The deviation of Universe expansion from general relativity

[consuli]

What is the deviation in the expansion of the universe exactly quantified, when I would assume general relativity and project it backwards?

As a statistician I am asking for data, for either the backwards projected general relativity case and either the real expansion case, as it is reconstructed by the observations of emissions (light, microwave, radio, ...), which traveled for millions of years to us.

And ideally this data would be already aggregated to a one dimensional figure already, for instance an acceleration measure, so that I do not need to handle with the masses and the distances of astronomic objects (galaxies, stars, ...) any more.

Consuli

 

[phyzguy]

You need to realize that such a comparison can only be done statistically. We can see the universe as it existed soon after the big bang by studying the cosmic microwave background radiation (CMB). This radiation has been mapped out in exquisite detail, and the Lambda-CDM model of cosmology, based on general relativity, fits it extremely well. The plot below shows the measured fluctuations of that radiation as compared to the Lambda-CDM model, and the fit is excellent. Ideally, we would then model those fluctuations forward and compare them to the fluctuations we measure in space around us now. The problem is that the CMB radiation reaching us was emitted by a volume which is now billions of light-years away. The light from the galaxies that formed in that volume won't reach us for many more billions of years. Similarly, the CMB radiation emitted from the space around us has now propagated billions of light-years away and is no longer accessible to us. So the best we can do is to compare the statistical properties of the fluctuations of the CMB (modeled forward in time) to the statistical properties of the density fluctuations we see around us. When we do this, we get a plot like the one attached. It looks good, but we'd like more data and smaller error bars, which is what most cosmologists are working on today.CMB measurements.

 

[consuli]

The problem is that the CMB radiation reaching us was emitted by a volume which is now billions of light-years away. The light from the galaxies that formed in that volume won't reach us for many more billions of years. (...)

I need to do a statistical summary of your points:
1. We constantly get CMB radiation observations from other galaxies.
2. For near galaxies these CMB observations belong to a short time ago.
3. For far away galaxies these CMB observations belong to long time ago.
4. Thus, CMB observations for near galaxies a long time ago and the CMB observations for far away galaxies a short time ago are missing. To be precise, we can only observe a distance-time-ago-frontier of events.

From my statistical point of view that just means these observations are a specially censored sample.

So the best we can do is to compare the statistical properties of the fluctuations of the CMB (modeled forward in time) to the statistical properties of the density fluctuations we see around us.

I do not understand this one. Why is not possible to project back in time applying general relativity (for the reason to track the deviation from it)? Especially, as in general relativity all objects obey to the same rules, thus it should be possible to conclude from far away objects to near objects and vice versa.

Consuli

 

[phyzguy]

1. We constantly get CMB radiation observations from other galaxies.
2. For near galaxies these CMB observations belong to a short time ago.
3. For far away galaxies these CMB observations belong to long time ago.
4. Thus, CMB observations for near galaxies a long time ago and the CMB observations for far away galaxies a short time ago are missing. To be precise, we can only observe a distance-time-ago-frontier of events.

No. The CMB radiation was emitted approximately 400,000 years after the Big Bang, when the universe first became transparent. There were no galaxies at that time. The radiation has then free-streamed to us from that time. It is no longer being emitted by galaxies.

Why is not possible to project back in time applying general relativity (for the reason to track the deviation from it)?

We could in principle take the distribution of matter density around us and project back to the CMB fluctuations that would have been emitted from our region of space 400,000 years after the Big Bang. But we do not have access to that CMB radiation. It is billions of light-years away. So we cannot do a comparison.

 

[kimbyd]

What is the deviation in the expansion of the universe exactly quantified, when I would assume general relativity and project it backwards?

As a statistician I am asking for data, for either the backwards projected general relativity case and either the real expansion case, as it is reconstructed by the observations of emissions (light, microwave, radio, ...), which traveled for millions of years to us.

