Cosmological constant as an eigenvalue of the Hamiltonian constraintin Horava-Lifshitz theory

Remo Garattini*

Universita degli Studi di Bergamo, Facolta di Ingegneria, Viale Marconi 5, 24044 Dalmine (Bergamo) Italyand I.N.F.N.-sezione di Milano, Milan, Italy

(Received 11 September 2012; published 6 December 2012)

In the framework of Horava-Lifshitz theory, we study the eigenvalues associated with the Wheeler-

DeWitt equation representing the vacuum expectation values associated with the cosmological constant.

The explicit calculation is performed with the help of a variational procedure with trial wave functionals

of the Gaussian type. We analyze both the case with the detailed balanced condition and the case without

it. In the case without the detailed balance, we find the existence of an eigenvalue depending on the set of

coupling constants (g2, g3) and (g4, g5, g6), respectively, and on the physical scale.

DOI: 10.1103/PhysRevD.86.123507 PACS numbers: 98.80.Qc, 04.50.Kd

I. INTRODUCTION

The cosmological constant represents one of the greatestscientific challenges of this century [1]. If one is temptedto compute it from the zero-point energy of some physicalfield, one discovers an enormous discrepancy with theobserved data. This discrepancy is of the order of 10120:this is known as the cosmological constant problem. Manyattempts have been made to explain such a discrepancy, butthey appear to be far from satisfactory. Recently, Horavaproposed a modification of Einstein gravity motivated bythe Lifshitz theory in solid state physics [2,3]. This modi-fication allows the theory to be power-counting ultraviolet(UV) renormalizable and should recover general relativityin the infrared (IR) limit. Nevertheless, Horava-Lifshitz(HL) theory is noncovariant. Indeed, in this approach spaceand time exhibit Lifshitz scale invariance of the form

t ! ‘zt and xi ! ‘xi; (1)

with z � 1. z is called the dynamical critical exponent andin the present case it is fixed to z ¼ 3. The breaking of thefour-dimensional diffeomorphism invariance allows for adifferent treatment of the kinetic and potential terms for themetric: from one side the kinetic term is quadratic in timederivatives of the metric, and from the other side thepotential has high-order space derivatives. In particular,the UV behavior is dominated by the square of the Cottontensor of the three-dimensional geometry by means of a k6

contribution to the propagator, leading to a renormalizablepower-counting theory. The original HL theory is based ontwo assumptions: detailed balance and projectability [4].1

The projectability condition is a weak version of theinvariance with respect to time reparametrization andtherefore to the Wheeler-DeWitt (WDW) equation [25].Motivated by these interesting features, we ask ourselves ifthe problem of the cosmological constant in HL theory has

a better chance of being solved than in ordinary Einsteingravity.2 Indeed, the modification of the gravitational fieldat short distances could produce new contributions thatallow what in ordinary Einstein gravity is forbidden. Inordinary Einstein gravity, there exists a well-acceptedmodel connected with the cosmological constant, namely,the Friedmann-Robertson-Walker (FRW) metric, whoseline element is

ds2 ¼ �N2dt2 þ a2ðtÞd�23: (2)

d�23 is the usual line element on the three-sphere and N is

the lapse function. In this background, we have simply

Rij ¼ 2

a2ðtÞ�ij and R ¼ 6

a2ðtÞ : (3)

For simplicity we consider the case in which k ¼ 1. Thegeneralization to k ¼ 0, �1 is straightforward. The WDWequation for such a metric is

H�ðaÞ ¼�� @2

@a2þ 9�2

4G2

�a2 ��

3a4��

�ðaÞ ¼ 0: (4)

It represents the quantum version of the invariance withrespect to time reparametrization. If we define the follow-

ing reference length a0 ¼ffiffiffiffiffiffiffiffiffi3=�

p, then Eq. (4) assumes the

familiar form of a one-dimensional Schrodinger equationfor a particle moving in the potential

UðaÞ ¼ 9�2a204G2

��a

a0

�2 �

�a

a0

�4�

(5)

with zero total energy and without a factor ordering.Equation (4) can be cast into the formR

Da��ðaÞ½� @2

@a2þ 9�2

4G2 a2��ðaÞR

Da��ðaÞ½a4��ðaÞ ¼ 3��2

4G2; (6)

*Remo.Garattini@unibg.it1Different aspects of HL theory are discussed in Refs. [5–24].

