# A relationship between flat vs. finite and (2) Hubble constant tension

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• Buzz Bloom
In summary: It makes sense that the universe would be very close to flat, and I get that it's a lot easier to do the math if it is exactly flat. But the observations are that it is not exactly flat.So my question is: "What is the practical difference between exactly flat and not exactly flat?"In summary, the conversation discusses the possibility of the universe being infinite or finite and the current understanding that it is close to being exactly flat due to the process of inflation. However, observations show that it is not exactly flat. Some argue that the universe is still flat despite this discrepancy, while others propose alternative solutions such as a slightly curved or open universe. The practical difference between a flat and
Buzz Bloom
Gold Member
TL;DR Summary
Many of the threads I see include posts which seem to be based on certainty that the universe is flat. I found an interesting article that one of several possible resolutions to the Hubble constant tension is that the universe may not be flat.
One recent example of a thread discussing flat or not is:

I found an interesting 2021 article regarding the Hubble constant tension.
It discusses three possible solutions. The second solution is:
A second adjustment would be for the curvature of the universe to depart very slightly from a flat geometry.​
The article gives the following reference for this.
Benjamin Bose and Lucas Lombriser, “Easing Cosmic Tensions with an Open and Hotter Universe,” Physical Review D 103, no. 8 (April 27, 2021): id. L081304, doi:10.1103/PhysRevD.103.L081304.​

I am unable to understand the thought process of many cosmologists who seem to reject this possible solution because they are completely convinced that the universe is flat. I hope that this thread will produce some interesting explanations regarding this issue.

NOTE: I accidentally included an attachment, and I do not know how to get rid of it.

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I believe the mit lecture slide attached below may solve your doubts, universe is said to be flat because of inflation, inflation drives the ratio of critical mass density and actual mass density to be 1

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frost_zero said:
I believe the mit lecture slide attached below may solve your doubts, universe is said to be flat because of inflation, inflation drives the ratio of critical mass density and actual mass density to be 1
No. We have no empirical evidence that it actually is exactly one. Even the article you posted does not say that it is 1, it says 1 plus/minus .0065.

Buzz Bloom
It is close to one with the accuracy of .0065, and only 13.82 billion years are predicted to have passed since big bang, as more time passes this value will get closer and closer to 1. James webb telescope might give a more accurate value of this.

frost_zero said:
It is close to one with the accuracy of .0065, and only 13.82 billion years are predicted to have passed since big bang, as more time passes this value will get closer and closer to 1. James webb telescope might give a more accurate value of this.
Yes. And none of that makes it exactly 1. You stated in post #2 that is is 1 and you have offered nothing to back that up.

Buzz Bloom
Buzz Bloom said:
A second adjustment would be for the curvature of the universe to depart very slightly from a flat geometry.
What are the practical consequences, if any, from the difference between flat and approximately flat?

Hi @anorlunda:

As I understand your question, there is no practical difference, except for the possibility that the universe may be finite. Sometime in the future its finiteness may produce an astronomical observation showing clearly that the size of the finite universe has become smaller than observable universe. Such an occurrence will permit an observation of identical detailed stuff at a far distance in opposite directions of the sky. Another observation might be that the origin of CMB will not be detectable because its distance from the Earth will have become greater than the radius of the observable universe.

However, whether practical or not, there seem to be quite a few PF posts in which the poster is saying with certainty that the universe is absolutely flat-infinite.

Regards,
Buzz

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Hi @frost_zero:

First, the distribution statement from the about Ω is
Ω = 1.0010 +/- 0.0065.​
1.0010 is the mean m of the distribution of Ω, and 0.0065 is the standard deviation σ. Ω is the density parameter, but the value 1 corresponds to a flat universe. The 1.0010 implies a finite hyper-sphere.

Second, your lecture slides are from 2013. An improved distribution statement from 2018 is the Planck result equation #47n on page 40 of
k = 0.0007 ± 0.0019​
k is the Friedmann equation term for the curvature parameter. A flat universe would have
k = 0.​
A positive value corresponds to a finite universe.

In either case, the shape of the distribution is not Gaussian, but I think within the given standard deviation, it is reasonably close to Gaussian for a rough calculation of the probability of flat and infinite, spherical and finite, and hyperbolic and infinite. However, I am not sure of how to make this calculation. If you would like me to try to make such calculations, I will do so. However, I would prefer for someone who has a better understanding regarding such probability calculations to point me in the right direction.