And ideally this data would be already aggregated to a one dimensional figure already, for instance an acceleration measure, so that I do not need to handle with the masses and the distances of astronomic objects (galaxies, stars, ...) any more.

Consuli

I'm not entirely sure what you're asking for, but it's highly unlikely for there to be any deviations from General Relativity in observations of the expansion of the universe.

The issue is that the expansion is a very large-scale phenomenon with very small space-time curvature. Typically we would expect to see deviations on small scales, in regions with high space-time curvature (such as near the event horizon of a black hole).

When the acceleration of the expansion was initially observed back in the late 90s, there was a flurry of activity which attempted to see if a long-distance modification of GR could potentially explain the acceleration, but this ultimately proved to be very difficult to do in a mathematically-consistent way that didn't also break short-range gravity experiments.

Still, many teams have attempted to measure potential deviations from General Relativity on large scales, though these measurements usually focus on structures like galaxies and galaxy clusters.

 

[consuli]

No. The CMB radiation was emitted approximately 400,000 years after the Big Bang, when the universe first became transparent. There were no galaxies at that time. The radiation has then free-streamed to us from that time. It is no longer being emitted by galaxies.

I guess, this is a misunderstanding. I thought you mentioned CMB radiation to track astronomic objects (galaxies, ...) back in time. But it seems like CMB radiation is about the temperature and the masses in the early universe.

Consuli

 

[kimbyd]

I guess, this is a misunderstanding. I thought you mentioned CMB radiation to track astronomic objects (galaxies, ...) back in time. But it seems like CMB radiation is about the temperature and the masses in the early universe.

Consuli

The CMB was emitted when the universe was a nearly-uniform plasma that cooled into a gas. The temperature was, at the time, about 3000K, with typical deviations of about 0.03K in temperature from place to place on the sky.

Compact objects only formed far later, when the very slightly more dense regions in the early universe collapsed further into galaxy clusters, galaxies, globular clusters, and star systems.

 

[consuli]

I'm not entirely sure what you're asking for, but it's highly unlikely for there to be any deviations from General Relativity in observations of the expansion of the universe.

I have thought, there is no general accepted knowledge regarding the exact cosmologic inflation/ exponential expansion of the universe respectively a cosmologic constant so far. Is there?

Still, many teams have attempted to measure potential deviations from General Relativity on large scales, though these measurements usually focus on structures like galaxies and galaxy clusters.

Exactly. And there has not been found one, not even in the early stages of the universe?

Consuli

 

[kimbyd]

I have thought, there is no general accepted knowledge regarding the exact cosmologic inflation respectively a cosmologic constant so far. Is there?

It's hard to parse what precisely you intend here, as I think you're using these terms slightly incorrectly.

Cosmic inflation is one possible explanation for the behavior of the very early universe. The cosmological constant is the simplest explanation for the late-time accelerated expansion which we have observed. Both are still areas of active research, and we'd need a far more precise question to provide you with any answer.

Exactly. And there has not been found one, not even in the early stages of the universe?

There are two signals from the very early universe:
1) The CMB, which, as phyzguy stated, was emitted when our universe was a few thousand years old. If there was any deviation from General Relativity in the early universe, it isn't yet detected there. One possible avenue of research here is the detection of what is known as "primordial B-mode polarization" in the CMB. This particular polarization, if detected, would rule out some of the modified-gravity alternatives to cosmic inflation (in particular, Loop Quantum Cosmology). This polarization has not yet been detected.
2) Primordial light element abundances. The elements which initially formed in the early universe (75% hydrogen, 25% helium, and trace amounts of a few other light elements) formed within the first few minutes. Current predictions of these light element abundances match quite precisely with the theoretical predictions (predictions which assume General Relativity), though there are some minor discrepancies which could either mean that our understanding of high-energy physics needs modification or our understanding of stars is a bit off (primordial abundances are measured by observing certain populations of stars that haven't processed much of their matter). These measurements aren't likely to inform us about discrepancies from General Relativity.

