Dark Energy Increases the Mass Density of the Universe in Accelerated Expansion to a Limiting Value

AN
6/2/2010

 

We explore the impact of dark energy in the form of the cosmological constant on mass density (visible matter and dark matter) during accelerated expansion of the universe.  Our model, based on general relativity with the cosmological constant, is validated with available data on the current mass density and the radius of the universe when the accelerated phase started.  According to our model, as the expansion of the universe started to accelerate 5 billion years ago, the mass density has been increased and will be approaching a limiting value of 0.4 ρcri  where ρcri  is the critical mass-energy density of the universe (the current mass density is 0.3 ρcri  ). It is suggested that this increase does not violate the conservation of energy since it is cancelled out by the increase in negative gravitational energy.  The predicted change in mass density with this model will need to be validated in future observations.  Future observations may also reveal the distribution of this increase in mass density between visible matter and dark matter.

 

 

Introduction

Mass is typically defined as the amount of matter in a body that interacts with gravity and it is the measure of inertia of a body that creates resistance to change of motion [3]. Mass is made quantitative in Newton's second law of motion and gravitation.  Beyond that, it is not known how the mass property is created.  Electroweak Higgs field, a quantum field permeating space, has been proposed to create mass by interacting with elementary particles [1].  To date, the electroweak Higgs field is still waiting to be confirmed by experimental data.  Researchers hope to find evidence of the Higgs boson in future experiments at the Large Hadron Collider of the European Organization for Nuclear Research [9].  It is estimated that the Higgs boson is at least heavier than 120 proton masses [1].

 

A striking difference between mass and charges of other forces is the lack of quantization of mass.  The charges of electromagnetic, weak and strong forces seem to come in parcels of definite size [8].  Although we have never observed an electric charge other than an integral number of the fundamental charge (that of an electron), we find all magnitudes of mass and no mass quantization at all.

 

The principle of equivalence [3] states that the inertial and gravitational masses of a body, with respect to acceleration and gravity respectively, are equal.  According to general relativity, one cannot distinguish between the effect of acceleration and a gravitational field [1].

 

Fritz Zwicky was the first to infer the existence of a phenomenon called dark matter in 1933 [4,5]. He estimated the total mass of the Coma cluster of galaxies based on the motions of galaxies near its edge. When he compared this estimate to one based on the number of galaxies and total luminosity of the cluster, he found that there was about 400 times more mass than expected from the observed galaxies. Based on these data, Zwicky inferred that there must be some non-visible dark matter which would provide adequate mass to hold the cluster together.  Similar discrepancy is also found in individual galaxy.  Rotation curve of a typical spiral galaxy shows the rotational velocity of V of a star in the galaxy as a function of radius R, the distance to the galaxy center.  It is predicted to show a decreasing rotational velocity with increasing distance.  However, the observed velocity shows a flat appearance out to large distance, an effect assumed to be due to dark matter occupying a dark halo surrounding the visible part of the galaxy. Data from a number of sources, including galaxy rotation curves, gravitational lensing, and structure formation, indicate that 85-90% of the mass in the universe does not interact with the electromagnetic force. This dark matter is only evident through its gravitational effect. Two general categories of dark matter have been postulated, baryonic dark matter and non-baryonic dark matter [4,5].  Baryonic dark matter contains things that we know already exist, such as black holes, neutron stars, faint stars, and planets.  They go under the heading of Massive Astrophysical Compact Halo Objects (MACHOs). Non-baryonic dark matter comprises entirely new types of particles that are collectively known as Weakly Interacting Massive Particles (WIMPs).  They are proposed to be superpartners of the known Standard Model particles and they are with the lowest mass (also known as the Lightest Superpartner Particles).  WIMPs are difficult to detect since they barely interact with normal visible matter.

