Testing Alternative Theories of Gravity with Atomic Clocks

Modelling the Dark Side of the Universe

Most our instruments are built to detect light (e.g., stars, dust, galaxies), and so we still know very little about the dark sector, which comprises more than 95% of the universe and neither emits nor absorbs light. The simplest type possible type of dark matter particles are those represented by a spin zero scalar fields. Scalar fields can arise from not-yet understood physics such as compactified extra dimensions. The Higgs particle is believed to be the first fundamental scalar field ever detected. In cosmology, the initial rapid inflation of the universe is also best modelled by a scalar field called the inflaton, while the quintessence scalar field generates a form of dark energy that may account for the accelerated rate of expansion of the universe. It is reasonable to believe that similar to the visible sector, the dark sector is rich and contains particles of different kinds with various masses.

Alternative theories of gravity and PPN parameters

The solid line corresponds to 10^-16 clock stability. 

Einstein's theory of gravity alone cannot account for dark matter or dark energy. Instead, a scalar field that may couple in different ways to different kinds of matter enters the theory of gravitation in addition to the metric tensor of general relativity. This field couples to matter and therefore violates General Relativity. The standard way to describe deviations from General Relativity in weak gravitational fields is via the Parametrized Post-Newtonian (PPN) formalism. The most commonly constrained PPN parameters are 𝛄 and β.

In Einstein's General Relativity 𝛄=β=1. When the scalar field is massless, the 𝛄 and β parameters are constant. The introduction of a mass term causes the 𝛄-parameter to become distance dependent. It is thus natural to expect that PPN parameters may depend on both the distance from and the composition of the mass they are tested around, e.g.,  their value measured around the Earth would be different than around the Sun. The strongest constraint on the 𝛄 PPN parameter comes from the Cassini spacecraft, which limits the size of the parameter around the Sun to 2.3 x 10-5 (1-sigma confidence level). Null results from planetary ephemerides place similar constraints on the  β parameter.

Testing Alternative theories of Gravity with Clocks in Space

Performing accurate timing experiments with satellites carrying state-of-the-art atomic clocks in Earth orbit can test alternative theories of gravity such as scalar-tensor theories. The Hydrogen Maser placed aboard the International Space Station as part of the Atomic Clock Ensemble in Space (ACES) mission is expected to reach  a frequency inaccuracy of 10^-16.  

We calculate the gravitational redshift for an eccentric Earth orbit induced by varying PPN parameters and compare it to the one predicted by GR.  We find that a clock with the stability that should be achieved by the ACES mission placed in an elliptical orbit would constrain 𝛄 and β to the 10-6 level (see figure). We take an eccentric orbit with the same parameters as that proposed for the orbit of the originally proposed STE-QUEST satellite.  Choosing an  eccentric orbit makes the signal stronger since the relativistic effects are larger than for a circular orbit due to the higher velocity at perigee.

PPN parameters for general scalar-tensor theories
We compute the PPN parameters γ and β for general scalar-tensor theories in the Einstein frame, which we compare to the existing PPN formulation in the Jordan frame for alternative theories of gravity. Note that there are infinitely many possible frames, but these two frames are the ones that are typically chosen. In the Einstein frame, the Ricci scalar appears alone and the matter fields couple to a conformally related metric, while in the Jordan frame, the scalar field multiplies the Ricci scalar and any matter fields present couple directly to the frame metric. This computation is important for scalar-tensor theories that are expressed in the Einstein frame, such as chameleon and symmetron theories, which can incorporate hiding mechanisms that predict environment-dependent PPN parameters. 

Extended Point Source?
The PPN parameters are typically calculated for a space-time consisting of a point source surrounded by vacuum, which is not accurate for experiments performed around extended objects like the Sun. Instead, we add a parameter that measures how much the exterior gravitational field deviates from that of a point source with the same mass.  To check our assumptions, we recompute the Cassini constraint modelling the Sun as a homogenous sphere instead of a point source for massive Brans-Dicke theories, and find that the 𝛄-PPN  constraint becomes more stringent.

The article this post is based on was led by PhD student Andreas Schaerer.  The paper is published in Physical Review D. I have also written a conference proceeding that describes tests of gravity with space-based atomic clocks and atom interferometers.