Hubble's law

A decade before Hubble made his observations, a number of physicists and mathematicians had established a consistent theory of an expanding universe by using Einstein field equations of general relativity. Applying the most general principles to the nature of the universe yielded a dynamic solution that conflicted with the then-prevalent notion of a static universe. Slipher's observations In 1912, Vesto M. Slipher measured the first Doppler shift of a "spiral nebula" (the obsolete term for spiral galaxies) and soon discovered that almost all such objects were receding from Earth. He did not grasp the cosmological implications of this fact, and indeed at the time it was highly controversial whether or not these nebulae were "island universes" outside the Milky Way galaxy. FLRW equations In 1922, Alexander Friedmann derived his Friedmann equations from Einstein field equations, showing that the universe might expand at a rate calculable by the equations. The parameter used by Friedmann is known today as the scale factor and can be considered as a scale invariant form of the proportionality constant of Hubble's law. Georges Lemaître independently found a similar solution in his 1927 paper discussed in the following section. The Friedmann equations are derived by inserting the metric for a homogeneous and isotropic universe into Einstein's field equations for a fluid with a given density and pressure. This idea of an expanding spacetime would eventually lead to the Big Bang and Steady State theories of cosmology. Lemaître's equation In 1927, two years before Hubble published his own article, the Belgian priest and astronomer Georges Lemaître was the first to publish research deriving what is now known as Hubble's law. According to the Canadian astronomer Sidney van den Bergh, "the 1927 discovery of the expansion of the universe by Lemaître was published in French in a low-impact journal. In the 1931 high-impact English translation of this article, a critical equation was changed by omitting reference to what is now known as the Hubble constant." It is now known that the alterations in the translated paper were carried out by Lemaître himself. Shape of the universe Before the advent of modern cosmology, there was considerable talk about the size and shape of the universe. In 1920, the Shapley–Curtis debate took place between Harlow Shapley and Heber D. Curtis over this issue. Shapley argued for a small universe the size of the Milky Way galaxy, and Curtis argued that the universe was much larger. The issue was resolved in the following decade with Hubble's improved observations. Cepheid variable stars outside the Milky Way Edwin Hubble did most of his professional astronomical observing work at Mount Wilson Observatory, home to the world's most powerful telescope at the time. His observations of Cepheid variable stars in "spiral nebulae" enabled him to calculate the distances to these objects. Surprisingly, these objects were discovered to be at distances which placed them well outside the Milky Way. They continued to be called nebulae, and it was only gradually that the term galaxies replaced it. Combining redshifts with distance measurements to Hubble's law. Various estimates for the Hubble constant exist. The velocities and distances that appear in Hubble's law are not directly measured. The velocities are inferred from the redshift of radiation and distance is inferred from brightness. Hubble sought to correlate brightness with parameter . Combining his measurements of galaxy distances with Vesto Slipher and Milton Humason's measurements of the redshifts associated with the galaxies, Hubble discovered a rough proportionality between redshift of an object and its distance. Though there was considerable scatter (now known to be caused by peculiar velocities—the 'Hubble flow' is used to refer to the region of space far enough out that the recession velocity is larger than local peculiar velocities), Hubble was able to plot a trend line from the 46 galaxies he studied and obtain a value for the Hubble constant of 500 (km/s)/Mpc (much higher than the currently accepted value due to errors in his distance calibrations; see cosmic distance ladder for details). A straight line of positive slope on this diagram is the visual depiction of Hubble's law. Cosmological constant abandoned After Hubble's discovery was published, Albert Einstein abandoned his work on the cosmological constant, a term he had inserted into his equations of general relativity to coerce them into producing the static solution he previously considered the correct state of the universe. The Einstein equations in their simplest form model either an expanding or contracting universe, so Einstein introduced the constant to counter expansion or contraction and lead to a static and flat universe. After Hubble's discovery that the universe was, in fact, expanding, Einstein called his faulty assumption that the universe is static his "greatest mistake". The cosmological constant has regained attention in recent decades as a hypothetical explanation for dark energy. == Interpretation ==