2See also Ref. [26] for a different approach to the cosmologi-cal constant problem in HL theory.

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which appears to be an expectation value. In particular, itcan be interpreted as an eigenvalue equation with a weightfactor on the normalization. The application of a varia-tional procedure in ordinary gravity with a trial wavefunctional of the form

� ¼ expð��a2Þ (7)

shows that there is no real solution of the parameter �compatible with the procedure. The purpose of this paper isto obtain an eigenvalue equation for the cosmologicalconstant like the one in Eq. (6) but in HL theory with thehelp of the WDWequation. Nevertheless, as pointed out byMukohyama [27], there are four versions of the theory:with/without the detailed balance condition, and with/without the projectability condition. In this paper, wewill consider the problem with and without the detailedbalanced condition.

II. HL GRAVITY WITH THE DETAILEDBALANCED CONDITION

With the assumption of the detailed balanced condition,the action for the Horava theory assumes the form

S ¼Z��I

dtd3xðLK �LPÞ; (8)

where

LK ¼ Nffiffiffig

p 2

�2ðKijKij � �K2Þ (9)

is the Lagrangian kinetic term. The extrinsic curvature isdefined as

Kij ¼ 1

2Nf� _gij þriNj þrjNig; (10)

where K ¼ Kijgij is its trace, and Ni is the shift function.

For the FRW metric, there is no shift function. TheLagrangian potential term LP is

Nffiffiffig

p ��2

2w4CijCij � �2�

2w2"ijkRilrjR

lk þ

�2�2

8RijRij

� �2�2

8ð1� 3�Þ�1� 4�

4R2 þ�WR� 3�2

W

��; (11)

where

Cij ¼ �iklrk

�Rjl �

1

4R�j

l

�(12)

is the Cotton tensor. By plugging the FRW metric (2) intoLP, we obtain

LP¼Nffiffiffig

p �2�2

8

�3

ð1�3�Þa4�6�W

ð1�3�Þa2þ3�2

W

ð1�3�Þ�;

(13)

and the action for the potential part reduces to

SP ¼ �Z��I

dtd3xLP

¼ � �2�2

8ð1� 3�ÞZIdtN2�2a3

�3

a4� 6�W

a2þ 3�2

W

�:

(14)

Concerning the kinetic part, one gets

LK ¼ Nffiffiffig

p 2

�2ðKijKij � �K2Þ

¼ a3sin2 sin3

N�2

�_a

a

�2ð1–3�Þ; (15)

and the corresponding action is

SK ¼Z��I

dtd3xLK ¼ZIdt2�2a3

3

N�2

�_a

a

�2ð1–3�Þ:

(16)

Now, we can compute the canonical momentum to find

�a ¼ �SK� _a

¼ 12�2

�2a _að1–3�Þ; (17)

where we have set N ¼ 1. The resulting Hamiltonian iscomputed by means of the usual Legendre transformation,leading to

H¼Z�d3xH ¼

Z�d3x½�a _a�L�

¼ �2�2a

12�2að1�3�Þþ2�2�2�2

8ð1�3�Þ�3

a�6�Waþ3�2

Wa3

�;

(18)

and the classical constraint can be read off quite straight-forwardly:

�2a þ 9�2�4½1� 2�Wa

2 þ�2Wa

4� ¼ 0: (19)

Equation (19) has been analyzed in Ref. [18]. However,we are interested in the WDW equation associated withEq. (19), which is given by

� @2�

@a2þ 9�2�4½1� 2�Wa

2 þ�2Wa

4�� ¼ 0; (20)

where we have promoted the canonical momentum �a toan operator �a ¼ �i@a. Note that the classical constraint(19) and the WDW equation (20) are independent of theparameter � for the FRW metric. This means that the limitof � ! 1, which is necessary to reproduce general relativ-ity, does not constrain �W to be negative. It is useful to

define the following dimensionless variable x ¼ ffiffiffiffiffiffiffiffi�W

pa.

