hep-th/0410119

MZ-TH/04-15

Quantum Gravity

[2.35mm] at Astrophysical Distances?

M. Reuter and H. Weyer

Institute of Physics, University of Mainz

Staudingerweg 7, D–55099 Mainz, Germany

Assuming that Quantum Einstein Gravity (QEG) is the correct theory of gravity on all length scales we use analytical results from nonperturbative renormalization group (RG) equations as well as experimental input in order to characterize the special RG trajectory of QEG which is realized in Nature and to determine its parameters. On this trajectory, we identify a regime of scales where gravitational physics is well described by classical General Relativity. Strong renormalization effects occur at both larger and smaller momentum scales. The latter lead to a growth of Newton’s constant at large distances. We argue that this effect becomes visible at the scale of galaxies and could provide a solution to the astrophysical missing mass problem which does not require any dark matter. We show that an extremely weak power law running of Newton’s constant leads to flat galaxy rotation curves similar to those observed in Nature. Furthermore, a possible resolution of the cosmological constant problem is proposed by noting that all RG trajectories admitting a long classical regime automatically give rise to a small cosmological constant.

## 1 Introduction

During the past few years, in the light of a series of investigations [1, 2, 3, 4, 5, 6, 7, 8, 9], it appeared increasingly likely that Quantum Einstein Gravity (QEG), the quantum field theory of gravity whose underlying degrees of freedom are those of the spacetime metric, can be defined nonperturbatively as a fundamental, “asymptotically safe” [10] theory. By definition, its bare action is given by a non–Gaussian renormalization group (RG) fixed point. In the framework of the effective average action [11, 12, 13] a suitable fixed point is known to exist in the Einstein–Hilbert truncation of theory space [1, 6, 3] and a higher–derivative generalization [5] thereof. Detailed analyses of the reliability of this approximation [3, 4, 5] and a conceptually independent investigation [14] suggest that the fixed point should indeed exist in the exact theory, implying its nonperturbative renormalizability.

The general picture regarding the RG behavior of QEG as it has emerged so far points towards a certain analogy between QEG and non–Abelian Yang–Mills theories, Quantum Chromo–Dynamics (QCD) say. For example, like the Yang–Mills coupling constant, the running Newton constant is an asymptotically free coupling, it vanishes in the ultraviolet (UV), i. e. when the typical momentum scale becomes large. In QCD the realm of asymptotic freedom, probed in deep inelastic scattering processes, for instance, is realized for momenta larger than the mass scale which is induced dynamically by dimensional transmutation. In QEG the analogous role is played by the Planck mass . It delimits the asymptotic scaling region towards the infrared (IR). For the RG flow is well described by its linearization about the non–Gaussian fixed point [4]. Both in QCD and QEG simple local truncations of the running Wilsonian action (effective average action) are sufficient above and , respectively. However, as the scale approaches or from above, many complicated, typically nonlocal terms are generated in the effective action [15, 8, 16]. In fact, in the IR, strong renormalization effects are to be expected because gauge (diffeomorphism) invariance leads to a massless excitation, the gluon (graviton), implying potential IR divergences which the RG flow must cure in a dynamical way. Because of the enormous algebraic complexity of the corresponding flow equations it is extremely difficult to explore the RG flow of QCD or QEG in the IR, far below the UV scaling regime, by purely analytical methods. In QCD lattice techniques can be used to study the IR sector, but despite recent progress on dynamical triangulations [17, 18] there exists no comparable tool for gravity yet.

In QCD we have another source of information about its small momentum or large distance regime. If we take it for granted that QCD is the correct theory we can exploit the available experimental data on the strong interaction, interpret them within this theory, and thus obtain information about the quantum dynamics of QCD, in particular its nonperturbative IR sector, from the purely phenomenological side. An example to which we shall come back in a moment are the non–relativistic quark–antiquark potentials extracted from quarkonium data (and confirmed on the lattice). They suggest that nonperturbative IR effects modify the classical Coulomb term by adding a confinement potential to it which increases (linearly) with distance:

(1) |

Here and the string tension are constants [19].

In this paper we are going to apply a similar “phenomenological” strategy to gravity. Under the assumption that QEG is the correct theory of gravity on all distance scales, we try to describe and characterize the distinguished RG trajectory which is realized in Nature as completely as possible. We use both observational input and the available analytical RG studies.

We shall start from the flow equations in the Einstein–Hilbert
truncation, determine which type of its RG trajectories the one
realized in Nature belongs to, identify a regime on it where standard
General Relativity is valid, and finally argue that this regime does
not extend to arbitrarily large distances. In fact, for smaller
than the momenta typical of the regime of standard gravity, the
truncation predicts a strong increase of with decreasing ,
which, at a certain critical value of , becomes infinite even. The
diverging behavior is clearly an artifact of an insufficient
truncation, but we shall see that the growth of Newton’s constant with
the distance can be understood on general grounds as due to a
potential IR singularity. It is the main hypothesis of the present
paper that a “tamed” form of this nonperturbative IR growth of
is a genuine feature of exact QEG.^{1}^{1}1Using different
methods or models, IR quantum gravity effects have also been studied
in refs. [20, 21, 22, 23].

