2007, Volume 5, Number 1
Vacuum energy density resolved, thus dark energy solved
Abstract
A constant for the vacuum energy density in the universe was resolved by creating a mathematical model of two average galaxy clusters at average distances as they existed at the beginning of the epoch of acceleration, and then setting the total vacuum pressure equal to the total gravitational pressure. These two clusters were then compared at their present distances with the new constant introduced that produced findings that coincide with current data that reveals the ratio of dark energy to gravitational energy to be 3:1. This paper finds that the positive but near-zero value of vacuum energy density is true while completely discounting the very flawed quantum field theory model altogether.
Introduction
In 1929 Edwin Hubble and Milton Humason let the world in on their startling discovery that the Universe was expanding at a rate of almost 500km/s per megaparsec (Mpc); a constant that became know as the Hubble constant (H0). Today, that number has shrunk considerably as technology and methods have improved greatly from those of the first half of the twentieth century. As of 2006, the Hubble constant has now been resolved down to 70.8km/s/Mpc +/-1.6km/s/Mpc (NASA, 2006).
Astronomers see that the Hubble constant actually slows over time due to the drag produced by gravity (Guth, 1997; Bennett, 2006; Goudarzi, 2006). The realization that the Universe was expanding and slowing raised many questions among cosmologists and astronomers. Chief among these were: will the Universe eventually slow and collapse in on itself (the Big Crunch) or did it achieve escape velocity and will continue to expand forever (the Big Whimper)? This debate was resolved (rendered mute) in 1998 by the perplexing and mind-bending discovery by two independent groups: the Supernova Cosmology Project at Lawrence Berkeley National Laboratory and the High-z Supernova Search Team from the Australian National University. These two teams of scientists threw conventional models of Universal expansion out the window by proving that the Universe was no longer slowing down in its expansion, but actually accelerating (Goldsmith, 2000).
The astronomical community was now raising new questions: How could the Universe be accelerating? What force could possibly be at work to pull apart the Universe beyond the momentum created by the initial Big Bang? This yet to be defined force was given the ominous name of “dark energy” and what causes it has been a matter of debate for the better part of the last decade.
It was this very topic I was discussing with my friend Dan Lafferty, an instructor of graphic design at the Art Institute of Seattle, around Christmas of 2004 that brought me to this paper. I had finished work at the bar back home in Seattle where I am employed during breaks from school and Dan and I were catching up and conversing about my time at school in Hawai’i, and he was peppering me with questions about the workings of the Universe. The topic of dark energy came up and Dan suggested that maybe the fact that the Universe is accelerating is due to the fact that the Universe is mostly vacuum. His reasoning was that we use vacuum chambers to paint everything from little pieces of plastic to motorcycle frames. Paint particles expand evenly in a vacuum so why couldn’t big galaxy clusters do the same? Now, the adiabatic expansion of paint particles in a vacuum chamber is fundamentally different from what happens in the Universe we see today. The Universe of old had gobs of adiabatic expansion and cooling due to the excited nature of the super-hot particles in the initial moments after the Big Bang. Dan’s idea intrigued me none-the-less and I decided to learn more about this area of cosmology and what vacuum pressure may have to do with it. After all, the volume of the Universe is dominated by vacuum. I set out on my mission and was surprised to learn that many had already started pursuing the possibility that energy present in the vacuum of space may be the contributing force that comprises dark energy. A main contributor of energy in a vacuum in the quantum field theory is thought to come from the particles and antiparticles that randomly appear, colliding and annihilating each other all over the Universe (Wright, 2004). Many articles concerning vacuum energy density in the Universe have recently been published with attempts to resolve the insanely vast differences between the quantum field theory model for vacuum energy density and that of what projects like the Wilkinson Microwave Anisotropy Probe (WMAP) are finding. The difference between these two alone is about 120+ orders of magnitude (Baez, 2006)!
Quantum field theory must be wrong because it calculates that the energy density (in terms of weight) of the vacuum of space is about 1096 kilograms per m3 (Wright, 2004). Common sense tells me that there is no way that one cubic centimeter of vacuum has 60+ orders of magnitude more energy in terms of weight than that of the entire Sun.
When Einstein created his general theory of relativity he believed the Universe to be static (this was before Hubble’s and Humason’s discovery) but his equations describing general relativity, now known as the Einstein field equations, kept making his model of the Universe expand at an everincreasing rate. To correct for this anomaly Einstein created what he called the “cosmological constant” to be an opposing force against the runaway universe that his equations suggested.
