Relativity and Cosmology

It is well-known fact that the one-way speed of light is not a measurable parameter, because it is actually defined to be the same as the two-way speed of light in all inertial frames. Put differently, we define the synchronization of two clocks (that are not co-located, but stationary relative to each other) in such a way that the one-way speed of light comes out the same as its two-way speed. Hence to measure the one-way speed of light using such clocks does not make much sense. The outcome is a given before the test.

However, people have not stopped trying to see if it could be done without synchronized clocks. All 'official' efforts have so far been refuted, proving that they are in fact two-way measurements. Physicist Don Lincoln of Fermi-lab has told me about a method they use that is (almost?) an irrefutable one-way test.

Here is a quote from his post on another Blog. The sketch is my own doing, upper is cable calibration and lower is laser pulse speed test:

[quote=Don Lincoln]

Take two photon detectors. These can be arbitrarily thin - less than a millimeter if necessary. Take the two detectors and place them side by side. From each detector take a cable of a convenient length. Put both of those cables into fast electronics (a modern digital oscilloscope will work just fine).

Fire a light pulse through both detectors. Since these two detectors are adjacent to one another, the transit time from one to the other is of order (1 mm)/(speed of light) = (1 x 10^-3 m)/(3 x 10⁸ m/s) = 3 x 10^-12 seconds. If sub- 3 picosecond speed is needed, there are ways.

Using your oscilloscope, you can calibrate your cables to establish what "simultaneous" means. In the abstract, the cables can be of identical length. This means that the signals from the two detectors will arrive simultaneously at your oscilloscope.

Now move one detector far away...maybe 1000 feet. Do not disconnect the cables, so you have identical conditions. Fire the light pulse (use a laser) through one detector to hit the other. The signals from the two detectors will transit the cables and hit your oscilloscope at a single spatial point. Since you have already established that the transit time in the cables of both detectors are identical, the only difference between the signal arrival time at your detector is the transit time of light from one to the other. If you have measured the distance exactly, you can then determine the speed of light by distance over time.

If you do not want to measure the distance between the two detectors, you can verify the isotropy of space (and consequently, the identical nature of the 1-way speed of light). First do as I said, and fire a laser that first hits detector 1 and then hits detector 2. Record the transit time seen in your oscilloscope. Now have a laser pointing in the opposite direction, hitting detector 2 and then detector 1. Again, record the transit time.

Since the distances are the same, and the only difference is the direction in which the light is travelling, you can establish that light going one way takes the same speed as the other way. I believe that within the uncertainties of your equipment, this detector configuration will establish that the speed of light is the same in either direction."

[/quote]

What do you think? Did Don measure the one-way speed of light, or has he got a 'hidden two-way' assumption in there somewhere?

-J

The thinking around cosmological models has undergone a shift during the last two decades. My prior blog post hinted at the importance of the Hubble radius and the 'emergent spacetime' view that is implied.

Models seem to have moved from the Einstein-de Sitter (matter dominated) view, through the ΛCDM (dark energy dominated) decade to the present 'emergent spacetime' views. The ΛCDM model can actually handle all the views, simply by setting its parameters appropriately. My old Cosmo Calculator^[1] does exactly that and spews out numerous calculated results for a given set of inputs - actually enough information to confuse everyone, except perhaps the experts.

In line with the new thinking, I have teamed up with Marcus from PhysicsForums^[2] to develop a more intuitively simple calculator that may be more appealing at beginners level, while still conforming to the concordant cosmic model of today. It is called CosmoLean and is presently at version A25. 'Lean', because it does not flood the user with results. The main output is a table giving the evolution of some useful parameters over time. Before you dig into it, please read on to get some perspective on it first.

CosmoLean starts with a premise that we can forget about dark energy if we accept that there exists an underlying constant spacetime curvature, but that space is perfectly flat.^[3] Without going into the "why is this so?" type questions, let me just say that it is perfectly in line with all present observations and paints a picture that is compatible with, but conceptually simpler than the dark energy paradigm. In effect it says that from Einstein's 1915 papers^[4] onwards, the laws of gravity had two constants: the local scale gravitational constant (G) and the large scale cosmological constant (Λ), both representing a curvature of spacetime, as applicable to their respective regimes.

Without going into this formidable equation, let's move on to what we need for a new calculator. As in my prior Blog, what we observe today is a (sort of) transient to a future constant Hubble radius (R_H), which only depends on the cosmological constant Λ. How close we are to this future constant (say R_{H_inf}) depends on how much the radiation and matter energies have been diluted by the cosmic expansion. We can completely specify this by three parameters, R_{H_now}, R_{H_inf} and the ratio between matter and radiation energy density. For the latter we can choose the redshift at which they were equal (z_eq) in the past.

