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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 21. Выпуск 2.
УДК 524.83 DOI 10.22405/2226-8383-2020-21-2-109-138
О нелокальной модифицированной гравитации1
И. Димитриевич, Б. Драгович, 3. Ракич, Е. Станкович
Димитриевич Иван — доктор наук, профессор, математический факультет Белградского университета (г. Белград, Сербия). e-mail: ivand<Smatf.bg. ас.rs
Драгович Бранко — доктор наук, профессор, Институт физики Белградского университета; Математический институт САНУ (г. Белград, Сербия). e-mail: dragovichMipb.ac.rs
Ракич Зоран — доктор наук, профессор, математический факультет Белградского университета (г. Белград, Сербия). e-mail: zrakic@matf.bg.ас.rs
Станкович Елена — доктор наук, профессор, педагогический факультет Белградского университета (г. Белград, Сербия). e-mail: jelenagg@gmail.com
Аннотация
За последние сто лет многие существенные гравитационные явления были предсказаны и обнаружены Общей теорией относительности (GR), которая до сих пор остается лучшей теорией гравитации. Тем не менее, из-за великих наблюдательных открытий 20-го века некоторые (квантовые) теоретические и (астрофизические и космологические) феноменологические трудности современной гравитации были мотивацией для поиска более общей теории гравитации, чем ОТО. В результате были рассмотрены многие модификации ТО. Одним из многообещающих недавних исследований является нелокальная модифицированная гравитация. В этой статье мы представляем обзор некоторых нелокальных гравитационных моделей с их точными космологическими решениями, в которых нелокальность выражается аналитической функцией от оператора Даламбера — Бельтрами □. Некоторые из полученных решений содержат эффекты, которые обычно присваиваются темной материи и темной энергии.
Ключевые слова: Нелокальная измененная гравитация, точные космологические решения, темная материя, темная энергия, общая теория относительности и конечная производная гравитация.
Библиография: 63 названия. Для цитирования:
И. Димитриевич, Б. Драгович, 3. Ракич, Е. Станкович. О нелокальной модифицированной гравитации // Чебышевский сборник, 2020, т. 21, вып. 2, с. 109-138.
1Эта статья частично поддержана сербским Министерством образования и науки, проект № 174012.
CHEBYSHEVSKII SBORNIK Vol. 21. No. 2.
UDC 524.83 DOI 10.22405/2226-8383-2020-21-2-109-138
On Nonlocal Modified Gravity
I. Dimitrijevic, B. Dragovich, Z. Rakic, J. Stankovic
Dimitrijevic Ivan — Doctor of Sciences, Professor, Faculty of Mathematics, University of Belgrade (Belgrade, Serbia). e-mail: ivand@matf.bg. ac.rs
Dragovich Branko — Doctor of Sciences, Professor, Institute of Physics, University of Belgrade; Mathematical Institute SANU (Belgrade, Serbia). e-mail: dragovich,@ipb. ac.rs
Rakic Zoran — Doctor of Sciences, Professor, Faculty of Mathematics, University of Belgrade (Belgrade, Serbia). e-mail: zrakic@matf.bg.ac.rs
Stankovic Jelena — Doctor of Sciences, Professor, Teacher Education Faculty, University of Belgrade (Belgrade, Serbia). e-mail: jelenagg@gmail.com
Abstract
In the last hundred years many significant gravitational phenomena have been predicted and discovered by General Relativity (GR), which is still the best theory of gravity. Nevertheless, due to the great observational discoveries of 20th century some (quantum) theoretical and (astrophysical and cosmological) phenomenological difficulties of modern gravity have been motivation to search more general theory of gravity than GR. As a result, many modifications of GR have been considered. One of promising recent investigations is Nonlocal Modified Gravity. In this article we present a review of some nonlocal gravity models with their exact cosmological solutions, in which nonlocality is expressed by an analytic function of the d'Alembert-Beltrami operator □. Some of obtained solutions contain effects which are usually-assigned to the dark matter and dark energy.
Keywords: Nonlocal modi
ed gravity, exact cosmological solutions, dark matter, dark energy, general relativity, in
nite derivative gravity.
Bibliography: 63 titles. For citation:
I. Dimitrijevic, B. Dragovich, Z. Rakic, J. Stankovic, 2020, "On Nonlocal Modified Gravity" , Chebyshevskii sbornik, vol. 21, no. 2, pp. 109-138.
1. Introduction
In 2015, General relativity (GR), known also as Einstein theory of gravity, celebrated its first hundred years is considered as one of the most profound and beautiful physical theories with great phenomenological achievements and nice theoretical properties. GR has important astrophysical implications predicting existence of black holes, gravitational redshift, gravitational lensing and
gravitational waves2, it has been tested and quite well confirmed in the Solar system, and it has been also used as a theoretical laboratory for gravitational investigations at other spacetime scales.
Despite of just mentioned phenomenological successes and many nice theoretical properties, GR is not complete theory of gravity. For example, attempts to quantize GR lead to the problem of nonrenormalizabilitv. GR also contains singularities like the Big Bang and black holes. In cosmology, it predicts existence of about 95% of additional new kind of matter, which makes dark side of the universe. Namely, if GR is the gravity theory for the universe as a whole and if the universe is homogeneous and isotropic with the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric at the cosmic scale, then it contains about 68% of dark energy, 27% of dark matter, 5%
so far seen in particle physics. If a physical theory contains singularities then it has to be modified in that domain. Because of that, there are many attempts to modify General relativity. Motivations for its modification usually come from quantum gravity, string theory, astrophysics and cosmology (for a review, see [55, 58, 15, 16, 57]). We are mainly interested in cosmological reasons to modify GR, i.e. to find such extension of Einstein theory of gravity which will not contain the Big Bang singularity and offer another possible description of the universe acceleration and large velocities in galaxies instead of mysterious dark energy and dark matter. In the case that dark energy and dark matter really exist it is still interesting to know is there a modified gravity which can imitate the same or similar effects.
