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Detecting Dark Energy with Atoms

By

James Eyres

Physics Research Project (F34PJM UK) (FYR 14-15)

MSCI Physics

Student Number: 4164849

Tutor: Dr J OShea

May, 2015

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Abstract In order to explain the accelerating expansion of the universe it seems that dark energy must be modelled as a new scalar field, quintessence, whose range is coupled to the surrounding matter energy density. This new field should mediate a fifth force which has not yet been detected. The chameleon mechanism of quintessence would explain the accelerating expansion of the universe and also explain why the field has not been detected since the range of the scalar field changes with its environment- making it small on earth but large in the cosmos. This project will ultimately use the principles of atom interferometry in order to theoretically determine the constraints needed to be able to detect the fifth force, while modelling this fifth force around a massive spherical object and a point source along the way. Introduction & Literature Review The universe is expanding. This expansion is a logical consequence of the big bang. However, what may be less apparent is that the universe is expanding at an increasing rate i.e. accelerating. This is equivalent to stating that the cosmic scale factor () has a second derivative which is above zero [1]. The accelerating expansion was first discovered in 1998 by the Supernova Cosmology Project by observing distant supernovae [2]. The existence of dark energy, which has negative pressure and is spread equally throughout the cosmos, is the most widely accepted explanation for this accelerating expansion. This is not, however, the only proposed explanation for the accelerating expansion of the universe, another model is quintessence. This type of dark energy would have a non-constant state equation and a density that decreased with time. Further explanations include dark fluid, which attempts to combine dark energy and dark matter [3] and repulsive gravitational interactions of antimatter [4]. Dark energy is an obscure type of energy permeating all of space and is the driving force behind the accelerating expansion of the universe [5]. It is estimated that the observable universe is made up of 26.8% dark matter, 68.3% dark energy and 4.9% ordinary matter [6], according to the Planck mission team. The density of dark energy is lower than that of ordinary matter and dark matter, with a value of 6.911027 3. There are two main proposed forms of dark energy. The most widely accepted form (as briefly mentioned above) is the cosmological constant, a homogeneous, constant energy density [7]. The other form includes scalar fields, for example moduli or quintessence, changing fields whose energy density varies in space and time. It is this second type of dark energy upon which this project will be based, and the reasons for this will be briefly summarised below. The cosmological constant can be formulated to be the value of the energy density of the vacuum of space. Einstein originally introduced the concept in 1917 [8] as an amendment to the theory of general relativity in order to achieve a stationary universe by supressing gravity. In 1929 Hubble discovered that all galaxies outside our local group are moving away from each other suggesting that the universe is expanding. Hence Einstein abandoned the cosmological constant and dubbed it his greatest blunder. Hereafter the cosmological constant was set at zero for around 80 years. As briefly mentioned the accelerating expansion of the universe was discovered in 1998 from numerous independent experimental observations. Furthermore the existence of dark energy was suggested and estimated to make up about 70% of the universe. Dark energy isnt very well understood on a fundamental level, however it is apparent that the properties of dark energy should include that it clusters less (or perhaps not at all) than matter and that it dilutes slower than matter

