Cosmological constant problem: deflation during inflation
Felipe Canales Benjamin Koch Cristobal Laporte Angel Rincon















    □
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        Pontificia Universidad Católica de Chile Instituto de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, ChilePontificia Universidad Católica de Chile Instituto de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, ChileInstitut für Theoretische Physik,  Technische Universität Wien,
         Wiedner Hauptstrasse 8-10, A-1040 Vienna, AustriaPontificia Universidad Católica de Chile Instituto de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, ChileInstituto de Física, Pontificia Universidad Católica de Valparaíso, Avenida Brasil 2950, Casilla 4059, Valparaso, Chile.angel.rincon@pucv.cl
        We argue that the discrepancy between the Planck mass scale and the observed value of the cosmological constant can be largely attenuated if those quantities are understood as a result of effective, and thus scale-dependent, couplings.
        We exemplify this mechanism for the early inflationary epoch of the universe by solving the corresponding effective gap equations, subject to an energy condition. 
        Several non-trivial checks and extensions are discussed.
        A comparison of our results to the renormalization group flow, obtained within the asymptotic safety program reveals a stunning agreement.
        Cosmological constant problem: deflation during inflation
            Ángel Rincón 
            December 2017 (Updated Feb 2018 and Dec 2018)
        =========================================================
        
        
        
        
        
        
        
        § INTRODUCTION
        
        The cosmological constant problem (CCP) seems to point towards deep
        misconceptions in our current understanding of the interplay between quantum field theory and general relativity <cit.>.
        In this article, we want to point out that one possible misconception is that,  one makes reference to gravitational couplings at different epochs of the universe,
        in particular for the most dramatic version of the CCP.
        This reference to different epochs of the Universe is however inconsistent if there is an epoch where the gravitational couplings are not constant, 
        as it is expected for example from quantum
        effects which are can induce an
        effective running of the fundamental couplings  (e.g. Newtons coupling G_0 → G_k).
        It is clear that for quantum effects to be notable one needs a violent epoch of the universe. Due to our current understanding
        the early inflationary period <cit.> is the best candidate for such non-classical corrections to be relevant.
        In particular, during inflation the quantum effects and thus the scale-dependence of effective couplings
        can be expected to be much stronger than during other later evolution periods.
        
        
        There are two arguments that support this idea.
        
        First, since the very early universe is typically related to Planck-time and Planck-distances it is natural to expect important quantum gravity 
        corrections <cit.>,
        such as scale-dependence of the effective couplings.
        Second, 
        in these very early times, other scales such as particle masses, which might moderate the scale-dependence are still irrelevant.
        During later times, to the contrary, there are no Planck distances involved
        and there are also logarithmically running particle masses contributing which could
         diminish the scale-dependence of the gravitational couplings further.
        
        
        
        
        
        
        
        
        Thus, it makes sense to focus this type of study on the first inflationary period of the Universe.
        Under such extreme conditions the gravitational couplings including the cosmological
        “constant” can not be expected to be constant <cit.> and thus one has to take
        care when realizing a comparison of those couplings at different times.
        
        
        In the following sections we will elaborate a scale-dependent vacuum dominated model which incorporates 
        the following features, which are in agreement with the ideas mentioned above:
        
        i) scale-dependence within effective gap equations
        ii) inflationary behavior of the cosmological scale factor a(t)
        iii) non–trivial evolution of the vacuum energy density and the gravitational coupling
        and
        iv) exponential suppression of the CCP during the inflationary period.
        
        
        
        
        
        
        
        
        
        
        This is achieved by
        applying methods that have  been successfully used in order to include quantum correction into gravitational theories,
        black hole backgrounds, wormholes, and particular cosmological problems <cit.>.
        
        
        
        
        
         §.§ Classical Λ dominated universe
        
        
        
        
        Based on the cosmological principle and assuming a flat Universe, one uses the line element
        
            ds^2 = - dt^2 + a(t)^2 
            [
            
            dr^2 
            + 
            r^2 dΩ^2_2
            ],
        
        as ansatz for the metric field, where dΩ^2_2 = dθ^2 + sin^2(θ) dϕ^2.
        
        The scale factor gives
        
            a(t)=a_i e^t/τ,
        
        where 
        τ = ±√(3/Λ_0 ).
        
        We will only use the positive sign in (<ref>),
        which corresponds to an expanding universe.
        
        
        
        
         §.§ The cosmological constant problem in the very early universe
        
        
        There are different versions of the CCP to be found in the literature.
        Most of them can be cast in the dimensionless ratio between the highest UV cut-off M_UV of
        the quantum field theory one is using to calculate the vacuum energy density and
        the energy density ρ=Λ_0/G observed due to the accelerated expansion of the universe <cit.>
        r≈M_UV^4/ρ≈M_UV^4 G/Λ_0.
        
        Figure <ref>  shows the exceedingly large values of (<ref>)
        for various emblematic energy scales M_UV from neutrino physics, proton mass, top mass, up to 
        the Planck scale.
        For a good theoretical prediction, the value of (<ref>) is supposed to be
        of order one, as indicated by the orange line.
        
        
        Clearly, there is a dramatic discrepancy. 
        
        As explained in the introduction, in this paper we will focus on the early expansion of the universe. During this epoch, 
         governed by pure gravity, the only dimensionful
        quantity is actually the Planck mass given by m_P^-1=√(8 π G).
        If one uses this scale as UV scale M_UV^2≈ 1/G to estimate the vacuum energy density,
        the ratio (<ref>) simplifies to
        
        r_infl≈1/G ·Λ_0≈ 10^122,
        
        which is the line labeled with “Planck” in figure <ref>.
        
        The fact that (<ref>) differs from one by more
        than one hundred orders of magnitude is known as the cosmological constant problem in its worst form.
        
        There have been numerous attempts to either solve, alleviate,
        or to deny this problem <cit.>, but up to now the quarrel continues and no agreement has been achieved.
        
        Since our purely gravitational approach ceases to be valid as soon as further mass scales such as the top mass
        appear, we only can attempt to explain the discrepancy of 66 orders of magnitude spanned by the value of the ratio (<ref>) 
        between the top mass and the Planck mass.
        
        
        
        
        
        § SCALE–DEPENDENCE IN GRAVITY
        
        
        A very powerful tool in quantum field theory is the use of effective actions, which incorporate quantum effects in a scale–dependent modification of the classical action.
        This type of scenario has been proposed and studied in the context of gravity in 
        different ways and with different methods 
        <cit.>.
        
        In the so-called Einstein-Hilbert truncation the effective action reads
        
        
        
        
            Γ[g_μν,k] = ∫d^4x √(-g)[
            1/16 π G_k(R-2Λ_k)  +  ℒ_M
            ],
        
        where Λ_k and G_k are the cosmological and Newton scale–dependent couplings, respectively.
        A great advantage of working with effective quantum actions like (<ref>) is
        that they already incorporate the effects of quantum fluctuations like those 
        being responsible for the CCP.
        Thus, if the problem is properly addressed in terms of a background solution of this action,
         no additional quantum
        corrections need to added.
        In order to obtain background solutions for this effective action
        one has to derive the corresponding gap equations.
        Thus, varying the effective action respect to the inverse metric field, 
        one obtains the corresponding Einstein field equations <cit.>
        
            G_μν =  T_μν^effec=- Λ(t) g_μν - Δ t_μν,
        
        where
        
        
            Δ t_μν = G(t)
            [
            g_μν∇^α∇_α-∇_μ∇_ν]
            G(t)^-1.
        
