Расширенный класс точных космологических решений на основе преобразований Дарбу

The theory of cosmological inflation, based on the assumption of the existence of an accelerated expansion, which precedes the hot universe stage in the Big Bang theory, is the most consistent approach to describing the evolution of the early universe [1–3]. To describe the inflationary stage of the early universe, various models are considered, based on both Einstein gravity and its various modifications [1– 3]. In this paper inflationary models based on Einstein gravity with a single scalar field will be considered.

The dynamic equations of a scalar field for a spatially flat Friedmann-Robertson-Walker (FRW) space-time can be noted as [1, 3]

H2 = 3 ( 2 ф2 + V(ф^ ,

H = - 2 ф 2,(2)

ф + 3Нф= -V ‘(ф).(3)

In this system of dynamic equations a(t) is the scale factor, H(t) = a(t)/a(t) is the Hubble parameter, where the Einstein gravitational constant к =1 , ф(1) is the scalar field, and V(ф) is a potential of a scalar field. A dot denotes the derivative with respect to the cosmic time t , and a prime denotes the derivative with respect to the scalar field.

In the system (1)–(3) one has two independent equations only, and to construct the exact cosmological solution one can solve two dynamic equations (1)–(2) only. There are a different methods to generate the exact solutions of system (1)–(3) (for the review, see [3–5]).

One of such approach based on the following representation of the dynamic equations [3–5]

V (ф(t)) = 3H ² + H, (4)

H = - 2 ф 2 , (5)

where the exact solutions can be generated by the choice of a scalar field evolution ф = ф(€) or dynamic of the universe’s expansion H = H (t) .

At present, cosmological models with different scalar field potentials V(ф) are considered, that determines a different ways of implementing the inflationary scenario [6]. Thus, the only true expression for the potential of a scalar field is unknown.

It should be noted that inflationary models must contain a combination of Friedmann solutions and (quasi) de Sitter solutions which present the basis of an actual description of the evolution of the universe. In the context of the inflationary paradigm, the early universe expands rapidly for some time and then it transits to a power-law expansion regime without acceleration corresponding to the Friedmann solution [1–3].

In this paper we consider the Schr¨odinger-like representation of the equation (5) and use the Darboux transformations to construct the chains of exact cosmological solutions corresponding to compound exponential power-law dynamic of the universe’s expansion. Such solutions define the explicit dependencies of a scalar field, Hubble parameter and the potential from cosmic time that implies the reconstructing the shape of an potential of scalar field.

1.    Schr¨odinger-like representation of cosmological dynamic equations

The Schr¨odinger representation of the first dynamic equation was proposed in [7–9] The basis of this approach is representation of equation (4) as the Schr¨odinger-like one in terms of cosmic time

Ф -

U (t)ф = 0,

with corresponding second dynamic equation

Ф 2 =

2 d2 T --In Ф

3 dt ’

where U (t) = 3V(t) and Ф = Ca3 (t) which can be considered as some partial solution of equation f - U(t)f = 0,

with the same potential, and C is an arbitrary constant.

Further, one can use the Darboux transformations

U = U - 2

ф = ф - ф

{ s'" ^{f (t)}}, { st ^{ln f ( t )} } ,

where function f (t) can be defined as f (t) = ^(t) (C1 + C2 У [Ф(t)]

² dt) ,

to construct the exact cosmological solutions of new Schr¨odinger-like equation

Ф - и (t)Ф = 0,

on the basis of some known ones.

The disadvantage of this approach is that often it is impossible to find the field evolution ф = ф(1) in explicit form equation (7) after applying Darboux transformations. Thus, the shape of the scalar field potential V(ф) for such solutions remains unclear.

The other approach based on the representation of equation (4) as Schr¨odinger-like one in terms of a scalar field is considered in [10], and dynamic equations were represented as

[- dlr ■ и(ф] ^o, (13) v=6 [1—3и («я] ффф, (14) ф= -2фф, (15) where ф(ф) = H(ф).

The Darboux transformations (9)-(12) give the new potential V and the Hubble parameter H , however, solutions of equation (15) in explicit form for the new Hubble parameter H can be obtained for a limited number of models only.