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as the universe expands. Due to the nature of a cosmological constant it is the simplest proposed form of dark energy, it is constant in time and space, which leads to the current, as of 2015, standard model of cosmology, the Lambda-CDM model. There is, however, a large problem with the cosmological constant, which has been called the worst theoretical prediction in the history of physics [9]. This problem is that the value of the quantum vacuum, as predicted by quantum field theories, is extremely high. Furthermore, the quantum vacuum can be argued to be equivalent to the cosmological constant [10]. There is a discrepancy between the predicted quantum vacuum (and hence predicted cosmological constant) and the experimentally measured cosmological constant of 120 orders of magnitude, where the experimental cosmological constant is the smaller of the two. To put this discrepancy into perspective the statement, the universe consists of exactly one elementary particle is more correct than the previous discrepancy, by at least 10 orders of magnitude. Some supersymmetry theories even predict a cosmological constant of zero. There is no known way to theoretically derive the observed cosmological constant from particle physics, this is the cosmological constant problem. This problem leads into the reason for the need of other models of dark energy, such as quintessence. Since the cosmological constant appears to be the wrong model for dark energy, introducing new models is the next logical step in trying to examine the fundamental properties of dark energy. Quintessence is a proposed form of dark energy, used to explain the accelerating expansion of the universe. Unlike the cosmological constant, quintessence is dynamic and changes over time, it can be attractive of repulsive depending on the ratio of its potential and kinetic energies. It is suggested that quintessence became repulsive when the universe was approximately 4 billion years old [11], and is suggested to be a fifth fundamental force, hence its name. A direct follow-on from quintessence is the chameleon, a scalar particle which couples to matter [12], that is a proposed dark energy candidate. The chameleon was first proposed in 2003 by Khoury et al. [18] with multiple articles building upon the mathematical model, for example a 2014 paper by Burrage et al. [19] both showing that the chameleon force obeys a Yukawa potential. The chameleon has a variable effective mass that increases as a function of the surrounding energy density; due to this property the range of the mediated force is predicted to be small where the mass density is large and large when the mass density is small. As an example, on earth the range of the force is predicted to be less than 1mm whereas in intergalactic regions it is predicted to have a range of thousands of parsecs. This property explains the observed accelerating expansion of the universe, however, it also makes the chameleon very difficult to experimentally detect. Current constraints on the earth-bound tests of gravity and general relativity are too large to detect this hypothetical fifth force, as will be examined below.

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Throughout this project linear differential equations will be solved. As an illustration of the solution to these types of equations Newtons law of universal gravitation has been derived starting from the Laplace and Poisson equations.

There are many different tests that probe general relativity, as described by a 2008 review article by Adelberger et al. The equivalence principle states that a homogeneous gravitational field is locally equivalent to a uniformly accelerated reference frame [13]. This principle can be tested, as described in the article and as pictured in Figure 1. The most direct test of this principle is that the trajectory of a point object freely falling in a gravitational field depends only on the objects initial condition and not on its composition. As shown in the figure, the torque of the wire describes the gravitational field and a torque that does not equal zero can only occur for two reasons. Either, the two masses are in

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different gravitational fields or the equivalence principle has been violated. If the force vectors on the masses arent parallel then a torque acts on the pendulum. This causes it to twist until the restoring torque of the fibre counteracts the initial torque. The twist of the pendulum can be measured by reflecting a collimated light beam from a mirror mounted on the pendulum. In this experiment, any violation of the equivalence principle could be attributed to the chameleon. However, no such violations are detected, to quite a significant precision.

The article goes on further to describe how no violations have been found of any tests of gravity and examines such experiments as short-range tests of the gravitational inverse-square law and spin-pendulum experiments. Both of these exploit the torque of massive objects in a system of wires to test their constituent principles, however, neither of which found any kind of violation or fifth force. It follows that there should be a reason why earth-bound experiments do not constrain the chameleon, if it is present, and there is. A partial explanation comes from the range of the chameleon on earth (as mentioned above this is below 1mm due to the large mass energy density) which leads to the thin shell effect. As depicted in Figure 2. The chameleon is screened by the size of the massive object, that is, most of the effects of a fifth field are blocked from detection due to the them not being able to escape the source. The outer layer of the object would allow some of the chameleon force to

escape and be detected, however, this is only a small proportion of the object because, in order to escape the source, the chameleon would have to be at or near the surface. The contribution from this surface chameleon force would be negligible when compared to the gravitational force of the source. Furthermore, the experimental limitations of these massive experiments produce larger errors than that of the force any surface chameleon would contribute. As a brief divergence from cosmology and gravity, the introduction of interferometry is prudent. Interferometry is any technique where waves, often electromagnetic, are superimposed and interfere with themselves in order to find out information about the waves [14]. It has a wide range of applications and is often used to measure small displacements to a good degree of accuracy. The most

Figure 1- the left graph shows the inertial

and gravitational forces on a test-body

mounted at a latitude = 4 7 . The ratio of

the forces is exaggerated by a factor 200.

