The time-dependent expansion of spacetime is characterized in the FLRW
equations as a function of redshift z by the Hubble parameter H(z).
Under the assumption of ΛCDM,
H(z) = H_{0} * sqrt(Ω_{m}(1+z)^{3} + Ω_{Λ} + Ω_{k}(1+z)^{2})
(e.g. Wei & Wu 2017, Chen, Kumar & Ratra 2017, Verde et al. 2014, Farooq & Ratra 2013).
The present-day (z=0) value of the expansion is referred to as the Hubble constant, H_{0}.
There is a long history in the literature toward determination of an unbiased
and accurate value of H_{0}, dating from the 1920's with the observations of
galaxies by Hubble. There are a number of reviews of this history; we cite
for example Freedman & Madore (2010), and online websites such as this one at CFA. Current values in the literature hover near
70 km/s/Mpc. However, a concordance value with associated uncertainty 1% or less has yet to be reached, and recent literature
has noted the moderate tension between values for H_{0} derived using a variety of methods.
Both unaccounted-for uncertainties in the data and potential inadequacy of the standard ΛCDM model
may explain the tensions.

We illustrate the recent history of Hubble constant determinations with two separate plots.
The first plot intercompares H_{0} values derived from a variety of methods. This plot is
designed to provide an overview of methods and tensions discussed in the literature.
Some measurements presented in this plot combine multiple datasets.
The second plot compares determinations of H_{0} through the use of only CMB data.

The first determinations of H_{0} involved measurements of distances and
radial velocities associated with objects
far enough away to not be bound gravitationally to our Local Group.
Distances determined on cosmological scales rely on the 'distance ladder', which
builds on the reliability of direct measurements close-by and the extension of those
measurements outward through the use of standard candles such as Cepheid variables and
Type Ia supernovae. We represent determinations of this type with three entries in the
first plot: the HST Key Project (Freedman et al. 2001),
Cepheids+SNIa (Riess et al. 2011)
and Distance Ladder (Riess et al. 2016).
Characterization of potential systematics affecting the distance determination is a key issue in this method.
Another standard candle method, based on the correlation between turbulent emission
lines velocity dispersion and their integrated luminosity, is employed by Fernández Arenas et al. (2018) to derive H_{0} from observations of H II galaxies and giant H II regions.
They report an initial result of 71.0 ± 2.8(random) ± 2.1(systematic) km/s/Mpc.

Recent advances in gravitational wave (GW) detection technology provide an
analysis approach independent of the cosmic distance ladder. The first
estimate of the Hubble constant using a GW source detection is a joint effort of
the LIGO/Virgo teams (Abbott et al. 2017), combined with collaborative
followup observations identifying the optical counterpart of the source.
In this determination, the amplitude
of gravitational waves resulting from the merger of a binary neutron star system are
analyzed to determine the luminosity distance to the system, and a cosmological redshift
obtained from optical identification of the source host galaxy. Degeneracy between the
computed luminosity distance and the binary orbital inclination angle is the primary source of uncertainty; the value obtained is H_{0} = 70.0_{-8}^{+12}.
Future GW source detections should provide tighter constraints.

In ΛCDM theory, a characteristic acoustic scale for
the oscillating primordial photon and baryon plasma is established at the time of decoupling,
when the plasma temperature has cooled sufficiently for recombination.
Acoustic features are detected not only in CMB radiation power spectra, but also in the
linear matter power spectra constructed from positional information of objects in large-scale structure surveys
(Baryon Acoustic Oscillations, BAO, e.g., Eisenstein, Hu & Tegmark 1998).
BAO data alone cannot be used to derive H_{0}, but combinations of BAO with CMB, large-scale structure (LSS)
clustering information
or baryon density may be used for this purpose (e.g., Addison et al. 2013).
A combination of this type, labeled 'BAO+D/H 2017' in the plot, is represented by the results of Addison et al. (2017),
who used BAO constrained with a baryon density determined from deuterium abundance combined with BBN theory. Aubourg et al. (2015) used BAO + SNIa in combination with CMB sound horizon scale (an 'inverse
distance ladder' technique) to infer H_{0} = 67.3 ±1.1 km/s/Mpc without the constraint of a flat ΛCDM cosmology.

