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hoava  lifshitz gravity has been proposed as a power - counting renormalizable gravity theory , and a potentially ultra - violet ( uv ) complete theory of quantum gravity @xcite .
the improved uv behaviour is achieved by giving up lorentz symmetry and adding terms to the action that are higher order in spatial derivatives , which suitably modifies the propagators .
the theory is most straightforwardly defined in a space - time decomposition ( adm splitting ) where a preferred foliation has been imposed and @xmath0 is the lapse , @xmath1 is the shift and @xmath2 the 3-metric induced on the spacelike hypersurfaces .
the action of the theory is then s = s_k - s_v , where the kinetic term @xmath3 is given by s_k = dtd^3x n k_ijg^ijklk_kl , @xmath4 is the extrinsic curvature of the spacelike hypersurfaces , @xmath5 a coupling constant with suitable dimensions and g^ijkl=(g^ikg^jl+g^ilg^jk)-g^ijg^kl , is the generalized dewitt metric .
the potential term can be written as s_v = dtd^3x n v[g_ij , n ] .
the action is required to be invariant under the subclass of diffeomorphisms that leave the foliation intact , _
@xmath6 and @xmath7 , so in principle @xmath8 should include all terms compatible with this symmetry .
in addition , @xmath8 should contain terms which are at least 6th order in spatial derivatives for the theory to have the desirable uv properties @xcite .
an unappealing feature of the full theory is that , without any further restrictions or symmetries , @xmath8 will contain a very large number of terms ( and independent couplings ) .
two restrictions have been considered as the means to limit the proliferation of couplings in the original proposal : _ projectability _ and _ detailed balance_. projectability sums up to the requirement that @xmath9 , whereas detailed balance requires @xmath8 to be derived from a  superpotential " ( see below ) .
an important characteristic of the theory is that it propagates not only a spin-2 , but also a spin-0 mode .
the assumption of projectability , with or without detailed balance @xcite leads to pathologies in the dynamics of the scalar mode , in particular instabilities and strong coupling at low energies @xcite .
for this reason , here we discuss a version of the theory which satisfies the detailed balance condition without assuming projectability .
this proceedings contribution is based on ref .
@xcite , where a more detailed discussion can be found .
see also ref .
@xcite for a brief review on the various version of hoava  lifshitz gravity .
detailed balance prescribes that the potential @xmath8 is derived from a superpotential @xmath10 as follows v = g_ijkl , [ db ] where @xmath11 is the inverse of the dewitt metric @xmath12 .
once @xmath10 has been chosen the action is fully determined .
@xmath10 can be a functional of the metric @xmath2 but also of the lapse @xmath0 , and in particular of the combination @xmath13 , if invariance under foliation preserving diffeomorphisms is imposed @xcite .
furthermore , in order to have a power - counting renormalizable theory one needs at least sixth order spatial derivatives in the action @xcite , thus the superpotential @xmath10 must contain at least third order spatial derivatives .
then , the most general @xmath10 containing all of the possible terms up to third order spatial derivatives is w=++ , [ w ] where @xmath14 is the gravitational chern - simons term , @xmath15 , @xmath16 , @xmath17 and @xmath18 have dimensions of a mass , whereas @xmath19 is a dimensionless coupling .
the corresponding action is [ faction ] s_h&=&dt d^3x n\ { k_ijk^ij - k^2+r-2+a^ia_i - r_ijr^ij+ r^2 + & & + - ( a^ia_i)^2+^ijkr_il_j r^l_k + & & + c^ija_ia_j - c_ijc^ij } , where @xmath20 is the cotton  york tensor and = , = ^2(1 - 3)m_4 ^ 2 .
