A Mathematical Connection between Cosmic Expansion and Exponential Growth

ricco
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I'm exploring whether exponential growth observed on cosmic scales (universal expansion) and smaller scales (technology, information) share a fundamental mathematical connection, or if their similarity is purely coincidental. I'm seeking mathematical/statistical methods and necessary data to test this hypothesis.
Hello everyone,
I'm currently exploring the hypothesis that exponential growth might be a universal principle manifesting across different scales—from the cosmic expansion of the universe (e.g., characterized by the Hubble constant and driven by dark energy) to microscopic, technological, informational, or societal growth processes.

My core question:

Is there any mathematical connection (such as correlation or even causation) between the exponential expansion of the universe (cosmological scale, described by the Hubble constant) and exponential growth observed at smaller scales (like technology advancement, information generation, population growth, etc.)?


Specifically, I’m looking for:

✔ Suggestions for mathematical methods or statistical analyses (e.g., correlation analysis, regression, simulations) to test or disprove this hypothesis.
✔ Recommendations on what type of data would be required (e.g., historical measurements of the Hubble constant, technological growth rates, informational growth metrics).
✔ Ideas about which statistical tools or models might be best suited to approach this analysis (e.g., cross-correlation, regression modeling, simulations).


My aim:
I would like to determine if exponential growth at different scales (cosmic vs. societal/technological) merely appears similar by coincidence, or if there is indeed an underlying fundamental principle connecting these phenomena mathematically.


I greatly appreciate any insights, opinions, or suggestions on how to mathematically explore or further investigate this question.


Thank you very much for your help!
Best regards,
Ricco
 
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Finding exponential behavior in nature shouldn't be very difficult. It follows the rule ##y'\sim y.## The Hubble parameter goes ##y'=H(t)y## which becomes the same equation if ##H(t)## is constant, but who knows?
 
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ricco said:
My core question:
Is there any mathematical connection (such as correlation or even causation) between the exponential expansion of the universe (cosmological scale, described by the Hubble constant) and exponential growth observed at smaller scales (like technology advancement, information generation, population growth, etc.)?


My aim:

I would like to determine if exponential growth at different scales (cosmic vs. societal/technological) merely appears similar by coincidence, or if there is indeed an underlying fundamental principle connecting these phenomena mathematically.
This question is disturbing to me. The mistake of thinking that statistical correlation implies a causal relationship is very common. We are always trying to correct those mistakes. You seem to be trying to make those mistakes, rather than avoiding them.
Just because two trends follow the same mathematical form (versus time) does not mean there is a causal relationship between them. In fact, you mention several examples that are, IMO, clearly unrelated to the expansion of the universe. But you are asking us to help you make the mistake of assuming otherwise.
Any causal relationship between the expansion of the universe and some other trend can not be proven statistically. It must come from subject matter expertise and logic.
 
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FactChecker said:
This question is disturbing to me. The mistake of thinking that statistical correlation implies a causal relationship is very common. We are always trying to correct those mistakes. You seem to be trying to make those mistakes, rather than avoiding them.
Just because two trends follow the same mathematical form (versus time) does not mean there is a causal relationship between them. In fact, you mention several examples that are, IMO, clearly unrelated to the expansion of the universe. But you are asking us to help you make the mistake of assuming otherwise.
Any causal relationship between the expansion of the universe and some other trend can not be proven statistically. It must come from subject matter expertise and logic.
I get where you're coming from, and I appreciate the caution against mistaking correlation for causation—it’s a trap that many fall into. But I think you might be slightly misinterpreting my intention. I’m not simply pointing at two trends that share a mathematical form and assuming a direct causal link. Instead, I’m asking whether there could be a deeper fundamental principle at play—one that underlies both the expansion of the universe and the exponential growth we see in information, technology, and intelligence.

We already know that exponential growth is not random; it consistently emerges in self-organizing systems, whether in biology, economics, or information theory. Systems that rely on positive feedback loops tend to exhibit exponential behavior. The question is whether the universe’s expansion might serve as a kind of boundary condition that influences these smaller-scale phenomena. If entropy is constantly increasing due to cosmic expansion, and entropy is also a key driver in the evolution of complexity, could it be that the two are connected on a deeper level?