The right figure shows the forces on a

simplified torsion pendulum where 2 is

greater than 1. The pendulum tips by an

angle so that the centre of mass lies

vertically below P . [13]

Figure 2- Illustration

of the thin shell effect

and explanation of

why this makes it

impossible to detect

chameleon forces

classically.

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famous interferometry experiment was perhaps the Michelson-Morley experiment where the duo attempted to detect the relative motion of matter through the aether [15]. A beam of light from a monochromatic source is split into two and then recombined, this recombination produces an interference pattern of bright and dark rings that change proportionally to the difference in length of the two paths the light took. This experiment is sensitive to sub-wavelength changes in distance. Taking the idea of interferometry and expanding it, one arrives at atom interferometry. An atom interferometer is similar to a classical interferometer, the difference comes from the wave source. An atom interferometer exploits the wave characteristics of atoms and can be used to make highly accurate comparisons of displacement. Atom interferometry can constrain fundamental constants like the gravitational constant and could be used to search for gravitational waves [16] or the fifth force, as will be explained below. As described in Figure 3 the chameleon is not screened by a point mass (an atom) and thus can be detected. This leads to the idea of using atom interferometry to constrain the chameleon. Atom interferometry would detect gravitational forces plus an extra, fifth, force if indeed there was a chameleon present.

A 2007 study from Dimopoulos et al. performed atom interferometry with the hope of detecting the fifth force. Below is a brief summarisation of their experiment through the use of one of their papers figures. Figure 4 [17] demonstrates the experiment succinctly. The wavy lines in the figure represent the lasers while the smooth lines represent the atom trajectories. The y-axis shows the height of the atom while the x-axis shows the progression of time. The solid lines represent atoms that are only being subject to the effects of gravity while the dashed lines are atoms who have received an increase in energy (have increased energy state) due to the interaction with the lasers. There are two lasers in each interaction simply so one can excite the atom while the other can stimulate the atom to emit light into the other. This setup allows control over the fine-structure splitting of the atoms.

Although Figure 4 looks complex, it becomes easier to understand by following the trajectories of the atom. There are three instances when the atom interacts with the lasers. Whenever this happens the velocity of the atom changes. At the first interaction there is a probability of 0.5 that the atom will receive a kick, thus there are two paths which an atom can take (shown by the dashed and solid lines, where the dashed line atoms are travelling upwards more rapidly than the solid line atoms). Due to

Figure 3- Illustration of the

chameleon escaping the atom

and being able to be detected.

Figure 4- Illustration of atom

interferometry as performed in

the 2008 study detailed in this

report [17].

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the quantum nature of the experiment the atom actually takes both paths. Gravity slows the atom, hence accounting for the curved (deceleration) lines and eventually they begin to descend. The second interaction (at time T) is chosen so that the atom in the dashed line trajectory is flipped and vice versa with the solid line thus bringing the atom back together for the third interaction where the lasers perform the same action as at time 0. Much like classical interferometry, an interference pattern is observed from the recombination of the atom which can be used to detect very small displacements. This experiment couldnt constrain the chameleon, however, it is clear to see that any deviation from the expected gravitational trajectory of the atom could be accounted for by the fifth force. Furthermore, atom interferometry is a relatively young field and so it is reasonable to predict that this type of experiment could constrain the chameleon in the future. A significant deviation from predicted gravitational interferometry results would constitute a smoking gun for the chameleon model of dark energy. Methodology It may be possible to detect chameleon forces using ultra-cold earth-bound experiments, or in other

words atom interferometry experiments. Although the chameleon field can be described by

relativistic quantum field theory, a much more manageable equation describing its non-relativistic

steady-state can be written as [18],

2 = ,+

. [1]

Where is the field, , is the first derivative of the potential of the field with respect to , is a

coupling constant, is the local density and is the Planck mass.