The discovery of the linear relationship between redshift and distance, coupled with a supposed linear relation between recessional velocity and redshift, yields a straightforward mathematical expression for Hubble's law as follows: v = H_0 \, D where • is the recessional velocity, typically expressed in km/s. • is Hubble's constant and corresponds to the value of (often termed the Hubble parameter which is a value that is time dependent and which can be expressed in terms of the scale factor) in the Friedmann equations taken at the time of observation denoted by the subscript . This value is the same throughout the universe for a given comoving time. • is the proper distance (which can change over time, unlike the comoving distance, which is constant) from the galaxy to the observer, measured in mega parsecs (Mpc), in the 3-space defined by given cosmological time. (Recession velocity is just ). Hubble's law is considered a fundamental relation between recessional velocity and distance. However, the relation between recessional velocity and redshift depends on the cosmological model adopted and is not established except for small redshifts. For distances larger than the radius of the Hubble sphere , objects recede at a rate faster than the speed of light (See Uses of the proper distance for a discussion of the significance of this): r_\text{HS} = \frac{c}{H_0} \ . Since the Hubble "constant" is a constant only in space, not in time, the radius of the Hubble sphere may increase or decrease over various time intervals. The subscript '0' indicates the value of the Hubble constant today. Redshift velocity and recessional velocity Redshift can be measured by determining the wavelength of a known transition, such as hydrogen α-lines for distant quasars, and finding the fractional shift compared to a stationary reference. Thus, redshift is a quantity unambiguously acquired from observation. Care is required, however, in translating these to recessional velocities: for small redshift values, a linear relation of redshift to recessional velocity applies, but more generally the redshift-distance law is nonlinear, meaning the co-relation must be derived specifically for each given model and epoch. Redshift velocity The redshift is often described as a redshift velocity, which is the recessional velocity that would produce the same redshift it were caused by a linear Doppler effect (which, however, is not the case, as the velocities involved are too large to use a non-relativistic formula for Doppler shift). This redshift velocity can easily exceed the speed of light. In other words, to determine the redshift velocity , the relation: v_\text{rs} \equiv cz, is used. That is, there is between redshift velocity and redshift: they are rigidly proportional, and not related by any theoretical reasoning. The motivation behind the "redshift velocity" terminology is that the redshift velocity agrees with the velocity from a low-velocity simplification of the so-called Fizeau–Doppler formula z = \frac{\lambda_\text{o}}{\lambda_\text{e}}-1 = \sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}-1 \approx \frac{v}{c}. Here, , are the observed and emitted wavelengths respectively. The "redshift velocity" is not so simply related to real velocity at larger velocities, however, and this terminology leads to confusion if interpreted as a real velocity. Next, the connection between redshift or redshift velocity and recessional velocity is discussed. Recessional velocity Suppose is called the scale factor of the universe, and increases as the universe expands in a manner that depends upon the cosmological model selected. Its meaning is that all measured proper distances between co-moving points increase proportionally to . (The co-moving points are not moving relative to their local environments.) In other words: \frac {D(t)}{D(t_0)} = \frac{R(t)}{R(t_0)}, where is some reference time. If light is emitted from a galaxy at time and received by us at , it is redshifted due to the expansion of the universe, and this redshift is simply: z = \frac {R(t_0)}{R(t_\text{e})} - 1. Suppose a galaxy is at distance , and this distance changes with time at a rate . We call this rate of recession the "recession velocity" : v_\text{r} = d_tD = \frac {d_tR}{R} D. We now define the Hubble constant as H \equiv \frac{d_tR}{R}, and discover the Hubble law: v_\text{r} = H D. From this perspective, Hubble's law is a fundamental relation between (i) the recessional velocity associated with the expansion of the universe and (ii) the distance to an object; the connection between redshift and distance is a crutch used to connect Hubble's law with observations. This law can be related to redshift approximately by making a Taylor series expansion: z = \frac {R(t_0)}{R(t_e)} - 1 \approx \frac {R(t_0)} {R(t_0)\left(1+(t_e-t_0)H(t_0)\right)}-1 \approx (t_0-t_e)H(t_0), If the distance is not too large, all other complications of the model become small corrections, and the time interval is simply the distance divided by the speed of light: z \approx (t_0-t_\text{e})H(t_0) \approx \frac {D}{c} H(t_0), or cz \approx D H(t_0) = v_r. According to this approach, the