I get that mathematical similarity alone doesn’t imply a direct relationship, but across many fields, we find that similar patterns—such as fractal structures or power-law distributions—often hint at shared underlying principles. Dismissing potential connections outright seems just as unscientific as assuming a causal link without evidence. If nothing else, wouldn’t it be valuable to explore whether this is mere coincidence or something more fundamental?

At the very least, I think this is a question worth asking, rather than shutting it down based on the assumption that the trends must be unrelated. What do you think?
 
ricco said:
Instead, I’m asking whether there could be a deeper fundamental principle at play
That sounds like New Age woo woo to me.
ricco said:
What do you think?
I think your concept sounds like New Age woo woo.

Yes, I know that sounds harsh, but hey ... you asked.
 
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fresh_42 said:
Finding exponential behavior in nature shouldn't be very difficult. It follows the rule ##y'\sim y.## The Hubble parameter goes ##y'=H(t)y## which becomes the same equation if ##H(t)## is constant, but who knows?
Finding exponential behavior in nature is not surprising, and I completely agree that many systems follow the form ## y' \sim y ##. However, the real question is not whether exponential growth appears in different domains, but whether there is a deep, underlying principle connecting these processes beyond mere coincidence.


The Hubble expansion equation follows ## y' = H(t) y ##, which, if ## H(t) ## is constant, has the same form as many other naturally occurring exponential growth laws. But what if this is more than a coincidence? If the universe provides the fundamental conditions for all physical and informational processes, wouldn't it be logical that its large-scale expansion influences growth patterns across different scales?


For example, if we generalize exponential growth as:

$$ y(t) = y_0 e^{\int H(t) dt} $$

then the integral of ## H(t) ## acts as a "scaling factor" for any process governed by similar growth dynamics. If we find that systems as diverse as biological evolution, technological progress, and cosmic expansion all align in their growth rates or constraints, wouldn't that suggest a deeper, structural connection?


So the question isn't just whether exponential functions appear often, but whether they are all tracing back to a common cosmological driver rather than being independent phenomena.
 
phinds said:
That sounds like New Age woo woo to me.

I think your concept sounds like New Age woo woo.

Yes, I know that sounds harsh, but hey ... you asked.
I appreciate your honesty, but dismissing it as 'woo woo' without addressing the actual question seems unfair. I'm asking whether the similarity in exponential growth across domains is purely coincidental or rooted in a deeper mathematical principle. If you disagree, could you explain why?
 
The underlying principle is the eigenvalue equation ##D(y)=y.## It is a result of how we describe nature. What do bacteria in a Petri dish have in common with Lie algebras, which we use to describe quantum field theory?
 
  • #10
ricco said:
Finding exponential behavior in nature is not surprising, and I completely agree that many systems follow the form ## y' \sim y ##. However, the real question is not whether exponential growth appears in different domains, but whether there is a deep, underlying principle connecting these processes beyond mere coincidence.
There is a "deep, underlying principle" in exponential growth that relates the derivative to the antiderivative. But that does not "connect" two different processes that both have that property.
To establish a connection, you should first specify the processes. Then use logic and subject-matter expertise to propose a connection. Do you have an example of that? Otherwise, it is just a random fishing expedition.
 
  • #11
FactChecker said:
There is a "deep, underlying principle" in exponential growth that relates the derivative to the antiderivative.
"God made the integers, man made the rest." ~ Leopold Kronecker

He may also have created the exponential function. However, I think a biological species cannot survive with only polynomial growth. And here we are again, at the anthropological principle: "It is as it is since otherwise we wouldn't be there to ask this question."
 
  • #12
FactChecker said:
There is a "deep, underlying principle" in exponential growth that relates the derivative to the antiderivative. But that does not "connect" two different processes that both have that property.
To establish a connection, you should first specify the processes. Then use logic and subject-matter expertise to propose a connection. Do you have an example of that? Otherwise, it is just a random fishing expedition.
I see your point, and I appreciate the emphasis on correctly defining processes before making connections. I’m not suggesting that just because two systems follow an exponential law, they must be causally linked. But what I’m questioning is whether there could be a deeper structural reason why so many different domains—cosmology, information growth, biological evolution, and technological progress—all exhibit similar exponential patterns.