First, the equation of motion will be solved. It should be noted that throughout this project it has been

assumed that objects and fluids are perfect and have homogeneous density and pressure. For

simplicity the equation of motion was solved for a non-moving, spherically symmetric source object

with non-relativistic matter that has been placed into a medium of smaller, non-zero, homogeneous

density. Assuming that the coupling term, , is in fact a constant and that the equation of

motion can be written as,

1

2

[2

()

] =

()

+

()

() . [2]

Where 2 has been re-written in spherical polar coordinates and =

has been defined for

continuity with the literature. This is congruent with Figure 5 which shows that the effective potential

can be written as,

() = () +

. [3]

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From Figure 5 it is clear to see where equation [3] comes from, since the effective potential is the

combined contributions from the actual potential which is of the runaway form and the coupling to

matter density which is dependent on the local density. The way in which the local density affects the

effective potential is shown, quite succinctly, in Figure 6. The effective potentials take on very different

forms depending on whether the local density is large or small.

In order to appreciate the meaning of the term density in this context Figure 7 has been included.

This shows the general system for which the equation of motion will be solved, where >

0 with the density of the source object, , and the density of the medium . The point at which

= 0 describes the centre of the source object, while the point at which = describes the surface

of the source object, where the local density drops from the value of the density of the source object

to the value of the density of the local medium, thus creating this step function graph.

Figure 5- The effective potential

(solid) is the sum of the two contributing

dashed/ dotted curves; the actual

potential () (dashed) and coupling to

matter density (dotted).

Figure 6- The effective potential in

regions of high local density (in an

atom for example) and low local

density (in the surrounding medium

for example). As decreases the

minimum shifts to larger values of .

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With anticipation of the results it is useful to classify two regimes that source objects can be a part of.

These are the thin shell regime and the thick shell regime. As described in the introduction, objects

that have a thin shell can be divided into infinitesimal volume elements of equal size and only those

elements which lie at the surface of the object will contribute to the exterior chameleon field. This is

not the case for objects in the thick shell regime and so the solution of the equation of motion is

dependent upon which regime is solved. Therefore the solutions to the equation of motion have been

split into two sections representing the two regimes. Not only will the equation of motion be solved

for two different regimes, it will be solved, much like any other second order differential, by splitting

each regime into adjacent areas which are connected through boundary conditions, specifically these

will be the interior of the source object, the thin shell of the source object (for the thin shell regime

only) and the exterior of the source object.

Thin shell regime

Exterior solution

In Figure 6 for small there is a distinct minima outside the source at , by introducing a massive

object such as the one described in the text the minima will be perturbed. Since the minima is

perturbed the effective potential can be modelled by simple harmonic motion meaning that a Taylor

expansion can be implemented to change the form of the equation of motion,

2

2+

2

= ( )

2()

2 . [4]

At first glance this form appears to be more complex than the original, however, by implementing

three substitutions the equation of motion can be simplified dramatically to the form of equation [8].

These substitutions in chronological order are as follows,

= ( ) , [5]

2()

2= 2 , [6]

() =()

. [7]

By applying the aforementioned substitutions the equation of motion now becomes,

2()

2= 2() . [8]

Equation [8] is indeed a much more simple form of the equation of motion and it is reminiscent of

equations of motion that describe simple harmonic motion. Therefore, this equation was solved in a

Figure 7- The function describing the local

density. Inside objects the density is high

while outside the density is low, leading

to a step function. > 0.

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way that one would solve a simple harmonic motion equation of motion, by a trial solution of the

form,

() = + . [9]

By taking into consideration what happens at the extremes of the model, namely noting that as

, the solution can be written as,

=

+ , [10]

where is a constant that will be determined by applying boundary conditions to the other areas in

this regime.

Shell solution

By crossing the boundary of the object the position of the minima, , is changed, as shown by

figure 8. This causes the field to roll to the new value of the minima and hence this part of the model

is continuous. In this region the potential only sees the coupling to matter potential contribution to

the effective potential, this means that the effective potential has a different value to that in the

exterior solution.

