<p>With your edited question, I think perhaps your misunderstanding is different than what I thought first, so here's another answer. I leave the first answer in the bottom.</p>
<h3>Redshifts vs. recession velocities</h3>
<p>First, although it's not uncommon to quote measured redshifts in terms of recession velocities — and even more so in the past — it's a bad idea. If our understanding of expansion and general relativity is correct, the measured redshift is not due to (e.g.) a galaxy receding from us, but due to space having expanded while the photon was traveling. Even in the case of a hypothetical universe that were static when the distant galaxy emitted a photon, and were static when you observe the photon, but somehow expanded by some factor in between, you would measure a redshift.</p>
<p>When people quote recession velocity, they often just mean the corresponding value of <span class="math-container">$cz$</span>, which is indeed a good approximation at low redshifts. But in reality, the recession velocity is not readily seen from a measured redshift; it requires a cosmological model.</p>
<h3>Current velocity</h3>
<p>With observations at various redshifts, we build up a cosmological model. The generally accepted model is, as I wrote in my first answer, that expansion is <em>homologous</em>, i.e. recession velocity is proportional to distance <span class="math-container">$d$</span>, as given by Hubble's linear law, <span class="math-container">$v_\mathrm{rec} = H_0 d$</span>. The distance here is the <em>current</em> distance, which is what you'd measure if you magically froze the Universe and started laying out measuring rods. Likewise, the velocity is the rate at which this distance is currently increasing.</p>
<p>The most distant observed galaxy, GN-z11, has a redshift of 11.1. That doesn't mean that it's receding at 11.1 times the speed of light, it means that the Universe has expanded by a factor of 11.1 size the light that we see today was emitted. The recession velocity of GN-z11 cannot be calculated from the redshift alone, but with a cosmological model, we can in fact say that GN-z11 is ~32 Glyr away, and hence is receding at <span class="math-container">$\sim2.2c$</span> (we can also say that, when it emitted the light we see, it was receding even faster, at some <span class="math-container">$4c$</span>, then decelerated to slightly below <span class="math-container">$2c$</span>, then accelerated again).</p>
<h3>Time dilation</h3>
<p>Distant phenomena such as the decline of a supernova lightcurve are indeed seen to be time dilated. General relativity predicts this time dilation, and if our understanding is correct, this does <em>not</em> have anything to do with the distant object receding. However, if space were not expanding, but galaxies were simply moving through space, special relativity would predict the exact same factor of time dilation (namely <span class="math-container">$(1+z)$</span>), so this doesn't tell us which interpretation is true.</p>
<hr />
<h3>My answer before your edited question follows here:</h3>
<p>I think perhaps you're mixing up two different things:</p>
<p>The fact that there is a distance beyond which recession is superluminal (the &quot;edge of the Hubble sphere&quot;) doesn't have anything to do with whether expansion is slowing down or speeding up. Even a steadily expanding universe would have such a distance. This is simply because the expansion is &quot;homologous&quot;, i.e. the velocity increases proportionally to distance. The velocity referred to here is the <em>current</em> velocity.</p>
<p>To understand how we know that the expansion rate of the Universe has changed through time, consider the following:</p>
<p>The observed redshift <span class="math-container">$z$</span> of a light source (e.g. a supernova) tells you by how large a factor the Universe has expanded since the observed light was emitted, namely a factor <span class="math-container">$1+z$</span> (if you believe that general relativity is a good description of the Universe).</p>
<p>Since we don't know the absolute size of the Universe, we use instead the <em>scale factor</em> <span class="math-container">$a$</span> which is normalized the <span class="math-container">$1$</span> today, but was e.g. <span class="math-container">$0.5$</span> when everything was half as close to each other. The relation between an observed redshift and the scale factor at the time the light was emitted is simply <span class="math-container">$a = 1/(1+z)$</span>.</p>
<p>If you observe <em>many</em> sources at various distances — and hence at various epochs in the history of the Universe — together with their redshifts (and hence scale factors) you build up a picture of the size of the Universe as a function of time.</p>
<p>What the observations of near and far sources have shown us is that the size of the Universe increased rapidly in the early Universe, slowly decreased in expansion rate, but then &quot;recently&quot; started speeding up again.</p>