Any well founded modification of the Einstein theory of gravity should be a generalization of the general theory of relativity, and consequently it should be verified at least on the dynamics of the Solar system. On the mathematical level it should be formulated within the pseudo-Riemannian geometry in terms of covariant quantities and take into account equivalence of the inertial and gravitational mass. It means that the Ricci scalar R in gravity Lagrangian Cg of the EinsteinHilbert action should be replaced by an appropriate function which may contain not only R but also some scalar covariant constructions (as norms of Ricci and curvature tensor, etc.) which are possible in the pseudo-Riemannian geometry. Since there are infinitely many possibilities for such functions (i.e. its modifications), and since so far there is no guiding theoretical principle which could make appropriate choice between all possibilities, our task is very complicated. In this context the Einstein-Hilbert action is the simplest one, i.e. it can be viewed as realization of the principle of simplicity in construction of Cg.
Modifications of GR were started a few years after its birth adding cosmological constant in the Lagrangian of Hilbert-Einstein action, later by adding Gauss-Bonnet invariant, replacing R with f (R) (/(R) modified gravity, see [45]) and recently nonlocal modifications, which are promising modern approaches towards more complete theory of gravity. Motivation for nonlocal modification of general relativity can be found in string theory which is nonlocal theory and contains gravity. We present here a review of our results on nonlocal gravity. In particular, we pay special attention to models in which nonlocalitv is expressed by an analytic function of the d'Alembert operator □ = dv like nonlocalitv in string theory.
In Section 2 we give some preliminaries on cosmology and mention a few different approaches to nonlocal modified gravity. Section 3 contains a general modified action with an analytic nonlocalitv and we derive the corresponding equations of motion for nonlocalitv of the form %(R) F(□) Q(R). In Sect. 4. we presented several different models of nonlocalitv which are special cases of general model from previous Section. In all models we found some cosmological solutions for appropriate scaling factor. We emphasize here the case when the nonlocalitv is given by %(R) = Q(R) = VR — 2 A with scaling factor of the form a(t) = At2 e 141 in the fiat universe. This model gives some effects usually attributed to the dark matter and dark energy. Finally, Sect. 5. is devoted to the perturbations over de Sitter background. We obtained some cosmological solutions and investigated stability of them.
2which were experimentally discovered in 2016. [1].
At the end we showed that equations of motion of gravitational waves in our nonlocal modified gravity coincide with the corresponding equations in GR.
2. Preliminaries 2.1. Metric
General theory of relativity, i.e. Einstein theory of gravity (EG) assumes that the universe is four dimensional homogeneous and isotropic pseudo-Riemannian manifold M with metric ) of signature (1, 3). There exist three types of homogeneous and isotropic simple connected spaces of dimension 3:
o flat space R3 (of curvature equal 0), o sphere S3 (of constant positive sectional curvature), o hyperbolic space H3 (of constant negative sectional curvature).
The universe is homogeneous and isotropic (observation data) manifold, the generic metric in these spaces is of the form (Friedmann-Lemaitre-Robertson-Walker metric, (FLRW)):
ds2 = -dt2 + a2(t)( i + r2dd2 + r2 sin2 , k e{-1, 0,1}, (1)
where a(t) is a scaling factor which describes the evolution (in time) of the universe and parameter k describes the curvature of the space. In the FLRW metric the Ricci scalar (scalar curvature) is
^('(io? k\
R = 6 - + -2 + 2
\a a2 a2 J
and
□ = -9t2 - 3Hdt. (3)
where H = a/a is the Hubble parameter, which describes the expansion of the universe. We use natural system of units in which speed of light is c = 1.
2.2. Einstein-Hilbert action.
GR is based on Einstein-Hilbert action:
5 = i (T^ + (4)
,/M V 16 -KG )
where E is scalar curvature of M, g = det(g^) is determinant of metric tensor, A is cosmological constant and Cm is Lagrangian of matter.
By variation of the action S with respect to we obtain Einstein equations of motion:
R^v RgßV + A gßV = 8 nGTßV, (5)
where is ^te energy momentum tensor, g^v is metric tensor, is tensor and R is scalar curvature.
2.3. Friedman equations.
The energy momentum tensor for an ideal fluid (matter in cosmology) is given by
T = d\ag(-pgoo,gnp,g22P,g33P), (6)
where p is energy density and p is pressure. Using the conservation law we get
0 = V^ = —p — 3^(p + p). (7)
Since in the cosmology holds p = wp, where w is usually a constant, we have that equation (7) has general solution p = Ca-3(1+W\
The basic types of matter in the universe are: cosmic dust- w = 0( pm = C a-3), and radiation - w = 1/3 (pr = C a-4 ). Nowadays, the ratio ^ & 106 , in the early universe radiation was dominant. Introducing the cosmological constant A (= 0) is equivalent to the (dark) energy of vacuum. From Einstein equation one can find energy-momentum tensor for vacuum,
= - sAg9*"', (8)
and see that p = — p = It is clear that energy density is a constant which does not depend a.
energy of vacuum will become dominant to the energy densities of matter and radiation. Now, Einstein equation (5) implies Friedmann equations
a 4xG. , A / a \2 8nG k A ^ = —■—(p+3P) + a, l^aj =—?—-2 + Ar (»)
There are several of cosmological parameters which describe the state of the universe, the most important are: Hubble parameter H; the deceleration parameter
aa a
q = — ^ , 10
a 2
which measures the cosmic acceleration of the universe; the parameters of density
^ 8KG , 3H2 . . . , , .
^ = q ui P = P/Pc, where pc = -—is a critical energy density, (11)
3 H- 8 G
= pi/pc-, and pi are mass densities of matter, radiation and dark energy, and others. From Friedmann equations (9) follows,
k
a —1 = wa < (12)
and the sign of constant k is determined by Q, and it shows which of three FLRW metrics describes the universe. More precisely we have,
o Q < 1, p < pc, k = —1.
o Q = 1, p = pc, k = 0.
o Q > 1, p > pc, k = +1.
Let us mention some cosmological parameters obtained from Planck 2018 [63] which describe
A
o H0 = (67.40 ± 0.50) km/s/Mpc - Hubble parameter;
o Qm = 0.315 ± 0.007 - matter density parameter;
o Qa = 0.685 - A density parameter;
o to = (13.801 ± 0.024) ■ 109 vr - age of the universe;
o w0 = —1.03 ± 0.03 - ratio of pressure to energy density.
2.4. Nonlocal modified gravity.
As we mentioned in Section 1, GR has certain deficiencies, and should be modified. One of the most prominent approaches is nonlocal modification.