Ultimately, I think you're approaching this question in the wrong way. You seem to be asking, "What is the probability that anything besides General Relativity is correct?" But that is an unanswerable question. In order to see the probability that some other model is correct, you have to, at the very least, be able to write down parameters for that model. There's no way to write down parameters for "something else" because it could refer to literally anything.

The way that advancements happen in science is typically as follows:
1) Narrow in on an unexplained observational discrepancy.
2) Come up with potential explanations for the observed discrepancy, explanations which could vary from measurement errors to simulation errors to revising our understanding of fundamental laws.
3) See if those potential explanations can be tested using independent means. The more independent ways we verify that the model matches reality, while the alternative model does not, the more confident we can become that the model is accurate.

Ultimately there's a lot of judgment calls to be made in the above. Scientists certainly try to be statistically rigorous, but it turns out that statistical inference cannot be done without making assumptions, assumptions which can significantly change the result.

With General Relativity specifically, there is no observational evidence that it is incorrect in any way. There have been some attempts to modify General Relativity to explain certain discrepancies not otherwise explained, but not one of those attempts has been confirmed by multiple independent tests.

 

[phyzguy]

I am also struggling to understand what you are asking. Many tests have been done to look for deviations of various astronomical measurements from GR. No deviations have yet been found - GR has passed every test. So our models of the universe are based on GR. We cannot watch the motions of specific objects as the universe expands because the motions are much too slow, so I tried to describe some of the statistical comparison that are done. But apparently that was not what you were asking about. Is there some specific test that you have in mind?

 

[Jean Tate]

I've two things to add, neither of which is directly related to the OP, but which I think may add some context.

1) the "CMB", is, as detected, "just" microwaves, ~the sort of electromagnetic radiation your microwave oven emits. As does dust in the Milky Way's (and Magellanic Clouds', and ...) interstellar medium; and ... There are some very cool all-sky maps from COBE's DIRBE instrument which show the detected microwaves (and mid-IR, and far-IR). The CMB signal can be extracted from these foregrounds by some wonderful techniques, including statistical ones.

There's also distortion of the CMB, as we observe it, by matter that is well and truly foreground to the CMB but also very much more distant than the Local Group; for example, scattering off hot electrons in the hot intergalactic medium of rich clusters of galaxies (the Sunyaez-Zel'dovich effect). This can be exploited to, for example, detect such clusters independently of what wide surveys (such as SDSS) find. It also creates a kind of CGRB (cosmic gamma-ray background), a distorted "mirror" of the CMB.

2) There is an equivalent CNB, Cosmic Neutrino Background; when the universe was very much hotter, denser, and younger, it suddenly became transparent to neutrinos (before then they "recombined" with nucleons, much as photons would continue to, with ions, until much later). As far as I know, there is at least one planned experiment to try to detect this, using tritium (the CNB should also show up, very indirectly, in the abundances of light, primordial nuclides).

 

[kimbyd]

1) the "CMB", is, as detected, "just" microwaves, ~the sort of electromagnetic radiation your microwave oven emits.

Sort of. The frequency used by Microwave ovens is lower at around 2.5GHz (most of the CMB radiation is between around 30GHz and 500GHz or so).

 

[consuli]

It's hard to parse what precisely you intend here, as I think you're using these terms slightly incorrectly.

That's just because I am only a bloody statistician, not a physician. ;-)

Let's keep it simple and stupid in the first step, just for the reason to understand each other.

On https://en.wikipedia.org/wiki/Metric_expansion_of_space there is a schematic picture of the expansion of the universe (which will depend on a certain theory and might be discussable to some degree).

What acceleration of an average galaxy would I get, when I would transform this expansion picture to an acceleration over time? Or, if more appropriate, to an effective gravitation of an everage galaxy (over time), or any other comparable aggregated one dimensional quantity of an average galaxy (over time)?

Consuli

 

[kimbyd]

That's just because I am only a bloody statistician, not a physician. ;-)

Let's keep it simple and stupid in the first step, just for the reason to understand each other.