Regards,
Buzz

Buzz Bloom said:
I am unable to understand the thought process of many cosmologists who seem to reject this possible solution because they are completely convinced that the universe is flat. I hope that this thread will produce some interesting explanations regarding this issue.
I didn't see the thread you linked - not sure what it's doing in General Discussion.

I think it's the prevailing view that the universe is flat, and my understanding is that inflation pushes it to be incredibly close to flat, and without inflation it's difficult to explain the large-scale homogeneity of the universe. So there's observation that the universe is indistinguishable from flat and homogeneous, and a strong theoretical notion that these two facts are linked. So I don't think anybody particularly objects to the notion of the universe not being flat, but as far as I'm aware there's some theoretical work to be done to explain homogeneity in a universe with any significant curvature. Note that the PDF reader on this phone is rubbish so I haven't looked at the papers you linked yet, so I'm not sure what size of curvature they are proposing.

One other thing to note about PF is that we do try to answer B threads at B level, and you may find a lack of qualifiers in those, such as saying that 'the universe is flat' rather than 'our current best models treat the universe as flat'. I try not to do it myself, but probably do from time to time.

Ibix said:
I think it's the prevailing view that the universe is flat, and my understanding is that inflation pushes it to be incredibly close to flat, and without inflation it's difficult to explain the large-scale homogeneity of the universe.
Hi Ibix:

Thank you for your post. I get that inflation is a plausible conjecture to explain how it came to be that all sufficiently large regions of the observable universe have very similar characteristics. Why is it not possible for a large but finite nearly flat universe to have been inflated in a manner similar to that conjectured regarding an infinite universe? Just how big would the finite universe have to be for inflation to have produced what is observed regarding uniformity of large regions?

Regards,
Buzz

What do you mean by a finite universe here - a closed FLRW spacetime with a very large radius of curvature? I don't think it's ruled out, but I think inflation makes it very very large - and then the question is if the deviation from flat can be enough to fix the Hubble tension issue. I'm not taking a position one way or the other (as I say I haven't read your citations yet), but it's the obvious challenge to a non-flat universe.

Ibix said:
What do you mean by a finite universe here - a closed FLRW spacetime with a very large radius of curvature?
Hi @Ibix:

I have unsuccessfully tried to find a reference discussing the difference in properties between a current finite universe of
(2) specified size of large regions of the observable universe having very similar characteristics,​
and corresponding properties at an early time soon after the big-bang. That is, I would like to have an understanding of the actual numbers involved that make the conjecture of inflation necessary. Do you know of any such reference?

Here is what I have found which does not include the numbers I am interested in understanding.
The inflationary epoch lasted from 10−36 seconds after the conjectured Big Bang singularity to some time between 10−33 and 10−32 seconds after the singularity.​

My questions are:
How would the "large regions" now compare with the same regions at 10−32 seconds after the singularity? This would be the condition after inflation.​
and
How does the above numerically compare with the same comparison at 10−36 seconds after the singularity? This would be before inflation.​

Regards,
Buzz

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Buzz Bloom said:
Why is it not possible for a large but finite nearly flat universe to have been inflated in a manner similar to that conjectured regarding an infinite universe?
It is possible. Inflation as a model does not rule out a closed universe with a finite spatial volume, as long as that volume is much, much larger than the volume of our observable universe. Cosmologists working on inflation typically discuss the flat universe case because it is the easiest to deal with conceptually, but the actual math of inflation does not say anything about the universe being exactly spatially flat (and infinite).

Buzz Bloom and timmdeeg
I'm not sure why inflation contributes to the believe that the universe is flat, which is my impression at least.
Flatness today requires flatness before inflation and thus inflation doesn't change it. If we assume that the spatial geometry of the universe is arbitrary and not due to a certain physically fundamental principle then there shouldn't be a preference for euclidean flatness.

