-resolution spectroscopy to measure the velocity dispersion from the
H$\beta$ (or [O iii] for comparison) line profile.

In figure 1, we show as an example the data for one HIIR in a nearby galaxy
used in the calibration of the zero-point of the $L-\sigma$ relation.

Figure 1: H$\alpha$ image obtained from NASA/IPAC Extragalactic Database
(NED), high-resolution profile for the giant HIIR (IC10: 101) and low-
resolution spectrum. Taken from Fernandez2018

To guarantee the best-integrated spectrophotometry and the youngest objects,
the following selection criteria are applied:

  * •

A lower limit for the equivalent width, EW(H$\beta$)$>50$Å(HIIGs and giant
HIIRs).

  * •

An upper limit to the Balmer line widths, $\log\sigma<1.8$ km s-1(HIIGs and
giant HIIRs).

  * •

Redshift range $0.01<z<0.2$ (HIIGs).

  * •

Petrosian diameter less than 6 arcsecs (HIIGs).

Figure 2 shows typical HIIG in the local universe and their high-resolution
spectrum dominated by strong emission lines.

Figure 2: A selection of colour images of HIIGs from the sample of Chavez2014
. The SDSS name and the index number are indicated in the stamps. The changes
in colour are related to the redshift of the object. In the right panel:
Examples of the high dispersion spectra obtained for the same object with
Subaru HDS (top) and VLT UVES (bottom), showing the region covering H$\beta$
and the [O iii] lines at $\lambda\lambda$ 4959,5007 Å. The instrumental
profile is shown in red at the left of each spectrum. Taken from Chavez2014 .

The above criteria allow us to exclude highly evolved regions, to diminish
contamination by an underlying older stellar population and to avoid objects
with a high rate of escape of ionizing photons. They also minimize the
possibility of including systems supported by rotation or with multiple young
ionizing clusters as it was discussed in melnick1988 . The lower redshift
limit in HIIGs is selected to avoid nearby objects that are more affected by
local peculiar motions relative to the Hubble flow and the upper limit was set
to minimize cosmological non-linearity effects.

In figure 3, we can see the evolution of the H$\beta$ equivalent width for a
simple instantaneous burst with metallicity $Z=0.004$. In the right panel, we
show the location of the HIIGs in the BPT diagrams Baldwin1981  which are
widely used to differentiate the excitation mechanism for star-forming
galaxies from AGN, the EW criterion clearly can separate between other similar
star-forming ionizing mechanism as Blue Compact Dwarfs and also show green
peas galaxies as a class of HIIGs discovered using different selection
criteria. Interesting, in recent years, making evolution studies of these
objects the interpretation has been that these young massive stellar clusters,
HIIGs and GHIIRs can evolve to form globular clusters and ultracompact dwarf
ellipticals in about 12 Gyr so that present-day globular clusters and
ultracompact dwarf ellipticals could have originated in conditions similar to
those observed in today GHIIR and HIIG Terlevich2018 . However, this is a
topic for another discussion.

Figure 3: The evolution of the H$\beta$equivalent width for an instantaneous
burst with metallicity $Z=0.004$ and a Salpeter IMF with an upper limit of 100
M⊙Leitherer1999 . The blue area marks the H$\beta$ equivalent width of 50Å,
corresponding to an age of $\sim 5$ Myrs. In the right plot: BPT diagram
showing the high excitation level of a sample of HIIGs selected mainly as
having high equivalent width in their Balmer emission lines. The solid line
represents the upper limit for stellar photoionization, from Kewley2001 .
Taken from Fernandez PhD Thesis, 2018.

###  0.3.1 Giant HII Regions and HII galaxies

HIIG are compact and massive systems experiencing luminous bursts of star
formation generated by the formation of young super clusters (SSCs) with a
high luminosity per unit mass and with properties similar, if not identical,
to giant HIIRs. The potential of GHIIRs as distance indicators was originally
realized from the existence of a correlation between the giant HIIR diameter
and the luminosity Sersic1960 ; Sandage1962  see also Kennicutt1979 .