Since the effective potential follows the coupling to matter density it can be written that,

=

. [11]

Furthermore, by using equation [2] and the simplifications from earlier, it can be written that,

2

. [12]

In order to solve this equation the rule = + will be used, where is the complimentary

function and is the particular integral. This is a standard method for solving second order

differential equations of this form and it is done so by using the trial solutions,

= , [13]

= , [14]

where and are constants to be determined. Taking into consideration dimensional analysis

regarding the radius, namely that = 2 for the dimensions to be consistent it follows that the general

solutions are:

Figure 8- As the field moves from the

surrounding medium into the object there

is a sharp increase in local density, moving

the position of the minima so that the field

has to roll to reach its new minima. Hence

why the effective potential is treated as

continuous.

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= 2 +1

, [15]

=

62 . [16]

By combining equation [15] & [16] it is found that the general solution for the shell in the thin shell

regime is,

= 2 +1

+

62 . [17]

As in the exterior solution, the constants 2 and 1 will be determined by applying the boundary

conditions at the appropriate regions.

Interior solution

For the interior solution, , where is simply the field of the chameleon inside the source

object. By studying the high density area of Figure 6 which describes the interior of the source object

it is clear to see that the field wants to sit and the bottom of the curve, since it take a lot of energy

for the field to move up the side of the steep curve. This minima is the point at which the field

minimises the potential, where the value of this field is . In other words it is the value of the field

for which the first derivative of the effective potential is zero,

() = 0.

Defining the constants

In order to define the constants 1 and 2 a common rule can be applied to the boundary of the

region, that is, at = , where describes the point inside the object at which the thin shell

begins. At this radius the field must be smooth and continuous ( = &

=

= 0). Therefore,

by taking the solution to the shell region, equation [17], the solution to the interior region and applying

boundary conditions it is found that,

= 2 +1

+

2

6 , [18]

1

2 +

3

= 0 . [19]

Equation [19] can be rearranged to give a value for 1 which can then be substituted into [18] to give

a value for 2. Now that the constants have been found the general solution for the shell in the thin

shell regime can be quantified,

=

3(

2

2+

3

)

2

2+ For < < . [20]

The final constant to be determined in the thin shell regime is the value of , this is found through a

similar method to the previous. The same boundary conditions are applied to the edge of the object

where = , between the solutions for the shell and the exterior leading to the following equations,

2 +1

+

62 =

+ , [21]

1

+

3 =

2

. [22]

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Values for 1 and 2 can be substituted into [22] and using simple algebra a value for can be found

and substituted into equation [10]. Furthermore, by using relations such as = =4

3

3 and

defining = where is the radius of the shell it is straight forward to find the solution

for the exterior of the thin shell regime,

=

4(

3

)

()

(

1

1+) + For < < . [23]

Where the substitution = has been made since the minimum value that the field can take is

the value of the field in the surrounding medium, which has been defined to be non-zero.

Thick shell regime

Exterior solution

The thick shell solution for the exterior is solved by following the same steps as the thin shell solution

for the exterior. However the constant for the general solution is calculated with the constraint

that 0. Therefore this region is most easily solved by applying this limit to equation [22]. This

constraint effectively changes the value of the constant and hence the total solution, which is now

written as,

= (

4)

()

(

1

1+) + For < < . [24]

Interior solution

The interior solution to the thick shell is solved in much the same way as the exterior, by applying the

constraint 0, it is easiest to take equation [20], the solution for the shell in the thin shell

regime, replace with and apply the constraint. describes the initial value of the field so that

= ( = 0) and . By applying these substitutions and constraints it follows that the

interior solution to the thick shell regime is,

=

2

6+ For 0 < < . [25]

Final solutions

Now that we have the solutions for each region of both systems we can write the solutions in a way

that eliminates the dependence on (), in other words we can equate the equations at the boundary

of the source object, = . In order to eliminate this dependence equations [24] & [25] are equated

thus eliminating (). By applying conditions such as in natural units 2 =

1

8 and the Newtonian

potential =

the equation describing the thick shell solution becomes,

= 3 . [26]

A similar method can be used to eliminate the () dependence in the thin shell regime. To do this

equations [20] & [23] are equated since the boundary is continuous, note that 1. From here

we can perform a number of operations that will simplify our equation such as, = =