Strong gravitational lensing of a variable source, resulting in multiple lensed images,
may be used to obtain H_{0} via
the observed time delay for an intrinsic source brightness change to
propagate between lensed images. The time delay function is proportional to H_{0}^{-1},
but also requires knowledge of the lens redshift and angular diameter distances involved,
and has some dependence on the assumed cosmology.
The technique presents challenges in observational precision and accuracy in modeling the lens mass distribution; for reviews, equations and error discussion see e.g.
Schechter (2005), Freedman & Madore (2010),
Suyu et al. (2010),
Linder (2011) and Wilson et al. (2017). Although the basic
theory was discussed as early as 1964 (Refsdahl 1964), improvements in observational sensitivity
and lens modeling have been a key to shrinking uncertainties. In the plot, we quote the most recent
results of the COSMOGRAIL project (H0LiCOW; Bonvin et al. 2017) based on observations of three
systems and the assumption of ΛCDM.

Observations of the thermal SZ effect (tSZ) have a number of cosmological applications
(e.g. determination of matter distribution, see also
Carlstrom et al. 2002,
Birkinshaw & Hughes 1994).
Determination of the Hubble constant independent of the cosmic distance ladder is one such application.
Combined tSZ decrement and X-ray emission observations of a galaxy cluster enable direct
determination of the angular diameter distance to that cluster, which along with its redshift
allow H_{0} to be computed using an assumed cosmology. Distance determinations for a
number of X-ray clusters and a range of spectroscopic redshifts provide reductions in statistical
uncertainties (Reese et al. 2002,
Carlstrom et al. 2002).
Bonamente et al. (2006) used CHANDRA X-ray observations and interferometric tSZ data for 38 clusters
to derive a value of H_{0} = 73.7_{-8.5}^{+10.5}; the quoted uncertainties are a quadrature sum of
statistical and systematic uncertainties. Systematics are a dominant issue in this determination:
subsequent analyses suggested larger absolute calibration uncertainties
for the X-ray instrument (Reese et al. 2010), and uncertainties in the modeling of the
cluster density and temperature profile can lead to systematic over- or under-estimates of
H_{0} (Bonamente et al. 2012, Batistelli et al. 2016).
We include the 2006 CHANDRA H_{0} result
in our plot as a reminder of the potential of the technique.

Finally, the value of the Hubble constant may be determined indirectly from CMB data through
the use of the ΛCDM model itself (the equation in the first paragraph shows the main parameter dependencies).
In addition to maps of
temperature and polarization CMB anisotropies, CMB observations on small angular scales
provide additional model constraints through the effect of weak gravitational lensing of the
CMB. In the first summary plot, WMAP and Planck data have been combined with data from other
sources, including BAO data (see data references), which reduces the quoted uncertainty.
In the CMB-only plot, we illustrate uncertainties obtained without combining with other datasets,
and a wider range of determinations, including suborbital measurements from ACTPol (Louis et al. 2017)
and SPTpol (Henning et al. 2017).
Because of smaller sky coverage, suborbital surveys lack large angular scale information and
must assume a prior for the optical depth to reionization, τ.

The Hubble Constant H_{0} characterizes the present-day expansion rate of the universe.
Its value may be determined using a variety of methods. The figure includes results
from distance ladder determinations
(labeled HST Key Project, Cepheids+SNIa, Distance Ladder),
indirect CMB measurements (WMAP9++, Planck_PR2++, both of which combine CMB with other data),
BAO in combination with baryon abundance (BAO+D/H), the thermal SZ effect (CHANDRA+tSZ), strong gravitational lensing
(Gravlens Time Delay) and gravitational waves (LIGO/Virgo grav waves).
More detailed descriptions of these methods are given in the text. Uncertainties in the
gravitational wave method are expected to decrease as the method matures. A tightly
constrained value for H_{0} with agreement across multiple methods has yet to be achieved.
The gray vertical line, representing the weighted average of WMAP and Planck data points, is positioned at 68.1 km/sec/Mpc.

The value of H_{0} may be inferred from fits of the ΛCDM model to observations of
the Cosmic Microwave Background (CMB).
Because the CMB data represent a snapshot
of the universe at high redshift, this method relies on the model to compute
the Hubble parameter at z=0 and is not a direct measurement of H_{0}.
How well low redshift direct measurements agree with model predictions based on fits
to high redshift data serves as a test of the standard ΛCDM model. This figure
illustrates recent H_{0} determinations using only CMB data, uncombined with lower redshift measurements.
Values with the smallest quoted uncertainties are those from the
Planck mission, which also show the most disagreement with the 2016 local distance ladder
determination shown in the preceding plot. The difference is an interesting topic requiring more
study. The gray vertical line at 68.1 km/sec/Mpc is taken from the first figure and shown as a fiducial guide.