Cosmology has been developed for thousands of years and there have been dozens of theories proposed

to describe the observable universe. Starting from the Babylonian cosmology 2300 BCE, which

suggested that the flat Earth is embedded into the ”waters of chaos” up to the present time when

extremely sophisticated cosmological models are being used. However, the most fast-paced development

of cosmology was observed during the last century, at the point when Albert Einstein introduced

his theory of Special and General Relativity (further – SR/GR). The notion of SR had various practical

applications – it became possible to derive numerous observables at speeds close to the speed of

light – where the well-known classical physics, developed back in the 17th century by Newton failed.

Afterwards, Einstein introduced GR by expanding SR to any curved spacetime. GR was the first

theory to successfully solve the Mercury perihelion problem and it was tested with the use of solar

eclipse (relating expected and observed positions of stars near the sun, i.e. observation of the effect of

bending of light) as well as with Shapiro delay by Cassini–Huygens probe and many others. Moreover,

the unimaginable at that time freedom of GR helped Karl Schwarzschild to develop the first simple

model of black holes and Nathan Rosen to introduce the first non-traversable wormhole solution, which

inspires sci-fi writers to this day. However, it was discovered that GR has numerous problems – just

geometric treatment of gravity could not explain the late-time accelerated expansion of the universe,

observed with the help of supernovae in distant galaxies by Nobel prize winner Adam Riess in 1998.

To introduce such a feature, later cosmological constant (dark energy in simpler terms, referred

to by Einstein as Λ term) was implemented into Einstein’s field equations (which in turn describe

the evolutionary dynamics of the universe), creating a de Sitter universe. Contrary to Einstein’s

static universe, de Sitter one could expand with acceleration, but such a universe still was filled only

with dark energy, and therefore it could not show most of the properties of the observable universe –

galaxy and star formation, primordial nucleosynthesis (formation of the first elements in the universe,

mostly Hydrogen and Helium) or reionization. Many of those properties can be implemented only by

using baryonic (normal, observable) matter, but there was another problem to consider, namely the

rotational curves of the galaxies. After the discovery that the Milky Way was not the only galaxy in

our universe by Hubble in 1924, scientists began exploring our galactic neighborhood and trying to

figure out how to model our galaxy by observing other ones, such as Andromeda Galaxy or Magellanic

Clouds. In 1957 and 1959 years respectively, the first evidence that the speed at which stars rotate

around the center of a galaxy is flat and does not change rapidly with the distance from the center was

found by the group of astronomers at Dwingeloo Radio Observatory. This finding was very surprising,

as it was expected that the velocity of stars would follow the same curve as the galaxy luminosity

and decrease by the end of the galactic disk, which was backed up by Keplerian laws of motion, that

worked perfectly within the Solar System. Later a new approach was proposed that could solve this

discrepancy, we just had to add additional matter that is uniformly distributed within the galaxy and

can therefore help stars to maintain their rotational speeds. This is how the theory of Dark Matter

was born!

It’s not hard to figure out that to solve acceleration, galaxy formation, nucleosynthesis, and rotational

curve problems we just have to pair the dark energy, dark and baryonic matter together into a

new model, the most complete theory of cosmology in the 20th century, namely Λ Cold Dark Matter

theory. This model was pronounced the fiducial, standard model of cosmology for decades to come

and had minor modifications (in early times, one should for example add the period of cosmological

inflation to solve horizon and graviton mass problems), but its main postulates remain unchanged. The very interesting fact is that ΛCDM is an extremely simple theory, it differs from GR one just by the

addition of −2Λ and of course by matter content, as mentioned above. On the contrary, the standard

model of particle physics has an absurdly complicated notion that takes almost a whole page to write

down. But why, while being one of the most mysterious forces, gravity could be described in such simple

terms (and with such great precision, even while using the data from the most recent cosmological

probes!) geometrically? Unfortunately, humanity has yet to address this question properly.

But, while ΛCDM is known to be a great theory of cosmology, it still has numerous flaws. For

example, the Lithium abundance problem states that Big Bang Nucleosynthesis (BBN) models predict

three times as much Lithium as being observed, which is a huge contradiction. In addition, the

collisional velocity of galaxies in the El Gordo cluster inferred from observations is too big and cannot

be described within the fiducial ΛCDM model. From the quantum standpoint, the ΛCDM theory is

also flawed. At the very early times t < 10−35s, it is known that quantum gravity effects dominated

and the viable theory of gravity must have a quantum gravity analog acting at the early times, but due

to its simplicity and the absence of higher-order terms, ΛCDM cannot show a viable quantum gravity

behavior (here the famous string theory comes into play, but it’s a very different story). Moreover,

there is no experimental motivation behind the choice of dark energy and dark matter ansatz, since

both of those ”substances” were never observed up to date and we have no idea what they are at

all, we can only make educated guesses and hope for the best. It is worth noticing that one of the

biggest puzzles of modern cosmology is the so-called Hubble tension. One constant, namely the Hubble

constant defined by a symbol H0 is extremely important and it represents the rate of expansion of the

universe at the present time and controls the values of many observable features of our universe. The

value of H0 was derived from both early universe (Cosmic Microwave Background, Baryon Acoustic

Oscillations) and late universe (supernovae of Ia class, masers) probes, but it was discovered that

results disagree at more than 5σ significance (such deviation was enough to make the discovery of

Higgs boson)! This is an indicator of strong evidence against ΛCDM and all of the efforts to get rid of

those issues while still staying under the framework of standard cosmology failed. Here, the so-called

modified gravity comes to the rescue.

Modified gravity arises when someone tries to add anything to the fiducial model – additional matter

field, new curvature invariant, non-minimal coupling between ordinary and dark matter, radiation or

gravity, etc. Therefore the concept of modified gravity by itself is very vague. Many of such theories by

themselves include higher-order terms, making it possible to explore them at the quantum level, and

at the present time almost a hundred models have been reported to solve Hubble tension or alleviate

it, some of them even made it possible to encode late-time accelerated expansion and cosmological

inflation, other eras into the model simultaneously. However, while solving some of the issues of

ΛCDM, there is always a price to pay – many of those theories are known to have pathologies such as

extreme fine-tuning required to hold cosmic coincidence valid.

So what are we even supposed to do? While trying to get away from the troubles in ΛCDM by

modifying gravity we encountered even more obstacles on our way. Some researchers believe that

modifications of gravity are not required and that standard cosmology by itself is sufficient and the

main problem lies within the systematics of data. This is a valid approach as well, indeed a significant

deviation in the value of H0 was observed while comparing results from two different hemispheres of

CMB data (the so-called dipole problem), but still, no significant results were reached in this field.

There is also a possibility that both Lambda CDM and modified gravity paradigms are wrong and we

need to completely change our understanding of cosmology to arrive at a better model (as it was done

in the transition from classical Newtonian mechanics to General Relativity).

In the following years, we hope to obtain tighter constraints on H0 from the upcoming cosmological

surveys, such as LSST, DESI, or Euclid which may help us to reduce the modified gravity landscape

and reach some kind of a conclusion on this dilemma, but this still remains an open question. Is

modified gravity an illusion?

January 30, 2024