My Science & Epistemology Researches
Cosmology is the scientific study of the universe’s origin, evolution, large-scale structure, and future, combining physics and astronomy to understand the cosmos as a whole. It uses observations and theories to explain how galaxies, stars, and planets form and change over vast timescales, addressing fundamental questions about existence.
A short paper, in english, on a fascinating development of General Relativity.
Abstract
Extended Theories of Gravity (ETG) represent a prominent framework in modern theoretical physics and cosmology, aimed at modifying or extending Albert Einstein’s General Relativity to address open phenomenological questions at both ultraviolet (early universe) and infrared (galactic and cosmological) scales.
Brussi 2026 Mathematical and historical foundations of ETG (305 KB)
Introduction
What we are about to explore is what I call ‘the stage of reality’, the place where we, and everything around us, move (a trivial side note: nothing is truly ‘at rest’ in our universe, as there is no absolute reference frame against which to measure motion). This stage is what we call spacetime, composed of the three familiar spatial dimensions plus time, which is simply an additional dimension. We cannot ‘visualize’ it unless we imagine removing one of the spatial dimensions, as seen in the famous Minkowski diagrams (see Figure, Hermann Minkowski, 1864-1909).
It was Albert Einstein (1879-1955), in his brilliant work in the early 1900s, who viewed the universe not as an empty space where time flows separately, but as a single four-dimensional context. These dimensions are inextricably linked by a metric governed by the laws of gravitation. In this scenario, time is not an external clock, but a real direction in which we move, just as we move North or South, to put it in geographic terms.
Gravity
Why is the force of gravity not a force? For centuries, thanks to Isaac Newton (1643-1727), we thought of gravity as a kind of ‘invisible hand’ that ‘pulls’ objects. But if we apply what Einstein called the equivalence principle (which he described as “the happiest thought of my life”), we discover a fundamental concept. He imagined being in an elevator in deep space, far from any gravitation, accelerating upwards (remember that acceleration is the variation of velocity). You would feel pressed to the floor exactly as you do on Earth. If you were to release Newton’s legendary apple, it would ‘fall’ toward the floor. Yet, there is no gravity ‘pulling’ it; there is only acceleration. Einstein realized that gravity is not a force acting in space; rather, the effect we observe and call ‘force’ is the consequence of the curvature of spacetime itself. In fact, the presence of mass —or rather, mass-energy, since they are the same thing except for a conversion factor (the famous E=mc² )— warps the fabric of spacetime like a weight on an elastic sheet. For example, the Earth, with its concentration of mass-energy, creates a geometric deformation: the apple doesn’t fall because a force pushes it, but because it is simply following the straightest possible line (called a geodesic) in a space that has become curved.
According to Newton’s laws of motion, a body with no forces acting upon it moves with uniform rectilinear motion (respecting the principle of inertia). According to Relativity, an apple falling in a gravitational field is, paradoxically, the only object experiencing no force at all (respecting the principle of inertia). To quote Einstein again: “Matter [mass-energy] tells spacetime how to curve, and spacetime tells matter how to move.”
Light
Now let’s consider light. We attribute a ‘speed’ to it (which we define as a change in position), but as physicist Leonard Susskind (1940-) explains, it is not a speed in the common sense of the term (like that of a car). Instead, it is a fundamental property of the geometry of spacetime —a simple ‘conversion factor’ between spatial and temporal coordinates.
In the four-dimensional universe in which we exist, space and time are different directions of the same thing, and c is the number that tells us how many meters are equivalent to one second. That is: time=space / c .
The value of the universal constant c is approximately 300,000 km per second, or just over a billion km per hour (approx. 1.08 billion km/h).
Susskind’s profound suggestion is that every object in the universe always and constantly moves at the exact same speed:
– When we are ‘still’ in space—for instance, sitting in our chairs—we are traveling at the speed of light along the axis of time
– If we start moving through space, we must ‘subtract’ speed from time to compensate for the spatial movement (as if paying a toll using the conversion factor, ). From this, the relativistic time dilation is derived (e.g., the twin paradox).