A nonlocal modified gravity model contains an infinite number of spacetime derivatives of the d'Alembert operator □, in the form of a power series expansion of it. In this article, we are mainly interested in nonlocalitv expressed in the form of an analytic function F(□) = ^0 fn^n, where coefficients fn should be determined from various theoretical and phenomenological conditions. Some conditions are related to the absence of tachvons and ghosts.
Here, it is worth to mention some other interesting approaches to the nonlocal gravity, as approaches containing □-1 (see e.g., [19, 18, 61, 56, 41, 62, 38, 39, 42, 43, 5, 53] and references therein), nonlocal models with power of the inverse d'Alembert operator, i. e. with □~n, which are proposed to explain the late time cosmic acceleration without dark energy. Such models have the form
5 = 16^/ (R + Lnl) d4x, (13)
where two typical examples are: Lnl = Rf (□-1E) (see a review [55, 19] and references therein), and Lnl = — 6m2R^~2R (see a review [31] and references therein).
<x
Nonlocal models with F(^) = ^ /«□"" are mainly considered to improve general relativity
n=0
in its ultraviolet region, unlike models with □ -1 and □~2 which intend to modify gravity in its infrared sector. It may happen that there will be more than one modification of general relativity, which are valid at the different scales. Namely, any physical theory has a domain of validity, which depends on some conditions, including spatial scale and complexity of the system. It is natural that validity of general relativity is also restricted. At very short and very large cosmic distances may act different gravity theories.
Some other aspects of nonlocal gravity models have been considered, see e.g. [14, 11, 12, 54, 13, 31, 46] and references therein.
Our motivation to modify gravity in an analytic nonlocal way comes mainly from string theory, in particular from string field theory (see the very original effort in this direction in [2]) and p-adic string theory [3, 4, 10, 34, 35, 36, 60]. Since strings are one-dimensional extended objects, their field theory description contains spacetime nonlocalitv expressed by some exponential functions
□.
At classical level analytic non-local gravity has proven to alleviate the singularity of the Blackhole type because the Newtonian potential appears regular (tending to a constant) on a universal basis at the origin [37, 8, 6]. Also there was significant success in constructing classically stable solution for the cosmological bounce [8, 9, 44, 47, 50].
Analysis of perturbations revealed a natural ability of analytic non-local gravities to accommodate inflationary models. In particular, the Starobinskv inflation was studied in details and new predictions for the observable parameters were made [17, 49]. Moreover, in the quantum sector infinite derivative gravity theories improve renormalization, see e.g. while the unitaritv is
still preserved [51, 52, 49] (note that just a local quadratic curvature gravity was proven to be renormalizable while being non-unitarv [59]). Later, we shall also investigate some conditions on the analytic function F(□) = ^0 fnon, in order to escape unphvsical degrees of freedom like ghosts and tachvons, and to have good behavior in quantum sector (see [5, 6, 7, 37]).
3. The equations of motion
Models of nonlocal gravity which we mainly investigate are given by the following action
5 = f (R - 2A + n(R)T(^)g(R) + Cm) V—gd4x, (14)
IbirG JM
oo
where M is a pseudo-Riemannian manifold of signature (1, 3) with metric (), T(^) = ^ fn^n,
n 0
% and £ are differentiate functions of the scalar curvature R and A is cosmological constant, and
C m
Now, we will give an overview of deriving of the equations of motion (EOM) for the action
C m
[26, 29]. Firstly, we are starting with some technical lemmas which are proved by using standard variational calculus and Stokes theorem. Note that variations of the metric tensor elements and their first derivatives are zero on the boundary of manifold M, i.e. \qM = 0 Sndxx\qM = 0,n £ N.
M
Sg = gg" = -gg^tg^, (15)
= -1 a^u , (16)
Ku = -2 {9uaV"6gXa + 9"aVu5gXa - g^gupVx6ga^ , (17)
Lemma 2. The variation of Riemman tensor, Ricci tensor and scalar curvature satisfy the following relations
S Ra,pv = V.^ - V vsr^p, (18)
SR,v = V л5Г, „ - V (19)
l,v5g,v - K,vl
5R = R,v5g,u -К^Г, (20)
5V,V^ = V^Vv6iP - VлФ^ЛlV, (21)
where = V^Vv - g^v □, and where ^ is a scalar function Lemma 3. For every scalar function %(R) holds
i Hg,uu(абд^)V- d4x = / g,v(UH)5g,v V- d4x, (22)
jm jm
f HV ,Vv5g,vV-d4x = f V,VuH 6g,uV- d4x, (23)
jm jm
[ HK,v5g,vV~9 d4x = / H 5g,vV- d4x. (24)
jm jm
Lemma 4. Let %(R) and Q(R) be scalar functions such that 5QIqm = 0. Then for all n <E N one has
n— 1
1 x ' ' " '—¡l^ r-l"—1—If
i m□nGV—~9 d4x = 1 V / (□ lH, an—1—lg)5g»v V=9 d4x
■Jm 2 l=0 J m
+ / □nH5g^—d4x, (25)
Jm
where (A, B) = g^vVaAVaB + g^A^B — 2Vp,AVvB.
Theorem 1. Let % and Q be scalar functions of scalar curvature, then
i m(V—) d4x = —1 / g^vmg^vV— d4x, (26)
Jm 2 Jm
I mRV—d4x = i (R^v% — K^H)5g»vV— d4x, (27)
Jm J m
[ m(^(□)g)V—9 d4x = i (R^ —Ka,) (g'?(□)%) Sg^V—d4x jm jm
ro n— 1 1 x ^ » x ^ ' ~ '—¡l^ r-i'n— 1—If
+ 2 E ^ E S^(□ %, □n—1—lG)5gv V~g d4x, (28)
2 n=1 i=o J M
where (A, B) = g^vVaAVaB + g^A^B — 2Vp,AVvB.