On https://en.wikipedia.org/wiki/Metric_expansion_of_space there is a schematic picture of the expansion of the universe (which will depend on a certain theory and might be discussable to some degree).

What acceleration of an average galaxy would I get, when I would transform this expansion picture to an acceleration over time? Or, if more appropriate, to an effective gravitation of an everage galaxy (over time), or any other comparable aggregated one dimensional quantity of an average galaxy (over time)?

Consuli

The first question I can provide the equations for fairly easy. The second one doesn't make as much sense. The equation for acceleration is the second Friedmann equation:

$${\ddot{a} \over a} = - {4\pi G \over 3} \left( \rho + {3p \over c^2}\right) + {\Lambda c^2 \over 3}$$

Each type of density that makes up our universe evolves in a a certain way over time, and has a certain amount of pressure. To a very good approximation, the contents of the universe for most of its 14 billion year history can be reduced to only two components: matter (normal or dark) and the cosmological constant. Matter has no pressure on cosmological scales (i.e., galaxies do not place any pressure on other galaxies they aren't colliding with). We can get rid of most of the annoying constants by using the concept of density fractions, and including how the densities change over time as the universe expands:

$$\Omega_m = {8 \pi G \rho(a=1) \over 3 H(a=1)^2}$$
$$\Omega_\Lambda = {\Lambda c^2 \over 3 H(a=1)^2}$$

This is using the notation where $a=1$ at the present time. $H(a=1)$ is typically referred to as $H_0$, the current rate of expansion. Using the above notation and taking into account that $\rho_m(a) = \rho_m(a=1) / a^3$ while $\Lambda$ is just a constant, the acceleration equation becomes:

$${\ddot{a} \over a} = H_0^2\left(-{\Omega_m \over 2 a^3} + \Omega_\Lambda\right)$$

Given that the distance to a far-away galaxy is given by $d(a) = a d(a=1)$, the acceleration of the far-away galaxy is $\ddot{d} = \ddot{a} d(a=1)$.

The only remaining pieces of the puzzle are the measured values of the three constants described above. The best-fit values from the Planck 2015 data release are:

$$H_0 = 67.3 km/s/Mpc$$
$$\Omega_m = 0.315$$
$$\Omega_\Lambda = 0.685$$

Unit note: yes, the traditional units for the rate of expansion are kilometers per second per megaparsec. If you want to use mks units (meters, kilometers, seconds), its value is $2.18 \times 10^{-18}$Hz.

 

[kimbyd]

That's just because I am only a bloody statistician, not a physician. ;-)

Let's keep it simple and stupid in the first step, just for the reason to understand each other.

On https://en.wikipedia.org/wiki/Metric_expansion_of_space there is a schematic picture of the expansion of the universe (which will depend on a certain theory and might be discussable to some degree).

What acceleration of an average galaxy would I get, when I would transform this expansion picture to an acceleration over time? Or, if more appropriate, to an effective gravitation of an everage galaxy (over time), or any other comparable aggregated one dimensional quantity of an average galaxy (over time)?

Consuli

Second post for the second question because the first one was already long.

The problem is that General Relativity drastically changes the notion of what gravitation even means. An observer on a far-away galaxy will estimate that their galaxy is experiencing no gravitation at all. This is related to the equivalence principle: acceleration and gravity are equivalent, in the sense that if you're in a closed box, it's not possible to tell by any measurement at a single location in the box the difference between that box sitting on the surface of the Earth, and it out in space being pushed by some means at a constant $9.8m/s^2$ acceleration.

The "single location" caveat is important, because there will be tidal forces due to the part of the box closer to the Earth experiencing more space-time curvature.

This discussion adds up to the fact that there is no one answer for how much gravitation (or, really, acceleration) the far-away galaxy is experiencing. The acceleration is defined using a peculiar definition for distance, a definition which is actually pretty arbitrary: there are other definitions of distance used in other contexts that are every bit as valid, and those definitions will change the answer for what the acceleration is. Despite the math I showed above, there is no one true answer for how much acceleration there is.