There are no variable speeds; there is only a distribution of the ‘total speed’ among different dimensions. Therefore, light isn’t ‘racing’: we could say that light spends its entire ‘allowance’ on space, leaving zero for time, based on the conversion factor c .
This constant —the speed of light— is the geometric limit of what can happen in the universe.
Conclusions
There is, therefore, no speed limit; there is only a global geometry in which we are all immersed. Gravity is the curvature of the road; the constant holds together the very fabric of the reality surrounding us. Physics does not describe ‘what happens’, but describes the geometric structure in which everything is already contained.
Understanding this means stopping looking at ‘things’ that move and starting to look at the shape of the stage on which they move.
A Minkowsky diagram (credits to medium.com)
The URD Framework: Cosmological Distances from Redshift in a Non-Standard Context
While I am still developing the project —including through parallel studies on the Tolman test and the quality of indirect data obtained from recent galaxy surveys— I am making my work public as a preprint. I always hope that someone might be interested in my conjectures or perhaps even willing to collaborate with me to address the critical issues of the URD Framework.
Abstract
A phenomenological framework for determining cosmological distances is proposed, based on a non-expanding spacetime interpretation consistent with current observational datasets. The model introduces a redshift mechanism associated with cumulative photon energy attenuation along the line of sight, formulated in a way that is not equivalent to standard tired light scenarios. The approach is constructed to remain compatible with established relativistic principles whereas adopting boundary conditions distinct from the metric expansion paradigm. Distance estimation is achieved through the URD equation (U energy density, Redshift, Distance), which incorporates two phenomenological parameters that describe conformal geometric dissipation in extended gravity and an effective relativistic path-length correction. Within this formulation, the model addresses the primary shortcomings traditionally associated with energy dissipation effects, specifically reconciling time dilation in supernova light curves and the Tolman surface brightness relation. The model is calibrated using the Pantheon+ supernova sample by comparing and cross-validating distances inferred from the proposed redshift mapping with redshift-independent indicators, with additional comparison performed against baryon acoustic oscillations, radio galaxy samples, and high-energy quasars. Across these datasets, the framework reproduces observational constraints over a wide redshift range within a phenomenological, non-standard cosmological setting. The results indicate that alternative, non-expansion-based mappings between redshift and distance can be constructed that remain consistent within the observational range considered, suggesting a phenomenological alternative worth further study.
Keywords
cosmological distance ladder; redshift-distance relation; non-expanding spacetime; cosmic microwave background; extended gravity; Pantheon+ sample; baryon acoustic oscillations; Tolman test; time dilation; geodetic correction
Here is the last version of the preprint (5.1 MB):
Brussi 2026_URD Framework Preprint
Version uploaded to Zenodo:
An in-depth analysis by Leonard Susskind of the constant c, commonly referred to as the “speed of light.” It is essentially a collection of familiar concepts, but reinterpreted in an illuminating way, that are well worth considering.
Abstract
This brief work, then became an Appendix to my URD Framework study, proposes a cosmological model characterized by a globally flat, spatially infinite, and non-time-limited spacetime. It challenges the standard application of the Friedmann-Lemaître-Robertson-Walker metric by reinterpreting general relativity as a strictly local phenomenon within a non-continuum matter distribution. In this framework, the observed flatness of the universe is a fundamental geometric property rather than a dynamical result of inflation, and cosmological redshift is modeled as a cumulative energy dissipation process rather than metric expansion.
Brussi 2026_Cosmology for a static universe
Ai generated image from free Adobe stock
Exploring beyond our galaxy, to be able to reach the edges of the known universe with measurements that can be perceived, it is necessary to change scale again compared with the previous post on the Milky Way. Over there, the distances involved become almost impossible to truly understand, they are so mindblowing!
Let us now consider our galaxy to be 1 millimeter across. We are part of a ‘local’ group of galaxies made up of more than a hundred smaller galaxies (of which only one is of significant size, the Triangulum Galaxy, known as M33, see Figure 1), and a large spiral galaxy broadly similar to ours, Andromeda (Figure 2), which is about 2.5 cm away. Andromeda is the most distant cosmological object that can be seen with the naked eye (see also my post References for laymen: angles in the sky).