Equation (20) then becomes

�@2�

@x2�W þ 9�2�4½1� 2x2 þ x4�� ¼ 0: (21)

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We compute the value of �W by adopting a variationaltechnique with a trial wave functional of the form

� ¼ expð��x2Þ: (22)

For this purpose, Eq. (21) must be transformed into anexpectation value computation:

Z 1

0dx��

���W

@2

@x2þ 9�2�4½1� 2x2 þ x4�

�� ¼ 0;

(23)

where the eigenvalue �W is implicitly defined by Eq. (23).We find

Z 1

0dx��� ¼

ffiffiffiffiffiffiffi�

8�

s; (24)

representing the normalization of the wave function.The other terms become

�Z 1

0dx�� @

2�

@x2¼Z 1

0dx��½2�� 4�2x2��

¼ 2�

ffiffiffiffiffiffiffi�

8�

s� 4�2

ffiffiffiffi�

8

r �1

4��3

2

�

¼ �

ffiffiffiffiffiffiffi�

8�

s; (25)

�18�4�4Z 1

0dx��x2� ¼ �18�4�4

ffiffiffiffi�

8

r �1

4��3

2

�; (26)

and

Z 1

0dx��x4� ¼

ffiffiffiffi�

8

r �3

16��5

2

�: (27)

By plugging Eqs. (24)–(27) into Eq. (23), one gets

�W

0@� ffiffiffiffiffiffiffi

�

8�

s 1Aþ9�4�2

ffiffiffiffiffiffiffi�

8�

s �1� 1

2�þ 3

16�2

�¼0: (28)

In the spirit of the variational procedure �W must bestationary against arbitrary variations of the parameter �.Thus, we impose

1

9�4�2

d�Wð�Þd�

¼0¼ 1

�4

��2��þ 9

16

�; (29)

and we find that the solutions exist only for complex �.Things do not change if we make an analytic continuationof the parameter � ! i� [5,18–22,28]. Therefore, itappears that in ordinary HL gravity we cannot find a realeigenvalue for the cosmological parameter �W . Thingscould change if we slightly modify the action satisfying

the detailed balanced condition [4] by adding an appropri-ate IR term of the form

LsmallP ¼N

ffiffiffig

p �2�2

8ð1�3�Þ½�W�1Rþ�2�2W�: (30)

The addition of LsmallP affects Eq. (18), which becomes

H¼ �2�2a

12�2að1�3�Þþ�2�2�2

4ð1�3�Þ��3

aþ6�Wð1��1Þaþð3þ�2Þ�2

Wa3

�: (31)

Simplifying somewhat, we obtain the new form of theWDW equation,

H� ¼ �@2�

@x2�W þ 3�4�2½3þ 6ð1� �1Þx2

þ ð3þ �2Þx4�� ¼ 0; (32)

where we have used the dimensionless variable x ¼ffiffiffiffiffiffiffiffi�W

pa. By adopting the same methodology of Eq. (23),

one gets

�W

0@� ffiffiffiffiffiffiffi

�

8�

s 1Aþ9�4�2

ffiffiffiffiffiffiffi�

8�

s �1þ1��1

2�þð3þ�2Þ

16�2

�¼0;

(33)

where we have used Eqs. (24)–(27). We now compute theextreme of �W � �Wð�Þ to obtain

1

9�4�2

@�W

@�¼ 0 ¼ 1

�4

��2 þ �ð1� �1Þ þ 3

ð3þ �2Þ16

�;

(34)

whose solutions are

�1;2 ¼ �1 � 1

2�

ffiffiffiffi�

p2

; � ¼ ð1� �1Þ2 � 3ð3þ �2Þ

4:

(35)

In order to have normalizable solutions, we must imposethe positivity of�1;2. Hence we have three cases: (i)�> 0,(ii) � ¼ 0, and (iii) �< 0. Case (iii) involves complexsolutions, and therefore a wave functional with an oscillat-ing part. For this reason, it will be discarded. We begin withcase (ii).Case � ¼ 0. We have two real coincident solutions

determined by

�1;2 ¼ �1 � 1

2: (36)

