# Dark matter in quantum gravity

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## Abstract

We show that quantum gravity, whatever its ultra-violet completion might be, could account for dark matter. Indeed, besides the massless gravitational field recently observed in the form of gravitational waves, the spectrum of quantum gravity contains two massive fields respectively of spin 2 and spin 0. If these fields are long-lived, they could easily account for dark matter. In that case, dark matter would be very light and only gravitationally coupled to the standard model particles.

While finding a unified theory of quantum field theory and general relativity remains an elusive goal, much progress has been done recently in quantum gravity using effective field theory methods [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. This approach enables one to perform model independent calculations in quantum gravity. The only restriction is that only physical processes taking place at energy scales below the Planck mass can be considered. This restriction is, however, not very constraining as this is the case for all practical purposes in particle physics, astrophysics and cosmology.

In this paper, we show that quantum gravity could provide a solution to the long standing problem of dark matter. There are overwhelming astrophysical and cosmological evidences that visible matter only constitutes a small fraction of the total matter of our universe and that most of it is a new form of non-relativistic dark matter which cannot be accounted for by the standard model of particle physics. Gravity could account for dark matter in two forms. The first gravitational dark matter candidates are primordial black holes, see e.g. [16] for a recent review. They have been investigated for many years, and although the mass range for such objects to account for dark matter has shrunk quite a bit, they remain a viable option for dark matter, in particular Planckian mass black hole remnants are good dark matter candidates. Here we discuss a second class of candidates within the realm on quantum gravity. Recent work in quantum gravity has established in a model independent way that the spectrum of quantum gravity involves, beyond the massless gravitational field already observed in the form of gravitational waves, two new massive fields [12]. Their properties can be derived from the effective action for quantum gravity. We will show here that these new fields are ideal dark matter candidates.

Remarkably, the values of the parameters \(b_i\) are calculable from first principles and are model independent predictions of quantum gravity, see e.g. [17] and references therein. They are related to the number of fields that have been integrated out. The non-renormalizability of the effective action is reflected in the fact that we cannot predict the coefficients \(c_i\) which, in this framework, have to be measured in experiments or observations. There will be new \(c_i\) appearing at every order in the curvature expansion performed when deriving this effective action and we thus would have to measure an infinite number of parameters. Despite this fact, the effective theory leads to falsifiable predictions as the coefficients \(b_i\) of non-local operators are, as explained previously, calculable.

*W*(

*x*) is the Lambert function and \(\kappa ^2=32 \pi G\),

*G*is Newton’s constant. The \(b_i\) for the graviton are known: \(b_1=430/(11520 \pi ^2),b_2=-1444/(11520 \pi ^2)\) and \(b_3=434/(11520 \pi ^2)\). The \(b_i\) are thus small and unless the \(c_i\) are large, the masses \(m_2\) and \(m_0\) will be close to the Planck mass \(M_P\) and the corresponding fields will decay almost instantaneously [12]. As we are interested in the case where the new fields are light, it is useful to consider the limit where the \(c_i\) (or one of them at least) are large and \(b_i\ll c_i\). In that case we can rewrite the masses as

*k*, we find

*V*, such as the W and Z bosons, is given by

*k*. Its partial width to massless vector fields is given by

*N*=1 for photons and \(N=8\) for gluons. In the case of massive massive vector fields, one has

While we have established that quantum gravity provides two new candidates for dark matter, it remains to investigate their production mechanism. Thermal production is a possibility, but we would have to consider all higher order operators as we would need to consider temperatures larger than the Planck mass \(T\ge M_P\) since these objects are gravitationally coupled to all matter fields. Also we may not want to involve temperatures above the inflation scale which we know is at most \(10^{14}\) GeV. The weakness of the Planck-suppressed coupling hints at the possibility of out-of-equilibrium thermal production as argued in [22]. However, the mass range allowed for the dark matter particles within that framework is given by TeV\(<m_{DM}< 10^{11}\) GeV [22] and it is not compatible with our ranges for the masses of our candidates. The fact that our dark matter candidates are light points towards the vacuum misalignment mechanism, see e.g. [23]. Indeed, in an expanding universe both \(\sigma \) and *k* have an effective potential in which they oscillate. The amount of dark matter produced by this mechanism becomes simply a randomly chosen initial condition for the value of the field in our patch of the universe. In [24], it was shown that the vacuum misalignment mechanism leads to the correct dark matter abundance \(\rho _{DM}=1.17\) keV/cm\(^3\) if the dark matter field takes large values in the early universe. For example, a dark matter field with a mass in the eV region would need to take values of the order of \(10^{11}\) GeV to account for all of the dark matter in today’s universe [24].

In summary, we have shown that gravity, when quantized, provides new dark matter candidates. As these fields must live long enough to still be around in today’s universe their masses must be light otherwise they would have decayed long ago. It is quite possible that gravity can account for all of dark matter in the form of primordial black holes and the new fields discussed in this paper without the need for new physics.

## Notes

### Acknowledgements

The work of XC is supported in part by the Science and Technology Facilities Council (Grant Number ST/P000819/1). XC is very grateful to MITP for their generous hospitality during the academic year 2017/2018.

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