Figure 1 – triangle galaxy M33 (credits: Nasa, Esa e M. Durbin, J. Dalcanton e B. F. Williams, University of Washington)
Figure 2 – Andromeda galaxy (image credit: Westend61 via Getty Images)
Andromeda is more extended but less massive than our galaxy. It’s getting closer to us (or rather we’re getting closer to each other) at about 400,000 km/h, on our new scale, less than four thousandths of a millimeter (about one hundredth the thickness of a hair) in a million years. In other words, in 4–5 billion years the two galaxies will merge, still in a collisionless way (we demonstrated this here, §1, not for newbies), and will probably turn into a giant elliptical galaxy (like NGC 1600 in Figure 3) after a few more billion years. However, recent measurements from the Gaia* satellite reduce the probability of this merger happening on such ‘short’ timescales, even though it is inevitable that it will occur.
* The Gaia satellite is an astrometric mission of the European Space Agency (ESA), launched in 2013 to map the Milky Way in 3D. It orbits around the Lagrange point L2 (see my post The three-body problem and the five Lagrangian points), and has created the most precise stellar catalog of more than 2 billion stars around us, measuring their positions, motions, brightness, and chemical composition.
Figure 3 – NGC 1600 by HST, it has a diameter of about 120.000 light -years, or about 1.2 mm on our scale (credits: A. Quillen, University of Rochester, G. Bower, CSC/STScI, and G. Rieke, Steward Observatory/University of Arizona)
The Local Group of galaxies (see the 3D schematic in Figure 4) has a radius of about 3 cm on the adopted scale, and is surrounded in an “homogeneous” way by similar structures. For example, in Figure 5 a radius of 33 cm is considered on the same scale, and in Figure 6 a radius of about 1.5 m. Proceeding in an analogous way, one can reach the limit of the observable universe (that is, before redshift completely prevents sources from being detected) which, on the adopted scale, can be taken to be at a distance of just over 4 meters.
Figure 4 – Local Group of galaxies (source starwalk.space)
Figure 5 – Virgo Supercluster, along with 100 other galaxy groups (source starwalk.space)
Figure 6 – Laniakea Supercluster which includes almost 100.000 galaxies more than ours (source starwalk.space)
We have reached the limit of the current theory of the standard cosmological model. It is a theory, based on creation from nothing through an initial Big Bang, which is strongly supported by observations but also has major gaps in explaining other evidence.
Our journey therefore stops right at the limit of ‘measurable’ findings grounded in commonly accepted theories; beyond this point, it becomes epistemology rather than cosmology.
Previous episodes:
Understanding the Sizes : 3 – Milky Way
Understanding the Sizes : 2 – Solar System
Understanding the Sizes : 1 – Earth and Moon
The Milky Way is our galaxy: a gravitationally bound structure that includes about 100 billion stars and an enormous amount of gas (essentially atomic hydrogen), with an approximate shape of a rotating disk, which, of course, we can only imagine and not actually see, since we are inside it (our view towards the galactic centre is known, see Figure 1, even if it is increasingly difficult to observe, due to atmospheric pollution, especially light pollution.). See Figure 2 for a photo of a similar example. The stars, gas, and dust that make up the disk rotate in the sense that they orbit around the roughly spherical central nucleus (the bulge), which instead is not rotating (its stars have chaotic motions).
Figure 1 -Milky Way as seen from Earth (my photo)
Figure 2 – example of a galaxy similar to the Milky Way: NGC 6744, taken at the European Southern Observatory’s La Silla Observatory in Chile (image credits: ESO)
To understand the sizes, it is necessary to change scale again compared with the previous post on the Solar System: the quantities involved become increasingly difficult to ‘grasp’. We have seen that Sun light takes about 4 hours to reach Neptune, the outermost planet of the Solar System. The star closest to us is Proxima Centauri (a red dwarf in a triple star system), about 4 light-years away. We therefore choose the light-year as the unit for this new distance scale, imagining 1 light-year as equal to 1 millimeter. Then Proxima Centauri is about 4 mm from the Sun (of course, since the difference is only a few light-minutes compared with 4 light-years, Earth and the Sun that Earth orbits can be considered at essentially the same distance).