relation is an approximation valid at low redshifts, to be replaced by a relation at large redshifts that is model-dependent. See velocity-redshift figure. Observability of parameters Strictly speaking, neither nor in the formula are directly observable, because they are properties of a galaxy, whereas our observations refer to the galaxy in the past, at the time that the light we currently see left it. For relatively nearby galaxies (redshift much less than one), and will not have changed much, and can be estimated using the formula where is the speed of light. This gives the empirical relation found by Hubble. For distant galaxies, (or ) cannot be calculated from without specifying a detailed model for how changes with time. The redshift is not even directly related to the recession velocity at the time the light set out, but it does have a simple interpretation: is the factor by which the universe has expanded while the photon was traveling towards the observer. Expansion velocity vs. peculiar velocity In using Hubble's law to determine distances, only the velocity due to the expansion of the universe can be used. Since gravitationally interacting galaxies move relative to each other independent of the expansion of the universe, these relative velocities, called peculiar velocities, need to be accounted for in the application of Hubble's law. Such peculiar velocities give rise to redshift-space distortions. Time-dependence of Hubble parameter The parameter is commonly called the "Hubble constant", but that is a misnomer since it is constant in space only at a fixed time; it varies with time in nearly all cosmological models, and all observations of far distant objects are also observations into the distant past, when the "constant" had a different value. "Hubble parameter" is a more correct term, with denoting the present-day value. Another common source of confusion is that the accelerating universe does imply that the Hubble parameter is actually increasing with time; since {{nowrap| H(t) \equiv \dot{a}(t)/a(t) ,}} in most accelerating models a increases relatively faster than {{nowrap|\dot{a},}} so decreases with time. (The recession velocity of one chosen galaxy does increase, but different galaxies passing a sphere of fixed radius cross the sphere more slowly at later times.) On defining the dimensionless deceleration parameter {{nowrap| q \equiv - \ddot{a} a / \dot{a}^2 ,}} it follows that \frac{dH}{dt} = -H^2 (1+q) From this it is seen that the Hubble parameter is decreasing with time, unless ; the latter can only occur if the universe contains phantom energy, regarded as theoretically somewhat improbable. However, in the standard Lambda cold dark matter model (Lambda-CDM or ΛCDM model), will tend to −1 from above in the distant future as the cosmological constant becomes increasingly dominant over matter; this implies that will approach from above to a constant value of ≈ 57 (km/s)/Mpc, and the scale factor of the universe will then grow exponentially in time. Idealized Hubble's law The mathematical derivation of an idealized Hubble's law for a uniformly expanding universe is a fairly elementary theorem of geometry in 3-dimensional Cartesian/Newtonian coordinate space, which, considered as a metric space, is entirely homogeneous and isotropic (properties do not vary with location or direction). Simply stated, the theorem is this: In fact, this applies to non-Cartesian spaces as long as they are locally homogeneous and isotropic, specifically to the negatively and positively curved spaces frequently considered as cosmological models (see shape of the universe). An observation stemming from this theorem is that seeing objects recede from us on Earth is not an indication that Earth is near to a center from which the expansion is occurring, but rather that observer in an expanding universe will see objects receding from them. Ultimate fate and age of the universe and ultimate fate of the universe can be determined by measuring the Hubble constant today and extrapolating with the observed value of the deceleration parameter, uniquely characterized by values of density parameters ( for matter and for dark energy).A closed universe with and comes to an end in a Big Crunch and is considerably younger than its Hubble age.An open universe with and expands forever and has an age that is closer to its Hubble age. For the accelerating universe with nonzero that we inhabit, the age of the universe is coincidentally very close to the Hubble age. The value of the Hubble parameter changes over time, either increasing or decreasing depending on the value of the so-called deceleration parameter , which is defined by q = -\left(1+\frac{\dot H}{H^2}\right). In a universe with a deceleration parameter equal to zero, it follows that , where is the time since the Big Bang. A non-zero, time-dependent value of simply requires integration of the Friedmann equations backwards from the present time to the time when the comoving horizon size was zero. It was long thought that was positive, indicating that the expansion is slowing down due to gravitational attraction. This would imply an age of the