# B Question pertaining to dark energy

1. Jul 26, 2016

### Physics1218

I had questions concerning the Hubble Sphere and while researching why the Hubble sphere doesn't shrink as the universe accelerates someone informed me that the universe isn't truly accelerating in its expansion as we might conventionally think. They informed me that the universe was "accelerating" because as space expands there is more space in between two objects. However the actual expansion has been slowing down since the big bang. Perhaps he articulates it in a more coherent manner.

"If we assume that the Big Bang happened everywhere and at once, at the start of the Universe, then we must understand that the entire Universe isn't expanding or moving away from a single point. Rather, every point is moving away from every other point, space is expanding at every point uniformly. So imagine that every point in space is doubling (or increasing at whatever rate - the rate is not important in this explanation). Every point is becoming more points as time progresses. Now realize the following. If we start with 20 points and after 1 second we get 40, then in the next second, 40 points are now doubling to give 80. In the first step the expansion consisted of 20 points but in the second step it is 40. This is the ABSOLUTE expansion of the Universe. You can see that the next step is then an expansion by 80 points and so on.

The part that is decreasing with time is the RELATIVE rate of expansion or what I simply called "doubling" in the previous paragraph. Of course the rate is not (x2), or doubling, but I'm just using it for simplicity. So this rate is in fact decreasing. The point for point expansion of space OR how much more space can one point create in the next second. This is what is decreasing. However you must realize that even though this rate is decreasing, since we have more space to multiply in the next step, the overall expansion or the ABSOLUTE expansion is still increasing. To demonstrate, let's use our same numbers from before but with a decreasing relative expansion. So in the first step we have 20 points that expand by (x2) to give 40. Our increase was 20. In the next step, these 40 points expand by, say (x1.9), which has decreased from 2 by 0.1. In this step we get an increase from 40 points to 40 x 1.9 = 76. So we have expanded by 36 points. You can see that the overall expansion or ABSOLUTE expansion has increased from step 1 (36>20) but the RELATIVE expansion, that is the point for point expansion, has decreased from 2 to 1.9."

With this is mind, it makes sense that the universe is accelerating. Why do we need dark energy to explain this rather than just saying that as there is more space then there is more space to expand into?

Last edited: Jul 26, 2016
2. Jul 26, 2016

### Chalnoth

If there were no form of matter/energy with $w < -1/3$, then the distances between any two far-away objects in an expanding universe would always be decelerating. Accelerating distances only happens when we have a form of matter with $w < -1/3$, which is what we call dark energy. The simplest model, a cosmological constant, has $w = -1$.

3. Jul 27, 2016

### Jorrie

Another way to look at it is that when the second derivative of the expansion (or scale-) factor
$\ddot{a} = a H_0 (\Omega_\Lambda - \Omega_m/(2a^3)) > 0$, it is accelerated expansion. Without a suitable positive value $\Omega_\Lambda$, the expansion is always decelerating.

Last edited: Jul 27, 2016
4. Jul 27, 2016

### Bandersnatch

I suspect the 'B' tag in the OP requires a more conversational answer.

We need dark energy, because the kind of expansion evolution that you wrote is impossible in a universe without dark energy.

First some terminology to make it more relatable:
What was used in your example as the total number of points, is usually referred to as proper distance, and the ratio of proper distance at some chosen time with respect to some other time (usually = now) is called scale factor - this is simply how many times larger are the distances when compared to now. The scale factor is denoted $a$, and you can see it in the equation provided by Jorrie above.
If the scale factor is growing, then we have expansion. The rate of growth is denoted by a dot: $\dot a$ - i.e. it's like velocity of growth.
If the rate of growth of the scale factor is itself growing - $\ddot a$, we have accelerated expansion.

In the example provided, the scale factor grows from 1 (=20 points) to 2 (40 points), and then to 3.8 (76 points).

What was used as the rate of expansion - i.e., x2, x1.9, is what is normally called Hubble constant. Hubble constant (H) is usually denoted in units of km/s/Mpc, but it can also be expressed in terms of percentage growth, where the present value (H0) as measured by the PLANCK mission: 68km/s/Mpc is equivalent to 1/144 % increase every million years (or x2 every 14.4 billion years).

With these in mind, we need to consider the dynamics of the expansion. In the BB theory you start with some initial impulse to make everything fly apart, and then let it evolve over time according to what and how dense the contents of the universe are. If we then imagine a universe that is completely empty - no matter or energy of any kind, but that still does have this initial expansion impulse imparted, then its scale factor will grow* by the same amount every equal interval of time, as there will be nothing gravitate towards each other thus slowing the expansion, nor anything to push stuff apart.

