Consequences of Symmetries and Consistency Relations in the Large-Scale Structure of the Universe for Non-local bias and Modified Gravity

Consistency relations involving the soft limit of the (n + 1)-correlator functions of dark matter and galaxy overdensities can be obtained, both in real and redshift space, thanks to the symmetries enjoyed by the Newtonian equations of motion describing the dark matter and galaxy fluids coupled through gravity. We study the implications of such symmetries for the theory of galaxy bias and for the theories of modified gravity. We find that the invariance of the fluid equations under a coordinate transformation that induces a long-wavelength velocity constrain the bias to depend only on a set of invariants, while the symmetry of such equations under Lifshitz scalings in the case of matter domination allows one to compute the time-dependence of the coefficients in the bias expansion. We also find that in theories of modified gravity which violate the equivalence principle induce a violation of the consistency relation which may be a signature for their observation. Thus, given adiabatic Gaussian initial conditions, the observation of a deviation from the consistency relation for galaxies would signal a break-down of the so-called non-local Eulerian bias model or the violation of the equivalence principle in the underlying theory of gravity.


Introduction
Symmetries play a crucial role in understanding the properties of a physical system and they have turned out to be quite useful in characterizing the cosmological perturbations generated during a de Sitter stage [1]. Since the de Sitter isometry group SO (1,4) acts like conformal group on R 3 when the fluctuations are on super-Hubble scales, the correlators of scalar fields, which are not the inflaton, are constrained by conformal invariance [2][3][4][5][6]. The fact that the de Sitter isometry group acts as conformal group on the three-dimensional Euclidean space on super-Hubble scales can be also used to predict the shape of the correlators involving the inflaton and vector fields [7]. Furthermore, if the inflationary perturbations are generated in single-field models of inflation, there exist conformal consistency relations among the inflationary correlators [8][9][10][11][12][13][14][15]. Consistency relations involving the soft limit of the (n + 1)-correlator functions of matter and galaxy overdensities have also been proposed by investigating the symmetries enjoyed by the Newtonian equations of motion of the non-relativistic dark matter and galaxy fluids coupled to gravity [16,17]. These consistency relations have been recently generalized to the relativistic limit [18] (see also [19]), based on the observation that a long mode, in single-field models of inflation, reduces to a diffeomorphism since its freezing during inflation all the way until the late universe, even when the long mode is inside the horizon (but out of the sound horizon).
The large-scale consistency relations have the virtue of being true also for the galaxy overdensities, independently of the bias between galaxy and dark matter. As such, they may serve as a guidance in building up a bias theory. Indeed, we will argue that the non-local Eulerian bias model can be seen as being built of quantities which are invariant under the symmetries enjoyed by the Newtonian fluid equations. Furthermore, they might be useful in testing theories of modified gravity where extra degrees of freedom appear mediating extra long-range forces (other than the gravitational one) and possibly leading to a violation of the Equivalence Principle (EP) in the late universe and therefore to a violation of the consistency relation. In fact, assuming adiabatic Gaussian initial conditions, an observed violation of the consistency relations would either indicate a break-down of the non-local Eulerian bias model (and also the presence of terms in the effective fluid equations for galaxies that break the aforementioned symmetries), or a violation of the EP in the underlying theory of gravity.
It is in the spirit of exploring these topics that in this paper we aim to investigate what the large-scale consistency relations may tell us about the galaxy bias and how they can be used to scrutinize modified gravity theories. In particular, we will show that the symmetries leading to the consistency relations allow the presence of what is commonly dubbed non-local bias, that is a relation between the galaxy and the dark matter overdensities which is not a simple function of the local dark matter abundance. We will identify a series of invariants (with respect to the symmetries) which should appear in the galaxy bias expansion, precisely because they are allowed by the symmetries of the problem. Furthermore, we will investigate under which conditions the consistency relations are valid in the case in which a modification of gravity is attained far in the infrared on cosmological scales.
