The Greatest Expansion Part-2

A question that has always created a lot of misconceptions among the lay people is that according to our current scientific understanding, no part of matter or matter itself can travel in or through space at the speed of light. But how can the universe itself expand faster than the speed of light aka FTL? How does it gain this ability to be FTL without breaking the known cornerstone of natural principles?

The question is simple, but the answer is difficult in itself to understand. Because The universe is not a matter or particle, hence it is not restricted to expansion up to the relativistic limit.

I will, from this point on, try to explain it so one even without a degree in STEM will be able to comprehend this idea of FTL expansion of the universe.

Speed ​​Limit of Universe: 2.99792458×10^8 m/sec is limited to matter, particles and their related information (such as momentum, position, energy, wave function etc.) The universe is not matter, it is a dynamic entity in itself which can propagate at the speed of Faster Than Light (FTL) without breaking the laws of physics.

• Dynamics means it is not static:
• It is constantly expanding.
• It evolves due to the explosion of a star, the formation of a black hole or the collision of planets. (Events shape the universe and how things interact with each other.)
• Matter, radiation and dark energy are effective cause from time to time.
• And that is why time is an effective cause too, since the universe itself has this past, present and a future.

That's why we called it dynamic above.

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THE Math : Now as a reader you must understand that without introducing a few mathematical terms I will only be doing injustice toward your intellectual and cerebrum capacity.

Let me introduce an equation from Einstein's General Relativity.

Friedmann-Lemaître-Robertson-Walker (FLRW) Metric is as important in cosmology as the peanut butter is important in a PB&J sandwich. There is a term called the scale factor a(t) which is a time dependent function that describes how the speed of expansion of the universe increases as the distance increases with time. [1]

Now if you have read the part 1 on this article about Big Bang you will also understand that the Big Bang is not an explosion but an expansion. you can call it The Big Inflation for a brief period of time  and that the expansion which is still going on and will continue.[2] and if you do not know this big inflation talk don’t worry you can visit the Part 1 of the article anytime.

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KNOW THY BALOON: Now you must have blown a balloon in your life. Imagine the universe was like this balloon in the very beginning and then you blow very very fast and the balloon suddenly inflates.

The FLRW metric says that this inflation of the balloon happens in a similar way in all directions and if you put 10 marks on this balloon with a marker, then this inflation will look exactly the same from each mark.

This is called Homogeneous meaning equal expansion and Isotropic meaning the universe looks the same from everywhere.[3]

Mathematically it looks something like this. 

Image Credit: De Gruyter Brill[4]

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Now suppose the balloon can become as big as your body by inflating, then as the balloon expands, the points you made with the marker will also expand. If the distance between the 3 points made with the marker is one inch, then first it increases by 2 inches and as the air keeps filling the balloon, the distance between the points will increase to 4, 8, 16 inches i.e. as much as the balloon can expand without bursting.

So now you tell me, the distance between the points you made with the marker on the balloon was just one inch and now when you have inflated the balloon and made it as big as you, then the distance between each point has become up to one foot.

So now you tell me ‘which measurement is wrong?’ None! Because the material from which the balloon is made, that material in itself is made for stretching.

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Galaxies and Expansion

Now understand that each point is a galaxy and the expansion between them depends on many things.

First, as the universe expands, if two galaxies are one megaparsec (Mpc) apart, then as the scale factor a(t) increases, the galaxies will move away from each other faster and faster, and their speed is determined by the formula v = H⋅d. This is called Hubble’s Law. [5]

    • H = a˙/a : This is called the Hubble parameter, which tells how fast the balloon is expanding.

    • d = distance between the galaxies.

And the math says that if the distance between the two galaxies is more than 4200 Mpc, then v > c because the universe is expanding much faster than the speed of light to cover this distance.

What is the reason that the scale factor a(t) increases?

Here we come to the Friedmann equation which tells us about the role of all the materials in the universe and who is playing what role.

    • ρ is all the present matter, heat, dark energy,

    • k is curvature,

    • Λ is dark energy which is driving this expansion,

The normal matter and radiation at the beginning of the universe resisted the expansion of the universe, due to which the value of a˙ fell, but as the universe cooled, dark energy (Λ) increased the speed of this expansion. [6]

And now this expansion has increased to become faster than the speed of light at the boundaries of the universe.

