Local Effects of the Cosmological Constant

© M. Nowakowski1,2   and  I. Arraut 1

1 Departamento de Fнsica, Universidad de los Andes, Bogota, Colombia
2 Email:  mnowakos@uniandes.edu.co

Abstract: If the positive cosmological constant  explains the recently discovered accelerated stage of the universe it is, at least effectively, beside the Newtonian constant GN=mPl-2, the second fundamental constant of gravity. 
Besides its cosmological implications, affects also astrophysical properties of large structures. We will probe into these matters by using virial equation, hydrostatic equilibrium and the Schwarzschild-de Sitter metric. It will be shown that the effects can be non-negligible. Moreover, it is legitimate to put forth the question how the mass scale  or combinations with the dimension of mass like , , etc. alter our views not only of large scale astrophysics but also our expectations for quantum gravity. We make use of the generalized uncertainty principle and black body radiation to investigate these issues. 

1.	Introduction: 

The resurrection of the cosmological constant  in an accelerated universe [1] (or, rather in rescue to explain the stage of the accelerated universe) has awakened old controversies about the true nature of this constant. Above all, the “to be or not to be” of  has been an eagerly discussed topic up to now. It is usually based on the contribution to vacuum energy from zero-point energy in quantum field theory. The latter is a divergent integral regularized by a cut-off chosen to be at Planck scale. The result is 10125 too large which prompted many physicists to search for a convincing mechanism forcing  to become zero. No such, generally accepted mechanism has been found.
Instead, as for now many physicists and cosmologists accept  as a possible explanation of the accelerated universe (and the related problem of the age globular clusters) which in terms of the one of the Friedmann’s equations reads:
             (1)

Note that without the inclusion of  would be always negative. The evidence for a new cosmology (dark energy as it is called) is gaining speed and comes not only from supernovae type Ia surveys [3], but also from cosmic microwave background [4], baryon acoustic oscillations [5] and weak lensing [6].
Having accepted  as a possible theory to explain these facts, we can go back to see what we had to modify in order to reach at eq. (1). Although this seems like a mere conceptual undertaking, it has to do with a misunderstanding sometimes encountered and rooted in the name “Cosmological Constant” (in other words, does  affect only Cosmology?). It is actually the Einstein tensor Gµн which got altered to:

                     (2)
and not the energy-momentum tensor Tµн connected to by Einstein’s equations:
                  (3)

This simple fact makes the modification universal as opposed to situation-specific which would be the case, had we changed Tµн for cosmology. This is to say that in spite of the name “Cosmological Constant”,  will not only affect cosmological features, but any local physics which emerges from now modified Einstein’s equations. In a way, this seems obvious (albeit not always appreciated) as e.g nobody doubts that with the inclusion of  the Schwarzschild metric becomes now the Schwarzschild-de Sitter metric, and the Newtonian limit receives modifications[7]:

                     (4)

The relevant question is rather: are the local effects of  important?       

2.  Local aspects of 

If  is the correct model to account for the accelerated universe, we ought to have сvac ≈ 0,7сcrit. With this value almost all scales set by  are of cosmological order of magnitude. Hence it seems that even if  affects, in principle, local physics, it does so in a negligible way. However, it can be immediately seen that such an argument is fallacious as it discards the existence of other local scales which in conjunction with can give rise to an observable of astrophysical relevance. Such an example is the motion in Schwarzschild-de Sitter metric [9].

          (5)

with 
     ;         ;                (6)
The motion of a test-body in such a metric is determined according to the equation:

                               (7)
with:

                        (8)

playing here the role of an effective potential. Three length scales enter the problem: rs,  and rl (angular momentum per unit mass). The first three terms conspire to give Ueff a local maximum close to the center and a local minimum (which is the standard minimum of planetary motion). In contrast, the Newtonian part   and ; combine to give Ueff  a local maximum beyond which Ueff is a monotone decreasing function. As a result, all bound orbits can happen only within r < rmax where rmax is the position of the last local maximum due to . We can estimate rmax by setting rl=0. The calculation gives:

                                (9)
Since  (Hubble radius) is a large number compared to rs, the resulting rmax is of astrophysical order of magnitude and its specific values in dependence of the mass M bear strange coincidences [9]. With M the mass of the black hole in the center of our galaxy (M≈106Msun), rmax≈10Kpc, which is roughly the order of magnitude of the extension of the galaxy.
Another example of the impact of  on local physics is gravitational equilibrium [10]. There the parameter of importance is the density of the object i.e сvac will play the decisive role. The adequate tool to probe into equilibrium matters is the virial theorem [10].
                (10)

Where Kij is the kinetic part, Wij the potential part and Iij the inertial tensor. Denoting the trace of the tensors by K, W and I and making use of the fact that K ≥ 0, we obtain the inequality:
                (11)
which is trivial in case of . To illustrate the relevance of this inequality, let us assume с=const. we obtain [10]:

                                    (12)

