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Is modified gravity an illusion?

by Oleksii Sokoliuk

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