S
the following auxiliary actions
50 = / (R — 2A) V— d4x, (29)
J M
51 = / H(R)F(□)£(R) V~9 d4x. (30)
Jm
So
5So = / (G^ + Ag^) d4x. (31)
Jm
S1
1
2 -/M
(32)
6S1 = —1 f g^v%( ( R)Sg^v V— d4x
2 J M
+ / ( R^uW — K^VW) Sg^ V— d4x Jm
ro ra— 1 „
+ 0 E ^ E / S,v(□1%(R), □n—1—lg(R))5g^vV—d4x,
2 ra=1 1=0 JM
where W = %!(R)F(^)g(R) + Q'(R)F(^)H(R). S1
5S1 = / f%(R№)0( R)5(V—g) + S%(R№)£( R) V=5 Jm v
+ %( R)S(.F(^)0( R)) V~9Sjd4x. (33)
All the terms in the previous formula are obtained by Theorem 1. In particular (26) yields
1
im 2 J M
Also, from equation (27) we get
/ Щ R)F(a)g( R)5(V—) d4x = - 1 i H(R)F (a)g (R) 5g»v d4x. (34) jm 2 jm
I S(H(R))F(^)g( R) V—d4x =1 %'(R)SR F(^)g(R) V—d4x jm jm
= t (r^v%'(R)F(□№(R) — K^ %(R)F(^)g(R))) 5g^V— d4x. (35)
Jm v j
The last term is calculated by (28).
/ %(R)5(F(^)g(R)) V—d4x Jm
= i (r^ug'(R)F(□)%(R) — K^ (g'(R)Fp)n(R))) V— d4x Jm v j
ro n—1 „
+ E fra E/ (□ %(R), □n—1—lg(R))5g»vV— d4x. (36)
ra=1 1=0 J M V ^
□
Theorem 2. Variation of the action (14) is equal to zero iff
0 = G^V = + - H(R)F(a)g(R) + RVW - K^VW) + , (37)
where
W = % (R)F(^)g (R) + g' (R)F(^)H(R), (38)
ro n—1
= £ fn£(□lH(R), □n—1—lg(R)) . (39)
n=1 1=0
□
Corollary 1. Under the assumptions of previous theorem holds:
(1) V»G^v = 0,
(2) Equations of motion (37) are invariant on the replacement of functions g and %.
Since, in the case of FLRW metric only two of four (diagonal) non-trivial equations are linearly independent trace and 00 component, the equation (37) is equivalent to the following system:
4.A — R — 2%(R) F(^) g (R) + RW + 3 □ W
ro n—1
+ E ^ E id^l%( R) (^□n—1—ig (R) + 2uiu(R)on—lg (r)) = 0, (40)
ra=1 l=0
G00 + Agcc - 2ffoon(R) F(a) g(R) + R00W - K00 W
те n—1
+ 2 E f™ E (900 gaf) daalH(R) dfian—1—lg(R)
n=1 1=0
- 280a1 П(R)d0an—1—eg( R) + 900 alH(R)an—eg(R)) = 0. (41)
4. Cosmological solutions of EOM
The search for a general solution of the scale factor a(t) of equations (40) and (41) is a very ambitious, and because of that we use the following ansatzs, see [22, 23, 24, 32, 27, 28, 45]:
• aR = rR + s, where r and s are constants.
• aR = qR2, where q is a constant.
• aR = qR3, where q is a constant.
• anR = cnRn+1, n > 1, where cn are constants.
• a(R + R0)m = p (R + R0)m, where m £ Q and R0,p are constants.
In fact these ansatzs make some constraints on possible solutions, but on the other hand they simplify formalism to find a particular solution.
We consider several models of nonlocal gravitv without matter which are described bv the action
(14),
5 = Tito JM (R - 2A + n(R) 7(a)9(R)) d'x, (42)
for the following choice of functions % and Q :
1. %( R) = R, Q(R) = R.
2. %( R) = R-\ g(R) = R.
3. %(R) = Rp, g(R) = Rq.
4. R = const.
5. %(R) = ( R + Ro)m, g(R) = (R + Ro)m.
6. %( R) = g(R) = VR - 2 A.
Let us mention that for all cases we consider different scale factors, and cases 1.. 2., 5. are not special case of 3. Now we will give short overview on the all six models.
4.1. Case 1: u(r) = r, q(r) = r.
In this model we use linear ansatz (fore more details see [23, 32, 28], with scaling factor a(t) = a0(aext + te-Xt), a0 > 0, X,a,T £ R and firstly, we have:
Lemma 6. (il) For n £ N, r,s £ R holds
anR = rn(R + -), n > 1, F(a)R = F(r)R + -(F(r) - fo).
(i2) For given scaling factor hold
X(aext -te-Xt) . 6 (2a20 X2 {a2e4tx + r2) +k e2tx)
H (t) = ^xt + , R(t) =
a R =
aext + te-xt a0 (ae2tx + r)2
12 X2e2ix (4 a2 X2ar -k)
a0 (ae2tx +r)2 (i3) aR = 2 X2R - 24X4, r = 2X2, s = -24X4.
Using previous Lemma, and EOM we have the following theorem.
Theorem 3. The scaling factor of the form a(t) = a0 (a ext + t e—Xt) is a solution of EOM in the following three cases:
Case 1. F (2 X2) = 0, F' (2 X2) = 0, Case 2. 3k = 4 a0 Лат.
1
8Л.
1 2
CaseS. F (2 X2) = —— + - f0, F' (2 X2) = 0, k = -4a0 Лат.
12 H 3
In all three cases holds 3 X2 = Л.
(43)
(44)
(45)
From previous theorem easily follows:
In the Case 1. there exist a solution for arbitrary a, t and a0
a(t) = ao(aeM + t e-M),
where F satisfies (43) and for arbitrary k = 0, This solution generalize the case a(t) =
= a0 cosh ^^ , which is obtained in [8].
In the Case 2. we obtained a family of solutions for arbitrary a = 0, a0 and arbitrary analytic function F
a(t) = a0[ аеxt +
3 k
4a0 Л а
The Case 3. also gives a family of solutions
a(t) = a^ aext -
k
-xt
4a0 Л a
F
Let us mention that in the Case 2. for k = 0 equation (44) and conditions (45) in Case 3.
a
see ([9]),
a\(t) = a0 ext, a2(t) = a0e
and that all solutions satisfies
a (t) = X2a(t) > 0.
-xt
4.2. Case 2: %(r) = r—\ q(r) = r.
In this model it is evident from action given by (42) that nonlocal term R—1F(^)R is invariant under the transformation R —> c R, c E R*. Let us remark that in this case f0 plays role of cosmological constant, and because of that we put A = 0. We are searching for the solutions of EOM with the scaling factor of the form a(t) = a0lt — t0la. For more details see [24, 32, 28]
Firstly we have a lemma.