The paper is organized as follows. In section 2 we discuss the symmetries of the non-relativistic fluid equations for both dark matter and galaxies and we derive galaxy consistency relations for the n-point correlators of short wavelength modes in the background of a long wavelength mode perturbation. In section 3 we provide the invariants under the symmetries of the galaxy and dark matter fluids and we discuss their implication for the non-local bias. We also check that the galaxy consistency relation holds at tree-and one-loop level in the bias model. In section 4 we show how to extend the galaxy consistency relations to redshift space where actual experiments are made. In section 5 we discuss the consequences of the symmetries for the theories of modified gravity and how such modifications are imprinted in the (n + 1)-point correlators in the squeezed limit. Finally, section 6 presents our conclusions.

Symmetries and consistency relation of galaxy correlation functions in real space
Galaxies (or more precisely, some population thereof), once formed, obey the following equations on sub-Hubble scales where we have denoted by x the comoving spatial coordinates, τ = dt/a the conformal time, a the scale factor in the FRW metric and H = d ln a/dτ is the conformal expansion rate. In addition, is the overdensity over the mean matter density density ρ, δ g ( x, τ ) and v g ( x, τ ) are the galaxy overdensity and peculiar velocity, and Φ( x, τ ) is the gravitational potential due to density fluctuations. Finally Ω m = 8πGρa 2 /3H 2 is the density parameter. Eq. (2.1) assumes number conservation [20]. Eventually, one would like to go beyond the treatment presented here in order to account for phenomena like formation and merging, which could be done for example by adding a source term to the right hand side of Eq. (2.1).
Dark matter is described by a similar set of non-relativistic fluid equations in the presence of gravity is invariant under the transformations (for a generic vector n(T )) where provided that one transforms the fields as follows This is true even if the we do not set v g ( x, τ ) = v( x, τ ), that is if we do not assume that the galaxy peculiar velocity is unbiased. Note that if one adds a source term to the right hand side of Eq. (2.1) to account for the change of the number density of galaxies in time, and such a source term depends only on quantities which transform as scalars, the equations of motion are still invariant under these transformations.
Consider the n-point correlation function of short modes of the density contrast. The symmetries of the Newtonian fluid equations imply, for instance, that (2.14) The points are supposed to be contained in a sphere of radius R much smaller than the long wavelength mode of size ∼ 1/q and centered at the origin of the coordinates. The non-relativistic equations of motion are invariant under the generic transformation τ → τ and x → x + n(T (τ )). This means that we can generate a long wavelength mode for the dark matter velocity perturbation v L (τ, 0) just by choosing properly the vector n(τ ) In other words, the correlator of the short wavelength modes in the background of the long wavelength mode perturbation should satisfy the relation [16] δ g (τ 1 , This is nothing else that the statement that the effect of a physical long wavelength galaxy velocity perturbation onto the short modes should be indistinguishable from the long wavelength mode velocity generated by the transformation with δx i = n i (τ ). In momentum space one therefore obtains The variation of the n-point correlator under the infinitesimal transformation is given by Then we find that In a ΛCDM model we have where D(τ ) is the linear growth factor and δ is the dark matter overdensity. We thus obtain the consistency relation where the primes indicate that one should remove the Dirac delta's coming from the momentum conservation. Notice that, if the correlators are computed all at equal times, the right-hand side of Eq. (2.21) vanishes by momentum conservation and the 1/q 2 infrared divergence will not appear when calculating invariant quantities. For the three-point correlator, we obtain Similarly, the dark matter correlators of the short wavelength modes in the background of the long wavelength mode perturbation should satisfy the relation leading to [16][17][18] δ( q, τ )δ( k 1 , τ 1 ) · · · δ( k n , τ n ) where P δ lin (q, τ ) = (D(τ )/D(τ in )) 2 P δ lin (q, τ in ) is the linear matter power spectrum. For the three-point correlator, we obtain (2.25) Once more, we stress that these relations are valid at beyond linear order for the short wavelength modes which might well be in the non-perturbative regime.