The Schr¨odinger-like representation of the second dynamic equation (5) is considered in [11] as сРа(т)    1

dr ² + 2

d _ d dt dr

Нф(т ) dr

а(т ) = 0,

In this case, the Darboux transformations (9)-(12) give a new scale factor a and kinetic energy ^,я²

2   ^dr    only but not the transformation of the field itself. Thus, in this case, the problem again reduces to finding solutions to equation (5) in terms of a new time parameter т.

Therefore, the use of Darboux transformations (9)–(12) for these methods for reducing the equations of cosmological dynamics to the Schr¨odinger-like equation seems to be an ineffective tool to find the evolution of a scalar field ф = Kt) in explicit form and the shape of its potential V(ф) as well.

Thus, to solve the problem of constructing the exact cosmological solutions for a complicated dynamics of the expansion of the universe, a new method is needed that will allow one to use the

Darboux transformations and combine them with other form-invariant transformations as well.

2.    Form-invariant transformations of the second dynamic equation

Form-invariant transformations (FIT) associated with various types of symmetries preserve the form of dynamic equations, and application of FIT to equations (4)–(3) can be written as

,             ,        , ~   ~   ~.

FIT : {ф,H,V}ч{ф,H,V},(18)

where { ф, H,V } are initial exact solutions of these equations and { ф, H,V } are new exact solutions.

To generate exact solutions, we consider form-invariant transformations of the second dynamic equation ф2 = -2H,(19)

• i .e. we will consider the transformations { ф, H } ч { ф, H } to obtain the solutions of equation

ф2 = -2H.(20)

т           1                 11-     11           11-       Гу тт1     1 Г у тт1       1    1 г* 1 г 11

In general case, the relations between solutions { ф, H } and { ф, H } can be defined as follow

ф(ф) = ффф + 6(t),                                       (21)

H (t) = H (t) + tit),                                     (22)

where 0(t) and r(t) are some function of cosmic time connected by equation

Г + Q 0 + ф) 0 = 0,

and form-invariant transformations of equation (19) define this connection in explicit form.

Such transformations can be applied an arbitrary number of times, which determines the chains of exact solutions of equation ф- = -2Hi,                                            (24)

where i denotes the order of the transformations.

The potential of a scalar field as a function of cosmic time can be obtained from expression

V(t) = 3H 2 + Н _г ,                                    (25)

for each i -solutions of equation (24).

The potential as a function of scalar field is defined in parametrical form by following way

• v._(ф.) = J Vi = ^V^ ^{г г}ф = = ф^).

2.1.    The shifts and dilations of a scalar field and Hubble parameter

The other way to obtain the expression for potential V (ф _i ) is to find the dependence H i = H i (ф _i ) from H _i = H _i (t) and t _i = t _i (ф _i ) .

Further, one can use expression

V i (ф i ) = 3К 2 (ф_г) - 2 ( ^H^ ) ,                           (26)

based on the Ivanov-Salopec-Bond approach [3] to obtain the potential of a scalar field in explicit form.

In the other cases, for solutions without the explicit inverse dependence t _i = t _i (ф _i ) , it is possible to reconstruct the shape of the potential V (ф _i ) from expressions Vi = V (t) and ф_i = ф_i (t).

Now, we consider some types of form-invariant transformations, which can be used to construct new exact cosmological solutions from known ones.

A simple transformations of equation (19) are the combinations of the shifts and dilations of the scalar field ф and the Hubble parameter H .

Shifts transformations can be defined as ф ч ф + c, H ч H + A.

Therefore, one can obtain the field evolution ф(€) and Hubble parameter H(t) from equation (19) up to the choice of constants c and A that corresponds to the following transformation of a scale factor

a(t) ч a(t) exp(At).

This result demonstrates that the simplest case of inflation for ф = 0 is a pure exponential expansion with a(t) к exp(At) which induced by cosmological constant Л = V = 3A - .

The other form-invariant transformations (combination of shifts and dilations) can be written as [3, 5]

ф ч ^^пф + c,   H ч nH + A, where 7 = ±1, n and A are an arbitrary constants.

The scale factor under these transformations changes as

a(t) ^ [a(t)] ⁿ exp(At),                                             (30)

that corresponds to generalized exponential power-law dynamics.

Also, the choice of the sign of the constant n allows one use the transition between exact solutions for canonical and phantom scalar fields [3, 5].