universe less than (which is about 14 billion years). For instance, a value for of 1/2 (once favoured by most theorists) would give the age of the universe as . The discovery in 1998 that is apparently negative means that the universe could actually be older than . However, estimates of the age of the universe are very close to . Olbers' paradox The expansion of space summarized by the Big Bang interpretation of Hubble's law is relevant to the old conundrum known as Olbers' paradox: If the universe were infinite in size, static, and filled with a uniform distribution of stars, then every line of sight in the sky would end on a star, and the sky would be as bright as the surface of a star. However, the night sky is largely dark. Since the 17th century, astronomers and other thinkers have proposed many possible ways to resolve this paradox, but the currently accepted resolution depends in part on the Big Bang theory, and in part on the Hubble expansion: in a universe that existed for a finite amount of time, only the light of a finite number of stars has had enough time to reach us, and the paradox is resolved. Additionally, in an expanding universe, distant objects recede from us, which causes the light emanated from them to be redshifted and diminished in brightness by the time we see it. then to write Hubble's constant as , all the relative uncertainty of the true value of being then relegated to . The dimensionless Hubble constant is often used when giving distances that are calculated from redshift using the formula . Since is not precisely known, the distance is expressed as: cz/H_0\approx(2998\times z)\text{ Mpc }h^{-1} In other words, one calculates 2998 × and one gives the units as Mpc  or  Mpc. Occasionally a reference value other than 100 may be chosen, in which case a subscript is presented after to avoid confusion; e.g. denotes  , which implies . This should not be confused with the dimensionless value of Hubble's constant, usually expressed in terms of Planck units, obtained by multiplying by (from definitions of parsec and Planck time|), for example for , a Planck unit version of is obtained. Acceleration of the expansion , 2021 A value for measured from standard candle observations of Type Ia supernovae, which was determined in 1998 to be negative, surprised many astronomers with the implication that the expansion of the universe is currently "accelerating" (although the Hubble factor is still decreasing with time, as mentioned above in the Interpretation section; see the articles on dark energy and the ΛCDM model). == Derivation of the Hubble parameter ==

Start with the Friedmann equation: H^2 \equiv \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3}\rho - \frac{kc^2}{a^2}+ \frac{\Lambda c^2}{3}, where is the Hubble parameter, is the scale factor, is the gravitational constant, is the normalised spatial curvature of the universe and equal to −1, 0, or 1, and is the cosmological constant. Matter-dominated universe (with a cosmological constant) If the universe is matter-dominated, then the mass density of the universe should be taken to include just matter so \rho = \rho_m(a) = \frac{\rho_{m_{0}}}{a^3}, where is the density of matter today. From the Friedmann equation and thermodynamic principles we know for non-relativistic particles that their mass density decreases proportional to the inverse volume of the universe, so the equation above must be true. We can also define (see density parameter for ) \begin{align} \rho_c &= \frac{3 H_0^2}{8 \pi G}; \\ \Omega_m &\equiv \frac{\rho_{m_{0}}}{\rho_c} = \frac{8 \pi G}{3 H_0^2}\rho_{m_{0}}; \end{align} therefore: \rho=\frac{\rho_c \Omega_m}{a^3}. Also, by definition, \begin{align} \Omega_k &\equiv \frac{-kc^2}{(a_0H_0)^2} \\ \Omega_{\Lambda} &\equiv \frac{\Lambda c^2}{3H_0^2}, \end{align} where the subscript refers to the values today, and . Substituting all of this into the Friedmann equation at the start of this section and replacing with gives H^2(z)= H_0^2 \left( \Omega_m (1+z)^{3} + \Omega_k (1+z)^{2} + \Omega_{\Lambda} \right). Matter- and dark energy-dominated universe If the universe is both matter-dominated and dark energy-dominated, then the above equation for the Hubble parameter will also be a function of the equation of state of dark energy. So now: \rho = \rho_m (a)+\rho_{de}(a), where is the mass density of the dark energy. By definition, an equation of state in cosmology is , and if this is substituted into the fluid equation, which describes how the mass density of the universe evolves with time, then \begin{align} \dot{\rho}+3\frac{\dot{a}}{a}\left(\rho+\frac{P}{c^2}\right)=0;\\ \frac{d\rho}{\rho}=-3\frac{da}{a}(1+w). \end{align} If is constant, then \ln{\rho}=-3(1+w)\ln{a}; implying: \rho=a^{-3(1+w)}. Therefore, for dark energy with a constant equation of state , {{nowrap|\rho_{de}(a)= \rho_{de0}a^{-3(1+w)}.}} If this is substituted into the Friedman equation in a similar way as before, but this time set , which assumes a spatially flat universe, then (see shape of the universe) H^2(z)= H_0^2 \left( \Omega_m (1+z)^{3} + \Omega_{de}(1+z)^{3(1+w)} \right). If the dark energy derives from a cosmological constant such as that introduced by Einstein, it can be shown that . The equation then reduces to the last equation in the matter-dominated universe section, with set to zero. In that case the initial dark energy density is given by \begin{align} \rho_{de0} &= \frac{\Lambda c^2}{8 \pi G} \,, \\ \Omega_{de} &=\Omega_{\Lambda}. \end{align} If dark energy does not have a constant equation-of-state , then \rho_{de}(a)= \rho_{de0}e^{-3\int\frac{da}{a}\left(1+w(a)\right)}, and to solve this, must be parametrized, for example if , giving H^2(z)= H_0^2 \left( \Omega_m a^{-3} + \Omega_{de}a^{-3\left(1+w_0 +w_a \right)}e^{-3w_a(1-a)} \right). == Units derived from the Hubble constant ==