R_{H_now} = cT_{H_now} , the speed of light multiplied by the Hubble time. This again is inversely proportional to the Hubble constant (H₀). If we choose units appropriately,^[5] then c=1 and T_H = 1/H₀, which has the value 13.9 Gy, according to today's best observations. The future value R_{H_inf} =16.3 Gly and the past value z_eq ~ 3500 have both been derived from those observations.

To make the cosmological equations^[6] as readable as possible, we have decided to use the symbol Y = T_H = c/R_H for Hubble time. This gets rid of one extra level of subscript. We also use "stretch factor" (S) in place of the usual redshift z, where S = z + 1, a factor that crops up all over the show. Stretch is simply the factor by which wavelengths have increased from the time that light has left an observed source, e.g. a stretch S=2 means wavelengths have been doubled by the cosmic expansion while the photons were in transit.

For the main calculator inputs, we simply have to tell it the values of Ynow, Yinf and S_eq. We may then also specify the output in tabular form in terms of S_upper, S_lower and the output steps in-between, either as step size, or as the number of steps (see the info-tooltip of the live user interface).

To make a connection back to the "old ways" of specifying inputs for cosmo-calculators, we also compute the conventional values and show them at the top-right. The page is pre-populated with default values and there are ample info-tooltips to make it (hopefully) easy to use. So, without further delay, please give it a try and tell us whether you think it is cool, whether it sucks, or anything in-between. It is a work in process, so you may actually still influence the tool.

Some further usage tips will follow...

Click here and take a cosmic dive: TabCosmo6.

Usage tips: see comments #3, #5 below.

-J

[1] http://cr4.globalspec.com/blogentry/20218/Cosmological-Calculator-Update

[2] http://www.physicsforums.com/forumdisplay.php?f=69

[3] It is presently thought that the universe is either spatially flat (or very near to flat), i.e. parallel lines 'here' are still parallel 'there', when considered on a large scale and viewed everywhere at the same cosmic time. However, due to the expansion, light rays that are sent out to be parallel, will not remain parallel over time; hence, spacetime is curved.

[4] Einstein (1915), "Die Feldgleichungen der Gravitation (The Field Equations of Gravitation)"

[5] The conventional Hubble constant is given in units Km/s/Mpc, i.e., a recession speed per distance. With years for time and light-years for distance, speed becomes dimensionless and the speed of light is 1. If we also convert Mega-parsec to billion light years, we get that the present H₀ = 70.36 km/s/Mpc becomes 70.36/978 = 0.0712 Gly^-1. Now if we invert that, we get R_{H_now}= 13.9 Gly and T_{H_now}= 13.9 Gy.

[6] Here are the simplified equations of CosmoLean, for those who can't live without them.

Our presently observable universe may be no more than a brief transient between two more dominant static states, a recent (quite credible) paper suggests.^[1] Granted, the term 'static state' has been redefined a little, but the author uses very convincing arguments for his ideas.

The first static phase is what we normally call 'inflation'. But how can inflationary expansion be static? The trick is to define 'static space' as a constant Hubble sphere radius (the cosmological event horizon), rather than as a constant proper radius.^[2] Parts of the universe that fall (or are shifted) outside this radius cannot have any effect on us whatsoever, not even gravitationally.

The standard inflation period had a very high, but constant expansion rate (H) and hence a small constant Hubble radius R_H = c/H, somewhere near the Planck scale (t~10^-43 seconds, r~10^-43 light-seconds). This (presumed) part of our cosmic history is not all that well understood in present gravitational theory^[3] - hence the quest to find a consistent theory of quantum gravity.

At around 10¹¹ Planck time units (10^-32 seconds), some form of symmetry breaking occurred, causing a gradual lowering in the expansion rate (H) and thus an increase in the Hubble sphere radius. More space became observable and could influence larger volumes - it is described as the emergence of space. Present observations suggest that we are in this 'emergent phase', but that the Hubble radius is now asymptotically approaching a constant value again.