Most HIIG were discovered in objective-prism surveys thanks to their strong
narrow emission lines. Currently, in spectroscopic surveys like Sloan Digital
Sky Survey (SDSS), they are selected by very large equivalent widths in the
Balmer lines. Since the luminosity of HIIG is dominated by the starburst
component they can be observed even at large redshifts becoming interesting
standard candles.

So far, we have analyzed a sample of HIIG and GHIIRs containing 217 objects,
which can be split into: the anchor sample (36 GHIIRs in 13 local galaxies
with distances from primary indicators, Fernandez2018 ), the local sample of
HIIG (107 with $z<0.16$, Chavez2014 ) and a high-$z$ sample based on 29 KMOS,
15 MOSFIRE, 6 XShooter and 24 literature objects (see Gonzalez2019 ;
Gonzalez2021  for more details). We used these data to constrain cosmological
parameters and to determine the local value of the Hubble constant by means of
the distance indicator defined by the $L-\sigma$ relation.

Figure 4: $L-\sigma$ relation for the GHIIRs. The adopted distances to the
parent galaxies come from Cepheids. The green solid line is the fit to the
data given in the inset and the dashed line is the fit. Taken from
Fernandez2018 .

##  0.4 Methodology

The Hubble constant is determined as follows: first we fix the slope of the
$L-\sigma$ relation using the velocity dispersion and luminosity of the HIIGs.
The slope is independent of the actual value of H0.

To estimate the Hubble constant we use the slope ($\alpha$) of the $L-\sigma$
relation of the HIIG and new GHIIR data (anchor sample) to calibrate the zero
point ($Z_{p}$) of the distance indicator (see figure 4) as follows:

$Z_{p}=\frac{\sum_{i=1}^{36}{{\it W_{i}}(\log L_{\rm
GHR,i}-\alpha\times\log\sigma_{\rm GHR,i})}}{\rm\sum_{i=1}^{36}{{\it W_{i}}}}$
(2)

where $L_{GHR,i}$ is the H$\beta$ luminosity of each GHIIR and $\sigma_{\rm
GHR,i}$ is the corresponding velocity dispersion. The statistical weights
$W_{i}$ are calculated as:

$W_{i}^{-1}=\Bigl{(}0.4343\frac{\delta L_{\rm GHR,i}}{L_{\rm
GHR,i}}\Bigr{)}^{2}+\Bigl{(}0.4343\alpha\frac{\delta\sigma_{\rm
GHR,i}}{\sigma_{\rm GHR,i}}\Bigr{)}^{2}+(\delta\alpha)^{2}(\sigma_{\rm
GHR,i}-<\sigma_{\rm HIIG}>)^{2}$ (3)

where $<\sigma_{\rm HIIG}>$ is the average velocity dispersion of the HIIG
that define the slope of the relation. Thus, the calibrated $L-\sigma$
relation or distance estimator is: $\log L(H\beta)=\alpha\log\sigma+Z_{p}$. To
calculate the Hubble constant we minimise the function,

$\chi^{2}(\textrm{H}_{0})=\sum_{i=1}^{N}[{W_{i}(\mu_{i}-\mu_{\textrm{H}_{0},i})^{2}-ln(W_{i})]}$
(4)

where $\mu_{i}$ is the logarithmic distance modulus to each HIIG calculated
using the distance indicator and the H$\beta$ flux F(H$\beta$) as

$\mu_{i}=2.5[Z_{p}+\alpha\times\log\sigma_{i}-\log F_{i}(H\beta)-\log 4\pi]$
(5)

and $\mu_{\textrm{H}_{0},i}$ is the distance modulus calculated from the
redshift using either the linear relation $D_{L}=zc/\textrm{H}_{0}$ or the
full cosmological prescription with $\Omega_{\Lambda}=0.71$.