        In order to get physical information out of those equations
        one has to set the renormalization scale in terms of the
        physical variables of the system under consideration k→ k(x, …). 
        
        
        
        
         §.§ Scale setting
        
        
        
        For most types of scale settings <cit.>
        the identification k→ k(x, …) breaks the reparametrization symmetry (<ref>).
        There are, however, approaches which allow to maintain the symmetries
        and thus the consistency of the system.
        We will invoke a procedure which invokes the variational principle for kδΓ_k/δ k=0.
        
        This approach
        has been proposed in <cit.> and
        it is a straightforward implementation of background scale 
        independence <cit.>.
        Relation (<ref>) is not a fixed point condition but rather a scale setting condition,
        it even can be used when the renormalization group flow has already 
        approached its fixed point <cit.>.
        
        
        Further, for highly symmetric systems with
        a small number of couplings one can attempt to solve (<ref>)
        for g_μν(x),  G(x), and Λ(x) directly by the use of 
        a special ansatz <cit.> motivated
        by the structure of the classical solution <cit.>,
        or by imposing an energy condition <cit.>.
        
        In this paper we will apply both strategies, the variational scale setting (<ref>) and the energy 
        condition mentioned above.
        
        
        
        
        
         §.§ Modified Friedmann equations
        
        
        Within the ansatz (<ref>) the dynamical variable a=a(t) does only depend on time and not on spatial directions,
        the same will be assumed for the two additional variables G=G(t) and Λ=Λ(t).
        With this, the gap equations (<ref>)
        take the form of generalized Friedmann equations
        
        
            (ȧ/a)^2
            - 
            Λ/3   =
            8 π G/3ρ(t),
            
            
            2 ä/a 
            + 
            (ȧ/a)^2 
            -
            Λ   =
            -
             (8 π G)p(t),
        
        where an effective density (ρ) and pressure (p), given by the scale–dependent framework have been identified
        with the density and pressure parameters of a perfect fluid
        
        
            8 π G/3ρ(t)    ≡(ȧ/a) 
            (Ġ/G),
            
            
            -  (8 π G) p(t)    ≡ -
            2 (Ġ/G)^2 
            + 
            (G̈/G)
            + 
            2 (ȧ/a) (Ġ/G).
        
        Further, one has the scale setting relation (<ref>)
        d/dt[
        1/16 π G(t) (R-2Λ(t)) 
        ]=0.
        
        There are three equations, namely the
        two cosmological gap equations (<ref>, <ref>) and
        the scale setting condition (<ref>). There are also  three unknown functions (a(t),  G(t), Λ(t)).
        However, the system respects general covariance, which implies that ∇_μ G^μν=0 and thus
        one of the three equations (<ref>, <ref>, <ref>) 
        is just a consequence of the other two.
        This means that
        one needs one additional relation or condition in order to determine a solution.
        In the next subsection, we will invoke an energy condition in
        order to provide the missing equation.
        
        
        
        
        
        
         §.§ Null energy condition (NEC)
        
        
        
        
        
        In an empty universe, a light-like signal traveling on a null geodesic with velocity vector ℓ^μ =d x^μ/dt should not
        perceive any energy density of the background.
        This should be true for other vacuum contributions to the effective stress energy tensor, such as Δ t_μν,
        introduced by the scale-dependence of the gravitational coupling
        
        
            T_μν^effecℓ^μℓ^ν  = -Δ t_μνℓ^μℓ^ν  =0.
        
        This is reminiscent of the null energy condition <cit.>.
        The above relation is the scale setting condition which we will impose in addition to
        the cosmological gap equations (<ref>, <ref>).
        
        The vector field ℓ^μ satisfying the geodesic equation can be written as 
        ℓ^μ = C_0  a^-1 (1,(1-κ r^2)^1/2 a^-1,0,0).
        
        
        
        
        Replacing this in  (<ref>) gives the NEC condition for the cosmological model
        
        
            - 
            2 (Ġ/G)^2
            +
            (G̈/G)
            - 
            (ȧ/a) (Ġ/G) 
            
            = 
            0.
        
        
        
        § RESULTS AND DISCUSSION
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        The solution of  (<ref>), (<ref>), and (<ref>) 
        that describes an expanding universe is
        
            a(t)    =  a_i e^t/τ,
            
            
            G(t)    =G_0 [1+ ξ̃∫_t_0^t a(t')dt' ]^-1= G_0/1 + ξ a(t),
            
            Λ(t)    = Λ_0 
            [
             1 + 2ξ a(t)/1 + ξ a(t)].
        
        It is direct to see that  (<ref>) takes the same form as the classical counterpart
        (<ref>).
        A generalization to non zero values of κ is straight 
        forward (see version one of <cit.>).
        
        The solution involves four integration constants which we have labeled 
        (a_i, G_0, Λ_0, ξ).
        They are set by the conditions
        
        lim_t→ 0a(t) = a_i,
        lim_ξ→ 0G(t) = G_0,
        lim_ξ→ 0Λ(t) = Λ_0.
        
        One notes that the parameter ξ plays the role of a control parameter,
        When setting ξ to zero, the time-dependent couplings G(t) and Λ(t) are reduced to their
        constant and thus “classical” counterparts (G_0 and Λ_0).
        Please note that the expansion rate is given by τ = ±√(3/Λ_0 ),
        which looks exactly like the classical rate (<ref>). There is however an important 
        conceptual difference
        since now, Λ_0 is one of the integration constants, while in (<ref>) it
        was given as input parameter.
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
         §.§ Equation of state of scale-dependent G(t)
        
        
        
        
        
        
        
        
        Is there a meaningful equation of state p=ωρ
        
        
        for the effective type of pressure and energy density?
        Subtracting (<ref>) from 3×(<ref>),
        one finds for the solution (<ref>, <ref>)
        
        p(t)+ρ(t)=0.
        
        Thus, the effective parameters take the form of a “classical" Dark Energy equation of state with ω = -1.
        Thus, the effective pressure and the effective energy density cancel each other, which explains why the scale–dependence parameter ξ does not appear in the solution for a(t).
        
        
        
        
        
         §.§ Deflation of the cosmological constant problem
        
        
        
        
        
        As explained in the introduction, the CCP 
        in a Planck era universe can be parametrized by
        the value of Λ· G (<ref>).
        However, the quantum gravity calculation makes reference
        to the very early Planck era t_i, while the measurement makes reference
        to the more recent history of the universe, long after the end of inflation t_f.
        scale-dependence can offer alleviation of this problem since it
        allows for a continuous evolution of Λ(t) · G(t)
        from the Planck era t_i to the end of inflation t_f as shown in the left panel of figure <ref>.
        