3 Consequences of the symmetries for the galaxy bias theory: non-local bias As the spatial gradients remain invariant, ∇ = ∇ ′ , one can easily realize that there are the following scalar quantities in the dark matter sector at our disposal is the growth factor as a function of the scale factor a), we have removed the trace part from ∂ i ∂ j Φ( x, τ ), which is nothing else than the dark matter overdensity δ( x, τ ), and t ij ( x, τ ) is vanishing at first-order in perturbation theory. Notice that these quantities are scalars beyond the linear perturbation theory as the symmetries identified in the previous section are valid at any order in perturbation theory. These symmetries are larger than the Galilean group identified in Ref. [21] for the large-scale dynamics. Furthermore, upon constructing the invariant one can construct two more scalar quantities but they are nothing else than the momentum conservation quantities for the dark matter and the galaxy, respectively. They identically vanish on-shell and therefore are trivial.
The set of invariants (3.2) are useful in constructing a galaxy bias theory which goes beyond the local bias model [22]. In the latter the galaxy overdensity δ g ( x, τ ) is written as a completely general function f [δ( x, τ )] of the mass density perturbation δ( x, τ ), and then the function is Taylor expanded, with the unknown coefficients in the series becoming the bias parameters This local expansion, even though it is consistent with the first invariant of the list (3.2), is expected to be valid only on very large scales and small times: as the symmetry dynamics allows the presence of more scalar quantities, there is no reason why they should not be generated along the subsequent evolution.
This logic is the same which applies in quantum field theory for operators: even though some of them are not present in the tree-level Lagrangian, they will appear at a certain order in perturbation theory unless they are forbidden by symmetry arguments. Therefore, assuming homogeneity and isotropy, one would expect a more general bias model of the form (where the coefficients should be intended to be the renormalized ones [23]) at quadratic order in the fields and the dots stand for the various other terms one can construct out of s ij ( x, τ ) and gradients. We see that an unavoidable consequence of the symmetries of the problem is that the bias model is a non-local bias model [23][24][25][26]; in fact the non-local expansion (3.6) has been first proposed in Ref. [23] where the same invariants have been employed based on general arguments on the homogeneous gravitational field and dark matter velocity. Some comments are in order: • The series does not contain a piece proportional to the gravitational potential Φ( x, τ ): it is simply forbidden by the symmetries of the problem as Φ( x, τ ) alone is not a scalar quantity.
• The non-local bias expansion (3.6) is not dictated solely by rotational invariance. Instead it is the more generic symmetry (2.7) together with isotropy which fixes the form of the expansion.
• The fluid equations during the matter-dominated period are also invariant under Lifshitz scalings of the form [16,27,28] for a generic Lifshitz weight z and Therefore, the Lifshitz weights of the bias coefficients should be (3.14) These Lifshitz weights fix the time-behaviour of the corresponding coefficients for the growing mode. The fact that the Lifshitz weights of b 1 and b 2 are vanishing tell us that their growing mode is constant in time. Indeed, it is well-known that at large times the system experiences the so-called debasing: b 1 converges to unity and b 2 goes to zero. Furthermore, the Lifshitz weights fix the corresponding time-behaviour of the remaining bias coefficients in their growing modes: c ∇ 2 , c s 2 , c s 2 ∇ 2 and c s 2 ∇ 4 should scale as τ 2/z , τ 4 , τ (4z+2)/z and τ (4z+4)/z , respectively. In particular, if one matches with the linear power spectrum of dark matter with spectral index n, one finds z = 4/(3 + n) ≃ 1 [21]. This explains why the non-local bias coefficients increase with time during the matter-dominated period. Furthermore, if one expresses the non-local invariant at second-order in terms of the product of the linear overdensities, one finds that the Lifshitz symmetry imposes that the overall time scaling is τ −2 in a matter-dominated universe (once one goes to momentum space). This is precisely the scaling found in Ref. [29] and leads to the so-called debasing, that is at late times the bias converges to unity and matter and galaxy density fields agree.
• As we already mentioned, galaxies form at a range of redshifts and merge. So it would be interesting to extend our results to the more realistic case when the number density of galaxies changes with redshift due to some arbitrary source including the effects of galaxy formation and merging.
However, if the effective source is a function of the scalar functions described above then our symmetry considerations will apply to this more complete galaxy description too. For instance, in Ref. [26] it was assumed that the effective source was of the form A(τ )j(ρ), where A(τ ) parametrizes the epoch of galaxy formation and j(ρ) the effects of dark matter on galaxy formation and merging.
