- Hello and welcome to this Gresham College lecture on the broken cosmic distance ladder. My name is Roberto Trotta, I'm a visiting professor of cosmology at Gresham College and a professor of astro-statistics at Imperial College London, currently on leave of absence, and visiting the International School for Advanced Studies in Trieste, Italy, from where I'm giving this lecture today. We're going to talk about how to measure distances in the universe, which is a surprisingly hard problem. In fact, if we look to the Moon in the sky, sometimes it appears so close, it feels as if almost we could just touch it, and reach out to it, and pluck it out of the sky, it appears so close to us, we can make out all sorts of things on its face when it's fully illuminated. Some people see a rabbit on the face of the Moon, some cultures see a toad, a lady, a man with a bundle on his back, even a man's face, which I never quite could understand and could never quite make out. So how far is the Moon, and how can we tell, especially in antiquity, when technology and science were not quite so advanced as they are today, and determining distances to objects in the sky was always a big challenge? And in fact, while the Moon seems tantalizingly close, sometimes, the stars, on the other hand, appear always to be forever out of reach, aloof and distant, and forever unreachable. In fact, so unreachable and so aloof that W.H. Auden, in his poem "The More Loving One"

"How should we like it were stars to burn"with a passion for us we could not return?"If equal affection cannot be,"let the more loving one be me."Admirer, as I think I am,"of stars that do not give a damn,"I cannot, now I see them, say,"I have missed one terribly all day." So stars are distant, stars are really out of reach of us, except when perhaps we see a shooting star, which of course is not a star at all, it's just a little piece of rock, a meteorite, burning up in the atmosphere, which however does give us the illusion that stars, for a fleetingly short moment, come to within our reach. So how far away is the Moon, and the Sun, and the thousands of stars that bejewel the night? And what about nebulae, those fuzzy, glowing patches of light that only the big telescopes of the 20th century could reveal with some clarity, how far are they? And can our human eye, combined with the powers of our imagination, and of science and technology, reach all the way to the far end of the universe and stretch a cosmic ruler to measure distances out in space? Can we build a cosmic distance ladder that, rung after rung, will take us all the way to the end of the visible universe? We will today explore all of those questions and see that, mysteriously, the cosmic distance ladder that we have built over centuries and millennia appears today to be broken. The earliest method to establish distances for a faraway object is one that remains very much useful today, and it is literally at your fingertips. If you stretch your arm in front of you and put up your index finger, like this, and you look at the finger by closing one eye, and then the other eye, you will see the position of the finger appear to jump with respect to a distant background. My finger is moving here, but if you just close, alternatively, the left and right eye, the appearance of movement will be due to the different vantage point, about 10 or so centimeters apart, from each one of your two eyes. This phenomenon is called parallax, parallax coming from an old Greek word which means a change. And if you move now your finger closer to the tip of your nose, and you do the same experiment again, by alternating the eye that you close, you will see that the parallax is now much increased. So the closer by the object, the bigger the apparent angle by which it jumps with respect to a distant background, and conversely, the farther away, the smaller the angle. So if, conceivably, we can measure the parallax angles of objects in the sky by looking at them from different vantage points, then we can, in principle, hopefully, measure their distance using simple trigonometry. So it is a great tool to measure distances to objects that are relatively nearby provided we have a backdrop by which to judge their apparent change in position. Also, we can already predict that if objects are too far away, the parallax angle will be there, but it will be too small to be measurable, so at some point, parallax will stop working. The first person to use parallax to try and measure the distance to the Sun and the Moon was the ancient Greek astronomer and mathematician Hipparchus of Nicaea, which is in modern-day Turkey. Hipparchus lived in the 2nd century BC, and he's the inventor of trigonometry, he's also considered the father of astronomy, by many, he's considered the greatest astronomer and observer in antiquity. He made many discoveries, which included the procession of the equinoxes, the fact that seasons are of an equal length, and he also perfected a method to predict eclipses more accurately than was possible before, he created the first astronomical catalog of stars, measuring the brightness and position of about 850 stars in the sky. And when considering the question of how far away is the Moon, Hipparchus used the fact that a total solar eclipse, which is one in which, as we know, the Earth, the Moon, and the Sun are lined up, like in this diagram here, the fact that the Sun was completely covered by the Moon, if you were looking at the solar eclipse from his hometown, Nicaea, so the line of sight to the rim of the Sun was the red one in this diagram, while the eclipse was partial, as seen from the city of Alexandria, in Egypt, which was on the same meridian as Nicaea, but nine degrees further south. So by using this fact, and the fact that the Sun was sufficiently far away so as to not