Hubble time The Hubble constant has units of inverse time; the Hubble time is simply defined as the inverse of the Hubble constant, i.e. t_H \equiv \frac{1}{H_0} = \frac{1}{67.8 \mathrm{~(km/s)/Mpc}} = 4.55\times 10^{17} \mathrm{~s} \approx 14.4 \text{ Gyr} This is slightly different from the age of the universe, which is approximately 13.8 billion years. The Hubble time is the age it would have had if the expansion had been linear, and it is different from the real age of the universe because the expansion is not linear; it depends on the energy content of the universe (see ). The Hubble time is sometimes said to set the scale for the age of universe. The present age of the universe can be related to the relative density of matter (\Omega_m), radiation (\Omega_\textrm{rad}), and dark energy (\Omega_\Lambda) as t_\textrm{age} = \frac{1}{H_0} \int_0^1 \frac{da}{\sqrt{ \Omega_{\rm m} a^{-1} + \Omega_\mathrm{rad} a^{-2} + \Omega_{\Lambda} a^{2} + (1- \Omega_{\rm m} - \Omega_\mathrm{rad} - \Omega_{\Lambda} )}} . The integral is close to 1 so the Hubble time factor in front sets the scale. Hubble length The Hubble length or Hubble distance is a unit of distance in cosmology, defined as — the speed of light multiplied by the Hubble time. It is equivalent to 4,420 million parsecs or 14.4 billion light years. (The numerical value of the Hubble length in light years is, by definition, equal to that of the Hubble time in years.) Substituting into the equation for Hubble's law, reveals that the Hubble distance specifies the distance from our location to those galaxies which are receding from us at the speed of light. Hubble volume The Hubble volume is sometimes defined as a volume of the universe with a comoving size of . The exact definition varies: it is sometimes defined as the volume of a sphere with radius , or alternatively, a cube of side . Some cosmologists even use the term Hubble volume to refer to the volume of the observable universe, although this has a radius approximately three times larger. == Determining the Hubble constant ==

in Rome. For most of the second half of the 20th century, the value of was estimated to be between . The value of the Hubble constant was the topic of a long and rather bitter controversy between Gérard de Vaucouleurs, who claimed the value was around 100, and Allan Sandage, who claimed the value was near 50. In 1996, a debate moderated by John Bahcall between Sidney van den Bergh and Gustav Tammann was held in similar fashion to the earlier Shapley–Curtis debate over these two competing values. This previously wide variance in estimates was partially resolved with the introduction of the ΛCDM model of the universe in the late 1990s. Incorporating the ΛCDM model, observations of high-redshift clusters at X-ray and microwave wavelengths using the Sunyaev–Zel'dovich effect, measurements of anisotropies in the cosmic microwave background radiation, and optical surveys all gave a value of around 50–70 km/s/Mpc for the constant. Precision cosmology and the Hubble tension By the late 1990s, advances in ideas and technology allowed higher precision measurements. However, two major categories of methods, each with high precision, fail to agree. "Late universe" measurements using calibrated distance ladder techniques have converged on a value of approximately . Since 2000, "early universe" techniques based on measurements of the cosmic microwave background have become available, and these agree on a value near . (This accounts for the change in the expansion rate since the early universe, so is comparable to the first number.) Initially, this discrepancy was within the estimated measurement uncertainties and thus no cause for concern. However, as techniques have improved, the estimated measurement uncertainties have shrunk, but the discrepancies have not, to the point that the disagreement is now highly statistically significant. This discrepancy is called the Hubble tension. An example of an "early" measurement, the Planck mission published in 2018 gives a value for of . and confirmed by the James Webb Space Telescope in 2023. The "early" and "late" measurements disagree at the >5 σ level, beyond a plausible level of chance. The resolution to this disagreement is an ongoing area of active research. Reducing systematic errors Since 2013, extensive checks for possible systematic errors and improvements in reproducibility have been undertaken. The "early universe" or inverse distance