        As argued in the introduction, we assume that the most dominant scaling effects
        occur in this period and that in later times, after t_f, the 
        scaling effects will be dominated by coupling to 
        external particle scales m_i, which only run logarithmically. 
        
        In order to show that the CCP can be addressed effectively within our framework,
        we impose six conditions
        on the scale-dependent solution (<ref>-<ref>) during inflation.
        One has to check whether those conditions
        can be met by a choice of the constants t_i, t_f, G_0, Λ_0,  a_i, ξ.
        
        
          a) At the initial time t_i≡ 0 one demands
        
        a(t_i)=1.
        
        and
        Λ(t_i)· G(t_i)≈ 1,
        
        which meets the theoretical expectation.
        
          b)
        At the end of inflation t_f one imposes that 
        the gravitational coupling has evolved to a value 
        close to the current measured value
        
        G(t_f)≈ G_N ,
        
        and that the cosmological constant problem has gravitationally evolved to
        Λ(t_f)· G(t_f)≈ 5 · 10^-66,
        
        around the magnitude dictated by the heaviest standard
        model particle.
        
          c)
        The observed flatness of the universe suggest further that 
        during this epoch (from t_i to t_f) of rapid growth, the universe expanded at least
        sixty e-folds <cit.>
        N_e ≥ 60.
        
        By imposing the conditions (<ref>, <ref>, <ref>) one finds
        
        a_i    =   1,
        
         
        t_f   =    N_e τ,
        
        
        Λ_0   =   (1+ξ)^2/G_N (1+2 ξ)(1+ξ e^N_e).
        
        In order to analyze the condition (<ref>), we put it into an order of magnitude inequality, accepting
        five orders of magnitude deviation from the value given in (<ref>)
        
         10^-71⪅Λ(t_f)· G(t_f)⪅ 10^-61.
        
        This double inequality puts conditions on the two remaining constants N_e and ξ.
        The region of allowed values for those two constants, according to (<ref>), is shown in the blue contour in figure <ref>,
        where we have defined ξ= e^ξ_V.
        
        
        
        
        
        
        
        
        
        
        From this figure one notes that the condition (<ref>) never gets in conflict with the observational bound (<ref>).
        
        It is thus shown that substantial alleviation of the cosmological constant problem within the presented framework can be achieved. For this one
        does need the inflationary period to last at least ln(10^66)≈ 150 e-folds.
        This is significantly larger than the experimental lower bound, but it does not cause tension
        with observational data since it is
        very hard to establish a distinctive upper bound for this quantity <cit.>.
        
        The scale-dependent evolution could be incorporated
        into more realistic models of inflation <cit.>, but this goes beyond the scope of this article.
        It is further tempting to speculate whether the suppression of G(t) in the
        first equality of (<ref>) can be used in combination 
        with the scale-dependence of other couplings in the post-inflationary universe <cit.>
        in order to contribute to the solution
        of the hierarchy problem <cit.>. This possibility will be explored in future studies.
        
        
        
        
        
         §.§ Comparison with the functional renormalization group
        
        
        
        
        Even though our calculation never makes use of specific quantum gravity
        beta functions, it is tempting to make a comparison with the findings in such an approach.
        The discussion below will be complementary to the findings in <cit.>,
        where strong effects of the evolution of couplings on the cosmological constant problem have been pointed out in context with a possible infra-red instability of asymptotically save gravity <cit.>
        
        Using functional renormalization group (RG) methods, in particular,
        the Wetterich equation <cit.>, it is possible to obtain
        a prediction for the scale-dependent couplings in gravitation actions such as (<ref>).
        The dimensionless couplings ĝ(k)=Ĝ(k)k^2 and λ̂(k)=Λ̂(k)/k^2 are found for the dimensionless renormalization scale
        t̂=log(k/k_0)ĝ(t̂)   =   g_0 e^2 t̂/1+g_0(e^2 t̂-1)/g^*,
        λ̂(t̂)   =   g^* λ_0 + e^-2 t̂(e^4 t̂ -1)g_0 λ^* /1+g_0(e^2 t̂-1)/g^*,
        
        where g^* and λ^* are the values of the ultra violet fixed point
        predicted by the Asymptotic Safety (AS) conjecture <cit.> and found within the functional RG approach <cit.>.
        Further,
        λ_0 and g_0 are initial values defined at an intermediate renormalization scale t̂=0.
        For the CCP in the very early universe the relevant quantity is the product of the dimensionful couplings Ĝ(t̂)·Λ̂(t̂).
        In four dimensions the 
        product of the dimensionful couplings is equal to the product of the dimensionless couplings (<ref>, <ref>)
        Ĝ(t̂)Λ̂(t̂)=g(t̂) λ(t̂).
        
        Therefore, a comparison of the naturally dimensionless product (<ref>) allows for much more robust statements
        than a comparison of the individual dimensionful or dimensionless couplings.
        
        
        The one million dollar question is whether there is a scale setting t̂=t̂(t) for which the product (<ref>) obtained from AS
        agrees, at least qualitatively, with the product G(t)·Λ(t) derived in our scale-dependent cosmological solution (<ref>, <ref>).
        In the ultraviolet (UV) and close to the separatrix [With “separatrix” it is referred 
        to the RG flow line
        which connects the trivial fixed point g=λ=0 with the non trivial UV fixed point (g^*, λ^*).] of the RG  flow one can approximate (<ref> and <ref>) for
        e^2t̂, g^*/g_0≫1, which gives
        
        
        
        
        
        ĝ(t̂) λ̂(t̂) = g^* λ^*
        ( 
        g^* λ_0/g_0 λ^*+e^2 t̂)
        ( e^2 t̂+g^*/g_0)^-2 .
        
        By inspection one realizes that the simple replacements
        
        
        
        
        
        
        
        
        
        
         g^* λ^*→ G_0 Λ_0,
        
        g_0 →G_0/a_i ξ,
        t̂→ -t/2 τ,
        
        together with the choice λ_0=2 λ^*, leads to a stunning exact matching
        
        .ĝ(t̂) λ̂(t̂) |^_=G(t) Λ(t).
        
        
        This means that for a proper scale setting and  a proper choice
        of the flow trajectory, both approaches
        show the same type of deflation of the cosmological constant problem.
        Graphically this behavior is shown in figure <ref>.
        
        
        
         Comparing our solutions to the AS predictions, on one hand, we observe that the behavior of the product G·Λ as a function of t is exemplified in the figure (<ref>) (LEFT). One notes that G(t)·Λ(t) experiences a strong suppression. In addition, for smaller values of ξ the evolution of the couplings starts at later times. Apart from this change, the functions are form-invariant under t→ t-t_i transformations. This means that the value of ξ can be reabsorbed in a shift in the initial or final conditions for a(t), G(t), and Λ(t). 
        On the other hand, the same features are shown in (<ref>) (RIGHT).
        