exhibit any noticeable parallax, he was able to estimate the angle between the red and the green line, and with some mathematics, therefore establish the distance between the Earth and the Moon, correctly attributing the difference in appearance of the solar eclipse to the parallax phenomenon as applied to the Moon. He eventually arrived at a figure for the distance between the Earth and the Moon of 63 Earth radii. In order to convert this distance, which was remarkably close to the right value, which is about 60 radii, in order to convert this value to an actual distance, one needed to know what was the radius of the Earth, and therefore the circumference of the Earth. This had been done already 50 years earlier by Eratosthenes, who had heard from a traveler that on the summer equinox, the Sun's rays at noon illuminated the bottom of a well in the city of Syene, in modern-day Egypt, without casting any shadows. This meant that, at that point in time, the Sun was directly overhead that city. On the same day, he measured the length of a shadow cast by a stick in Alexandria, and therefore, by comparing the angle that the Sun made in Alexandria and in Syene, he was able to find the difference in latitude between the two cities, which he estimated as being 1/50 of a full circle. If he had the difference in latitude, all he needed to do was to measure the physical distance between the two cities, and so he hired professional surveyors who walked with equally-spaced steps between the two cities, reporting a distance of 5,000 stadia. And he concluded, therefore, that the circumference of our planet was 250,000 stadia, 50 times as much, which is a number that's a little difficult to translate in today's units, because a stadium as a unit of measure was not uniform throughout antiquity. If we put 10 stadia to the mile, then Eratosthenes estimates, for the circumference of the Earth will be 25,000 miles, which puts the Earth radius, if we assume the Earth to be a perfect sphere, which it isn't, to 3,980 miles, or 6,400 kilometers, which is very close to the actual value of 6,378 kilometers. Given this estimate of the size of the Earth, it is clear that 63 Earth radii, Hipparchus' estimate for the distance of the Earth to the Moon, did put the Moon out of reach of even the tallest ladder on top of the tallest of mountains. This measurement established, 200 years BC, the first rung in our distance ladder, which, today, sits at about 384,000 kilometers, that's the mean distance between the Earth and the Moon, the actual distance varies as the orbit of the Moon is not a perfect circle. So this was our first rung, what about the next rung, the distance to the Sun? Hipparchus tried to measure the distance to the Sun as well, but his method gave a hopelessly low estimate, much, much lower than what it is, he estimated it as 490 Earth radii, which is too small by a large factor, the actual distance, Earth-Sun, is about 23,400 times to the Earth radius, or 150 million kilometers. So how could we then measure the distance to the Sun? By the 17th century, the Copernican and Keplerian revolutions in terms of our understanding of the orbits and the structure of the solar system had been completed and was in full swing. In particular, Kepler told us that the orbits of planets going around the Sun are ellipses, and those ellipses, which are a kind of stretched circle, had several properties, that are encapsulated in Kepler's "Laws of Planetary Motion," that Newton would, a few decades later, reinterpret and explain in terms of gravity. In particular, Kepler's third law tells us that the orbital period squared of a planet going around the Sun divided the semi-major axis of the ellipse, which is the long part of the ellipse, in blue in this diagram, the orbital period squared divided by the semi-major axis cubed is a constant for all planets, indeed for all bodies orbiting another body. And that, in turn, because we know the period of all the planets, at least all the planets that were known at Kepler's time, that in turn gives us the relative distances of the planets to the Sun. So if we take the Earth to define the standard of distance, the AU, the astronomical unit, and we say that the Earth-Sun average distance is one AU, then you see, on the right, that all of the other planets, thanks to Kepler's third law, can be put at relative distances from the Sun, Mercury, 0.39 AU, Venus, 0.72 AU, Mars, 1.5 AU, and so on, and so forth. So Kepler gave us a way of measuring the relative distances of the planets to the Sun, and therefore, to measure the relative size of the solar system, but we still didn't know what the AU was, how big was this astronomical unit in units of which all of the other distances of the solar system could be measured. And that's where Edmund Halley comes in. In 1716, the famous astronomer issued a real call-to-arms to all the diligent searchers of the heavens to take advantage of a very rare astronomical phenomenon that would happen a few decades later in order to finally pinpoint down the AU, the astronomical unit. The phenomenon that Halley was interested in was a transit of Venus. Venus is one of the inferior planets, one of the planets that orbit inside the orbit of the Earth, and as such, it goes around the Sun faster than the Earth does, and as it does so, the Earth and Venus line up several times in the course of their orbits around the Sun. However, because the orbit of Venus is tilted by three degrees with respect to the orbit of the Earth, as you see in this diagram, the only times when the Earth and Venus alignments lead to Venus actually being in front of the Sun, a so-called transit of Venus on the face of the Sun, is when the alignment between the Earth and Venus happens at the point of intersection of the two orbits, which is called a node. This is an