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50
UDC 530.1; 539.1
Big bounce and inflation from gravitational four-fermion interaction
I. TI. Khriplovich
Budker institute of Nuclear Physics, Novosibirsk, 630090, Russia. 11 Lavrentjev pr., Russia
E-mail: khriplovich@inp.nsk.su
If the torsion exists (i.e. if the ChristofFel symbols are not symmetric), it induces the four-fermion gravitational interaction. This interaction is dominating below the Planck scale. Its regular, axial-axial part by itself does not stop the gravitational compression. However, the anomalous, vector-vector interaction results in a natural way both in big bounce and in inflation.
Keywords: Planck scale, gravitational four-fermion interaction, big bounce, inBation.
1. The observation that, in the presence of torsion, the interaction of fermions with gravity results in the four-fermion interaction of axial currents, goes back at least to [1,2].
We start our discussion of the four-fermion gravitational interaction with the analysis of its most general form.
As has been demonstrated in [3], the common action for the gravitational field can be generalized as follows:
S' = - less I d4x (-e) (RJ -1; <!>
here and below G is the Newton gravitational constant,
I, J = 0,1, 2, 3 (and M, N below) are internal Lorentz indices, ^, v = 0,1,2, 3 are space-time in dices, is the tetrad field, e is its determinant, and e^ is the object inverse to eU. The curvature tensor is
= -d+dvwIJM+wIKM ukJv-uIKv ^kJm,
here wJ is the connection. The first term in equation
(1) is in fact the common action of the gravitational field written in tetrad components.
The second term in equation (1), that with the dual curvature tensor
RiJ = 1 Rkl
Ruv = 2 £KLr^v ,
does not vanish in the presence of spinning particles generating torsion.
As to the so-called Barbero-Immirzi parameter 7, its numerical value
Y = 0.274 (2)
was obtained for the first time in [4], as the solution of the "secular" equation
CO ______
E (2j + 1)e-2nVj(j+1) = 1. (3)
j=1/2
Interaction of fermions with gravity results, in the presence of non-propagating torsion, in the four-fermion action which looks as follows:
3 Y 2 r ________
Sff = 2nGY2 + 1 J d4xV'Zg[niJaIA'J
+ aniJ(V1AJ + A1 VJ) - a2 niJV1VJ]; (4)
7
here and below g is the determinant of the metric tensor, A1 mid V1 are the total axial and vector neutral currents, respectively:
Al = E A = E V’a Y5 Y1 ^a ;
a a
V1 = E Vi = E ^a y1 ^a; (5)
aa
the sums over a in (5) extend over all sorts of elementary fermions with spin 1/2.
AA
(up to a factor) to the action derived long ago in [1,2]. Then, this contribution was obtained in the limit
Y ^ to in [5] (when comparing the corresponding result from [5] with (4), one should note that our convention nIJ = diag(1, —1, —1, —1) differs in sign from that
AA
given in (4), was derived in [6].
VA VV
in [7] as follows. The common action for fermions in gravitational field
Sf = J d4x V—g ^[ Y1 eu iVM^ — i Vm^y 1 eu^] (6) can be generalized to:
Sf = J d4x V—g1 [(1 — ia) Y1 eu iVM^
— (1 + ia) i Vu^Y1 e^]; (7)
here Vu = — 4 wIJu yIYJ j wIJu is the connection.
a
quence, generating only a total derivative, if the theory is torsion free. However, in the presence of torsion this constant gets operative. In particular, as demonstrated VA VV itational four-fermion interaction (4).
Simple dimensional arguments demonstrate that interaction (4), being proportional to the Newton conG
gets essential and dominates over the common interactions only at very high densities, i.e. on the Planck scale and below it.
The list of papers where the gravitational four-fermion interaction is discussed in connection with cosmology, is too lengthy for this short note. Therefore, I refer here only to the most recent one [8], with a quite extensive list of references. However, in all those papers the discussion is confined to the analysis of the axial-axial interaction.
VA VV
AA
argument could be as follows. Under these extreme conditions, the number densities of both fermions and antifermions increase, due to the pair creation, but the total vector current density V1 remains intact.
By itself, this is correct. However, the analogous line of reasoning applies to the axial current density A1. It is in fact the difference of the left-handed and right-handed axial currents: A1 = AL — AR. There is no reason to expect that this difference changes with temperature and/ or pressure.