(*let's conveniently forget about the issue with measuring distances with a universe that has nothing in it)

Your $a$ is then a straight line, growing from 1 to 2 to 3 to 4... (or 20 points, 40, 60, 80...). The growth rate $\dot a$ is constant; the acceleration $\ddot a$ is 0.
This straight line in empty universe puts a constraint on what the evolution of H can be, since for the scale factor to grow like it does, H must go down as 1/a. If you start with H=100% per unit time (from 20 to 40 points), then in the next period of expansion must have H=50% to net extra 20 points of growth, and the next 33%, and so on.

Once again, this is the limiting case for a universe without any contents but just the initial impulse to expand. If you then add any sort of regular energy to it (matter, radiation), this additional content will act gravitationally on itself to slow down the expansion to less than the limiting case. The graph below illustrates this

(scale factor vs time)
Depending on how much energy you add to the universe, it will expand (or expand and then contract) at various rates, lower than the limiting straight line case. The acceleration is negative for every case, all the time (on the graph this means that the slope is curved downwards at every point).
In all of those cases, the Hubble constant must go down faster than 1/a.

However, if you were to add to that empty universe not regular energy, but dark energy - i.e. some inherent property of space that pushes everything apart, something that causes space to 'gain' extra space over time, then you'll have your scale factor grow faster than the straight line. How it'd grow depends on the properties of this new energy - is it constant? is it growing over time or going down? (this is what Chalnoth referred to in his post)

In the case of a constant dark energy, you get exponential growth of the scale factor, as H is constant - rather than being limited to the initial impulse, now every bit of space 'creates' some set percentage amount of space every set period of time (so 20 points may become 40 points after sufficient period of time has passed, and then 80, and so on).

Combining the two types of content in a single universe produces an additional type of evolution of the scale factor:

(same type of graph as the one before; scale factor vs time; red line shows regular energy + dark energy; you can see various values for matter and energy densities Ω included - these are the values present in the equation provided by Jorrie in post #3)

In this situation, you have a period of deceleration (slope bends downward, H goes down faster than in the limiting case), and then acceleration (slopes bends upward after the inflection point, H goes down slower than the limiting case, trending towards a constant value as matter gets diluted by the expansion).

Looking again at the equation in post #3, you can see how this slope of the graph is produced - with sufficiently high matter density Ωm when compared to dark energy density ΩΛ (Ων on the second graph), or sufficiently early time (meaning very low value of the scale factor a), the negative term in the brackets is higher than the positive one, and you get a deceleration. At some point, however, the growing value of a dilutes matter Ωm so much, that it will become lower than ΩΛ, and you get accelerated expansion.

Now, let's get back to the expansion example you provided. Since the next step after a 100% growth of the scale factor (i.e., the 'x2' step) is something more than 50% (x1.9 is 90%), which results in a slope of the scale factor that is bent upwards, then you already know that this is only possible in a universe with dark energy.

Depending on what is the next step in the evolution of H - e.g., 1.8, or 1.85?, the value of dark energy would have to be either falling down with time or constant respectively for the type of expansion to make sense, because it will go down to 0 in ten steps and then reverse in the first case, and it will stabilize at a constant value in the latter - which is also what happens in our universe.

Since the actual shape of the slope of the scale factor does seem to match this one:

I.e. it does slope upward beginning approx. 5 billion years ago, it's an indication that we do need dark energy.

5. Jul 27, 2016

### Physics1218

Thanks for that post, Bandersnatch. It was really elaborate and insightful. I would just like to ask you two questions. As you explained, contemporary evidence points out that the Hubble constant will eventually evolve into 1/144% every million years and stay there. Currently the Hubble constant is decelerating less than it was previously due to the continual diluting of matter. Does this mean that the Hubble Sphere will eventually stop growing and will remain at a certain place? Lastly can you explain why in the conventional expansion of H through matter it must abide by 1/a.

Last edited: Jul 27, 2016
6. Jul 28, 2016

### Bandersnatch

This is all correct, with the exception of the value of 1/144%/My - this is the current Hubble constant. It evolves asymptotically towards 1/173%/My (as per the 2013 PLANCK data).
I'll just add, since this often pops up in these questions, that the Hubble sphere is not a horizon, nor the size of observable universe, as it only marks where the recession velocity is c at a given time.
Also, 'eventually' here means after infinite time has passed, so for all practical purposes the Hubble sphere will be always growing, albeit ever more slowly.

I'm not sure what you mean here.
Do you mean why it must be less than 1/a in a non-empty universe (with matter)? Or, do you mean why is it 1/a in an empty universe in the first place?

7. Jul 28, 2016

### Physics1218

Sorry, I phrased that question strangely. After reading your response, I came to the conclusion that the primary evidence for dark energy is that the the evolution of H is not constrained.
Why does H have to evolve like this in a universe without matter or dark energy?