extremely rare astronomical phenomenon, with a cycle of 243 years. What happens for an observer on the Earth is that this transit, Venus passing in front of the disc of the Sun, happens twice in June, then nothing in the course of eight years, then nothing for 105 years, then another two transits eight years apart in December, and then nothing again for the next 121 years, after which the cycle recommences. Halley, writing in 1716, predicted a pair of transits happening in 1761 and 1769, and he knew that by looking at Venus from different vantage points on the surface of the Earth, and looking at the transit of Venus on the face of the Sun, let's say, if you were able to look at Venus transiting the Sun from a northerly location, such as Canada, and then from a southerly location, such as Tahiti, in the Pacific Ocean, your line of sight to the transit would be slightly different, and parallax would mean that the transiting planet on the face of the Sun, which is shown by the green and dashed red lines on the right-hand side, would cut a slightly different path. So by timing the transit from two wildly different locations on the surface of the Earth and using the phenomenon of parallax, one could measure the distance to Venus, and therefore the AU, effectively, because we know that Venus is so many AUs away from the Sun. So this was a great opportunity, and a rare opportunity, to see a transit, and therefore, all the natural scientists of the time geared up to make that observation. In fact, this was not the first time that anybody had seen the transit of Venus, because the English astronomer Jeremiah Horrocks had been one of only two people in recorded history, together with his friend William Crabtree, to observe the earlier transit of Venus in 1639, which Horrocks himself had predicted against Kepler's prediction of a near-miss. In fact, there is even a memorial tablet that commemorates him inside Westminster Abbey, saying,"Having in so short a life"detected the long inequality"in the mean motion of Jupiter and Saturn,"discovered the orbit of the Moon to be an ellipse,"determined the motion of the lunar apse,"suggested the physical cause of its revolution,"and predicted, from his own observations,"the transit of Venus,"which was seen by himself and his friend William Crabtree"on Sunday the 24th of November, in the old calendar, 1639."This tablet, facing the monument of Newton,"was raised after a lapse of more than two centuries,"December the 9th, 1874," a fitting memorial to a great astronomer whose life was tragically cut short at the young age of 22. In fact, when Horrocks saw the spectacle of the transit of Venus, he was so elated that he wrote in his diary

"Thy return, posterity shall witness,"years must roll away,"but then, at length,"the splendid sight"again shall greet our distant children's eyes." Sadly, Horrocks would not have any children who would witness the 1761 or 1769 transits. In 1761, over 120 missions in 62 countries were dispatched to observe it, the grandest international astronomical effort ever attempted. In fact the Royal Astronomer, Nevil Maskelyne, for example, sailed all the way to St. Helena, only to have his observations scuppered by clouds. So evidently, Halley's good wishes, which he expressed in 1716,"And I wish them luck,"and pray above all"that they are not robbed of the hoped-for spectacle"by the untimely gloom a cloudy sky," well Halley's wishes didn't work out for Maskelyne. In fact, despite the many measurements during the 1761 transit, the measurements were too variable, the quality wasn't sufficient to pin down the AU and the parallax angle in a reliable fashion. A second chance would present itself in 1769, and this time, that chance could not be missed, because no other chance would present itself for another century, or more. In England, the Royal Society mounted an ambitious scientific expedition. They had convinced the Admiralty to buy and outfit, at great cost, a sturdy merchant ship, which they renamed the Endeavour, and they gave its command to the ablest navigator and cartographer of His Majesty's fleet, Lieutenant James Cook. They also petitioned the king, George III, to bankroll the mission with over a million pounds, in today's money, of his own money, and they equipped the Endeavour with the finest telescopes, and the finest instruments, the finest clocks that money could buy at the time, before sending it to the other side of the world, in the middle of the Pacific Ocean, to the island of Tahiti, that had recently been conveniently discovered, quote-unquote, because, of course, Tahiti had been inhabited for centuries already by Tahitians, coming, presumably, from other Polynesian islands, perhaps even from New Zealand, had been discovered, like I said, for the king, quote-unquote, by Captain Wallis two years previously. When the transit occurred, the British expedition found itself ready. On June the 3rd, 1769, they were in Tahiti, they had built themselves a fort to be undisturbed by the locals during their observations, which they called Fort Venus on Tahiti, which you can see pictured here in this lithography by Sydney Parkinson, one of the artists onboard James Cook's first voyage. This time, Cook, Charles Green, who was the second astronomer, because Cook himself could be counted as an astronomer onboard, the two astronomers found themselves in Tahiti in perfect conditions, and Cook wrote in his diary,"The day proved as favorable to our purpose"as we could wish,"not a cloud was to be seen the whole day,"and the air was perfectly clear"so that we had every advantage we could desire"in observing the whole of the passage of the planet Venus"over the Sun's disc." The data that Cook brought back to England allowed Thomas Hornsby, a Savilian professor of astronomy at Oxford, to derive a value of the