We Don’t Know 96% of Our Universe-2: Darth Vader Explains Dark Energy (confronted by Bane and Batman) [Audio Available]

(This article would not have been possible without help from a lot of people. Author would like to thank hardworking Cornell undergrads Benjamin Roberts, Elizabeth Donoway and Jesse Hoke for pointing out inconsistencies in first drafts.  Author is also indebted to his dear cosmologist friend Cornell grad Soumyajit Bose for resolving all the doubts and sitting down and working out all the tiny details mathematically, despite being very busy with his exams and otherwise busy schedule. As usual for an audio version, please go to the bottom of the article.)

Hello Earthlings,

We meet again! I hope you have been thinking about the importance of the Dark Matter in stabilizing the universe and making all existence possible. Today I am here to reveal the secrets of the life of the Universe itself. How the Universe was born? Is the Universe going

Ignorant as you humans are, many of you used to think that the Universe is static. A place which always existed and will keep existing forever.

About 100 years ago, Astronomer Edwin Hubble using a 100 inch telescope measured the speed of galaxies and proved that they are all moving away from each other. Further the galaxy, faster it is moving away from its neighbors.


(Image Source: https://map.gsfc.nasa.gov/universe/bb_tests_exp.html)

[Fig.1 Edwin Hubble and the telescope he used to measure the speed of galaxies. The plot shows the velocity of galaxies in km/s plotted against the distance in Megaparsec = 30000000000000000000 km (~3E10+19 km)! It was thought that the velocity v is linear in distance r as v=Hr. But with later measurements, the ‘Hubble Constant’ was found to be varying as will be explained later]

Since everything is moving away from each other, if you turn the clock back, everything must be all together at a single point, at the beginning of the time! The whole Universe, the space-time exploded into existence from a huge explosion, the Big Bang.

Where and when did the Big Bang happen, you ask? It happened everywhere, about 13 billion years ago! There was no space before the Big Bang. There is no center of this expansion. As the Universe expanded, things cooled down and gravitational force clumped matter together to form galaxies, stars, planets like your earth and eventually the life, like you humans to observe it all!

The expansion of the Universe can be thought of as a rising loaf of raisin bread. The raisins in the bread are galaxies and the loaf is the Universe. The universe expands and as a result, galaxies move away from each other like raisins in a rising raisin-bread.


(Image Source: https://map.gsfc.nasa.gov/universe/bb_tests_exp.html)

[Fig.2 Image showing the expanding Universe as a loaf of raisin bread. Raisins are galaxies in Universe moving away from each other as the bread, the Universe expands]

Now imagine throwing a ball upwards in the sky.  It should slow down, stop and eventually fall back because of gravitational pull of earth. Similarly the expanding Universe, due to the gravitational pull of the matter inside it, should slow down, stop and eventually collapse, ending in the Big Crunch, the opposite of the Big Bang.

Is Big Crunch the fate of Universe? Is it going to die?

In 1998, a team of scientists found a result which changed our present understanding of the Universe. They studied supernovae, dying stars which explode like powerful thermonuclear bombs and emit huge amount of light. In the weeks, star explodes, it may shine brighter than the whole galaxy and act as a standard candle for us to measure the distance. Its like having a known 100W bulb, further it is, weaker its brightness. So from its brightness, you can figure out its distance.


[Fig.3 The light curve for the brightness of multiple supernovae. You can see that different supernovae, denoted by different colored dots have same light curve: a standard candle, whose brightness can be considered fixed.]

However, bear in mind that the Universe is expanding. Hence when comparing distances on the scale of the Universe, one should filter out the effect of expansion. [see Appendix, for an explanation by Soumyajit in the language of Physics, Maths!] This can be done by setting the scale factor of the Universe to be unity today, and talk about distances today. You may ask, but what about tomorrow ? Would the distance not change due to expansion ?  Your tiny human life-span is nothing compared to the cosmic time scales, so it introduces negligible error.

You know that the Universe is expanding, thanks to Hubble and others. When light reaches us from such far far away supernovae, the light gets stretched due to the expansion of Universe. This stretching of light can be measured and is called the redshift. Hence measuring the redshift of a supernova can tell us how much the Universe has expanded in the time it took for the light to reach us from that supernova. The brightness of supernova, a standard candle, is a measure of its distance today. The distance today is a net effect of expansion in the past. If the expansion of the Universe is accelerating today, then it was slower in the past when the light started traveling from the supernova. Hence from past time to present, to accumulate given redshift with slower expansion, the distance today of the supernova must be more than in the case of decelerated expansion. In the case of a decelerating expansion today, expansion was faster when the light left the supernova. Hence to accumulate the same redshift, the distance today of the supernova must be less than in the case of accelerating expansion. [see Appendix, for an explanation by Soumyajit in the language of Physics, Maths!]