Lemma 7. Let us consider the nonlocality of the form, %(R) = R—1, Q(R) = R, with the scaling factor a(t) = a0lt — t0la, and for A = 0. Then
(%1) R(t) = 6(a(2a — 1)(t — h)—2 + 4 (t — h)—2a).
(i2) □ R = q R2, where q depend on a.
R
condition a R = q R2, we find
a(2a - 1)(qa(2a - 1) - (a - 1))(i - to)-4 + ^(t - to)-4a
ok a0 (46)
+ Ok(1 - o + 6q(2a - 1))(t - h)-2a-2 = 0.
ULt0
1. k = 0,o = 0, q£ R, 4. k = -1, o = 1,q = 0,a0 = 1,
2. k = 0,o = 1, q£ R, , 2 k = 0 o = 0 = 0
3. k = 0, a = a = 2)
a— 1 a(2a— 1)
q = -M—h), 6- k = 0 a = U = 0.
In the cases (1), (2) and (4) we have R = 0 and therefore R 1 is not defined. The case (5) yields
a
The cases (3) and (6) from above lemma need additional investigations (for details, see [24]) which we will omit here, giving the following solutions.
Theorem 4. The scale factor a(t) = a0lt- t0la for A = 0 is a solution of EOM in the following cases:
(3) For k = 0, o = 0, o = 1 and £ N, the coefficients of F are
3o (2° - 1)
f0 = 0 h = - 2 (3 o - 2) ,
fn = 0 for 2 <n < 3 o - 1
n £ R n >
2
3 a — 1
(6) For k = 0, the coefficien ts of F are
fo = 0, h = - 4 , fn e R, n > 2,
where s = 6(1 + .
Let us remark here that in the both cases k = 0, a = 0,1/2, the obtained solutions have not as its background Minkowski space, and as a special case of (6) is the solution a(t) = It — i0| for k = - 1
4.3. 3. Case: %(r) = rp, q(r) = rq.
We consider the model given by %(R) = Rp, Q(R) = Rq with ^te scaling factor a(t) = a0 e, 7 £ R, fa details see [25, 20] Let us mention if 7 = 0 then M is a Minkowski space which is a solution of EOM (37) for A = 0. In the following analysis is independent of the sign of 7, and one can obtain models in which the universe is expanding (7 < 0) and collapsing (7 > 0). Using appropriate formulas, firstly we have technical lemma.
Lemma 8. In the model of nonlocality given by %(R) = Rp, Q(R) = Rq with metric 1 +2
a(t) = a0 e-121 , 7 E R, hold
^ t2
, • J С шли
1 1 , о 1
(1) H (t) = — 6-it, R(t) = 3-y (112 — 3), R00 = 4(l — R). (47)
(2) □RP = p^Rp — ^(4p — 5) 72RP-1 — 4p(p — 1)r3Rp-2. (48)
Equation (48) implies that linear space Vp = span{1, R, R2,..., Rp} is invariant under the □.
polynomial in R of degree p + q. If we consider only leading coefficients of both equations, forp = q we obtain an linearly dependent system
pF (qj)( q — p + 2)+ qF (pj)( q — p — 2) = 0, (49)
(—Q — 1 p(q — P))F (qi) + (—2q(q — p) + q)F(pi) = 0. (50)
Similarly, in the case when p = q we will obtain linearly dependent system. So, we have the following theorem.
Th
eorem 5. Let us consider nonlocality given by "H.(R) = Rp, Q(R) = Rq with scaling factor a(t) = a0 e-12f2, 7 e R. Then:
(11) for any p,q E N trace and 00 equation are equivalent.
(12) The trace equation is of polynomial type of degree p + q in R, with coefficients depending on f0 = F(0), F(7), ..., F(P1), F'(7), ..., F'(q7).
(13) for p = q = 1, trace equation is satisfied iffj = —12A, F' (7) = 0 and f0 = 3 — 8F(j). In this case system has infinitely many solutions.
Remark 1. 1. In this model, for the most simple case p = q = 1, the scaling factor of the form a(t) = a0eA 1 , firstly was considered by Koshelev and Vernov in the paper [44J-
2. The exact solutions, for 1 < q < p < 4, are found by I. Dimitrijevic in his PhD thesis. It is
4.4. Case 4: r = r0 = const.
In this case we do not have any condition on functions % and Q, for details see [26, 27, 28, 21]. R = R0
6(a + (a )2 + 4) = R0. (51)
a a a
After the change of variable b(t) = a2(t) we obtain a second order linear differential equation with constant coefficients
3b — R0b = —6k. (52)
Depending on the sign of R0 there exist the following solutions for b(t),
6 k / Ro < / Ro <
R0 > 0, b(t) = ^~+aev^ + te V R0
R0 = 0, b(t) = —k t2 + at + t, (53)
™ w ч 6k l-Ro l-Ro
Ro < 0, b(t) = — +a cos у—^t + т sm у1.
122
И. Димитриевич, Б. Драгович, 3. Ракич, Е. Станкович
If we now replace R = Ro into trace equation (40) and 00 equation (41), we obtain a system of equation defined by the coefficient fo
—2 U + RoW = Ro - 4 A, 1u + Roo W =A -Goo, (54)
where U = fo H(Ro) g (Ro) and W = fo H(R) g (R)) |r=r0.
U
( Ro + 4 Roo) ( W + 1) = 0. (55)
In the case when Ro + 4Roo = 0, the following conditions on the parameters a and r hold:
Ro > 0, 9k2 = RlaT,
Ro = 0, a2 + 4kr = 0, (56)
2
o
Ro < 0, 36 k2 = R?0 (a2 + т2).
Finally, we get the following theorem.
Theorem 6. Let, R = Ro = constant. Then, we have solutions in the following cases:
1. If Ro > 0 then for k = 0 there is a solution with constant Hubble parameter, for k = +1 the solution is a(t) = ^R cosh 2 (^^ft + and for k = —1 it is a(t) =
sinh § ft +
where a + t = R cosh p and a — t = R^ sinh p
2. If Ro = 0 then for k = 0 the solution is a(t) = yfr = сonst and for k = —1 the solution is a(t) = It + §
3. If R0 < 0 then for k = —1 the solution is a(t) = ^ —§ cos 1 — W^t — pj
where
a = -W6 cos p and т = -в66 sin p.