As one can see, the repeated transformations (29) does not change the character of the dynamic, so their using to construct the exact cosmological solutions is limited by the law of expansion (30). Thus, to generate the exact cosmological solutions with different model’s parameters { ф, H,V } one must use the other types of a form-invariant transformations of equation (19).

• 3.    The Darboux class of exact cosmological solutions

In order to find the other type of form-invariant transformations of equation (19) we consider the representation of this equation as following system ф — u(t)ф = 0,

u(t) = ф — 2H,

ф(*) = /1 ехр(фф)),(33)

where (31) is the Schrodinger-like equation and / is an arbitrary constant.

Now, we consider two Schr¨odinger-type equations ф — u(t)ф = 0, and ф — u(t)$ = 0, which can be represented as the following systems

u(t) = ф — 2H,

P(t) = /1 exp(ф(t)),(37)

ф2 = —2H, and

U(t) = ф — 2H,

P(t) = 12 exp(ф(t)), ф2 = —2H, where ф and ф are partial solutions of the equations (34) and (35) respectively.

The connection between these solutions can be obtained from the Darboux transformations

d2        , , u = u — 2;wln f ^Ф dt2 (42) ф = ф—ф {dln f(t)}, (43) where f (t) is the general solution of the equation (34)

f — u(t)f = 0.

Based on transformations (42)–(44) one can write the connection between the exact solutions of equations (38) and (41) in following form

ф(Ф) = Ф(t) + x(t)’

f

Н (t) = Н (t) + f

⁺ 2 ^x’

X^(t) = In

ф^ -

М2 \ f)

’

f - фф - 2H^f = 0.

On the basis of the expression (38) equation (48) can be represented as f - (ф + Ф ² )f = 0.

General solution of equation (49) is f (t) = exp[ф(t)] |ci + C2 У exp[-2^(t)]dtj ,

where C 1 and c ₂ are an arbitrary constants.

After substituting the solution (50) into expressions (45)–(48) and redefining the constants as

^C 3 =

^^ 1 ^ 2 we obtain transformation from equation (38) to (41)

ф² = - 2Н’ ^ ф ²

- 2H’

which is defined by following connections

ф(^ = ф(t') + ln

C 3 !

Н (t) = Н (t) +

C 1 + С 2СГ J ’ C 2 d

2(C 1 + C 2 Cr) ’

Thus, these transformations define the Darboux class of exact cosmological solutions.

4.    Extended Darboux class of exact cosmological solutions

Firstly, from the definition (54) we can write

ф(Ф) = - |ln! ’ and represent expression (52) as

ф(ф) = -1 ln ^ + 1 ln [T—C3J—«] = 1 In [T---C3a 42] .

2        2   L(C 1 + C 2 J)²]    2   L(C 1 + C 2 J)²]

Secondly, after redefinitions of functions and corresponding constants

where q is an arbitrary constant, from (38) and (41) we obtain the following relations

H(t) = q2H (t) + c2+B ’(59)

0 = exp [qф(t)] , where constant A = 0.

For the case C = 0 , q =1 , A = B² one has ф = ф and H = H, i.e. the initial solutions of equation (38). Inverse transformations towards (58)–(60) are written similarly to direct ones up to the choice of constants that can be verified by trivial calculations.

Finally, after combining transformations (58)–(60) with shifts and dilations (29) one can define the recurrence relations

фк+1(t)—-fV^k inL    . R^l,                           (61) L (Ck ck + вк )2 j Hk+1(t) = ^k \qkHk(t) + r    ' R ] + At,                       (62) Ck ск + вк] Ck = exp[qkфк (t)] ,                                                       (63) which connect the exact solutions of the cosmological dynamic equations Vt (t) = 3H2 + Ht,                                  (64) фк = -2Hk,                                        (65) and Vt+i(t) = 3H2+1 + Hk+1,                               (66) фк+i = -2Hk+1,                                       (67) where index к = 0,1,2, 3,... defines the order of the transformations (61)-(63).