The Log-Log plot (above)^[4] shows the proper radius curve of our observable universe in grey and the Hubble radius in red, both against time. Note the dramatic increase in proper radius during the inflationary epoch up to 10^-32 seconds, yet the Hubble radius stayed near the Planck scale. After the symmetry breaking, radiation energy decelerated the very rapid expansion rate until around 300,000 years, when radiation energy ran out of steam, being red-shifted out of contention. Matter energy took over as decelerating agent until around 10 billion years, when vacuum (dark) energy started another (milder) inflation epoch. This resulted in our present constant Hubble radius phase.^[5]

The paper goes further and shows that both spacetime and gravity are phenomena that emerge from underlying (poorly understood) degrees of freedom, hidden in the mysterious quantum-gravity 'underworld'. It is pictured in broad terms on the right. For a better resolution graphic, check the main link in end note [6].

The circle represents the Hubble sphere in 2-D. The 'inside' area is called the 'bulk', essentially space that has already emerged. N is the number of degrees of freedom (independent variables) represented in that space - N is presently different for the bulk and the surface.^[7] As matter density gets diluted with the increase in the Hubble radius, while vacuum energy density remains constant, N_sur will eventually equal B_bulk and the Hubble radius will become constant again.

As far as I understand, the mechanisms are similar to thermodynamics, where e.g. temperature and pressure are emergent phenomena that are useful without having to know the underlying decrees of freedom (e.g. molecular and atomic dynamics). Today we think we know the underlying facts for matter, but we do not understand the 'atoms' of spacetime yet. However, like with thermodynamics, we can quite confidently use the emergent spacetime phenomena, as long as we get results that agree with observations.

Amazingly, thermodynamics features strongly in present quantum gravity studies. The referenced paper shows how the entropy (hence the temperature) of flat spacetime can be determined, even in the presence of radiation and/or matter. Theorists say that gravity emerges from this entropy by means of quantum entanglement across any event horizon.^[8]

The bottom line seems to be that nature dislikes an imbalance between degrees of freedom of the bulk and the Hubble surface, a situation essentially caused by the presence of radiation and matter. By creating more space inside the Hubble surface, the influence of these irritations will effectively be diluted away (energy density becomes smaller). Once this 'transient' is over and N_sur= N_bulk again, the vacuum may perhaps rule undisturbed forever.

One may perhaps ask: is this of any importance and if so, what is the use of it all? Well, since we do not really understand gravity, dark energy etc., any credible new angle on them should be taken seriously, I think.

-J

[1] T. Padmanabhan: Emergent perspective of Gravity and Dark Energy, http://arxiv.org/abs/1207.0505. There were some earlier works in this direction; see e.g. http://en.wikipedia.org/wiki/Entropic_gravity. It does however appear as if Padmanabhan expanded the ideas quite a bit further than previous treatments. His section 5 (page 28) illustrates the newer (AFAIK) insights and is IMO quite brilliant.

[2] The proper radius/distance is what one would measure if the expansion could have been stopped instantaneously. During inflation, proper distances between locations increase and less space falls within the constant Hubble radius. After inflation, the Hubble radius started to increase at a faster rate than the expansion and some of the space that was 'lost' during inflation emerged again (inside the horizon).

[3] There are theories that avoid inflation altogether, e.g. brane cosmology, de Sitter expansion, etc. None of them are without its own set of problems, so cosmologists tend to stick to the one making the fewest assumptions (Occam's razor), although they do not fully understand inflation theory either. Note that even in inflation theory, time did not necessarily start at 10^-43 seconds; it is customary to start plots at the smallest time with any meaning, but the constant radius could have lasted for an arbitrarily long time. It simply enters as a time constant into the equations.

[4] Log-log graphic adapted from fig. 15.4 of Relativity-4-Engineers. You can read chapter 15 'Inflation', linked from here. Note that on log-log diagrams, curved lines represent an exponential law and straight lines represent a power-law of some sorts, not as linear law. The sharp changes in log-log slopes represent gradual changes in slope (change in expansion law).

[5] The present Hubble constant, H~70 km/s/Mpc (giving R_H~14 Gly) is still mildly decreasing, but it is already quite close to the final constant rate of H~60 km/s/Mpc (R_H~16 Gly).

[6] A much clearer picture is available in the source, Figure 1, page 19 of the pdf from http://arxiv.org/abs/1207.0505, also linked to in [1] above.

[7] This has to do with the way vacuum energy works - both pushing and pulling on the expansion rate. Radiation and matter only 'pulls' gravitationally and this causes the imbalance, which can only be restored by diluting the effects of their gravity through increase in the Hubble radius.

[8] This is similar to the gravity of a black hole, which emerges from the 'hidden inside', but can be observed from outside the surface (event horizon).

-=-

I have often referred to the Cosmo-Calculator in the past. Here is a Blog post dedicated to this ''workhorse''. The picture below is a cropped screenshot, so rather look at the real-thing under the link.

You may wonder: why an update? Is there something new?