In such a case the symmetry (2.7-2.13) holds.
• If the fluid equations are not invariant under the set of transformations (2.7-2.13), as it happens for example in some modified theories of gravity to be discussed below, one expects other terms to appear in the bias expansion as the bias is scale-dependent. The possibility of testing the Poisson equation with a scale-dependent bias was discussed in [31].

Consequences of the symmetries for the galaxy bias theory: independence from the smoothing scale
The galaxy consistency relation also holds for smoothed quantities as the smoothing operation commutes with the coordinate transformation (2.7). Indeed, suppose we perform a smoothing operation with a window function around a sphere of radius where W is the appropriate window function. Then we have the correlation functions on some scale r. This is because the smoothen scale R L is not physical, it is just a tool for the effective description and an arbitrary ultra-violet cute-off [30].
The symmetries at our disposal provide a simple and straightforward way to show that the galaxy correlation functions do not depend on the smoothing scale R L . Indeed, suppose we work in Fourier space and that we change the smoothing scale R L by an infinitesimal amount δR L . Correspondingly, the Fourier transformed window function will be where the prime stands for the differentiation with respect to the variable qR L . We can perform now an infinitesimal coordinate transformation x ′ = x + n(τ ). According to the relation (3.16), both tracers and dark matter overdensities will transform in momentum space as and therefore We see that if we choose the infinitesimal vector n(τ ) to be we can compensate the infinitesimal change of the smoothing radius R L and obtain that Since the correlators in the old and the new coordinate system have to be the same, we conclude that the dependence on the smoothing radius R L drops off. Physically, this is due to the fact that changing the large-scale smoothing radius by some amount amounts to include (or exclude) more momentum modes into the smoothed overdensity. This addition (or subtraction) of momentum modes can be compensated by going to a coordinate system where these long wavelength modes have been removed (or added). This argument holds in all epochs, included the Λ-dominated epoch. During the matter-dominated epoch we have another tool to reach the same conclusion: the Lifshitz symmetry. Indeed, The change in the smoothing scale R L can be compensated by a scaling transformation x ′ = λ x, or q ′ = q/λ. In such a case we have and again we conclude that the smoothing scale dependence drops off when correlators are considered.

Galaxy bispectrum consistency relation at tree-level
Since the bias model (3.6) respects the symmetries (2.7-2.13), the three-point function of galaxies computed in this model should satisfy the consistency relation. In the next two subsections we explicitly verify that this is the case in perturbation theory at the tree and one-loop levels. Let us start with the tree-level case. The equal time DM-galaxy cross-correlation at second order in perturbation theory is (3.24) while the unequal time power spectrum is The bispectrum of the galaxies at fourth order for unequal times is S ( k 1 , k 2 ) + b 2 (τ ) + c s 2 (τ )S( k 1 , k 2 ) + cyclic permutations of (τ, q), (τ 1 , k 1 ) and (τ 2 , k 2 ), (3.26) where (3.27) In the squeezed limit q → 0, k 1 ≃ − k 2 we find We observe that the consistency relation is trivially satisfied at linear order. One should note that nonlocal terms are sub-leading. We shall therefore ignore them in the one-loop computation and consider only the local-bias model in the following.

Galaxy bispectrum consistency relation at one-loop
The check the consistency relation at one-loop, or more precisely at order 6 in perturbation theory, we have to evaluate the following expression We first consider the right-hand side where one should be careful when expanding the square parenthesis.
Indeed, even when δ is in the linear regime, δ g might be non-linear and higher order corrections to δ g have to be taken into account. The square parenthesis at order 4 in perturbation theory is therefore g ( q, τ ) is the third order contribution to δ g ( q, τ ). The first term on the right-hand side can be written using the bias model as where while the second term is where we defined the linear variance Let us now compute the left-hand side of Eq. (3.30) with the help the expressions one can find in Ref. [34] and check that the equality is satisfied. The unequal-time bispectrum δ g ( q, τ )δ g ( k 1 , τ 1 )δ g ( k 2 , τ 2 ) ′ is composed by several terms which, for compactness, we will denote analogously to what done in Ref. [34] by the notation In the following, we compute each term identifying the ones which behave at least O(q −1 P δ lin (q)) as q → 0.