[ cc ] for @xmath21 , which are the values these parameters have in general relativity , the bare cosmological constant , @xmath22 , is negative @xcite .
moreover , the magnitude of @xmath22 is directly related to @xmath17 @xcite which is the mass scale at which lorentz - violating effects become manifest as modifications to the dispersion relation . the mildest constraint on the value of @xmath17 , which comes by the fact that the gravitation interaction has been tested down to mm scales , is @xmath23 .
but depend on the details of how lorentz violations percolate into the matter sector , see _
e.g. _  ref .
thus , the bare cosmological constant is at best 60 orders of magnitude bigger than the observed value
.    the overall value of the cosmological constant would be the sum of the bare cosmological constant and the contribution from the vacuum energy of matter fields .
therefore , the size and not the sign of the bare cosmological constant is the main concern here . the contribution of the vacuum energy is not known with any certainty , but one can only hope for a miraculous cancelation if the overall value is to match the observed value . there is no known reason to expect such a cancelation ( see , however , ref .
in order to highlight that the magnitude of the cosmological constant is the only real concern and that the action constructed here describes a dynamically sensible theory , we will proceed as follows : we will set @xmath24 by fiat ( to simplify the analysis ) and consider perturbations around a flat background .
we focus in particular to scalar perturbations , as the main concern is the dynamics of the scalar mode .
thus we have n=1 + , n_i=_i y , g_ij = e^2_ij , and at quadratic order the action becomes [ quad ] s^(2)=dt d^3x \{^2 + 2(-1)^2-(^2)^2 } , where we have integrating out the nondynamical degrees of freedom @xmath25 and @xmath26 .
the corresponding dispersion relation for the scalar is ^2=(-1)p^2+()^2p^4 .
[ disp ] then we can conclude that there is a suitable choice of the parameters such that the scalar has positive energy ( the right sign is determined by the sign of the kinetic term of the spin-2 graviton ) , and at the same time is classically stable : 2>>0 .
the coefficient of the @xmath27 term in eq .
( [ disp ] ) is manifestly positive , so there are no short - wavelength instabilities .
however , the dispersion relation does not contain any sixth order terms .
this is because the only sixth order term in action ( [ faction ] ) does not contribute to the dynamics of the scalar more .
the absence of sixth order terms in the dispersion relation poses a threat for renormalizability .
however , a straightforward resolution is to add to the superpotential @xmath10 fourth order terms , so as to generate sixth ( and eight order terms ) in the action , which would restore the appealing renormalizability properties of the theory .
the minimal consistent prescription would be to add to @xmath10 all possible fourth order terms that respect invariance under foliation preserving diffeomorphisms , but also impose parity invariance .
then the terms added would be r^2 , r^r _ , r ^i a_i , r^ija_i a_j , r a_i a^i,(a_i a^i)^2 , ( ^i a_i)^2 , a_i a_j ^i a^j .
one would be left with 12 free couplings , which is an appreciable improvement with respect to the theory without detailed balance .
we have put together an action for hoava  lifshitz gravity that satisfies detailed balance without assuming projectability .
this version appears to lead to sensible dynamics for the scalar mode and be free of other pathologies that plague the projectable version of hoava  lifshitz gravity .
additionally , it has the appealing renormalizability properties of the most general non - projectable theory , with about an order of magnitude less independent couplings .
however , imposing detailed balance leads to a large and negative bare cosmological constant . barring a miraculous cancelation with the ( yet to be understood ) vacuum energy contribution to the cosmological constant ,
this is obviously hard to reconcile with low energy gravitational phenomenology .
an additional caveat in the use of detailed balance is the fact that it is not clear if or why it should be robust against radiative corrections .
the implementation of detailed balance presented here leads to a consistent theory which is free of pathologies and , therefore , allows to consider this question rigorously .
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hoava  lifshitz gravity with `` detailed balance '' but without the projectability assumption is discussed . 
 it is shown that detailed balance is quite efficient in limiting the proliferation of couplings in hoava  lifshitz gravity , and that its implementation without the projectability assumption leads to a theory with sensible dynamics . 
 however , the ( bare ) cosmological constant is restricted to be large and negative .