The problem of the missing mass or “dark matter” is one of the most puzzling mysteries of modern astrophysics and cosmology [24]. It has been known for a long time that the luminous matter contained in a galaxy, for instance, does not provide enough mass to explain the gravitational pull the galaxy exerts on “test masses” in its vicinity. Typically their rotation curves , the orbital velocity as a function of the distance, are almost flat at large distances rather than fall off according to Kepler’s law [25]. Similar, but even stronger mass discrepancies are observed on all larger distance scales and in particular in cosmology. The recent high–redshift supernova and CMBR data show very impressively that the known forms of baryonic matter account only for a small percentage of the matter in the Universe. A possible way out is the assumption that the missing mass is due to some sort of “dark matter” which would manifest itself only by its gravitational effects. However, as to yet it has not been possible to convincingly identify any dark matter candidate, and so it might be worthwhile to think about alternatives.

It is a very intriguing idea that the apparent mass discrepancy is not due to an unknown form of matter we have not discovered yet but rather indicates that we are using the wrong theory of gravity, Newton’s law in the non–relativistic and General Relativity in the relativistic case. In fact, Milgrom [26] has developed a phenomenologically very successful non–relativistic theory, called MOdified Newtonian Dynamics or “MOND”, which explains many properties of galaxies, in particular their rotation curves, in a unified way without invoking any dark matter. In its version where gravity (rather than inertia) is modified, a point mass produces the potential

(2) |

where and are constants. The second term on the RHS of (2) is responsible for the flat, non–Keplerian rotation curves at large distances. So far no wholly satisfactory relativistic extension of MOND is known.

Also the relativistic theory proposed by Mannheim [27] where the Lagrangian is the square of the Weyl tensor tries to explain the rotation curves as due to a non–Newtonian force. The corresponding potential is of the form

(3) |

The resulting rotation curves do not become flat but still seem to be in accord with the observations.

For a detailed discussion of other attempts at modifying gravity at astrophysical distances and a comprehensive list of references we refer to [28]. A possible relation to quintessence has been speculated about in [29].

In the present paper we are going to explore the idea that the IR
quantum effects of QEG, in particular the growth of at large
distances, induces a modified Newtonian potential similar to (2)
or (3). If so, one can perhaps solve the missing mass problem in
a very elegant and ‘‘minimal’’ manner by simply quantizing the fields
which are known to exist anyhow, without having to introduce ‘‘dark
matter’’ on an ad hoc basis.^{2}^{2}2See refs.
[22, 23] for a related analysis within a
perturbatively renormalizable higher derivative gravity.

It is particularly intriguing that the potentials we would like to derive within QEG are strikingly similar to the nonperturbative quark–antiquark potentials generated by (quenched) QCD. In particular eqs. (1) and (3) are mathematically identical, describing “linear confinement”, while the MOND potential increases slightly more slowly at large distances. In view of the many similarities between QCD and QEG it is hard to believe that this should be a mere coincidence.

The purpose of the present paper is to learn as much as possible about the gravitational RG trajectory Nature has chosen and to investigate the possibility that the IR renormalization effects of QEG are “at work” at galactic and cosmological scales.

The remaining sections of the paper are organized as follows. In Section 2 we discuss the Einstein–Hilbert truncation of theory space with an emphasis on the strong IR renormalization effects it gives rise to. In Sections 3 and 4, we analyze the RG trajectory realized in Nature, first using mostly analytical results (Section 3) and then also phenomenological input (Section 4). In Section 5 we employ a plausible model of the trajectory in the deep IR to demonstrate that an extremely tiny variation of Newton’s constant would explain the observed flat rotation curves. The results are summarized in Section 6.

##
2 Towards the infrared with the

Einstein–Hilbert truncation

### 2.1 Structure of the RG flow

In this subsection we discuss some properties of the Einstein–Hilbert truncation, in particular the classification of its RG trajectories. The emphasis will be on their behavior in the IR. We refer to [1] and [4] for further details.

Our basic tool is the effective average action , a free energy functional which depends on the metric and a momentum scale with the interpretation of a variable infrared cutoff. The action is similar to the ordinary effective action which it approaches for . The main difference is that the path integral defining extends only over quantum fluctuations with covariant momenta . The modes with are given a momentum dependent and are suppressed therefore. As a result, describes the dynamics of metrics averaged over spacetime volumes of linear dimension . The functional gives rise to an effective field theory valid near the scale . Hence, when evaluated at tree level, correctly describes all quantum gravitational phenomena, including all loop effects, provided the typical momentum scales involved are all of order .

Considered a function of , describes a RG trajectory in the space of all action functionals. The trajectory can be obtained by solving an exact functional RG equation. In practice one has to resort to approximations. Nonperturbative approximate solutions can be obtained by truncating the space of action functionals, i. e. by projecting the RG flow onto a finite–dimensional subspace which encapsulates the essential physics.