If we assume that the universe provides the fundamental framework for all physical and informational processes, then its large-scale dynamics (such as expansion) might act as a boundary condition that indirectly shapes how complex systems evolve. For example, if entropy is constantly increasing due to cosmic expansion, and entropy also plays a crucial role in the emergence of complexity in self-organizing systems, could there be a deeper connection?


I’m not claiming that we already have the answer, but rather that this is a question worth exploring rather than dismissing outright. If we find a way to mathematically demonstrate that ## H(t) ## could act as a universal scaling factor, affecting systems across scales, then that would be a significant insight. Do you see any potential approach that could help clarify whether this is a meaningful connection or just a coincidence?
 
  • #13
fresh_42 said:
The underlying principle is the eigenvalue equation ##D(y)=y.## It is a result of how we describe nature. What do bacteria in a Petri dish have in common with Lie algebras, which we use to describe quantum field theory?
I see your point about how the eigenvalue equation ## D(y) = y ## plays a fundamental role in describing nature, but I think there's a slight distinction in what I'm exploring. My question isn't just about shared mathematical structures across different fields, but whether these structures emerge due to a common underlying driver. The fact that exponential growth appears in diverse systems—whether in biology, economics, or cosmic expansion—suggests a deeper organizational principle at work.


You mention Lie algebras and bacteria in a Petri dish—both obey mathematical principles, but their connection is abstract. My question is whether something like ## H(t) ##, which governs cosmic expansion, might act as a universal scaling factor influencing growth dynamics across different domains. If that's true, then it's not just a case of similar equations appearing independently, but rather a sign of a deeper, unifying principle.


Would you agree that identifying such a link, if it exists, would be valuable in understanding the fundamental structure of growth in the universe?
 
  • #14
ricco said:
My question isn't just about shared mathematical structures across different fields, but whether these structures emerge due to a common underlying driver.
See my post #11.
ricco said:
My question is whether something like ## H(t) ##, which governs cosmic expansion, might act as a universal scaling factor influencing growth dynamics across different domains.
No. This is simply far fetched and reflects what @FactChecker has already addressed: you confuse cause and correlation. It boils down to the question of whether exponential behavior is necessary or not. My answer is because otherwise we wouldn't have evolved. Religious people might see God as reason. I think the universe is as it is because we were able to emerge. Whether this has a cause other than by chance is a religious question and cannot be answered by science. Not that famous people wouldn't have tried, cp. https://en.wikipedia.org/wiki/Existence_of_God.
 
  • #15
fresh_42 said:
See my post #11.

No. This is simply far fetched and reflects what @FactChecker has already addressed: you confuse cause and correlation. It boils down to the question of whether exponential behavior is necessary or not. My answer is because otherwise we wouldn't have evolved. Religious people might see God as reason. I think the universe is as it is because we were able to emerge. Whether this has a cause other than by chance is a religious question and cannot be answered by science. Not that famous people wouldn't have tried, cp. https://en.wikipedia.org/wiki/Existence_of_God.
I see your point about avoiding a confusion between correlation and causation. My goal isn’t to claim a direct cause, but rather to explore whether there might be a deeper structural relationship between different types of exponential growth—one that isn’t just coincidental.

You mentioned that exponential behavior is necessary for evolution, but that doesn’t explain why so many different systems—from biology to technology to cosmic expansion—follow the same mathematical form. If it’s just random, shouldn’t we expect more variation in how different domains exhibit growth?

I’m not arguing for a metaphysical cause, but rather asking: Is there a measurable, mathematical reason for this structural similarity? If not, how would we statistically differentiate between ‘just coincidence’ and an actual underlying scaling factor? If the answer is purely probabilistic, how would we quantify it?
 