However, � ¼ 0 means that

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1� �1 ¼ � 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð3þ �2Þ

q; (37)

which implies that

�� ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð3þ �2Þ

p4

: (38)

Since only the positive root leads to a normalizable solu-tion, we take �þ with the further condition of �2 >�3;otherwise, we have a solution which is not normalizable.By plugging �þ into Eq. (33), one gets

�W

9�4�2¼ � 1

3�þ; (39)

and therefore it will be discarded when � < 1=3 andaccepted when � > 1=3. This is particularly importantwhen we approach the general relativity limit where� ¼ 1 and we are forced to have a negative �W .

Case�> 0. Here we have two real distinct solutions forthe parameter � expressed by (35). We find the followingsubcases:

Subcase �1 ¼ 1, �2 >�3. For this choice of theparameters we find that �2 < 0 and

�1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð3þ �2Þ

p2

; �2 >�3: (40)

By plugging �1 into Eq. (33), we obtain

�W ¼ � 9�4�2

�1

�1þ ð3þ �2Þ

16�21

�; (41)

and since its value is negative, this solution will bediscarded when � < 1=3 and accepted when � > 1=3.

Subcase �1 � 1, �2 ¼ �3. We find that

�1 ¼ 0 and �2 ¼ �1 � 1; (42)

with the further condition of having �1 > 1. �1 must bediscarded because it leads to a non-normalizable solution.By plugging �2 into Eq. (33), we obtain

�W ¼ � 9�4�2

2ð�1 � 1Þ ; (43)

which is negative, and therefore it will be discarded when� < 1=3 and accepted when � > 1=3.

In the next section, we abandon the detailed balancedcondition to see if we can obtain different solutions.

III. HL GRAVITY WITHOUT THE DETAILEDBALANCED CONDITION

In Refs. [29,30], a more general form of the potentialthat avoids the detailed balanced condition has been pro-posed. Its expression is

~LP ¼ Nffiffiffig

p ½g0�6 þ g1�4Rþ g2�

2R2 þ g3�2RijRij

þ g4R3 þ g5RðRijRijÞ þ g6R

ijR

jkR

ki

þ g7Rr2Rþ g8riRjkriRjk�: (44)

The couplings gaða ¼ 0 . . . 8Þ are all dimensionless, andpowers of � are necessary to maintain such a property of

ga. By plugging the FRW background into ~LP, one gets

~LP ¼ Nffiffiffig

p �g0�

6 þ g1�4 6

a2ðtÞ þ12�2

a4ðtÞ ð3g2 þ g3Þ

þ 24

a6ðtÞ ð9g4 þ 3g5 þ g6Þ�: (45)

The term g0�6 plays the role of a cosmological constant. In

order to make contact with the ordinary Einstein-Hilbertaction in 3þ 1 dimensions, we set (without loss of general-ity) g0�

6 ¼ 2� and g1 ¼ �1. In case one desires to studythe negative cosmological constant, the identification will(trivially) be g0�

6 ¼ �2�. After having set N ¼ 1, theLegendre transformation leads to

H ¼ �a _a�LK þLP; (46)

and the Hamiltonian becomes

H ¼Z�d3xH

¼ � �2�2a

12�2að3�� 1Þ þ 2�2a3ðtÞ

��2�� 6�4

a2ðtÞ þ12�2b

a4ðtÞ þ 24c

a6ðtÞ�; (47)

where (c3g2 þ g3 ¼ b;

g4 þ 3g5 þ g6 ¼ c:(48)

The WDW equation can be easily extracted to give

�2a�þ ð3�� 1Þ

�224�4a4ðtÞ

���2�þ 6�4

a2ðtÞ �12�2b

a4ðtÞ � 24c

a6ðtÞ�� ¼ 0; (49)

and by adopting the same procedure as Eq. (32) we canwrite

Z 1

0da��

��2

a�þ ð3�� 1Þ�2

24�4a4ðtÞ

���2�þ 6�4

a2ðtÞ �12�2b

a4ðtÞ � 24c

a6ðtÞ��� ¼ 0: (50)