Bo ^^ tt"u ' - Bo
PROOF. The proof directly follows from (53) and conditions (56), for more details SGG a
Remark 2. The second case of the equation (55) gives the solution
W = -1, U = 2A -R0. (57)
0
d
%(R0) a( R0) - ( R0 - 2A) — (%(R) 0(R)) lR=Ro =0. (58)
4.5. Case 5: %(r) = (r + r0)m, q(r) = (r + r0)m.
Here we consider the action (14) given by % = Q = ( R + R0)m, where R0 and m are real
a( )
a(t)=A tne-?2t2, (59)
for more details see [28, 29]
Let us consider the ansatz in the form
n( R + Ro)m = r(R + Ro)m + s, (60)
where R0, r, s, m are n real constants. In the case when s = 0, the anzatz gives the following system of equations:
0 = -648mn2(2n - 1)2(2m - 3n + 1),
0 = —324n(2n — 1) (—7m + 6jmn2 — 4jmn — mnR0 + mR0 + 2n2r — nr) , 0 = 18n(2n — 1) (8j2m2 — 2m + 12j2mn — 3jmR0 + 24jnr + 6jr — 6rR0) , 0 = 3m — 24j3mn2 — 14j3mn + 6j2mnR0 + 2j 2mR0 + 72j2 n2r + 12j2nr
— 24jnrRo + 3j2r — 6jrRo + SrRfi, 0 = — j2 (4j2m2 + j2m + 18j2mn — 3jmRo — 24jnr — 6jr + 6rR0) , 0 = — j4( r — 7m). This system has five solutions:
1. r = mj, n = 0, R0 = 7, m = 2
2. r = mj, n = 0, R0 = -3, m = 1
3. r = mj, n = 2, Ro = 37, m = 1
4. r = mj, n = I R0 = 37, m = — 4
5. r = mj, n = 2m3+1, R0 = 337, m = 2.
The case 2. is considered in the Subsection 3 and case 3. is known and considered in ([33]). Let us consider now the case 1: n = 0, m = 1, then the anzatz is reduced to DyR+7 = = IjVR + 7, and we have the following con sequence: T (O)^R + 7 = T (.
R
in T(2) Mid T' (2): * *
2 2 2 3
j2 j 2 j2 j 3 j
7 + 4A — TT(2) — lT'(2) = ° ^ — Ttnp + l= ci)
2
—A + 2 T(7) + T2 T'(7) = 0, 72 + 72 T(7) — 73 T'(7) = 0. (62)
The solutions of these systems are the same for 7 = —2 A and equal to:
T( 2) = —1, T' (T) = 0. (63)
It is clear from (63) that considered case is a generalization of the Case 3. Here we allow that p, q are rational numbers. In this case we have p + q = 1, and similarly as in Theorem 5 there exist a unique solution in T(—) and T' (2 )■
Taking the similar analysis it was shown that in the Case 4. there are no solutions which satisfy EOM.
Finally, in the Case 5. the parameters are: r = —¡, n = |, R0 = 77, m = 2- Since, the action is the same as in the 3. and similar calculations, using trace and 00 equations give
T( T) = —1, T' (2) = 0, A = — 6 7. (64)
Let us remark that in (64) the solution could be given in terms of F(f) and F'(f) with a constrain which connects cosmological constant A, and 7. Similar case was considered in the Case 3 for p = q = l.
4.6. Case 5: h(r) = q(r) = /r—2k.
In this nonlocal model (for more details see [29, 28, 30]) we can rewrite the action (14) in more compact form
S = f /R — 2A F(□) /R — 2A d4x, (65)
16-^G J
<x
where F(□) = 1 + ^ fn □n.
n=1
The related Friedmann equations (9) are
a 4ttG , 0 , A a2 + k 8ttG A
H = ~aG(' + ap)+3 • ^^ = HTp + A • (66)
where p and p are analogs of the energy density and pressure of the dark side of the universe, respectively. Denote the corresponding equation of state as p(t) = w(t) p(t).
In this case we consider several subcases defining by different scaling factors. We mention three of them, and emphasize the first one since this case is the most important.
4.6.1. Cosmological solution a(t) = At3 en12 , k = 0.
Theorem 7. In the case of nonlocality %(R) = Q( R) = y/R — 2 A with scaling factor of the form a(t) = At2 ei4i2 and k = 0 holds:
4 22 12 2 1
R(t) = -t-2 + 272A + -A2t2, H (t) = -t-1 + -At. (67)
There exists a cosmological solution for ansatz □VR — 2A = — 7A\/R — 2A, and conditions
3
^ — —1, F'{ — 7A)=0 • A = 0.
Let us remark that in this case we have,
R00 = 33 i-2 — A — -|A2i2 , G00 = 3 i-2 + -A + |A2i2 , (68)
and
(h+98A2t2—i-A) • p(()=— <2—1). <69>
This cosmological solution for a(t) = A t2 en1 can be viewed as a product of ¿3 factor, related to the matter dominated case in Einstein's gravity, and en1 which is related to an acceleration. Moreover, the Hubble parameter consists of two terms: ft-1 is just H(t) in Einstein's theory
term 7At corresponds to an acceleration for A > 0, which is dominant role for larger times. Time dependent expansion acceleration is given by
2 A A2
a (t) = ( — ^t-2 + A + A/Wi). (70)
Also, according to expressions (69) follows that w(t) ^ —1 when t ^ what corresponds to an analog of A dark energy dominance in the standard cosmological model. Therefore, one can say that nonlocal gravity model (65) with cosmological solution a(t) = Aei41 describes some effects usually attributed to the dark matter and dark energy. This solution is invariant under transformation t ^ —t and singular at cosmic time t = 0. Namelv, R(t), H(t) and p(t) tend to when t ^ 0, while p(0) is finite.
Taking the above Planck results for t0 and H0 in the second formula of (67) one obtains A = 1,05 ■ 10-35 s-2 (in c = 1 units). This is close to A = 0.98 ■ 10"35s-2 calculated by standard formula A = 3 H^Qa- From the same formula (67) one can also find time (tm) for which the Hubble parameter has minimum value Hm, i.e. tm = 21,1 ■ 109 yr and Hm = 61, 72 km/s/Mpc.
From (70) one can find that beginning of the universe expansion acceleration was at ta = 7, 84 ■ 109 yr, or in other words at 5.96 billion years ago.
