Pure Gravitational Dark Matter, Its Mass and Signatures

In this study, we investigate a scenario that dark matter (DM) has only gravitational interaction. In the framework of effective field theory of gravity, we find that DM is still stable at tree level even if there is no symmetry to protect its longevity, but could decay into standard model particles due to gravitational loop corrections. The radiative corrections can lead to both higher- and lower-dimensional effective operators. We also first explore how DM can be produced in the early universe. Through gravitational interaction at high temperature, DM is then found to have mass around TeV $\lesssim m_X \lesssim 10^{11}$GeV to get the right relic abundance. When DM decays, it mostly decays into gravitons, which could be tested by current and future CMB experiments. We also estimate the resulting fluxes for cosmic rays, gamma-ray and neutrino.


I. INTRODUCTION
Evidence for the existence of dark matter (DM) is compelling, supported from astrophysical length to cosmological scale. Despite its convincing inferences in cosmic microwave background (CMB), big-bang nucleosynthesis, large scale structure (LSS) and other astrophysical observables, DM's particle identity is still a mystery since all the confirmed evidence simply suggests DM should have gravitational interaction.
From the current experimental searches for DM, we have already known that the interaction between DM and the standard model particle should be weak. DM is stable or at least has a very long lifetime, much longer than the age of our Universe t U ∼ 10 17 s, otherwise it can give rise observable signatures in CMB, LSS, cosmic rays, gamma-ray and neutrino experiments.
It is a logical possibility that DM might have only gravitational interaction. It is usually expected there would be no way to produce DM since its interaction with standard model (SM) particle is super weak, not to mention how to search for it. This is true if we only consider classical theory of gravity where DM is stable even though there might be no symmetry to protect its longevity. However, as we shall show in this paper, if perturbative gravitational loop corrections are taken into account, effective operators are induced and can make DM decay.
To estimate the quantum correction from graviton, a consistent quantum theory of gravity is needed. For our purpose in this study, we may investigate properties of DM within the framework of effective field theory of gravity [1,2]. This framework is justified if we only consider low-energy processes or weak gravity approximation which are satisfied in the problem of our interest.
Recently, it has been shown that the general theory of relativity can be derived as an effective field theory of gravitational quantum field theory with spin and scaling gauge symmetries [3].
This paper is organized as follows. In Sec. II we establish the conventions and definitions. Then in Sec. III, we explore a possible production mechanism of DM and obtain its viable mass range. In Sec. IV, we illustrate how effective operators that induce interactions between DM and SM particles could arise. In Sec.V we present possible decay channels for DM and its signatures. Finally, we give a summary.

II. GRAVITATIONAL DARK MATTER (GDM)
We start with a minimal setup with one scalar dark matter (DM) field X and one non-DM scalar φ. Here φ may be denoted collectively as any non-DM fields. Explicit extensions with fermionic and gauge fields will be discussed later. In the flat spacetime, the general action S would be an integration over Lagrangian density L, We explore our investigation with the following potential, where m i and λ i are masses and quartic couplings, respectively. There is a discrete Z X 2 symmetry, X → −X, which can protect the stability of DM X. When there is no any direct or indirect interactions other than gravity arXiv:1604.04701v3 [hep-ph] 5 Jun 2016 between X and φ, we shall refer X as Gravitational Dark Matter (GDM).
In flat Minkowski spacetime without gravity, φ-X system with potential V are renormalizable in the sense that all ultraviolet (UV) divergences from loop corrections can be absorbed into fields, masses and couplings, and that no other counterterm needs to be introduced. So φ and X are completely decoupled at all scale even if after renormalization group flow. However, as we shall see soon, the above picture will be changed dramatically after including perturbative gravitational effects and non-zero terms, such as λ Xφ φ 2 X 2 , can be induced, with Planck scale suppressed λ Xφ ∼ m 2 X m 2 φ /M 4 P . Now let us include gravity with standard Hilbert-Einstein action. The Lagrangian would be modified to where R is the Ricci scalar, G is the Newton's constant, g(x) is the determinant of spacetime metric tensor g µν and g µν is the inverse matrix of g µν with g µρ g ρν = δ µ ν , Note that our framework is similar to Ref. [4], but different from those [5,6] where non-minimal coupling between R and X is introduced, and also different from Refs. [7,8] which rely on dynamics of quantum field theory on curved background spacetime [9] . We consider the weak gravity case and express the metric field around the flat Minkowski background spacetime as follows, where κ = √ 16πG ≡ 1/M P and h µν is identified as quantum field for spin-2 massless graviton, propagating in flat background spacetime. This expression is useful and justified when we are only interested in environment without strong gravity and in low-energy physics if the energy is smaller than Planck scale [1,2].
The above expression of g µν is general, but the expansions of inverse metric and determinant are approximate with ignoring higher-order terms. For our purpose, it is enough to keep terms up to second order in κ only, where h ≡ η µν h µν . At this stage, we have already seen that there are infinite operators in the expansion series, which partially shows the non-renormalizability of gravity. This is not a problem in effective field theory where one can only keep terms up to κ n and n is determined by the concerned precision. To quantize h µν , we need to fix the gauge. We choose the harmonic gauge-fixing condition, then we have the graviton propagator with a simple form in momentum space, Note that the corresponding ghost in this gauge is irrelevant for our calculations of one-loop gravitational corrections, which is similar to quantum electrodynamics with Feynman gauge. The Lagrangian now can be rewritten as where δL (h µν , X, φ) at order of κ 2 is given by The above formulas would be the main Lagrangian or framework for discussions in next section. At this point, it is easy to check that X is still stable at tree level. Even if we include a Z X 2 symmetry breaking term in the potental V X , X is still stable since there is no available interactions to decay through. However, as we shall show in Sec. IV, once we take loop corrections into account, X shall decay. The above cubic term can be induced if the discrete Z X 2 symmetry is spontaneously broken. For example, if a scalar X has a potential with µ > 0. Then X would get a vacuum expectation value, X = 6µ 2 /λ ≡ v X . Substitute X = v X + X, we get the potential for X, with m X = λv 2 X /3 and µ X = √ 3λ.