CC BY-SA 4.0
2022-01-18T08:35:34.403Z
Thank you very much @pela for providing a valuable justification regarding the interpretation of redshifts in terms of scale factor evolution; this definitely helps keep a check on the Universe's expansion if it has slowed down with time, or if it has sped up.
Karan R.Takkhi
CC BY-SA 4.0
2022-01-19T10:56:46.763Z
I updated my answer to take into account your edited question. This site is not the place for personal hypotheses, it's solely for "mainstream physics". Sharing a link in a comment might be okay though, I'm not exactly sure about the rules.
pela
CC BY-SA 4.0
2022-01-21T09:16:01.390Z
I am not emphasizing that a galaxy or a quasar is moving/receding through space, however, when it comes to the determination of the expansion rate, it seems that the researchers use recession velocities (corresponding value of cz) over the entire redshift domain (https://astro.ucla.edu/~wright/DL-vs-z-26Mar2015.gif). Taking values as they are, nearby supernovae accurately yield ≈70 km/s/Mpc, whereas distant ones yield ≈45 km/s/Mpc. This lower value is in agreement with deep determination of expansion rate (48 km/s/Mpc) (Kochanek & Schechter 2004); 30% lower than the local value of 70 km/s/Mpc.
Karan R.Takkhi
CC BY-SA 4.0
2022-01-21T10:08:39.427Z
$H_0$ from gravitational time delays are only starting to become competitive with SNe and CMB, so don't put too much faith into a paper from 2004 (despite the second author). I don't see where you get the 45 km/s/Mpc from?
pela
CC BY-SA 4.0
2022-01-27T12:41:36.110Z
Luminosity distance (DL) is the directly-observable distance to type Ia supernovae that shows that remote supernovae are 10% to 25% dimmer as they are further away than expected, and, since observables play an important role in cosmology, therefore, I have taken into account those directly-observable quantities here to determine the expansion rate from the plot – luminosity distance and redshift. H = cz (as per the plot)/DL. In fact, this scales perfectly with respect to the slope of the plot; nearby supernovae yield a steeper slope (higher expansion rate) as compared to the distant ones.
Karan R.Takkhi
CC BY-SA 4.0
2022-01-17T16:30:58.280Z
2022-01-19T10:54:16.163Z
2022-01-19T10:54:16.163Z
pela
<p>Davis and Lineweaver correctly point out some persistent mistakes in others' descriptions of FLRW cosmology, but they make errors of their own. One of those is the idea that there is superluminal expansion in cosmology and that this has something to do with &quot;expansion of space itself&quot;.</p>
<p>Recession speeds larger than <span class="math-container">$c$</span> appear even in the <span class="math-container">$ρ=p=Λ=0,\, a'&gt;0$</span> special case of FLRW cosmology. That FLRW spacetime is isomorphic to a patch of Minkowski spacetime, and has a global speed limit of <span class="math-container">$c$</span> when speeds are defined in the usual special-relativistic way. But the recession speed, defined in the usual cosmological way, works out to be <a href="https://en.wikipedia.org/wiki/Rapidity" rel="nofollow noreferrer">rapidity</a> times <span class="math-container">$c$</span>, and it exceeds <span class="math-container">$c$</span> when <span class="math-container">$z&gt;e-1\approx 1.72$</span>, not terribly different from the real-world ΛCDM value. Nothing special happens at this speed (which is equivalent to about <span class="math-container">$0.76c$</span> in <span class="math-container">$dx/dt$</span> speed). A Hubble sphere can be defined but has no physical meaning.</p>
<p>In ΛCDM the numbers are different but the principle is the same. Recession speeds are defined in such a way that <span class="math-container">$c$</span> has no special meaning. The Hubble sphere has no physical meaning (except that in the distant future, when <span class="math-container">$ρ\approx 0$</span>, it coincides with the cosmological horizon).</p>
<p>I think that none of this is related to the evidence for accelerating expansion.</p>
<hr />
<p>In the plot in your question (which was added after pela's answer), the data points on the right are from earlier times. If the expansion is accelerating then the expansion rate at earlier times should be lower, and they are.</p>
<p>But that plot strikes me as misleading because it doesn't show what was actually measured; both axes depend on the cosmological parameters that they're supposed to be fitting to the data. Here's a better one, from <a href="https://astro.ucla.edu/%7Ewright/sne_cosmology.html" rel="nofollow noreferrer">this page</a> of Ned Wright's cosmology tutorial:</p>
<img src="https://astro.ucla.edu/~wright/DL-vs-z-26Mar2015.gif">
<p><sup>(If you're color blind: &quot;Closed Matter Only Model&quot; is the topmost curve, &quot;de Sitter Model&quot; is the bottommost, and the other colored lines are fairly good fits to the data.)</sup></p>
<p><span class="math-container">$cz$</span> is the measured redshift times <span class="math-container">$c$</span>, not the recession speed. (It's possible that the horizontal axis of your plot is also <span class="math-container">$cz$</span>; it has two inconsistent labels.) Luminosity distance is defined as a function of relative and absolute magnitude independently of the FLRW parameters.</p>
<p>It's not so easy to understand in Wright's diagram why the curves should be shaped as they are, but I think that's fine because it's how science usually works. You don't see fundamental parameters in the data; you tweak fundamental parameters until the predictions of the model fit the data.</p>
<p>That's also the answer to your concern about time dilation. The cosmological model is based on general relativity, and calculations within general relativity automatically take time dilation into account. The term &quot;time dilation&quot; isn't precisely defined, and you can argue about whether it should apply here, but the predictions of the model are unambiguous.</p>

