Time-Delay Cosmography and Hubble Tension - 2 - Trinity Option - Science

We continue the story about the Hubble Tension that we started in the last issue - a problem associated with the inconsistency between different methods of measuring the Hubble constant, which determines the current rate of expansion of the Universe. The first part explained the essence of this constant, its role in modern cosmology, and examined various methods for its numerical evaluation. The first were measurements of currently observable objects, such as Cepheid variable stars, supernovae and other “standard candles”, i.e. objects with similar luminosity, thanks to which their distance, and therefore the Hubble constant, can be determined. However, with the advent of an alternative class of methods related to the study of CMB fluctuations, the same inconsistency arose: the previous methods give values ​​grouping around 72–74, and the new ones - 67–68 (km/s)/Mpc. How to “reconcile” these results?

Looking for a way out

It can be noted that the bimodality of the numerical values ​​of the Hubble constant is based on the simple fact that they are obtained on the basis of two fundamentally different approaches. Estimates of the upper mode are derived by recording photon fluxes generated at a redshift value z not exceeding 2 - that is, in an already formed Universe filled with galaxies and galaxy clusters. They carry information about events that, on the cosmological time scale, are only moderately distant from our era. But the relict radiation came to us from the era of recombination, which took place only 379 thousand years after the Big Bang at z = 1089. That is, at least a hundred million years before the birth of the first stars. So the spectral analysis of the cosmic microwave background radiation contains information about the value of the Hubble parameter in that distant era. Its modern value (that is, the Hubble constant H0 ) is then derived using the mathematical apparatus of the cosmological model adopted as the basis.

Of course, one can assume that Hubble tension is nothing more than an artifact of systematic errors that have not yet been taken into account that arise when performing a great variety of astronomical and astrophysical measurements. Otherwise, we will have to admit that this disconnect is absolutely real and requires not methodological, but physical explanations. It is not surprising that many different hypotheses have been put forward for this. For example, some experts propose modifying the standard history of the early expansion of the Universe by accepting as a hypothesis the existence of unknown relativistic particles or the influence of dark energy unforeseen by modern models. There are other versions of the theoretical elimination of Hubble tension, and there are many of them.

But there is another way out. Let's assume that we can find a new way to measure the Hubble constant with sufficient precision that does not use previously used methods at all. The results obtained could be compared with those already available and see if they shift the scales in one direction or another. With such an outcome, the discussion of the reasons for the Hubble undocking would at least become more meaningful.

As already mentioned in the first part of the publication, this method is not new in theory. It was proposed in 1964 by the Norwegian astrophysicist Sjur Refsdal 1 , who was then a graduate student at the University of Oslo. In his article, he proposed using the effect of gravitational lensing of supernova explosions 2 to measure the Hubble constant. In this version, Refsdal's idea was put into practice only in 2017, eight years after his death and three years after the first observation of the supernova named after him, which became the object of gravitational lensing 3 . As the reader has probably already guessed, it was the technique proposed by Refsdal that formed the basis of time-delay cosmography.

The method proposed by Refsdal for measuring the Hubble constant was first tested long before 2017. The initial successes of the method became possible thanks to the use of much more powerful cosmic lamps - quasars - as radiation sources, rather than supernovae. However, this also took a lot of time. The first strongly lensed quasar was discovered back in the late 1970s, and by the end of the last century their number was already measured in dozens. However, the earliest estimate of the Hubble constant using Refsdal's scheme was published only in 2002, and it was based on observations of a single quasar. Then the number of “reference” quasars began to grow. In December 2025, the TDCOSMO collaboration released its most reliable gravitational lensing measurements of H0 to date, based on observations of eight quasars so far. I'll get back to them later.

How does this work

Now let's talk about time-delay cosmography. There is hardly any need to explain in detail what cosmic gravitational lensing is - it is now well known to everyone who is even the slightest bit interested in astronomy. Let me just remind you that this effect consists in the bending of light rays coming to Earth from a distant source if, on the way to our planet, they pass near an area with strong local gravitational fields. It is called a gravitational lens. If the luminosity of the source is constant, the action of gravity will only lead to the appearance of false images of the observed object, which will disappear when the gravitational deflector leaves the line of sight from the Earth to the object.

The situation becomes much more interesting if the brightness of the source changes over time, which always happens with the radiation of supernovae and quasars. Let us assume for certainty that earthly observers are watching a lensed supernova explosion. Since its rays will reach the Earth along paths of different lengths, the flare will be observed not only in different parts of the sky, usually separated by distances of about one arcsecond (remember, the lens creates false images!), but also at different times. This is due both to the different geometric lengths of photon trajectories and to the gravitational time dilation arising from the general theory of relativity, which manifests itself the more strongly the deeper the photons are immersed in the gravitational field of the deflector. As a result, observers may believe that they have discovered several different supernovae, but this, of course, will be just an optical illusion. A supernova actually explodes only once.