astronomical unit in this paper here, of 93,726,900 English miles, which is an error of less than 1% with respect to the actual value, an absolutely brilliant measurement that they performed using a Gregorian telescope like the one that's shown here, which they stood on barrels, like the one that's shown in this reconstruction by the Royal Observatory, Greenwich, which they filled with wet sand to ensure stability in the scorching heat of a Tahitian June. And you see here at the center, the observations of the transit, the planet impinging onto the solar disc, by the astronomer Charles Green, who, tragically, would not make it back to England, he would die in Jakarta during the second leg of the voyage. But thanks to the observations of James Cook, Charles Green, and many others, the AU was finally pinpointed down to 1% of its actual value, and the second rung was established, at 150 million kilometers' distance. By the end of the 18th century, the size of the solar system had been reliably established thanks to the measurement of the AU, the astronomical unit. Uranus, discovered in 1781, stood at an average distance of the Sun of 2.8 billion kilometers, and Neptune, when it was discovered in 1846, meant that that already enormous amount of empty space almost doubled to almost 5 billion kilometers, giving a really huge size for the extent of the solar system. But what about the stars? Because the stars had not been pinned down to a distance yet. Everybody agreed that they had to be almost unfathomably far away, but how far exactly, nobody could quite say. Parallax could, in principle, be used to establish their distance by exploiting the fact that our vantage point with respect to the universe changes as we go around the Sun, and so if we measure the position of a relatively nearby star six months apart, as we go around the Sun, its apparent location on the background of even more distant stars will change by a certain angle. If we can only measure this angle, the parallax formula will give us the distance, but those angles are incredibly small, even though the distance from one end to the other of our change of vantage point is a mighty 300 million kilometers. Let's get a sense for those angles. Remember, the Moon in the sky, the Moon in the sky subtends an angle of about 1/2 a degree. Each degree is further subdivided in 60 parts, which we call minutes of arc, or arc minutes, and each arc minute is further subdivided in 60 arc seconds, equal parts. And the reason why we use this factor of 60 goes back all the way to 5,000 years ago, to Babylonian time, it's a fascinating story, but not one that we can tell today. The unaided human eye can discern angles as small as one arc minute, so 1/60 of a degree. This is the same angular size as a one pound coin seen from 80 meters away. An arc second, remember, is 60 times smaller, so this is the angular size of a pound coin seen from 4.8 kilometers away. Today, we have space-based observatories that can measure tiny, tiny angles, and therefore measure the distance to stars in the Milky Way very reliably. The best observatories we have have achieved accuracy of one millionth of an arc second. This is equivalent to the angle subtended by a coin on Neptune seen from the Earth. The parallax is, in fact, so fundamental that it led astronomers to use it to define another unit of measure for distances, the parsec, a word that's a contraction of the words parallax and second. A parsec is the distance at which a star would exhibit a parallax of one arc second, and that corresponds to 3.26 light years, and given how big a light year is, that amounts to 30,000 billion kilometers. In order to measure the distance to the stars, then, we needed to be able to measure tiny, tiny angles of parallax. Nobody had been able to build an instrument capable of doing that until 1838, when the mathematician and astronomer Friedrich Bessel, the director of Konigsberg Observatory, in today's Kaliningrad, on the Russian Baltic, did it. What Bessel did was to use a heliometer, an instrument that had been originally conceived to measure the diameter of the Sun, but then perfected. A Heliometer is a telescope with a lens that's sliced in half, like you can see here, and the two relative halves of the lens can be adjusted horizontally by moving a micrometer screw. And so if you have two stars in the sky, and you see them through a telescope with this kind of sliced lens, that will give you four different images, here, and by moving the micrometer thumbscrew, you can make those double images of the stars align, and if you are able to do so by reading off by how much you've moved the relative position of the lenses with respect to each other from the micrometer, in this scale up here, you can establish very, very small angle differences between the stars. Bessel carefully monitored the position of the star 61 Cygni over the course of a year with respect to the background stars using his heliometer, finally announcing a parallax of 0.314 arc seconds. He pinned this star to being about three parsecs away,

