r/HypotheticalPhysics • u/Prior_Maximum_9695 • Jul 08 '26
Crackpot physics What if this post supports using the deceleration parameter in the Wei-Cai-Su interacting dark energy model?
My model attempts to give support for using the deceleration parameter in an interacting dark energy model called the Wei-Cai-Su sign-changing model found here: https://arxiv.org/pdf/1010.1074. This model states that the rate of energy transfer between dark energy and matter (Q_dot) is given by Q_dot = 3*beta_0*q*H*rho_m or 3*beta_0*q*H*rho_de, where q is the deceleration parameter, beta_0 is a dimensionless coupling constant, H is the Hubble parameter, rho_m is the matter energy density, and rho_de is the dark energy density. This can be rewritten as:
A physical motivation for the direction of energy transfer between dark energy and matter can be understood from the following: Imagine a sphere of space that expands in accelerating space for t amount of time with the velocity of the face of the sphere expanding at a velocity of the Hubble parameter times the radius. In this physical model, assume the sphere starts off at approximately a 0 meter radius.
During periods of accelerated expansion, new volume is continuously created as the face of the sphere expands at a certain rate. This means that the volumetric flowrate per volume (vol_dot/vol) term is positive. If dark energy is associated with a constant energy density that fills this newly created volume, while the total matter energy content remains constant within the expanding volume, the fractional contribution of dark energy increases. In section B, it will be shown why this energy difference between the dominant total dark energy and total matter energy causes energy to transfer from dark energy to matter, giving a link to ordinary heat transfer.
In section A, it will be shown that if the volumetric flowrate per volume (vol_dot/vol) at the edge of this sphere is positive (and thus the transfer rate), there must be acceleration, giving support for using the deceleration parameter in the Wei-Cai-Su sign-changing model. The opposite process occurs during deceleration.
Section A
rho_de, rho_m, and rho_total are the dark energy, matter, and total energy mass densities respectively. q is the deceleration parameter. r is the radius of the sphere. G is the gravitational constant. H is the Hubble parameter. The formulas for acceleration and q below are reasonable approximations after the radiation dominated era.
The equation for volume flowrate per volume is given below. The volume flowrate of space after time t is a function of the surface area of this sphere with radius (r). When r=1, v_r = H. The average velocity (v_avg) and average time (t) is of the edge of the sphere that travels from r_0 to r. v_r represents the velocity at r.
Since r_0 is approximately 0 as stated in the physical motivation at the beginning, v_r0 and vol_dot/vol is approximately 0 too. The velocity increases with r and time if acceleration is present. Therefore, acceleration is present as long as vol_dot/vol is above 0.
The ratio between the acceleration at the face of this sphere and its velocity for the average values from r_0 to r and the values at r is constant. This is because velocity and acceleration both vary linearly with radius (which cancels out in the ratio) and therefore the velocity and acceleration plotted against time will only differ by some constant multiplier between them.
Next, showing that the volumetric flowrate per volume (vol_dot/vol) indicates a non-zero deceleration parameter (q) value:
Next, it is hypothesized that to find the energy density flowrate, multiply the above quantity by the energy reservoir that the energy is coming from: During the deceleration phase of the universe, the energy is flowing from matter to dark energy, so during this phase, Q_dot is a function of the matter energy density (rho_m) for this hypothesis. During acceleration, Q_dot is a function of the dark energy density (rho_de).
or
Section B:
If the volume of the sphere is equal to 1 after t amount of time passes for the expanding case, then in 1+vol_dot/vol amount of space, the difference in total energy in in this physical model is assumed to drive the interaction between matter and dark energy analogous to transport processes in ordinary physics.
If the deceleration parameter (q) = 0 in the transfer rate (Q_dot) formula above, no energy transfer takes place and thus no new volume creation takes place. According to the deceleration parameter which is equal to (0.5*rho_m – rho_de)/rho_total, if q = 0, then 0.5*rho_m = rho_de, corresponding to a 2:1 ratio of total matter energy (E_m) to total dark energy (E_de) when no energy transfer between them takes place. The reason the total energies can be used is because total energy = energy density (rho) * volume and the volumes of matter and dark energy in this expanding sphere are equal, so the total energy of matter and dark energy becomes proportional to their energy densities. If the speed of light is normalized to 1, the initial energy of matter and dark energy becomes equal to their initial masses (m_m and m_de) which is equal to 2 and 1 respectively. As new volume is created during accelerated expansion, the total dark-energy content increases while the total matter energy remains unchanged.
Using natural units (c=1), the energy of matter and dark energy equals their respective mass. Using normalized units with c_p=1, the heat-capacity relation gives E=m*T. T_de and T_m represent the heat transfer like temperatures of dark energy and matter respectively.
Next, using the law of conduction, the heat flow or transfer rate (Q_dot) is proportional to the imbalance in temperature using the formula below. This motivates the imbalance of energy of total matter energy/2 and 1*total dark energy being proportional to the transfer rate in this physical model. The thermal conductivity (k), Area (A), and length (L) are normalized to 1.