We can rearrange the previous expression to obtain

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R10 da��f�2

a�þ ð3��1Þ�2 144�4½�4a2ðtÞ � 2�2b� 4ca�2ðtÞ�g�R1

0 da��a4ðtÞ� ¼ ð3�� 1Þ�2

48�4�: (51)

The role of � is now clear. As a trial wave functional, we adopt the same form as Eq. (22) and, using the results ofEqs. (24)–(27), we can write

�ffiffiffiffiffi�8�

q þ ð3��1Þ�2 144�4

h�4

ffiffiffi�8

p �14�

�32

� 2�2b

ffiffiffiffiffi�8�

q þ 4cffiffiffiffiffiffiffiffiffiffi2��

p iffiffiffi�8

p �316�

�52

¼ ð3�� 1Þ�2

48�4�; (52)

where we have used the following relationship:

�cZ 1

0da��a�2�¼�c

Z 1

0

dx

2ffiffiffiffiffix3

p expð�2�xÞ

¼�c

2�

��1

2

� ffiffiffiffiffiffiffi2�

p ¼ cffiffiffiffiffiffiffiffiffiffi2��

p: (53)

By simplifying somewhat, one gets

ð1þ 16~ccÞ�3 þ ~c

��4

4�� 2�2�2b

�¼ 9�4 ð3�� 1Þ

�2�;

(54)

where we have defined

~c ¼ ð3�� 1Þ�2

144�4: (55)

By applying the variational procedure, one obtains

d�

d�¼ 3ð1þ 16~ccÞ�2 � 4~cb�2�þ �4~c

4¼ 0: (56)

We now discuss the set of solutions, taking each of thedifferent cases into consideration. We begin with thespecial case where 1þ 16~cc ¼ 0, b � 0. The solution ofEq. (56) is

�0 ¼ �2

16b: (57)

By plugging �0 into Eq. (54), one gets

�6

8b¼ � ¼ g0�

6

2; (58)

which implies

1

4b¼ g0: (59)

Another special case is 1þ 16~cc � 0, b ¼ 0. In thissituation, we find

�� ¼ � �2ffiffiffi~c

p

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1þ 16~ccÞp : (60)

�þ must be rejected because it leads to solutions which arenot normalizable. By plugging �� into Eq. (54), one gets

8ffiffiffi~c

p�6

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1þ 16~ccÞp ¼ � ¼ g0�

6

2; (61)

which implies

16ffiffiffi~c

p

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1þ 16~ccÞp ¼ g0: (62)

If c ¼ 1, we can approximate the previous result to obtainffiffiffi3

p9

¼ g0: (63)

Instead, for the particular case where c ¼ 0, we find fromEq. (60) that

�� ¼ � �2ffiffiffi~c

p

2ffiffiffi3

p ; (64)

and by plugging �� into Eq. (54), one gets

8ffiffiffiffiffi3~c

p�6

9¼ � ¼ g0�

6

2; (65)

which implies

16ffiffiffiffiffi3~c

p9

¼ g0: (66)

Namely, there remains a dependence on �.When the parameters satisfy the condition 1þ16~cc�0

and b � 0, the general solution is

�1;2 ¼ �22~cb�

ffiffiffiffi~�

p3ð1þ 16~ccÞ ; (67)

with

~� ¼ ð2~cbÞ2 � 3ð1þ 16~ccÞ ~c4: (68)

As in Sec. II, we can distinguish between three cases:

(i) ~�> 0, (ii) ~� ¼ 0, and (iii) ~�< 0. Case (iii) is dis-carded by imposing the requirement that we have two real

solutions. We begin with the case where ~� ¼ 0. We havetwo real coincident solutions,

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�1;2 ¼ � ¼ �2

8b; (69)

when 4~cb ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1þ 16~ccÞ~cp

. For the positive sign infront of the square root, � > 1=3, b > 0, and c >�1=16~c. By plugging Eq. (69) into Eq. (54), we find therelationship

�6

6b¼ � ¼ g0�

6

2) 1

3b¼ g0: (70)

On the other hand, when � < 1=3, then necessarily b < 0,c > 1=16j~cj, and

�1;2 ¼ � ¼ �2

8b< 0; (71)

and in this case we have no normalizable solutions.

