qBounce

# Motivation

The last fourteen years of experiments and observation have produced an entirely new map of the universe. Cosmological parameters describe the properties and the global dynamics of the universe. Observational cosmology has determined these parameters to one or two significant figure accuracy. Examples include:

- a determination of the mass content of the universe. The majority of matter consists of 23% unknown dark matter.
- the known particles of the Standard Model. They account for only 4% of energy density of the universe.
- an unknown form of energy. It is responsible for the acceleration of the universe and accounts for about 73% of the energy density of the universe.

In addition, there are indications that all known basic forces, the weak force, the strong force, electromagnetism and gravity, might be unified at high energies. Such new information needs entirely new concepts, language, theoretical and experimental approaches for their understanding.

Our line of attack are novel gravity experiments using a quantum interference technique, which is also “dark” in that sense that it is neither uses, nor is influenced by electromagnetic forces, fields or potential ^{[1]}. We build an apparatus designed to carry out “Spectroscopy of Gravity”, which only depends on natural constants. We then plan to clarify the following three concepts and approaches with an absolute frequency standard together with the most precise measuring technique, the Ramsey method of separated oscillating fields.

Firstly, it is of highest importance to reveal the nature of dark matter. The dark matter has only been observed indirectly, through the influence of its gravity on surrounding ordinary matter. It has evaded more conventional observation using optical, radio or other telescopes, so it evidently only has a feeble coupling to photons. We do not yet know precisely what this dark matter is, but the leading theoretical candidates (axions and weakly interacting massive particles, WIMPs) are massive particles that do not acquire their mass by interacting with the electroweak Higgs condensate. One eye-catching concept is the idea of detecting very light bosons through the macroscopic forces which they mediate. These particles may be responsible for the dark matter or part of the dark matter in the universe. These particles must be very light with a mass of about 10 µeV to 10 eV to be compatible with astronomical observation. This corresponds to a range of 0.2 µm and 2 cm they can mediate, leading to a deviation from Newton’s law at short distances, exactly in the range of this experiment.

Secondly, distinct from dark matter is the concept of dark energy - interpreted as negative pressure – that is causing the universe to accelerate. A particularly attractive possibility is that a scalar field, usually called quintessence, is responsible for inflating the universe. A competing possibility is that the acceleration is due to a modification of gravity, i.e., the left-hand side of Einstein’s equation rather than the right. Observations, which can distinguish these two possibilities are desirable, since measurements of expansion kinematics alone are not able to. Remarkably, the best limits for such scalar fields (chameleons) come from our experiment with ultra-cold neutrons, as we will show.

Thirdly, there are indications that all known basic forces, the weak force, the strong force, electromagnetism and gravity, might be unified at high energies. One approach are string-theories, which are naturally defined in ten or eleven space-time dimensions. The hope is that a future theory of quantum gravity can be constructed in such a way that all forces, including gravity, could be combined within it. Such a theory cannot be constructed in a consistent way if space-time is limited to four dimensions, so additional spatial dimensions are needed to accommodate such a theory in a reasonable way. Extra-dimensions have not been observed in everyday life, because these extra-dimensions might be “curled-up” with a possibly very small radius of curvature. This should lead to deviations from Newton’s gravitational law at very small distances. The focus of these has shifted over the years and it has been realized, that the string scale, which controls the strength of gravity as well as the strength of other forces, is not necessarily related to the Planck size of 10^{-35} m, instead it can be considered as a free parameter and we might live with large extra dimensions, again in the range of our experiments.

The idea of finding a common ground to the problems related to string theories, dark energy and dark matter is aesthetically appealing to us. That these problems might have a common solution can be seen that these approaches are strongly interrelated, not only by a unified physics aim, like the role of gravity in modern gauge theory for cosmology, but also by the common methodical approach addressed by this experiment.

These three concepts are all linked to Newton’s gravity law at short distances being probed by a newly developed resonant spectroscopy technique. The power dependence 1/r^{2} is purely geometrical and caused by the Gauss law in 3–dimensional space. Such a choice, i.e. the Inverse Square Law (ISL), is not well–motivated theoretically.

Our solution is to use the interaction of a macroscopic system, here a mirror for neutron reflection, with a purely quantum mechanical system, i.e. the excitation of bound quantum states of a neutron in the gravity potential of the Earth. The aim is to establish an absolute frequency reference for these measurements. This reference is only based on natural constants: the mass of the neutron *m*, the Planck constant *h*, and the acceleration of the Earth *g*. Neutrons are trapped by gravity, but the motion is unrestricted parallel to a mirror. The lowest energy eigenvalues En, (n = 1, 2, 3, 4, 5), are 1.41 peV, 2.46 peV, 3.32 peV, 4.09 peV, and 4.78 peV, see Fig. 2. The neutron matter waves can be excited by an oscillator coupled to these states in order to drive transitions between state |p> and state |q>. A frequency measurement determines the energy difference with unprecedented accuracy.