        
        Clearly, this agreement can still be a coincidence but it appears sufficiently
        interesting to us to mention it here in the discussion. 
        There are several comments about the result (<ref>) and the identifications 
        (<ref>)
        
        
        to be made.
        The result (<ref>) is not the product of a 
        very particular cut-off identification. It is straightforward to check that the same agreement can
        be obtained from different beta functions arising from different cut-off prescriptions.
        From  (<ref>)
        one sees from the UV/AS perspective that the control parameter ξ
        is not necessarily small, but a rather large number.
        This means that scale-dependence in this regime of high energies is a dominant effect and not just a small correction to classical behavior. 
        One further sees from (<ref>)
        that the product  Λ·  G in the UV is predicted to
        be just the product of the two fixed points g^* ·λ^* of the RG flow.
        The values of those fixed points
        are typically found to be of order∼ 0.2, the product is thus not dramatically far from
        unity, just as we had supposed 
        in our working assumptions.
        Finally, one notes from (<ref>)
        that the scale identification 
        is in agreement with 
        our intuitive expectations, since it associates very early times t→ -∞ to very 
        large RG scales k→∞ and late times t→∞ 
        to small RG scale k→ 0.
        
        We can not resist to insist that the identity (<ref>)
        is absolutely not trivial. It 
        links two completely different and independent
        calculations.
        One is implementing quantum effects into a gravitational theory by
        means of effective quantum field theory and the other
        solving the modified Friedmann  equations (<ref>, <ref>)  subject to a simple energy condition (<ref>).
        The support of both approaches is mutual.
         One can read the agreement of (<ref>, <ref>)
        with (<ref>, <ref>) as support of the approach and the solution presented in this paper.
        One can also conclude that AS is in good
        shape to contribute to the solution of the CCP, if one chooses
        a flow line which is in sufficiently close vicinity of the separatrix line (g_0≪ 1).
        
        
        
        
        
        § CONCLUSIONS
        
        
        
        
        
        In this article, we presented a study on the early inflationary universe in the context of scale-dependence.
        We give reasons, why it can be expected that the leading gravitational couplings experience scale-, in particular time-dependence
        in this early epoch.
        It is then argued that, if one accepts possible scale-dependence of the couplings, then
        the CCP is ill-defined since it makes an identification of the product
        Λ(t) · G(t) at two different times of the cosmological evolution. We conjecture that
         this misconception and thus the CCP can be resolved if
         one takes into account a proper scale-dependence of those couplings.
        
        As a concrete example, we solve the gap equations (<ref>) of the scale-dependent gravitational action in the Einstein-Hilbert truncation
        for the homogenous isotropic line element (<ref>) subject to a null energy condition (<ref>).
        It is found that within this solution (<ref>-<ref>) 
        the scale factor a(t) grows exponentially. 
        Further, the couplings Λ(t) and G(t) have the expected 
        strong time-dependence. 
        It is shown that the time evolution of Λ(t) · G(t) indeed allows accommodating
        a substantial alleviation of the CCP in agreement with the conditions (<ref>-<ref>) and the observational
        lower bound (<ref>).
        
        We note two particular features of our results.
        First, the contribution of the scale-dependent coupling G(t),
        can be written as an ideal fluid obeying an equation of state, which is usually associated with vacuum energy density.
        This is a strong support for the chosen energy condition (<ref>), because it shows
        that the condition guaranties a vacuum type equation of state for the scale dependent contribution Δ t_μν
        of the stress energy tensor.
        Second, a comparison between the product Λ(t) · G(t), deduced in this paper, with
        the product λ̂(t̂) ·ĝ(t̂), obtained from functional RG methods
        within the AS scenario, reveals highly non-trivial similarities. 
        
        
        
        
        
        
         Acknowlegements.
        B. K. was supported by the Fondecyt 1161150 and Fondecyt 1181694. 
        Á.  R. acknowldeges DI-VRIEA for financial support through Proyecto Postdoctorado 2019 VRIEA-PUCV.
        
        
        
        99
        Adler:1995vd 
          R. J. Adler, B. Casey and O. C. Jacob,
          
          Am. J. Phys.  63, 620 (1995).
          doi:10.1119/1.17850
          
          
        Martel:1997vi 
          H. Martel, P. R. Shapiro and S. Weinberg,
          
          Astrophys. J.  492, 29 (1998)
          doi:10.1086/305016
          [astro-ph/9701099].
          
          
        Weinberg:1988cp 
          S. Weinberg,
          
          Rev. Mod. Phys.  61, 1 (1989).
          doi:10.1103/RevModPhys.61.1
          
          
        Guth:1980zm 
          A. H. Guth,
          
          Phys. Rev. D  23, 347 (1981)
          [Adv. Ser. Astrophys. Cosmol.  3, 139 (1987)].
          doi:10.1103/PhysRevD.23.347
          
          
        Linde:1981mu 
          A. D. Linde,
          
          Phys. Lett.  108B, 389 (1982)
          [Adv. Ser. Astrophys. Cosmol.  3, 149 (1987)].
          doi:10.1016/0370-2693(82)91219-9
          
          
        Mukhanov:1990me 
          V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger,
          
          Phys. Rept.  215, 203 (1992).
          doi:10.1016/0370-1573(92)90044-Z
          
          
        Gasperini:1992em 
          M. Gasperini and G. Veneziano,
          
          Astropart. Phys.  1, 317 (1993)
          doi:10.1016/0927-6505(93)90017-8
          [hep-th/9211021].
          
          
        Novello:2002nz 
          M. Novello, J. Barcelos-Neto and J. M. Salim,
          
          Class. Quant. Grav.  19, 3107 (2002)
          doi:10.1088/0264-9381/19/11/323
          [hep-th/0202169].
          
          
        Ashtekar:2004eh 
          A. Ashtekar and J. Lewandowski,
          
          Class. Quant. Grav.  21, R53 (2004)
          doi:10.1088/0264-9381/21/15/R01
          [gr-qc/0404018].
          
          
        Thiemann:2002nj 
          T. Thiemann,
          
          Lect. Notes Phys.  631, 41 (2003)
          doi:10.1007/978-3-540-45230-0 3
          [gr-qc/0210094].
          
          
        Rovelli:2011eq 
          C. Rovelli,
          
          PoS QGQGS  2011, 003 (2011)
          doi:10.22323/1.140.0003
          [arXiv:1102.3660 [gr-qc]].
          
          
        Wetterich:2017ixo 
          C. Wetterich,
          
          Phys. Lett. B  773, 6 (2017)
          doi:10.1016/j.physletb.2017.08.002
          [arXiv:1704.08040 [gr-qc]].
          
          
        Rubio:2017gty 
          J. Rubio and C. Wetterich,
          
          Phys. Rev. D  96, no. 6, 063509 (2017)
          doi:10.1103/PhysRevD.96.063509
          [arXiv:1705.00552 [gr-qc]].
          
          
        Houthoff:2017oam 
          W. B. Houthoff, A. Kurov and F. Saueressig,
          
          Eur. Phys. J. C  77, 491 (2017)
          doi:10.1140/epjc/s10052-017-5046-8
          [arXiv:1705.01848 [hep-th]].
          
          
        Bosma:2019aiu 
          L. Bosma, B. Knorr and F. Saueressig,
          
          arXiv:1904.04845 [hep-th].
          
        Afshordi:2015iza 
          N. Afshordi and E. Nelson,
          
          Phys. Rev. D  93, no. 8, 083505 (2016)
          doi:10.1103/PhysRevD.93.083505
          [arXiv:1504.00012 [hep-th]].
          
