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Gravity in Extra-dimensions, Manyfold and Pre-Big Bang Universes


Contents

Gravity in Extra-dimensions
Black Hole in LHC
Manyfold Universe
Pre-Big Bang Universe

Gravity in Extra-dimensions

It is known that even the feeble subatomic force responsible for electroweak interactions is 17 orders of magnitude stronger than gravity. It means that quantum gravity is not accessible above length scales of 10-33 cm. Such a miniscule distance cannot
Gravitational Force Coupling Strength be probed or the consequences of quantum gravity experimentally tested. Only in the very early universe are energies on the Planck scale encountered, which is why particle physicists and cosmologists have placed such emphasis on understanding conditions immediately after the Big Bang. A further consequence of gravity's weakness is that the inverse square law of attraction has only been verified at distances greater than 0.1 cm. It is now suggested that the huge disparity in interaction strength is caused by the "leaking" of gravitational field into the extra dimensions as envisioned by the superstring theory. As it will be shown in the followings, these extra dimensions can be as large as 0.1 cm without violating known laboratory, astrophysical, or cosmological

Figure 01a Gravitational Force [view large image]

Figure 01b Coupling Strength [view large image]

data. In this picture, the particles and forces are stuck to a
3-D (3 spatial dimensions) brane sitting inside the extra dimensions, the only force in the higher-dimensional bulk is
KK Graviton gravity. The Large Hadron Collider (LHC) (at CERN, to be completed by 2007) should be able to detect strong quantum gravitational effects; for example, the high-energy particle beam at the LHC can cool by boiling off gravitons into the extra dimensions. More exotic gravitational objects, such as small black holes, can also be produced at the TeV energies available with LHC. In fact the LHC now becomes a quantum-gravity machine, which can look into these extra dimensions of space through apparent violations of energy conservation, as well as the appearance and disappearance of graviton's KK particles from extra dimensions (see Figure 02).

Figure 02 KK Graviton
[view large image]

In 3-dimensional space the gravitational force between two mass m1 and m2 is (see Figure 01a):

F = -Gm1m2 / r2 ---------- (1a)

where r is the separation between the two masses, and

G = 6.6742x10-8 cm3/sec2-gm = 6.7087x10-39 c / (Gev / c2)2 = c / ( 1.22x1019 Gev / c2)2

is the gravitational constant in the 3-dimensional space expressed in various forms.

If the gravitational force spreads over to n additional dimensions, which curl up into circles with radius R, the generalized gravitational force can be expressed as:

F = -gLnm1m2 / rn+2      for r < R      ---------- (1b)

where g is the gravitational constant for the case of extra dimension, and L is a length scale (distance scale) associated with g. The length scale is related to the energy scale by the formula for the de Broglie wavelength :

= c/E = [2x10-14/E(Gev)] cm ---------- (2)

For the example in Figure 01b, the shielding or antishielding from the virtual particles surrounding the target particle in addition to the real charge itself combine to yield an effective coupling strength, which is now a function of the energy transfered or its corresponding length scale. Eq.(2) shows that more energetic probe can resolve finer detail of the target. Similarly for the case of gravity, the gravitational constant (coupling strength) can vary with the length scale as well (not shown in Figure 01b).

Extra-dimension Starting from the source, the lines of force spread apart rapidly through all the dimensions. At distance larger than R, the force lines have filled the extra dimensions, which then cease to exert further influence (see Figure 03). Thus, we can equate Eq. (1a) and (1b) at the distance R and obtain:

Figure 03 Extra-dimension [view large image]

G = g (Ln/Rn) ---------- (3)

which states that the value for the gravitational constant G is the result of modifying the original value g by a factor (Ln/Rn). In natural units§ which set = 1 and c = 1, then G ~ 1/( 1019 Gev )2. It can be translated into the length scale: K2 ~ (10-33cm)2. To eliminate the hierarchy problem of huge energy gap between the Planck mass and the electro-weak mass, it is assumed that the strength of the gravitational interaction in the extra dimensions is g ~ 1/( 1 Tev )2, then g can be similarly expressed in length scale as k2 ~ (10-17cm)2. Since L = k by definition, Eq.(3) can be re-written as:

Rn ~ kn+2 / K2 ---------- (4a)      or      R ~ 10(32/n) - 17 cm ---------- (4b)

Some interesting cases of n and R are listed in Table 01 below.

# of Extra Dimensions n Curled-up Size R (cm) Comments
0 Infinity No extra dimension
1 1015 ~ Size of the Solar system
2 0.1 At detectable limit (in the 1990's). The limit is down to 4.4x10-3 cm in 2012.
3 4.6x10-7 Experiments in 2000's seem to rule out curled-up size larger than 4.4x10-3 cm, this is the next viable possibility for large extra dimension
6 2.15x10-12 Upper limit for the # of extra dimensions in superstring theory
Infinity 10-17 Infinite # of extra dimensions

Table 01 Curled-up Size as a Function of n

A different view of Eq.(4a) is to impose a certain curled-up size such as R = 0.1 cm, and to consider the interaction strength (in term of the length scale k) as a function of the number of extra dimensions n. With the observed gravitational length scale (for the case of no extra dimension) K = 10-33 cm as quoted above, Eq.(4a) becomes:

k ~ 10-(n+66)/(n+2) ---------- (4c)

Table 02 lists some of the more interesting cases.