The number of such flares depends on the strength and spatial structure of the gravitational field of the deflector. When a cluster of galaxies acts in this capacity, such a structure can be very complex and spatially extended (let me remind you that the masses of the largest clusters reach 1015 solar masses, and their scales are measured in tens of millions of light years). Therefore, the intervals between outbreaks can be measured in days, months and even years. For example, in 2016, the Hubble telescope recorded three appearances and disappearances of a type Ia supernova with the index AT 2016jka (also referred to as SN Requiem). It exploded over 10 billion years ago in the massive galaxy MRG-M0138 at redshift z = 1.95. The deflector was the vast cluster MACS J0138–2155, which includes at least 84 galaxies. Researchers working on this supernova expect to observe a fourth explosion in the second half of 2026 or 2027. These expectations are based on their model of the gravitational field of the cluster MACS J0138–2155. It should direct photons from the explosion of the supernova AT 2016jka along one more path, already the fourth in a row. As calculations show, it will be an order of magnitude longer than the three trajectories along which the light fluxes observed in 2016 took place. It is assumed that this flare will first be detected by the Hubble Space Telescope, and then the James Webb 4 will be involved in monitoring the observations. Of course, there are no guarantees yet that this forecast will come true, but the wait won’t be long.

During the first quarter of this century, astronomers reported the reliable identification of several highly lensed Type Ia supernovae. One of them, SN H0pe 5 with a redshift z = 1.78, was discovered by the James Webb Space Telescope, which began operating in the summer of 2022. The white dwarf explosion that created this supernova took place when the Universe was 3.5 billion years old. It was detected while observing the galaxy cluster PLCK-G1165.7+67.0 at a redshift of 0.35, which acted as a deflector of supernova radiation. Information about this discovery appeared in 2023. It made it possible to recalculate the value of the Hubble constant, which will be discussed in the next section.

There is no doubt that other similar discoveries will follow in the coming years. Perhaps they will “appear” already in the first data array from the European space observatory Euclid, launched from Cape Canaveral to the second Lagrange point L₂ on July 1, 2023. The publication of this array is expected this year. Astronomers also have high hopes for the ten-year space scanning program of the American Vera Rubin Observatory, located in northern Chile on the El Piñon peak of Cerro Pachon, and for the upcoming launch of the Nancy Grace Roman Telescope, which NASA expects to carry out before the end of this year.

The mathematics of space lanterns

How can one obtain information about the numerical value of H 0 from the observation of two or more lensed appearances of photons from the same cosmic event in our sky? You just can’t explain this in words, you need to refer to mathematics. In principle, it is not very complicated, but it is not suitable for presentation in a popular article. However, I will try to illustrate the general idea - of course, using a very simple example.

To begin with, let's imagine two flashes of an electric flashlight located not far from an ordinary earthly observer. Let's call the first event A, and the second - event B. If the owner of the flashlight pressed the button with a time interval Δ t , the observer will see a second flash with the same delay compared to the first. If Δt is multiplied by the speed of light c , we obtain a value with the dimension of distance. In our case, it does not represent anything interesting, it is simply the distance that light travels in a given period of time. Purely formally, let's call it time delay distance and denote D Δt (or completely D Δt(A, B)). It is obvious that Δt = D Δt/s. There is nothing more to be gained from this thought experiment.

Now consider as event A the first appearance of light from a lensed cosmic source in time for an earthly observer, and as event B the second, which is recorded after time Δt (for simplicity, we will assume that both events are instantaneous). It turns out that in this case there is a similar relation Δ t = D Δt/s […]. Here D Δt is again a certain parameter with the dimension of distance (let us again denote it time delay distance), and in square brackets there is a certain formula that includes the angular coordinates of both the lensed and unlensed source (that is, its coordinates in the absence of a gravitational deflector), as well as an additional term describing the gravitational potential of the deflector (which in this case can be considered an analogue of the optical characteristics of a conventional lens).

Now comes the fun part. For D Δt there is a fairly simple algebraic expression that I can give without scaring the reader too much: D Δt = (1 + z d) D d D s / D ds. Here z d is the redshift of the deflector, D d is its so-called angular distance from the Earth, D s is the goniometric distance of the lensed source, and D ds is the goniometric distance of the gap between the source and the gravitational lens.

In order not to make the article heavier, I will not explain what the term “angular distance” means in cosmology. This information is easy to find on the Internet. What is important for us is that all three distances depend on the expansion rate of the Universe, which, in turn, is determined by the Hubble constant. Since each distance is inversely proportional to H 0, D d D s / D ds, as is easy to see, is also inversely proportional to the same Hubble constant. More precisely, it is inversely proportional to a first approximation. Taking into account the multiplier (1 + z d) gives easily calculated corrections, but there are other disturbing factors. Perhaps this is all that can be said at this stage. The specific dependence of goniometric distances on redshifts is calculated within the framework of one or another cosmological model and is most often written using rather long formulas.