• The first term is where we used the consistency relation for matter. This is exactly the term proportional to P 11 g in Eq. (3.32) in the right-hand side of the consistency relation.
• We express the trispectrum in the integral of the following term using the consistency relation This is equal to the term proportional to P 12 g in Eq. (3.33). In the second line we ignored the permutation containing a bispectrum not in the squeezed limit.
• The following contribution reproduces the term proportional to P 22 g in Eq. (3.34).
In the second equality we kept the only permutation enhanced in the squeezed limit and in the third we used the consistency relation for matter. Finally, we used the fact that which can be deduced simply by doing the shift p → k 1 − p.
• The term below is enhanced in the squeezed limit as it contains a bispectrum at unequal times. It reproduces the term proportional to P 13 g in Eq. (3.35) together with the term in Eq. (3.36) • The term is not dominant because the O(δ (4) ) corrections to P (q) are at most O(P δ lin (q)) when q → 0.
• The following terms are not relevant because they involve either terms that are proportional to the non-squeezed bispectrum, which makes them at most O(P δ lin (q)), or terms containing the bispectrum in the squeezed limit at equal times, which vanish due to the consistency relation. B denotes the bispectrum of matter • The following terms are not enhanced in the squeezed limit as they are just products of linear power spectra at this order • Finally, the two-and three-loops corrections ignored in Ref. [34] are at most constant in the squeezed limit such that our result is fully correct at sixth order. Overall, we conclude that the galaxy consistency relation is satisfied at tree-and one-loop level.

Consistency relation of galaxy correlation functions in redshift space
Let us discuss now discuss how the galaxy consistent relations are modified when going from real space to redshift-space where experiments are performed. The mapping from real-space position x to redshiftspace s is given by [32] 1) and the density field in redshift-space is obtained by imposing mass conservation In Fourier space the condition (4.2) reads By performing a spatial coordinate transformation x → x ′ = x + n(τ ) we know that, if δ g ( x, τ ) and v( x, τ ) satisfy the fluid equations, then δ ′ g ( x, τ ) = δ g ( x ′ , τ ) and v ′ ( x, τ ) = v( x ′ , τ ) −˙ n do as well. This implies that for the new solution we have This expression is exact. Expanding for small n(τ ), we get If we start from this expression, upon integrating by parts we find This gives At this point we can use the distant observer approximation, that is take the direction of the vector x fixed, since it varies little from galaxy to galaxy: galaxies are relatively close to each other on the plane orthogonal to the line-of-sight. This amounts to taking ∇x ≃ 0 and we finally obtain Note that here the first line corresponds to the field transformation that gives rise to the consistency relation, which in redshift space will contain new terms induced by the second line of this expression.
Using the explicit expression for˙ n˙ we obtain, where µ k is the cosine between the vectork andx, and we used the distant observer approximation to take the cosines out of the integral in the second equality. We therefore obtain that in redshift space the consistency relations reads δ g,s ( q, τ )δ g,s ( k 1 , τ 1 ) · · · δ g,s ( k n , τ n ) ′ q→0 δ g,s ( q, τ 1 ) · · · δ g,s ( k n , τ n ) where we have used the linear relation [28] δ g,s ( q, τ ) = b 1 (τ ) + f (τ )µ 2 q δ( q, τ ). (4.11) In particular, the consistency relation for the bispectrum in redshift space explicitly reads δ g,s ( q, τ )δ g,s ( k 1 , τ 1 ) · · · δ g,s ( k 2 , τ 2 ) 5 Consequences of the symmetries for the modified theories of gravity Theories that (attempt to) explain the observed cosmic acceleration by modifying general relativity all introduce a new scalar degree of freedom that is active on large scales, but is screened on small scales to match experiments. All these theories introduce an extra light scalar field to modified gravity in the infrared. Typical examples are represented by the f (R) theories [33], which are equivalent to classic scalar-tensor theories [35] and the screening effect takes place through the so-called chameleon mechanism [36], and by Galileon theories [37] where the extra degree of freedom is appropriately dressed through higher-derivative interactions which decouple it form short-scale physics in accordance with solar system tests.