The “Einstein–Hilbert truncation”, for instance, approximates by a linear combination of the monomials . Their prefactors contain the running Newton constant and the running cosmological constant . Their –dependence is governed by a system of two coupled ordinary differential equations. and

The flow equations resulting from the Einstein–Hilbert truncation are most conveniently written down in terms of the dimensionless “couplings” and where is the dimensionality of spacetime. Parameterizing the RG trajectories by the “RG time” the coupled system of differential equations for and reads , , where the –functions are given by

(4) |

Here , the anomalous dimension of the operator , has the representation

(5) |

The functions and are defined by

(6) |

The above expressions contain the “threshold functions” and . They are given by

(7) |

and a similar formula for without the –term. In fact, is a dimensionless version of the cutoff function , i. e. . Eq. (7) shows that becomes singular for . (For all admissible cutoffs, assumes its minimum value at and increases monotonically for .) If , the ’s in the –functions are evaluated at negative arguments . As a result, the –functions diverge for and the RG equations define a flow on a half–plane only: , .

This point becomes particularly clear if one uses a sharp cutoff [4]. Then the ’s either display a pole at ,

(8) |

or, in the special case , they have a logarithmic singularity at :

(9) |

The constants parameterize the
residual cutoff scheme dependence which is still present after having
opted for a sharp cutoff. In numerical calculations we shall take them
equal to the corresponding --value of a smooth
exponential cutoff^{3}^{3}3For this purpose we employ the exponential
cutoff with “shape parameter” . In , the only
’s we need are and . See ref. [4] for a detailed discussion
of the sharp cutoff., but their precise value has no influence on
the qualitative features of the RG flow [4]. The
corresponding ’s are constant for the sharp cutoff:
.

From now on we continue the discussion in dimensions. Then, with the sharp cutoff, the coupled RG equations assume the following form:

(10a) | ||||

(10b) | ||||

(10c) |

The RG flow is dominated by two fixed points : a Gaussian fixed point (GFP) at , and a non–Gaussian fixed point (NGFP) with and . There are three classes of trajectories emanating from the NGFP: trajectories of Type Ia and IIIa run towards negative and positive cosmological constants, respectively, and the single trajectory of Type IIa (“separatrix”) hits the GFP for . The short–distance properties of QEG are governed by the NGFP; for , in Fig. 1 all RG trajectories on the half–plane run into this point. The conjectured nonperturbative renormalizability of QEG is due to the NGFP: if it is present in the full RG equations, it can be used to construct a microscopic quantum theory of gravity by taking the limit of infinite UV cutoff along one of the trajectories running into the NGFP, thus being sure that the theory does not develop uncontrolled singularities at high energies [10]. By definition, QEG is the theory whose bare action equals the fixed point action .

The trajectories of Type IIIa have an important property which is not resolved in Fig. 1. Within the Einstein–Hilbert approximation they cannot be continued all the way down to the infrared () but rather terminate at a finite scale . At this scale they hit the singular boundary where the –functions diverge. As a result, the flow equations cannot be integrated beyond this point. The value of depends on the trajectory considered.

In ref. [4] the behavior of and close to the boundary was studied in detail. The aspect which is most interesting for the present discussion is the following. As the trajectory gets close to the boundary, approaches from below. In this domain the anomalous dimension (10c) is dominated by its pole term:

(11) |

Obviously for , and eventually at the boundary. This behavior has a dramatic consequence for the (dimensionful) Newton constant. Since , the large and negative anomalous dimension causes to grow very strongly when approaches from above. This behavior is sketched schematically in Fig. 2.

At moderately large scales , well below the NGFP regime, is approximately constant. As is lowered towards , starts growing because of the pole in , and finally, at , it develops a vertical tangent, . The cosmological constant is finite at the termination point: .

By fine–tuning the parameters of the trajectory the scale can be made as small as we like.

Since it happens only very close to , the divergence at is not visible on the scale of Fig. 1. (Note also that and are related by a decreasing factor of .)

The phenomenon of trajectories which terminate at a finite scale is not special to gravity, it occurs also in truncated flow equations of theories which are understood much better. Typically it is a symptom which indicates that the truncation used becomes insufficient at small . In QCD, for instance, thanks to asymptotic freedom, simple local truncations are sufficient in the UV, but a reliable description in the IR requires many complicated (nonlocal) terms in the truncation ansatz. Thus the conclusion is that for trajectories of Type IIIa the Einstein–Hilbert truncation is reliable only well above . It is to be expected, though, that in an improved truncation those trajectories can be continued to .

We believe that while the Type IIIa trajectories of the Einstein–Hilbert truncation become unreliable very close to , their prediction of a growing for decreasing in the IR is actually correct. The function obtained from the differential equations (10) should be reliable, at least at a qualitative level, as long as . For special trajectories the IR growth of sets in at extremely small scales only. Later on we shall argue on the basis of a gravitational “RG improvement” [30, 31, 32, 33, 34, 35] that this IR growth is responsible for the non–Keplerian rotation curves observed in spiral galaxies.