a new rung had been added to the cosmic distance ladder, one that was described by the Royal Astronomical Society President John Herschel as the greatest and most glorious triumph which practical astronomy has ever witnessed, when he conferred the Gold Medal of the Royal Astronomical Society to Bessel in 1841. 61 Cygni is not the nearest star to the Earth, Alpha Centauri and Proxima Centauri actually are, they stand at about four light years, or just over a parsec way, but still, this is the new distance scale, which puts stars, generally, tens, hundreds, thousands of light years away from us. John Herschel also said that thanks to this measurement, finally, the sounding line of distances in the universe had touched bottom by measuring, finally, the distance to the stars, but of course, that wasn't true, because further and further realms of discovery beckoned immediately afterwards, the question was merely displaced from the province of the stars, whose distances now could routinely be measured to thousands of light years away, to the mysterious realm of the nebulae. Those were those faint puffs of haze, sometimes showing a hint of some twirling and spiral structure that nobody could quite understand what they were. Were those just puffs of gas nearby the Milky Way, but they're quite small, or were they mighty galaxies in their own right, which were reduced to faintness merely by virtue of their great distances? Nobody could tell. Maybe, if you could measure distances to us, that could give us a hint about their true nature. With the turn of the 20th century, a new, powerful tool came in aid to the astronomers, the photographic plate. One of the pioneers of the technique had been the amateur astronomer, and medic by profession, Henry Draper, who, together with his wife Mary Anna Draper, was among the first to use photographic plates to capture stars and nebulae, and spectra, even, of those objects. His untimely death in 1882 spurred his widowed wife to generously support the efforts of Harvard Observatory to continue her husband's work and map out the stars, and for several decades, she generously funded the establishment of the Henry Draper Catalogue that would eventually consist of over half a million photographic plates. As the observatory kept on measuring and photographing all parts of the sky, a great deal of effort was needed to look at the plates, measure the position and brightness of the ever-increasing number of stars that the photographic plates were revealing, and that's where the Harvard Observatory computers come in, a team made exclusively of women who were entrusted the difficult and painstaking job of going through the plates, measuring position and brightness of the stars. You can see some of them here, the lady sat in the middle is Mary Anna Draper herself. The most gifted and prolific of the lady astronomers was undoubtedly Ms. Annie Jump Cannon, who worked for 40 years at the observatory, and she personally classified over 200,000 stars, and she not only classified them according to their spectrum and measured their brightness, she did so using an instrument which appears quite strange, but was very, very useful, the so-called fly spanker. It's nothing else but a glass plate with a little handle, and on the glass plate, you see marked in ink calibration stars, little stars that were used as comparators to establish the magnitude, the brightness of the stars of interest. It was called the fly spanker because it was too small to be a fly swatter. A contemporary of Ms. Cannon was Ms. Henrietta Swan Leavitt, who had excelled in her studies at Radcliffe College, one of the few women's colleges at the time, and she was a particularly gifted in mathematics, algebra and calculus, and she therefore landed, initially, an unpaid assistantship at the Harvard Observatory in 1895, and at that point, Edward Pickering, the director of the observatory, charged her with assessing the brightness, and therefore the magnitude, of stars, from photographic plates, and she used the fly spanker to do the job. She later left the observatory for a stint in Europe, and then in Wisconsin, and she returned in 1903, when she was paid 30 cents an hour to first identify variable stars, stars whose brightness changes over time, in the Magellanic Clouds, in the Southern Hemisphere, and then measure the changes in the brightness over time, a painstakingly difficult job. By 1908, she had amassed 777 variable stars, and she noticed a strange pattern among 16 of them in the Small Magellanic Cloud. In the paper she wrote in 1908, she stated,"It is worthy of notice"that the brighter variables have the longer periods." By 1912, here she is, Henrietta Leavitt, in 1921, by 1912, she had found another nine variables exhibiting the same pattern. The period of variability of those remarkable stars appeared to reflect their brightness. You see here a sketch of what the brightness over time of those stars would do, it would grow, and then it would rapidly decline, and then grow rapidly again in this characteristic pattern. We now know that those stars, which we call Cepheid variables, because of the name of the prototypical example of those stars, which is a star found in the constellation of Cepheus, those stars are actually pulsating stars, they blow up, and they trap their light inside them as they blow up, and then their opacity drops, and therefore the star contracts, they release the light, and they dim again in a periodic fashion. Effectively, those stars are stars that breathe light in and out. But what Henrietta Leavitt discovered was that the period of a pulsation, for the period of variation in brightness, seemed to be related to their intrinsic brightness. So by establishing the existence of 25 such stars in the Magellanic Cloud, and because they were all in the same cloud, so presumably at the same distance from Earth, she could arguably say that their apparent brightness was a function of their period, and it wasn't due to their different distance, they were all at the same distance. And when she plotted the brightness here, the logarithm of the brightness, as a function of the logarithm of the period, all 25 stars snapped on a line. That meant that if you could measure the period of the stars, by looking at this line, you would be able to work the brightness of the stars, and from that, you could work out distance. Leavitt's Law gave us a new way of measuring distances to the universe. And guided by this very law, Edwin Hubble, working hundreds of nights at the 100-inch Hooker telescope at Mount Wilson Observatory in