CC BY-SA 4.0
2022-01-19T08:25:45.093Z
Thank you @benrg for your answer. According to this Science News article, using SNe Ia as standard candles, researchers measured their recession velocities & distances; this helped them conclude that the Universe was expanding 10% - 15% more slowly in the past (https://www.sciencenews.org/archive/astronomy-138). The High-Z plot (illustrated in the question) is the result of these measurements. As per the plot, further-away-than-expected SNe Ia on the R.H.S. lie in the region where the expansion is slower, however, there is another aspect (other than acceleration) that can mimic this perfectly.
Karan R.Takkhi
CC BY-SA 4.0
2022-01-19T08:27:26.427Z
https://www.researchgate.net/profile/Karan-Rtakkhi/publication/343484700_Shedding_Light_on_an_Undiscovered_Aspect_that_Perfectly_Mimics_Cosmic_Acceleration/links/61d9a21fd450060816992df4/Shedding-Light-on-an-Undiscovered-Aspect-that-Perfectly-Mimics-Cosmic-Acceleration.pdf
Karan R.Takkhi
CC BY-SA 4.0
2022-01-18T21:41:08.793Z
2022-01-18T21:41:08.793Z
benrg
<p>Type Ia supernovae discoveries at high redshifts (<em>z</em> &gt; 1) support past deceleration (Riess et al. 2001). This past deceleration has also been confirmed using quasars (<em>z</em> ≈ 6) (Risaliti and Lusso 2015) as well as gamma-ray bursts (<em>z</em> ≈ 9) (Demianski et al. 2017).</p>
<p>According to Davis and Lineweaver (2004), “Recession velocities exceed the speed of light in all viable cosmological models for objects with redshifts greater than <em>z</em> ~ 1.5”. In fact, in the ΛCDM concordance model itself, all galaxies beyond a redshift of <em>z</em> = 1.46 are receding faster than the speed of light (Davis and Lineweaver 2004).</p>
<p>Therefore, my question is, how can every superluminal remote expansion (<em>z</em> = 1.7 (SN Ia), <em>z</em> = 6 (QSO), <em>z</em> = 9 (GRB)) be justified as “slowing down” under the gravitational influence of matter as compared to subluminal local expansion (<em>z</em> = 0.01 (SN Ia))?</p>
<p>Isn’t this is a paradox?</p>
<p>I would like to add some additional details to the above question as it still appears paradoxical when discoveries at high redshifts are taken as an indication of &quot;slowing down&quot; particularly when those redshifts are interpreted in terms of recession velocities.</p>
<p>Take for instance the following plot by the High-Z Supernova Search Team; the researchers have clearly interpreted redshifts in terms of recession velocities.</p>
<p><a href="https://i.sstatic.net/kbu2m.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/kbu2m.png" alt="Supernovae observations by the High-Z Supernova Search Team that confirmed cosmic acceleration. Credit: High-Z Supernova Search Team." /></a></p>
<p>According to this plot, local recession velocities (just 1% of speed of light; <em>z</em> = 0.01) indicate a faster rate of expansion (acceleration) as compared to remote recession velocities (60% of speed of light; <em>z</em> = 0.6) that indicate a slower rate of expansion (deceleration) - a clear-cut paradox.</p>
<p>Here is an excerpt that shows recession velocities from observed redshifts do play a role in determining the expansion rate, “It has been noted by Zehavi et al. (1998) that the SNe Ia out to 7000 km/s (108 Mpc) exhibit an expansion rate (65 km/s/Mpc) that is 6% greater than that measured for the more distant objects” (Riess et al. 1998). It becomes obvious that redshifts of the more distant objects might have also been interpreted in terms of recession velocities to compare the expansion rates.</p>
<p>Moreover, significant time dilation is observed at high redshifts than at low redshifts. Now, time dilation is based on the fact that a moving clock ticks slowly as compared to a stationary one, that is, an event on a &quot;receding&quot; emitter appears time dilated. The fact that there is significant time dilation at high redshifts than at low redshifts suggests that the expansion at high redshifts (<em>z</em> &gt; 1) is definitely not a feeble expansion as compared to the expansion at low redshift (<em>z</em> = 0.01) which is as good as &quot;almost stationary&quot;.</p>