          
        Bonanno:2001xi 
          A. Bonanno and M. Reuter,
          
          Phys. Rev. D  65, 043508 (2002)
          doi:10.1103/PhysRevD.65.043508
          [hep-th/0106133].
          
          
        Reuter:2003ca 
          M. Reuter and H. Weyer,
          
          Phys. Rev. D  69, 104022 (2004)
          doi:10.1103/PhysRevD.69.104022
          [hep-th/0311196].
          
          
        Reuter:2004nx 
          M. Reuter and H. Weyer,
          
          JCAP  0412, 001 (2004)
          doi:10.1088/1475-7516/2004/12/001
          [hep-th/0410119].
          
          
        Weinberg:2009wa 
          S. Weinberg,
          
          Phys. Rev. D  81, 083535 (2010)
          doi:10.1103/PhysRevD.81.083535
          [arXiv:0911.3165 [hep-th]].
          
          
        Tye:2010an 
          S.-H. H. Tye and J. Xu,
          
          Phys. Rev. D  82, 127302 (2010)
          doi:10.1103/PhysRevD.82.127302
          [arXiv:1008.4787 [hep-th]].
          
          
        Bonanno:2010bt 
          A. Bonanno, A. Contillo and R. Percacci,
          
          Class. Quant. Grav.  28, 145026 (2011)
          doi:10.1088/0264-9381/28/14/145026
          [arXiv:1006.0192 [gr-qc]].
          
          
        Eichhorn:2010tb 
          A. Eichhorn and H. Gies,
          
          Phys. Rev. D  81, 104010 (2010)
          doi:10.1103/PhysRevD.81.104010
          [arXiv:1001.5033 [hep-th]].
          
          
        Litim:2011cp 
          D. F. Litim,
          
          Phil. Trans. Roy. Soc. Lond. A  369, 2759 (2011)
          doi:10.1098/rsta.2011.0103
          [arXiv:1102.4624 [hep-th]].
          
          
        Grande:2011xf 
          J. Grande, J. Sola, S. Basilakos and M. Plionis,
          
          JCAP  1108, 007 (2011)
          doi:10.1088/1475-7516/2011/08/007
          [arXiv:1103.4632 [astro-ph.CO]].
          
          
        Benedetti:2012dx 
          D. Benedetti and F. Caravelli,
          
          JHEP  1206, 017 (2012)
          Erratum: [JHEP  1210, 157 (2012)]
          doi:10.1007/JHEP06(2012)017, 10.1007/JHEP10(2012)157
          [arXiv:1204.3541 [hep-th]].
          
          
        Hindmarsh:2012rc 
          M. Hindmarsh and I. D. Saltas,
          
          Phys. Rev. D  86, 064029 (2012)
          doi:10.1103/PhysRevD.86.064029
          [arXiv:1203.3957 [gr-qc]].
          
          
        Eichhorn:2012va 
          A. Eichhorn,
          
          Phys. Rev. D  86, 105021 (2012)
          doi:10.1103/PhysRevD.86.105021
          [arXiv:1204.0965 [gr-qc]].
          
          
        Dona:2013qba 
          P. Doná, A. Eichhorn and R. Percacci,
          
          Phys. Rev. D  89, no. 8, 084035 (2014)
          doi:10.1103/PhysRevD.89.084035
          [arXiv:1311.2898 [hep-th]].
          
          
        Copeland:2013vva 
          E. J. Copeland, C. Rahmede and I. D. Saltas,
          
          Phys. Rev. D  91, no. 10, 103530 (2015)
          doi:10.1103/PhysRevD.91.103530
          [arXiv:1311.0881 [gr-qc]].
          
          
        Bonanno:2015fga 
          A. Bonanno and A. Platania,
          
          Phys. Lett. B  750, 638 (2015)
          doi:10.1016/j.physletb.2015.10.005
          [arXiv:1507.03375 [gr-qc]].
          
          
        Rodrigues:2015hba 
          D. C. Rodrigues, B. Chauvineau and O. F. Piattella,
          
          JCAP  1509, no. 09, 009 (2015)
          doi:10.1088/1475-7516/2015/09/009
          [arXiv:1504.05119 [gr-qc]].
          
          
        Modesto:2015ozb 
          L. Modesto and I. L. Shapiro,
          
          Phys. Lett. B  755, 279 (2016)
          doi:10.1016/j.physletb.2016.02.021
          [arXiv:1512.07600 [hep-th]].
          
          
        Rodrigues:2015rya 
          D. C. Rodrigues, B. Koch, O. F. Piattella and I. L. Shapiro,
          
          AIP Conf. Proc.  1647, 57 (2015).
          doi:10.1063/1.4913338
          
          
        Rodrigues:2016tfm 
          D. C. Rodrigues, S. Mauro and Á. O. F. de Almeida,
          
          Phys. Rev. D  94, no. 8, 084036 (2016)
          doi:10.1103/PhysRevD.94.084036
          [arXiv:1609.03613 [gr-qc]].
          
          
        Koch:2016uso 
          B. Koch, I. A. Reyes and Á. Rincón,
          
          Class. Quant. Grav.  33, no. 22, 225010 (2016)
          doi:10.1088/0264-9381/33/22/225010
          [arXiv:1606.04123 [hep-th]].
          
          
        Platania:2017djo 
          A. Platania and F. Saueressig,
          
          Found. Phys.  48, no. 10, 1291 (2018)
          doi:10.1007/s10701-018-0181-0
          [arXiv:1710.01972 [hep-th]].
          
          
        Bonanno:2017gji 
          A. Bonanno, S. J. Gabriele Gionti and A. Platania,
          
          Class. Quant. Grav.  35, no. 6, 065004 (2018)
          doi:10.1088/1361-6382/aaa535
          [arXiv:1710.06317 [gr-qc]].
          
          
        Rincon:2017ypdÁ. Rincón, B. Koch and I. Reyes,
          
          J. Phys. Conf. Ser.  831, no. 1, 012007 (2017)
          doi:10.1088/1742-6596/831/1/012007
          [arXiv:1701.04531 [hep-th]].
          
          
        Rincon:2017gojÁ. Rincón, E. Contreras, P. Bargueño, B. Koch, G. Panotopoulos and A. Hernández-Arboleda,
          
          Eur. Phys. J. C  77, no. 7, 494 (2017)
          doi:10.1140/epjc/s10052-017-5045-9
          [arXiv:1704.04845 [hep-th]].
          
          
        Modesto:2017hzl 
          L. Modesto, L. Rachwal and I. L. Shapiro,
          
          Eur. Phys. J. C  78, no. 7, 555 (2018)
          doi:10.1140/epjc/s10052-018-6035-2
          [arXiv:1704.03988 [hep-th]].
          
          
        Contreras:2017eza 
          E. Contreras, Á. Rincón, B. Koch and P. Bargueño,
          
          Int. J. Mod. Phys. D  27, no. 03, 1850032 (2017)
          doi:10.1142/S0218271818500323
          [arXiv:1711.08400 [gr-qc]].
          