4.1.    Chaotic inflation with quadratic potential

The differences between solutions (21)–(22) can be defined by expressions (52)–(54) and (25) in following form

Аф = Ф — Ф = 0(t) = ln [ ^^ 1 ,

L C 1 + C 2 cr J

AH = H - H = ,,(f)=      '   ,

2(C 1 + С 2 СГ)

AV = V - V = 3ф2(€} + ф + 6y(t)H(t).

with following connection c = exp[-2ф(t)], corresponding to fulfillment of the condition (23).

Also, from equations (68) and (69) one has following expression

V(t) = |2- exP(0(t)),

2c3

which characterize the connection of differences between initial and new parameters of cosmological models for the Darbox class of exact solutions.

In principle, it is possible to generate exact cosmological solutions by specifying the differences 0 or ф . As the example, we consider the case 0 = 0 , and from equation (68) we obtain

c(t) = co(t) = — exp [-C2(C4—t) 1 - —,

С2               C3     J where C4 is the constant of integration.

From equations (69) one has

H(t) = H (t) + 2|, and from (71) we obtain the linear evolution of a scalar field

Ф(1) = ф(Ф) = Ф о (t) = at + ф,

where we redefine the constants as a and ф , and the index 0 means that we will consider this evolution of the scalar field as the first in the chain of exact solutions.

For this law of the evolution of scalar field (75) from equations (19) and (25) (for к = 0 ) one has following exact solutions

Н о ^ф = — —^t + p, o , a o( t ) = Q o exp [— 4 (a ² t — 4м о ) j , „       /a²      \ 2 a ²

V 0 ^{( t} )=3^— ^t + MoJ     2,

К > ( ^ф о ) = ³ [ 2^{(ф — ф} о ) + м о ] —2 1

where ц _о and Q _o are the constants of integration.

These solutions correspond to chaotic inflation with quadratic potential and massive scalar field [13]. Also, solutions (76)–(79) was considered in [3].

Thus, applying transformations (68)-(71) к -times, one can construct a chain of exact cosmological solutions.

5.    Degenerate form-invariant transformations

After redefinition of function a(t) and combining the result of the Darboux transformations (52)(54) with shifts and dilations (27), (29) one can define the relations for extended Darboux class of exact cosmological solutions as

Ф 1 (t) = — In a k ,

^H k = ^—

q k

(0 1 } 2 2 q k V kJ ’

Ф k +1(t) = уу/п к ln

A k a к

. ^{( C} k ^a k + Sk)2_| ’

H k +i (t) = n k q k H k (t) +

2C k 0 k

^C k ^a k + ^B k

+ ^X k .

Thus, the exact solutions ^_k ,H _k } and { ф _k + ₁ ,H _k + ₁ } of equation (19) are connected by relations (80)–(83) that corresponds to a chain of exact cosmological solutions induced by some initial solutions { Ф о ,Н о } for к = 0 .

Also, from equation (81) and the definition of Hubble parameter H = a/a one can obtain the transformation of a scale factor ak+i(t = Qk+1 [ak(1)ПкЧк [Ckak(t) + Bk]2nk exp (Xkt),

where Q k +1 is the constant of integration.

Thus, each application of transformations (80)–(82) induces a cosmological model with a new type of dynamics of expansion of the universe, in contrast to (27) and (29). The difference from the dynamics obtained using transformations (29) is determined by the function a _k (t) .

From the expression (82) for к + 1 element in chain of exact solutions ak+i = exp [qk+iфk+1(t)] ,

using equation (80), one can obtain the following connection

A ak+1 = exp I 7^ qk+1 ln I (q—

k a k p ■                          (86)

+ ^B k

As one can see, from (82)–(83), the transformations

a(t) ^ C i a(t) + C 2 ,                                           (87)

where C 1 = 0 and c ₂ are some constants, don’t change the character of the scalar field evolution and cosmological dynamic. Thus, one can define this function from connection (86) up to these transformations.

In general case, the connection between a _k +1 and a _k form equation (86) can be expressed in quadratures only.

We also note, that under condition

7 y/ ⁿ k ^q k +i = 1, 7    ^q k +i =    ,                               ⁽⁸⁸⁾

+    nk one can find from (86) the connections between ak+1 and

Ak ak+i = -~^r (Ck^k + Bk)   + Ek,                             (89)

bk where Ek is the constant of integration.