 

The field equation of general relativity predicts a dynamic universe in which space can expand or collapse, but can not maintain a fixed size. The cosmological constant was introduced by Einstein in 1917 to counteract gravity in order to build a static universe, a prevailing belief at the time.  It was abandoned in 1927 after Edwin Hubble discovered that the universe is expanding.  In early 1998, two independent groups using supernova explosions in galaxies as standard candles made an unanticipated discovery that in the last 5 billion years, the expansion of the universe has been in deed accelerated [11, 12].  This discovery has been further verified by the Hubble Space Telescope.  Since then, the cosmological constant has been refashioned to accommodate for this acceleration [1].  In recent years, evidence is mounting that the cosmological constant may, in fact, be required to describe the dynamics of our universe [5].  Many physicists now expect the cosmological constant  to provide the key to moving beyond general relativity to a deeper understanding of space, time, and gravity [1].  According to general relativity, space-time curvature can cause accelerated expansion of the universe.

 

The cosmic microwave background (CMB) is the most distant source of light that we can detect from earth.  It was generated when the universe was about 380,000 years old due to decoupling of photons from electrons and protons which combined the first time to neutral hydrogen when the temperature was sufficiently low [9].  The universe appears so homogenous and isotropic according to measurements of the CMB, consistent with flat space.  A flat universe, in turn, requires a critical energy density.  The matter in the galaxies and galaxy clusters (visible and non-visible), determined with gravitational lensing, account for only 30% of the critical density.  An unseen form of energy, called dark energy, must account for the remaining 70% of the critical energy density.  Data from distance supernovae indicated that the universe is accelerating-the signature of dark energy [15].  Since dark energy permeates the cosmos and does not clump like mass energy, it cannot be detected with gravitational lensing which measures light shear in images of distant galaxies due to gravitational distortions. Our only hope of learning more about dark energy is to find its impact on the architecture of the cosmos and the evolution of spacetime.  Dark energy counteracts attractive gravity and causes accelerated expansion of the universe.  If one can obtain detailed measurements of the evolution of structures, the imprint of dark energy can be extracted [15].  Current approaches to measurements of dark energy impact include four key techniques: supernovae observations, baryon acoustic oscillations, gravitational weak lensing, and galaxy cluster count using Sunyaev-Zeldovich effect on the CMB spectrum [9,15].

 

Besides the cosmological constant as the source of dark energy, three other theories have also been proposed [1,3,9].  The first one is called quintessence, which involves an unidentified energy field of varying strength and filling up space, similar to the inflation Higgs field driving inflation.  The second proposal suggests that dark energy is in fact an artifact due to a breakdown in Einstein’s theory of gravity over the largest distance scales.  In the third proposal, dark energy is suggested to be a form of vacuum energy that exists in space even when devoid of matter. The vacuum energy is deduced from the concept of virtual particles (pairs of particle-antiparticle), which are themselves derived from the energy-time uncertainty principle. Special relativity predicts that energy is equivalent to mass, and therefore, if the vacuum energy is present, it should exert a gravitational force. Essentially, a non-zero vacuum energy is expected to contribute to the expansion of the universe. However, many workers think that the vacuum energy may not be a viable concept since it is mathematically infinite without renormalization.  Virtual particles are either fermions (electrons, neutrinos, and quarks) or bosons (photons, gluons, and Higgs particles).  The virtual fermions carry negative energy whereas the virtual bosons carry positive energy.  The energy contribution from each one of these particles is different and the net energy of all virtual pairs in the universe comes out an enormous number (denoted as Λp , one Planck mass per cubic Planck length or 1097 kg/m3) which could tear the cosmos with repulsion [1]. The only way for the vacuum energy to get a smaller value is through a proposed supersymmetry configuration in the early universe.  Experimental measurements of the accelerate expansion requires that the vacuum energy contributions cancel out to 120 decimal places (vacuum energy at around 10-120 Λp), a fine-tuning with an unrealistic accuracy without invoking the anthropic principle [13].

 

In this article, we attempt to derive equations from general relativity with the cosmological constant to calculate the mass density of the universe (combined of both visible matter and dark matter) in accelerated expansion.  Our model is validated with available data on the current mass density and the radius of the universe when the accelerated phase started. These equations will provide insight on how the mass of the matter in the universe changes with time.

 

Further background on general relativity with the cosmological constant

Since 1998, accumulated data have shown that the universe expansion has gone through the following phases: inflation, followed by deceleration, and lastly acceleration. 