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That is why when we simplify these equations by making time negative, i.e. reversing it, it takes us back to a hot and dense point-like universe. And when we come to this point and measure the scale factor a(t), we find that during the initial expansion of the universe, the value of a(t) increases exponentially and later slows down.

Light moves at the speed of c in the universe, but when the distance between two points gets rapidly increasing, then light cannot cover this distance at its speed and that is why we say that after the Hubble Horizon, which is about ~14 billion light years, the universe is expanding so fast that on that edge, the galaxies are going faster than us, i.e. v > c, i.e. faster than the speed of light.

So in this way, no law of relativistic physics is broken and the universe is also expanding faster than the speed of light.

Image Credit: ASTROMICCA[7] 

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The De-Sitter Universe

So this is what we uhave understand that if dark energy continues to play a role in the expansion of the universe, then a(t) will keep increasing due to Λ>0 and our universe will become an empty, ever-expanding universe, with galaxies moving away from us faster than the speed of light, never to be seen again. This kind of universe is called a De Sitter space.[8]

But Mr. Writer Sir I still do not know what is this Dark Energy?  

Dark energy is a completely different concept. It is the component responsible for the acceleration of the expansion of the universe as it continues to expand. It is not a substance or matter; it is a completely new dimension that began in 1998.

Image Credit: Earthsky[9] 

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American astronomers Adam Riess and Saul Perlmutter, and Australian astronomer Brian Schmidt independently proposed this theory, and this happened about 28 years after Vera Rubin (the gal who proposed dark matter). Their supernova observations suggest the universe's expansion began to accelerate about 10 billion years ago, and cosmologists have estimated that this energy is about two-thirds of the cosmic energy density, is gravitationally repulsive, and is not directly visible in galactic clusters. [10]

The concept of an energy that exerts a negative pressure against gravity causing the universe to expand is based on the fact that dark matter only interacts with gravity, so it is called dark energy.

If you really want to know and get a good grasp on dark matter do read this article: https://idineshmandora.blogspot.com/2025/03/dark-matter-my-headache.html 

Dark energy is the opposite of this, as it works against gravitational collapse on a cosmological scale. It is different from dark matter, which is dark stuff only in name.

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Mathematical Understanding of the Universe's Expansion

Now we will dive deep into the mathematical understanding of the expansion of the universe through math, and if you are not comfortable you can skip because you won’t be missing a thing just some mathematical gap in your understanding.

To understand the expansion of the universe through the lens of dark energy and acceleration equations mathematically, we have already discussed the Friedmann equations. We consider the universe to be homogeneous, which means uniform expansion and isotropic, which means the universe looks the same from everywhere. (e.g.: FLRW metric)

Since  Blog-post Doesn't allow the latex code without any editing in HTML and i do not want to loose the integrity of Blog I will be using images to show this math-work.   

$\mathrm{ds2=-dt2+a(t)2[dr21-kr2+r2d\theta 2+r2sin2\theta d\varphi 2]\; --------(1)ds2=-dt2+a(t)2[dr21-kr2+r2d\theta 2+r2sin2\theta d\varphi 2]}$

$a(t)=\; scale\; factor\; and$

$k=\; curvature\; constant\; (k\; =\; 0,\; +1,\; -1)$

$\mathrm{Einstein’s\; Field\; Equations}$

Similarly, the field equations from Einstein’s theory of relativity are

${\mathrm{G\mu \nu =R\mu \nu -12Rg\mu \nu =8\pi Gc4T\mu \nu \; ----------(2)\; G\mu \nu =R\mu \nu -12Rg\mu \nu =8\pi Gc4T\mu \nu }}^{}$

Now if 

$c=1$

$G_{\mu \nu }=8\pi G{T}_{\mu \mathrm{\nu \; ————-(3)}}$

 Perfect Fluid Energy-Momentum Tensor : Similarly, if our homogeneous universe is considered as a perfect fluid, then the Perfect Fluid Energy-Momentum Tensor:

$T_{\mu \nu }=(\rho +p)u_{\mu }{u}_{\nu }+p{g}_{\mu \mathrm{\nu \; ———-\; (4)}}$

Here 

$\rho \; =\; energy\; density, p\; =\; pressure$

 $u^{\mu }$=(1,0,0,0) (our-velocity in co-moving) coordinates.)

Friedmann Equation– 00 Component

This equation also shows the expansion of energy density with the expansion of the universe.