8. Jul 28, 2016

### Chalnoth

The evolution of the scale factor is given by the Friedmann equations, in particular the first one, which can be written as:

$$H^2 = {8 \pi G \rho \over 3} + {\Lambda c^2 \over 3} - {k c^2 \over a^2}$$

The terms on the right side of the equation are, in order, contributions from matter/energy density, the cosmological constant, and spatial curvature. When there is no matter or energy, and no cosmological constant, then this equation reduces to:
$$H^2 = -{k c^2 \over a^2}$$

Here, $k$ is a constant that defines the spatial curvature, and in this situation $k$ has to be negative. Take the square root, and you get:
$$H = \sqrt{-k}{c \over a}$$

...which indicates an expansion that scales with the inverse of the scale factor.

9. Jul 29, 2016

### Jorrie

One can say that in the standard cosmological model $H$ could remain permanently zero in a hypothetical universe with zero radiation, zero matter, zero dark energy and zero curvature, e.g. Minkowski spacetime. In other words the model allows $H$ to not evolve at all.
However, quantum fluctuations would probably not allow such a situation to exist. We also know that it is not the case in our universe.

10. Jul 29, 2016

### Physics1218

Thank you for the response guys. I would lastly just like to confirm that I'm correct in believing that the primary evidence of dark energy is in that the Hubble sphere is not constrained to 1/a.\
Furthermore, I would like to thank you again Bandersnatch for the explanation.

11. Jul 29, 2016

### Jorrie

No. Even without dark energy, in a universe with matter only, $H$ is also not constrained to 1/a.
Look at Chalnoth's 1st eq. in post 8. The value of $\rho$ scales with $1/a^3$, while dark energy density does not scale with $a$ at all. Fitting to observations require both of those terms to have positive values. This is essentially the evidence for dark energy.

Last edited: Jul 29, 2016
12. Jul 29, 2016

### timmdeeg

I think $$H^2 = -{k c^2 \over a^2}$$ describes the empty FRW-universe which is hyperbolic and expanding and which is equivalent to flat Minkowski spacetime after coordinate transformation.

13. Jul 29, 2016

### Physics1218

Thanks for the response. Although from what I can see, the post by Bandersnatch is in conflict with yours. Bandersnatch said,
(It should be noted that Bandersnatch was referring to a universe in which there wasn't matter and energy.) In a universe with matter, then H would have to be lower than 1/a

14. Jul 29, 2016

### Jorrie

I'm not sure that one can transform a uniformly expanding space into Minkowski spacetime. Doesn't that require k to be zero?

15. Jul 29, 2016

### Bandersnatch

There's no conflict - with only matter you've got it going down faster than 1/a, and with only dark energy you get slower than 1/a. The evolution of H is limited to 1/a only in the case of a completely empty universe.

Last edited: Jul 29, 2016
16. Jul 29, 2016

### Jorrie

Then you need to add that in the case of a universe with matter + dark energy, H would evolve above the 1/a scale. Recall that in such a case, H would eventually reach a constant value which is larger than zero. This seems to be the simplest answer to your question regarding evidence for dark energy.

I do not think there is any conflicting view with Bandersnatch, but just about the interpretation of what k=0 means in the Friedmann equations. Obviously, if there is any expansion, k cannot be zero, but I don't think the equations exclude the k=0 case.

I'll wait for some expert advice, because I might be wrong...

17. Jul 29, 2016

### timmdeeg

In post #8 Chalnoth said "in this situation $k$ has to be negative". I suspect that in the case of the 'empty universe' spatially flatness, $k=0$, is excluded, because the ratio of actual density to critical density can't be 1. Remarkably the empty FRW-universe expands even though $\Lambda=0$.

Regarding the coordinate transformation see the thesis of Tamara Davis https://arxiv.org/pdf/astro-ph/0402278v1.pdf , Chapter 4, The empty universe.

Last edited: Jul 29, 2016
18. Jul 29, 2016

### Staff: Mentor

You can't transform the uniformly expanding "empty universe" into all of Minkowski spacetime; but you can transform it into a "wedge" consisting of all points within the future light cone of the origin. The surfaces of constant "comoving time" in Minkowski coordinates are then hyperboloids of constant proper time from the origin, and the "comoving" worldlines are just all possible timelike geodesics that come from the origin.

19. Jul 29, 2016

### Chalnoth

You need R to be zero, which is different from k. R is the total space-time curvature at every point, and is independent of whatever coordinates you choose.

k is the spatial curvature only. And it turns out that in the FRW universe, R has a contribution from both H and k, so that it's possible to write down a universe where the two cancel to get R = 0 (in fact, I believe it's simply $R = H^2 + k c^ / a^2$).

20. Jul 29, 2016

### Jorrie

Thanks guys, I now properly understand the meanings of $k$ and $R$.
But I still do not get it why for an empty universe, $k$ and $H$ cannot both go to zero in $H^2=-kc^2/a^2$ for an arbitrary constant $a > 0$.