# of Extra Dimensions n Interaction Strength k (cm) Comments
0 10-33 No extra dimension
1 0.46x10-22 In the middle of the "energy desert"
2 10-17 Electroweak interaction scale at TeV
3 0.16x10-13 Strong interaction scale at GeV
6 10-9 Upper limit for the # of extra dimensions in superstring theory
~ electromagnetic interaction in atomic scale
Infinity 10-1 Infinite # of extra dimensions

Table 02 Interaction Strength as a Function of n

For the case of n = 2, k = 10-17 cm, and incident energy ~ 1 TeV, the Schwarzschild radius is:

rs = 2 g m / c2 = 2 G (k / K)2 m / c2 = 2.4 x 10-17 cm > k

Black Hole, Simulation Thus, a black hole can be created with such energy packed into the corresponding length scale. Such mini black hole will evaporate in 10-88 seconds, losing most of its mass by Hawking radiation. It is estimated that the final burst should radiate a large number of particles in all directions with very high energies. The decay products include all the particle species in nature. The LHC could provide the first evidence for Hawking radiation from such signatures of the black holes. Figure 04a depicts the simulated decay of a black hole inside a particle detector. From the center of the accelerator pipe (black circle) emerge particles (spokes) registered by layers of detectors (concentric colored rings). The sequence from birth to death of a mini black hole with an initial mass of 10 Tev is shown schematically in Figure 04b. It is created by the collision of two energetic particles (a). The scenario suggests that it will emit gravitational and electromagnetic waves as it settles

Figure 04a Black Hole, Simulation [view large image]

down. It becomes an almost featureless body, characterized solely by charge, spin and mass (b). Even the charge quickly leaks away as the black hole gives off charged particles. At first, the black hole emission comes at the expense of spin (c), so it slows down and
Black Hole, Evaporation relaxes into a spherical shape, which is characterized solely by its mass (d). Finally, the mass bursts away in the form of radiation and massive particles (e). The remnant of the black hole approaches the Planck mass and disappears into nothingness.

Figure 04b Black Hole, Evaporation [view large image]

Note that if there is no leaking of gravity into the hypothetical extra-dimensions, the Schwarzschild radius would be 10-49 cm (for an incident energy of 1 TeV), which is much smaller than the smallest conceivable length scale of 10-33 cm. No black hole can form under this circumstance (a black hole can form only when the Schwarzschild radius rs is outside the central object) - there is no danger for the LHC to create a blackhole devouring the Earth and the Solar system.

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Black Hole in LHC

High Energy Collision at LHC There were many people who worried about the Earth devoured by a black hole created within the LHC (Figure 04c). In order to forestall the LHC's operation, at least one lawsuit had been filed against CERN and a host of government agencies. Straight forward calculation shows that no black hole would be created by the maximum collision energy of 14 Tev. However, if the universe does possess extra-dimensions as predicted by the theory of superstring, black holes will be materialized depending on the number of extra-dimensions. Even then such miniature black holes will evaporate long before

Figure 04c High Energy Collision at LHC

they are able to suck anything inside. Following is a more quantitative analysis to supplement the assurance (of safety) provided by so many physicists.

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Manyfold Universe

Manyfold Universe Superstring theory predicts the universe has ten or eleven dimensions. Why don't we see these extra dimensions? Perhaps we are living on a brane (short for membrane) - floating in a space of five, six or more dimensions, like a soap bubble in the bathroom. The "manyfold universe" theory asserts that the brane we live on could be folded over on itself many times, accordion-fashion (Figure 05). Light could travel only on the brane, but gravity could take a shortcut by jumping from one fold to the next. Nearby matter on other folds can be detected gravitationally as unseen dark matter, since its emitting light takes a long time to reach us traveling around the fold (see Figure 05).

Figure 05 Manyfold Universe [view large image]

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Pre-Big Bang Universes

Two Views of BB The concept of branes floating in higher dimensional space has offered some new ideas about condition at the moment of Big Bang and further back to the time before the event (see Figure 08). According to general relativity, density and temperature become infinity at the moment of Big Bang. The nonzero size and novel symmetries of strings set upper bounds to physical quantities that increase without limit in conventional theories, and they set lower bounds to quantities that decrease. The string theory expects that the curvature of space-time increase as the history of the universe is re-winded backward in time. But instead of going all the way to infinity, it eventually hits a maximum and shrinks once more (see Figure 06). The string theory also proposes some hazarded guesses about the pre-big bang universe. There are two popular models floating around - the Pre-Big Bang and the Ekpyrotic scenarios.