So, the general scheme is clear. Measurements of the time delays of lensed photon fluxes and their angular coordinates on the earth's sky in combination with modeling the gravitational fields of deflectors allows one to calculate the time delay distance, and then obtain an approximate estimate of H 0. The results obtained from the analysis of successive flares of one or several sources are compared, analyzed, summed up and reduced to one denominator using very serious statistical models. The flares themselves are searched for and recorded during long-term observations using telescopes located on both earth-based and space platforms. Naturally, only numerous international collaborations can solve all these problems. I believe that it is now clear why this method is called time delay cosmography.

Of course, D Δt depends not only on H 0, but also on other cosmological constants that appear in models of the evolution of the Universe. However, these additional dependencies are quite weak and, in principle, can be taken into account. This circumstance, as well as the autonomy of time delay cosmography in relation to other methods of cosmological measurements, makes it a very promising method for accurately determining the Hubble constant. Of course, it has its technical difficulties (and considerable ones!), but they are gradually being overcome. More on this in the next section.

The results obtained and their interpretation

First of all, let's look at the article by Jesus Vega-Ferrero and his three co-authors, which was mentioned in the section “Finding a Way Out.” It contains their estimate of the Hubble constant, based on observations of the explosion of the Refsdal star, whose redshift was 1.49. It was lensed by the gravitational field of the galaxy cluster MACS J1149.5+2223 with a redshift of z = 0.544. In 2014, there were four appearances of this supernova with very short intervals (only a few days), which did not allow these observations to be used to estimate H 0. However, on December 11, 2015, the Hubble telescope recorded the fifth image of its outburst, which has already proven useful. The authors of this article concluded that H 0 lies in the range from 62+4–4 (km/s)/Mpc to 64+9–11 (km/s)/Mpc. This is quite close to the results of the Planck collaboration, but noticeably lower than the above result of the SH0ES collaboration and other similar estimates. In 2023, the results for the Refsdal star were revised, the new value of H0 was 66.6+4.1–4.3 (km/s)/Mpc. Finally, calculations of the Hubble constant based on observations of the lensed supernova SN H0pe gave a much larger value for H 0 - 75.4 + 8.1–5.5 (km/s)/Mpc.

In the 2010s, active work began to measure the Hubble constant based on observing changes in the brightness of quasars. At the end of this decade, reports were published from two collaborations, one focusing on six quasars and the other on seven. Their estimates of the Hubble constant were very close: 73.3 ± 1.8 (km/s)/Mpc and 74.2 ± 1.6 (km/s)/Mpc. Finally, the already mentioned “eight-quasar” result of the TDCOSMO (Time-Delay COSMO) collaboration formed in the early 2020s, which became known at the very end of 2025, gave for this constant a value of 71.6 + 3.9–3.3 (km/s)/Mpc, estimated with a two percent confidence 6 . No more recent estimates have appeared at the time of writing this article. The TDCOSMO collaboration plans to increase the reliability of its results to one percent in the future.

Nowadays, time-delay cosmography typically works with light sources with redshifts of one to three, with typical deflector placements at z = 0.5. This is more or less the same region of the Universe that is accessible when Type Ia supernovae are used as “standard candles.” Farther space is not yet accessible to her.

So where have we come to? Последние численные значения постоянной Хаббла, полученные на базе time-delay cosmography, хорошо согласуются с результатами метода стандартных свечей и сильно отличаются от оценок, основанных на анализе анизотропии реликтового излучения. Так что Hubble tension никуда не исчезла — по-прежнему цветет и пахнет. Что это означает, пока приходится только гадать. Констатацией этого несомненного факта я и закончу.

Алексей Левин

1 www.trv-science.ru/2016/01/dejavu-supernovy-refsdala/

2 Refsdal S. On the possibility of determining Hubble's parameter and the masses of galaxies from the gravitational lens effect // Mon. Notices of the Royal Astron. Soc. 128, 307 (1964).

3 Vega-Ferrero J. et al. The Hubble Constant from SN Refsdal // The Astrophysical Journal Letters. Vol. 853, Issue 2 (2018).

4 См. Clery D. Cosmic Illusions // Science. 392 (6793), 2 April 2026, P. 17–21.

5 Название дано с умыслом: H0 — постоянная Хаббла, hope — надежда, то есть сверхновая дает надежду разобраться со значением постоянной.

6 TDCOSMO 2025: Cosmological constraints from strong lensing time delays // Astronomy&Astrophysics. 704, A63 (2025).