California, started to hunt for Cepheid variables in nebulae, and no other nebula was as prominent as the Andromeda Nebula, which we now know as the Andromeda Galaxy. In this historic plate, that's preserved at the Carnegie Observatories, he put down his exclamation of joy when he identified, with red ink, in the top-right, a new variable star that he later understood was precisely a Cepheid variable of the kind identified by Henrietta Leavitt, and therefore, by observing the period of that star, he was able to measure the distance to the Andromeda Nebula, which we see here. At this point in time, nobody knew how distant the Andromeda Nebula was. Some, including the Harvard Observatory director Harlow Shapley, thought that the Andromeda Nebula was just a mass of gas, relatively nearby our own Milky Way, and that there were no other galaxies in the universe. Others stated that Andromeda was actually a galaxy in itself, but deciding the matter was crucially dependent on how far away the nebula was, and Hubble, by measuring this light curve, which is a characteristic pattern of Cepheid variables exactly of the kind discovered by Henrietta Leavitt, was able to put the actual distance to Andromeda to 1 million light years, that's half the right value that we know today, less than half, but still substantial enough to firmly place Andromeda outside the Milky Way, and therefore put it into the realm of the galaxies, and determine its nature as a galaxy. So it was only fitting, then, that when Hubble sent this letter, including this diagram, to Harlow Shapley at Harvard, writing,"Dear Shapley,"you will be interested to hear"that I have found a Cepheid variable"in the Andromeda Nebula," this was in 1924, it was only fitting that when Shapley happened to open Hubble's letter, which he knew would deal a fatal blow to his theory of the Andromeda Nebula being part of our own galaxy, the person who witnessed Shapley's reaction was Cecilia Payne, a Cambridge graduate who had taken over Ms. Leavitt's old desk in the observatory as a computer herself, and Ms. Payne later said that when Shapley read Hubble's letter, said, "This is the letter that has destroyed my universe." Hubble's discovery, then, thanks to Ms. Leavitt's Law, added a fourth rung to our cosmic distance ladder, its top end now reaching firmly into the millions of light years, thus the distance ladder started to seriously stretch far into the distance. Cepheid variables gave us the means of measuring distances to galaxies millions of light years away, but as telescopes became bigger, and fainter and fainter galaxies came into view, even Cepheid variables became unobservable. Those new galaxies were simply too far away, too distant for Cepheid variables to be observable, and so a new, even more distant rung was needed if we were to extend our ladder all the way to the end of the visible universe. Another thing that came into view in the 1920s is that the distance between us and galaxies is growing due to the expansion of the universe, as predicted and described by Einstein's general theory of relativity, a topic that I describe in more detail in my other Gresham lecture"Einstein's Blunder." The discovery of the expansion of the universe is usually attributed to Edwin Hubble, but that's an historical mistake because, in fact, the Belgian priest Georges Lemaitre had made that discovery two years earlier, in a paper that was only published in 1931 in English, therefore the credit went to Hubble, but really belongs, equally at least, to Lemaitre and Hubble. As the universe expands, the light coming to us from distant galaxies is uniformly shifted towards the red end of the spectrum, a phenomenon we call redshift. Redshift tells us how much the universe has expanded since the light left the galaxy, but says nothing about the distance that the galaxy is at that has emitted it. In fact, if we multiply redshift by the speed of light, we get the velocity, and that velocity can be interpreted as the recession velocity of the galaxy, the velocity at which the galaxy is flying away from us, and as an aside, that velocity can and does exceed the speed of light for distant galaxies. That is an apparent contradiction with the special theory of relativity, but it's actually allowed under the general theory of relativity because those galaxies are not moving away from us in space, it is space itself that's stretching faster than the speed of light between us and the distant galaxies. Einstein's theory of general relativity, like I say, predicted this relationship, and in fact, for relatively small distances, still measured in millions of light years, that relationship is a line, and this is the so-called Discovery Plot by Hubble in 1929 that shows that, indeed, further galaxies appear in this plot fly away from us at a faster and faster speed than nearby ones. The slope of this line, the relationship between distance and recession velocity is today called the Hubble-Lemaitre constant, and it's denoted by this symbol here, H0. It used to be called the Hubble constant, and in 2018, rightly, the International Astronomical Union decided to call it Hubble-Lemaitre constant in recognition of Georges Lemaitre's fundamental contributions to cosmology early on in its history. We need to dwell a second on this constant because it's one of the fundamental quantities in cosmology today. Its physical interpretation is that it tells us the expansion speed of the universe today. It's expressed in funny units, the units are kilometers per second, per megaparsec, which means that if the Hubble-Lemaitre constant is, let's say, 70 kilometers per second, per megaparsec, it means that for every megaparsec of distance, that's to say every 3.26 million light years of distance, galaxies speed away 70 kilometers per second faster. So for example, if you