Differentiating The First Friedmann Equation (5) with respect to time on both sides gives:

${(\dot{}aa)2=8\pi G3\rho -ka2(a\dot{}a)2=8\pi G3\rho -ka2———(8)ddt((\dot{}aa)2)=ddt(8\pi G3\rho -ka2)Left-hand\; Side\; (LHS): ddt(\dot{}a2a2)=2\dot{}aa(¨aa-(\dot{}aa)2)ddt(a\dot{}2a2)=2a\dot{}a(a¨a-(a\dot{}a)2) ———--(8.1)Right-hand\; Side\; (RHS): ddt(8\pi G3\rho )=8\pi G3\dot{}\rho ddt(8\pi G3\rho )=8\pi G3\rho \dot{}————(8.2)}^{}$

Derivative of $\frac{k}{a^2}$:

$\frac{d}{dt}\left(\frac{k}{a^2}\right)=-2\frac{k}{a^3}\dot{a}$————-(8.3)

Substituting into Equation (5)

Substituting the value of $\dot{\rho }$ from equations (8.1), (8.2), (8.3) and continuity equation (7) in equation (5), we get: 

$\frac{8\pi G}{3}\dot{\rho }=-8\pi G\frac{\dot{a}}{a}(\rho +p)$ — From equation (7) 

so 

Congratulations! The above equation (9) is Friedmann's equation for acceleration.

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Expanding Friedmann’s Equation

Now if we break down the total energy density ρ and pressure p into the components of matter (m), radiation (r) and dark energy (Λ), then

ρ=ρm+ρr+ρΛ

p=pm+pr+pΛ=0+13ρr−ρΛ

Putting these values in the acceleration equation (9):

$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}({\rho }_m+2{\rho }_r-2{\rho }_{\Lambda })\; -------\; (10)$

In the late-time universe, dark energy will be very dominant. The density of matter (ρm) and radiation 

(ρr) will keep on decreasing in the expanding universe.

Therefore, as time increases, the value of dark energy density $ρ_\Lambda$ will keep on increasing due to its negative pressure, and the speed of expansion of the universe will increase.

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Final Question: Is There an End to the Universe?

There is a term called Observable Universe in cosmology. We can see where the object is in the universe from where the light travels to reach an observer. Now because the speed of light is finite (i.e., it covers a fixed distance in a fixed time) and it is reaching us from all directions, we consider the universe to be a sphere.

1. The radius of this Observable Universe is 46.5 billion light-years.
2. Spherical Universe is called only because it is easy to understand, otherwise the flatness of the universe is more helpful for data compilation. However, the shape and structure of the universe depend on the curvature (k) in flat or open or closed relativity.
3. The diameter of our spherical universe is 93 billion light-years, which is not the size of the entire universe.This is the maximum distance in the universe from where light can reach us. We call it the Observable Universe, which means we can observe what is at that end.

Image credit: explainingscience[11] 

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Expansion and the Cosmic Event Horizon

Because the universe is not static, it means it is expanding. We call it a Dynamic Universe, and it has some key properties:

• It is continuously expanding.
• It evolves (events shape it).
• Matter, radiation, and dark energy show their effects from time to time.
• That is why time is effective here, because the universe has a history, a present, and a future.

That is why we have called it dynamic above.

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Space is expanding, and as time passes, every galaxy is not only moving away from us but also from each other.

1. The light which is travelling towards us from these distant galaxies has to cover more distance as time increases. 
2. This is not the final boundary from where light is able to reach us today. Rather, this is the dynamic boundary at which the speed of expansion of the universe is equal to the speed of light. That is, v = H₀⋅d = c, which is constantly changing.
3. This is not any kind of physical boundary; in fact, it is the limit of observing our universe at v = c, because after this, v > c is observed.
4. Hubble Horizon is the point after which the universe itself is expanding faster than the speed of light, that is, v > c. 
5. This expansion will keep on increasing in the future as well because it is getting acceleration from dark energy (Λ > 0).
6. After a certain time, the value of d_H will start to stabilize, assuming that our universe enters the de Sitter phase. [12]
7. Very distant galaxies will fall outside our observable range at this new fixed boundary, which we will call the cosmic event horizon (about 16–17 billion light-years).
8. Galaxies beyond the cosmic event horizon will be so far away from us that these galaxy will eventually become permanently unobservable.