The other trajectories with , the Types Ia and IIa, do not terminate at a finite scale. The analysis of ref. [4] suggests that they are reliably described by the Einstein–Hilbert truncation all the way down to .

### 2.2 What drives the IR renormalization?

Next we discuss a simple physical argument which sheds light on the dynamical origin of the expected strong IR effects. As we shall see, they are due to an “instability driven renormalization”, a phenomenon well known from many other physical systems [16, 36], spontaneous symmetry breaking being the prime example.

For an arbitrary set of fields , and in a slightly symbolic
notation^{4}^{4}4We ignore possible complications due to gauge
invariance. They are inessential for the present discussion., the
exact RG equation for the effective average action reads
[11, 12]

(12) |

It contains the fully dressed effective propagator where denotes the Hessian of and the cutoff operator. It is instructive to rewrite (12) in the form

(13) |

where the derivative acts on the –dependence of only. The RHS of (13) represents a “–functional” which summarizes all the infinitely many ordinary –functions in a compact way. Obviously its essential ingredient is a kind of a “one–loop determinant”. It differs from that of a standard one–loop calculation by the presence of the cutoff term and by the use of the dressed inverse propagator rather than the classical . More importantly, in the standard situation one expands the classical action about its minimum , in which case the one–loop effective action sums up the zero–point energies of the small stable oscillations about . On the RHS of (13), instead, is a prescribed external field, the argument of on the LHS. It can be changed freely, eq. (13) holds for all , so that is not in general a stationary point of .

Thus we may conclude that the basic physical mechanism which drives the RG flow is that of quantum fluctuations on arbitrary off–shell backgrounds . They determine the –functional, and depending on how “violent” those fluctuations are, the RG running is weaker or stronger.

It is helpful to consider two extreme cases. Let us first assume that,
for a certain fixed , the operator is
positive definite; then is positive,
too,^{5}^{5}5In order to capture the essence of the argument it is
sufficient to use a mass–type cutoff [12] for which
and for all
. and the quadratic action governing the fluctuations
about given by

If, on the other hand, has one or several negative eigenvalues with , then is positive only in presence of the IR regulator, for . Without the IR regulator there is a real physical instability. Within the linear approximation the fluctuation modes grow unboundedly; beyond the linear approximation they would try to “condense” in order to turn into a stable ground state. Following a well behaved RG trajectory one stays in the regime where is positive. However, when approaches from above, the lowest eigenvalue of gets very close to zero, and the RG flow is strongly affected by the presence of the nearby singularity. The effective propagator becomes very large, and typically this leads to an enormous growth of the (standard) –functions when . They give rise to comparatively strong “instability induced” renormalizations. We shall see in a moment that the IR effects of QEG are precisely of this type.

In order to find the RG flow on the full theory space the above stability analysis and the “summation of zero–point energies” has to be performed for infinitely many different backgrounds ; they are needed in order to “project out” all the possible field monomials which constitute the functional . On a truncated theory space just a few ’s might be sufficient.

In order to illustrate the relationship between the (ordinary) –functions and the instability presented by let us look at a scalar model (on flat spacetime) in a simple truncation:

(14) |

Here denotes a real, –symmetric scalar field, and the truncation ansatz (14) retains only a running mass and –coupling. In a momentum basis where we have

(15) |

Always assuming that , we see that is positive if ; but when it can become negative for small enough. Of course, the negative eigenvalue for , for example, indicates that the fluctuations want to grow, to “condense”, and thus to shift the field from the “false vacuum” to the true one. By the mechanism discussed above, this gives rise to strong instability induced renormalizations. In fact, the standard –functions for and can be found by inserting (15) into (12), taking two and four derivatives with respect to , respectively, and then setting in order to project out and . As a result, the –functions are given by –integrals over (powers of) the propagator

(16) |

In the symmetric phase () this (euclidean!) propagator has no pole, and the resulting –functions are relatively small. In the broken phase (), however, there is a pole at provided is small enough: . For the –functions become large and there are strong instability induced renormalizations.

In a reliable truncation, a physically realistic RG trajectory in the spontaneously broken regime will not hit the singularity at , but rather make run in precisely such a way that is always smaller than . This requires that

(17) |

This strong instability induced mass renormalization is necessary in order to evolve an originally –shaped symmetry breaking classical potential into an effective potential which is convex and has a flat bottom. (See [12] for a detailed discussion of this point.)

Unfortunately the two–parameter truncation (14) is too rudimentary for a reliable description of the broken phase. Its RG trajectories actually do run into the singularity. They terminate at a finite scale with at which the –functions diverge. Instead, if one allows for an arbitrary running potential , containing infinitely many couplings, all trajectories can be continued to , and for one finds indeed the quadratic mass renormalization (17) [12].

Let us return to gravity now where corresponds to the metric.
In the Einstein–Hilbert truncation it suffices to insert the metric
corresponding to a sphere of arbitrary radius
into the flow equation^{6}^{6}6We stress, however, that the
–functions do not depend on the choice of
background used for projecting onto the various invariants
[12, 1, 3]. in order to disentangle the
contributions from the two invariants and . Thus we may think of the Einstein–Hilbert flow
as being a manifestation of the dynamics of graviton fluctuations on
. This family of backgrounds, labeled by , is
“off–shell” in the sense that is completely arbitrary and not
fixed by Einstein’s equation in terms of .