Figure 06 Two Views of the Big Bang
[view large image]

The Pre-Big Bang model was developed in 1991 by combining T-duality with the symmetry of time reversal. The combination gives rise to a cosmological model in which there was a period of acceleration before the Big Bang and then the deceleration.
Pre-Big Bang Ekpyrotic Model The Big Bang was simply a violent transition from on phase to another (see Figure 07). The other model is the Ekpyrotic (conflagration) scenario. It relies on the idea that our universe is one of many D-branes floating within a higher-dimensional space. The branes exert attractive forces on one another and occasionally collide. The Big Bang could be the impact of another brane into ours as shown in Figure 08. The idea has been extended to a cyclic model, in which the two branes are held together by a spring-like force (proportional to separation). The moment of collision (zero separation) corresponds to the Big Bang when matter and radiation are produced. The spring-like force manifests itself as dark energy, which is weak at the moment of Big Bang but becomes increasingly

Figure 07 Pre-BB [view large image]

Figure 08 Ekpyrotic Model
[view large image]

important as the separation getting larger. The dark matter is just the material on the other brane interacting with the visible matter on our brane only through gravitation.

In another application of the String Theory, a model universe without the Big Bang and inflation has been constructed by squeezing a 3D-brane (our universe) into the throat of the Calabi-Yau space (Figure 09a) according to the following scenario:
1. The spinning brane containing our universe is squeezed as it drops down the throat of the Calabi-Yau space.
2. The brane is reflected near the end of the throat.
3. The universe expands as it flies back up the throat.
There is no Big Bang event and no requirement for inflation as the universe is ever-existing.
CalabiYau Space Branes Collision A variance of this inflation inside the Calabi-Yau throat posits that the annihilation of the incoming brane and an antibrane at the end of the throat triggered the Big Bang (Figure 09b). According to this scenario, inflation occurred as the branes approached each other. Annihilation of the branes released vast energy to heat up the universe and to initiate the Big Bang. It is purported that different Calabi-Yau configuration will yield different signature on the

Figure 09a Calabi-Yau Space [view large image]

Figure 09b Branes Collision [view large image]

cosmic microwave background radiation (CMBR). Thus theoretically, the CMBR pattern will determine uniquely the Calabi-Yau manifold for our universe.

CalabiYau Space Variation Yet another stringy scenario found that the configuration of the extra dimensions can change over from one form to another as shown in Figure 09c. This general process is called moduli inflation because the moduli fields, which describe the geometry (the size and shape of the hidden space dimensions in huge number of possible configurations), act as the inflatons driving the inflation of the observed three dimensions. This model predicts un-observably weak gravitational waves. Thus, it would be ruled out if primordial gravitational waves are detected by observation in the future.

Figure 09c Mo- duli Variation

There are other models, which do not depend on the theory of Superstring. One of these involves a sea of microscopic black holes before the Big Bang (see Figure 10). The uncertainty principle of quantum mechanics creates some ordinary space between them. It is those slow moving spaces with low-entropy that can expand quickly imitating the Big Bang. Another model starts with the assumption that the amount of dark energy in a given volume increases as the universe expands until at the last moment called the "big rip" in which the universe's expansion rate becomes infinite and each part of this mother universe now
Black Hole Sea detached from all the others; then the attractive aspect of the dark energy takes over. This causes each island universe to contract, but eventually it gets so dense that its radiation reverses the contraction. We are left with innumerable expanding little universes - of which ours may have been one (see Figure 11).

Figure 10 Black Hole Sea [view large image]

Figure 11 Fragmenting Universe [view large image]

Remarkably, these models of Pre-Big Bang universe offer prediction that can be verified by observation. Since it proposes an ever increasing cosmological constant, the fragmenting universe requires the w parameter ~ -1.05. By contrast, in the Ekpyrotic model, dark energy results from the potential energy between the two branes, which depends on how far apart they are. As the branes move apart, dark energy's strength decreases. This corresponds to a w ~ -0.95. While the potential evidence for black hole models would be associated with the detection of primordial black holes. Right now, the observed w parameter is very close to -1.0 and there are no observed primordial black holes of size < 100 gm. Thus, none of the models will be proved correct any time soon.

See more Pre-Big Bang Theories.

§ The natural units are defined by = c = 1. Setting c = 1 means that seconds and centimeters are to be treated on the same footing, such that exactly 3x1010 centimeters is equivalent to 1 second. Thus, the second and the centimeter are to be treated as if they were expressions of the same unit. Likewise, setting = 1 means that the erg-sec is now dimensionless with 1.054x10-27 erg = 1/sec, so the erg and the second are inverses of each other. This also means that the gram or Gev is inversely related to the centimeter or 1 cm = 1 / (2x10-14Gev). With these conventions, the only unit that survives is the centimeter, or equivalently, the gram or Gev.