have a galaxy 100 million light years away, which is about 30 megaparsecs, it would be moving away from us at a speed of 2,100 kilometers per second. Measuring this constant is, therefore, one of the key endeavors of cosmology, and Hubble and Lemaitre, early on in the '20s and '30s, had only sketchy data to go by, and that put the value of H0 to somewhere in the region to 500 or 600 kilometers per second, per megaparsec. As we shall see, this number has steadily gone down, but we also realized that the relationship between distance, on this axis, and recession speed, or redshift, on this axis, is linear only for nearby objects, millions or hundreds of millions of light years away, those are the first four rungs of our ladder. If you go to even further distances, this relationship begins to deviate from a straight line, and it deviates in a way that depends on the matter-energy content of the universe, how much dark matter, how much cosmological constant you have in the universe. And so by measuring recession speeds, therefore redshifts, and distances for very distant objects, one can pinpoint down the cosmic recipe, and more about this is described in my "Einstein's Blunder" Gresham lecture. But the point is that to peer even further out in space than the four measured distances, to even greater separation from us, Cepheid variables could not be used, they were simply too faint, we needed a much more powerful light beacon, and one came to us in the form of powerful explosions of stars at the end of their lives called supernovae, type Ia. Here is a Hubble Space Telescope picture of one such explosion. You will see here the supernova exploding and then dimming away over time. This is a star that is made of carbon, and oxygen, a very compact, dense star, a white dwarf at the end of its life, accreting mass from a companion star, and the gravitational pressure increases the temperature in the core of the star, and that creates a thermonuclear runaway reaction, which unbinds the star, makes it explode in a fraction of a second, and then the afterglow of the star, powered by radioactive elements, lasts for a few weeks in a very bright phase, during which the star can be as bright as 10 billion suns. Such bright explosions are very, very useful for us because they can be seen from very far away, in very distant galaxies, and they're all uniform because of the threshold mass at which the star explodes, the light emitted by the star is not exactly identical for all of them, but can be made uniform by looking at the shape of the light curve of the star. So by using a similar trick as Leavitt's Law, we can not only standardize the supernovae, make them all look equal to each other, but also tell how much distance there is between us and them simply by the fact that if they all emit the same amount of light intrinsically, the amount of light that we receive and observe must be diluted by distance, and therefore we can work out the distance to very faraway objects in the universe. And that has been done over the past 20 or 30 years, and you can see the distance ladder, the relationship between velocity, or redshift, and distance to extragalactic objects, to distant galaxies, has been increased mightily in distance. Look at the hundreds of megaparsecs in the x axis, and the original Hubble range of observations is the tiny, little red square in the bottom-left. So the distance ladder, thanks to the supernovae, type Ia, has been increased to achieve a new rung, a rung that now firmly puts the top end of the ladder to 2 billion light years, so really a big ladder reaching very, very far into the universe. In December, 2021, the SHOES program reported their new results. This program had been going on for over 15 years using 1,000 orbits of the Hubble Space Telescope in order to observe the three distance ladder rungs in sequence, and with overlapping distance indicators. They spent a great deal of effort in improving the distance ladder by looking at parallax measurements, and then Cepheid variables, and then supernova explosions of the kind that we just saw, and making sure that at each rung of the ladder, they would have at least two distance indicators in order to make sure that no errors and offsets would be introduced between one rung and the other. They also used the Hubble Space Telescope because of its unique stability and unique capability of seeing clearly and measuring brightness very, very precisely from orbit, where conditions are stable and do not change over time. Furthermore, the Hubble can also see in near-infrared light, a type of light with longer wavelength than visible light, which is less subject to absorption by interstellar and intergalactic dust, and therefore those observations give a more precise and accurate measurement of the brightness of objects. The final results are summarized in these three boxes here, each one of them shows one rung of the ladder, from the bottom-left, parallax, to the middle one, Cepheid variables, and the top-right, the supernovae stellar explosions, and you can see they all line up beautifully. And from this very precise data, they could derive a value for the Hubble constant, which comes out to 73.04 kilometers per second, per megaparsec, with a margin of uncertainty of only one kilometer per second, per megaparsec, so a very, very highly precise measurement of the value of the Hubble-Lemaitre constant. And that is wonderful, except while the astronomers were busy building and improving their distance ladder from the ground up, cosmologists, such as myself, were actually doing it in the other way, building a distance ladder from the opposite end of the universe by observing the leftover light, the leftover radiation from the Big Bang itself, the relic radiation that is the afterglow of the Big Bang, here in this diagram, which