Image credit: explainingscience[13] 

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So, How Big Is the Hubble Horizon?

If we use the formula: $d_H = \frac{c}{H_0}$

• The speed of light (c ≈ 3 × 10⁸ m/s)
• The Hubble constant is the rate of expansion of the universe, which is currently 70 × 10⁻³ km/s/kpc

Calculation of Hubble Horizon:

dH=3×108m/s70km/s/Mpc=3×108m/s70×103m/s/Mpc

Converting parsecs to mega parsecs:

$d_H\approx 4.29\text{Gpc}$

Finally is $d_H\approx 13.9\text{billion light-years}$

1. So the Hubble Horizon is a horizon at 13.9 billion light-years,
2. from where light is able to cover this space at its speed to reach us.
3. This is not a physical limit, but a dynamical limit.
4. Before this speed, the galaxy is moving away from us at the speed of light (v = c).
5. After this, the universe is moving away from us faster than the speed of light (v > c).

That is why the universe before this limit is observable for our instruments, and the galaxies after this horizon limit are impossible for us to observe with scientific resources.

I repeat: The galaxies beyond the cosmic event horizon limit will be so far away from us that eventually they will become permanently unobservable.

Image credit: explainingscience[14] 

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Conclusion

This is, in a way, the last observable journey, which you can call the end of the universe if you want. This is poetically accurate, but scientifically, it is only a perceptual limit.
The inability to observe is not proof of the end of the universe.
For me, this is only proof that this great universe—
which is filled with continuous infinity, is
amazingly black, vast, detached, strange, non-specific,
expressed yet unexpressed,
patient yet roaring,
possessing duality and non-duality equally,
digesting the infinite soundless hum,
alive and resonating with its own melody—
is also destructible and ephemeral.

— m. dinesh

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M.Dinesh

M.दिनेश© 

-Dinesh Mandora     

Dinesh Mandora All rights reserved ©

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References and Footnote For People with degree in Physics/Math/Engineering

1. Phalungsongsathit, Sattha. Cosmological Principles, Maximally Symmetric Space, and FLRW Metric. 22 Apr. 2025, Indico Global, https://indico.global/event/8336/contributions/79192/attachments/36978/69112/FLRW%20metric.pdf.
2. https://qr.ae/pYzEBh
3. Roy Choudhury, Tirthankar. Cosmology: Lecture 2 – The FLRW Metric. 2021, National Centre for Radio Astrophysics, Tata Institute of Fundamental Research. link: http://www.ncra.tifr.res.in:8081/~tirth/Teaching/Cosmology-2021/Lectures/lecture-02-web.pdf
4. Melia, Fulvio. The Friedmann–Lemaître–Robertson–Walker Metric and the Principle of Equivalence. 21 Apr. 2023. https://www.degruyterbrill.com/document/doi/10.1515/zna-2022-0307/html?lang=en
5. nasa.gov 'Imagine The Universe' link: https://imagine.gsfc.nasa.gov/features/yba/M31_velocity/hubble_law/hubble_meaning.html
6. Mandora, D. (2025, March 30). Dark Matter: My Headache. Retrieved April 22, 2025, from https://idineshmandora.blogspot.com/2025/03/dark-matter-my-headache.html   
7. Where is the Edge of the Universe? link: https://astronomicca.com/2018/07/11/where-is-the-edge-of-the-universe/
8. Galante, D. A. (2023). Modave lecture notes on de Sitter space & holography (Version 2). arXiv. Link: https://arxiv.org/pdf/2306.10141 
9. Illustration showing snapshots from a simulation by astrophysicist Volker Springel of the Max Planck Institute in Germany. Link: https://earthsky.org/space/definition-what-is-dark-energy/
10. Caldwell, Robert. Dark Energy. n.d. PDF file. https://sites.astro.caltech.edu/~george/ay21/readings/CaldwellDarkenergy.pdf  
11. Illustration from explaining science link: https://explainingscience.org/wp-content/uploads/2021/04/horizon-particle.png
12. Carneiro, S., & Borges, H. A. (2017). Dynamical system analysis of interacting models. arXiv. https://doi.org/10.48550/arXiv.1704.07825
13. Illustration from explaining science link: https://explainingscience.org/wp-content/uploads/2021/04/horizon-hubble.png
14. Illustration from explaining science link: https://explainingscience.org/wp-content/uploads/2021/04/image.png