The first of Friedmann equations, ^ = + ^, combined with expression (67) for the Hubble parameter, gives the critical energy density pc and the energy density of the dark matter p for the
, • a 2 A+2 " " " "
solution a(t) = At3 ei41 :
* = icH° =8 51 ■10-330 ^ (T1>
p= (P-2 — 7 + dh =2'261°-W & (72>
From above formula, it follows that Q = = 0,265, and since Qv for the visible matter is
Pc _ _
approximativelv Qv = 0,0^, then the value of Qa = 1 — Q — Q = 0, 685 coincides with the value obtained from Planck 2018 mission.
4.6.2. Another cosmological solution.
In this model we consider different scale factors and different type of the universe. Results are given in the following theorem.
Th eorem 8. (il) In the case of nonlocality "H.(R) = Q(R) = y/R — 2 A with scaling factor of the form a(t) = A e^t2 and k = 0 holds:
R(t)=2A(1 + 2A t2), H (t) = \A t. (73)
33
There exists a cosmological solution for ansatz O^R — 2A = —A^R — 2A, and conditions
F(—A) = E ^(—A)n = —1, P(—A) = E fn n(—A)n~1 = 0. (74)
n=l n=l
(i2) In the case of nonlocality %(R) = Q(R) = VR — 2 A with scaling factor of the form,
a(t) = A e v 6 * and k = ±1 holds:
m = ^' + 2A, H = . (To)
There exists a cosmological solution for ansatz D\/R — 2A = j^VR — 2A, and conditions
?(A)=—i, (A)=o- (76)
Remark 3. (il) In this case one can calculate Roo and Goo,
A2 A2
Roo = - -3-12 - A, Goo = y t2 . (77)
Also, we have
A / A 2 N A
8-G <A2 -^' m = -2TiC
m = ^Cit2 -1), m = -d^(A*2 -1). P8)
Solution a(t) = Ae^1 is nonsingular with R(0) = 2A and H(0) = 0. There is acceleration expansion a(t) = + 4r t2) a(t) which is positive and increasing with time.
(i,2) Similarly to the previous case we find
v A 3k t . A
Roo = -7T, Goo = -TK ^V 3 + -
2- — A2^ 3 +2. ^
m = — (- A + *) p(t) = — (A - k i) (80)
= 8iG 2+A26 ^ P(t) = 8iG 2 A2 6 ^ l8Uj
We have two solutions: (1) a(t) = Ae v 6 and (2) a(t) = A e v 6 ; for frofh k = +1 and
± / — t
k = -1. They are similar to the de Sitter solution a(t) = A e v 3 , k = 0, but have time dependent R(t),p(t) and p(t). When t ^ +<xi, parameter w(t) ^ -1 in the case (1) and w(t) ^ -3 for (2)
5. Perturbations
This section is based on the papers [27, 30, 21, 48].
5.1. Conformal time.
In this section we are searching the cosmological spatially flat de Sitter which can be written as
ds2 = -dt2 + a20e2Htdx 2 = -dt2 + a(t)2dx 2, (81)
where H is constant, t the cosmic time and x is the 3-dimensional vector. The last equality shows that this metric is a particular case of a spatially flat FLRW metric.
a d = d .
ds2 = a(r)2(-dr2 + dx2), (82)
and connection between cosmic and conformal time is given by
r = -aoHe-Ht * a(T) = -H~T .
So when t goes from past to future infinity, t goes from to 0— t = 0 corresponds to r = -^h-
a
a
5.2. Perturbations.
The variation of the metric is as usual
5V = iW + fy^ • (83)
where bars denote the back ground metric, and we will use this convention for background quantities. Perturbations of equations of motion (37) around the de Sitter vacuum are
—m2ÔG^ + R — )v(a)SR = 0, (84)
where m2 = 1 + fo(H'Q + Ç'H) and v(a) = —((H"Q + G"H)fa — 2UÇ'F(□)), where ' denotes derivative with respect to R. Since the variation of the □ acting on a scalar function is a pure differential operator and all background curvatures are constants, we have
(S □) f = [—W (d,dv — T^dp) — rvrf„dP] f• (85)
Here Tf^, denotes the Christoffel symbol. If we start with (83), then we get
= -ht»>, ÔTPiv = ^ = -(y^hP + y,Щ - raV*h^). (86)
So, for f be a constant we get (5□) f = 0. The same is true for Kd
(5 K£) f = [—h^ (dadv — t%vdp) — raYa„dp — 6^5 □] f, (87)
Taking the trace of (40) one gets
[ m2 + (R + 3 □)«(□)]<$R = U(^R = 0. (88)
R
of Weierstrass factorization
U(□)5R = fl^ —u2) el(a)5R = 0 , (89)
i
where u2 are the roots of the equation U(u2) = 0 and since 7(□) is taken to be an entire function and ei(uj ) has no roots. Let assume that there are no multiple roots. Such roots complicate the story, but can be treated analogously, see [40]. Then we can solve (89) for each separately
(□ — u2)5R = 0. (90)
The latter differential equation can be written explicitly as
& 2Я , 1,2 ш2
— -9r + k2 + H^J SR = 0, (91)
where we have taken the de Sitter form of the background. The solution yields
5 Ri = (—kr)3/2 (CuJVi(—kr) + C2iYUi(—kt)) , (92)
where J, Y are the Bessel functions of the first and second kinds, respectively, with ^ = \J9 — ^ and Cu,2i are the integration constants.
For small values of t which correspond to large cosmic times t the Bessel functions have the following asymptotic behavior
Jv (z) — ,
Yv(z) — z-Revl for Rev = 0,
Yv(z) — \nz for Rev = 0 .
From this we conclude that 5Ri are bounded,
3
|Revi| < ^. (93)
R
5 R = ^S Ri. (94)
Ri R
5.3. Scalar perturbation and Bardeen potentials.
Since the behavior of vector and tensor classical perturbations remain the same as in GR, we will focus on scalar classical perturbations.
The metric for the scalar perturbations around a FLRW background is given by
ds2 = a(r)2 [-(1 + 2<)dr2 - 2difidTdxi + ((1 - 2ip)+ 2didj^)dxidxj] . (95)
where <, P, ^ and 7 are scalar functions.