          
        Calmet:2017rxl 
          X. Calmet, S. Capozziello and D. Pryer,
          
          Eur. Phys. J. C  77, no. 9, 589 (2017)
          doi:10.1140/epjc/s10052-017-5172-3
          [arXiv:1708.08253 [hep-th]].
          
          
        Toniato:2017wmk 
          J. D. Toniato, D. C. Rodrigues, Á. O. F. de Almeida and N. Bertini,
          
          Phys. Rev. D  96, no. 6, 064034 (2017)
          doi:10.1103/PhysRevD.96.064034
          [arXiv:1706.09032 [gr-qc]].
          
          
        Merzlikin:2017zan 
          B. S. Merzlikin, I. L. Shapiro, A. Wipf and O. Zanusso,
          
          Phys. Rev. D  96, no. 12, 125007 (2017)
          doi:10.1103/PhysRevD.96.125007
          [arXiv:1711.02224 [hep-th]].
          
          
        Rincon:2018sgdÁ. Rincón and G. Panotopoulos,
          
          Phys. Rev. D  97, no. 2, 024027 (2018)
          doi:10.1103/PhysRevD.97.024027
          [arXiv:1801.03248 [hep-th]].
          
          
        Hernandez-Arboleda:2018qdo 
          A. Hernández-Arboleda, Á. Rincón, B. Koch, E. Contreras and P. Bargueño,
          
          arXiv:1802.05288 [gr-qc].
          
          
        Contreras:2018dhs 
          E. Contreras, Á. Rincón, B. Koch and P. Bargueño,
          
          Eur. Phys. J. C  78, no. 3, 246 (2018)
          doi:10.1140/epjc/s10052-018-5709-0
          [arXiv:1803.03255 [gr-qc]].
          
          
        Rincon:2018lydÁ. Rincón and B. Koch,
          
          Eur. Phys. J. C  78, no. 12, 1022 (2018)
          doi:10.1140/epjc/s10052-018-6488-3
          [arXiv:1806.03024 [hep-th]].
          
          
        Rincon:2018dsqÁ. Rincón, E. Contreras, P. Bargueño, B. Koch and G. Panotopoulos,
          
          Eur. Phys. J. C  78, no. 8, 641 (2018)
          doi:10.1140/epjc/s10052-018-6106-4
          [arXiv:1807.08047 [hep-th]].
          
          
        Calmet:2018elv 
          X. Calmet,
          
          Phys. Lett. B  787, 36 (2018)
          doi:10.1016/j.physletb.2018.10.040
          [arXiv:1810.09719 [hep-th]].
          
          
        Contreras:2018gct 
          E. Contreras, Á. Rincón and J. M. Ramírez-Velasquez,
          
          Eur. Phys. J. C  79, no. 1, 53 (2019)
          doi:10.1140/epjc/s10052-019-6601-2
          [arXiv:1810.07356 [gr-qc]].
          
          
        Bonanno:2018gck 
          A. Bonanno, A. Platania and F. Saueressig,
          
          Phys. Lett. B  784, 229 (2018)
          doi:10.1016/j.physletb.2018.06.047
          [arXiv:1803.02355 [gr-qc]].
          
          
        Contreras:2018gpl 
          E. Contreras and P. Bargueño,
          
          Mod. Phys. Lett. A  33, no. 32, 1850184 (2018)
          doi:10.1142/S0217732318501845
          [arXiv:1809.00785 [gr-qc]].
          
          
        Contreras:2018swc 
          E. Contreras and P. Bargueño,
          
          Int. J. Mod. Phys. D  27, no. 09, 1850101 (2018)
          doi:10.1142/S0218271818501018
          [arXiv:1804.00988 [gr-qc]].
          
          
        Sahni:2002kh 
          V. Sahni,
          
          Class. Quant. Grav.  19, 3435 (2002)
          doi:10.1088/0264-9381/19/13/304
          [astro-ph/0202076].
          
          
        Prokopec:2006yh 
          T. Prokopec,
          
          gr-qc/0603088.
          
          
        Brodsky:2009zd 
          S. J. Brodsky and R. Shrock,
          
          Proc. Nat. Acad. Sci.  108, 45 (2011)
          doi:10.1073/pnas.1010113107
          [arXiv:0905.1151 [hep-th]].
          
          
        Kaloper:2013zca 
          N. Kaloper and A. Padilla,
          
          Phys. Rev. Lett.  112, no. 9, 091304 (2014)
          doi:10.1103/PhysRevLett.112.091304
          [arXiv:1309.6562 [hep-th]].
          
          
        Stojkovic:2014lha 
          D. Stojkovic,
          
          Mod. Phys. Lett. A  28, 1330034 (2013)
          doi:10.1142/S0217732313300346
          [arXiv:1406.2696 [gr-qc]].
          
          
        Padilla:2015aaa 
          A. Padilla,
          
          arXiv:1502.05296 [hep-th].
          
          
        Bass:2015yaa 
          S. D. Bass,
          
          Mod. Phys. Lett. A  30, no. 22, 1540033 (2015)
          doi:10.1142/S0217732315400337
          [arXiv:1503.05483 [hep-ph]].
          
          
        Novikov:2016hrc 
          E. A. Novikov,
          
          Mod. Phys. Lett. A  31, no. 15, 1650092 (2016).
          doi:10.1142/S0217732316500929
          
          
        Nojiri:2016mlb 
          S. Nojiri,
          
          Mod. Phys. Lett. A  31, no. 37, 1650213 (2016)
          doi:10.1142/S0217732316502138
          [arXiv:1601.02203 [hep-th]].
          
          
        Biemans:2016rvp 
          J. Biemans, A. Platania and F. Saueressig,
          
          Phys. Rev. D  95, no. 8, 086013 (2017)
          doi:10.1103/PhysRevD.95.086013
          [arXiv:1609.04813 [hep-th]].
          
          
        Hossenfelder:2018ikr 
          S. Hossenfelder,
          
          arXiv:1801.02176 [physics.hist-ph].
          
          
        Weinberg:1976xy 
          S. Weinberg,
          
          doi:10.1007/978-1-4684-0931-4 1
          
          
        Hawking:1979ig 
          S. W. Hawking and W. Israel,
          
          
          
        Wetterich:1992yh 
          C. Wetterich,
          
          Phys. Lett. B  301, 90 (1993)
          doi:10.1016/0370-2693(93)90726-X
          [arXiv:1710.05815 [hep-th]].
          
          
        Morris:1993qb 
          T. R. Morris,
          
          Int. J. Mod. Phys. A  9, 2411 (1994)
          doi:10.1142/S0217751X94000972
          [hep-ph/9308265].
          
          
        Bonanno:2000ep 
          A. Bonanno and M. Reuter,
          
          Phys. Rev. D  62, 043008 (2000)
          doi:10.1103/PhysRevD.62.043008
          [hep-th/0002196].
          
          
        Litim:2002xm 
          D. F. Litim and J. M. Pawlowski,
          
          Phys. Rev. D  66, 025030 (2002)
          doi:10.1103/PhysRevD.66.025030
          [hep-th/0202188].
          