<cosmology><dark-energy>
How can it be justified that every discovery at z > 1 is an indication of "slowing down" or past deceleration?
Karan R.Takkhi
# How can it be justified that every discovery at z > 1 is an indication of "slowing down" or past deceleration?

(Asked by: username_0 on 2022-01-17)

Type Ia supernovae discoveries at high redshifts (*z* > 1) support past deceleration (Riess et al. 2001). This past deceleration has also been confirmed using quasars (*z* ≈ 6) (Risaliti and Lusso 2015) as well as gamma-ray bursts (*z* ≈ 9) (Demianski et al. 2017).

According to Davis and Lineweaver (2004), “Recession velocities exceed the speed of light in all viable cosmological models for objects with redshifts greater than *z* ~ 1.5”. In fact, in the ΛCDM concordance model itself, all galaxies beyond a redshift of *z* = 1.46 are receding faster than the speed of light (Davis and Lineweaver 2004).

Therefore, my question is, how can every superluminal remote expansion (*z* = 1.7 (SN Ia), *z* = 6 (QSO), *z* = 9 (GRB)) be justified as “slowing down” under the gravitational influence of matter as compared to subluminal local expansion (*z* = 0.01 (SN Ia))?

Isn’t this is a paradox?

I would like to add some additional details to the above question as it still appears paradoxical when discoveries at high redshifts are taken as an indication of "slowing down" particularly when those redshifts are interpreted in terms of recession velocities.

Take for instance the following plot by the High-Z Supernova Search Team; the researchers have clearly interpreted redshifts in terms of recession velocities.

[![Supernovae observations by the High-Z Supernova Search Team that confirmed cosmic acceleration. Credit: High-Z Supernova Search Team.](https://i.sstatic.net/kbu2m.png)](https://i.sstatic.net/kbu2m.png)

According to this plot, local recession velocities (just 1% of speed of light; *z* = 0.01) indicate a faster rate of expansion (acceleration) as compared to remote recession velocities (60% of speed of light; *z* = 0.6) that indicate a slower rate of expansion (deceleration) - a clear-cut paradox.

Here is an excerpt that shows recession velocities from observed redshifts do play a role in determining the expansion rate, “It has been noted by Zehavi et al. (1998) that the SNe Ia out to 7000 km/s (108 Mpc) exhibit an expansion rate (65 km/s/Mpc) that is 6% greater than that measured for the more distant objects” (Riess et al. 1998). It becomes obvious that redshifts of the more distant objects might have also been interpreted in terms of recession velocities to compare the expansion rates.

Moreover, significant time dilation is observed at high redshifts than at low redshifts. Now, time dilation is based on the fact that a moving clock ticks slowly as compared to a stationary one, that is, an event on a "receding" emitter appears time dilated. The fact that there is significant time dilation at high redshifts than at low redshifts suggests that the expansion at high redshifts (*z* > 1) is definitely not a feeble expansion as compared to the expansion at low redshift (*z* = 0.01) which is as good as "almost stationary".

**Tags:** cosmology, dark-energy


---

## 2 answers

### Answer by username_1 on 2022-01-17. Score: 3

With your edited question, I think perhaps your misunderstanding is different than what I thought first, so here's another answer. I leave the first answer in the bottom.