It is convenient to decompose the fluctuation on the sphere into irreducible (TT, TL, ) components [3] and to expand the irreducible pieces in terms of the corresponding spherical harmonics. For in the transverse–traceless (TT) sector, say, the operator equals, up to a positive constant,

(18) |

with the covariant Laplacian acting on TT tensors. The spectrum of , denoted , is discrete and positive. Obviously (18) is a positive operator if the cosmological constant is negative. In this case there are only stable, bounded oscillations, leading to a mild fluctuation induced renormalization. This is precisely what we observe in the IR of the Type Ia trajectories: there is virtually no non–canonical parameter running below . The situation is very different for where, for sufficiently small, (18) has negative eigenvalues, i. e. unstable eigenmodes. In fact, expanding the RHS of the flow equation to orders and the resulting –functions are given by traces (spectral sums) containing the propagator [1]

(19) |

The crucial point is that the propagator (19) can have a pole
when is too large and positive. It occurs for , or equivalently , at . Upon performing the –sum this pole
is seen to be responsible for the terms and in the
–functions which become singular at . The allowed part of the -–plane ()
shown in Fig. 1 corresponds to the situation where the singularity is avoided thanks to the large
regulator mass. When approaches from above
the --functions become large and strong
renormalizations set in, driven by the modes which would go
unstable^{7}^{7}7From the propagator (19) it is obvious that
the smaller the eigenvalue of a fluctuation mode, the higher
is the scale at which this particular mode starts
contributing significantly, and the more important is its impact on
the RG flow. A galaxy–size fluctuation is more important than a
solar system–size fluctuation, for example. at .

In this respect the situation is completely analogous to the scalar theory discussed above: Its symmetric phase () corresponds to gravity with ; in this case all fluctuation modes are stable and only small renormalization effects occur. Conversely, in the broken phase () and in gravity with , there are modes which are unstable in absence of the IR regulator. They lead to strong IR renormalization effects for and , respectively. The gravitational Type Ia (Type IIIa) trajectories are analogous to those of the symmetric (broken) phase of the scalar model.

As for the behavior of the RG trajectories near the boundary the crucial question is whether, when is lowered, decreases at least as fast as or more slowly. In the first case the trajectory would never reach the singularity, while it does so in the second. In the Einstein–Hilbert approximation the trajectories of Type IIIa indeed belong to the second case; since does not decrease fast enough the RG trajectory runs into the pole at a certain where .

The termination of certain trajectories is not specific to gravity. We saw that it happens also in the scalar model if we use the over–simplified truncation (14). This simple ansatz has similar limitations as the Einstein–Hilbert truncation. In the scalar case the cure to the problem of terminating trajectories is known [12]: If one uses a more general truncation, allowing for a non–polynomial , the RG trajectories never reach the singularity and extend to , with strong renormalizations, however, in particular the quadratic running (17). In a certain sense the Einstein–Hilbert truncation has a similar status as a polynomial truncation for : it is not general enough to be reliable down to for a positive cosmological constant or in the broken phase, respectively. While there are computationally manageable truncations of sufficient generality in the scalar case it is not known which truncations would allow for a reliable continuation of the Type IIIa trajectories below . They are likely to contain nonlocal invariants [16, 8, 39] which are hard to handle analytically.

In view of the scalar analogy it is a plausible and very intriguing speculation that, for , an improved gravitational truncation has a similar impact on the RG flow as it has in the scalar case. There the most important renormalization effect is the running of the mass: . If gravity avoids the singularity in an analogous fashion the cosmological constant would run proportional to ,

(20) |

with a constant . In dimensionless units (20) reads , i. e. is a fixed point of the –evolution. If the behavior (20) is actually realized, the renormalized cosmological constant observed at very large distances, , vanishes regardless of its bare value. Clearly this would have an important impact on the cosmological constant problem [40].

##
3 The RG trajectory realized in Nature:

the Einstein–Hilbert domain

How can we find out which one of the RG trajectories shown in Fig.
1 is realized in Nature? As in every quantum field theory,
one has to experimentally determine the value of appropriate
‘‘renormalized’’ quantities.^{8}^{8}8Above we considered pure
gravity, while the experimental data include renormalization effects
due to matter fields. Since in this paper we are interested in order
of magnitude estimates only, and we anyhow do not know the exact
matter field content of Nature, we assume that the inclusion of
matter does not change the general qualitative features of pure
gravity. As for the nonperturbative renormalizability it is known
that there exist matter systems with this property and that they are
“generic” in a sense [7]. In Quantum
Electrodynamics (QED), for instance, one measures the electron’s
charge and mass in a large–distance experiment, thus fixing
and at . In QCD the point is
inaccessible, both theoretically and experimentally, so one uses a
“renormalization point” at a higher scale . In QEG the
situation is similar. In the extreme infrared () we have
neither theoretically reliable predictions nor precise experimental
determinations of the gravitational couplings and .