The best value of $\textrm{H}_{0}$ is then obtained minimising $\chi^{2}$ with
statistical weights
$W_{i}^{-1}=\delta\mu_{i}^{2}+\delta\mu_{\textrm{H}_{0},i}^{2}$ calculated as,

$W_{i}^{-1}=6.25[(\delta Z_{p})^{2}+\Bigl{(}0.4343\frac{\delta
F_{i}}{F_{i}}\Bigr{)}^{2}+\Bigr{(}0.4343\alpha\frac{\delta\sigma_{i}}{\sigma_{i}}\Bigl{)}^{2}+(\delta\alpha)^{2}(\sigma_{i}-<\sigma>)^{2}]$
(6)

##  0.5 Systematics

Genuine systematic errors are difficult to estimate. However, we have explored
and listed a range of parameters to quantify at least part of the systematic
error component. In particular, we present different systematic errors that
can be taken into account in order to determine the value of H0 using giant
HIIRs and HIIGs, these effects are described in the following:

###  0.5.1 Age Effects

Even in our sample of HIIGs and giant HIIRs, chosen to be the youngest
systems, it is crucial to verify that the rapid luminosity evolution of the
stellar cluster does not introduce a systematic bias in the distance
indicator. This would happen, for example, if the average age of the giant
HIIRs is different from that of the HIIGs or if luminous and faint HIIGs have
different average ages. In general, if velocity dispersion measures mass,
younger clusters will be more luminous than older ones for a given velocity
dispersion. The evolution effect can be scrambled by the superposition of
bursts of different ages along the line of sight. Nevertheless, even small
systematics can have a sizable effect on the value of H0 so it is important to
remove the evolution effect from the data in a similar fashion as for the dust
extinction.

The equivalent width of H$\beta$ [EW(H$\beta$)] is a useful age estimator
dottori1981 ; sta96 ; mar08  or at least it provides an upper limit of the age
of the burst terlevich2004 . Indeed there is some empirical evidence for this
as discussed by Melnick2000  and Bordallo2011 . Chavez2014  explored the
possibility that the age of the burst is a second parameter, using the
EW(H$\beta$) as an age estimator in the $L(\mathrm{H}\beta)-\sigma$ relation
for HIIGs, and found a rather weak dependence. Different evolution corrections
have been explored in recent years, e.g. Fernandez2018  and Telles2018 . This
correction must be taken into account in order to improve the estimates of the
value of H0 but it must still be explored in more detail with better
observations especially to characterize the underlying older population.

###  0.5.2 Extinction Effects

One of the major contributions to the systematic errors is related to
extinction correction, in particular the use of different extinction laws. In
particular,calzetti2000  law yields larger values of H0 than Gordon2003
typically by about 1.5$\sim$$\rm{km~{}s^{-1}}~{}Mpc^{-1}$. As the calzetti2000
law was derived from a sample of eight heterogeneous starburst galaxies where
only two, Tol 1924-416 and UGCS410 are bonafide HIIGs and the rest are evolved
high metallicity starburst galaxies, we prefer to use the Gordon2003
extinction curve which corresponds to the LMC2 supershell near the 30 Dorado
star-forming region, the prototypical giant HIIR, in the Large Magellanic
Cloud.

###  0.5.3 Metallicity Effects

In 1981, Terlevich et. al. Terlevich1981  proposed that oxygen abundance is a
good indicator of the long-term evolution of the system. They proposed a
simple ‘closed box’ chemical evolution model with many successive cycles of
star formation. In it, for each cycle, evolution is traced by the EW(H$\beta$)
whereas the long-term evolution of the system, spanning two or more cycles,
could be traced by the oxygen abundance, which then becomes a plausible second
parameter in the $L(\mathrm{H}\beta)-\sigma$ correlation.