Not really, but there are a few extra bits and pieces of information in the updated calculator that may be of interest to some readers. The change was motivated by a comment (on another Blog) that enquired about using the standard Special Relativity (SR) Doppler shift to explain cosmological redshift. My immediate reaction was that it is invalid because SR only applies in flat (Minkowski) spacetime. The cosmos is in all probability 'flat' in term of space, but not in terms of spacetime. The simple reason is that there are gravity and vacuum energy (possibly) operating on the cosmic scales.

Then I remembered that in my eBook Relativity-4-Engineers, I used exactly that: the SR formula for Doppler shift as related to cosmic redshift. In chapter 16,^[1] I noted

The question is: why is this a good approximation of a cosmological recession speed? My best answer is that it is partly due to a coordinate system choice and partly due to the quasi-flatness of the present values of the ΛCDM cosmic model.

The solid curve represents ΛCDM and the dotted line is for a simple linear expansion law, where a = t/T_Hubble, with T_Hubble the inverse of the Hubble constant H₀ (in suitable units). From the recession speed v/c, one gets distance D, due to Hubble's Law: v/c = H₀D.

H₀ is normally expressed in units of km/s per Megaparsec (Mpc),^[2] i.e. recession speed per unit distance. Distances in Mpc normally means co-moving or 'proper' cosmological distances, telling us how far the source is from us today, assuming that one could hypothetically stop expansion completely and then see how long light would take to traverse the distance (or measure it with our standard meter sticks).

However, in the SR-Doppler-redshift interpretation, that distance is look-back distance, derived directly from how long it took light to travel the distance in an expanding space. It usually has the units light-year, for obvious reasons. It is simply a different coordinate system than the usual cosmological choice, but there is nothing wrong with it.

In a way, this SR interpretation is easier to comprehend than the usual co-moving interpretation. The SR value is usually qualified with the term apparent (i.e. apparent recession speed/velocity). The latter is just referred to as recession speed or sometimes proper recession speed, due to its connection with proper distance. At the risk of confusion, I will measure both distance types in million light-years (Mly), but will name them appropriately.

Now back to the cosmo-calculator, as pictured above, or under the hotlink provided. The relationship between proper recession speed and apparent (SR) recession speed is immediately visible. For z=1088 (CMB redshift, the farthest we can optically observe), the present proper distance is 45,918 - i.e. the region of early space were the radiation originated, is presently that far from us. Its proper recession speed 'now' is 3.3c. The proper recession speed 'then' was about 65c. The present look-back distance is 13,757 Mly, i.e. less that 0.4 Mly from the theoretical look-back distance of the BB. The apparent recession speed of the region is 0.999998 c.

As I said above, the latter is perhaps the more comprehensible value. The more distant the source (higher z), the closer to c the apparent recession speed becomes, yet it never reaches c. The proper recession speed 'now' levels off at around 3.4c, as we reach the horizon of the observable universe. However, the proper recession speed 'then' increases without limit against z,^[3] according to the standard ΛCDM model.

To wrap up the output, I have added the average temperatures of the CMB 'now' and 'then', which are simply related by a factor 1/a = (1+z). You can check by inputting a redshift of 9 and note that the temperature 'then' is 10 times larger than 'now'.

Any feedback would be appreciated.

-J

[1] You can read chapter 16 here; you will find the complete Relativity-4-Engineers here.

[2] One Mpc is about 3.26 light-year. See Wiki for more definition on cosmic distances.

[3]A successful quantum gravity theory may avoid these infinities - and it may imply a 'bounce' rather than a 'bang'.

Dutchman Willem de Sitter developed an expanding cosmological solution^[1] to Einstein's field equations (EFEs) in the same year that Einstein made his "biggest blunder" - 1917.

The "blunder" was adding a cosmological constant (Λ) to the EFE's in order to keep his cosmos from collapsing or expanding. On the other hand, de Sitter's expanding model sported only Λ and nothing else - no matter, no radiation, or at least the latter two had no effect on the expansion dynamics. What's more, de Sitter's model was a rigorous solution to the EFEs.

Although Einstein could not find any mathematical error in de Sitter's work, he argued that it could not possibly have any physical meaning and must hence be just an artifact of the math. After all, the real cosmos had both matter and radiation energy and it was static, as far as observations went at the time. Further, his general theory was all about matter curving spacetime and how curved spacetime affects the movements of radiation and matter.