All that said, the way the expansion over time is measured, which is what I think you were trying to ask, is to compare the relationship between redshift and distance over a range of redshifts and using a variety of types of ways of measuring distance. Redshift is a measure of how much the universe has expanded: the expansion of the universe impacts the wavelengths of light in the same exact way that it impacts the distances between galaxies. Twice the distance between galaxies means twice the wavelength. So light that is observed with a redshift of $z=1$, which has twice the wavelength now as when it was emitted (redshift is defined using $\lambda = (z+1)\lambda_0$), was emitted at a time when galaxies were, on average, half the distance from one another as they are today.

Redshift is easy to measure very accurately using spectroscopy (by matching emission or absorption lines for various gases). That is then correlated with a measure of distance. As I state above, there are actually multiple possible ways to define distance. David Hogg put together a good summary of the main ones used in cosmology here. So for Type 1a supernovae, which all have similar brightnesses at their source, we would compare the luminosity distance (as defined in that paper) for many hundreds of supernovae which occurred at a variety of different redshifts.

The typical process then used is to use some kind of maximum-likelihood method to determine the best-fit model for that list of distances and redshifts, or, more commonly, to use Markov Chain Monte Carlo to estimate the full probability distribution for a set of parameters in a model given the data.

 

[consuli]

Thanks a lot.

Sorry for the little "physicians" language fallout. In former latin physics refers to examination of anorganic objects and pyhsiology refers to the examination of organic objects. Thus, calling medcial doctors physicians is very strange. Maybe you have already noticed this irregularity in the english language yourself. mathematics -> mathematician, statistics -> statistician, physics -> physicist.

You have mentioned H0=67.3km/s/Mpc for the present Hubble "constant" derived from billion years old CMB radiation and some model.

However the true Hubble "constant" is H0=74 km/s/Mps derived from present observations of galaxies and stars. Compare: Scientists Still Don't Know How Fast The Universe Is Expanding.

If you tell me, the Hubble "constant" is the appropriate metric for the expansion of the universe, I would want to model the expansion of the universe represented by the Hubble "constant" back in time using general relativity only. Which means starting with H0=74 km/s/Mps.

As the expansion of the universe has not been constant at all, I still have doubt, that a quantity called constant is the appropriate metric to do that.

Consuli

 

[phyzguy]

Thanks a lot.
Sorry for the little "physicians" language fallout. In former latin physics refers to examination of anorganic objects and pyhsiology refers to the examination of organic objects. Thus, calling medcial doctors physicians is very strange. Maybe you have already noticed this irregularity in the english language yourself. mathematics -> mathematician, statistics -> statistician, physics -> physicist.
You have mentioned H0=67.3km/s/Mpc for the present Hubble "constant" derived from billion years old CMB radiation and some model.
However the true Hubble "constant" is H0=74 km/s/Mps derived from present observations of galaxies and stars. Compare: Scientists Still Don't Know How Fast The Universe Is Expanding.
If you tell me, the Hubble "constant" is the appropriate metric for the expansion of the universe, I would want to model the expansion of the universe represented by the Hubble "constant" back in time using general relativity only. Which means starting with H0=74 km/s/Mps.
As the expansion of the universe has not been constant at all, I still have doubt, that a quantity called constant is the appropriate metric to do that.
Consuli

There are a large number of cosmologists working to understand whether the discrepancy in H0 derived from the CMB and that derived from local measurements is a real discrepancy that would implay a need for new physics or whether it simply represents error in the measurements. As of today, nobody knows. Anyone who works in this field is aware of this issue. Did you have something to add, or did you have a question?

 

[timmdeeg]

If you tell me, the Hubble "constant" is the appropriate metric for the expansion of the universe, I would want to model the expansion of the universe represented by the Hubble "constant" back in time using general relativity only. Which means starting with H0=74 km/s/Mps.