was emitted when the universe was a mere 380,000 years old, and that has been traveling through the cosmos ever since, until it ends up in our telescopes and observatories 13.8 billion years later. This map shows a snapshot of the baby universe, and the distribution of that cosmic light as it was 380,000 years after the Big Bang. And as I discussed in my dark universe and my weighing the universe Gresham lectures, a great deal of information can be extracted from this leftover light from the Big Bang, in particular, the value of the Hubble-Lemaitre constant, while not measured directly by the relic radiation, is important because dialing the value of H0 puts this data in and out of focus as well, so by choosing a specific value for the Hubble-Lemaitre constant, the data from the relic radiation snap into focus, and that is an indirect method of measuring the value of H0. The cosmologists had another means, a very distant means, in fact, of measuring the current expansion of the universe, and the hope was, and the expectation certainly was that the two ladders, the one built from the SHOES program, from the ground up, and the other coming all the way from the end of the visible universe, from the relic radiation, dangling down, as it were, from the end of the universe, that the two ladders would meet and give us the same value of the Hubble-Lemaitre constant. But the fact is they don't. And you can see that the relic light measurement of the Hubble-Lemaitre constant gave a lower value of 67.4 kilometers per second, per megaparsec, compared to the 73 value given by SHOES. You might think, well what's the big deal? It's a 10% difference, what's a few kilometers per second, per megaparsec between friends? Well that's true, but the margin of error is important, the margin of error is now only one kilometer per second, per megaparsec according to the SHOES program, and this margin of error has been really very precisely established, and the way those margins of error work is that a margin of one kilometer per second, per megaparsec means that there is a probability of about 68% that the true value of H0 is between 72 and 74, there is a probability of about 95% that it is between 71 and 75, and so on, and actually, there is a probability of less than one in a million that the true value of the Hubble-Lemaitre constant is between five times the error, which is a range between 68 and 78, and therefore, the fact that the margin of error is so tight means that the value measured by the relic radiation, of 67, is really, really improbably, it's really, really improbable that the two measurements are both correct, and they differ only by random noise, by measurement uncertainty, something is up in the universe. The fact that these two measurements, with very tight errors, nevertheless are far away from each other in terms of theirs margins of error, means that there is something we don't understand in the distance scale. What could it be? There are three possible solutions to this conundrum. One, the SHOES distance ladder is somehow in error. That's possible, because it's made up of many different measurements from geometry and parallax, Cepheid variables, supernova explosions, and all of them are very difficult to make, and they rely on certain assumptions, they rely on the physics of the objects, some observational error could have crept in somehow. But the SHOES team has been incredibly careful in trying different variants of the analysis, in making sure they have checked for all possible errors in their distance rung measurements, it's really, really difficult to see how their analysis could be improved. What about the distant end of the ladder, maybe that is in error? That's possible too, but really, the relic radiation from the early universe is very well understood, and it's in exceedingly good agreement with many, many predictions of general relativity, we think we really understand the baby universe almost better, in fact, than we understand the present-day universe, so it's really hard to see how this 67 kilometers per second, per megaparsec measurement could be in error. And so if both measurements are right, then the third option is that our model for the universe might be wrong, or at least missing a piece. It could be, for example, that in the early universe, right after the Big Bang, something else is going on, some new physics is at work that makes it so that the data that we receive from the relic light from the Big Bang are somehow in a different focus, and therefore they lead to the wrong value of H0 simply because we're missing a piece of physics that goes on very early on after the Big Bang, that we don't know about. So it could well be that this discrepancy is pointing at some fundamental new component in the makeup of the universe and our understanding of the physics right after the Big Bang, we simply don't know. Many, many theoretical explanations have been put forward by the community, none of them, so far, is clearly in the lead, and so I think that the resolution of this big conundrum, the bringing together of the two distance ladders, will have to await, unless we find a mistake in the current measurements, will have to await new, independent measurements of H0 that do not rely on the distance ladder as we've known it so far, that do not rely on the relic radiation, if we do get a third measurement, for example using gravitational waves, then that could be the way to establish which one of the two, 67 or 73, is right, and that could give us a hint as to what the universe is up to. It is a fascinating mystery, and one that will keep cosmologists and astronomers busy for the next few years, and finally, we hope to be able, one day, to extend our cosmic distance ladder, uninterrupted, all the way to the end of the visible universe. Thank you.