=

3

43,

applying a Taylor series since = where 1, in natural units 2 =

1

8 and the

Newtonian potential =

. By following this logic the equation describing the thin shell regime is,

=

6

. [27]

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Thin shell regime

In order to investigate what types of object have a thin or thick shell equation [27] can be used, since,

for a thin shell

1 and for a thick shell this is not true. Therefore equation [27] can be investigated

numerically to determine whether a chosen system has a thin or thick shell but first it must be turned

into a form with quantifiable terms. We can eliminate immediately by applying the condition

that . In order to eliminate a value must be chosen for the potential of the chameleon

field. A common choice for this potential is [19],

() =5

. [28]

From Figure 5 it is clear to see that the effective potential of our system is given by the summation of

the constituent parts making up the contributions to the effective potential, one of which is (),

therefore we can substitute our value for the potential into the equation describing the effective

potential,

() =5

+

. [29]

Due to the fact that is the term that must be eliminated we must investigate the effective potential

in the exterior region of the problem. By investigating the exterior region can be rewritten as ,

likewise can be rewritten as , where these terms describe the local density and field of the

surrounding medium respectively. Much like what was done with the equation of motion the

condition that the coupling is constant and is applied to the effective potential. By

differentiating this equation with respect to the field and by setting the result (())

= 0 a value

for the exterior field is obtained,

= = (5

)

1

2 . [30]

In order to arrive at the final solution is eliminated from equation [27], as above, and then equation

[30] is substituted into the remaining equation to eliminate ,

=

4

3(

3

)

1

2(

5

3)

1

2

. [31]

This equation is useful since it describes the ratio of the radius of the thin shell of the source object to

the radius of the source object. If this ratio is much smaller than 1 then the object has a thin shell,

otherwise it has a thick shell. The next section will discuss the procedure of classifying objects as either

in the thin shell regime or the thick shell regime depending on the value of this ratio.

Classification of source objects

Equation [31] describes the solution to the thin shell regime and it does so in quantifiable terms. That

is to say that the mass and radius of the source object can be chosen as well as the background density

of the medium in which the source object has been placed. As discussed the ratio describing the thin

shell regime is

1 and from this we can investigate the values for which the parameters and

are valid. First, however, equation [31] will be rewritten using the condition =

and defining the

radius and mass of the source as such,

14

=

4

3(

35

)

1

2

. [31]

The fraction describing the ratio of the radius of the thin shell to the radius of the object has been

chosen to be 1

100. This essentially means that for

1

100>

4

3(

35

)

1

2

the source object has a thin shell

while for 1

100 3 . [49]

The first term in equation [48] is the contribution to the acceleration from gravity and the second is

the contribution from interaction with the chameleon. When the source object is large and heavy the

terms go to 0, thus explaining the lack of detection of the chameleon on macroscopic levels.

Conversely, when the source object is small and light, like an atom, the terms go to 1, hence the

possibility of detection of the chameleon in these experiments. The term describes the self-

interaction of the field from the test particle, while the term describes the interaction of the field

from the source object. When 2 < 3 the field is unsuppressed, and 1. When

2 >

3 the field is suppressed inside the object except for a thin shell at the surface of the object, this

means that the chameleon is negligible when compared to the contribution of the gravitational term.

To be able to quantify the second possible value of must be investigated. To quantify we

consider in this vacuum chamber which has large, dense, spherical walls with radius . At the walls

of the chamber the field is near zero, however, the field value rises to that of at the centre of the

chamber, given by equation [51]. If this chamber is large enough then the field reaches its

equilibrium value. Clearly for small chamber the field may not reach its equilibrium value.

Hence, , in equation [49] can essentially take on two values the first is [19],

= (52)

1

3 . [50]

There is an unknown in equation [50] that must be determined. By applying a Matlab code which uses

a Runga-Kutta method for solving differential equations the constant is found and the equation

becomes,

= 0.69(52)

1

3 . [51]

The second value that can take is given for when the field reaches its equilibrium value when the

chamber is large enough, this can be written as [19],

= = (5

)

1

2 , [52]

where describes the local density of matter. In essence can have three different values depending

on the location in the vacuum chamber, this can be written as,

20

= min (1,3

2 ,

3

2 ) , [53]

where and have already been defined.