However, when 4~cb ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1þ 16~ccÞ~cp

, we can havetwo further subcases, � > 1=3 and b < 0, but both of thesewill be rejected because they lead to a non-normalizablesolution. The other subcase is given by � < 1=3 and b > 0.Here we can get solutions provided c > 1=16j~cj.

We have two real distinct solutions,

�1¼�22~cbþ

ffiffiffiffi~�

p3ð1þ16~ccÞ and �2

2~cb�ffiffiffiffi~�

p3ð1þ16~ccÞ¼�2; (72)

when 4~cb >ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1þ 16~ccÞ~cp

. The general form of �1 and�2 is not very illuminating. It is more useful, even in thiscase, to consider particular choices of the parameters.

Subcase b � 0, c ¼ 0. With this combination of theconstants we get

�1 ¼ �22~cbþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2~cbÞ2 � 3~c=4

p3

and

�22~cb� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2~cbÞ2 � 3~c=4

p3

¼ �2:

(73)

If 1 b and � < 1=3, we can write

�1 ’ �2�4j~cjbþ ffiffiffiffiffiffiffiffi

3j~cjp6

and

�2 ’ �2�4j~cjb� ffiffiffiffiffiffiffiffi

3j~cjp6

:

(74)

Since �2 < 0, this solution will be discarded. By plugging�1 into Eq. (54) and taking linear terms in b, one gets

2

3

2ffiffiffiffiffiffiffiffi3j~cjp3

� j~cjb!�6 ¼ � ¼ g0�

6

2; (75)

which implies

g0 ¼ 4

3

2ffiffiffiffiffiffiffiffi3j~cjp3

� j~cjb!: (76)

On the other hand, when 4ffiffiffi~c

pb ffiffiffi

3p

, then necessarily� > 1=3, which means that �1 and �2 become

�1 ’ �24~cb

3and �2 ’ 0: (77)

By plugging �1 into Eq. (54), we find that

9�

144�6’ � 32

27b3~c2; (78)

which implies

g0 ¼ � 1024

27b3~c2: (79)

When c is large and negative and jcj b, ~� can beapproximated with

~� ’ ð2~cbÞ2 þ 12~c2jcj: (80)

Then the solutions in (72) become

�1 ’ ��2

b

24jcj þffiffiffiffiffiffiffiffi3jcjp

24jcj!

and

�2 ’ ��2

b

24jcj �ffiffiffiffiffiffiffiffi3jcjp

24jcj!:

(81)

�1 is always negative and therefore will be discarded,while �2 can be further reduced to

�2 ’ �2

24

ffiffiffiffiffiffi3

jcj

s: (82)

By plugging �2 into Eq. (54), we find

1

9

ffiffiffiffiffiffi3

jcj

s�6 ¼ � ¼ g0�

6

2; (83)

which implies

2

9

ffiffiffiffiffiffi3

jcj

s¼ g0: (84)

Subcase b c 1. In this approximation we find

~� ’ ð2~cbÞ2 � 12~c2c: (85)

Then the solutions in (72) become

�1 ’ �2�b

12c� 1

64b

�’ �2b

12cand �2 ’ �2

64b: (86)

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Both �1 and �2 can be accepted to estimate �. Byplugging �1 into Eq. (54), we find

�b

3c� 2b3

27c2

��6 ¼ � ¼ g0�

6

2; (87)

which implies

2b

3c� 4b3

27c2¼ g0: (88)

Instead, by plugging �2 into Eq. (54), we find

7�6

128b¼ � ¼ g0�

6

2; (89)

which implies

7

64b¼ g0: (90)