  • #16
ricco said:
You mentioned that exponential behavior is necessary for evolution, but that doesn’t explain why so many different systems—from biology to technology to cosmic expansion—follow the same mathematical form. If it’s just random, shouldn’t we expect more variation in how different domains exhibit growth?
There are two reasons in my opinion: biology and our desire to recognize patterns, which in a way is biology, too.

Exponential behavior is a biological necessity. The alternative is extinction. This is my opinion, and would require a closer examination. However, many biological systems are constrained by the existence of predators. A subexponential growth would be difficult to ensure existence in the presence of predators. It is how biology is setup.

The second reason is simple mathematics. We use terms that automatically lead to eigenvalue considerations. We could alternatively use power series. That wouldn't make the exponential solutions vanish, but it wouldn't be seen so easily. Eigenvalue equations in all their variants are - again in my opinion - due to our motivation to see patterns, things that do not change qualitatively.

We do not need to write ##y'=y.## We could as well write ##y=\displaystyle{\sum \dfrac{t^n}{n!}.}## Our way to summarize it leads to ##t\mapsto e^t## and ##Dy=y.## It always ends up with us and how we have chosen to describe nature. It worked. And it works better than operating with power series alone.

There is no ultimate reason, only biology. The behavior of the universe is currently completely unknown and there are dozens of proposed models. There is no reason why the universe should behave like biology does.
ricco said:
I’m not arguing for a metaphysical cause, but rather asking: Is there a measurable, mathematical reason for this structural similarity?
This IS metaphysics!
 
  • #17
ricco said:
what I’m questioning is whether there could be a deeper structural reason why so many different domains—cosmology, information growth, biological evolution, and technological progress—all exhibit similar exponential patterns.
The change of ##x,\ \Delta x,\ ## is proportional to ##x## itself. I can think of hundreds of examples that are very natural and obvious. This is a relationship between a variable and itself that doesn't require any other variable or explanation.
CORRECTION: There are complicated interactions with many variables that lead to the exponential growth of a variable. So it is possible to find more complicated systems with exponential behavior of some variables.
 
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  • #18
Everything grows exponentially, unless it doesn't.

The list of things that don't grow exponentially is endless.

This thread illustrates why we generally don't allow discussion of personal theories.
 
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  • #19
fresh_42 said:
There are two reasons in my opinion: biology and our desire to recognize patterns, which in a way is biology, too.

Exponential behavior is a biological necessity. The alternative is extinction. This is my opinion, and would require a closer examination. However, many biological systems are constrained by the existence of predators. A subexponential growth would be difficult to ensure existence in the presence of predators. It is how biology is setup.

The second reason is simple mathematics. We use terms that automatically lead to eigenvalue considerations. We could alternatively use power series. That wouldn't make the exponential solutions vanish, but it wouldn't be seen so easily. Eigenvalue equations in all their variants are - again in my opinion - due to our motivation to see patterns, things that do not change qualitatively.

We do not need to write ##y'=y.## We could as well write ##y=\displaystyle{\sum \dfrac{t^n}{n!}.}## Our way to summarize it leads to ##t\mapsto e^t## and ##Dy=y.## It always ends up with us and how we have chosen to describe nature. It worked. And it works better than operating with power series alone.

There is no ultimate reason, only biology. The behavior of the universe is currently completely unknown and there are dozens of proposed models. There is no reason why the universe should behave like biology does.

This IS metaphysics!
I see your point, but I think you might be misinterpreting the nature of my question. I’m not claiming that the universe "should" behave like biological systems, nor am I suggesting a metaphysical explanation. Rather, I’m questioning whether the similarity in how exponential growth manifests across different domains could have a deeper mathematical origin—beyond just coincidence or the way we choose to describe nature.


You argue that biology requires exponential growth for survival, while in physics, we use mathematical tools that naturally lead to eigenvalue solutions and exponential functions. But isn't that precisely what makes this question interesting? If the same mathematical structures emerge across vastly different scales—cosmology, biology, economics, technology—why is that?