The method works because a quantum mechanical system that is described by two states can be understood in analogy to a spin ½ system (assuming two states of a fictitious spin in the multiplet, similarly to spin up and spin down states). The time development of such systems is described by the Bloch equations. In magnetic resonance of a standard spin ½ system, the energy splitting results in the precession of the related magnetic moment in the magnetic field. Transitions between the two states are driven by a transverse magnetic radio frequency field. Similar concepts can be applied to any driven two level system, e.g. in optical transitions with light fields. Here we apply this picture to quantum states in the gravity field.

**Impact:** The clarification of the role of dark matter and dark energy will have decisive impact on the future research in particle physics, astrophysics and cosmology. Such fundamental research addresses key questions asked by humankind: What is the universe made of? What was the past and what will be the future of the universe? What are the forces of nature that keep the world together? Is gravitation an entropic force and is it compatible with quantum theory [Ver11, Chai12,Kob11]?

## Motivation

The deviation from the Newton gravitational potential energy may have the following origins:

**Extra Dimensions: **The string and *D _{p}*-brane theories predict the existence of extra space-time dimensions [Joh03, Str01]. These new dimensions cannot be seen by Standard Model particles, since they are confined to a three-dimensional subspace of the higher-dimensional theory. If the Standard Model particles are all confined to such a

*D*-brane, one cannot feel additional dimensions except the modification of the gravitational force law.

_{3}**Infinite-Volume Extra Dimensions: **The analysis of extra dimensions, carried out by Randall and Sundrum [Ran99] is based on the phenomenology of the graviton zero mode that is localised about a *D _{3}*-brane, embedded in a non-compact, infinite extra dimension. The shape of the potential energy may be obtained by summing up the contributions of the bound-state modes and the continuum Kaluza-Klein mode spectrum [Ade09, Ran99].

**Exchange Forces from Conjectured New Bosons: **Even if new dimensions are absent or small, a deviation from the ISL can be induced by the exchange of new (pseudo)scalar and (pseudo)vector bosons. The (pseudo)scalar and (pseudo)vector bosons may mediate attractive and repulsive Yukawa interactions, respectively.

– **Axion**: One of the candidates for a new pseudo-scalar boson, responsible for an additional Yukawa deviation of the Newton gravitational potential, is the axion. The axion as a hypothetical elementary particle has been postulated by the Peccei-Quinn theory in 1977 to resolve the problem of strong CP-violation in Quantum Chromodynamics (QCD) [Pre78]. Since the axion cannot be treated as a true Nambu-Goldstone boson, it should be massive [Wei78, Wil78]. It would mediate long-range forces, which are severely constrained by “Fifth Force” experiments [Fis92]. Those looking for new mass-spin couplings provide significant constraints on pseudoscalar bosons [Moo84, You96]. However, they do not yet cover realistic parameters for the axion because they are only sensitive for small m_{A}. Experimental, astrophysical, and cosmological limits have been refined and indicate that axions, if they exist, very likely have very low mass, 10 µeV < m_{A} <10 eV, suggesting that axions are a non-negligible fraction of the cosmic dark matter [PDG10]. This corresponds the axion interaction range within 0.2 µm < λ < 2 cm, exactly in the range of our experiment.

– **Scalar boson. Cosmological consideration:** Cosmology with light scalar bosons can be made acceptable by invoking a very late stage of the inflation with the Hubble constant *H*_{0} less or approximately equal to the mass *m*_{Φ} of the scalar boson. This gives the estimate of the scalar boson mass *m*_{Φ} ~ *H*_{0} ~ 10^{-33} eV [PDG10]. In turn, in the cosmological setting, the chameleon field settles down at the bottom of its time-dependent effective potential very early in the universe, thus preventing large mass variations during big bang nucleon synthesis. Meanwhile, on cosmological scales the mass of the chameleon field can be of order *H*_{0 }that allows to the chameleon field to evolve cosmologically today [Kho04a, Kho04b, Mot07a, Mot07b]. Scalar fields might couple to ordinary matter and might mediate a new long range force, which show up in our fifth force searches or equivalence principle tests [Bra11].

– **Bosons from Hidden Supersymmetric Sectors**: One of the candidates for a new physics is supersymmetry [Ram08]. A typical scale is of order of 1 TeV. Unbroken supersymmetry predicts an unobservable degeneracy between fermions and bosons. The most popular scenario involves the supersymmetry breaking at the scale of order of *M*_{S} ~ 10^{11}GeV in a ”hidden” sector that couples to our visible world only via gravity and interactions of similar strengths. A scale of the supersymmetry breaking in the visible world should be of order of *M*^{2}_{S/}*M*_{P} ~ 10^{3 }GeV, where *M*_{P} is the Planck scale [PDG10]. The scalar particles from the “hidden” sector can have a mass of order of a few meV and effectively change the gravitational potential energy at the scales of order of 100 mm [Ade09]. At the same scale of 100 mm deviations of Spin-1 bosons are expected leading to a repulsive correction of Newton's law [Ark99, Ade09]. We would like to emphasise that the proposed neutron interference experiment is sensitive to this scale.

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