All these stars orbit around the nucleus but also have their own independent motions, which are collisionless (we demonstrated this here, §1, not for newbies), a remarkable fact given the number of bodies, and in contrast, for example, with a collection of gas molecules in a container, which instead undergo continuous collisions (which determine its pressure and temperature). But compared with a gas, the distances between the components in a galaxy are immensely larger.
In the new scale we have adopted, the radius of the galactic disk is about 105 meters (about 105,000 light-years), and our position is about 26 meters from the center. In Figure 3 the drawing schematically outlines one possible configuration and the proportions. Our linear speed is about 790,000 km/h and we complete one orbit in about 250 million years, so on our scale, despite the crazy speed, we move only about 1.5 cm in a thousand years, in this 105-meter-radius disk. Since the Sun and Earth formed, about 4.5 billion years ago, we have made about 18 revolutions around the galactic nucleus.
Figure 3 – schematic drawing of the Milky Way and its proportions (gases beyond the star limit are not drawn but they are gravitationally relevant; source starwalk.space)
The disk of our galaxy is made up of several arms, where the concentration of stars is higher, and a more rectilinear component that originates from the central nucleus, from which the arms branch out, as schematized in Figure 3. A galaxy of this kind is called a ‘barred spiral’ and its shape when seen edge-on is similar to the example in Figure 4.
In a future post we will talk about the central black hole and the star-forming regions.
Figure 4 – Example of a spiral galaxy seen edge-on, ESO 121-6 (source: HST by ESA/Hubble & NASA)
As a final remark, note that nearly all the stars visible to the naked eye belong to the Sun’s local stellar neighborhood, as highlighted in Figure 3. It’s a tiny region compared with the size of the whole Milky Way. Most of the Galaxy’s other stars are too faint and/or too obscured by interstellar dust to be seen individually, and instead contribute to the Milky Way’s diffuse glow: a blend of millions to billions of unresolved stars.
Next episode:
Understanding the Sizes : 4 – galaxy clusters and beyond
Previous episodes:
Understanding the Sizes : 2 – Solar System
Understanding the Sizes : 1 – Earth and Moon
Second episode of the deep dives into the measurements of the universe around us. The first one was about the Earth and the Moon.
This one is about the Solar System, that is, the bodies that orbit the Sun, our star.
The planets classified as such are, in order of increasing orbital radius: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune (see Figure 1). Besides these main bodies there are many secondary ones, such as dwarf planet, moons, asteroids, and comets, but we will focus only on the main objects.
Figure 1 – drawing of the sequence of planets of the solar system
Since Kepler (1571–1630) we know that orbits are ellipses (see Figure 2), with the Sun located at one of the foci. In reality, eccentricity (i.e. a – b with reference to Figure 2) is quite low, about 3% for Earth and modest for all the planets except Mercury. Other celestial bodies orbiting in the Solar System, such as asteroids (larger ‘rocks’ measuring a few hundred kilometers), also exhibit significant eccentricity (which measures how much their orbit deviates from a circle). For simplicity we will consider circular orbits and a single radius.
Figure 2 – An ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of both distances to the two focal points is a constant ( FP + PF’ = constant for each P on the curve); in this example, the eccentricity is very large, while for the planets it is much smaller; the F points are called foci, a is the semi-major axis, b is the semi-minor axis.
We will not list all of the measurements for these objects, but only a few reference ones. It is important to understand the need to change the reference scale, moving to a larger one, we can no longer use the soccer ball from the previous post. Now let us imagine Earth as a tiny grain of fine sand (0.09 mm), and its distance from the Sun as 1 meter (the radius of its orbit, usually called the astronomical unit, AU). On this scale, the planet closest to the Sun is Mercury (with a radius 0.38 that of Earth), at about 38 cm. Mars (with a radius 0.52 that of Earth) orbits at about 1.5 m; the gas giants Jupiter (radius 11 times that of Earth, i.e., 0.9 mm on the adopted scale) at about 5 m, and Saturn (radius 9.5 times that of Earth) at about 9.5 m. Uranus and Neptune are very far away, at about 19 m and 30 m respectively. See Figure 3 for a scale diagram.