After Hubble’s and Humason’s discovery of the steady-rate expanding Universe, Einstein abandoned the cosmological constant and it was thought to have died. Rapture for the constant came in the late 1990s after the discovery of the accelerating Universe by the groups from Berkeley and Australia. Einstein can now be credited as being more of a genius than even he realized.
Figure 1. A vector field representing the expansion of the Universe since the Big Bang and showing the acceleration since the Epoch of Acceleration.
The cosmological constant can now be used as a positive, neutral or negative force and depending on its value predicts whether the Universe will collapse, stay static or expand forever (Guth, 1997; Spark, Gallagher, 2005).
The problem now is that the cosmological constant fudge factor used in general relativity is much, much smaller than that of the quantum field theory I mentioned above, but is still positive and probably very small. Small enough to be treated as zero on small scales (Carroll, 1998). I personally believe that since the quantum field theory suggests a constant that would essentially tear the universe apart so fast that galaxies, stars, planets and ourselves would never even have had the opportunity to exist, then there certainly must be something wrong with the way the equation is applied. There are many who will stand by the quantum field theory to answer the question as to just what dark energy is, but I am still waiting for quantum mechanics to be applicable to anything larger than an atom before I apply it to masses 75 orders of magnitude bigger than that of a lowly proton.
I decided that since these two models cannot agree it would be best to independently calculate a constant for vacuum energy density by creating my own plausible model and seeing if it fit with the 3:1 ratio of dark energy to gravitational energy that we see around us today.
After the Big Bang the Universe exploded outward and inflated rapidly transmitting huge amounts of momentum to the matter it heaved forth. For about 8-9 billion years this momentum-driven Universe slowed under the drag of gravity produced by it massive parts, that is until about 5 billion years ago (see figure 2). Just recently, astronomers looking back to a point in time where z = 0.36 (~5Gya) discovered that this is around the time the epoch of acceleration began (Goudarzi, 2006). This is where my idea came from on how to calculate the energy contained in each chunk of vacuum throughout the Universe.
Methods
Today, the average galaxy cluster is about 10Mpc from another average galaxy cluster, and about 1 to 10Mpc in diameter (this figure isn’t that important as I am going to treat the clusters as point particles) and about 1015 Solar masses (Cambridge University, 2006; Chaisson, 2005). Knowing this, I concluded that if the epoch of acceleration began about 5Gya then it is at this time when dark energy equaled gravitational energy or, in this case, gravitational pressure equaled vacuum pressure. To calculate the attractive acceleration due to gravity as well as the repulsive acceleration due to the vacuum between two average galaxy clusters I would need to define a constant for the component of dark energy. To obtain my constant I have to go back in time and see how these two average clusters existed 5Gya. This way I can resolve a constant for the acceleration component contained within each meter of vacuum and make the difference between the two acceleration components equal to zero:
,
where DEacc is the repulsive acceleration component from dark energy and Gacc is that same but attractive component provided by gravity.
Before I can use the above equation I have to calculate the distance two average galaxy clusters were away from each other 5Gya. To make things easy I assume a H0 of 70km/s/Mpc and then go back in time with two clusters that today are 10Mpc apart. I must mention that by assuming a H0 of 70km/s/Mpc I am going to have small and hopefully negligible error in my end result because the Hubble constant is anything but constant as the Universe is accelerating so this value will have actually changed over the past 5Gyr. I am sure that later works will let the technicalities ooze forth and correct my simple assumptions and help reduce any error in the end result.
I will use this equation to determine the past separation of the two clusters:
, (Crowe, Heacox, 2006),
where D0 is the distance the clusters were from each other at the beginning of the Epoch of Acceleration, Df is the distance they are apart from each other today (10Mpc), t represents the 5Gyr of time since the beginning of the Epoch of Acceleration and the Hubble constant can be treated as 0.072Mpc/Gyr/Mpc. I then get:
.
Now, I need to calculate the gravitational acceleration between these two clusters as they were 5Gya:
This is the equation I am using to define the acceleration due to dark energy:
,
where CDE is the repulsive constant associated with the vacuum. The calculations stay very simple from this point forward. All we need to do now is solve for our vacuum constant.
,
.
Now that I have my constant defined I need to compare it with the acceleration values for the galaxy clusters in their current state of being 10Mpc apart. The repulsive acceleration experienced by the clusters due to the vacuum is now:
.
The attractive acceleration due to gravity at 10Mpc is:
.
To calculate the ratio of vacuum force to that of the gravitational force we just divide the two acceleration components into each other, and low and behold:
, wow!