It has been recently realized that in the modified gravity models where there is an efficient screening phenomenon to make the set-up experimentally consistent there might also be order unity violation of the EP [38]. The galaxy and dark matter consistency relations are based on a coordinate transformation 1 (in a matter-dominated period) [16,18] τ ′ = τ, we are basically removing the time-dependent, but homogeneous gravitational force via a change of coordinates. This corresponds to an homogeneous acceleration transformation which allows to go to a free-falling observer, precisely the essence of the EP. Therefore, one expects a violation (or a spatial dependence) of the galaxy consistency relation in modified gravity models where the screening mechanism is in action.
Let us therefore consider modifications of gravity that violate EP. As we said, in these models there exist extra light scalar fields which effectively screen the scalar charge of objects as compared to unscreened objects of the same mass. In other words, different objects of the same mass may have different scalar charge and they can move differently in the same environment violating the EP. The chameleon for example, has a potential such that it has long range forces outside of objects while it is massive in their interior. Therefore, the existence of such field is consistent with solar system and fifth force tests but still can modify gravity at large distances. At any rate, we will assume here that there is such a screening mechanism and, irrespectively of its origin, that it violates the EP. The latter may be implemented by modifying the energy-momentum conservation as For the non-relativistic dark matter fluid one therefore finds where we assumed that with ϕ( x, τ ) the scalar field that has environmental couplings that causes violation of the EP and q is the scalar charge density of the fluid. We follow the parametrization introduced in Ref. [38] and we assume q = ǫαρ, where α is a constant and ǫ is a parameter that describes the degree of screening (ǫ =0 for screened objects and ǫ = 1 for unscreened ones). We should supplement the above equations with the Poisson equation for the gravitational potential Φ( x, τ ) and a corresponding equation for ϕ( x, τ ) which we write as where V (ϕ) is the scalar potential of the chameleon-like field. We may now consider perturbations ρ( x, τ ) = ρ(1 + δ( x, τ )) and ϕ( x, τ ) = (ϕ + δϕ( x, τ )) around the corresponding background values ρ and ϕ and we find that these perturbations satisfy the equations where is the mass of the scalar field. Restricting ourselves to the matter-dominated case, it can be checked that the Eqs. (5.8-5.11) are invariant under the transformations As a result, it is still possible to to remove a long wavelength mode for the velocity perturbation v L (τ, 0) by properly choosing the vector n(τ ) in order. Indeed, in the linear regime in momentum space the dynamical equations are given by q 2 δϕ( q, τ ) = −(m 2 δϕ)( q, τ )a 2 − 3αH 2 Ω m δ L ( q, τ ), (5.21) where (m 2 δϕ)( q, τ ) is the Fourier mode of m 2 ( x, τ )δϕ( x, τ ). In particular consider the configurations shown in figure 1, in a region outside the spherical over-density of radius R 0 where the chameleon-like field is not screened and its mass may be neglected, one has where the equation for the linear matter overdensity satisfies the equation 2 is the growth function for (5.23). On the contrary, in a screened region, where the the field δϕ is massive enough so that ϕ is not excited, but fixed to some constant background value within a sphere of radius R 0 . In such a case the equation for the overdensity is given byδ and it is solved by δ < L (τ ) = D < (τ )/D < (τ in )δ L (τ in ). Therefore v L will be different in the two regions r < ∼ R 0 and r > ∼ R 0 . As a result, two different vectors n's will be needed to generate (or remove) the long wave velocity perturbation, one for r < ∼ R 0 and the other r > ∼ R 0 : in the presence of modified gravity exploiting the screening effect, it is not possible to find a spatially independent vector n(τ ) and the consistency relations must be violated for objects which are unscreened. The vector n(τ ) is chosen such as to have a free-falling frame, defined bÿ n + H˙ n + ∇Φ = 0. (5.25) The solution to this equation iṡ Then, by using Eqs. (5.23) and (5.24), we find that the free-falling frame is specified by where we have indicated by δ > L and δ < L the dark matter overdensities in the two