          
        Reuter:2004nv 
          M. Reuter and H. Weyer,
          
          Phys. Rev. D  70, 124028 (2004)
          doi:10.1103/PhysRevD.70.124028
          [hep-th/0410117].
          
          
        Bonanno:2006eu 
          A. Bonanno and M. Reuter,
          
          Phys. Rev. D  73, 083005 (2006)
          doi:10.1103/PhysRevD.73.083005
          [hep-th/0602159].
          
          
        Niedermaier:2006ns 
          M. Niedermaier,
          
          Class. Quant. Grav.  24, R171 (2007)
          doi:10.1088/0264-9381/24/18/R01
          [gr-qc/0610018].
          
          
        Percacci:2007sz 
          R. Percacci,
          
          In *Oriti, D. (ed.): Approaches to quantum gravity* 111-128
          [arXiv:0709.3851 [hep-th]].
          
          
        Bonanno:1998ye 
          A. Bonanno and M. Reuter,
          
          Phys. Rev. D  60, 084011 (1999)
          doi:10.1103/PhysRevD.60.084011
          [gr-qc/9811026].
          
          
        Emoto:2005te 
          H. Emoto,
          
          hep-th/0511075.
          
          
        Reuter:2006rg 
          M. Reuter and E. Tuiran,
          
          doi:10.1142/9789812834300 0473
          hep-th/0612037.
          
          
        Ward:2006vw 
          B. F. L. Ward,
          
          Acta Phys. Polon. B  37, 1967 (2006)
          [hep-ph/0605054].
          
          
        Koch:2007yt 
          B. Koch,
          
          Phys. Lett. B  663, 334 (2008)
          doi:10.1016/j.physletb.2008.04.025
          [arXiv:0707.4644 [hep-ph]].
          
          
        Hewett:2007st 
          J. Hewett and T. Rizzo,
          
          JHEP  0712, 009 (2007)
          doi:10.1088/1126-6708/2007/12/009
          [arXiv:0707.3182 [hep-ph]].
          
          
        Litim:2007iu 
          D. F. Litim and T. Plehn,
          
          Phys. Rev. Lett.  100, 131301 (2008)
          doi:10.1103/PhysRevLett.100.131301
          [arXiv:0707.3983 [hep-ph]].
          
          
        Burschil:2009va 
          T. Burschil and B. Koch,
          
          Zh. Eksp. Teor. Fiz.  92, 219 (2010)
          [JETP Lett.  92, 193 (2010)]
          doi:10.1134/S0021364010160010
          [arXiv:0912.4517 [hep-ph]].
          
          
        Falls:2010he 
          K. Falls, D. F. Litim and A. Raghuraman,
          
          Int. J. Mod. Phys. A  27, 1250019 (2012)
          doi:10.1142/S0217751X12500194
          [arXiv:1002.0260 [hep-th]].
          
          
        Casadio:2010fw 
          R. Casadio, S. D. H. Hsu and B. Mirza,
          
          Phys. Lett. B  695, 317 (2011)
          doi:10.1016/j.physletb.2010.10.060
          [arXiv:1008.2768 [gr-qc]].
          
          
        Basu:2010nf 
          S. Basu and D. Mattingly,
          
          Phys. Rev. D  82, 124017 (2010)
          doi:10.1103/PhysRevD.82.124017
          [arXiv:1006.0718 [hep-th]].
          
          
        Reuter:2010xb 
          M. Reuter and E. Tuiran,
          
          Phys. Rev. D  83, 044041 (2011)
          doi:10.1103/PhysRevD.83.044041
          [arXiv:1009.3528 [hep-th]].
          
          
        Cai:2010zh 
          Y. F. Cai and D. A. Easson,
          
          JCAP  1009, 002 (2010)
          doi:10.1088/1475-7516/2010/09/002
          [arXiv:1007.1317 [hep-th]].
          
          
        Falls:2012nd 
          K. Falls and D. F. Litim,
          
          Phys. Rev. D  89, 084002 (2014)
          doi:10.1103/PhysRevD.89.084002
          [arXiv:1212.1821 [gr-qc]].
          
          
        Becker:2012js 
          D. Becker and M. Reuter,
          
          JHEP  1207, 172 (2012)
          doi:10.1007/JHEP07(2012)172
          [arXiv:1205.3583 [hep-th]].
          
          
        Becker:2012jx 
          D. Becker and M. Reuter,
          
          doi:10.1142/9789814623995 0405
          arXiv:1212.4274 [hep-th].
          
          
        Koch:2013owa 
          B. Koch and F. Saueressig,
          
          Class. Quant. Grav.  31, 015006 (2014)
          doi:10.1088/0264-9381/31/1/015006
          [arXiv:1306.1546 [hep-th]].
          
          
        Gonzalez:2015upa 
          C. González and B. Koch,
          
          Int. J. Mod. Phys. A  31, no. 26, 1650141 (2016)
          doi:10.1142/S0217751X16501414
          [arXiv:1508.01502 [hep-th]].
          
          
        Torres:2017ygl 
          R. Torres,
          
          Phys. Rev. D  95, no. 12, 124004 (2017)
          doi:10.1103/PhysRevD.95.124004
          [arXiv:1703.09997 [gr-qc]].
          
          
        Knorr:2017fus 
          B. Knorr and S. Lippoldt,
          
          Phys. Rev. D  96, no. 6, 065020 (2017)
          doi:10.1103/PhysRevD.96.065020
          [arXiv:1707.01397 [hep-th]].
          
          
        Eichhorn:2017egq 
          A. Eichhorn,
          
          Found. Phys.  48, no. 10, 1407 (2018)
          doi:10.1007/s10701-018-0196-6
          [arXiv:1709.03696 [gr-qc]].
          
          
        Pawlowski:2018swz 
          J. M. Pawlowski and D. Stock,
          
          Phys. Rev. D  98, no. 10, 106008 (2018)
          doi:10.1103/PhysRevD.98.106008
          [arXiv:1807.10512 [hep-th]].
          
          
        Alkofer:2018zze 
          N. Alkofer,
          
          arXiv:1810.03132 [hep-th].
          
          
        Eichhorn:2018yfc 
          A. Eichhorn,
          
          arXiv:1810.07615 [hep-th].
          
          
        Koch:2010nn 
          B. Koch and I. Ramirez,
          
          Class. Quant. Grav.  28, 055008 (2011)
          doi:10.1088/0264-9381/28/5/055008
          [arXiv:1010.2799 [gr-qc]].
          
          
        Domazet:2012tw 
          S. Domazet and H. Stefancic,
          
          Class. Quant. Grav.  29, 235005 (2012)
          doi:10.1088/0264-9381/29/23/235005
          [arXiv:1204.1483 [gr-qc]].
          
          
        Koch:2014joa 
          B. Koch, P. Rioseco and C. Contreras,
          
          Phys. Rev. D  91, no. 2, 025009 (2015)
          doi:10.1103/PhysRevD.91.025009
          [arXiv:1409.4443 [hep-th]].
          
          
        Manrique:2009uh 
          E. Manrique and M. Reuter,
          
          Annals Phys.  325, 785 (2010)
          doi:10.1016/j.aop.2009.11.009
          [arXiv:0907.2617 [gr-qc]].
          