Buzz Bloom
timmdeeg said:
Flatness today requires flatness before inflation and thus inflation doesn't change it.
More precisely, exact flatness requires exact flatness before inflation and inflation doesn't change it. But the "flatness problem" in cosmology that inflation purports to solve is not a problem about exact flatness. See below.

timmdeeg said:
If we assume that the spatial geometry of the universe is arbitrary and not due to a certain physically fundamental principle then there shouldn't be a preference for euclidean flatness.
Yes, but our actual observations indicate that our universe is very, very close to Euclidean flatness spatially, so close that we can't observe any deviation from flatness. And by the very argument you give in this quote, those observations require some explanation. Inflation provides one possible explanation: that before inflation, the universe's spatial geometry could have been anything, but whatever it started out as, inflation would drive it towards flatness, and if there were enough e-foldings of inflation, then by the end of inflation the spatial geometry would be close enough to flatness to explain our current observations. (Note that from the end of inflation until a few billion years ago, the universe was either radiation dominated or matter dominated, and those conditions cause the spatial geometry to evolve away from flatness if the universe is not exactly flat. So the conditions we observe now are not identical, as far as spatial geometry goes, to the conditions at the end of inflation: we have had about 10 billion years of evolving away from flatness, and then a few billion years of dark energy dominated evolution, which does drive the spatial geometry back towards flatness, but extremely slowly, so it doesn't really make much difference overall.)

timmdeeg
PeterDonis said:
Note that from the end of inflation until a few billion years ago, the universe was either radiation dominated or matter dominated, and those conditions cause the spatial geometry to evolve away from flatness if the universe is not exactly flat.
Hi @PeterDonis:

I do not understand how the Friedmann equations would be consistant with this quote. Are you saying that the size of a finite spherical universe (a being the scale factor which is proportional to the radius) got smaller for this period of ten billion years. What are the values of the H0 and the four Ω parameters during the radiation dominated period? I would assume that
Ωr + Ωk = 1, and​
Ωm = ΩΛ =0.​
I also assume that at time
t0 = 10−32 seconds, and​
a = 1.​

Regards,
Buzz

Buzz Bloom said:
Are you saying that the size of a finite spherical universe (a being the scale factor which is proportional to the radius) got smaller for this period of ten billion years.
Not at all. "Increasing deviation from flatness" is not at all the same as "decreasing size of a finite spherical universe".

"Deviation from flatness" as the term is used in this particular context means the absolute value of the curvature density parameter ##\Omega_k##. For an expanding radiation dominated or matter dominated universe, this absolute value will increase with time; for an expanding dark energy dominated or inflationary universe, this absolute value will decrease with time. To see why, just note that the sum of all the density parameters (including the curvature one) must always be ##1##, at any time, and look at how the relevant density parameters vary with scale factor. (Heuristically, the "curvature density" dilutes more slowly with scale factor than matter or radiation--##1 / a^2## vs. ##1 / a^3## or ##1 / a^4##--but dark energy doesn't dilute at all.)

PeterDonis said:
Yes, but our actual observations indicate that our universe is very, very close to Euclidean flatness spatially, so close that we can't observe any deviation from flatness. And by the very argument you give in this quote, those observations require some explanation.
Yes, inflation solves the homogeneity- and "almost flat" problem. It just doesn't refer to Euclidean flatness.
PeterDonis said:
Inflation provides one possible explanation: that before inflation, the universe's spatial geometry could have been anything, but whatever it started out as, inflation would drive it towards flatness, and if there were enough e-foldings of inflation, then by the end of inflation the spatial geometry would be close enough to flatness to explain our current observations. (Note that from the end of inflation until a few billion years ago, the universe was either radiation dominated or matter dominated, and those conditions cause the spatial geometry to evolve away from flatness if the universe is not exactly flat. So the conditions we observe now are not identical, as far as spatial geometry goes, to the conditions at the end of inflation: we have had about 10 billion years of evolving away from flatness, and then a few billion years of dark energy dominated evolution, which does drive the spatial geometry back towards flatness, but extremely slowly, so it doesn't really make much difference overall.)
According to our present model. I am very curious if it will hold in view of the 'Hubble tension'.

PeterDonis said:
"Deviation from flatness" as the term is used in this particular context means the absolute value of the curvature density parameter . For an expanding radiation dominated or matter dominated universe, this absolute value will increase with time;
Hi @PeterDonis:

Thank you for the clarification. I still have a bit of confusion regarding
PeterDonis said:
the sum of all the density parameters (including the curvature one) must always be , at any time,
I understand that the actual values of the Ωr, Ωm, and Ωk variables vary with the value of a negative power of the variable a. However, to get the sum of the four Ωs to be unity, a revised value of a is needed. That is, as I understand it, the sum of the four Ωs equaling unity for time t requires that a=1 for this value of t. (I need to do a bit of math to explain in detail how this works. At my advanced age, my math skills have slowed down.)