However, connections (89) lead to change of constants {A_k,B_k,C_k} 7 {A_k+1, B_k+1, C_k+1} only in expression for the scalar field (82), and, therefore, such a transformations don’t change the type of cosmological dynamics (83). Thus, to generate a new (к + 1)-solutions in a chain it is necessary to solve equation (86) for each к-solutions.

• 6.    Л—chains of exact cosmological solutions

• 6.1.    Л(exp)—chain of exact cosmological solutions

Now, we consider the examples of cosmological exact solutions which follow from the inflationary models with flat potential V = Л = const or cosmological constant Л corresponding to de Sitter solutions [3]. Thus, the transformations (80)–(83) induce the deviations of the potential from a flat shape. Since, through inverse transformations towards (80)–(83) all these solutions are reduced to models with a cosmological constant, we will call them as Л-chains of exact cosmological solutions.

In such a chain for i = 0 from equations (24)-(25) one has following initial solutions

^0 = 0,    Ho = :o,    ao(t) = Co exp^t),    Vo = 3^0, which one can obtain from (19) and (25).

For the first order transformations к = 0 from (80)-(83) one has ao(t) = t,

м) = 7^

H^t) = no A:^ +  2

⁺^o,

\       Go* + Bo / ai(t) = Qi [Co* + Bo]2”0 exp [(noq^o + M*] .

From equation (92) one has the dependence

(Cot + Bo)^-1

:

After substituting dependence (95) into (93) one has ф1        dHi^h)        (

Н_х (ф1) = a; + Д exp -—— ,   ——--= Wo exp -—— ,          (96)

A 27М0/       dф1          \27упо/ where ao = noq2Mo + Ao, Д = 2^^'/ and W = Co ^|0

From equation (26) for i = k +1 one has

МФ1) = (3^0 - 2w2)exp

PM

MMo?

+ 6ao^o exp

A 27Mo/

+ 3a2.

Thus, as the result of the first-order transformations in Л(exp)-chain we obtain the exact solutions for exponential power-law (EPL) inflation [12].

In partial case ao =0 or Ao = -noq^o this model is reduced to the case of power-law inflation []. Also, for the case 372 — 2w^ =0 or n₀= 1/6 one has the following potential

МФ1) = 6a_o7o exp

(> a

+ 3a0.

For the second-order transformations with k = 1 from (80)-(83) we obtain

MP=aP^(%-1),                                            (99)

^ф-^(,)= ■ ^ln[ад^М +_B1?] ’                      ⁽¹⁰⁰⁾

Mp)= ni

2C,M1 qiHi(t) +ca^i-1) +B1J +A1,                   (101)

MP = Q2 ЫРГ’2 'C| a 7 ²■■ ¹+ Bi] ²ⁿ1 exp (Ait),               (102)

where £ = ^(t) = (Cot + Bo) 1, ai = '(j-)^ and 7i = y^n^i.

As one can see, for 71 = 1 or B_i = 0, solutions (99)-(102) are reduced to (91)-(94) up to the choice of the constants. In general case, one can define the potential V2 = У₂(ф₂) only in parametrical form.

For the case yq1 ^n^ = 1/2 from equation (86) one has

MP = ^i lnMP,

where u_i(t) = Cot + Bo and w_iis an arbitrary constant.

For this function, from equations (82)–(83) one has

Ф2(р=7Mi in        i¹;^^,

_Mi(t)(CiWi inui(t) + Bi)²

H2(t) = П1

Co   +         2Ci^iCo

2u1(t)    u_i(t)(C_i^_i inw_i(t) + B_i)

^{+ A},

^a2^(t) = ^Q2^(C0^{t + B}0) ^{2 exp}[c^(C0^{t + B}0)] [^Ci^wi ^in(Cot ^{+ B}o) ^{+ B}i] 1 ,

where all constant terms (shifts) in expression for the Hubble parameter are denoted as A = £ const.

Thus, at the second order of transformations one has the modification of EPL dynamic by the third multiplier in expression (106). Such type of a model can be called as EPL–logarithmic inflation. Now, we find the corresponding potential for this inflationary model.

From equation (104) one has

»i(« = ' exp (—4=) W [aiexp (— MM.           (107)

4C2Wi    A 7 Mo 7   L A 27Mo/J where W denotes the Lambert function, (31 — - A^0 exp ( 2еффШг) and - — ±1.