AV
2. Let us consider the energy-momentum tensor (EMT) Tuv generated by action (4). Therein, the expression in square brackets has no explicit dependence at all either on the metric tensor, or on its derivatives. The metric tensor enters action Sff via a/—g only, so that the corresponding EMT is given by relation
2 V g Sff •
Thus, with identity
1 dV—9 = _1 V—g dguv = 2 9uv ,
we arrive at the following expression for the EMT:
T = -—G y2
= 2 G 72 + 1
[nuA A
+ - niJ (V1AJ + A1 VJ) - a2 nijV1 VJ].
The nonvanishing components of this expression, written in the locally inertial frame, are energy density
Too = £ ^d pressure T11 = T22 = T33 = p (for the correspondence between £ p and EMT components see [9], §35).
Thus, the equation of state is here
£ = — p
Y
.2
p2 [(3 — 11 Z) — a2(60 — 28 Z)]. (11)
(8)
= — — G 2
48 y 2 + 1
In this expression, p is the total density of fermions and antifermions, and Z = < ffaff& > is the average value of the product of corresponding ^-matrices, presumably universal for any a Mid b. Since the number of sorts of fermions and antifermions is large, one can neglect here for numerical reasons the contributions of exchange and annihilation contributions, as well as the fact that if ff^d ab refer to the same particle, < ffatrb >= 3. The parameter Z j just by its physical meaning, in principle can vary in the interval from 0 (which corresponds to the complete thermal incoherence or to the antiferromagnetic ordering) to 1 (which corresponds to the complete ferromagnetic ordering).
It is only natural that after the performed averaging over all momenta orientations, the P-odd contributions VA £ p
3. Though for a ~ 1 the VV interaction dominates numerically the result (11), it is instructive to start
a=0
least, for the comparison with the previous investigations. We note in particular that, according to (11), the contribution of the gravitational spin-spin interaction to energy density is positive, i.e. the discussed interaction is repulsive for fermions with aligned spins. This our conclusion agrees with that made long ago
in [5].
To simplify the discussion, we confine from now on to the region below the Planck scale, so that one can neglect effects due to the common fermionic EMT, which originates from the Dirac Lagrangian.
A reasonable dimensional estimate for the temperature t of the discussed medium is
(12)
(9)
(here and below mPl is the Planck mass). This temperature is roughly on the same order of magnitude as the energy scale w of the discussed interaction
(13)
(10)
Numerically, however, t and w can differ essentially. Both options, t > w and t < w, are conceivable.
If the temperature is sufficiently high, t ^ w, it destroys the spin-spin correlations in formula (11). In the opposite limit, when t C w, the energy density (11) is minimized by the antiferromagnetic spin ordering.
Thus, in both these limiting cases the energy density and pressure simplify to
y2
16 y2 + 1
Gp
2.
y2
16 y2 + 1
Gp2 . (14)
£
p2 p
p
p
of the fermionic matter, and thus make the discussed system unstable.
A curious phenomenon could be possible if initially the temperature is sufficiently small, t < w, so that equations (14) hold. Then the matter starts compressing, its temperature increases, and the correlator Z = < ffaffb > could arise. When (and if!) Z exceeds its critical value Zcr = 3/11, the compression changes to expansion. Thus, we would arrive in this case at the big bounce situation.
However, I am not aware of any physical mechanism which could result here in the transition from the initial antiferromagnetic ordering to the ferromagnetic one with positive Z = < ffaff& >•
Here one should mention also quite popular idea according to which the gravitational collapse can be stopped by a positive spin-spin contribution to the energy. However, how such spin-spin correlation could survive under the discussed extremal conditions? The naive classical arguments do not look appropriate in this case.
4. Let us come back now to equation (11). In this
VV
action, the big bounce takes place if the energy density (11) is positive (and correspondingly, the pressure is negative). In other words, the anomalous, VV interaction results in big bounce under the condition
a2 >
3 — 11Z
4(15 — 7Z) .
(15)
2 1
a2 > — . “ 20
The next remark refers to the spin-spin contribution to energy density (11)
£C =
Y
48 y2 + 1
Gp2 (28a2 — 11) Z.
5. Equation (11) could have serious cosmological implications. It is rather well-known that this equation of state £ = —p results in the exponential expansion of the Universe. Let us consider in this connection our problem.
We assume that the Universe is homogeneous and isotropic, and thus is described by the well-known Friedmann-Robertson-Walker (FRW) metric
ds2
dt2 — a(t)2[dr2 + f (r)(dtf2 + sin2 tfd^2)]; (18)
here f (r) depends on the topology of the Universe as a whole:
f (r) = r2, sin2 r, sinh2 r
for the spatial flat, closed, and open Universe, respectively. As to the function a(t), it depends on the equation of state of the matter.