Results
By calculating my own constant for vacuum energy density with this mathematical model I found that vacuum energy (dark energy) is about 2.96 times that of the gravitational energy or 74.75% of the total energy at play in the present Universe. This is pretty darn close to the 75-76% measured by astronomers (Chandra X-ray Observatory, 2006) and I did nothing too technical or exacting. I just took two average clusters at average distances from each other as they are seen today and used an approximation of the Hubble constant to find out how far apart they were from each other at the beginning of the epoch of acceleration.
The results of this simple model do imply that distance and mass, not time, are the important factors in the increasing speed at which the Universe is expanding. By this I mean that two average-sized clusters, like the ones I chose for my model, that were more than 6.97Mpc apart 5Gya were already experiencing acceleration away from each other.
Conclusion
The basic premise behind the method I used should hold true when determining a more exact constant for the vacuum energy density. There are several ways that one can get a more precise answer and eliminate the error present in my ratio. I would suggest using the gravitational equations from general relativity to calculate the acceleration between the two clusters and to create an integral that would account for any change in the Hubble constant over the last 5Gy rather than my imprecise approximation. Of course, the most important step one should take to eliminate error from this model would be to conduct accurate measurements of distant, average galaxy clusters separated by average distances around z=0.36 and use those instead of my assumed cluster distances and sizes and then compare projected positions according to the current 3:1 ratio of dark energy to gravitational energy.
I cannot help but wonder just what Einstein would make of the recent discoveries of the runaway, Universe and how he would have approached the entire conundrum of dark energy. And how he would solve it–we all know he would.
Figure 2. A representation of the two clusters and their separation from each other over the last 9Gyr and how their accelerationscomponents due to gravity (green) and dark energy (red) may have changed in that time.
WORKS CITED
Baez, John. What is the Energy Density of the vacuum? [Internet]. University of California Riverside; 11/8/2006 [date of citation: 11/12/2006]. Available from http://math.ucr.edu/home/baez/vacuum.html.
Bennet, Charles L. 2006. Cosmology from start to finish. Nature 420: 1126-1131. Cambridge University. Galaxy Clusters and Large-Scale Structure [Internet]. Cambridge University; [date of citation: 11/12/2006]. Available from http://www.damtp.cam.ac.uk/user/gr/public/gal_lss.html.
Carroll, Sean. The Cosmological Constant [Internet]. University of Chicago; 12/14/1998 [date of citation: 11/12/2006]. Available from: http://pancake.uchicago.edu?~carroll/encyc/.
Chaisson, Eric J. Cosmic Evolution [Internet]. Boston, Mass: Tufts University; 2005 [date of citation: 11/12/2006]. Available from http://www.tufts.edu/as/wright_center/cosmic_evolution/docs/splash.html.
Chandra X-ray Observatory. 2006. Galaxy Clusters and Dark Energy: Chandra Opens New Line of Investigation on Dark Energy [Internet]. Harvard University; 8/30/2006 [date of citation: 11/20/2006]. Available from: http://chandra.harvard.edu/photo/2004/darkenergy.
Crowe, Richard. 2006. University of Hawai’i at Hilo; email correspondence 11/11/2006.
Heacox, William D. 2006. University of Hawai’i at Hilo; email correspondence 11/13/2006.
Goldsmith, Donald. 2000. The Runaway Universe. 1st edition. Cambridge, Mass: Perseus Publishing. 232.
Goudarzi, Sarah. 2006. The History of Dark Energy Goes Way, Way Back [Internet]. Space.com; 11/16/2006 [date of citation: 11/20/2006]. Available from: http://space.com/scienceastronomy/061116_darkenergy_infantuniverse.html.
Guth, Alan H. 1997. The Inflationary Universe. 1st edition. Reading, Mass: Perseus Books. 358.
NASA. 2006. How Fast is the Universe Expanding? [Internet]. NASA; 10/18/2006 [date of citation: 11/20/2006]. Available from: http://wmap.gsfc.nasa.gov/m_uni/uni_101expand.html.
Spark, Linda S. Gallagher, John S. 2005. Galaxies in the Universe. New York: Cambridge University Press. 379.
Wright, Edward L. Vacuum Energy Density, or How Can Nothing Weigh Something? [Internet]. University of California Los Angeles; 9/10/2006 [date of citation: 11/12/2006]. Available from http//www.astro.ucla.edu/~wright/cosmo_constant.html.
This paper is a final research paper written for English 225 (Writing for Science and Technology)