 

In the early universe with inflation, a process known as supercooled inflation Higgs field occurred.  The pressure was enormously high with negative value due to rapid growth of the bubble of broken symmetry in the false vacuum [13].  This negative pressure by the bubble, which takes the false vacuum to the true vacuum, causes the universe to expand exponentially in a hyper-expansion process called inflation. The universe doubled in size every  10-37 seconds. The inflation phase lasted for a mere 10-35 seconds to 10-32 seconds, but it was so powerful that the universe was expanded by a factor of 1030 to 10100, depending on the precise configuration of the inflation field’s potential energy before inflation [16]. This inflation explains why the universe today is so incredibly flat and uniform on a large scale.  It also solves the horizon problem since the currently observed universe is only a tiny portion of the entire space region which had reached thermal equilibrium prior to this inflation.  The inflation model has been confirmed with a high degree of accuracy by experimental data. According to this model, the spectrum of density perturbations should have density waves with shorter wavelengths and with smaller amplitudes toward the end of inflation due to decreasing energy.  Scientists call this phenomenon a scale-invariant spectrum with tilt.  This fingerprint of inflation is in remarkable agreement with measurements by the Cosmic Background Explorer (COBE) mission [13].

 

As the inflation field slid down its potential energy curve (inflation Higgs field),  it released the pent-up energy to the production of radiation and matter [16]. Radiation and matter subsequently took turn to be the dominant energy and both caused the expansion to decelerate due to gravitational attraction in the standard Big-Bang model.

 

The current acceleration phase is estimated to begin about 5 billion years ago [9].  Since all masses and photons have positive pressure, a new form of cosmic energy is thought to create gravitational repulsion.  Einstein found a means to achieve this without introducing an exotic media.  He noted that the space-time curvature allows for addition of a constant termed cosmological constant Λ in his field equation of general relativity:

Gμν   +  Λ gμν  = 8 π G Tμν     (Equation 1)

Where   Gμν: curvature tensor

              gμν : space-time metric tensor

              G: gravitational constant

              Tμν: stress-energy tensor

 

The cosmological constant corresponds to the curvature of space-time and does not have association with any physical medium.  Many workers have proposed that it is associated with dark energy, an unknown type of energy discussed earlier.  Note that empty space still has curvature in the absence of matter.  The existence of curvature in such empty space-time is proved by Willem de Sitter [3].  This solution of the general relativity equations for empty space-time is called de Sitter universe.  The cosmological constant is thus an intrinsic and uniform property of space-time [9], i.e. it is a type of energy that is constant throughout space and over time.  This particular property was originally introduced by Einstein, and is consistent with limited observations to date [17].

 

The expansion history of the universe is governed by the dominant type of energy in a particular phase [15].  All three types of energy-false vacuum, radiation, matter, and the cosmological constant –appear to participate in this history.  False vacuum energy created the first phase of inflation in 10-35 seconds with expansion of space in a supraluminal fashion.  Radiation immediately followed and dominated the first 50,000 to 75,000 years before handling the reins to matter.  Density perturbations occurred in the newly-created particles due to quantum fluctuations in the inflation field.  From these ripples in the density of the early universe, matter created galaxies and clusters for billions of years before the cosmological constant started the accelerating phase.

 

Let us introduce the second Friedman equation [10], which was derived from the field equation of general relativity

  d2 r/ dt2 = -(4/3) π G (ρ + 3 p) r         (Equation 2)

Where  d2 r/ dt2 : acceleration of the universe expansion

            p: pressure (predominant during inflation and also in a radiation-dominated
               universe)

            ρ:  mass-density (of both visible mass and non-visible mass, predominant in a
                 matter-dominated universe)

            r: distance from the center of the universe at time t

The above equation can be rearranged as:

(1/r) d2 r/ dt2 = -(4/3) π G (ρ + 3 p)                (Equation 3)

Adding the cosmological constant would modify the Friedman equation (Equation 3) as follows:

(1/r) d2 r/ dt2 = -(4/3) π G (ρ + 3 p) + Λ /3     (Equation 4)

 