### 3.1 Exploiting experimental information

We know that all gravitational phenomena at distance scales ranging from terrestrial experiments to solar system measurements are well described by standard General Relativity (GR). Since this theory is based upon the Einstein–Hilbert action with constant values of and , we can conclude that the RG evolution of those parameters for between the related typical mass scales and , say, is negligibly small. In this “GR regime” the renormalization group flow is essentially the canonical one, i. e. and which follows from , when . The corresponding RG trajectories are the hyperbolas depicted in Fig. 3. During the –interval defining the GR regime the true RG trajectory realized in Nature must be very close to one of those hyperbolas.

Which class does this trajectory belong to? Recent CMBR and high redshift supernova data show that the present Universe is in a state of accelerated expansion which, in a Friedmann–Robertson–Walker framework, can be explained by an nonzero positive cosmological constant. In the RG context this should mean that since the relevant scale is set by , the present Hubble parameter. Among the trajectories of the Einstein–Hilbert truncation only those of Type IIIa and IIIb run towards positive ’s for . Trajectories of Type IIIb correspond to a negative and are excluded therefore. Thus the RG trajectory realized in Nature, as long as it remains in the domain of validity of the Einstein–Hilbert truncation, belongs to Type IIIa.

We saw that trajectories of Type IIIa cannot be continued below a certain , and in the present paper we are going to argue that is roughly of the order of typical galaxy scales. As a result, the Einstein–Hilbert truncation is probably insufficient to describe the (continuation of the) Type IIIa trajectory realized in Nature at the cosmological scale where was actually measured. Nevertheless it seems to be clear that, for large enough, the true RG trajectory selected by Nature is a Einstein–Hilbert trajectory of Type IIIa. The reason is that the other alternatives, Type Ia and Type IIa, can be computed reliably down to , and they do not give rise to a positive cosmological constant in the infrared.

For our picture to be correct, the prospective Type IIIa trajectory must contain a sufficiently long “GR regime” where it runs on top of one of the hyperbolas of Fig. 3. The situation is sketched qualitatively in Fig. 4.

The Type IIIa trajectory spirals out of the NGFP, approaches the separatrix, runs almost parallel to it for a while, then “turns left” near the GFP, and finally runs towards the singularity at . After the turning point where , but before it gets too close to , this trajectory is an almost perfect hyperbola of the canonical RG flow. In Fig. 4 we indicate the latter by the dashed line which, between the points and , is indistinguishable from the Type IIIa trajectory. It is this segment between and which can be identified with the realm of classical GR. As we shall see in a moment, the variation of and along the true Type IIIa trajectory is unmeasurably small between and .

Which one of the infinitely many Type IIIa trajectories did Nature pick? We can answer this question if, from experiments or astrophysical observations, we know a single point the trajectory passes through. Let us assume we measure the (dimensionful) Newton constant and cosmological constant in a “laboratory” with a typical linear dimension of the order . We interpret the result of the measurements as the running couplings evaluated at this scale: , . Knowing those two values, as well as the pertinent “laboratory” scale , we can compute the dimensionless couplings:

(21) |

The pair uniquely fixes a trajectory in the Einstein–Hilbert approximation. If one uses a more general truncation, further parameters need to be measured, of course. The first one of the eqs. (21) can be rewritten in the following suggestive form:

(22) |

Here we defined the Planck length and mass in the usual way in terms of the measured Newton constant according to .

Newton’s constant has been measured at length scales ranging from the size of terrestrial experiments to solar system dimensions. Within the errors the result has always been the same: . We can now calculate according to (21). At the typical scale of a terrestrial laboratory one finds

(23) |

while at the solar system scale of ,

(24) |

In any case is an extremely small number for any in the GR regime, . Its precise value will not matter in the following; for the sake of clarity we shall use the example and for numerical illustration. Throughout the discussion the length scale is assumed to lie in the GR regime, ranging from terrestrial to solar system distances.

The determination of the associated is difficult; in fact, rather than at “laboratory” scales , was actually measured at cosmological distance scales. For a first qualitative discussion the following estimate is sufficient, however. According to the effective vacuum Einstein equation at the scale , a cosmological constant of magnitude leads to the spacetime whose radius of curvature is of the order

We know that in absence of matter, at , say, spacetime (when observed with a “microscope” of resolution ) is flat with a very high precision, i. e. that is much larger than the size of the “laboratory”: . As a consequence, must be very small compared to unity:

(25) |

This means in particular that, at and in the entire GR regime, the trajectory realized in Nature is still very far away from the dangerous singularity at where the Einstein–Hilbert approximation breaks down.

### 3.2 Approximate RG flow near the GFP

Since in the GR regime we may neglect higher order terms in the eqs. (10) and obtain the following flow equation linearized about the GFP:

(26a) | ||||

(26b) |

In the “linear regime” where (26) is valid, only the cosmological constant shows a non–canonical running, while the dimensionful Newton constant does not evolve in the approximation (26b).