ladder measures the observable consequences of spherical sound waves on primordial plasma density. These pressure waves – called baryon acoustic oscillations (BAO) – ceased once the universe cooled enough for electrons to stay bound to nuclei, ending the plasma and allowing the photons trapped by interaction with the plasma to escape. The subsequent pressure waves are evident in very small perturbations in the density imprinted on the cosmic microwave background, and on the large-scale density of galaxies across the sky. Detailed structure in high-precision measurements of the CMB can be matched to physics models of the oscillations. These models depend upon the Hubble constant such that a match reveals a value for the constant. Similarly, the BAO affects the statistical distribution of matter, observed as distant galaxies across the sky. These two independent measurements produce similar values for the constant from the current models, giving strong evidence that systematic errors in the measurements themselves do not affect the result. years before the first strongly lensed object was observed. The first strongly lensed supernova to be discovered was named SN Refsdal in his honor. While Refsdal suggested this could be done with supernovae, he also noted that extremely luminous and distant star-like objects could also be used. These objects were later named quasars, and to date (April 2025) the majority of time-delay cosmography measurements have been done with strongly lensed quasars. This is because current samples of lensed quasars vastly outnumber known lensed supernovae, of which <10 are known. This is expected to change dramatically in the next few years, with surveys such as LSST expected to discover ~10 lensed SNe in the first three years of observation. For example time-delay constraints on H0, see the results from STRIDES and H0LiCOW in the table below. In October 2018, scientists used information from gravitational wave events (especially those involving the merger of neutron stars, like GW170817), of determining the Hubble constant. In July 2019, astronomers reported that a new method to determine the Hubble constant, and resolve the discrepancy of earlier methods, has been proposed based on the mergers of pairs of neutron stars, following the detection of the neutron star merger of GW170817, an event known as a dark siren. Their measurement of the Hubble constant is (km/s)/Mpc. Also in July 2019, astronomers reported another new method, using data from the Hubble Space Telescope and based on distances to red giant stars calculated using the tip of the red-giant branch (TRGB) distance indicator. Their measurement of the Hubble constant is . In February 2020, the Megamaser Cosmology Project published independent results based on astrophysical masers visible at cosmological distances and which do not require multi-step calibration. That work confirmed the distance ladder results and differed from the early-universe results at a statistical significance level of 95%. In July 2023, an independent estimate of the Hubble constant was derived from a kilonova, the optical afterglow of a neutron star merger, using the expanding photosphere method. Due to the blackbody nature of early kilonova spectra, such systems provide strongly constraining estimators of cosmic distance. Using the kilonova AT2017gfo (the aftermath of, once again, GW170817), these measurements indicate a local-estimate of the Hubble constant of . and there are many possible proposed solutions. The most conservative is that there is an unknown systematic error affecting either early-universe or late-universe observations. Although intuitively appealing, this explanation requires multiple unrelated effects regardless of whether early-universe or late-universe observations are incorrect, and there are no obvious candidates. Furthermore, any such systematic error would need to affect multiple different instruments, since both the early-universe and late-universe observations come from several different telescopes. In particular, we would need to be located within a very large void, up to about a redshift of 0.5, for such an explanation to conflate with supernovae and baryon acoustic oscillation observations. There are very many theories in this category, for example, replacing general relativity with a modified theory of gravity could potentially resolve the tension, as can a dark energy component in the early universe, dark energy with a time-varying equation of state, or dark matter that decays into dark radiation. A problem faced by all these theories is that both early-universe and late-universe measurements rely on multiple independent lines of physics, and it is difficult to modify any of those lines while preserving their successes elsewhere. The scale of the challenge can be seen from how some authors have argued that new early-universe physics alone is not sufficient; while other authors argue that new late-universe physics alone is also not sufficient. Nonetheless, astronomers are trying, with interest in the Hubble tension growing strongly since the mid 2010s. == Measurements of the Hubble constant ==