Thus, from (105) one has

~

H2(2) = nipW (p) /1 +

^{u (}p).

+ ^,

dH2(ф2) Лф2

( ^n1 ) pW 3/2(9?)

= \ - ) T+W¹/²^

/31 +

C3i     33i^iCi

U(p)    U2(p) J ’

where

( Ф2 \ 9 = ^exp —= ,

\-Mo J

W (p) = W a₁ exp

-

Ф2

²—Vⁿo

)l'

U (p) = -Ciw1 In

l( 'CA-'A

+ Bi,  £i —8

CM

^Ai

,

/31 —

2 / .

Ai

Therefore, the potential for EPL–logarithmic inflation can be defined

as

follows

V2(ф2(p)) — 3 (nip W(p) pi +

^i

^u Ы

¹     ;     2nip2W²(p)¹3

J 4  [1+wv^pp ⁺

3i

u M

Cji^i^Ci | ^—

.

At the third order of transformations (к — 2) for the function (103) under condition -q₂pn! — -1/2 from equation (88) one has

^2^{(t) —}3 ^(C0^t+ Bq)² |^^C1W1 ^ln(CQi ⁺Bq)⁺3^(3B1 ^{— 2C}1W1)j .

Also, at the third order of transformations for the function (103) under condition -q₂^n1 — —1 from equation (88) one has the other type of the function a₂(t), namely

&2(t) — ^u2(t) ln²(Coi + Bq) + V2(t) ln(Co* + Bq) + a2t2 + fat + ^2,

where

/ A    1 “1C? 2

^{Mt) —}2 MT* ⁺

-iC2b)

MV — (-

1 W1C2

2 Aq

+

AqCq² C1B1 ^Ao

1 ^iC²B²

*⁺2 A₀C²’

)t2+(

W1C2B2 , „cbb) , AqCq^{2 + 2}AqCq г

1 -C В    C1B1B2

2 A₀C^{2 +}A₀C²’ ⁽¹¹6)

1 wiCl a2 — —--:--

4 ^Aq

1C1B1 2 Aq

, 1 Bi²

⁺2 A_owi ’

_ 1 ^iC²Bq

^{/2 —}2 Ac

CBiBo A₀Cq

+

B_oB2 ^1^AO^C0

The functions (113) and (114) give the different expressions for scalar field evolution ф₃ — ф₃(*) and cosmological dynamics H₃ — H₃(t) in explicit form from expressions (82)-(83) for к — 2, and the potential V₃— У₃(ф₃) can be defined parametrically.

6.2.    Λ(sinh)–chain of exact cosmological solutions

In such a chain for i — 0 from equations (24)-(25) one has following initial solutions

MV^—^ln [^tanh(|^] ’

Hq(V — ¹coth(at),    ao(t) — Cq sinh^1/3(at),    Vo — y, (118)

where a is a some constant.

From expression (80) for к — 0 one has mo — J H(/•   * <«.

The function a₀(t) can be obtained explicitly for different values of the constant q₀. Choosing q₀= yjm^w where m is an arbitrary constant, for the case m =1 one has

^0^(t)= Iln [^cosh(2^t)] ’

and for the other values m = 1 one can find the function U0(t) from expression

2 Г      /— \ q m—1    Г г /— \ q m —2

'^ol/,    .    Hhc *)]   cHC *)]   ^dt.

Therefore, one has a set of exact solutions corresponding to different dynamics and different evolution of a scalar field at the first order of transformations (80)–(83).

Also, the expressions function (119) for each m induce the new chains of exact solutions of the second Einstein-Friedmann equation in higher orders of transformations (80)–(83).

6.3.    Λ(cosh)–chain of exact cosmological solutions

In such a chain for i = 0 one has following initial solutions ф0(1) = ^ - - arcsin (tanh (—t)),    H0(t) = — tanh(—t),    a0(t) = C0 cosh1/3 (—t),

a2 V = —, 3

From expression (80) for к = 0 one has

^0(t) = У exp [q0 У — - arcsin (tanh (—t))j dt.

The function cr₀(t) can be obtained explicitly for q₀ = m^6 and q₀ = m^2, where ±1, ±2, ±3,...

m

As the result, one has complex function cr₀(t) = cr^⁶(t) + ctq”^(t).