In the next to leading order runs according to with proportional to . In any cutoff scheme is a positive constant of order unity, for the sharp cutoff. Hence

(27) |

The smallness of these numbers explains the success of standard General Relativity based upon the approximation , and it confirms our interpretation of the segment between the points and in Fig. 4 as the realm of classical gravity.

To the right of the point in Fig. 4, at scales lower than those of the GR regime, the growth of due to the infrared instability sets in.

Let us return to the linearized flow equations (26). They are applicable whenever the trajectory is close to the GFP (), not only in the GR regime. We may use them to derive a relation between the coordinates of the trajectory’s turning point at which it switches from decreasing to increasing values of . (See Fig. 4.) Setting in (26a) we obtain

(28) |

The constant is cutoff scheme–, i. e. –dependent, but it is of order unity for any cutoff. Therefore

(29) |

Eqs. (28), (29) are valid provided . Later on we shall see that this is actually the case in Nature.

After the trajectory has passed the turning point, keeps decreasing and increases. As a result, the second term on the RHS of (26a), , gradually becomes negligible compared to the first one, , so that the flow equation becomes the canonical one. This marks the beginning of the GR regime at .

It is easy to solve the coupled differential equations (26) exactly. They allow for two free constants of integration which we fix by requiring and . By definition, is the scale at which the trajectory passes through the turning point. The solution reads

(30a) | ||||

(30b) |

The corresponding running of the dimensionful parameters is given by

(31a) | ||||

(31b) |

Since the linear regime contains the GR regime, we may identify the constant in (31a) with . This entails the important relation , or

(32) |

We observe that a small will lead to a large hierarchy .

We can use (32) in order to eliminate from (30):

(33a) | ||||

(33b) |

For later use we note that for any in the linear regime

(34) |

Looking at eqs. (31a,b) we see that, while does not run at all in the linear regime, the scale dependence of is entirely due to the factor . Once has become much smaller than , after the trajectory has “turned left”, this factor approaches unity, and effectively stops to run. By definition, this happens at , the starting point of the GR regime.

Let us make this statement more precise. Denoting by the scale at which the trajectory passes through , the requirement is that . We quantify the precision with which the –term is negligible in the GR regime by means of an exponent . In terms of , we define by

(35) |

As a result, is smaller than for all scales in the GR regime (). A value such as should be sufficient in practice. It makes sure that in the GR regime is constant with a precision better than .

For any scale in the GR regime we obtain from (33), in very good approximation,

(36a) | ||||

(36b) |

Similarly (31) yields the following constant values of the dimensionful quantities:

(37a) | ||||

(37b) |

Remarkably, in the GR regime, differs from its value at the turning point precisely by a factor of . As a trivial consequence,

(38a) | |||

and (34) reduces to | |||

(38b) |

We shall come back to this relationship later on.

Next we derive various estimates for and at and
. If we evaluate (30) at , neglecting the
–term in (30b), and use (35) we find^{9}^{9}9In
order–of–magnitude equations such as (39) we suppress
inessential factors of order unity.

(39a) | ||||

(39b) |

Obviously has a – (–) coordinate which is smaller (larger) than the corresponding coordinate of by a factor (), as it should be according to Fig. 4. Now we exploit eq. (29). Neglecting factors of order we have which yields when combined with (39)

(40) |

Since increases along the trajectory we know that where is any “laboratory” scale in the GR regime. Therefore, as a consequence of the experimental result (25),

(41) |

Since and are almost equal, is small, too:

(42) |

According to (40), is even smaller than . Hence, with from (41), it follows that

(43) |

Eqs. (42) and (43) show that for the RG trajectory realized in Nature the points and are located at an extremely short distance to the GFP. The trajectory starts at the NGFP with coordinates [3, 5]. Then it follows the separatrix until, at very tiny values of and , it gets ultimately driven away from the GFP along its unstable –direction. In pictorial terms we can say that the trajectory is squeezed deeply into the wedge formed by the separatrix and the –axis. As a consequence, it spends a very long RG time near the GFP because the –functions are small there. In this sense the RG trajectory which Nature has selected is highly non–generic or “unnatural”. It requires a precise fine–tuning of the initial conditions, to be posed infinitesimally close to the NGFP.

###
3.3 Existence of a GR regime and the

cosmological constant problem

Why did Nature pick a trajectory which gets so “unnaturally” close to the GFP? Why not, for example, one of those plotted in Fig. 1 which always keep a distance of order unity to the GFP and require no special fine–tuning? Of course questions of this kind cannot be answered within QEG, for the same reason one cannot compute the electron’s charge or mass in QED.

However, it is fairly easy to show that a Universe based upon one of the generic trajectories would look very different from the one we know. The main difference is that, along a generic trajectory, no sufficiently long GR regime would exist where classical gravity makes sense at all. According to our previous discussion, the GR regime is located in between a regime with strong UV renormalization effects (spiraling around the NGFP related to asymptotic safety) and, most probably, a second regime with a significant running of the parameters in the IR. For classical GR to be applicable the UV and IR regimes must be well separated. For a generic trajectory this is not the case, however. The generic Einstein–Hilbert trajectories computed in [8] leave the UV regime at and soon after they terminate at a not much smaller than ; there is no GR regime which would last for a few orders of magnitude at least.