The data taken by teams of scientists shows that the supernovae for high redshifts are much weaker in brightness than they should be for a slowing down Universe. Hence the conclusion is inescapable: The expansion of the Universe is accelerating!


(Image Source: Saul Perlmutter, Physics Today, April 2003, p53)

[Fig.4 A plot showing the cosmic history for the different models of the Universe. Notice that for “Today” we set the brightness of supernovae to be 1, the redshift (expansion) to be 0 and the linear scale of the Universe or its size to be 1. As we get light from distant supernovae, surely light took time to reach us, hence they happened in past. Further they are in past, fainter they are. The bluer (redder) curves represent an accelerating (decelerating) Universe. The brightness of Supernovae for certain redshifts clearly shows that they lie on the blue side of the plot, hence the expansion of the Universe is accelerating.]

Now you know that the Universe is expanding, and expanding faster and faster as time goes on defying the pull of gravity of the matter of the Universe. What causes this accelerated expansion? The Dark Energy!

No one currently knows what it is! But there are speculations, some better than others.

Einstein’s theory of gravity allows for energy to be associated with space. This is called a cosmological constant, though it may or may not be a constant as our understanding develops. As space rolls out in expansion, the associated energy, due to cosmological constant also keeps growing and the Universe keeps on expanding at a faster and faster rate. Some scientists have proposed an energy fluid called quintessence, which keeps powering up this expansion.

It is the Dark Energy, which will save the Universe from the fate of the Big Crunch and instead, eventually all the matter will evaporate and ionize into its elementary particles like electrons, protons and neutrons. The Universe will end in a Big Rip!



Bane: “Hold your ship,  Alien! The Dark Energy is yet not well understood. If it changes its behavior in future, the Universe will end up in The Big Crunch, and from its

ashes, a new Universe will begin. The Big Bang will follow the Big Crunch! The Big Crunch has not been ruled out conclusively from the data. “


(Image Source: http://science.howstuffworks.com/dictionary/astronomy-terms/big-crunch6.htm)

[Fig.5 This is what Bane is talking about, some theories predict a Big Bounce: Big Bang, expansion, Big Crunch which leads to a singularity and then again Big Bang ]

[Suddenly a lot of bats start appearing]

Lord Vader: “You puny humans don’t understand the way of the Dark Energy…..what are these black small flying ships, are they drones?”

Bane: “Aah….The Batman.  Never lets dark villains enjoy a fun conversation.  Be glad Vader, that this bat does not have a spaceship, else the villains in the whole Universe would be in constant depression!”


Batman: “Will you stop predicting doomsday and stop creating panic? The Universe, does not care about your beliefs. It runs on laws of Physics. Science is always humble and as new facts and theories emerge, the understanding changes. The theory of gravity is not complete, it does not work at atomic scales. So our best theories, the quantum theory at atomic scales and the gravitational theory at larger scales are not compatible. We need to be hopeful to work hard and understand this puzzle, we don’t need your doomsday predictions! It is a work in progress and importance should be at more data collection from observations and more work in theory, rather than armchair speculations!”

[We don’t know what happened next. Before things heated up, we ran away with whatever recording we could get! If you really want a taste of fate of the Universe, we highly recommend this article and the references therein! http://math.ucr.edu/home/baez/end.html ]


I did not want to get my hands messy with maths, so my dear friend Messi (Somyajit Bose) gratefully figured out everything and shared his knowledge generously with us.

Distances of Supernovae: The Luminosity distance

For a source of light/radiation, the intensity received in an area, called the flux, falls of as inverse square of the distance from the source, as shown nicely in this picture.