Constrained Parameter Space

On substitution of equation [53] into equation [48] we have a full description of the acceleration felt

by atoms in an atom interferometer. In February 2015 a research group at the University of California,

Berkeley did an atom interferometry experiment with the same setup as in the beginning of this

section, with caesium atoms and aluminium source object. The group concluded that the atom

experienced an acceleration of 5.5 2 at the 95% confidence level [23]. By using this value as an

upper bound for the acceleration in equation [48] a Matlab code similar to the thin and thick shell

regime section can be implemented. This code is described in Appendix 2, however, in summary it

creates an inequality equation where equation [48] holds true if its theoretical value for the

acceleration is below the experimental value for the acceleration, 5.5 2 and false otherwise. This

allows the parameter space for & to be plotted, as in Figure 13, where the unphysical results

(white region) have been excluded, leaving the remaining constrained parameter space (black region)

as physical values for & .

Figure 13 shows the constrained parameter space of the chameleon. Here the white area has been

excluded since, given the acceleration from the group at Berkeley, these values of & are

Figure 13- The constrained parameter space available to the chameleon. The white region

has been proved unphysical given 5.5 2 as an upper bound at the 95% confidence

level of the acceleration felt by atoms in an interferometry experiment. The black region

describes all of the values of & for which the chameleon theory is still valid.

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unphysical. The remaining black region of the parameter plot is the area for which the values of &

are physical and thus still valid under the chameleon theory. As atom interferometry experiments

become more accurate the excluded region of the parameter space will become larger. Interferometry

experiments with lighter atoms (lithium for example) could improve the sensitivity of these results.

Furthermore, constraints on the ranges of & could become more accurate, restricting the axis

and constraining the chameleon further. This constraining will continue until the theory has been

falsified, meaning the whole available parameter space is unphysical (white), or until there are a

distinct set of values for & for which the chameleon theory is valid. As the precision of atom

interferometry increases there is the possibility that chameleon forces could be detected, these would

present themselves as an additional acceleration than a non-chameleon theory would predict (i.e.

without the second term in equation [48]). Thus being a smoking gun for the chameleon theory.

Conclusion

In summary, a scalar field describing dark energy has been introduced since the current accepted

model of dark energy in the model has a number of problems associated with it as it assumes

is a constant. It has been proven that atoms do in fact have a thick shell in this description of the

chameleon through the use of a Matlab code. This code will determine whether any spherical object

in a given environment falls within the thin shell or the thick shell regime. Since atoms have a thick

shell it may be possible to detect chameleon forces by performing experiments on these atoms. With

this knowledge in mind it has been shown how atom interferometry can determine the acceleration

felt by atoms in the experiment. Finally, the acceleration from a recent atom interferometry

experiment has been used to impose, to date, the most accurate limits on the parameters & in

the chameleon theory.