From 4 scalar modes only 2 are gauge invariant. The convenient gauge invariant variables, known also as Bardeen potentials, are introduced as
$ = < - = < -X, ^ = ^ + = $ + Hx, (96)
where x = aft + a27) $ = P + 7') H(r) = 0!/a. The prime denotes the differentiation with respect
Now, we want to determine the Bardeen potentials introduced by (96). To do this we need two equations: the first one is given by the formulation of 5R in terms of $ and ^ accounting that the R
S R =
a2
a"
k2($ - - 3-$' - 6—$ - - 9-tf'
(97)
where SR = 5R - R'((3 + 7') is a convenient gauge invariant analog of 5R. This expression is valid a
5R = -6 H2 (4$ - r($' + 3V) + r2 V) + 2t2H2k2 ($ - . (98)
The second equation can be obtained from the system of equations (84). We find the i = j component of the system (84) which becomes
-m2($ - ^) + v(a)5R = 0. (99)
(0 )
2m2(^' + H$) + («(□ )SR.)' - Hv(□ )SR = 0, (100)
and finally, we deduce the (00) equation of system (84) which yields
-2m2(k4 + 3W + 3^2Ф) - m(v(n)SR)' - (^k2 - v(a)5R = 0, (101)
1/ 2
space. If we multiply (99) bv k2, (100) by 3 H and summing these results with (101) and taking in account that for the de Sitter space-time H2 = 1/t2 we finally get
-m2k2 ($ + tf) = 0, (102)
which Js our second equation. Obviously, it simplifies the succeeding computations considerably. R
2$ = m_ ^ v(w2. )§Ri. (103)
R R R
(Newtonian) gauge P = 7 = 0. If we take into account (93), we see that Bardeen potentials are
| Re i| < 3/2 Re i = 0
At last, in the case that for at least one i we have |Revi| > 3/2 then perturbations grow. This is in perfect agreement with [9]. In that reference a more general class of solutions was studied which asymptote to the de Sitter background at late times while the nonlocalitv is given by RF(^)R. In our case when H( R) = Q( R) = R we obtained exactly the same result as in Section 4 of [6].
m2 = 0,
R = 0 ( □ )
All equations coming from (84) since all of them do not carry information about individual Bardeen
m2 = 0
one reduces the number of propagating degrees of freedom.
5.4. Stability for constant curvature background.
The main result of Section 5.2 implies the natural question of stability of the de Sitter vacuum solution of eq. (93). It turns out that v depends on the structure of the nonlocal operator U(□) such that
УЛ. (104)
4 H2' and U(w2) = 0.
Moreover, the system does not lead to ghosts demands that there is no more than one such a root w2 for the operator U, for more details see [27].
Firstly, let us make the analysis of nonlocalitv given by [21]
U( R) = Rp, G(R) = Rq, (105)
R - 2 A + foRp+q (2 -p- q)=0. (106)
This equation can be solved w.r.t R in general for p, q integers and —3 < p + q < 4 . It is necessary to analyze U to see whether the stability condition can be reached. Indeed, U is analytic by construction but a compatibility condition must be fulfilled
l + RP+q-1 (p + q)(2 -p- q)fo = -w2e7(0). (107)
It is obvious that as long as u2 is real it should be at least positive in order to satisfy (93). The constrain (107) clearly shows that u2 is real and therefore reduces to the following necessary inequality
1 + Rp+q-l(p + q)(2 - p - q)fo < 0. (108)
Satisfactory solution of previous relation is a necessary stability condition. Obviously we have two special cases, namely p + q = 0 and p + q = 2, and in both cases there is no stable solutions.
In a general situation we have to understand equation (106) together with the inequality (108). One can simplify (108) using the background equation (106) to
R (p + q - 1) > 2A(p + q). (109)
In the case p + q = 1 one can have a stable solution for a negative A. The latter condition is possible as long as Afa < 0.
In an attempt to solve the system (106) and (108) one can rewrite it as
1 - s + u = 0, 1 + uz< 0, (110)
where s = , z = p + q, u = faRz-l(2 - z). This latter system looks simple but unfortunately does not provide new interesting solutions from the physical point of view.
5.5. Perturbation of Minkowski space.
Let us recall that in 2018 the gravitational waves were experimentally discovered following results of GR, see [1]. In this subsection we want to investigate if our nonlocal modified gravity predict gravitational waves and their explicit description.
In GR the equations of gravitational waves were obtained as perturbations of EOM for Minkowski metric, i.e. in the form
= 0, = 0, (111)
where = - 1 gpv h, = 5gpv, h = gpvand Ih^l ^ 1. These equations are similar to expansion of electromagnetic waves with Lorentz condition. In this subsection we assume that metric gpv is Minkowski metric, given by gpv = = diag (-1,1,1,1).
The covariant derivative is equal to the partial derivative and d'Alambert operator is given by
□ = -d2tt + d2xx + d2yy + d2zz.
The perturbations of EOM (37) up to the linear degree of Minkowski metric are in the following form
- 1 (9n»U(R)F(^)g(R) - g^vU(R) faQ(R) SR - h^v fag(R)H(R))
21 + 5R„^ - K^5W + (5Rp„ - ^g^vSR + h^A),
where 5 W = -2Q'(R)U'(R)F(^)5R - fo(Q''(R)U(R) + U''(R)Q(R)) SR and where the variation
IfAV
of curvature tensor is 5R = -Kpvhpv, 5Rpv = V a7Au - VP7AAU.
If we use tensor (112) becomes 1
<3'(0)U(0) - 2Q'(0)U'(0)K^ ?(□)§R
+ Qv-faU(0)g(0) - fa(G''(0)u(0) + n"(0)g(0))k^v + 0 sr - hp v (a - 2 fag (0)U(0)^ + 5 R^ (fo(gn)'(0) + 1) =0,
^ r = v pv + 2 □(d ^ )■
Let us remark that if the equations (111) are satisfied then variations of 5R and 5R^V are vanishing and previous equation is reduced to
A = 2 /00(0) H(0), (114)
and gravitational waves in nonlocal modified gravity with nonlocalitv given bv HF(□) Q have the same behavior as in the GR. So, we prove the following theorem.
Th eorem 9. Let M be a Minkowski manifold with nonlocality given by H(R) F(^) Q(R), then A = 2 foS(0) H(0), the equations of gravitational waves are:
□^u = 0, = 0. (115)
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Получено 22.12.2019 г. Принято в печать 11.03.2020 г.