Regards,
Buzz

Buzz Bloom said:
I understand that the actual values of the Ωr, Ωm, and Ωk variables vary with the value of a negative power of the variable a.
This is the case if you insist on writing the first Friedmann equation in terms of values "now", including the Hubble constant "now", as well as values at whatever other time you are interested in. Then ##a## appears explicitly in the equation, with different powers in the different terms. In this version, yes, the value on the LHS is not ##1##, it's ##H^2 / H_0^2##, the (squared) ratio of the Hubble constant at the other time you are interested into the Hubble constant "now".

It is not the case, however, if you just take the first Friedmann equation and divide by ##H^2##, which puts it in terms of all of the ##\Omega##s taken at one time only. Then you just get that the sum of all the ##\Omega##s at any given time is ##1##. The different ways in which the ##\Omega##s vary with ##a## changes their relative ratios as a function of time, but ##a## does not appear explicitly anywhere in the equation in this form; it is only implicit in the definitions of each of the ##\Omega##s.

Buzz Bloom said:
to get the sum of the four Ωs to be unity, a revised value of a is needed.
Yes, and if you just take the first Friedmann equation and divide it by ##H^2##, so that the LHS is ##1##, then the "revised value of ##a##" is taken into account automatically, since you just have a relationship between all of the ##\Omega##s at one particular time, which will implicitly take into account the value of ##a## at that time.

Buzz Bloom said:
as I understand it, the sum of the four Ωs equaling unity for time t requires that a=1 for this value of t.
Dividing the first Friedmann equation by ##H^2##, as I described above, ends up doing this implicitly, if you want to think about it that way.

Buzz Bloom
PeterDonis said:
Dividing the first Friedmann equation by , as I described above, ends up doing this implicitly, if you want to think about it that way.
Hi @PeterDonis:

Thank you very much. You post greatly simplifies what I was trying to do.

Regards,
Buzz

I looked through the preprint for the journal article referenced:
https://arxiv.org/abs/2006.16149

It's an interesting idea, but I don't understand how they can justify a change in the CMB temperature estimate. The curvature difference they suggest to explain the results is 0.02, which is quite a bit larger than current estimates of curvature but still small enough I wouldn't consider it absurd on its face.

But they're also suggesting a temperature ##T_0=2.95##. This contrasts with the FIRAS measurement of ##T_0 = 2.72548 \pm 0.00057##, which makes for a whopping 400 standard deviation difference. I really don't understand how they're justifying this large discrepancy between the observed ##T_0## and the ##T_0## inferred from the other data. This paper seems like they're taking one source of tension and replacing it with a much, much greater tension.

Edit: Reading a little more carefully, it sounds like they're saying that the temperature difference can be explained by us residing in an underdense region, one that isn't particularly extreme (it's explained by the natural density variations from place to place in the universe). Interesting idea. If this turns out to be correct, then the explanation for the Hubble tension may just be that we've made incorrect assumptions about being able to ignore whether we live in a somewhat less dense or more dense region of the universe.

## 1. What is the flat vs. finite debate in relation to the Hubble constant?

The flat vs. finite debate refers to the shape of the universe and whether it is infinite or has a finite size. This debate is relevant to the Hubble constant because it affects how we measure the expansion rate of the universe.

## 2. How does the shape of the universe affect the Hubble constant?

The shape of the universe affects the Hubble constant because it determines the geometry of space and how light travels through it. This, in turn, affects our measurements of the distance to objects in the universe and their apparent velocities, which are used to calculate the Hubble constant.

## 3. What is the current tension between the flat vs. finite debate and the Hubble constant?

The current tension between the flat vs. finite debate and the Hubble constant is due to discrepancies in the measurements of the Hubble constant using different methods. Measurements from the cosmic microwave background suggest a lower value for the Hubble constant, while measurements from Type Ia supernovae suggest a higher value. This tension may be resolved by considering the shape of the universe.

## 4. How can the shape of the universe be determined?

The shape of the universe can be determined through observations of the cosmic microwave background radiation, which is the leftover radiation from the Big Bang. By studying the patterns and fluctuations in this radiation, scientists can infer the curvature of space and thus determine the shape of the universe.

## 5. What are the implications of resolving the tension between the flat vs. finite debate and the Hubble constant?

Resolving the tension between the flat vs. finite debate and the Hubble constant could have significant implications for our understanding of the universe and its evolution. It could also help us refine our models and theories of the universe, and potentially lead to new discoveries and insights about the nature of our cosmos.

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