References

[1]- An Introduction to Galaxies and Cosmology. Jones, Mark H., Robert J. Lambourne (2004). Cambridge University Press. p. 244. ISBN 978-0-521-83738-5 [2]- http://www.bbc.co.uk/news/science-environment-15165371, "Nobel physics prize honours accelerating Universe find". BBC News. 4th October 2011. [3]- HongSheng Zhao, Anaelle Halle, Baojiu Li, 2008, "Perturbations in a non-uniform dark energy fluid: equations reveal effects of modified gravity and dark matter [4]- D Hajdukovic, Quantum vacuum and virtual gravitational dipoles: the solution to the dark energy problem, Astrophysics and Space Science 339(1), 1-5, 2012 [5]- Ratra, Bharat & Peebles, P. J. E. 2003, "The cosmological constant and dark energy". Reviews of Modern Physics 75 (2): 559606. arXiv:astro-ph/0207347, Bibcode:2003RvMP.75.559P, doi:10.1103/RevModPhys.75.559 [6]- Ade, P. A. R.; et al. (Planck Collaboration) 22 March 2013, "Planck 2013 results. I. Overview of products and scientific results Table 9." Astronomy and Astrophysics. arXiv:1303.5062.Bibcode:2014A&A.571A.1P, doi:10.1051/0004-6361/201321529 [7]- Carroll, Sean, 2001. "The cosmological constant", Living Reviews in Relativity 4. [8]- Einstein, A (1917). "Kosmologische Betrachtungen zur allgemeinen Relativitaetstheorie". Sitzungsberichte der Kniglich Preussischen Akademie der Wissenschaften Berlin. part 1: 142152 [9]- General Relativity: An introduction for physicists, MP Hobson, GP Efstathiou & AN Lasenby (2006). Cambridge University Press. p. 187.ISBN 978-0-521-82951-9 [10]- Rugh, S, 2001 "The Quantum Vacuum and the Cosmological Constant Problem". Studies in History and Philosophy of Modern Physics 33 (4): 663705.doi:10.1016/S1355-2198(02)00033-3 [11]- Christopher Wanjek; "Quintessence, accelerating the Universe?" http://www.astronomytoday.com/cosmology/quintessence.html

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[12]- J. Khoury and A. Weltman, Phys. Rev. D 69, 044026 (2004) [13]- Torsion balance experiments: A low-energy frontier of particle physics, E.G. Adelberger, Center for Experimental Nuclear Physics and Astrophysics, 20 August 2008, doi:10.1016/j.ppnp.2008.08.002 [14]- Bunch, Bryan H; Hellemans, Alexander (April 2004). The History of Science and Technology. Houghton Mifflin Harcourt. p. 695. ISBN 978-0-618-22123-3 [15]- On the relative motion of earth and the luminiferous ether; by Michelson and Morley [16]- Dimopoulos, et al. "Gravitational wave detection with atom interferometry" PL B 678, 1 (2008). [17]- Testing General Relativity with Atom Interferometry, Savas Dimopoulos, Peter W. Graham, Jason M. Hogan, and Mark A. Kasevich, February 6, 2008, arXiv:gr-qc/0610047v2 [18]- Chameleon Cosmology, Khoury and Weltman, 1 Dec 2003, arXiv:astro-ph/0309411v2 [19]- Probing Dark Energy with Atom Interferometry, Clare Burrage, Edmund J. Copeland, and E. A. Hinds, 6 Aug 2014, arXiv:1408.1409v1 [astro-ph.CO] [20]- Upadhye, Dark energy fifth forces in torsion pendulum experiments. Phys. Rev. D 86, 102003 [21]- Burrage, C. Atomic precision tests and light scalar couplings. Phys. Rev. D 83, 035020 (2011). [22]- Jain, B., Vikram, V. & Sakstein, J. Astrophysical tests of modified gravity: Constraints from distance indicators in the nearby universe. Astrophys. J. 779, 39 (2013). [23]- P. Hamilton et al. Atom-interferometry constraints on dark energy, arXiv:1502.03888 (2015) [24]- P. Achim, High precision gravity measurements using atom interferometry, (1998) Appendix 1

Define range over which & are possible

Link & by using the 'meshgrid' command

Define an array, = 5 3 which

describes all points in the parameter space

Define the system; radius of source, mass of source,

local density

Take each value of & , substitute into

= 0.01 >4

3

1

2

If equation is true return '1' and save in an array the same size and at the same

position as

If equation is false return '0' and save in an array the same size and at the same

position as

Plot this logical array to find the parameter plot for the specified system

23

Appendix 2

Define range over which & are possible

Link & by using the 'meshgrid' command

Define an array, = 5 3 which

describes all points in the parameter space

Define the system including vacuum

chamber dimensions and properties as well as

source and test particle

Find and save the values of & at each point of

by using the function 'min' and applying it to the 3 possible values

Substitute the values for & into

= 5.52 >

21 + 2

2

If equation is true return '1' and save in an array

the same size and at the same position as

If equation is false, return '0' and save in an array

the same size and at the same position as

Plot this logical array to find the constrained

parameter space according to this

particular acceleration