### Redshifts vs. recession velocities

First, although it's not uncommon to quote measured redshifts in terms of recession velocities — and even more so in the past — it's a bad idea. If our understanding of expansion and general relativity is correct, the measured redshift is not due to (e.g.) a galaxy receding from us, but due to space having expanded while the photon was traveling. Even in the case of a hypothetical universe that were static when the distant galaxy emitted a photon, and were static when you observe the photon, but somehow expanded by some factor in between, you would measure a redshift.

When people quote recession velocity, they often just mean the corresponding value of $cz$, which is indeed a good approximation at low redshifts. But in reality, the recession velocity is not readily seen from a measured redshift; it requires a cosmological model.

### Current velocity

With observations at various redshifts, we build up a cosmological model. The generally accepted model is, as I wrote in my first answer, that expansion is *homologous*, i.e. recession velocity is proportional to distance $d$, as given by Hubble's linear law, $v\_\mathrm{rec} = H\_0 d$. The distance here is the *current* distance, which is what you'd measure if you magically froze the Universe and started laying out measuring rods. Likewise, the velocity is the rate at which this distance is currently increasing.

The most distant observed galaxy, GN-z11, has a redshift of 11.1. That doesn't mean that it's receding at 11.1 times the speed of light, it means that the Universe has expanded by a factor of 11.1 size the light that we see today was emitted. The recession velocity of GN-z11 cannot be calculated from the redshift alone, but with a cosmological model, we can in fact say that GN-z11 is ~32 Glyr away, and hence is receding at $\sim2.2c$ (we can also say that, when it emitted the light we see, it was receding even faster, at some $4c$, then decelerated to slightly below $2c$, then accelerated again).

### Time dilation

Distant phenomena such as the decline of a supernova lightcurve are indeed seen to be time dilated. General relativity predicts this time dilation, and if our understanding is correct, this does *not* have anything to do with the distant object receding. However, if space were not expanding, but galaxies were simply moving through space, special relativity would predict the exact same factor of time dilation (namely $(1+z)$), so this doesn't tell us which interpretation is true.

---

### My answer before your edited question follows here:

I think perhaps you're mixing up two different things:

The fact that there is a distance beyond which recession is superluminal (the "edge of the Hubble sphere") doesn't have anything to do with whether expansion is slowing down or speeding up. Even a steadily expanding universe would have such a distance. This is simply because the expansion is "homologous", i.e. the velocity increases proportionally to distance. The velocity referred to here is the *current* velocity.

To understand how we know that the expansion rate of the Universe has changed through time, consider the following:

The observed redshift $z$ of a light source (e.g. a supernova) tells you by how large a factor the Universe has expanded since the observed light was emitted, namely a factor $1+z$ (if you believe that general relativity is a good description of the Universe).

Since we don't know the absolute size of the Universe, we use instead the *scale factor* $a$ which is normalized the $1$ today, but was e.g. $0.5$ when everything was half as close to each other. The relation between an observed redshift and the scale factor at the time the light was emitted is simply $a = 1/(1+z)$.

If you observe *many* sources at various distances — and hence at various epochs in the history of the Universe — together with their redshifts (and hence scale factors) you build up a picture of the size of the Universe as a function of time.

What the observations of near and far sources have shown us is that the size of the Universe increased rapidly in the early Universe, slowly decreased in expansion rate, but then "recently" started speeding up again.

> **Comments:**
> **username_0**: Thank you very much @username_1 for providing a valuable justification regarding the interpretation of redshifts in terms of scale factor evolution; this definitely helps keep a check on the Universe's expansion if it has slowed down with time, or if it has sped up.  
> **username_1**: I updated my answer to take into account your edited question. This site is not the place for personal hypotheses, it's solely for "mainstream physics". Sharing a link in a comment might be okay though, I'm not exactly sure about the rules.  
> **username_0**: I am not emphasizing that a galaxy or a quasar is moving/receding through space, however, when it comes to the determination of the expansion rate, it seems that the researchers use recession velocities (corresponding value of cz) over the entire redshift domain (https://astro.ucla.edu/~wright/DL-vs-z-26Mar2015.gif). Taking values as they are, nearby supernovae accurately yield ≈70 km/s/Mpc, whereas distant ones yield ≈45 km/s/Mpc. This lower value is in agreement with deep determination of expansion rate (48 km/s/Mpc) (Kochanek & Schechter 2004); 30% lower than the local value of 70 km/s/Mpc.  
> **username_1**: $H_0$ from gravitational time delays are only starting to become competitive with SNe and CMB, so don't put too much faith into a paper from 2004 (despite the second author). I don't see where you get the 45 km/s/Mpc from?  
> **username_0**: Luminosity distance (DL) is the directly-observable distance to type Ia supernovae that shows that remote supernovae are 10% to 25% dimmer as they are further away than expected, and, since observables play an important role in cosmology, therefore, I have taken into account those directly-observable quantities here to determine the expansion rate from the plot – luminosity distance and redshift. H = cz (as per the plot)/DL. In fact, this scales perfectly with respect to the slope of the plot; nearby supernovae yield a steeper slope (higher expansion rate) as compared to the distant ones.  