Equation 4 closely describes all the phases in a quantitative manner [15].  In early phase of the cosmic history,  the pressure p due to false vacuum was enormous and with negative value.  It was solely responsible for inflation phase of the universe just prior to the standard Big-Bang model.  After inflation, radiation became the dominant energy.  Radiation energy exerted a positive pressure p and consisted of photons and relativistic particles that moved at nearly the speed of light.  The pressure p outweighed the mass density ρ  and the cosmological constant Λ.  The positive value of pressure p effectively created an attractive gravitation force that was responsible for the deceleration phase of expansion. As the universe expanded and temperature dropped, the energy of the relativistic particles decreased until they slowed down enough to become non-relativistic.  These slow-moving particles effectively switched from radiation to matter, i.e. motion energy transformed to mass energy.  Moreover, the wavelength of light became longer and caused radiation energy to decrease as the universe expanded.  With this development, matter energy became more dominant than radiation energy.  Both radiation energy and matter energy exerted gravitational attraction which decelerated the universe expansion between the time right after inflation and 5 billion years ago.  The attractive gravity, which depends on distance, is expected to decrease to a very small value when the galaxies separate to a large enough distance whereas dark energy, which does not depend on distance, remains unchanged [13].  Equation 4 shows that as pressure p and mass density ρ fell continuously with increasing radius r as the universe expanded in the decelerating phase, the cosmological constant Λ would eventually be dominant and the acceleration term d2 r/ dt2 became positive, i.e. to make the universe accelerate [5].   The gravitational repulsion due to the cosmological constant is therefore responsible for the current acceleration phase of expansion.  This repulsion is also responsible for putting a brake on the growth of galaxy clusters by counteracting gravitational attraction, a finding confirmed by NASA’s Chandra X-ray Observatory in December 2008 [9].

 

Calculation of the mass density of the universe in expansion with acceleration

 

In our current universe, the contribution of pressure is negligible [3], i.e. p ~ 0.  Equation 4 would be simplified to

(1/r) d2 r/ dt2 = -(4/3) π G ρ  + Λ /3     (Equation 5)

With  d2 r/ dt2 =  c2/r   [6,8] , where c is the speed of light, Equation 5 yields

ρ = (Λ - 3 c2/r2)/(4 π G)   (Equation 6)

With known values of Λ, c, G, r; ρ can be calculated [from Numerical Data at the end of this article]

ρ= 0.30 x 10-26 kg/m3

 

We call ρcri the critical mass-energy density of the universe  [4].  If the average mass-energy density of the universe has this value, then the space is spatially flat.  Since our observed universe is very close to being spatially flat according to precise measurements of the CMB by NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) in 2006, ρcri is very close to the current mass-energy density of the universe.

ρcri =10-26 kg/m3        [4]

The ratio of  ρ to ρcri  is

ρ / ρcri  = 30%

It has been determined through gravitational lensing measurements that about 30% of the universe's mass-energy comes from matter (5% from visible matter and 25% from dark matter).  The remaining 70% to achieve critical density for a flat universe is in the form of dark energy (associated with the cosmological constant) [9].  The ratio ρ/ρcri  of 30%, as calculated above, is the same as the value of 30% measured in experiments for this ratio.

 

From Equation 6, it follows that when the radius of the observable universe approaches infinity, the mass-density will increase and approaching a limiting value of

ρlim = Λ/(4 π G)   (Equation 7)

       =  0.4 x 10-26 kg/m3   =  0.4  ρcri     

 

The transformation of energy by the cosmological constant (due to curvature of spacetime) into new matter particles in the accelerated expansion is similar to that seen in the inflation phase [16].  If the mass-density is increased in an expanding universe, the principle of conservation of energy would be violated if there were no other means to decrease the total energy.  This is in deed provided in the form of negative gravitational energy. In a universe that is approximately uniform in space and in acceleration phase, the negative gravitational energy cancels out the positive mass-energy represented by matter [14].  Note that the current mass density of the universe is 0.3 ρcri 

 

Also, when the cosmological constant Λ became dominant at the end of the deceleration phase, i.e.  Λ >> ρ, Equation 5 is approximated as

 (1/r) d2 r/ dt2 = Λ /3     (Equation 8)

With  d2 r/ dt2 =  c2/r   [6,8] , it follows that

r = c (3/Λ)1/2    (Equation 9)

       =  2.63 x 1026 m     

This was the radius of the universe around the time the accelerated phase started about 5 billion years ago.  The ratio of this radius to the current radius is

(2.63 x 1026 m)/ 4.65 x 1026 m) = 0.57 = 57%

This ratio is close to the value 60% based on measurements [18,19].             