Where A is a dimensionless ratio of two integrals depending only on the geometry (shape) of the body. As сvac≈сcrit again it seems hopeless that (12) is of any phenomenological importance. Indeed, for spherically symmetric objects A=2. But for non-spherical objects A can be quite sizable (factor 102 or 103 if the deviation from sphericity is large). In passing we note that the dark matter density around astrophysical objects might be ~ 200сcrit.
If the dark matter distribution follows the non-spherical shape of the luminous part (oblate and prolate galaxies and even galaxy clusters are known), then the relevance of (12) is obvious. Other local effects were discussed in [11]. 
Interestingly, Newtonian hydrostatic equilibrium and its relativistic counterpart via Tolman-Oppenheiner-Volkoff equation, gives for spherical symmetry < 2сvac, where:

      (13)

And R is the size of the object (see [9] and the references therein). We can demonstrate the effect of  in gravitational equilibrium using a slightly different approach.
In place of the inequality, we can maintain K≠0 and solve the scalar virial equation for the mean velocity of the components in the astrophysical object under consideration. For an ellipsoid of constant density we obtain:

     (14)

where Г for an prolate type is:

    (15)

with:

    (16)

If for instance, с/сcrit is 103, it suffices to have , in order that the mean velocity approach to zero, an effect due to . This is an explicit example of the effect of combined with non-spherical geometry of the object. 
 
In deriving (11) and (12) we relied on the Newtonian limit of the Einstein’s equations with . This limit is the static version for the perturbation h00 (gµн=зµн+hµн, зµн the Minkowski metric) of the linearized Einstein’s equations :

        (14)
 
Besides the Newtonian limit which follows from it, is there any other effect in connection with (14)? The first what comes to mind are gravitational waves. Eq. (14) is like Fierz-Pauli equation for spin-2 with two sources: Sµн and . It goes without saying that (i) we cannot drop ad-hoc the term in (14) while studying gravitational waves and (ii) the solution of (14) will contain the standard oscillatory terms and new terms due to . Imposing the harmonic gauge condition (eq. (14) is gauge covariant under local gauge transformation (hµн→ h’µн= hµн+∂µен+∂неµ) and demanding the special solution of (14) , which is proportional to  to be asymptotically to de Sitter metric (up to a diffeomorphism), we arrive at:

                        (15)

where гµн is the standard retarded solution (oscillatory for away from the source) and:

      ;              ;               (16)

With еij= xixj for  and 0 otherwise. The solution is in agreement with [12] when we let m→0. No observable can be calculated without the full solution i.e without the inclusion of оµн. For instance, inserting hµн into the gravitational energy-momentum pseudo-tensor tµн and calculating the averaged components <toi>, (the averaged gravitational Poynting vector entering the expression for the power), we can infer when the part of <toi> proportional to becomes comparable to the standard oscillatory part for the case of monochromatic sources, this happens if the wave is produced at distances larger than:

                (17)

Where f is the frequency of the incoming wave and ĥ is the amplitude of the oscillatory part (wave) as arriving in the detector. Typical values of ĥ are 10-20-10-23 which converts  into an astrophysical distance in spite of the cosmological suppression factor . 

3.	 Two scales of gravity: GN  and .

The cosmological constant  makes the gravity theory a two scales theory. There is a dual interplay between GN and  which emerges in upper and lower bounds on some observables or in range of validity of certain approximations. For instance, in order for the Newtonian limit to be valid, one has to impose [7]:

       ;           (18)

In the hydrostatic equilibrium to ensure global solution, the so-called Buchdald inequalities [13] require:

              (19)
  
This implies:
 and       (20)

Equation (20) is valid up to numerical factors of order one; the scales  and  enter again into these restrictions.
Another example is Ueff (See eq. (8)), from the equation of motion (7) in the de Sitter metric. To avoid that the first local maximum and the local minimum coincide to form a saddle point, one has to respect:

                    (21)

If we use this result in the expression for elliptical orbits for elliptical orbits for non-relativistic (celestial) mechanics, we obtain:

    (22)

which is now a non-relativistic restrictions on the size of possible bound orbits with non-zero angular momentum. 
Since  allows a second local maximum beyond the minimum, we can repeat the procedure for this pair of extreme. The result reads [9]:

        (23)

Again, the cosmological constant results into a new bound. Can we see this kind of duality also in other places? Yes, through another unexpected connection. In 1966 Andrei Sakharov proved in a general context that the maximal temperature of a black body radiation can not exceed [14]:

         (24)

This result seems too extra-orbitant large, Tmax~1032K, to be of any use…, until combined with Hawking’s formula for black hole evaporation:

              (25)

Clearly a maximal temperature means here a minimal mass Mmin:

        (26)

Which means that Sakharov’s result implies a black hole remnant of the mass Mmin this is consistent with results obtained from Generalized Uncertainty Principle where the mass of the black hole remnant is of the order mpl [15]. How can we invoke  along similar lines? Using  from the Schwarzschild- de Sitter metric and demanding g00 > 0 to interpret  as proper time and trading the mass M in rs for the constant density с for which we can use the Stefan-Boltzmann law с=уT4, we finally obtain [16]:

          (27)

Obviously, we have to have , which together with [17] gives:

                   (28)

As before, we can use this in conjunction with Hawking’s formula. Now, Tmax defines a maximally possible mass:

      (29)

One can check that the modifications which the Hawking’s formula will receive due to , will not substantially change (29).
Whereas Mmin in (26) defines a minimum mass of a black hole, Mmax in (29) apparently defines the maximum mass. From equations (26) and (29), (24) and (28) one can see the dual nature of GN and .

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