III. PRODUCTION MECHANISM FOR GDM
In this section, we discuss how to produce dark matter particle in the early Universe. The dominant contribution to produce gravitational dark matter is through the tree-level s-channel process by mediating a graviton, φ+φ → X +X (φ can be any other particle in the thermal bath). The production cross section from interactions in Eq. 2.9 has the following form where T is the temperature of thermal bath in the universe. Due to the weakness of gravity, the interacting rate n φ σv (n φ T 3 is number density of φ) is much smaller than the expansion rate of universe, so X is not in thermal equilibrium with other particle. Now let us calculate how much X can be produced. The Boltzmann equation [10] that describes the changes of number density n X is given by where a is the scale factor, H ≡ȧ/a is the Hubble parameter, ∼ T 2 /M P , n eq is the equilibrium number n eq ∼ T 3 . Define the yield Y ≡ n X /s, s is the entropy density, we have where we can ignore Y in the right-handed side of the first equation due to Y Y eq . Therefore Y would be a power-law function of T with positive index after we put in σv ∼ κ 4 T 2 and n eq ∼ T 3 . We also should sum over all particles with gravitational interaction, which means we can replace Y eq with ∼ 1. Integrate Eq. 3.3 over T from O (m X ) to the maximal temperature T max , then to get the right relic abundance of X, we would need where T max may refer as the maximal temperature of our universe after inflation, or reheating temperature, Ω b and Ω X are the energy density fractions of baryon and dark matter, respectively, Ω X /Ω b 5, m p 1GeV is proton mass and η 6 × 10 −10 is baryon-to-photon ratio. From Eq. 3.4, we obtain We also need check whether the universe can be hot enough to produce X, namely T max O (m X ). With Eq. 3.5 we have which gives the lower bound for T max and can be definitely satisfied without violating any experimental limits.
The reason why we take T max O (m X ) as a constraint is that when the temperature is much lower than m X there would be an exponential suppression of particles that are energetic enough to produce DM X. Only particles at the very high energy tail of Bose-Einstein/Fermi-Dirac distributions can contribute to the production. The upper bound for T max , on the other hand, depends on the details of cosmic evolution. It is reasonable to expect that DM X is mostly produced after inflation since otherwise it would be extremely diluted by the exponential expansion. If we take T max as high as the inflation scale 10 −4 M P which is constrained by non-observation of primordial gravitational wave [11], we can have an upper bound on T max 10 −4 M P .
With the above bound 10 −7 T max /M P 10 −4 , then from Eq. 3.5, we would have a finite range for the mass of gravitational dark matter, 10 3 m p m X 10 −7 M P , TeV m X 10 11 GeV. (3.7) This is one of our main results, which predicts the mass range for GDM. This result is consistent with the case m X < T max in [4] which also considered T max > m X case, in the context of reheating process. Note that the above discussions do not depend on whether GDM is a scalar [12], fermion or vector. In later sections, we shall focus on scalar case for further investigations.

