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**Title:** Eppur è piatto? The cosmic chronometer take on spatial curvature and cosmic concordance

**Authors:** Sunny Vagnozzi, Abraham Loeb, Michele Moresco

**First Author’s Institution:** Kavli Institute for Cosmology, University of Cambridge, United Kingdom

**Status:** Submitted to *ApJ*

Though astronomers have been studying the universe for hundreds of years, there are still a lot of things we do not know about it. We do not know whether it is finite or infinitely large, and we cannot determine its overall shape. Nevertheless, we know that we can describe the universe with a four-dimensional spacetime, the combination of our three-dimensional space and time. This spacetime is not rigid, but can be distorted and deformed by the content of the universe, like a bowling ball distorts a spandex sheet. The matter (and energy) also changes how the space part of spacetime is curved — and we can measure this curvature.

There are three possibilities for the curvature of the universe, illustrated in Figure 1: the universe can be closed, flat, or open. A closed universe would be shaped like a sphere (although with a three-dimensional surface), meaning that if you would walk along a straight line you would inevitably end up at the position on which you started. Also, if you and a friend start walking on parallel paths, your paths will cross at some point. An open universe is the “opposite” of this: the distance between you and your friend will increase with each step and you will never end up near each other. A flat universe is exactly in between these two cases: parallel paths stay at the same distance and never cross.

We characterize the curvature with parameter Ω_{k}. Using the sign convention of the authors of today’s paper, a negative Ω_{k} indicates a closed universe, a positive Ω_{k} an open one. If the universe is flat, Ω_{k} is exactly zero.

_{k}= 0). This is not only suggested by a variety of measurements, but also a key prediction of the theory of cosmological inflation. Inflation describes a brief period during which the universe expanded exponentially (see this astrobite for more on inflation). The strong expansion of the universe decreased the curvature, in the same way that inflating a small balloon to the size of the Earth makes it appear flatter. Still, there is an ongoing debate on this issue. The Planck satellite tried to measure Ω

_{k}using the cosmic microwave background (CMB), remnant light from the early universe, which travelled through our potentially curved universe. The results suggest a Ω

_{k}between –0.095 and –0.007, so this measurement method points to a closed universe instead of flat. A reanalysis of Planck data confirmed this preference for a curved universe using the CMB.

However, the CMB on its own is not a sensitive probe for Ω_{k}. It determines a combination of Ω_{k}, the matter density in the universe Ω_{m}, and the expansion rate H_{0}, i.e., the Hubble constant. A strongly curved universe with a low value of H_{0} and a high value of Ω_{m} can have the same CMB as a flat universe with a high H_{0} and a low Ω_{m}. The fact that we can only measure H_{0}, Ω_{m}, and Ω_{k} together and not individually from the CMB is called the *geometrical **degeneracy*.

Cosmologists combine the Planck measurement with other probes, such as baryon acoustic oscillations (BAOs) or Type Ia supernovae. Combining the Planck data with BAO measurements from the Dark Energy Survey leads to Ω_{k }= 0.0007 ± 0.0019, which is consistent with a flat universe.

The authors of today’s paper, though, believe that this combination of Planck and BAOs is not valid. They argue that the Ω_{k }parameters inferred by each dataset on its own disagree so strongly, that the results of a combination of the data sets can be unreliable. If the results of two datasets are in strong tension, this could indicate that one or both include unknown systematic errors, or they need different models to be described. They should therefore not be combined. In case of the curvature of the universe, a different data set should be used to break the geometrical degeneracy. The choice of today’s authors: cosmic chronometers, the universe’s standard clocks.

Cosmic chronometers are objects whose time evolution we know (or can at least model very well), for example specific types of galaxies. We observe some of these objects at different redshifts, which indicate how far away they are. From the differences in their evolutional state, we then infer how much time has passed between the redshifts. This time difference tells us how fast the universe has expanded between the redshifts and gives the expansion rate H(*z*) at each redshift *z*. H(*z*) depends on the cosmological parameters, including Ω_{k}, so from this we can infer the cosmic curvature.

Which objects can we use as chronometers? The best choice are passively evolving galaxies. These are galaxies that have exhausted their gas reservoir and form only a few new stars. Since blue stars die earlier than red stars, the galaxies become redder with time. From the galaxies’ spectral colours (more precisely, their spectral energy distributions) and sophisticated models of stellar evolution, we can infer how much time has passed since they exhausted their gas and stopped star formation. When we compare two galaxies that formed at the same time but are at different redshifts, the difference in their evolution tells us how much time has passed between the redshifts. We have found our cosmic clocks!

Today’s authors use 31 measurements of H(*z*) with cosmic chronometers between redshift *z *= 1.965 (approximately 10 billion years ago) and *z *= 0.07 (approximately 1 billion years ago). Figure 2 shows these measurements, along with the best fit for H(*z*) and the prediction from the Planck measurements. Planck underpredicts H(*z*), but the tension between the cosmic chronometers and Planck is much smaller than the disagreement with the BAO measurements. Therefore, the authors argue that combining the Planck and the cosmic chronometer data set is justified.

_{m}, Ω

_{k}and H

_{0}shown in Figure 3. The combination of Planck and cosmic chronometers prefers a higher value of H

_{0}than the Planck data on its own. However, this is not enough to alleviate the famous Hubble tension. Most important, though, the combined data finds Ω

_{k }= –0.0054 ± 0.0055. This value is consistent with a flat universe for which Ω

_{k }= 0, as predicted by cosmological inflation. In conclusion, the authors of today’s paper argue that the universe is most likely not curved. Their result fits other measurements that combined Planck data with other probes, such as BAOs, but their choice to use cosmic chronometers produces a result that they consider more reliable, because the individual datasets did not disagree strongly. This result could be a notable step forward in solving the controversy around Planck’s curvature measurement. More measurements of cosmic chronometers are undoubtedly due in the future — so look out for more results from the universe’s clocks.

*Original astrobite edited by Haley Wahl.*

## About the author, Laila Linke:

I am a third year PhD Student at the University of Bonn, where I am exploring the relationship between galaxies and dark matter using gravitational lensing. Previously, I also worked at Heidelberg University on detecting galaxy clusters and theoretically predicting their abundance. In my spare time I enjoy hiking, reading fantasy novels and spreading my love of physics and astronomy through scientific outreach!