Therefore, if the inherent luminosity (power) of a Supernova (like 100W for a 100W bulb) is L, and it is at a distance d_{L} then, by measuring the flux F on earth, we can figure out the distance d_{L} by using

F = \frac{L}{4\pi {d_{L}}^2}

This is directly measurable quantity

Redshift z

Redshift or measure of expansion of the Universe, can be measured for a light directly by measuring how much the light has stretched. This is denoted usually by z. Redshift today is set to be z=0. This is also a directly measurable quantity

The scale factor a(t):

The scale factor represents the size of the Universe relative to today. It is usually denoted by a(t), notice that it is dependent on the time t! The scale factor for today is set to be a(t_0)=1. If a(t)=0.2 in past this means that the Universe was 0.2 times its size today.  The Hubble parameter H (or earlier called as Hubble constant, later found to be time dependent!) which was discussed above, relating the speed of galaxies to their distance depends on the scale factor as:


The Hubble constant now is defined as Hubble parameter today


Scale factor is related to redshift z by definition as:

a(t) = \frac{1}{1+z}

Scale factor as  a function of time usually comes from various cosmological models and data can be used to constrain them or rule them out, hence improving our understanding.

Distance today: Comoving Distance \chi

As we discussed, the expansion makes comparing distances non-trivial, hence a measure of distance independent of expansion is desirable. Comoving distance solves this issue by definition as:

r(t) = a(t)\chi

Where r(t) is the separation between two objects at time t. Now for t=t_0, today, the scale factor a(t_0)=1 hence the distance today r(t) is the comoving distance \chi

Comoving distance is also something which is not directly measurable as any light reaching us must have started traveling in past, not today. But this can be calculated for a given model by having a scale factor a(t) for that model:

Distance  r(t) = \int c{dt} where c is the speed of light. Cosmologists measure distances in units of time! (e.g. Lightyears) so they take c=1. So r(t) = \int {dt} Now by definition, comoving distance is \chi = r(t)/a(t) Hence

\chi = \int_{t_0}^{t_e}\frac{dt}{a(t)} :  Get distance today by integrating on all the past expansion since time of emission t_e to today t_0

\chi = \int_{t_0}^{t_e}\frac{dt}{a(t)} = \int_{1}^{a'}\frac{da'}{a'\dot{a'}} = \int_{1}^{a'}\frac{d(lna')}{a'(\dot{a'}/a)} = \int_{1}^{a'}\frac{d(lna')}{a'H}=\int_{1}^{a'}d(lna')H_R

Where (a'H)^{-1}=H_R, the Hubble radius. Now

\frac{d}{dt}H_R = \frac{1}{H_R^2}\ddot{a}

It is clear now then that by integrating over the Hubble Radius, we are taking into account the acceleration of the Universe in the past because the rate of change of Hubble radius is acceleration of scale factor \ddot{a}

Once we have this comoving distance from a given model, we can get the luminosity distance d_L from comoving distance and measured redshift by using this formula:

d_L = \frac{\chi}{1+z}

Hence we can have luminosity distance from direct measurement, which can now be compared with luminosity distance obtained from cosmological models with various accelerations. These comparisons will constrain or rule out various models. In Supernova project, decelerated expansion is ruled out by a good probability amount. (3 sigma ~ 99.7% by latest analysis)

Conclusions from The Supernova Project

The luminosity distances of supernovae and redshifts were measured. From various models, the comoving distance was calculated and using redshifts, the luminosity distances predicted from models were calculated as outlined above. The measurements showed that the measured luminosity distances were greater than they should be for the models with decelerated expansion and more consistent with models with accelerated expansion, as shown in Fig. 4

Use Supernova data to calculate and verify the existence of the Dark Energy on your own!

The apparent magnitude of brightness m of a source of absolute brightness M with a measured flux F is m = 5/2log(F)+constant. Since Flux goes as d_L^{-2}, we have

m = 5log(ad_L)+b

where a and b are constants.

Supernova 1997ap, found at a high redshift of 0.83 had an apparent magnitude of 24.32 and Supernova 1992P found at a low redshift of 0.026 had an apparent magnitude of 16.08. Since absolute brightness of Supernovae are same (standard candles), the difference in apparent magnitude will be only due to difference in luminosity distance:

24.32 - 16.08 = 5log(ad_L(z=0.83)) - 5log(ad_L(z=0.026)) = 5log(\frac{d_L(z=0.83)}{d_L(z=0.026)})

For a low redshift supernova, effects of expansions are negligible and its luminosity distance is simply z/H_0=0.026/H_0

\Rightarrow 8.24/5 = log(H_0d_L/0,026)+b

\Rightarrow d_L = 0.026\times10^1.648H_0^{-1} = 1.15H_0^{-1}

Now matter dominated Universe (100% matter) model calculations (as outlined above) for a redshift of  z = 0.83  give d_L = 0.95H_0^{-1} and a Universe with model of 70% dark energy and 30% matter  gives d_L = 1.23H_0^{-1}, closer to what our observed values from measurements are!







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