Figure 3 – a scale diagram of distances in the Solar System
But what does this one meter of distance, taken as a reference for Earth’s orbit, correspond to? It is about 149 million km, a distance that would take roughly 170 years to cover by car traveling at 100 km/h. Light, which travels at 300,000 km/s (more than 1 billion km/h), takes about 8 minutes to go from the Sun to Earth, and more than 4 hours to reach Neptune.
And how big is the Sun? Its radius is about 670,000 km, so compared with the Earth as a soccer ball model, it would be a sphere with a radius of 11.60 m, roughly the volume of a a five-story building of 400 square meters per floor. Compared with the Earth as a grain of fine sand model, it would be a ‘grain’ of almost 1 cm (9.3 mm).
Earth has one large moon, the Moon; Mars has two small ones, Phobos and Deimos. Jupiter has four major moons (Io, Europa, Ganymede, Callisto, discovered in 1609 by Galileo Galilei, 1564–1642) and 91 smaller ones. Saturn has as many as 146 in total! Practically all moons are in synchronous rotation, meaning they always show the same face to their planet, like our Moon.
Saturn’s rings (Figure 4), made up of countless particles of ice and rock, are more than 30,000 km wide (three times Earth’s diameter) but extremely thin, ranging from a few tens of meters to a few hundreds of meters. For this reason, when they are seen edge-on (about every 15 Earth years), they reflect almost no sunlight and are not visible from Earth.
Figure 4 – Saturn photo by HST (source: NASA, ESA, STScI, Amy Simon NASA-GSFC)
Beyond the orbit of Neptune, or on our scale between 30 and 50 meters, lies the Kuiper Belt (named after Gerrit Pieter Kuiper, 1905-1973), which contains thousands of icy bodies, remnants of the formation of the Solar System, including dwarf planets like Pluto, essentially distributed in a volume squashed on the plane of the ecliptic. The so-called heliosphere ends here. An even more external region, the Oort Cloud (named after Jan Oort, 1900-1992), between 2 and 200 km on our scale, has been hypothesized to contain an immense diffusion of ice and rocks, the reservoir from which comets are drawn by the Sun. We can consider it as the outer boundary of the Solar System.
Next episodes:
Understanding the Sizes : 3 – Milky Way
Understanding the Sizes : 4 – galaxy clusters and beyond
Previous episode:
Understanding the Sizes : 1 – Earth and Moon
In everyday life, we don’t stop to think about the ‘sizes’ of the planet we live in, the Solar System, and the galaxy (Milky Way) in which we orbit with our mothership, Earth.
In a series of insights, we aim to understand the distances and some parameters that characterize these systems. This first one concerns the Earth and its satellite, the Moon (Figure 1).
Figure 1 -the Earth and the Moon
The average radius of the Earth (slightly smaller at the poles than at the equator) is approximately 6,370 km, meaning its circumference is approximately 40,000 km. Therefore, an electromagnetic signal traveling at 300,000 km/s would circle the planet 7.5 times in one second.
Here are some interesting measurements to compare:
– The thickness of the troposphere, where virtually all meteorological phenomena occur and almost all water vapor is concentrated, varies from about 8 km (at the poles) to about 20 km (at the equator), or, on average, 14 km. That is, if the Earth were the size of a soccer ball (radius about 11 cm), the troposphere would be roughly the thickness of two sheets of paper (about 0.24 mm).
– The highest mountain (Everest, about 8.8 km) and the deepest ocean (Mariana Trench, about 10.9 km) are of the same order of magnitude as the height of the troposphere. As if to say that if you held that soccer ball in your hand you would almost not notice their presence.
– The Moon, which has a radius of 1,737 km, would be slightly smaller than a tennis ball (radius about 3 cm) compared to the soccer ball sized Earth, see Figure 2. Its average distance from Earth is about 380,000 km, so an electromagnetic signal takes about 1.3 seconds to arrive; in the proportions calculated for the balls, the distance between them would be about 6.5 meters.