          
        Manrique:2010am 
          E. Manrique, M. Reuter and F. Saueressig,
          
          Annals Phys.  326, 463 (2011)
          doi:10.1016/j.aop.2010.11.006
          [arXiv:1006.0099 [hep-th]].
          
          
        Manrique:2010mq 
          E. Manrique, M. Reuter and F. Saueressig,
          
          Annals Phys.  326, 440 (2011)
          doi:10.1016/j.aop.2010.11.003
          [arXiv:1003.5129 [hep-th]].
          
          
        Becker:2014qya 
          D. Becker and M. Reuter,
          
          Annals Phys.  350, 225 (2014)
          doi:10.1016/j.aop.2014.07.023
          [arXiv:1404.4537 [hep-th]].
          
          
        Dietz:2015owa 
          J. A. Dietz and T. R. Morris,
          
          JHEP  1504, 118 (2015)
          doi:10.1007/JHEP04(2015)118
          [arXiv:1502.07396 [hep-th]].
          
          
        Labus:2016lkh 
          P. Labus, T. R. Morris and Z. H. Slade,
          
          Phys. Rev. D  94, no. 2, 024007 (2016)
          doi:10.1103/PhysRevD.94.024007
          [arXiv:1603.04772 [hep-th]].
          
          
        Morris:2016spn 
          T. R. Morris,
          
          JHEP  1611, 160 (2016)
          doi:10.1007/JHEP11(2016)160
          [arXiv:1610.03081 [hep-th]].
          
          
        Ohta:2017dsq 
          N. Ohta,
          
          PTEP  2017, no. 3, 033E02 (2017)
          doi:10.1093/ptep/ptx020
          [arXiv:1701.01506 [hep-th]].
          
          
        Koch:2014cqa 
          B. Koch and F. Saueressig,
          
          Int. J. Mod. Phys. A  29, no. 8, 1430011 (2014)
          doi:10.1142/S0217751X14300117
          [arXiv:1401.4452 [hep-th]].
          
          
        Rincon:2017ayrÁ. Rincón and B. Koch,
          
          J. Phys. Conf. Ser.  1043, no. 1, 012015 (2018)
          doi:10.1088/1742-6596/1043/1/012015
          [arXiv:1705.02729 [hep-th]].
          
          
        Jacobson:2007tj 
          T. Jacobson,
          
          Class. Quant. Grav.  24, 5717 (2007)
          doi:10.1088/0264-9381/24/22/N02
          [arXiv:0707.3222 [gr-qc]].
          
          
        Visser:1999de 
          M. Visser and C. Barcelo,
          
          doi:10.1142/9789812792129 0014
          gr-qc/0001099.
          
          
        Canales:2018tbn 
          F. Canales, B. Koch, C. Laporte and Á. Rincón,
          
          arXiv:1812.10526 [gr-qc].
          
          
        Patrignani:2016xqp 
          C. Patrignani  et al. [Particle Data Group],
          
          Chin. Phys. C  40, no. 10, 100001 (2016).
          doi:10.1088/1674-1137/40/10/100001
          
          
        Banks:2003pt 
          T. Banks and W. Fischler,
          
          astro-ph/0307459.
          
          
        Kolb:1990vq 
          E. W. Kolb and M. S. Turner,
          
          Front. Phys.  69, 1 (1990).
          
          
        Capozziello:1996xq 
          S. Capozziello, R. de Ritis and A. A. Marino,
          
          Nuovo Cim. B  112, 1351 (1997)
          [astro-ph/9605176].
          
          
        Calmet:2001nu 
          X. Calmet and H. Fritzsch,
          
          Eur. Phys. J. C  24, 639 (2002)
          doi:10.1007/s10052-002-0976-0
          [hep-ph/0112110].
          
          
        Carneiro:2003as 
          S. Carneiro,
          
          doi:10.1142/9789812704030 0204
          gr-qc/0307114.
          
          
        Reisenegger:2009cq 
          A. Reisenegger, P. Jofré and R. Fernández,
          
          Mem. Soc. Ast. It.  80, no. 4, 829 (2009)
          [Highlights Astron.  15, 314 (2010)]
          doi:10.1017/S1743921310009518
          [arXiv:0911.0190 [astro-ph.HE]].
          
          
        Yunes:2009bv 
          N. Yunes, F. Pretorius and D. Spergel,
          
          Phys. Rev. D  81, 064018 (2010)
          doi:10.1103/PhysRevD.81.064018
          [arXiv:0912.2724 [gr-qc]].
          
          
        Uzan:2010pm 
          J. P. Uzan,
          
          Living Rev. Rel.  14, 2 (2011)
          doi:10.12942/lrr-2011-2
          [arXiv:1009.5514 [astro-ph.CO]].
          
          
        Fritzsch:2010zzc 
          H. Fritzsch,
          
          Nucl. Phys. Proc. Suppl.  203-204, 3 (2010).
          doi:10.1016/j.nuclphysbps.2010.08.002
          
          
        Anderson:2015bva 
          J. D. Anderson, G. Schubert, V. Trimble and M. R. Feldman,
          
          EPL  110, no. 1, 10002 (2015)
          doi:10.1209/0295-5075/110/10002
          [arXiv:1504.06604 [gr-qc]].
          
          
        Kantha:2016ylw 
          L. Kantha,
          
          Adv. Astron.  2016, 9743970 (2016).
          doi:10.1155/2016/9743970
          
          
        Calmet:2017czo 
          X. Calmet,
          
          Eur. Phys. J. C  77, no. 11, 729 (2017)
          doi:10.1140/epjc/s10052-017-5324-5
          [arXiv:1707.06922 [gr-qc]].
          
          
        Martin:1997ns 
          S. P. Martin,
          
          Adv. Ser. Direct. High Energy Phys.  21, 1 (2010)
          [Adv. Ser. Direct. High Energy Phys.  18, 1 (1998)]
          doi:10.1142/9789812839657 0001, 10.1142/9789814307505 0001
          [hep-ph/9709356].
          
          
        ArkaniHamed:1998rs 
          N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali,
          
          Phys. Lett. B  429, 263 (1998)
          doi:10.1016/S0370-2693(98)00466-3
          [hep-ph/9803315].
          
          
        Meissner:2006zh 
          K. A. Meissner and H. Nicolai,
          
          Phys. Lett. B  648, 312 (2007)
          doi:10.1016/j.physletb.2007.03.023
          [hep-th/0612165].
          
          
        Reuter:1996cp 
          M. Reuter,
          
          Phys. Rev. D  57, 971 (1998)
          doi:10.1103/PhysRevD.57.971
          [hep-th/9605030].
          
          
        Reuter:2001ag 
          M. Reuter and F. Saueressig,
          
          Phys. Rev. D  65, 065016 (2002)
          doi:10.1103/PhysRevD.65.065016
          [hep-th/0110054].
          
          
        Litim:2003vp 
          D. F. Litim,
          
          Phys. Rev. Lett.  92, 201301 (2004)
          doi:10.1103/PhysRevLett.92.201301
          [hep-th/0312114].