corresponding regions. Consider for example n-galaxies within a sphere of radius R > ∼ R 0 much smaller than the long wavelength mode of size ∼ 1/q and centered at the origin of the coordinates. Then, if all points are at distances r < ∼ R 0 , then the consistency relation for the n-point correlator is the one we already described If instead all points are at r > ∼ R 0 , we will have in this case The case in which galaxies are both screened and unscreened is more complex 3 , however we expect a violation of the consistency relation due to the difference in the growth factor. Indeed, consider those configurations in which m-galaxies are at r > ∼ R 0 and (n − m) are at r < ∼ R 0 , the consistency relation will be written as Notice that the right-hand side for the configuration (5.31) is not vanishing even for correlators at equal time for the n-points. This is due to the fact that the long wavelength chameleon-like field correlates only with the overdensity located in the unscreened region, the one in the screened region being completely independent from the chameleon-like perturbation. For instance, for n = 2 and m = 1, the corresponding bispectrum reads where δ L = (δ < L + δ > L )/2 and ∆δ L = (δ < L − δ > L ). The latter is also suppressed by α 2 ǫ which therefore gives an estimate of the violation of EP. Consider, for instance, a cluster of galaxies of mass M ∼ 10 14.5÷15 M ⊙ and radius R 0 ∼ (2 ÷ 10) Mpc. Inside it Φ cl = −GM/R 0 ∼ −10 −5 and one has [38] ϕ 2α where ϕ is the asymptotic background value of the scalar ϕ and the upper bound comes from the solar system [39]. In such a dense object the scalar field is screened, ǫ ≃ −ϕ/(2αΦ cl ) < ∼ 10 −1 , and we may take (n − m) galaxies residing there. Away from the cluster there might be small m galaxies with Φ g ∼ −10 −8 which are unscreened (therefore preferably residing in voids) and ǫ ≃ 1. For this configuration, one expects to see a violation of the consistency relation as predicted by Eq. (5.32). Notice also that our considerations hold as long as the Compton wavelength m −1 associated to the chamaleon-like field is larger than the scale where perturbations may be considered in the linear regime. At redshift z = 0, there is a strong upper bound of about 1 Mpc on such Compton wavelength m −1 (a 0 ) coming from the solar system tests [40,41], implying that the desired effects on the large scale structure are restricted to nonlinear scales. However, at higher redshifts a Compton wavelength of the form m −1 (a) = m −1 (a 0 )(a/a 0 ) p with p < −3 satisfies the experimental constraints and can lead to a modified gravity regime on large linear scales [40]. This scaling of m is faster than the one deduced from the Lifshitz scaling of the scale k NL (a) ∼ a −2/(n+4) (during matter-domination) at which cosmological perturbations become nonlinear [16,42] and the condition m(a) < k NL (a) is easily attained going back in time. Notice also that the bound describes in Ref. [40,41] does not hold in theories which screen by the Vainshtein mechanism, like the Galileon model, in which the scalar non-linearities result from derivative interactions because the screening condition only holds up to the Vainshtein radius [41]. In this class of theories though the violation of the EP for extended objects is tiny [38]. So, an interesting question is how well one can measure a violation of the EP through the galaxy consistency relation. Though an accurate estimate is beyond the scope of this paper, let us try to make a simple back-on-the-envelope computation by noting that the form of the bispectrum (5.32) is almost the same one one obtains in the galaxy local bias model in the presence of a primordial local non-Gaussianity [43] (see also Ref. [17]). Supposing that the combination α 2 ǫ is smaller than unity, one needs basically to identify (barring coefficient of order unity and assuming redshift z = 0) α 2 ǫ( q · k 1 ) with f NL H 2 0 , where f NL is the non-linear coefficient parametrizing the level of non-Gaussianity and H 0 is the present Hubble rate. The Fisher matrix analysis applied to the galaxy (reduced) bispectrum performed in Ref. [43] has shown that one can measure f NL up to O(10) for k 1 ∼ k max ∼ 0.1 h Mpc −1 , being k max the smallest scale scale included in the analysis. Therefore, again very roughly, we expect to be able to measure deviation from the EP at redshifts z > ∼ 1 of the order of O(10)(H 0 / k max ) ∼ 10 −3 (0.1 h Mpc −1 /k max ), where we have taken q ∼ 10 2 H 0 .