 

Conclusion

Until 1998, cosmologists had thought that the universe was slowly decelerating as the gravity of all matter within it slowed down the expansion.  Since then, accumulated data have shown that the expansion began an acceleration phase about 5 billion years ago.  The data also showed that this acceleration will continue forever [9].  Could this expansion eventually disperse everything in the cosmos into a cold, ever-expanding void?

 

In this article, our model shows that as the expansion of the universe was accelerated in the last 5 billion years by dark energy in the form of the cosmological constant, the mass density has in fact been increased and will be approaching a limiting value of 0.4 ρcri  where ρcri  is the critical mass-energy density of the universe. This increase does not violate the conservation of energy since it is cancelled out by the increase in negative gravitational energy.

 

Various projects are currently in progress or will be launched in the near future to study dark energy including the Joint Dark Energy Mission (JDEM), the Sloan Digital Sky Survey Data (SDSS), the Large Synoptic Survey Telescope (LSST), and the Square Kilometer Array [9]. The predicted change in mass density by our model will need to be validated or rejected with future measurements in these projects.  Future observations may also reveal the distribution of this increase in mass density between visible matter and dark matter.  Note that our model is based on general relativity with dark energy in the form of the cosmological constant.  This model would not be validated if dark energy is found to vary with space or vary over time.

 

Numerical Data

Gravitational constant      G = 6.67 x 10-11 Nm2Kg-2      [2]

Radius of the observable universe      R= 4.65 x1010 light-years = 4.65 x1026 m  [9]

Critical mass-energy density   ρcri =10-29 gm/cm3  = 10-26 kg/m3  [4]

Speed of light c = 3 x 108 m/s [2]

Cosmological constant    Λ =  3.9 x 10-36 s-2        [13]

One Planck mass per cubic Planck length    Λp = 1097 kg/m3     [1]

 

 

References

1.      The Frontier of Physics.  Scientific American, Feb 20, 2006. p34,37, 41-49, 67,68,73

2.      Fundamental Constants and Terrestrial Data. In: Physics Chart. Bar Charts, Inc., 2004.

3.      Stenger, V.  The Comprehensible Cosmos. Prometheus Books, 2006. p47,66,285,289

4.      Silk, J. A Short History of the Universe. Scientific American Library, 1995. p30-42,105,116-120

5.      Maoz, D. Astrophysics in a Nutshell (Princeton University Press, 2007. p148-162,203,204,205

6.      Smolin, L. The Trouble with Physics. First Mariner Books, 2007. p209

7.      Krauss, LM. Cosmological Antigravity. In: Scientific American, Jan 1999.

8.      Adair, RK. The Great Design: Particles, Fields, and Creation. Oxford University Press, 1987. p218,322

9.      Cosmology’s Greatest Discoveries. Astronomy, Special Edition, 2009. p8,37-41,77,80,180

10.  Davies, P. The New Physics. Cambridge University Press, 1989. p59

11.  Reiss, A et al. Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. Astronomical Journal, 116 (1998), p1009-38

12.  Permutter, S et al. Measurements of Omega and Lambda from 42 High-Redshift Supernovae. Astrophysical Journal, 517 (1999), p565-86

13.  Calle, C. The Universe: Order Without Design. Prometheus Books, 2009. p87-92,95-98,155,156,191-192,277

14.  Hawking, S. A Brief History of Time.  Bantam Books, 1988, p129

15.  Gates, E. Einstein’s Telescope: the Hunt for Dark Matter and Dark Energy in the Universe. W.W. Norton & Co, 2009, p16,66,198,208,226,244

16.  Greene, B. The Fabric of the Cosmos. Knopf, 2004, p284,301

17.  The Large Synoptic Survey Telescope (LSST),  last accessed 6/26/2010, http://www.lsst.org/lsst/public/dark_energy

18.  Bound and unbound universes and the closure density, last accessed 6/26/2010, http://www.britannica.com/EBchecked/topic-art/139301/123380/How-the-relative-size-of-the-universe-changes-with-time

19.  The Expanding Universe: Core Concepts for Astronomy 102, last accessed 6/26/2010, http://brahms.phy.vanderbilt.edu/a102//handouts/lecturenotes/20061127.pdf