When Hubble discovered that the universe was indeed expanding, Einstein dropped Λ from the EFEs, but the fact that his own theory did allow a solution with zero matter remained a puzzle. This was 'resolved' by the fact that Friedman and Lemaitre had already found other workable expanding solutions to the EFEs, which did not only allow radiation and matter to enter the model, but they did not need Λ at all.

The de Sitter cosmic model was all but forgotten, until in the 1980s, when cosmologists attempted to solve the 'flatness' and the 'horizon' problems of the standard model that cropped up out of observations.^[2] Alan Guth's inflation theory solved both problems and it was found that it is compatible with de Sitter cosmology, with Λ of course. When accelerating expansion was discovered in the late 1990s, it was soon realized that as time goes on, the cosmos may be heading more and more towards a de Sitter type expansion. The de Sitter model was then taken very seriously, because it seemed to describe the 'opening' and the 'end games' pretty adequately. (Picture on right from http://www.science20.com/hammock_physicist/geometry_big_bang-90461. See final paragraph below.)

But what about the 'middle game', which is all we can really observe? We surely see a lot of radiation, especially in the early mid-game and we observe a lot of matter. Well, maybe not all that much, because today the radiation energy density is negligible and the normal matter density is about 4% of what is needed for the 'flatness' we observe. The rest is dark matter and dark energy - unobserved, with only "circumstantial evidence".

Since our observational accuracy on things like large-scale distances (and hence the Hubble constant) does not really make the ±4% mark, it makes one uncomfortable, to say the least. Granted, there are many corroborating pieces of evidence that point towards the 'best-buy' values used by cosmologists today. But, they are invariably all interpreted along the lines of the currently preferred Lambda Cold Dark Matter (ΛCDM) model. What if they would be interpreted using the de Sitter model?

To do such an interpretation is beyond my knowledge, but fortunately there are more capable people around, e.g. the Blog from where I ripped the picture above. Johannes Koelman did an excellent job of explaining the science at the heart of the present thinking in an accessible way. I suggest you give it a read and if you feel intimidated by the heavyweights on that Blog, you are welcome to comment here.

-J

[1] http://en.wikipedia.org/wiki/De_Sitter_universe

[2] The 'flatness problem' centers on the question: why is space (not spacetime) of the cosmos appearing to be so near flat on the large scale? The 'horizon problem' is about answering the difficult question: why do we observe the CMB to have virtually the same temperature everywhere we look?

"25 May 2012, Amsterdam, the Netherlands - The Members of the SKA Organisation today agreed on a dual site solution for the Square Kilometre Array telescope, a crucial step towards building the world's largest and most sensitive radio telescope.

The ASKAP and MeerKAT precursor dishes will be incorporated into Phase I of the SKA which will deliver more science and will maximise on investments already made by both Australia and South Africa."

This agreement was reached by the Members of the SKA Organization who did not bid to host the SKA (Canada, China, Italy, the Netherlands and the United Kingdom). The Office of the SKA Organization (in the UK) will now lead a detailed definition period to clarify the implementation.

Photo: KAT-7 precursor array (7 dish), operational in South Africa.^[1]

So what is the SKA supposed to be?

In brief it is a system of three different antenna types (frequency bands), packed into clusters, which are then spread out over a considerable area. The total RF collecting area is planned to be about a square km, with the original maximum baseline about 3000 km. With the decision to split the system between Africa and Australia, the baseline could now potentially increase to some 9000 km, at least for some observations.

The preliminary frequency ranges are: 70 to 200 MHz (low), 200 to 450 Mhz (mid) and 0.45 to 10 GHz (dishes)^[2]. These ranges are nothing new in radio astronomy, but the resolution and baseline are ground-breaking for the frequency range.

What do we hope to achieve?

For me the most exiting prospect is that the period between 300 thousand years and 500 million years after T0 may become resolvable. This is from the horizon of the observable cosmos to the formation of the first stars and galaxies, an epoch where many new insights may lurk.

The improved performance will obviously refine many previous observations and hence make it possible to constrain (or even rule out) various models. Dark matter, dark energy, cosmic magnetism and gravitational waves (GWs)^[3] will be under close scrutiny.

Some speculative potential findings include extra terrestrial life and intelligence. We may perhaps find ET phoning us, here at home...

-J

[1] http://www.ska.ac.za/ ; http://www.skatelescope.org/

[2] 0.45 to 2 GHz in phase 1, up to 10 GHz or more in phase 2.

[3] The SKA will not directly detect GWs, but will be able to find and observe binary pulsars more accurately, thus enabling tests of General Relativity's prediction of the loss of orbital energy to higher precision. In some sense, the SKA plus a binary pulsar is a gigantic GW detector.

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