Similar considerations apply also to more conventional modifications of gravity induced by scalars, like Brans-Dicke theory, or dilaton gravity. In these theories, in spite of the fact that there is a universal coupling of the scalar to matter, there is a violation of the EP because different objects of the same mass may have different gravitational binding energies. However, this violation is subleading in the post-Newtonian approximation for non-relativistic matter and it can only give order one effects in strongly bound systems as binary systems and black holes [44]. To be more precise here, let us consider an action of the general form where f (R, φ, X) is a function of the Ricci scale R, a scalar φ and its kinetic term and X = − 1 2 ∂ µ φ∂ µ φ. This form of the action describes many models of modify gravity like Brans-Dicke theory, dilaton gravity, f (R) and many others. In this general class of models, the non-relativistic matter still satisfies Eqs.(2.4-2.6), where now Ω m = 8πG effρ a 2 /3H 2 and G eff is an effective Newton constant which encodes the modification of gravity given by [45] G eff (τ ) = 1 8πF Therefore, when the effective Newton constant is only time dependent and it just modifies the temporal dependence of the local growth function of the overdensity evolution. In this case, still, one may generate a long wavelength velocity mode by a vector n(τ ) as in Eq. (2.15). In the opposite case, Eq. (5.36) is not satisfied and G eff turns out to be space-dependent. The overdensity δ turns out to be also space-dependent as well and there may be no n(τ ) to generate a long wavelength velocity mode within the sphere of radius R 0 . To see when this is possible, let us mention that there is a crossover scale when the k-dependance of G eff starts become strong and which is defined by If R < ∼ R 0 , one may still define n(τ ) and so long wavelength modes may be generated. On the other side if R > ∼ R 0 , i.e. modification of gravity appears within the sphere, then there is no globally defined n(τ ) inside the sphere of radius R, which will cause a modification of the consistency relation. So the lesson here is that violation of the consistency relations is a signal of the spatial dependence of the effective Newton constant G eff and of a modification of gravity at large scales.
We should also note that we have not considered here intrinsic violation of the EP, i.e. at the microscopic level [46][47][48][49]. One for example may consider the case of extra scalar, vector or tensor couplings to only one component, say baryonic matter or dark matter. Such possibility has been considered recently in Ref. [50] where it has been pointed out the interesting feature that if a large scale velocity bias exists between the different components new terms appear in the consistency relations with respect to the single species case.

Conclusions
In this paper we have discussed the implications of the symmetry enjoyed by the Newtonian equations of motion describing the dark matter and galaxy fluids coupled through gravity. The fact that such symmetry applies to both galaxies and dark matter is particularly welcome because one can reach conclusions which are independent from the galaxy bias. On the contrary, one can use the power of the symmetry to deduce relevant informations on the theory of galaxy bias. In particular, we have shown that an unavoidable consequence of the symmetries at our disposal is that the bias is expected to be non-local. Furthermore, we have studied the modification (or violation) of the consistence relation in the case in which gravity is modified because of the presence of extra degrees of freedom propagating unscreened at large cosmological distances. Let us reiterate that our results are based on the assumption that the galaxy number is conserved. Eventually, one would like to extend our considerations by accounting for phenomena like halo formation and merging, nevertheless if the modification in the proper equations are such that the symmetries studied in this paper are preserved, e.g. if the new terms are a local function of the dark matter density, then our considerations remain valid. Also, apart from applying to non-linear scales and directly to galaxies, our results have the virtue of not being sensitive to the single stream approximation and to be valid also in the presence of velocity bias and/or vorticity (which is generated at higher-order in perturbation theory). Therefore, assuming that primordial perturbations satisfy the consistency relations of [8], the observation of a deviation from the consistency relation for the bispectrum of galaxies, Eq. (2.21), would signal either the inapplicability of the Eulerian bias model even including "non-local" terms as in Eq. (3.6) or the violation of the EP in the underlying theory of gravity.