The basic mechanism which allows for the emergence of a GR regime is to fine–tune the trajectory in such a way that it spends a lot of RG time near the GFP. In this manner the onset of the IR regime, within the Einstein–Hilbert truncation characterized by the value of , gets enormously delayed, being much smaller than . If classical GR is correct up to a length scale , the trajectory must be such that its is larger than since at the IR effects are likely to become visible.

A quantitative estimate of can be obtained as follows. In the linear regime the RG flow is explicitly given by eqs. (30). Once is sufficiently low we leave this regime and the full nonlinear Einstein–Hilbert flow equations have to be used. At even smaller scales, the truncation breaks down and we should switch to a more general one. Outside the linear regime, the –functions are no longer small, hence the trajectory has a comparatively high “speed” there. As a result, it takes the trajectory much longer to go from to the boundary of the linear, i. e. GR regime than from there to a point close to . Therefore we may use eq. (36b) for in the GR regime in order to derive an estimate for . We identify with the scale where, according to (36b), the value is reached: . In this rough approximation,

(44a) | ||||

(44b) |

or, in terms of length scales,

(45) |

This equation shows explicitly that by making small, can be made as large as we like.

Thus we have demonstrated that only a “non–generic” trajectory with an “unnaturally” small does give rise to a long GR regime comprising many orders of magnitude.

In (32) we saw that the turning point is passed at . Ignoring the –factor in (44b) this entails . As a result, there exists an exactly symmetric “double hierarchy” among the three mass scales , , and :

(46) |

Therefore, on a logarithmic scale, is precisely in the middle between and . Thanks to the smallness of it is many orders of magnitude away from either end.

The emergence of a long regime where gravity is essentially classical is one of the benefits we get from the unnaturalness of the trajectory chosen by Nature. Another one is that, in this classical regime, the cosmological constant is automatically small. Inserting (32) into (37b) we obtain the following in the GR regime:

(47) |

Again thanks to the smallness of and , the cosmological constant is much smaller than . Up to a factor of order unity,

(48) |

Thus we may conclude that the very fine–tuning which gives rise to a long GR regime at the same time implies a large hierarchy between the in this GR regime and , which often is considered its “natural” value.

This observation provides a solution to the “cosmological constant problem” by realizing that it is actually part of a much more general naturalness problem. Rather than “Why is so small?” the new question is “Why does gravity behave classically over such a long interval of scales?”.

The basic reason for this connection is very simple. Denoting the cosmological constant in the GR regime by we have there. This approaches unity so that the IR renormalization effects become strong once is of the order . Hence, roughly,

(49) |

which shows that is small if, and only if, is small.

This is precisely what one finds by numerically solving the flow equations: There do not exist any Type IIIa trajectories which, on the one hand, admit a long classical regime, and on the other hand, have a large cosmological constant. As a consequence, the smallness of the cosmological constant poses no naturalness problem beyond the one related to the very existence of a classical regime in the Universe.

Strictly speaking this resolution of the cosmological constant problem is only a partial one for the following reasons. (1) We analyzed the flow equations of pure gravity only, but we believe that the inclusion of matter fields (forming symmetry breaking condensates, etc.) will not change the general picture. Since anyhow only massless particles contribute to the –functions for , it is hardly possible that the matter fields destroy the IR renormalizations near . But if they survive the estimate (49) remains intact, again implying that a long GR regime requires a small . (2) Our argument refers to rather than the cosmologically relevant . Because of the IR effects, and differ in principle, but we shall see later on that in Nature this difference is small compared to the notorious 120 orders of magnitude one has to cope with.

In fact, the upper boundary of the linear regime is roughly the Planck scale since, according to (33), and are of the order of and slightly below . Between this scale and the turning point the change of is enormous. From (31b) with (32) and (42) we obtain . In the next section we shall see that , whence .

### 3.4 Is the Hubble scale within the classical regime?

The observational (CMBR, supernova, etc.) data, when interpreted within standard cosmology, show that the present vacuum energy density of the Universe is very close to the critical one, implying that is of the order of . (The general relationship is , and the data favor ).

We can now ask whether, if is as large as , the Hubble scale is still within the GR regime. If we interpret as a “laboratory” value, , the estimate (49) yields

(50) |

This is a very remarkable and intriguing result: On the RG trajectory Nature has picked, the Hubble scale is precisely at the boundary of the GR regime. At distances small compared to the Hubble length classical GR is a good approximation, but on length scales the IR renormalization effects become important so that its use is questionable there.

Since is just at the boundary of the GR regime it is plausible to assume that the cosmological values and do not differ too much from and , respectively. (Later on we shall see that the difference should not be more than one or two orders of magnitude.) If so, we are in a position to completely fix the parameters of the RG trajectory because we can fit the measured –value to the trajectories of the linearized flow. Inserting , , and into (38b) we find