Figure 2 – Soccer ball and tennis ball in the proportions of Earth and Moon
– The altitude at which airliners fly, about 10 km, is little more than the thickness of a sheet of paper. Seen from space they appear to crawl more than fly. The International Space Station (ISS) is maintained at an altitude of about 400 km, about 7 mm above the soccer ball, at a speed of about 28,000 km/h, or it would move about 8 mm per minute.
– Considering the Earth’s rotation in 24 hours, the tangential velocity of a point at the equator is almost 1,700 km/h; at our latitudes of about 45°, the speed is just under 1,200 km/h. For this reason, spacecraft launch pads are positioned at low latitudes, near the equator, to take advantage of the higher linear velocity. The escape velocity required to overcome gravitational pull (about 40,000 km/h on Earth) is more easily achieved by exploiting the velocity of the launch point (with a starting direction toward the east for maximum effect, considering that the Earth rotates towards the east and the speeds can add up).
Note that escape velocity is not tied to ‘up’; it’s the minimum speed at a given altitude to be unbound, in any direction, as long as you don’t collide with the planet.
Next episodes:
Understanding the Sizes : 2 – Solar System
Understanding the Sizes : 3 – Milky Way
Understanding the Sizes : 4 – galaxy clusters and beyond
Why is the night sky dark?
If universe were infinite, eternal, and static, as Giordano Bruno (1548-1600) claimed, as did cosmology until the early 20th century, adding that it is also populated by stars in a homogeneous manner, the night sky should be bright and not dark. This contradiction was already noted by Johannes Kepler (1571-1630) in 1610 in Dissertatio cum Nuncio Sidereo.
However, the principle was clearly formulated in 1826 by Einrich Wilhelm Olbers (1758-1840), who mathematically demonstrated (see below, optional reasoning which can be skipped) why the night sky should shine according to the cosmology of the time, highlighting a paradox with respect to observation.
Imagining the universe as consisting of spherical shells concentric with respect to our point of observation (see Figure 1), the intensity of radiation received (flux f ) from a source ( L ) at a distance ( r ) is inversely proportional to the square of the distance:
[1] f = L / 4 π r²
The number of sources contained in the shell is proportional to the volume, which increases with the square of the distance (by infinitesimal increments
dr = R – r with reference to Figure 1):
[2] dn ∝ dV ≈ 4 π r² dr
Therefore, the effects of [1] and [2] compensate each other and each shell contributes with the same intensity, regardless of distance.
The infinitesimal intensity ( dI ) coming from the infinitesimal shell, multiplying [1] and [2], is:
[3] dI = f dn = n L dr
Integrating [3] gives the total intensity:
[4] I =∫₀ᴿ n L dr = n L R
Therefore, for R → ∞, the intensity [4] should become infinite. So something is wrong with the reasoning (hence the paradox).
Figure 1 – sketch of a spherical shell
The observation of the dark night sky is explained in modern terms (based on cosmological standard model) because the universe has existed for a finite time (Big Bang hypothesis) and is expanding (interpretation of the redshift of radiation from remote sources).
In the first case, it is considered that the light from remote sources, which has a finite speed, simply has not reached us yet.
In the second case, redshift, or the shift of light towards longer wavelengths (for example, in the infrared band, beyond the range visible to the eye), prevents the observation of remote sources.
It should also be remembered that the distribution of the intensity of radiation from stellar sources follows a Planckian curve (see Figure 3), which has its maximum in the visible light band. In other words, there are no significant intensities of radiation in a typical stellar source that, moving due to the effect of redshift, could affect this visible band.
In fact, we emphasize that the range of radiation visible to our eyes is very small compared to the entire spectrum of radiation, as shown in Figure 2.
In reality, redshift alone would explain the dark night sky, even if the universe were infinite and without assuming a Big Bang. And even without expansion, if the observed redshift had a different cause.
Figure 2 – full spectrum of radiation with visible band
Figure 3 – A sketch of a Planckian functions of brightness (intensity of radiation emitted per unit solid angle) for stellar sources with different surface temperatures. That of the Sun is approximately 5700 K.