6.4.    Λ(sin)–chain of exact cosmological solutions

In such a chain for i = 0 one has following initial solutions

^0(t) = V -arctanh (cos(—t)),    H₀(t) =

— cot(—t),     bq (t) = Cq sin^1/3(at),

V = —  ,

From expression (80) for к = 0 one has

^0

(t) = У exp q₀ У^arctanh (cos(—t)) dt.

The function ^(t) can be obtained explicitly for q0 = у 2m or q0 = m^6, where m = ±1, ±2, ±3, ... Therefore, one has a set of exact solutions corresponding to different dynamics and different evolution of a scalar field at the first order of transformations (80)–(83).

Also, the expressions function (125) for each m induce the new chains of exact solutions of the second Einstein-Friedmann equation in higher orders of transformations (80)–(83).

6.5.    Λ(cos)–chain of exact cosmological solutions

In such a chain for i = 0 one has following initial solutions

^0^(t)= X^ln v 6

1 + sin(—t)

1 — sin(—t)

H0 (t) = — ^—tan(—t),    &0(t)=C0 cos^1/3(—t),    V0 = — ^—-, (126)

From expression (80) for k = 0 one has

(0= /■ ■          ■'

J |_ cos(at)

The function j₀(t) can be obtained explicitly for qo = m^6 and qo = m^J2, where m = ±1, ±2, ±3,... Therefore, one has a set of exact solutions corresponding to different dynamics and different evolution of a scalar field at the first order of transformations (80)–(83).

Also, the expressions function (127) for each m induce the new chains of exact solutions of the second Einstein-Friedmann equation in higher orders of transformations (80)–(83).

7.    <^2—chain of exact cosmological solutions

Now, we consider solutions for the extended Darbox class based on the solutions (75)–(79) for chaotic inflation with quadratic potential.

For the first order transformations k = 0 from (80) one has ao(t) = ^° exp[qo(at + £)] = ^°e2“(t),                          (128)

qoa                   qoa where u(t) = 2qo(at + ^).

From (81)–(83) one has

^i(t) = -27^0ln ^e“^(t)+ |°e^—“C)] ,

Я1 (t) = no

-aqou(t) +

2CoAo

B_oe^-2^ + C

+ ^,

^a1(t) = Q1 exp

(-4 [«noq2(at + 2^) - 4A]) [Ce’°<“^t+^>+ Bq]²”° ,

where C = ^a°^c° .

q°&

Without loss of generality we consider the function a₀(t) as follows

Jo(t)=exp[qo(«t + Я] ,                                (132)

to generate the function O-1 (t) at the second order of transformations (k = 1).

From equation (86) for q1 = ^^^, where m = 0, ±1, ±2, ±3,... (m G Z) for some specific value of m one has two types of solutions, namely

— 2m+1

^i^ (t) = ^ ^svS (t),        for m > 1, s=—m ai—)(t)= emt + ]T Ms^s(t),  for m < 0, s=—m where

v(t) = Co exp[qo(at + ^)] + Bq,(135)

and one has the exact expressions for the constants {w_s, ^_s,e_m} for each s and m.

After substituting function (133)–(134) into (82)–(83) taking into account (130) one has the exact solutions for any value of the constant m.

Thus, we have an arbitrary number of exact cosmological solutions at the second order of transformations, which are defined by the choice of the value of constant m.

Conclusion

In this paper, we considered form-invariant transformations of the second equation of cosmological dynamics based on the Darboux transformations. Also, the Darboux transformations were combined with shifts and dilations. The result of this approach is the possibility of constructing an arbitrary number of exact solutions of the Einstein-Friedman equations.

As an example, we considered Л-chains and <^²-chain of exact cosmological solutions. The advantage of this approach is the ability to generate exact solutions corresponding to the complicated dynamics of the expansion of the early universe. Also, one can consider generalization of these exact solutions for the case multi-field cosmological models [14] and inflationary models based on the Einstein-Gauss-Bonnet gravity [15] as well.

Nevertheless, it is necessary to analyze cosmological solutions for the correspondence of the scalar field potential to the physical mechanisms of realization of the inflationary scenario, and the correspondence of the parameters of cosmological perturbations to the current observational constraints [16, 17].