###  0.5.4 Aperture Effects

The sensitivity to H$\beta$ photometry could be crucial for the determination
of the luminosity especially when multiple bursts of star-forming regions and
the nearest HIIGs are present. This effect was explored by Fernandez2018  by
comparing SDSS and Chavez2014  photometric measurements. The results reveal
that the values of H0 are on average systematically lower by about
4.9$\sim$$\rm{km~{}s^{-1}}~{}Mpc^{-1}$for SDSS fluxes and no evolutionary
corrections. This systematic difference is reduced to
1.7$\sim$$\rm{km~{}s^{-1}}~{}Mpc^{-1}$when evolutionary corrections are
included. The smaller value of H0 for the SDSS photometry is related to the
systematically steeper slope of the $L(\mathrm{H}\beta)-\sigma$ relation
compared with that obtained when using Chavez2014  photometry, both with and
without evolution correction. Since the lower luminosity HIIGs are also the
closer ones, the steepening is a consequence of the smaller SDSS aperture,
compared with that of Chavez2014 , underestimating the line fluxes of the
nearest galaxies. This effect may also introduce a bias in luminosity or
velocity dispersion due to the multiplicity seen in some objects (e.g.
J084220+115000) as presented by Fernandez2023 .

###  0.5.5 Redshift Effects

An upper redshift cutoff of $z=0.1$ was set (S2) instead of $z=0.16$ (S1) for
the HIIGs. Comparing S1 (in which the distances were computed using a flat
cosmology with $\Omega_{m}=0.29$) with sample S2, using the linear Hubble
relation for the distances, we found that using $z<0.1$ reduces the value of
H0 typically by about 2 $\sim$$\rm{km~{}s^{-1}}~{}Mpc^{-1}$with a range from
1.1 to 3.3 $\rm{km~{}s^{-1}}~{}Mpc^{-1}$. The sensitivity of H0 to the actual
value of $\Omega_{m}$ is low, amounting in our case to an uncertainty of about
0.1% in H0 for an uncertainty in $\Omega_{m}$ of 0.02 Betoule2014 . This,
together with the larger uncertainty when using S2 drives us to use S1 with
the distance determined using a flat cosmology with $\Omega_{m}=0.29$. This
reinforces the point that we are not biasing the results by using additional
cosmological parameters.

Figure 5: Left: The $L-\sigma$ relation for the Chavez2014  sample using the
velocity dispersions in the original paper; the fluxes have been corrected
using Gordon2003  extinction law. The solid line is the fit to the HIIGs. The
inset equation is the distance indicator where the slope is obtained from the
fit to the HIIG and the zero point is determined following the procedure
described in the text. In the right panel: Our main result incorporating the
evolution correction, is H0=$71.0\pm 3.5$
$\rm{km~{}s^{-1}}~{}Mpc^{-1}$(random+systematic) a value that is between the
most recent determination from Planck and SNIa. Taken and updated from
Fernandez2018 . We also plot the most recent results derived from the
cosmological constrain using HIIGs from Gónzalez-Morán et. al. Gonzalez2021 .

##  0.6 Cosmological constraints

To calculate the parameters of the $L-\sigma$ relation in a unified way
including HIIGs and GHIIRs, we define the following likelihood function:

$\mathcal{L}\propto\textrm{exp}(-\frac{1}{2}\chi^{2}_{HII})$ (7)

where:

$\chi^{2}_{HII}=\sum_{n}\frac{(\mu_{0}(\log
f,\log\sigma|\alpha,\beta)-\mu_{\theta}(z|\theta))^{2}}{\epsilon^{2}}$ (8)

where $\mu_{0}$ is the distance modulus calculated from a set of observables
as:

$\mu_{0}=2.5(\alpha+\beta\log\sigma-\log f-40.08)$ (9)

$\alpha$ and $\beta$ are the $L-\sigma$ relation’s intercept and slope,
respectively, $\log\sigma$ is the logarithm of the measured velocity
dispersion and log f is the logarithm of the measured flux. For HIIG the
theoretical distance modulus, $\mu_{\theta}$, is given as:

$\mu_{\theta}=5\log d_{L}(z,\theta)+25$ (10)

where $z$ is the redshift, $d_{L}$ is the luminosity distance in Mpc and
$\theta$ is a given set of cosmological parameters. For GHIIRs, the value of
$\mu_{\theta}$ is inferred from primary indicators and finally $\epsilon^{2}$
are the weights in the likelihood function.