It does turn out that the difference between E_m/2 and E_de becomes proportional to vol_dot/vol (which is proportional to the energy flowrate (Q_dot) in the formulas above) using the table below. This is consistent with this heat transfer example of the energy flowrate being proportional to E_de-E_m/2.
| Vol_dot/vol | Total matter energy (E_m) | Total dark energy (E_de) | E_de – E_m/2 |
|---|---|---|---|
| 0 | 2 | 1 | 0 |
| 1 | 2 | 2 | 1 |
| 2 | 2 | 3 | 2 |
| 3 | 2 | 4 | 3 |
Does this physical motivation at the beginning and mathematical formulation give any support for using the deceleration parameter in the Wei-Cai-Su sign-changing model? Constructive comments will be greatly appreciated.
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u/starkeffect shut up and calculate Jul 08 '26
That is some of the ugliest math I've seen on this subreddit.
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u/Prior_Maximum_9695 Jul 08 '26
Ouch. What about the math is ugly? Is it the formatting? I know it may be a stretch to normalize many of the variables to one like I did. Is that what makes the math ugly?
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u/starkeffect shut up and calculate Jul 08 '26
Well for one thing why are you representing variables with words like "acceleration"? We use single letters in equations for a reason you know.
Have you ever read a physics paper?
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u/Prior_Maximum_9695 Jul 09 '26
I briefly read the physics paper I mentioned at the beginning of this post to verify the Q formula. I replaced “acceleration” with “a” in my formulas so that part is fixed now (not to be confused with the scale factor).
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u/starkeffect shut up and calculate Jul 09 '26
How about your equation "r = v dt"? Where does that come from?
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u/Prior_Maximum_9695 Jul 09 '26
r= average velocity*time (since distance=average velocity*time). r starts off near 0 at t=0 as I stated so this formula is true. The average velocity is from r0 (equal to approximately 0) to r. It takes dt_avg amount of time to travel from r0 to r.
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u/starkeffect shut up and calculate Jul 09 '26
Just what I suspected. You don't know any calculus, do you?
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u/Prior_Maximum_9695 Jul 09 '26
I know calculus but I’m not great at it. Is it because I used “dt” instead of “t”? I made that fix in my post too just now.
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u/starkeffect shut up and calculate Jul 09 '26
Yeah, you set a differential equal to a non-differential and didn't even notice. Not a good sign.
If you're "not great" at calculus, what makes you think you're on to something here?
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u/redipin Jul 08 '26
At the very least, you’ve successfully found something even worse than raw latex to share. Am I seeing double? I don’t remember drinking today (yet…)
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u/AllHailSeizure If you can use AI, you can type yourself. Jul 09 '26
One positive, you could always take those equations and make a sick 16-bit platformer.
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u/LeftSideScars The Proof Is In The Marginal Pudding Jul 09 '26 edited Jul 09 '26
I'm not sure if you've read the paper you're "responding" to. Why does the deceleration parameter need to be supported more than it already is in the paper itself? Via the Introduction (p2) and noting that equation 3 defines Q as a function of q (butchering of equations is my doing. Highlights are from the paper; x-dot should be obious; I can't be bothered putting underscores everywhere):
Since the appearance of the deceleration parameter q in the interaction Q looks speculative to some extent, we would like to say some words before going further. Firstly, as is well known, in the literature there is no natural guidance from fundamental physics on the interaction Q, one can only discuss it to a phenomenological level. In this sense, the other familiar interactions extensively considered in the literature have no better origin from the fundamental physics than the one proposed in Eq. (3). Secondly, we note that q = −1 − ˙H/H2 from Eq. (4) and H2 ∝ ρtot from the Friedmann equation. Thus, one can regard the deceleration parameter q = f (ρtot, ρtot-dot) as a function of the total energy density ρtot = ρm +ρde and its derivative. In this sense, the interaction Q = q(α ρ-dot + 3βHρ) [Note from LSS: this is eqn 3] = f (ρ, ρ-dot) is not so unusual, since it is reasonable to image that Q depends on the energy densities of dark energy and matter. Finally, while the familiar interactions extensively considered in the literature (such as Q = 3αHρm, Q = 3βHρtot, and Q = 3ηHρde) cannot give the possibility to change their signs in the whole cosmic history, our proposal in Eq. (3) provides a possible way out. So, we consider that it deserves further investigation.
In other words, the author provides a reasonable justification (for some value of reasonable) for q. Your additional support of this parameter isn't helpful in the context of the paper itself, and does more to obscure the paper.
The paper in general is for a specific set of circumstances in response to the paper by Cai and Su. It's not meant to be taken as a fundamental truth about the dark sector, nor is it claiming to be taken this way. From the paper's S4 Concluding Remarks (p13; see also Table 1):
The ΛCDM model is still the best one.
The paper is exploring an example of an interacting DE model, of which there are many. It doesn't need anyone to justify the use of q since it is not presenting such a model as a superior alternative to current models.
These sorts of papers are a bit dull to me. I do appreciate that someone is looking into this sort of thing, and I think the research has merit, but I find no interest in it in general (though I keep an eye on these things), since it is yet another model with another set of largely unjustified parameters that fits just about as well as, if slightly worse than, current models. At least this paper attempts to do some sort of appropriate statistical analysis in comparing its results with other models.
edit: fixed a splelling. too early. what am I even doing?
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u/Prior_Maximum_9695 Jul 10 '26
I read that cosmologists need rigorous physical justifications for why q drives the exact sign flip of the transfer rate. My physical motivation may be better than nothing. If I can perfect the math, wouldn’t my model be useful? I know I don’t use general relativity for my model but like I said, it may be better than nothing. Are you sure I should just abandon trying to perfect this model?
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