As the expansion of the universe has not been constant at all, I still have doubt, that a quantity called constant is the appropriate metric to do that.

Indeed, $H$ is often called Hubble Parameter because it is not constant over time. $H$ is decreasing and will approach a constant value in the very far future when the energy density of the universe consists of dark energy only (the matter density will be negligible then).
The appropriate physical quantity which determines the dynamics of the universe is given by the time dependence of the scale factor. You might have a look at the Friedmann equations to verify that.

 

[consuli]

Indeed, $H$ is often called Hubble Parameter because it is not constant over time. $H$ is decreasing and will approach a constant value in the very far future when the energy density of the universe consists of dark energy only (the matter density will be negligible then).
The appropriate physical quantity which determines the dynamics of the universe is given by the time dependence of the scale factor. You might have a look at the Friedmann equations to verify that.

Brilliant.

Where can I find data for the scale factor a ( t ) over time for different theories and ideally also for a plain vanilla back projection of the universe applying general relativity, only?

And I need the same data over time for the Hubble parameter then, as it is fully constant.

Thanks
Consuli

 

[timmdeeg]

Where can I find data for the scale factor a ( t ) over time for different theories and ideally also for a plain vanilla back projection of the universe applying general relativity, only?

And I need the same data over time for the Hubble parameter then, as it is fully constant.

It seems advisable that you make yourself familiar with the https://en.wikipedia.org/wiki/Friedmann_equations. To obtain $H$ "as it is fully constant" look at the first equation and set the matter density $\rho$ and the curvature parameter $k$ to zero (as the universe is spatially flat).

 

[phyzguy]

Brilliant.
Where can I find data for the scale factor a ( t ) over time for different theories and ideally also for a plain vanilla back projection of the universe applying general relativity, only?
And I need the same data over time for the Hubble parameter then, as it is fully constant.
Thanks
Consuli

You will not find "data" for these quantities, as they are not directly observable. The available data includes things like the spectrum of the CMB radiation as a function of direction, and the directions and redshifts of galaxies. You can find how the Hubble parameter and the scale factor vary with time in different GR-based models by studying the solutions to the Friedmann equations, as timedeeg has suggested.

 

[kimbyd]

Thanks a lot.

Sorry for the little "physicians" language fallout. In former latin physics refers to examination of anorganic objects and pyhsiology refers to the examination of organic objects. Thus, calling medcial doctors physicians is very strange. Maybe you have already noticed this irregularity in the english language yourself. mathematics -> mathematician, statistics -> statistician, physics -> physicist.

You have mentioned H0=67.3km/s/Mpc for the present Hubble "constant" derived from billion years old CMB radiation and some model.

However the true Hubble "constant" is H0=74 km/s/Mps derived from present observations of galaxies and stars. Compare: Scientists Still Don't Know How Fast The Universe Is Expanding.

If you tell me, the Hubble "constant" is the appropriate metric for the expansion of the universe, I would want to model the expansion of the universe represented by the Hubble "constant" back in time using general relativity only. Which means starting with H0=74 km/s/Mps.

As the expansion of the universe has not been constant at all, I still have doubt, that a quantity called constant is the appropriate metric to do that.

Consuli

There are a large number of measurements of the rate of expansion, and quoting one as "true" is simply misleading. Each measurement has errors on it, and the potential for those errors to be underestimated due to systematic effects that weren't properly accounted for.

I actually trust the CMB measurements to be more accurate in this case, because the CMB is a much cleaner signal with fewer systematic effects. It may seem counter-intuitive that a signal emitted so long ago might be a better measurement than nearby galaxies, but the rate of expansion is closely tied to other parameters measured far more accurately by the CMB, and the CMB really is that clean of a signal. Still, that might be my own bias showing through. It's very possible that the discrepancy between nearby measurements and the CMB indicates that we're missing something important in our theories.

 

[PeterDonis]

The OP question has been sufficiently addressed. Thread closed.