IV. CONCLUSIONS

In this paper, we have considered the recent proposal ofHorava to compute the cosmological constant in a modifiednoncovariant gravity theory at the Lifshitz point z ¼ 3. Wehave used the WDWequation as the backbone of all of ourcalculations, with the cosmological constant regarded as aneigenvalue [31]. We have analyzed the situation with andwithout the detailed balanced condition. A variationalapproach with Gaussian wave functions was used toperform practical calculations. The background geometryis a Friedmann-Robertson-Walker metric with k ¼ 1.Concerning the detailed balanced condition, we have foundan approximate eigenvalue not in the original configura-tion, but rather with a slight modification in the IR region.The main reason for this is related to the fact that theoriginal parameter �W cannot be considered as an eigen-value in this formulation. What we have found is that weonly have results which are in agreement with a bigcosmological constant, and therefore with Planck eraestimations. This means that the cosmological constantproblem in this approach is not solved. It is interesting tonote that only the dimensionless ratio �W=�

2 comes intoplay. As regards the case without the detailed balancedcondition, we obtain four main cases, described byEqs. (59), (70), (62), and (63):

(a) g0 is large and determined by the set of couplingconstants ðg2; g3Þ. For example, it could be fine-tuned to Planck era values and therefore to the orderof 10120. This implies that b ¼ 6g2 þ g3 1. Thiscan be achieved if g2, g3 1 or 6g2 ’ �g3. Wheng2, g3 1 we fall into a perturbative regime, butwhen 6g2 ’ �g3 this could not be the case. In this

respect, this version of HL theory and the versionwith the detailed balanced condition behave in thesame way except (eventually) for the smallness ofthe coupling constants g2 and g3.

(b) g0 is of the order of unity or less and determined bythe set of coupling constants ðg2; g3Þ. It can be fine-tuned to the values obtained from observation. Thisimplies that b ¼ 6g2 þ g3 ’ 1, which means thatthe set ðg2; g3Þ can be in the perturbative region.For example, it is sufficient to take the couple(g2 ¼ 1=12, g3 ¼ 1=2). Indeed, since in case(a) we started with g0 ’ 10120, we do not need toobtain g0 ’ 10�120 as a final result.

(c) g0 is large and determined by the set of couplingconstants ðg4; g5; g6Þ in Eq. (62). It can be fine-tunedto be of the order of 10120. This can be realized whenthe combination c ¼ 9g4 þ 3g5 þ g6 < 0, and evenif 1þ 16~cc � 0 nothing prevents us from consider-ing the case in which 1þ 16~cc is small. Since weare dealing with three coupling constants, it is nottrivial to have a discussion similar to the cases in(a) and (b), as we have more combinations.However, when one of the constants vanishes onecan repeat the same analysis as case (a), and onediscovers three other sub-subcases

(d) g0 is small and determined by the set of couplingconstants ðg4; g5; g6Þ in Eq. (63). This can be

achieved if c ! 1; then, g0 < 1. Of course, even in

this case, we do not need to obtain g0 ’ 10�120 as a

final result. The same considerations as in case

(c) can be applied here. Nevertheless, to have a

better comparison with observation one should

adopt the units in which ½dx� ¼ ½dt�, where the

speed of light is equal to 1 [29,30]. We can observe

that a big difference between HL theory with and

without the balanced condition is that the latter

admits a possible transition from big values to small

values of the cosmological constant. However, it is

likely that an improvement could be obtained by

introducing a renormalization group equation for

the various coupling constants, and eventually a

running�. Another improvement for having bounds

on parameters can be extracted from the equation of

state p ¼ ! with ! ¼ �1. Using the Gaussian

wave function’s value of the exponent � obtained

by the variational procedure into the equation of

state, we can extract more information about the

coupling constants. Even if a further and deeper

investigation is needed, it is interesting to observe

that the set of coupling constants ðg2; g3Þ and the

set ðg4; g5; g6Þ seem to be disconnected from each

other when we estimate g0. It goes without sayingthat the analysis has been done without a matter

field, and that adding a matter field could improve

the final result.

COSMOLOGICAL CONSTANT AS AN EIGENVALUE OF THE . . . PHYSICAL REVIEW D 86, 123507 (2012)

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International Journal of Modern Physics: Conference

Series Vol. 14: Proceedings of the 10th Quantum Field

Theory Under the Influence of External Conditions

(QFEXT11), edited by M. Asorey, M. Bordag, and

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p. 326, http://www.worldscientific.com/doi/pdf/10.1142/

S2010194512007441.

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