The luminosity distance $d_{L}$ of the sources tracing the Hubble expansion is
employed to calculate the theoretical distance moduli. We define, for
convenience, an extra parameter independent of the Hubble constant as:

$D_{L}(z,\theta)=(1+z)\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime},\theta)}$
(11)

i.e., $d_{L}=cD_{L}/\textrm{H}_{0}$. $E(z.\theta)$ for a flat Universe is
given by:

$E^{2}(z,\theta)=\Omega_{r}(1+z)^{4}+\Omega_{m}(1+z)^{3}+\Omega_{w}(1+z)^{3y}\textrm{exp}\left(\frac{-3w_{a}z}{1+z}\right)$
(12)

with $y=(1+w_{0}+w_{a})$. The parameters $w_{0}$ and $w_{a}$ refer to the dark
energy equation of state (DE EoS) the general form of which is:

$p_{w}=w(z)\rho_{w}$ (13)

with $\rho_{w}$ the pressure and $\rho_{w}$ the density of the postulated Dark
Energy fluid. Different DE models have been proposed and many are parametrized
using a Taylor expansion around the present epoch:

$w(a)=w_{0}+w_{a}(1-a)\Longrightarrow w(z)=w_{0}+w_{a}\frac{z}{1+z}$ (14)

The cosmological constant is just a special case of DE, given for
$(w_{0},w_{a})=(-1,0)$, while the so called wCDM models are such that
$w_{a}=0$ but $w_{0}$ can take values $\neq-1$.

Figure 6: Hubble diagram connecting our local and high redshift samples up to
$z\sim 2.6$. Red circles represent averages of the distance moduli in redshift
bins. The continuous line corresponds to $\Omega_{m}=0.249$ and $w_{0}=-1.18$
(our best cosmological model using only HIIG). The insets show the
distribution of the residuals of the fit that are plotted in the bottom
panels. Taken from Gonzalez2021 .

##  0.7 Conclusions

The accurate determination of the Hubble constant, H0, is considered one of
the most fundamental tasks in the interface between astronomy and cosmology
and the $L(\mathrm{H}\beta)-\sigma$ relation is a powerful independent
distance indicator that has to be explored with more precise data in the
future. Here we summarise the main results of the last year in the use of the
HIIRs and HIIGs, the Hubble constant and the cosmological constraints.

This chapter presents the determination of the local value of the Hubble
constant and observational constraints of the cosmological parameters making
use of the GHIIRs in nearby galaxies and HIIG local and at high-$z$, the most
recent results are summarized in the Figures 4,5, 6, where we describe the
local $L(\mathrm{H}\beta)-\sigma$ relation followed by GHIIRs, the
determination of the local value of the Hubble constant, the local
$L(\mathrm{H}\beta)-\sigma$ relation, the Hubble diagram tracing local GHIIRs
up to high-$z$ HIIG, the space of solutions in the cosmological parameters and
the joint analysis of the other tracers of the Hubble expansion.

From the analysis of different systematic effects, we can infer that they
cause changes in the $L(\mathrm{H}\beta)-\sigma$ relation that translate into
r.m.s. variations in the value of H0 of 1.5 to 2.5
$\rm{km~{}s^{-1}}~{}Mpc^{-1}$.

On the other hand, combining HIIGs (the $L-\sigma$ relation as distance
indicator), CMB and BAO yields: $\Omega_{m}=0.298\pm 0.012$ and
$w_{0}=-1.005\pm 0.051$, fully consistent with the $\Lambda$CDM model. It is
clear that the solution space of HIIG/CMB/BAO, although less constrained, is
certainly compatible with the solution space of SNIa/CMB/BAO.

###### Acknowledgements.

D.F.A acknowledges the support from the National Science Foundation under
grant 2109124 for SIGNALS: Unveiling Star-Forming Regions in Nearby Galaxies.
R.C. gratefully acknowledges support from the CONAHCyT of Mexico under the
grant CF2022-320152. D.F.A and R.C. would like to thank professors Roberto
Terlevich and Elena Terlevich for their constant support throughout these
years and for their valuable suggestions, which have allowed us to improve the
chapter that we present here.

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