---

### Answer by username_2 on 2022-01-18. Score: 1

Davis and Lineweaver correctly point out some persistent mistakes in others' descriptions of FLRW cosmology, but they make errors of their own. One of those is the idea that there is superluminal expansion in cosmology and that this has something to do with "expansion of space itself".

Recession speeds larger than $c$ appear even in the $ρ=p=Λ=0,\, a'>0$ special case of FLRW cosmology. That FLRW spacetime is isomorphic to a patch of Minkowski spacetime, and has a global speed limit of $c$ when speeds are defined in the usual special-relativistic way. But the recession speed, defined in the usual cosmological way, works out to be [rapidity](https://en.wikipedia.org/wiki/Rapidity) times $c$, and it exceeds $c$ when $z>e-1\approx 1.72$, not terribly different from the real-world ΛCDM value. Nothing special happens at this speed (which is equivalent to about $0.76c$ in $dx/dt$ speed). A Hubble sphere can be defined but has no physical meaning.

In ΛCDM the numbers are different but the principle is the same. Recession speeds are defined in such a way that $c$ has no special meaning. The Hubble sphere has no physical meaning (except that in the distant future, when $ρ\approx 0$, it coincides with the cosmological horizon).

I think that none of this is related to the evidence for accelerating expansion.

---

In the plot in your question (which was added after username_1's answer), the data points on the right are from earlier times. If the expansion is accelerating then the expansion rate at earlier times should be lower, and they are.

But that plot strikes me as misleading because it doesn't show what was actually measured; both axes depend on the cosmological parameters that they're supposed to be fitting to the data. Here's a better one, from [this page](https://astro.ucla.edu/%7Ewright/sne_cosmology.html) of Ned Wright's cosmology tutorial:

![](https://astro.ucla.edu/~wright/DL-vs-z-26Mar2015.gif)

(If you're color blind: "Closed Matter Only Model" is the topmost curve, "de Sitter Model" is the bottommost, and the other colored lines are fairly good fits to the data.)

$cz$ is the measured redshift times $c$, not the recession speed. (It's possible that the horizontal axis of your plot is also $cz$; it has two inconsistent labels.) Luminosity distance is defined as a function of relative and absolute magnitude independently of the FLRW parameters.

It's not so easy to understand in Wright's diagram why the curves should be shaped as they are, but I think that's fine because it's how science usually works. You don't see fundamental parameters in the data; you tweak fundamental parameters until the predictions of the model fit the data.

That's also the answer to your concern about time dilation. The cosmological model is based on general relativity, and calculations within general relativity automatically take time dilation into account. The term "time dilation" isn't precisely defined, and you can argue about whether it should apply here, but the predictions of the model are unambiguous.

> **Comments:**
> **username_0**: Thank you @username_2 for your answer. According to this Science News article, using SNe Ia as standard candles, researchers measured their recession velocities & distances; this helped them conclude that the Universe was expanding 10% - 15% more slowly in the past (https://www.sciencenews.org/archive/astronomy-138). The High-Z plot (illustrated in the question) is the result of these measurements. As per the plot, further-away-than-expected SNe Ia on the R.H.S. lie in the region where the expansion is slower, however, there is another aspect (other than acceleration) that can mimic this perfectly.  
> **username_0**: https://www.researchgate.net/profile/Karan-Rtakkhi/publication/343484700_Shedding_Light_on_an_Undiscovered_Aspect_that_Perfectly_Mimics_Cosmic_Acceleration/links/61d9a21fd450060816992df4/Shedding-Light-on-an-Undiscovered-Aspect-that-Perfectly-Mimics-Cosmic-Acceleration.pdf  

---