Yes, we could represent growth in other ways, but the fact remains: exponential laws dominate in systems driven by feedback loops and self-replication. The universe itself is expanding at a rate proportional to its size. That’s not just an arbitrary choice of description; it’s a fundamental feature of its dynamics. Could it be that a constraint—perhaps something like ## H(t) ##)—acts as a universal scaling factor for growth in different domains?


I’m not saying this must be the case. But dismissing the question outright as "metaphysics" seems premature when we have not yet explored whether a measurable, physical principle might be at play. Wouldn’t it be more scientific to test whether such a connection exists rather than assume there is none?
 
  • #20
FactChecker said:
The change of ##x,\ \Delta x,\ ## is proportional to ##x## itself. I can think of hundreds of examples that are very natural and obvious. This is a relationship between a variable and itself that doesn't require any other variable or explanation.
CORRECTION: There are complicated interactions with many variables that lead to the exponential growth of a variable. So it is possible to find more complicated systems with exponential behavior of some variables.
I see your point, and I fully agree that exponential growth naturally arises when a variable’s rate of change is proportional to itself. However, that alone doesn’t fully address the question I’m exploring.


I'm not just asking why exponential growth appears—that part is well understood mathematically. My question is whether there could be a shared underlying structural constraint that governs these processes across different domains. In other words, is there something like a universal scaling factor (e.g., ## H(t) ## in cosmology) that influences growth dynamics on a broader scale?


If ## H(t) ## acts as a fundamental driver of cosmic expansion, and we find structurally similar equations governing growth in biological systems, information theory, and technological progress, wouldn't it be worth investigating whether these similarities emerge due to a common governing principle rather than mere coincidence?


Otherwise, we're left assuming that these patterns simply arise independently in vastly different contexts—which might be true, but also might not be the full picture.


What do you think?
 
  • #21
PeroK said:
Everything grows exponentially, unless it doesn't.

The list of things that don't grow exponentially is endless.

This thread illustrates why we generally don't allow discussion of personal theories.
I understand that not everything grows exponentially, and I am not claiming that it does. My question is whether systems that do exhibit exponential growth might share a deeper structural cause, rather than this being purely coincidental. That’s why I’m asking for existing mathematical/statistical methods to analyze this. If there’s already research refuting such a connection, I’d be happy to read it.
 
  • #22
ricco said:
I understand that not everything grows exponentially, and I am not claiming that it does. My question is whether systems that do exhibit exponential growth might share a deeper structural cause, rather than this being purely coincidental. That’s why I’m asking for existing mathematical/statistical methods to analyze this. If there’s already research refuting such a connection, I’d be happy to read it.
What have statistics got to do with it?

Many common mathematical patterns are seen in disparate systems. This is why mathematics is so important and sometimes called the language of physics. And, to some extent, science generally.

Richard Feynman once said: " the same equations have the same solutions".

There is nothing to refute.

The issue is that by "deeper structural cause" you seem to implying some metaphysics underpinning all of nature. Metaphysics by definition cannot be refuted because it deals with a hypothetical essence that cannot be tested or analysed. You either take it or leave it.

That's why it was, rather harshly perhaps, described as "cosmic woo".
 
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  • #23
PeroK said:
That's why it was, rather harshly perhaps, described as "cosmic woo".
No, I would never call something "cosmic woo". That sounds much too scientific :smile: I called it, and still call it, "New Age woo woo", which to me equates with nonsense.

I see it as sort of the intellectual equivalent of numerology. Seeing connections where none exist.
 
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  • #24
If you try hard enough, you can find something that suits your assumption. This seems more like a sales pitch than an attempt to understand phenomenon from all angles.

Population growth could help explain most of the phenomenon you are talking about; more brains, more discovery. As for the universe, a lot of what is known is from the light that we can see it with. Light has so many properties that are still unknown that trying to relate something so manifest like a population to something like expansion wouldn't hold much weight.

I'm curious, do any AI text generators use "I see your point" for every beginning sentence? Just skeptical. But I do enjoy trying to figure out how people arrive at where they are.
 
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  • #25
mayflowers said:
I'm curious, do any AI text generators use "I see your point" for every beginning sentence?
I thought that was odd as well.
mayflowers said:
Just skeptical.
Same here.
 
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