IV. EFFECTIVE OPERATORS OUT OF GRAVITY
In this section, we consider loop contributions and show how they induce both lower-and higherdimensional operators. Some typical Feynman diagrams are shown in Fig. 1. Calculations of these diagrams are involved with UV divergences which can be handled with regularization. To cancel these UV divergences, we must introduce counterterms and perform renormalization. This is equivalently to say that, renormalization group flow shall necessarily introduce non-zero coefficients for the operators that are allowed by the sym-metry. Below we just list some operators with explicit mass-dependent coefficients, with common factor κ 4 /16π 2 . The finite coefficients C i in front of these operators are of the following form, after we introduced the counterterms to cancel the divergences, where µ is the renormalization scale, E is the energy, "..." refers to finite O(1) constant. Note that discrete Z X 2 symmetry is still maintained if µ X = 0 in the potential V. Importantly, as seen above, non-zero λ Xφ φ 2 X 2 term is induced with coupling which is inevitable once we include gravity. In principle, those induced operators would also contribute to the production of GDM in the early universe. However, their contributions are further suppressed by κ 4 m 4 X , in comparison with the leading one in Eq. 3.1. Now let us discuss the case without discrete Z X 2 symmetries or µ X = 0 in Eq. 2.10. At two-loop level, as shown in Fig. 1 (e) and (f), we have operators as following, with a common factor, κ 4 µ X m 3 X /256π 4 . Both operators are not present in the starting Lagrangian and can lead X to decay into two φs if m X > 2m φ . X can also decay into two gravitons due to one-loop diagram from Fig. 1

(d) with effective operators like
Xh αβ h αβ , Xh 2 , (4.5) with prefactor κ 2 µ X m 3 X /16π 2 . These two operators could arise from covariant term √ −gXR after renormalization and using equation of motion for X.
We are now in a position to discuss the connections between GDM and SM particles. The Lagrangian can be written collectively as where D is the covariant derivative, and in the second line, terms in the bracket corresponds to fermion, gauge, Yukawa interaction and Higgs terms, respectively. φ is the physical higgs boson m φ 125GeV. Similarly, we can get effective operators like with a common factor κ 4 m 2 X /16π 2 . Again if µ X = 0 operators that induce DM decay would also arise, with a common factor κ 4 µ X m 3 X /256π 4 .

V. SIGNATURES OF GDM
Now we discuss the possible signatures of GDM. If Z X 2 symmetry is not broken, DM X is stable but can pairannihilate into other particles. The differential flux for particle i is given by where σv ∼ κ 4 m 2 X is averaged annihilation cross section, dN i /dE is energy spectrum for particle i, r is the distance to galaxy center, r c ∼ 20kpc for Milky Way and ρ ∼ GeV/cm 3 is DM density. We estimate the total flux is around 10 −42 cm −2 s −1 , which is independent of the DM mass. This flux seems too small to be probed by any known techniques. For example, current gamma-ray experiments are only sensitive to flux down to 10 −4 cm −2 s −1 for TeV photons [13]. It is therefore necessary to look for exotic astrophysical compact objects with high DM density ρ 10 20 GeV/cm 3 , which might not be so surprising since we have already known the nucleon density can be as high as 10 30 GeV/cm 3 in white dwarf stars and 10 38 GeV/cm 3 in neutron stars, respectively.
When Z X 2 symmetry is broken for µ X = 0, DM X can decay. The lifetime of X should be longer than the age of Universe, which puts a constraint on its dominant decay width Γ h X , If µ X 1, the upper bound for m X is 10 7 GeV. The resulting flux is estimated as [14] Then the total flux of energetic graviton or highfrequency gravitational wave would be around 10 −3 cm −2 s −1 × µ 2 X × m X /10 7 GeV 4 . So far no experiment searches for gravitons with such high energies.
However, decay of DM can also change the evolution of our late Universe by decreasing the matter component and increasing radiation part, which can be probed by CMB with enhanced late integrated Sachs-Wolfe effect at large scale or low l, as shown in Fig. 2. Current bound on decaying DM is Γ X 0.1t −1 U [15]. GDM can also decay into to SM particles, such as X → φφ, γγ, ZZ, W W, gg, ψψ. The partial decay width can be estimated as For m X TeV, we also calculate the decay branch ratios where N c = 1, 3 for leptons and quarks, respectively. We then can make predictions for the spectra shapes of p, e ± , γ, ν from X decay, as shown in Fig. 3. Gamma line is also produced with energy E = m X /2. Unfortunately, the fluxes for SM particles are highly suppressed at least by a factor of 10 −48 for m X 10 7 GeV, compared to graviton flux. These fluxes would be too small for searches in near future unless there are exotic astrophysical objects with very high density.

VI. SUMMARY
In this paper, we have discussed a scenario that dark matter (DM) has only gravitational interactions. We have investigated how DM can be produced in the early universe and shown its mass range should be around TeV m X 10 11 GeV to have the correct relic abundance. We have also considered DM decay if the discrete symmetry that protects DM's longevity is broken, and calculated the resulting graviton flux. Fluxes of other decaying products, like cosmic rays, gamma ray and neutrinos, are intrinsically very small, which is far below