The Einstein equations for the FRW metric (18) reduce to
H2 = ( -
a
8nG£ k
a
a
4nG
“3“
3
(£ + 3p ).
-j2
(19)
(20)
They are supplemented by the covariant continuity equation, which can be written as follows:
£ + 3 H (£ + p ) = 0; H = — .
(21)
The energy-momentum tensor (11) dominates below the Planck scale. Since it results in £ = —p, equation (21) reduces to
£ = 0, or £ = const. (22)
It complies with EMT (10), which can be rewritten
Z
simplifies to
(16)
as
T£ with
-4g-
Y 2
48 y 2 + 1
p2 [3 — 11Z — a2 (60 — 28Z )].
(23)
(24)
As long as this contribution to the total EMT dominates below the Planck scale, it should be conserved. Then, with = 0, we arrive at
(17) dv £ = 0.
(25)
It could result in the ferromagnetic ordering of spins if
a2 > 11/28
Gp
ature, both of which are on the order of magnitude of mpi
Thus, the energy density and pressure, £ = —p, are here both time-independent and coordinate-independent. As to equation (20), it simplifies now to
—
8nG£
const.
(26)
£
2
£
£>0
pansion law:
,------ I am grateful to D.I. Diakonov, A.D. Dolgov,
a ~ eXp(Ht), where H = ./8nG£ = const (27) A.A. Pomeransky, and A.S. Rudenko for useful discus-
3
(as usual, the second, exponentially small, solution of The investigation was supported in part by the Rus-eq. (26) is neglected here). sian Ministry of Science, by the Foundation for Ba-
Thus, the discussed gravitational four-fermion in- sic Research through Grant No. 11-02-00792-a, by
teraction, induced by torsion, results in the inflation the Federal Program "Personnel of Innovational Restarting below the Planck scale. sia" through Grant No. 14.740.11.0082, and by the
Somewhat more detailed discussion of the problem Grant of the Government of Russian Federation, No.
can be found in [10]. 11.G34.31.0047.
References
[1] Kibble T. W. B. 1961 J. Math. Phys. 2 212
[2] Rodichev V. I. 1961 J. Exp. Theor. Phys. 40 1469
[3] Holst S. 1996 Phys. Rev. D53 5966 [arxiv:9511026[gr-qc]]
[4] Khriplovich I. B. and Korkin R. V. 2002 J. Exp. Theor. Phys. 95 1 [arxiv:0112074[gr-qc]]
[5] Kerlick G. D. 1975 Phys. Rev. D12 3004
[6] Perez A. and Rovelli C. 2006 Phys. Rev. D73 044013 [arxiv:0505081[gr-qc]]
[7] Freidel L., Minic D„ and Takeuchi T. 2005 Phys. Rev. D72 104002 [arxiv:0507253[hep-th]]
[8] de Berredo-Peixoto G., Freidel L., Shapiro I. L., and de Souza C. A. 2012 JCAP 06 017 [arxiv:12015423[gr-qc]].
[9] Landau L. D. and Lifshitz E. M. 1975 The Classical Theory of Fields (Butterworth - Heinemann)
[10] Khriplovich I. B. 2012 Int. J. Mod. Phys. D, in press [arxiv:12035875[gr-qc]]
Received 01.10.2012
if. В. Хриплович
БОЛЬШОЙ СКАЧОК И ИНФЛЯЦИЯ ВСЛЕДСТВИЕ ГРАВИТАЦИОННОГО ЧЕТЫРЕХ-ФЕРМИОННОГО ВЗАИМОДЕЙСТВИЯ
Если существуют торсионные поля (т.е. символы Кристоффеля несимметричны), то возникает гравитационное четырех-фермионное взаимодействие. Это взаимодействие является доминирующим ниже планковского уровня. Его нормальная аксиально-аксиальная часть сама не может остановить гравитационное сжатие. Однако аномальное вектор-векторное взаимодействие естественным образом приводит к большому скачку и инфляции.
Ключевые слова: планковский уровень, гравитационное четырех-фермионное взаимодействие, большой скачок, инфляция.
Хриплович И. Б., доктор физико-математических наук, профессор.
Институт ядерной физики им. Г.И. Вудкера Сибирского отделения Российской академии наук.
Просп. Лаврентьева, 11, Новосибирск, Россия, 630090.
E-mail: khriplovich@inp.nsk.ru
